Studies on the Arrangement of Repeated Sequences in DNA Citation Pearson, William Raymond (1977) Studies on the Arrangement of Repeated Sequences in DNA. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5v4r-gs82. https://resolver.caltech.edu/CaltechTHESIS:12072025-095838407 Abstract Parameters of repetitive DNA sequence organization have been measured in the rat, Drosophila and the pea genomes. Experiments using melting, hydroxyapatite binding and single strand nuclease digestion have been used to measure the number length and arrangement of repeated DNA sequences in the rat. About 20% of rat DNA is repeated 3000 fold. Half of the sequences are 200 - 400 nucleotides long while the remainder are longer than 1500 nucleotides. Rat repeated DNA sequences are interspersed among 2500 nucleotide long single copy sequences. Studies on the long and short repeated DNA sequences of the rat show that the long repeated sequences are also 3000-fold repeated. Cross-hybridization of isolated long and short repeated sequences and hydroxyapatite binding interspersion experiments indicate long and short repeated DNA may share sequences, although this may be due to cross-contamination. Self-renaturation, melting and electron microscopy of long repeated DNA fragments suggest some long fragments may be composed of arrays of shorter repeated sequences. A similar sensitive search has been made in Drosophila melanogaster for short repetitive sequences interspersed with single copy sequences. Five kinds of measurements all yield the conclusion that there are few short repetitive sequences in this genome: 1) Comparison of long and short fragment reassociation kinetics; 2) reassociation kinetics of long fragments driven by an excess of short fragments; 3) measurement of the size of repeated fragments after S-1 nuclease digestion; 4) measurement of the hyperchromicity of repeat sequence bearing fragments of different lengths; 5) direct assay by kinetics of reassociation of the amount of single copy sequence present on 1200 nucleotide long fragments which also contain repetitive sequences. Renaturation of pea DNA has been used to estimate the size of the pea genome and the fraction of pea DNA containing repeated DNA sequences. Pea DNA renaturation and single copy tracer hybridization indicate the size of the pea genome is 0.45 pg. More than 70% of pea DNA is repeated from 100 to 5000 times. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Biochemistry) Degree Grantor: California Institute of Technology Division: Biology Major Option: Biochemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Bonner, James Frederick Thesis Committee: Bonner, James Frederick (chair) Attardi, Giuseppe Davidson, Eric H. Revel, Jean-Paul Defense Date: 24 February 1977 Record Number: CaltechTHESIS:12072025-095838407 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12072025-095838407 DOI: 10.7907/5v4r-gs82 ORCID: Author ORCID Pearson, William Raymond 0000-0002-0727-3680 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17789 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 10 Dec 2025 22:28 Last Modified: 10 Dec 2025 22:43 Thesis Files PDF - Final Version See Usage Policy. 58MB Repository Staff Only: item control page

STUDIES ON THE ARRANGEMENT OF REPEATED SEQUENCES IN DNA Thesis by William Raymond Pearson In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1977 (Submitted February 24, 1977) - i - Acknowledgements I would like to thank my advisor, Dr. James Bonner, for his guidance, patience and understanding during my graduate career. thank Dr. E. Davidson, for his invaluable discussions, encouragement during this work. committee members, G. Attardi, N. I I also com.ments and appreciate the work of my other Davidson and J. P. Revel. Much of the work described in this Thesis was done in collaboration with others. Chapter I was done in collaboration with J. assistance from J. with Chapter II . R. Posakony. J. R. Wu and J. R. Wu with Posakony also helped Chapter IV was done with assistance from S. Smith and Wilson . I am especially grateful for the discussions, encouragement and understanding of my ~ood friends, Jung-Rung Wu and Margaret Chamberlin. ii A.bstract Parameters of repetitive DNA sequence organization have been measured in the rat, Drosophila and the pea genomes. Experiments using melting, hydroxyapatite bindin~ and single strand nuclease digestion have been used to measure the number length and arrangement of repeated DNA sequences in the rat. is repeated 3000 fold. About 20% of rat DNA Half of the sequences are 200 - 400 nucleotides long while the remainder are longer than 1500 nucleotides. DNA sequences are Rat repeated interspersed among 2500 nucleotide long sin~le copy sequences. Studies on the long and short repeated 'DNA sequences of the rat show that the long repeated sequences are also 3000-fold repeated. Cross-hybridization of isolated long and short repeated sequences and hydroxyapatite binding interspersion experiments indicate long and short repeated DNA may share sequences, cross-contamination. although this may be due to Self-renaturation, melting and electron microscopy of long repeated DNA fragments suggest some long fragments may be composed of arrays of shorter repeated sequences. A similar sensitive search has been made in Drosophila melanogaster for short reoetitive sequences interspersed with single copy sequences. Five kinds of measurements all yield the conclusion that there are short repetitive sequences in this genome: short fragment reassociation kinetics; long fragments 2) few 1) Comparison of long and reassociation kinetics of driven by an excess of short fragments; 3) measurement - iii of the size of repeated fragments measurement S-1 nuclease digestion; 4) of the hyperchromicity of repeat sequence bearing fragments of different lengths; the after amount 5) direct assay by kinetics of reassociation of of single copy sequence present on 1200 nucleotide long fragments which also contain repetitive sequences. Renaturation of pea DNA has been used to estimate the size of the pea genome and sequences. indicate the fraction of pea DNA containing Pea DNA renaturation and single the size copy tracer hybridization of the pea genome is 0.45 og. DNA is repeated from 100 to 5000 times. repeated DNA More than 70% of pea - iv - Table of Contents Chapter I Analysis of Rat Repetitive DNA Sequences Chapter rt Analysis of Sequences in Rat Repetitive DNA 55 A preliminary Report Chapter III Absence of Short Period Interspersion of 10b Repetitive and Non-repetitive Sequences In the DNA of Drosophila melanogaster Chapter IV Kinetic Determination of the Genome Size of the Pea 125 Appendix A A Program for Least Squares Analysis Of Reassociation and Hybridization Data 149 - 1 - CHAPTER I ANALYSIS OF RAT REPgTITIVE DNA SEQUENCES - 2 - Introduction Evidence differential transcription. is accumulating that one of the mechanisms for gene expression is the sequence specific regulation of RNA Recent work on a variety of organisms has demonstrated an almost universal highly ordered pattern of repetitive sequence organization in DNA (see Davidson et. al., 1975a for review). The proximity of subsets of repeated sequences to transcribed and translated DNA sequences supports a regulatory function. These structural observations on sequences near functional DNA sequences suggest that repeated important role sequences m~y play an in DNA sequence recognition during the re~ulation of transcription. Repeated DNA sequences display strikingly similar, structures across a wide range of organisms from insects to mammals (Davidson et. al., 1973a, 1973b; Bonner et. al., al., Angerer et. 1975; 1974; al., al., 1975; al., 1976). With a few exceptions (Manning et. al., 1976), repeated among 1973; Schmid and Deininger, 1975; sequences 200 to 1000 to 2000 Graham et. Chamberlin et. Goldberg et. interspersed highly ordered 400 al., Efstratiadis al., 1975; 1975; et. Crain et. nucleotides long are nucleotide non-repeated sin~le copy sequences in more than 65% of the DNA of all organisms studied. This organization was predicted by Britten and Davidson (1969; also Davidson and Britten, repeated DNA 1973) in a model which sequences can coordinately control copy genes. • suggested that the expression of adjacent single - 3 - While the similarity of organizational patterns across the evolutionary spectrum is striking evidence for repeated DNA function in gene expression, experiments demonstrating specific subsets of repeated sequences adjacent to transcribed and translated genes provide even stronger support for the regulatory function of repeated sequences. Experiments on large nuclear transcripts in the rat (Holmes and Bonner, 1974b) and sea urchin (Smith et. - nuclear RNA is sequences. repeated transcribed al., 1974) show that most large from DNA containing interspersed repeated More recent experiments suggest that a select subset of sequences are near transcribed and translated sequences. Gottesfeld has reported (1976) that a fraction of DNA from chromatin with increased transcriptional activity contains not only a subset of single copy DNA but also a subset of all repeated sequences. Experiments using sea urchin messenger RNA have shown that 80% of the translated sequences are adjacent to repeated DNA sequences et. al., (Davidson 1976b) and that these adjacent sequences are a subset of the repetitive sequence population (Klein et. al., in prep.). Selected repeated sequences are adjacent to single copy DNA sequences which are transcribed into message and translated. In this paper we describe experiments which measure the structural parameters of repeated DNA sequence organization in a mammal, the rat. We have made three basic measurements. First, the fraction of the DNA which contains repeated sequences has been determined. measured the length and thus Third, Second, we have the number of the repeated sequences. we have studied the distribution of repeated sequences in single copy DNA. - 4 - Materials and methods Preparation of DNA DNA used in these experiments was prepared from two sources. Unlabeled DNA was isolated from a rat Novikoff ascites cell line grown intraperitoneally and transfered at seven day intervals. Cells removed °c. For the DNA from the rats were frozen and stored at -80 preparation, cells were thawed and a crude nuclear oellet isolated by the method of Dahmus and McConnell (1969). The crude nuclei were resuspended in 0. 1 M Tris, 0.1 M EDTA pH 8.5 and lysed with the addition of 20% sodium dodecyl sulfate to a concentration of 2%. The viscous lysate was heated to 60 °c for 15 minutes and then digested with pronase (Calbiochem, grade B) (usually two hours). at 50 ug/ml until the DNA went into solution The DNA solution was then extracted with an equal volume of 1 : 1 phenol:IAC (24:1 chloroform:isoamyl alcohol), the aqueous phase removed and the interphase re-extracted with buffer and phenol:IAC. The aqueous phases were pooled and phenol:IAC followed by 2-3 extractions with IAC. re-extracted with The DNA was then precipitated with 2.5 volumes of 95% ethanol at room temperature and spooled. at The DNA was dissolved in 10mM Tris 10 mM EDTA pH 8.5 overnight 4 °c, then brought to 0.1 M Tris 0.1 M EDTA pH 8.5 and digested with 50 ug/ml RNase A (Worthington) for 90 minutes (after 10 min RNase preincubation at 95 °c) after which 200 ug/ml preincubated pronase (90 min, 37 °c) was added. After a second 90 minute incubation at 37 °c the DNA was extracted three times with IAC, precipitated with ethanol and spooled. SW27 The DNA was redissolved and spun at 27,000 RPM in rotor a Beckman to remove undissolved material, and then reprecipitated and - 5 - spooled. 3H labeled DNA was prepared from Novikoff hepatoma cells grown suspension culture in flasks on a shaker. modified Swimm's 67 medium in The cells were grown on a (Plageman and Swimm, 1966; Plageman, personal communication) and had a doubling time of 12 hours. Cells were labeled with 3H thvmidine by the addition of 0.1 uCi/ml mCi/umole) four times at 12 hour intervals. (Schwartz, 46 Under these conditions, we were able to isolate DNA with specific activities around 100,000 cpm/ug. After growth to a concentration of 1 x 106 cells/ml the cells were centrifuged at 2000g for 2 min and resuspended in lysis buffer, 10 mM Tris, 10 mM NaCl, 1.5 mM MgC1 2 , 1 mM CaC1 2 , 1 mM triethanolamine pH 7.4 (B4; Plageman, 1969), pelleted at 2K for 5 min, resuspended in B4 and allowed to stand for 5 min. The cells were lysed with 10 strokes of a glass homogenizer and the nuclei spun down at 2000g for 5 minutes. crude pellet was then extracted as described above for This the unlabeled ascites DNA. Preparation of DNA fragments DNA fragments of various sizes were prepared by shearing Virtis 60 homogenizer (Britten et. al., 1974). in a 4000 nucleotide fragments were prepared by shearing at 7500 RPM and 800-1100 nucleotide fragments were sheared at 30,000 RPM for 45 min. 350 nucleotide DNA was prepared by shearing at 50,000 RPM in 66% glycerol Britten et. al. (1974). by Broader distributions of fragment lengths for interspersion studies were prepared by shearing at 2000 from as described to 40,000 RPM 30 sec to 5 min and individual sizes fractionated from preparative - 6 - alkaline sucrose filtered gradients. After shearing, all preparations were and passed over Chelex-100 (BioRad) and then precipitated from 0.3 M NaAcetate with 2.5 volumes of 95% ethanol. Sizing DNA fragments Single stranded DNA sedimentation fragment lengths were through alkaline sucrose gradients. determined by Isokinetic sucrose gradients (Noll, 1967) were formed in SW41 tubes in 0.1 N NaOH using a Vmix of 10.4 ml, Cflask = 16 %w/v, Cres = 43% w/v. centrifuged from 16 to 24 hours at 40,000 RPM. markers of known molecular weight times. Gradients were All tubes contained two and samples were run at least two Molecular weights were calculated from sedimentation rates using the Studier (1965) equations. S-1 nuclease renaturation and digestion DNA samples to be digested with S-1 nuclease were incubated in 0.3 M NaCl 0.01 MPipes pH 6.8 at 65 °c (Smith et. al., 1975; Britten et. al., 1977). The equivalent Cot was calculated using a factor of 2.31 to correct . for the 0.3 M Na+ concentration. Long (>500 nucleotide) DNA samples were alkaline denatured to ensure complete denaturation. After incubation, samples to be nucleased were diluted with an equal volume of 0.05 M NaAcetate 0.2 m.M znso 4 pH 4.2 and dithiothreitol was added to 5mM. The final reaction mix was 0.05 M Pipes 0.025 M NaAcetate 5 mM pipes 0. 1 mM Znso 5mM DTT at oH 4.4. The pH was checked 4 in each experiment. The S-1 nuclease preparation used was the gift of Dr . F. Eden and D. Painchaud 1nd ha8 been extensively characterized - -7- (Britten et. 15 ul/mg al., 1977). (DIG = .78; The standard enzyme to DNA ratio used Britten et . fraction repetitive in the genome. nuclease for 45 min at 37 °c. was al., 1977) for measurements of the DNA samples were incubated with S-1 The reaction was terminated by addition of 2.0 M PB to a final concentration of 0.12 M and the over hydroxyapatite. fraction bound to For most experiments, the sample passed hydroxyapatite was denatured at 100 °c and eluted with 0.12 M PB. The DNA was not denatured before A-50 chromatography but eluted with 0. 5 M PB. The size distribution of S-1 nucleased duplexes was measured on a 110x1.3 cm column of agarose A-50 (Bio-rad). The gel bed was poured around a support of 5 mm glass beads (Britten et. al., 1974). Samples were chromatograohed in 0.12 M PB using 32Po 4 as an inclusion marker. DNA renaturation Samples which were not incubated in 0 to be digested .12 M PB at by S-1 6o 0 c or in 0.48 M PB at 70 °c. incubation, samples were frozen in dry ice-ethanol. and diluted 0 nuclease were After Samples were thawed to 0.12 or 0. 14 M PB and passed over hydroxyapatite at 60 c. The fraction bound was eluted after thermal denaturation at 100 in all experiments except the melt presented in Figure 4. °c The fraction and rate parameters for the renaturation curves were calculated using a non-linear least squares fitting program (Pearson et. al., 1977a). - 8 - Melting DNA samples were melted in 0.12 M PB in a Gilford model 2400 spectrophotometer equipped with a model 2527 thermal cuvette. Samples were melted at a rate of 0.5 °c/min and the A260 automatically sampled at 0.5 °c intervals. Hyperchromicity was calculated from the formula H _ A260(98°9) - A260(60°C) - A260(98 C) after subtraction of the buffer absorbance at each temperature. Results The renaturation of rat DNA Repeated sequences can be characterized renaturation of short DNA. nucleotide long fragments. Figure from 1 shows the renaturation of 350 The fraction of the duplex was determined by hydroxyapatite binding. fits to these data are summarized in Table I. qualitatively distinct components: between Cot 200 and 20,000; the kinetics of fragments containing Least squares computer The data display three a single slow component renaturing a more rapidly renaturing component binding between Cot 0.02 and 200 and a very rapidly reannealing component bound by the earliest times in the data. The slowly renaturing component containing 55-65% of the DNA is single copy non-repeated DNA. Table I shows two different fits of the data analysing this component. The first fit, a "free fit", allows all parameters to vary to find the best fit. This fit does not terminate but will continue beyond these parameters to a parameter set with more - 9 - Figure 1: Renaturation of 350 nucleotide rat DNA fragments These data have been collected by a number of investigators (Holmes and Bonner, 1974a; Gottesfeld et. al., 1976). Denatured 350 nucleotide DNA was renatured in .12 M PB at 60 °c or in .48 M PB at 70 fraction The single stranded was measured by hydroxyapatite chromatography. Equivalent Cot is the Cot (moles/liter-seconds) concentration factor (Britten et. single copy rate times a salt al., 1974). The solid lines show the least squares fit 1968). °c. of the data using a fixed at 0.00034 for a genome size of 2.9 pg (Sober, The actual coefficients for this fit are shown in Table IB. - 10 - --------------.,..----,-~---,------- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 u t-- z 0 w _J <( > ::) Qo -w d 0 d 0 0 d 0 N V ~ 00 q d d d d d 3111\:!d\:!AXO~OAH 01 aNnOB \:!NO NOllJ\:1~~ - 11 - Table I Renaturation of 350 nucleotide Rat DNA fragments A. Unconstrained fit: (no parameters fixed) Goodness of fit: 3.0% Component Fraction Rate Cot 1/2 Repetition Frequency 1 0.0883 > 100 0.01 >350,000 2 0. 177 (+0.015) 1. 06 (±0.39) 0.9434 3500 3 0. 191 C±0.098) 0.00381 (±0.00308) 262.5 11 4 0.537 (±0.079) 0.000292 C.1:0.000132) 3424 Final fraction unreacted: B. Constrained fit: 0.0067 ±0.0453 (single copy rate fixed at 0.00034 for 2.9 p~ genome) 1 0.0882 > 100 0.01 >350,000 2 O. 176 (±0.014) 1 . 10 (±0.39) 0 . 909 3235 3 0. 181 (+0.042) 0.00421 (±_0.00211) 238 12 4 0.525 (±0.051) Q.!..QQ.Q~ 2941 1 Final fraction unreacted: 0.0298 ±0.0194 - 12 - C. Constrained fit: Single copy and repetitive rates fixed 1 0.0729 > 100 0.01 2 0. 176 (+0.011) 2-L.9.1 a 0.337 8727 3 0.213 (±0 , 020) Q.&Q.1ilia 243 12 4 0.499 (±0,030) Q&Q.Q.ll Final fraction unreacted: 2941 >350,000 ' 1 0.0391 ±0,0165 FootRotes to Table I Fixed repetitive rate constants calculated from fit of the data in Fi~ure 2. the - 13 than 100% reaction (Pearson et. al., 1977a). In the second fit, the rate constant for the single copy component was appropriate For a rat genome size of for the genome size of the rat. fixed 2.9 picograrns (Sober, 1968) the single copy rate constant 0.00034 liter/mole-sec corresponding to at a number is a Cot1/2 of 2900. fixed at This fit terminates properly and we can calculate the fraction in the single copy component and the fractions and repetition frequencies of the repetitive components. The renaturation from Cot 0.02 to 200 is due to sequences repeated from the reaction of 10 to 100,000 times in the rat genome. 25% to 30% of the 350 nucleotide long fragments contain duplex region of the curve. in the Due to the scatter in the data and the fraction of the genome involved, accurate estimation of the repetition the different this The DNA fragments bound to hydroxyapatite contain single strand tails, and the real fraction of repeated sequences in genome is lower. About components is difficult. frequencies for This is reflected in the high parameter error estimates. For better resolution of the fraction of the genome was isolated. fragments were renatured sequences repetitive components a repetitive to Cot 3H labeled 350 nucleotide long 100 and separated on hydroxyapatite. the duplexed repetitive This repetitive fraction was then hybridized with a 100 - 1000 fold excess of whole 350 nucleotide long fragments. The data are shown in Figure 2. This analysis of the repetitive fraction provides a more accurate estimate of the reaction rate constant. repetitive About half of the repetitive reaction takes place with a rate constant of 3.0 liter-sec/mole, corresponding to a - 14 - Figure 2: Renaturation of selected repetitive sequences 350 nucleotide 3H labeled rat DNA was renatured to Cot 100 and double strand containing fraction (30%) separated on hydroxyapatite. This fraction was driven by unfractionated 350 nucleotide unlabeled DNA. the rat The mass ratio of driver to tracer DNA was 100 from Cot 0.001 to Cot 0.5 and 1000 from Cot 1.0 to Cot 1000. The best least squares fit to the data gives 0.45 of the DNA reannealin~ with a rate of 2.97 liter-sec/mole and 0.23 reannealing with a rate of 0.00412 . These 0.12 of the DNA had not reacted at the highest Cots. rate values were used for the third fit in Table I. drawn throu~h the data displays this fit. are also Figure 1. shown. The line The two repetitive components The line above the data is the whole rat DNA fit from - 15 - I .,,, .,,, .,,, ,,, ,,, I I I / / / .,,, ,,, ,,, ,,, ,,, / I 0 0 0 I 6 0 0 0 0 0 / 0 0 0 ,,,."' / / I I 0 u ~ z w 0 I I - 0 0 I I I . _J I <l; > :::) 0 I . aw I I I I d I I I I I q 0 I I 0 q 0 0 N ~ W 00 0 0 0 0 0 0 • 3111v'dv'AXO~0AH 01 0NnO8 'vN0 NOll:)'v~~ - 16 repetition frequency of constant of 0.004 frequency of 10. 10,000. The remainder hybridizes with a rate liter-sec/mole, corresponding in a interpretation the rat genome in Table I. of the kinetic There is little difference in the goodness of fit criterion for each of the three fits, use the second repetition These values can be used in the least squares fit of the whole rat data and provide a third fractions to interpretation (B) as our best and model we will for rat DNA renaturation. Ten percent of rat DNA has renatured by the earliest times shown on the curve. These sequences may contain very highly repeated (>500,000 fold) DNA sequences and fold-back self-complementary sequences. A.gain, some of this fraction is due to single strand tails on duplexes bound to HAP. This paper will concentrate on the repeated fraction of the renaturing before Cot 50. It is difficult to study the low (10 to 50 fold) repeat fraction because of substantial copy sequences. These fractions for the purposes contamination by single sequences have been classed with non-repeated of this paper. The sequences are included in most of our analysis. et. al. in prep.) that the fold-back are moderately repeated sequences in the genome. genome sequences rapidly There is evidence (Wu, complementary repeat similar reannealing to the sequences other repeated - 17 Melting experiments The number and length of the repeated sequences can be calculated from measurements on the fraction of the genome in true duplex after renaturation to repetitive Cots. We have used optical melting and nuclease digestion to measure the S-1 fraction of the genome containing repeated sequences and the length of the duplexed repeats. To measure the repeated nucleotide fraction of rat DNA optically, 1000 long fragments were renatured to Cot 5 and Cot 50 and melted in a spectrophometer. Figure 3 shows a sample melt, native DNA standard and a remelt of the melted DNA. which includes a The remelt shows the contribution of single strand tails and very rapidly reannealing sequences to the hyperchromicity of the duplex. Table II summarizes the measurements on Cot 5 and These values are very sensitive for single strand hyperchromicity. In addition, some of the apparent single strand hyperchromicity is due to renaturation to Cot 50 duplexes. • corrections of rapidly transition above 80 °c. 0.02 reannealing sequences indicated by the The hyperchromicity in the transition, 0.015 to is consistent with 5% rapidly reannealing DNA. When this 5% fraction is added to the values in Table II, we find 18% of the genome is in duplex at Cot 5 and 24% is in duplex at Cot 50. An average length for repeat duplexes can be measured by melting DNA fragments of different length containing duplexes (Graham et. 1974). al., DNAs of different lengths were renatured to Cot 50 and duplex containing strands isolated on hydroxyapatite. The optical melt of the duplex containing fragments is shown in Figure lt and summarized in Table III . the results are The hyperchromicity of each strand length is a - 18 - Figure 3: Melt of rat DNA fragments renatured to Cot 5. DNA was sheared to 1000 nucleotides, denatured and renatured to Cot 5 in .12 M PB. The sample was then melted in a spectrophotometer equiped with a thermal cuvette. 0.5 °c/minute to 98 °c. The temperature was raised at a rate of After melting, the samples were cooled to 60 °c and remelted. DNA. (□) 1000 nucleotide DNA renatured to Cot 5. (◊) native DNA. <+> remelt of the - 19 - ◊ ◊ -u 0 ◊ ◊ 0.9 ◊ OJ ◊ en ..__.. ◊ 0 <.O N ◊ ~ 0 <.O N <I ◊ ◊ 0.8 ◊ ◊ 0.7 _____.___ _ _ _ ____.....__ _......___ ___.__ _ 40 50 60 70 80 90 100 ~ TEMPERATURE °C - 20 - Table II Melts of 1000 nucleotide rat DNA fragments Renatured to Cot 5 and Cot 50 Sample Hyperchromicitya Tm oc Fractionb native Cot 5A 0.0800 77. 0.1818 B 0.0687 75.5 O. 1305 C 0.0586 75. 0.0764 D 0.0677 76. 0. 1259 Average fraction native 0.1287 ± 0.0431 Cot 50A 0.0788 74.5 0. 1764 B 0.0821 75.5 0.1914 Footnotes to Table II a Hyperchromicity measured from 60 °c to 98 °c. b Fraction native is corrected for 0.04 hyperchromicity after remelt and uses 0.2600 for the native hyperchromicity . The calculation is: H.yQe r . - O_._Q.!i 0.2600 - 0.04 - 21 - Figure 4: Melt of rat DNA fragments containing a repeated sequence . DNA was sheared to three lengths, 350, 1100 and determined by alkaline sucrose . 1600 nucleotides The samples were then renatured to Cot 50 and passed over hydroxyapatite. A double strand fraction eluted by 0 . 5 M PB was dialysed to 0. 12 M PB and melted as in Figure 3. A melt of native DNA is included in the fi~ure as a reference . <+) 350 nucleotide fragments . DNA frafslllents . ( □) (X) 1600 DNA nucleotide fragments. 1100 nucleotide (◊) native DNA. DNA - 22 _ ◊ I u 0 ro (J) ---- ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ 0.7 ~ - ~ - - I 40 50 60 70 TEMPERATURE~~ 90 100 - 23 - Table III Melts of rat DNA fragments Bearing reoeated sequences Input fragment length 350 1100 1600 Fraction bound to HAP 0.26 0.44 0.47 Hyperchromicity 0. 175 0. 110 0.090 Fraction duplexa 0.646 0 . 375 0 . 292 Duplexb length 226 413 467 Fraction of genome in duplexc 0 . 168 0 . 165 0 . 137 Native 0.260 Footnotes to Table III a Fraction duplex is calculated as in Table II for fraction native . b Duplex length is the product of the fraction duple x and t he input fragment length C Fraction genome in fraction duplex (hyd r oxyapatite) duplex is the pr oduct of the and the fraction bound to HAP - 24 function of the average length of the strand in duplex. have lower hyperchromicity because the duplex fraction of the molecule (Davidson et. al., 1973b). Longer strands region is a smaller These measurements indicate the average length of the duplex is 400 nucleotides. These data also provide an estimate of the amount of repetitive sequences in the genome. Although not all of the duplex molecules can be removed from hydroxyapatite by elution with the 0.5 M phosphate buffer used in these experiments , if the fraction retained has the same hyperchromicity as the duplex fraction eluted, about 17% of the DNA is duplex at Cot 50. Nuclease digestion of repeated sequences An independent measure of the fraction duplex is provided by digestion with the single strand specific nuclease S-1. Fragments 1000 nucleotides long were renatured to Cots from 0.05 and duplexed molecules fractionated to 50 on hydroxyapatite with and without S-1 nuclease digestion (Table IV). The difference between binding with and without nuclease the is due to absence or presence of single strand tails. Using the S- 1 criterion, 6% of the DNA contains repeats renaturing by Cot 0.05 while 16% is repetitive at Cot 5 and 24% at Cot 50. To make certain that the enzyme was behaving properly, the fraction of duplex as a function of enzyme to DNA ratio was investigated at Cot 5. The 16% of the DNA in duplex at Cot 5 is not a concentration over the range 15 ul/mg to standard concentration used for a DIG= .78; and function of enzyme 30 ul/mg (15 ul/mg is the Britten et . al., 1977) is not affected by a two fold increase in the salt concentration. 0 . 215 ±0,017 (n=2) 0.307 ±0.028 (n=2) 0 . 282 ±0,032 (n=3) 0.403 ±0,051 (n=5) 0 . 05 0.5 5. 50. 0.239 ±0,010 (n=4) 0.156 ±0 , 031 (n=5) 0.134 ±0,032 (n=2) 0.080 ±0,046 (n=3) Fraction duplex b after S-1 digestion b Fraction of DNA bound to hydroxyapatite after 45 min digestion nuclease (15ul/mg , DIG=0.78 Britten et. al . , 1977) with a Fraction of DNA bound to hydroxyapatite before S-1 nuclease digestion Footnotes to Table IV Fraction containing duplex before digestiona Cot Nuclease digestion of 1000 nucleotide DNA fragments Table IV S-1 I\.) I \Jl - 26 - Figure 5: Profile of rat repeated DNA duplexes on ag3rose A-50 DNA sheared to 3000 nucleotides was denatured and renatured to 5, digested with S-1 nuclease and bound to hydroxyapatie. strand fraction (15%) was eluted with 0.5 M PB and Cot The double chromatographed on Bio-gel agarose A-50. The size of the sucrose Figure 6. fraction sedimentation. indicated was determined by alkaline The fractions marked were used for the melt in - 27 - w u 250 z<{ 32 P il CD 0:: 0 Cf) CD <{ 26 10 20 39 44 n n 30 40 50 FRACTION NUMBER 60 70 - 28 - Table V A-50 fractionation of repeated DNA fragments Cot 0.05 5.0 Fraction Fraction excluded S-1 resistant from A-50 0.0368 0.484 0.0558 0.683 0. 150 0.453 0. 140 0.489 0. 171 0.481 O. 167 0.553 0. 177 0.534 - 29 Lowering the enzyme concentration or raising should stabilize poorly matched duplexes. the salt concentration We do not believe the enzyme is digesting slightly mismatched regions as these changes do not affect our results. Fragments from nuclease digestion can length of the be used duplex by direct sizing. to determine Duplexes from 3000 nucleotide DNA fragments isolated on hydroxyapatite after nuclease digestion fractionated by size on Biogel agarose A-50. elution pattern for this DNA renatured excluded material was electron microscopy . sized nucleotides long while nucleotides. to in Cot 5. the were Figure 5 shows a typical The on alkaline sucrose The material the Table V summarizes hydroxyapatite and and gradients and by included the excluded material included peak is 200-300 is longer than 1500 A-50 fractionation data at Cot 0.05 and Cot 5. To make certain that long duplexes did not contain single strand tails, duplexes fractionated on A- 50 were melted. three fractions indicated in Figure 5 is fractions show more shown A sample melt of in Figure 6. than 90% of native hyperchromicity. All The melting temperatures range from 2. 5 °c below native for the excluded material to 15°c below for the 200 nucleotide fraisments. The melts of both the long and short repetitive fragments display a number of distinct melting regions. These regions can be seen most clearly in the melt of the lorn~ A-50 excluded material where there · are °c, 82 °c and 86 °c. The short fragments also show components with T's of 65 °c , 75 °c and 83 °c. m components melting with T's of 70 m - 30 - Figure 6: Melt of repeated DNA duplexes of different fragment lengths. Repeated sequence DNA duplexes were isolated and described in Figure 5. fractionated as Fractions 26, 39 and 44 were melted as described in Figure 3. ( +) fraction 44; native DNA reference. ( □ ) fraction 39; ( X) fraction 26; (◊) - 31 - - ~ 0.9 co (j) ...__ 0 <.D N ~ 0 <.D N <( 0.8 0.7 ' - - - - - ' - - - - ~ - - - ' - - - - - - - ' - - - - - - - L - - ~ 40 50 60 70 80 90 100 TEMPERATURE °C - 32 - Interspersion of repeated sequences The data on melting and nuclease digestion of long DNA fragments presented earlier indicate that many short surro1mded by sin~le copy DNA. of the repeated sequences are In this section we present two measures quantity of interspersed repeated experiment gives a qualitative measure of the containing interspersed repeats; sequences. fraction The first of the genome the second gives quantitative data on the fraction of the DNA interspersed, the length of the interspersed single copy sequences and the length of the repeated sequences. When 300, 1500 and 3000 nucleotide long DNA fragments are renatured and the duplex containing fraction separated on hydroxyapatite, the repeated fraction appears to increase with the length of the renatured Figure 7 shows fragments. lengths. the renaturation of these three fragment Thirty percent of 300 nucleotide long fragments are bound at Cot 50 but 70% of 3000 nucleotide DNA fragments hydroxyapatite at this Cot. strand tails on The increased short duplexes. binding are retained on is due to single These tails were seen as loss of hyperchromicity in the melting experiments and loss of hydroxyapatite binding after S-1 nuclease digestion. The precise fraction repeats and of the DNA containing short interspersed their spacing can be measured by hybridizing trace quantities of varying length DNA fragments with a vast excess of unlabeled Whole rat DNA sheared to 350 nucleotides was used short DNA. to drive labeled fragments from 250 to 8000 nucleotides in length. fraction bound The to hydroxyapatite at zero time (<Cot 0.005), Cot 5 and Cot 50 was measured. ~igures 8A and 88 show the binding of the DNA at - 33 - Figure 7: Renaturation of different DNA fragment lengths DNA sheared to 300, 1500 and 3000 nucleotides determined alkaline sucrose sedimentation was denatured and renatured. were passed fraction over hydroxyapatite and eluted by The samples the double strand containing at 100 °c with .12 M PB. The fraction single stranded as a function of equivalent Cot is plotted. ( 0 ) 300 nucleotide DNA; 3000 nucleotide DNA fraf;;lllents. ( □) 1500 nucleotide fragments; (A ) - 34 - FRACTION DNA BOUND TO HAP • 0 • 0 0 0 p p p 00 O'> ~ 9 N 0 0 p ,, 0 c_ ~ rrr, z -to ,() r--+ 0 0 0 0 0 ..o 0 0 0 (JJ 0 0 0 z -f z -f 0 - 35 - Figure 8: The fraction of rat DNA sequences containing a repeated DNA sequence as a function of fragment length. Labeled DNA fragments of various sizes were driven by whole rat DNA to Cot 5 and Cot 50. Bindin~ to hydroxyapatite without driver at very short times (zero time binding) was also measured. fragments containing E was duplex measured by hydroxyapatite Values of E were corrected for chromatography. time binding sequences z. The fraction of the the fraction of zero The zero time binding correction was calculated from a linear fit of the data in Figure BC. The values plotted in Figures BA and Bare F- Z R _ - 1.0 - Z The solid line drawn to the data represents the best least fit to the data. of the data. squares The dashed lines provide alternative interpretations A summary of the interpretations is given in the text. BA: Corrected binding of fragments driven to Cot 5. fit with fraction repetitive (frep) = 0.140, interspersion period best (Lint)= 2315 nucleotides (NT), fraction interspersed at (fint) = 0.573, Solid line: short period repetitive sequence length (Lrep) = 835 NT, fraction bound by 8000 nucleotides (f8000 ) = 0.745. One dashed line shows the fit when the fraction repetitive is fixed at 0. 17 while the other shows the best interspersion distance is set to 1000 NT. fit of the data when the - 36 - best BB: Corrected binding of fragments driven to Cot 50. fit with Solid line: frep = 0.219, Lint= 2680, fint = .730, Lrep = 1270 NT, fsooo = .836. The dashed lines show the best fits when the fraction repetitive is set to 0.25 and the interspersion period is set to 1000 NT. BC: Binding of fra~ments incubated without driver or for very short times (zero time binding). The line drawn to the data was used to correct the data in Figures 8A and 8B. - 37 - 2000 0 4000 6000 8000 A 0.9 --- --- --- --- _,.: ~---~__...::,,==--~ D 0.7 □g -~L....L...ol--~-0 0.5 D 0.3 0.1 B R 0.9 g 0.7 ,,, 0.5 - ~-..6--- _,,,....-! - D D D D D I 0.1 I ± +- - - - - - - + + 0.1 0 2000 4000 6000 FRAGMENT LENGTH (nucleotides) 8000 - 38 Cot 5 and Cot 50 corrected for the zero time binding fraction. Figure 8C shows the zero time binding used to correct lines the data. The plotted show the range of curves which can be used to fit the data. The curves show that more than 80% of the - 8000 nucleotide DNA contains repetitive sequences hybridizing by Cot 50. The spacing between repeats (the interspersion period) is about 2300-2700 nucleotides and does change significantly between Cot 5 and Cot 50. At Cot 5, 57% of the DNA is bound at a fragment length of 2300 nucleotides; 73% of the DNA is bound at a fragment length of 2700 nucleotides at Cot 50. The fraction of the DNA in repeated duplexes is the Y-intercept of the data, 14% at Cot 5 and 22% at Cot 50. not about A length estimate for the repeat duplex is shown at the X-axis intercept and is about 835 nucleotides at Cot 5 and 1300 nucleotides at Cot 50. Discussion We have fraction presented measurements on the repetition frequency, of the genome and length of repeated DNA sequences in rat DNA. In this section we will summarize the different lines of evidence which support a consistent model for repeated sequence organization in the rat genome. The kinetic data from the hydroxyapatite Cot curve are consistent with the reported genome size for the rat (2.9 pg, Sober, 1968). A variety of combinations of rate constants and repetitive and single copy fraction quantities describe the data equally well. by the standard error of the parameters in Table I. This is indicated Some of this - 39 - uncertainty is due for the reaction. 1% changes to the difficulty of getting good termination data A change from a final fraction unreacted from 5% the apparent rate constant for single copy reaction from 0.00032 to 0.00024 liter-sec/mole . ten-fold repeated to fraction In addition, the presence of a makes accurate determination of the single copy rate constant more difficult. The slave "mini-Cot" reaction (Britten et . more accurate al., 1974) provides information about the moderately repeated fraction. Isolation of the fraction excludes many of the sequences repeated less than 10 times, however. The mini-Cot analysis certainly suggests there are two definite frequency classes of repeated DNAs. Measurements on repeated sequence elements The number, size and spacing of the repeated sequences have been measured by melting, nuclease digestion and interspersion experiments. Each of these experiments measures one number well, and information which should be consistent. provides other In this section we will try to collate the data from each of the experiments to find agreement on the fraction of the genome which is repetitive and on the length of the repeated sequences. The fraction repetitive Melting presents the best data on the fraction repetitive . Nuclease digestion may destroy mismatched duplexes which are considered repeated by standard hydroxyapatite fractionation. The interspersion data do not give precise values for the fraction repetitive or length of - 40 the repeated sequences. of the Melting data are difficult to interpret because large contribution of collapsed single strand tails when only a small fraction (<20%) of the DNA is duplexed. We have tried to control for the problem by immediately remelting the samples and subtracting the hyperchromicity due to single strands. hyperchromicity is due Some of the single strand to rapidly reannealing sequences, however, and the simple calculation: single strand hyperchromicity - duplex hyper 1 single strand hyper. - native hyper. gives 13% duplex at Cot 5 and 19% at cot 50. The single strand hyperchromicity were made at 60 °c; this excludes corrections contributions from very short duplexes which melt below 50 rapidly renaturing fraction fraction presented al., 1977). not attack slightly mismatched duplexes Our studies with a range of enzyme ratios and salt concentrations suggest that poorly matched duplexes are exposed the in The enzyme ratios used in these experiments selectively clip single strand tails and do (Britten et. total 18% at Cot 5 and 24% at Cot 50. These numbers agree closely with the nuclease values Table IV. A 5% subtracted out by the single strand correction must be added to the 13% and 19% to find the in duplex: °c. to same criterion with nuclease digestion as with optical melts and hydroxyapatite fractionation. The interspersion curves at Cot 5 and Cot 50 are also consistent with 18% of the genome in duplex at Cot 5 and 24% in duplex at Cot 50. - 41 Repetitive sequence length The best measure of repetitive sequence length comes from digestion studies. on agarose A-50. excluded from These studies display the real distribution of sizes About 40% to 50% of the A-50 while 200-300 base pairs. urchin repeated DNA sequences al., 1973b; Britten et. al., 1975), a host of marine invertebrates al., and distribution of insect (Efstratiadis et. excluded/included material the sea al., 1976), Aplysia (Angerer et. an are the remainder are included with a length of These lengths are similar to results from (Davidson et. 1975) nuclease (Goldberg et. 1976). The al., is variable. This distribution may be much more sensitive to enzyme concentration than the fraction of DNA bound to material may include hydroxyapatite short nuclease susceptible regions. and in some cases excluded sequences attached by poorly matched, The variability may also be due complementary long and short sequences. to Short duplex hybrids on long molecules may have a much higher sensitivity to slight changes in incubation conditions. If 60% of the mass of the repeated sequences are 275 nucleotides long and the remainder 1500 nucleotides long, the weight average length is (0.6*275)+(0.4*1500) = 765.0 nucleotides. Under ideal conditions the duplex melting experiment would provide the weight average nu~ber while the interspersion data would provide a number average estimate. number from the melting experiments ranges from 225 nucleotides duplex length on 350 nucleotide fragments to 470 nucleotide The fragments. nucleotides duplex on 1600 This may be due to the relatively short (300, 1100 and 1600 nucleotide) DNA fragment lengths used in the experiment. - 42 On the average, even the longest fragment length, 1600 nucleotides, could only form an 800 nucleotide duplex (Smith et . long repetitive sequences renature. account for their longer than full 3000 al., 1975) when two Long repetitive sequences would not fraction of hyperchromicity until nucleotides were technically difficult; 3000 nucleotide reacted. long fragments Such reactions are fragments form duplexes which cannot be eluted from hydroxyapatite with 0.5 M phosphate and must be denatured. The interspersion long tracer/short driver experiments provide a number average repetitive sequence length near 1000 nucleotides. difficult to interpret interspersion data to these values; the extrapolation It is of the zero fraction bound is dependent on assumptions about the interspersion of the long fragments. A consistent model for repeated sequences must nucleotide hydroxyapatite binding data . This strands bound on hydroxyapatite melting data of Figure 4 (65% are about the This Cot due to Davidson, 1976) . implies 75% of the are duplexed , in agreement with the in duplex). If 0.6 of the repeated same length as the renatured fragments , (300 nucleotides from A-50 fractionation) 50% of those duplexed duplexes at contrasts with 18% duplex at Cot 5 and 24% duplex at Cot 50 measured by melting and nuclease digestion. sequences 350 According to these data, 24% of the fragments contain duplex at Cot 5 and 31% contain 50 . explain the random overlap (Smith et. fragments al. , 1975; should be Britten and The same argument gives 1350/1500 or 90% of the 1500 long fragments in duplex. This implies 0.6*0.5 + 0.4*0.9 = 0.66 of the strands should be duplexed . The agreement between the 75% fraction - 43 duplex after hydroxyapatite binding and repetitive sequence length is good. more than 50% covered short repeats. the 66% expected from the The more repeated sequences will be but we may have overestimated the fraction of In any case the agreement with our estimates hydroxyapatite data are in excellent for length of the the fraction and repetitive duplex. These estimates for measurements made on sequence length 1000-3000 nucleotide are also fragments visualized in the electron microscope (Bonner et. et. al., 1977). Three al., similar to renatured and 1973; Wilkes lengths of DNA were stripped of rapidly renaturing sequences and renatured to Cot 50. The duplexes formed from 900 nucleotide fragments had a number average length of 329 nucleotides and a weight average of 542 nucleotides. nucleotide Duplexes formed from 1700 strands had nu.mber and weight average lengths of 376 and 675 nucleotides, while 450 and 990 nucleotide duplexes were formed nucleotide strands. The increase in average duplex length with fragment length is due to the long repeated sequences. longer structures by 2500 Longer fragments can form identifiable as duplexes in the electron microscope. Duplexes formed by 2500 nucleotide fragments have a weight average close to the 770 nucleotides predicted from our nuclease experiments. Duplexes formed by shorter fragments display the repeat fraction, by Chamberlin et. 200-400 nucleotides. al. length of the short Electron microscopy experiments (1975) find the short repetitive sequence length in Xenopus is 350 nucleotides, in good agreement with our findings. fragment size used the Xenoous long duplex sequences. The experiments precluded measurements on - 44 - While the data on the fraction of long and short repetitives (Table V) are variable, we believe they reflect distinct lengths of repetitive sequences in rat DNA. Our numbers suggest the rat genome can be separated into the repetitive and single copy classes shown in Table VI. Seventy percent of the DNA is single copy, 25% repeated from 100-100,000 fold and 5% contains sequences in duplex before Cot 0.05. The 16% in duplex by Cot 5 can be divided into a long fraction with 6.4% of rat DNA and a short 200-300 nucleotide fraction with 9.6% of the DNA. The long sequences differ from shorts not only in size but also fidelity of matching. The melting data in Figure 6 show the same melting temperature dependence on length reported by Davidson et. (1973b), and Goldberg et. al. (1975). al. The Tm differences between long and short repeated sequences are not simply due to the differences length. in in The 650/n equation expressing dTm as a function of nucleotide length n (Britten et. al., 1974) would only predict a 3 degree decrease in Tm for 200 nucleotide sequences while the differences are 15 degrees. This corresponds to 12% mismatch in short repeats assuming 1.0% mismatch per degree decrease in Tm (Britten et. al., 1974). fragment melting curves show more than one The long and short transition however. There are long sequences with low precision regions and high precision short repeated sequences. Interspersion of repeated sequences in rat DNA Almost every one of the results we have presented shows qualitative evidence for interspersion of short repeated sequences with single copy DNA in the rat . If reoeat sequences were not interspersed, we could not - 45 measure the duplex length by digesting away adjacent single strands. This is not a trivial result; (1976) including S-1 similar experiments by Crain et. al. nuclease digestion and melting failed to detect interspersed repetitive sequences in Drosophila. The best data on repeated sequence short driver/long labeled range of least squares indicated in the from the tracer experiments shown in Figure 8. The fits figure. interspersion come which can be imposed on the data is Changes in the fraction repetitive move the inflection point for the short interspersion oeriod. When the nuclease and melting data are used to constrain the fit to 16% and 25% repetitive at zero single strand tail length, the short interspersion periods are 2400 and 2700 nucleotides at Cot 5 and Cot 50. interspersion periods at those two Cots repeats are suggests different probably not arranged differently. periods, 57% and 73% of the DNA is organized repeats while another 15% and The similarity of frequency At these interspersion as short interspersed 10% of the DNA is interspersed with a period longer than 6000 nucleotides. Electron microscopy experiments mapping short duplex very long (10,000-50,000 nucleotides) DNA have also determine the interspersion period in the Wilkes et. al., 1977). fragments rat (Bonner on been used to et al., 1973, These data cannot show the same division of interspersion periods shown in Figure 8 but show a large class of spacings between 1000 and 3000 nucleotides. These data are in excellent agreement with measurements on human DNA (Schmid and Deininger, from measurements on Xenopus 1976) and are not significantly different (Davidson et. al., 1973a), sea urchin - 46 (Graham et. al., 1974) or Aplysia (Angerer et. in Figure 8 are open to interspersion al., 1975). a range of interpretations. The data A model of in the rat which included three interspersion periods, at 800, 4000 and >6000, nucleotides would fit the data equally well . Conclusions Table VI presents a model for the different sequence fractions in rat DNA. copy, the exact fraction is obscured repeated fraction number and length of the About 7oi of the DNA is single by the low frequency of the DNA seen in Figures 1 and 2. slow repeat fraction contains 10% of the DNA. (10-fold) We estimate the The major repetitive component is repeated about 3000 fold and includes about 20% of the DNA. From S-1 nuclease digestion and agarose chromatography, we can measure the length of the repeated sequences . Sixty percent of the moder ately repetitive sequences are 300 nucleotides long and 40% are longer than 1500 nucleotides . 50 different 3000- fold From this length distribution we calculate there are families and of sequences 1500 nucleotides long repeated 350 different 300 nucleotide sequences with the same repetition frequency . The interspersion measurements also allow us length and number of the single copy sequences. to calculate the There are 560,000 single copy sequences 2700 nucleotides long and another 50,000 sequences longer than 6000 nucleotides . There are enough short repeated sequences (1,050 ,000) to bound each of the short single copy sequences. 0.02b 0.20C 0.08d 0.70 2 3 4 0.25 0 . 75 0.6 >6000 2500 200-400 O. 4 > 1500 Fraction Size of DNAa Distribution 1 Comp 1. 12. 5 3000 >100,000 1.85x10 9 1.69x10 7 1.76x10 5 530. <80,000 560,000 350 50 Reoetition Comolexity Number of frequency (nucleotides) Sequences Seauence distribution in rat DNA Table VI .l= -..:i - 48 - Footnotes to Table VI a ~raction of DNA is our best estimate of the true fraction duplex. These values differ from those in Table I because of single strand tails bound to hydroxyapatite. b This fraction assumes that .67 of the 0.06 of the duplex bound before Cot 0.05 is 3000 fold repeated sequence (Wu et . al., in prep . ) c This fraction includes 0.04 from Cot 0.05. the binding before d This is a rough estimate. None of the nuclease digestion studies separated the low repeat fraction from single copy DNA. From the hydroxyapatite data of Table I, 0.12 of the genome consists of slow repeats. We have assumed 0.67 of that DNA is true duplex. - 49 The highly structured sequences, the interspersion of the short universality of this repeated DNA pattern of organization and the proximity of expressed sequences to subsets of repeated sequences all suggest an important functional role for repeated sequences. and Davidson (1969) have presented a model for repeated DNA sequences which Britten regulatory function of is consistent with a wide range of structural and functional features of DNA sequences discovered since the model was proposed. The distribution and function of the long repeated DNA sequences is unclear. Some of this DNA must contain the known repetitive gene families such as the ribosomal and histone genes 1974; Birnsteil et. al., 1974; (Brown and Sugimoto, for discussion see Galau et. 1976) and must be distinct from short repeated sequences. al., Long repeated DNA may contain other sequences which are shared with short repeated sequences. Some repeated sequences may sometimes be bounded by single copy sequences and other times be bounded by other repeated sequences as part of a long repeated sequence. Although we have begun to explore the sequence relationships between long and short repeated DNA sequences (Pearson et. al., 1977b), measurement of sequences shared by these size classes is technically difficult. derived from long repeated two Many short fragments may be sequences because of mechanical shear. Purification of long sequences is easier but it is difficult to design experiments which are insensitive to 5- to 10% contamination by short repeated sequences. demonstrated, If sequence it may provide evidence overlap can for the sequences postulated by Britten and Davidson (1969). unambiguously be "integrator" isene - 50 - References cited Angerer, R. C., Davidson, E. H. and Britten, R. J. (1975) "DNA sequence organization in the mollusc Aplysia californica" Cell 6:29-39 Bonner, J., Garrard, W. Sevall, J. S. T., Gottesfeld, J. and Wilkes, M. the mammalian genome" M. M., Holmes, D. S., (1973) "Functional organization of Cold ~Pring Harb. Quant, Biol. and Stafford, D. ( 1974) ~ 38:303-310 Birnstiel, M., Telford, J., Weinberg, E. "Isolation and some properties of the proteins" Proc. ~ AQfill,_ J. and Davidson, 8. Britten, R. higher cells: QS 71 :2900-2904 H. (1969) "Gene regulation in A theory" Science 165:349-357 Britten, R. acid Sci. genes coding for the histone J. and Davidson, E. H. (1976) "Studies on nucleic reassociation kinetics: empirical equations describing DNA reassociation" Proc. Nat.. ./\cad, Sci, T,!S 73: 415-419 E. and Britten, R. 11 J . , Graham, D. Neufeld, B. Analysis of repeating DNA sequences by reassociation" Enzymology (L. Grossman and K. Britten, R. and Davidson, J., Graham, D. E. H. (1977) R. (1974) Methods in Moldave, eds.) Vol 29 part E 363-418 E., Eden, F. C., Painchaud, D. M. "Evolutionary divergence and length of repetitive sequences in sea urchin DNA" .J.... MQ.L_ .E_vol. in press - 51 Brown, D. D. and Sugimoto, evolution of ribosomal mulleri: Quant. and 5S DNAs (1974) Biol. Symp. E., Britten, R. in Xenopus J. and Davidson, E. H. ( 1975) studied by electron microscopy" .L__ 96:317-333 Crain, W. Britten, Q.Q.ld Spring Harb. 38:501-505 "Sequence organization Biol. "The structure and in Xenopus laevis and Xenop!J§. the evolution of a gene family" Chamberlin, M. Mol. K. R., Eden, F. R. J. repetitive and (1976) C., Pearson, W. R., Davidson, E. H. and "Absence of short period interspersion of non-repetitive sequences in the DNA of Drosophila melanogaster" Chromosoma (Berl.) 56:309-326 Dahmus, M. E. and McConnell, . D. J. ( 1969) "Chromosomal ribonucleic acid of rat ascites cells" Biochemistry 8:1524-1534 Davidson, E. transcription and H. and Britten, R. J. (1973) "Organization, regulation in the animal genome" Quart. Rev. Biol. 48:565-613 Davidson, E. J. (1973a) H., Hough, B. "General R., A.rnenson, C. Davidson, E. H., Graham, D. Amenson, S., C. and Britten, R. interspersion of repetitive with non-repetitive sequence elements in the DNA of Xenopus" J__._ Mol. E., S. Hough, .B iol. 77: 1-23 R., Chamberlin, E., Neufeld, B. B. R. and Britten, R. J. (1973b) "Arrangement and characterization of repetitive sequence elements animal DNAs" Cold QQ.ring Harb. SyfilQ_._ Quant. Biol. 38:295-301 M. in - 52 Davidson, E. J. (1975a) H., Galau, G. A., Angerer, R. C. and Britten, R. "Comparative aspects of DNA sequence organization in metazoa" Chromosoma (Berl,) 51:253-259 Davidson, E. (1975b) H., Hough, B. "Structural R., Klein, W. genes adjacent to R. and Britten, R. interspersed J. repetitive DNA sequences" Cell 4: 217-238 Efstratiadis, A., Crain, W. and Kafatos, F. MA: H. E., Hough, B. R., Britten, R. J. (1976) "Evolution of repetitive and non-repetitive 12 of Molecular Evolution (F. J. Ayala, ed.) Sunderland, Sinauer Assoc. pp 200-224 Goldberg, R. B., Crain, W. R., Ruderman, J. Higgins, R. C.' Gelfand, and Davidson, E. H. (1975) "DNA sequence organization Barnett, J. Britten, R. in H. N.aL. Acad. Sci, QS 73:2289-2293 A., Chamberlin,~. and Davidson, E. DNA" Chap. J., Davidson, E. (1976) "ON.A. sequence organization in the lepidopteran Antherea pernyi" Proc. Galau, G. H., Britten, R. the R., J. genomes of five marine V., R. More, G. p. ' A., Galau, G. A., invertebrates" Chromosoma (Berl.) 51:225-251 Gottesfeld, J. M., Bagi, G. , Berg, "Sequence composition of the B. and Bonner, template-active J. ( 1976) fraction of rat liver chromatin" Biochemistry 15:2472-2483 Graham, D. J. E. , Neufeld, B. R., Davidson, E. H. and Britten, R. (1974) "Interspersion of repetitive and non- repetitive DNA sequences in the sea urchin genome" Cell 1:127-137 - 53 Holmes, D. S. and Bonner, J. (1974a) "Sequence composition of rat nuclear deoxyribonucleic acid and high molecular weight ribonucleic acid" Biochemistry 13:841-848 Holmes, D. S. and Bonner, J. (1974b) "Interspersion of repetitive and single copy sequences in nuclear ribonucleic acid of high molecular weight:' Proc. Manning, J. E., Nat._ Acad. Sci. US 71: 1108-1112 Schmid, W. and Davidson, C. N. (1975) "Interspersion of repetitive and non-repetitive DNA in the Drosophila melanogaster genome" Gell 4:141-155 Noll, H. (1967) "Characterization of molecules by constant velocity sedimentation" Nature 215:360-363 Pearson, W. program for R., Davidson, E. H. and Britten, R. J. (1977a) "A least squares analysis of reassociation and hybridization data" submitted to Nuc. Pearson, W. Acids~ R., Wu., J. "Analysis of sequences in R. , Posakony, J. and Bonner, J. rat repetitive DNA: (1977b) A preliminary report" Thesis, California Institute of Technology Plageman, P. menogovirus I: G. W. and Swimm, H. E. (1966) "Replication of Effect on synthesis of macromolecules by host cell" J..... Bact, 91:2317-2326 Plageman, P. growing G. W. (1969) rat hepatoma cells I. "RNA synthesis in exponentially A caution in equating pulse labeled RNA with messenger RNA" Biochirn.,_ Biophys. ~ 182: 46- 56 - 54 Schmid , C. W. and Deining;er, P. L. ( 1975) "Sequence organization of the human genome" Cell 6:345-358 Smith, M. H. (1974) J., Hough, 8. "Repetitive and R., Chamberlin, M. non-repetitive heterogeneous nuclear RNA" J..:._ Mol. Smith, M. J., Britten, R. Biol. J. E. and Davidson, sequences in E. sea urchin 85: 103-126 and Davidson, "Studies on nucleic acid reassociation kinetics: stranded tails in DNA-DNA renaturation" Proc. E. H. (1975) reactivity of single ~ Acad. Sci. US 72:4805-4809 Sober, H. ( 1968) "Deoxyribonucleic acid content IL per cell of various organisms: Table X Mammals" Handbook of Biochemistry Cleveland: Chemical Rubber Co. p. Studier, F. W. shape of DNA" 4..:.. Mol. Wilkes, M. H-58 (1965) "Sedimentation studies on the size and Biol. 11 : 373-390 M., Pearson, W. R. and Bonner, J. (1977) "Rat DNA sequence analysis by electron microscopy" Thesis, California Institute of Technology - 55 - CHAPTER II ANALYSIS OF SEQUENCES IN RAT REPETITIVE DNA A PRELIMINARY REPORT - 56 Introduction et. Recent studies on repetitive DNA sequence organization (Davidson al., al., 1975; 1973a; Efstratiadis et. are Goldberg et. al., 1976; al., 1975; Pearson et. Angerer et. al., 1977b . ) two classes of repeated DNA sequences. 30% et. of the repeated to 50% of the sequences are substantially longer, over 1500 nucleotides . sequences there In many organisms, 300 nucleotide interspersed sequences comprise 50% to 70% DNA while indicate Early studies on the (Davidson et. al., 1973a; arrangement of repeated DNA Chamberlin et. al., 1973) did not suggest long repeated al., 1975; sequences. Bonner These long sequences are consistent with a model for regulation of gene expression proposed by Britten and Davidson (1969), however, and may serve the function of "integrator genes" proposed in the model. The existence of two size classes of repeated sequences poses a number of interesting questions. kinetically different? First, are long and short sequences This question is answered by measuring the repetition frequency and complexity of purified long and short repeated sequences. Another question is perhaps more interesting: Do sequences which appear in short DNA sequences also appear in long sequences? question is more relevant to the "integrator gene" and Davidson, 1969). (Britten If some sequences in long "integrator" sequences are also present thoughout the genome as short interspersed sequences, batteries of control elements can be easily Davidson, 1969; hypothesis This Davidson and Britten, 1973). imagined (Britten and - 57 While these are intriguing questions, substantial technical difficulties hamper studies of large classes of random repetitive sequences. The technical problems stem from the difficulties isolating pure preparations of long and short repeats. in The short repeat fraction must always be contaminated by long sequences digested into short While fragments because of overlap errors during renaturation. one can be confident of much higher purity in the long repeat sequences, it is difficult to design experiments which can exclude the effects of 1% to 10% contamination of short sequences with high repetition frequency . We will examine these problems in more detail in the discussion. In this paper we present a preliminary set of experiments determine the kinetic to parameters of long and short repeated DNA sequences and to examine possible sequence relationships between the two classes of sequences. Long and short repeated DNAs have been isolated by nuclease digestion and agarose A-50 fractionation. has been driven by whole DNA to determine self-renatured to determine its complexity, repeated DNA to Long repeated DNA repetition frequency, and used examine cross-hybridization. to drive short These criteria show no kinetic difference between long repeated DNA and all repeated sequences in rat DNA and little repeated D~A sequences. to look for sequence difference between long and short Two more sensitive experiments have been used short sequences internal to long repeated DNA fragments . Long repeat DNA h~s been used to drive varying length whole DNA tracers to determine whole DNA. fraction the interspersion period of the sequences in long DNA in And long DNA fragments have been self-reacted and the of single strands on duplex containing molecules determined by - 58 hydroxyapatite fractionation before and after single strand nuclease digestion. Electron microscope visualization of these renatured duplexes indicate they contain branched structures. All of these data are consistent with the presence of short repeated sequences within long repeated sequences. Materials and methods Preparation of DNA Unlabeled DNA was extracted from rat ascites cells and DNA labeled and extracted from Novikoff hepatoma cells as described in Pearson et. al., (1977b). DNA fragments 3000 to 4000 nucleotides long were prepared by shearing for 45 min at 7500 rpm in a Virtis 60 homogenizer (Britten et. al., 1974) in 0.03 M NaAc pH 6.8. nucleotides to 350 at 50,000 RPM in 66% glycerol (Britten et. al., 1974). The DNA was then passed over Chelex 100 DNA was (Bio-rad) sheared and filtered and determined by precipitated with EtOH. Sizing DNA fragments Single sedimentation stranded DNA fragments through alkaline lengths were sucrose gradients. Isokinetic sucrose gradients (Noll, 1967) were formed in Sw41 tubes in 0 . 1 N NaOH using a Vmix of 10.4 ml , Cflask = 16 .0% w/v, Cres = 43% w/v. centrifuged from 16 to 24 hours at 40,000 RPM . makers of known molecular weight and times. Gradients were All tubes contained two samples were run at least two Molecular wei~hts were calculated from sedimentation rates using - 59 the Studier (1965) equations. Preparation of long repeated DNA fragments DNA shared to 4000 nucleotides was denatured and incubated at 65 °c in 0.3 M NaCl 0.01 M Pipes pH 6.8 to an equivalent Cot 5 using a factor of 2.31 for the reaction rate increase due After incubation, to the Na+ concentration. samples were diluted with an equal volume of 0.05 M NaAc 0.2 mM Znso4 pH 4.2 and concentration of 5mM. The dithithreitol was added final to a final reaction mix was 0.15 M NaCl, 5mM Pipes, 0.025 M NaAc 0.1 mM Znso 4 5 mM DTT pH 4.4. DNA samples were incubated with S-1 nuclease for 45' at 37 the reaction mix chilled on ice and made 0.12 M PB. °c and Duplex DNA strands were separated by hydroxyapatite chromatography, eluted with 0.5 M PB and chromatographed on Biegel agarose A-50. The long unlabeled DNA used in cross-hybridization and self-reaction and long/short interspersion experiments was isolated from 10 mg of 4000 nucleotide DNA. incubated the The DNA was denatured for 5 min at 100 °c, to Cot 5 (13 min) and digested with 250 ul of an S-1 nuclease preparation (the gift of Dr. Francine Eden). T his nuclease preparation has been extensively characterized (Britten et. al., 1977). concentration used (25ul/mg) corresponds to a DIG= 0.85 which times the standard incubation. is The 1.7 After hydroxyapatite chromatography, 16.8% of the DNA was found duplexed (30 A260 units) and this duplex DNA was passed over agarose A-50. 55% of the DNA was excluded. - 60 The 3H labeled DNA used as tracer in the repetition cross-hybridization experiments was prepared as frequency above. preparation, the enzyme to DNA ratio was 25ul/mg, 19% In and this of the DNA was bound to hydroxyapatite and 47% of the repetitive duplexes were excluded from agarose. A third preparation of DNA was used in the melting and renaturation experiments displayed in Figures 7 and 8. 4000 nucleotide DNA was alkaline denatured, incubated to Cot 5 and digested with 25ul/mg of nuclease. The fraction resistant duplex was 14%; 58% of the DNA was excluded from A-50. Renaturation Samples which were not to be digested by S-1 incubated in O .12 M PB at 6o 0 c or in 0.48 M PB at 70 °c. incubation, samples were frozen in dry ice-ethanol. and diluted 0 nuclease were After Samples were thawed to 0.12 or 0.14 M PB and passed over hydroxyapatite at 60 c. The fraction bound was eluted after thermal denaturation at 100 °c. The fraction and rate parameters for the renaturation curves were calculated using a non-linear least squares fitting program (Pearson et. al., 1977a). Melting DNA samples were melted in 0.12 M PB in a Gilford model 2400 spectrophotometer equipped with a model 2527 thermal cuvette. were melted at a rate of 0.5 °c/min and the A260 at 0.5 °c intervals. Samples automatically sampled Hyperchromicity was calculated from the formula H _ A.2,QQ(98°9) - A260(60°C} - A260(98 C) - 61 after subtraction of the buffer absorbance at each temperature. Electron microscopy of DNA DNA was dialyzed against 0.1 M Tris 0.01 M EDTA pH 8.5, made in 0.05 mg/ml cytochrome c, 40% formamide and spread for microscopy by the modified Kleinschmidt technique of Davis et. (1971). The hypophase was 10% formamide in 0.01 M Tris 0.001 M EDTA pH 8.5. Uranyl acetate (5 x 10-5 M in parlodian-coated ethanol) was used al. for staining. The or carbon-coated 300 mesh grids were rotary shadowed with Pt-Pd (80:20) and observed under a Philips 300 electron microscope . Results Figure 1 presents data on the renaturation of 350 rat DNA assayed on hydroxyapatite and the hybridization of a selected repetitive fraction. these data The repetitive nucleotide long Table IA is a summary of a least squares fit to using a single copy rate determined from the genome size. fraction of the genome is overestimated by these measurements beause the duplex fraction bound to hydroxyapatite contains single strand tails. Table IB indicates the true repetitive fraction of the DNA determined by melting and nuclease digestion experiments (Pearson et. al., 1977b). When 4000 nucleotide DNA is renatured to Cot 5, nucleased and duplexes sized the on agarose A-50, repetitive sequences are fractionated into two classes. A typical column profile is shown in Figure 2. (4000 fragments nucleotide) were used Long to minimize creation of short fragments by random shear and overlap of long repeated sequences. The - 62 - Figure 1: Renaturation of 350 nucleotide rat DNA fragments This data has been collected by a number of investigators and Bonner, 1974; Gottesfeld et. al., 1976; Pearson et. al., 1977b). Denatured 350 nucleotide DNA was renatured in. 12 M PB at 60 .48 M PB at 70 hydroxyapatite °c. al., fraction chromatography. (moles/liter-seconds) al., 1974). The (Holmes °c or in single stranded was measured by Equivalent Cot is the Cot times a salt concentration factor (Britten et. The solid lines show the least squares fit (Pearson et. 1977a) of the data using a single copy rate fixed at 0.00034 for a genome size of 2.9 pg (Sober, 1968). The actual coefficients for this fit are shown in Table I~. Also included is the hybridization of a 3H labeled repetitive DNA fraction renaturing before Cot 100 (Pearson et. al., 1977b) driven by whole rat DNA. (□) Renaturation of whole rat DNA. repetitive rat DNA driven by whole rat DNA. ( •) Hybridization of - 63 - 0 0 0 0 0 0 0 u rz w _J <{ > ::::> Qo -w d 0 d 0 0 d 0. 0 . O N . O v W. O . 00 0 0. 3111\1d\1AXOtiOAH 01 ONnos \1NO N011.::n1ti~ - 64 - Table IA Renaturation of 350 nucleotide Rat DNA fragments Constrained fit: (single copy rate fixed at 0.00034 for 2.9 pg genome) Goodness of fit: 3.1% Component Fraction Rate Cot 1/2 Repetition Frequency 0.0882 > 100 0.01 >350,000 1 . 10 0.909 3235 238 12 2941 1 2 (±0.39) 3 0. 181 (±0.042) 4 0.525 (±0.051) 0.00421 (+0.00211) Final fraction unreacted: 0.0298 ±0.0194 0.02b 0.20c o.oect 0.70 2 3 4 0.25 0.75 O. b >6000 2500 200-400 0.4 > 1500 fraction Size of DNAa Distribution 1 Comp 1. 12.5 3000 >100,000 1.85x10 9 1.69x1o7 1.7bx10 5 530. <~0 , 000 560,000 350 50 Repetition Complexity Number of frequency (nucleotides) Sequences Sequence distribution in rat DNA Table IB a, I \Jl - 66 - Footnotes to Table I a Fraction of DNA is our best estimate of the true fraction duplex. These values differ from those in Table I because of single strand tails bound to hydroxyapatite. b This fraction assumes that .67 of the 0.06 of the duplex bound before Cot 0.05 is 3000 fold repeated sequence (Wu et. al., in prep.) c This fraction includes 0.04 from the binding before Cot 0.05. d This is a rough estimate. None of the nuclease digestion studies separated the low repeat fraction from single copy DNA. From the hydroxyapatite data of Table I, 0.12 of the genome consists of slow repeats. We have assumed 0.67 of that DNA is true duplex. - 67 - Figure 2: Profile of rat repeated DNA duplexes on agarose A-50 DNA sheared to 3000 nucleotides was denatured and renatured to 5, digested with S-1 nuclease and bound to hydroxyapatie. The double stranded fraction (15%) was eluted with 0.5 M PB and chromatographed Bio-gel agarose A-50. Cot on The size of the fraction indicated was determined by alkaline sucrose sedimentation. - 68 - w u z<{ 250 32 P i~ CD a:: 0 Cf) CD <{ 10 20 30 40 50 FRACTION NUMBER 60 70 - 69 incubation was carried out to Cot 5 to prevent renaturation of single copy sequences at a higher effective Cot because of the longer fragment length. 1500- 2000 The long DNA excluded nucleotides long . from agarose A-50 is The short included DNA is 200-300 nucleotides long. Hybridization of long repeated DNA sequences Long labeled repeated DNA fragments were sheared to 350 nucleotides and driven by whole 350 nucleotide long DNA to determine the repetition frequency of the long DNA. the long DNA. data. Figure 3 shows the hybridization curve for Table II shows the results of a least squares fit to the A large fraction of the DNA (60%) hybridizes by Cot 10 with a Cot 1/2 of 0.167. The remaining 20% of the DNA which hybridizes by Cot 1000 contains less frequently repeated sequences from the slower repeat class shown in Figure 1. This part of the reaction may also include some single copy DNA sequences . The complexity of the long DNA was determined by self-reaction of this preparation. This curve is shown in Figure 4. preparation was sheared to 350 nucleotides to exclude In length effects. this preparation, 12% of the DNA was in duplex at Cot 5 of which 45% (or 5.4% of total rat DNA) was excluded from A-50. for Again the DNA Our best estimate the true fraction repetitive at Cot 5 is 16%, so the long sequences should represent a 1/( . 45*. 16)=13 . 9 fold purification of sequences whole DNA . Table III presents t wo fits of the data. best "free" fit with no parameters fixed. from The first is the In the second fit we assumed the long sequences are a fraction purified from whole repeat DNA . Two components with rates of 13 . 5 x 2.97 and 13 .9 x 0.0404 were The fit . - 10 - Hybridization of 3H long repetitive sequences driven Figure 3: by whole rat DNA 3H labeled fractionation long repeated on Bio-gel DNA fragments were isolated agarose A-50 and sheared to 350 nucleotides. These fragments were driven by a 100-fold excess from Cot 0.01 0.1, after to Cot a 200-500 fold excess from Cot 0.2 to 0.5, a 1000-fold excess from Cot 1.0 to 1000. and a 10,000- fold excess of unlabeled 350 nucleotide whole rat DNA at Cots greater than 1000 in 0.12 M PB at 60 °c. (e) fraction single stranded after binding to hydroxyapatite 0.12 M PB at 60 °c. in The line drawn through the data represents the best least squares fit presented in Table II. - 71 - ---'------,,----.----~--r--~---,,----~---r--, 0 0 0 6 0 0 0 • Q 0 0 0 0 0 0 0 u .,_ z 0 w _J <t > ::) 0. 0 w d 0 d 0 0 d 0 d N o ~ d W d 00 o 0 • 3111':Jd"vAXOt:J0AH 01 0Nnos 'vN0 NOllJ"vt:J~ - 72 - Table II Hybridization of sheared long repeated DNA sequences Driven by whole rat DNA A. Unconstrained fit: (no parameters fixed) Goodness of fit: 2.1% Component Fraction Rate Cot 1 /2 Repetition Frequency 1 0.592 (±0.020) 6.0 (±_0.86) 0. 167 17,600 2 0. 17 3 (±0.017) 0.00286 (±0,00309) 350 10 Final fraction unreacted: 0.132 ±0.014 B. Constrained fit: (slow repetitive and single copy rates set at values from Table I) Goodness of fit: 2.2% 1 0.600 (±0.020) 5.32 0 . 188 15,600 2 0. 178 (±0,017) ~Q.041.!!_ 242 12 3 0.032 (+0.050) Q.00Q.ll 2941 1 Final fraction unreacted: 0.010 ±0,029 - 73 - Figure 4: Renaturation of sheared long repeated sequence DNA fragments 3tt labeled and unlabeled long repeat DNA was isolated as in Figure 3. described The two DNA fractions were mixed and incubated to the Cot values shown. (e) fraction of 3H labeled counts single stranded after binding to The hydroxyapatite. line through the data is the least squares fit of two components with rates calculated from the rates of the fractions in the rat components (14-fold). repetitive times the expected purification of the long DNA - 74 - 0 0 0 0 . 0 0 0 • 0 0 o -0 u ~ z qw - __J <{ > :) -: a Ow 0. 0 0 0 d - ...__._______,__ 0 __.__--"--~-._____.__ N ~ W 0 0 0 __.__.........__--L--_....__. d 00 0 d o o o o • 311 l'Jd'JA XO~OJ..H 01 ON nos '1NO NOll:)'J~_:j - 75 - Table III Renaturation of sheared long repeated DNA fragments A. Unconstrained fit: (no parameters fixed) Goodness of fit: 0.9% Component Fraction Rate Cot 1/2 1 0.615 (±0.009) 47.6 ( ±3 . 2) 0.021 2 0.112 (±0.009) 0.114 (±0.040) 8.77 Final fraction unreacted: Repetition Frequency 10,oooa 0.216 ±0.008 B. Constrained fit: (rate constants set to values appropriate for 13.9 fold purification from whole DNA) Goodness of fit: 1.0% 0.618 (±0.085) 2 0. 107 (±0.009) 41.3 0. 02.12. Final fraction unreacted: 10,oooa 17.4 0.207 ±0.007 Footnotes to Table III a Repetition frequency whole DNA of the purified sequences in - 76 - Figure 5: Cross-hybridization of short repeated DNA sequences driven by long repeated DNA sequences The unlabeled long repeat fraction used in Figure drive sheared 4 was used to short DNA fragments from the included peak in Figure 2. The driver to tracer ratio was 10 for Cot 0.002, 20 for Cot 0. 01 and 100 for Cots greater than 0. 5. ( • ) fraction hydroxyapatite . of 3H counts The line drawn single stranded after binding to through the data is the best least squares fit of the data with coefficients presented in Table IV. - 77 - ...--~--,,---r---,---~--r--r---~-------~ 0 0 0 0 .. 0 0 0 0 0 ou0 J- z C?· w - _J <( > ::) ~a Ow 0 d 0 0 d 0 N V W 00 0 d d o o d • 3lll'vd'vAXO~OAH 01 ONnOB \tNO NOll:)\f~.:J - 78 - Table IV Hybridization of sheared short repeated DNA sequences Driven by sheared long repeated DNA sequences A. Unconstrained fit: (no parameters fixed) Goodness of fit: 0.6% Component Fraction 1/2 Repetition Rate Cot Frequency 1 10,oooa 2 18a 0.678 44.5 (±0.010) (±2.91) 0. 132 0.0843 (±0.010) (t0.0282) Final fraction unreacted: 0.0225 11. 8 0.130 ±0.010 Footnotes to Table IV a This repetition frequency assumes the same purification factor used in Table III 13.9 fold - 79 fits are virtually identical, implying the long DNA may be a pure fraction. The simplest way to look for cross homology between long and short DNA sequences is to drive one preparation by the other. The results of an experiment in which sheared long repetitive DNA drove sheared short repetitive DNA is shown in Figure 5. presented in Table IV. The kinetics of the reaction are Shorts were not used to drive longs because of the expected cross homology. A large fraction of the short DNA may be derived from long sequences by simple mechanical shear. Interspersion of long repeated DNA sequences Long repetitive sequences should be relatively free repetitive sequence contamination. It is difficult to of short imagine a mechanical shear, hybridization or nuclease artifact which would create long repeated sequences from short ones. to drive varying We have used long repeated DNA lengths of whole DNA tracers to determine the interspersion period for sequences in long repeated DNA. Figure 6 shows the hybridization at Cot 0.5 of whole DNA tracers as a function tracer length, corrected experiment is similar to period done al., 1974; for zero those done time binding in the tracer. to determine the Efstratiadis et. ~l., al., 1976; 1975; al., 1977b). lines plotted are interspersion curves which can be drawn from DNA tracers driven by whole rat Interspersion of long sequences Graham et. Schmid and Deininger, Pearson et. is similar This interspersion in other organisms (Davidson et. al., 1973a; Angerer et. of 1975; The dashed through data DNA at Cot 5 and Cot 50. to interspersion of all - 80 - Figure 6: The fraction of rat DNA containing long repeated DNA sequences as a function of fragment len~th. Labeled whole rat DNA fragments of various sizes were prepared and driven by the sheared long repeat DNA fraction used in Figures 3 and 4. The fraction of the fragments (F) containing duplexes binding to hydroxyapatite. was measured by The value plotted (R) is the value of (F) corrected for the amount of zero- time binding in the long tracers (Z) . The values plotted are: The values of (Z) were calculated from a linear binding data from Pearson et . al. (1977b). fit of the zero-time - 81 - ''\ \ \ \ \ \ \ \ \ \ \ \ \ \ . 0 (X) o_ • •• • • • □ O en (D - 0 Cl) •• (.) • • ::::, • • • • • • • • • • • □ C 0 0 \ ' ' '' ' . 0 LcJ •• • • • • ' '' ' '□ ' ' . V 0 0::: t- z • •• •• • ' '' ' (D _J • • □ '' □ ~ 0 (..!) 0 <t 00::: N • •• •• • • • • • •• •• • N 0 (..!) w •• •• □ I t- ~z •• \ \ Cl) -0 0 • \ \ \ \ \ \ \ '' (X) • • • •• • •• • \ \ \ \ . [] \ ' • • • • • • • □ \ \ \ \ \ \ 0 .•• \ \ 0 0 0 00 LL - 82 repeated DNA sequences in the rat. The increase in hybridization from 12% at zero length to 34% at 1800 nucleotides cannot be due to of 1500 nucleotide sequences. reaction The difference in the single strand tail length for a 1500 nucleotide sequence would only account for a 20% change from 12% to 14.4% if only long sequences had hybridized. Some of the sequences in the long repeated DNA preparation must be able to hybridize with short sequences interspersed throughout the genome. The 3200 nucleotide "best fit" interspersion period is slightly longer than the 2500 nucleotide values measured in whole rat DNA. is an encouraging result. This If some of the long sequences were not shared by short repeated sequences, the interspersion period of those sequences would be longer. Self-reaction of unsheared long repeats If a long repetitive sequence contains individual repeat sequences, self-reaction of unsheared long internal short repeats should generate short duplex regions with single strand tails containing other repeated sequences. We have visualized long and short repeated DNA fragments in the electron microscope after nuclease digestion and . fractionation. A-50 While most of the molecules are linear, a number of more complex circular and branched structures are found (Plate IA) . The structures displayed are more than 95% double stranded as measured by optical hyperchomicity. Thus some of the structures must be due sequence arrangements internal to the long repeat sequences ~ to - 83 - Plate IA: Long repetitive DNA fragments 4000 nucleotide DNA was denatured and incubated to Cot 5, with S-1 nuclease hydroxyapatite. and This double the double strand strand treated fraction bound to fraction was chromatographed on agarose A-50 and an excluded fraction (approx. fraction 20 in Figure 2) spread for electron microscopy. The bar in the lower right of the photograph is 0.2 microns (about 600 nucleotides). The number average length of the excluded fragments is 2400 nucleotides. Plate IB: Twice renatured long repetitive DNA fragments Long repeated DNA fragments isolated as in Plate IA were denatured, allowed to microscopy. renature a I second time to Cot 5 and spread for electron The bar in the lower right is 0.2 microns. The number average length of the structures nucleotides in a range from 750 to 5000 nucleotides. formed was 2000 - 84 - - 85 Long repeat visualized fragments can be denatured, by microscopy or assayed for renatured and single hydroxyapatite fractionation before and after S-1 strand then tails by nuclease digestion. When isolated long fragments were denatured, renatured to Cot 5 a second time and melted, the duplexes showed 80% hyperchromicity (Figure 7). native DNA standard run in parallel is also shown. A The twice renatured material has more than 80% of the hyperchromicity of native DNA with a Tm of 81 °c. the This may not indicate significant single strand regions; slight decrease hyperchromicity may be due in to impurities introduced during the preparation and renaturation of the DNA. This twice renatured DNA was also spread for microscopy using Kleinschmidt formed. technique. Plate IB displays some of the structures A large fraction of these structures are not linear and contain branches and tails. This suggests that when the long sequences renature a second time internal repeat duplexes may form. that the the molecules melt as It is not surorising if they were all duplex; a second renaturation to Cot 5 is equivalent to Cot 50 because of the decreased complexity of the micrographs must isolated also be long fraction. at least 80% The branches seen in the duplex and may reflect hybridization of one ordering of short sequences with a different ordering of short sequences on other strands. To look for single strands formed by renaturation of internal short repeats, long repeated sequences were renatured through a range of Cots Half of each sample from 0.0005 to 10 and the samples divided. treated with S-1 nuclease regions were measured to digest was single strands and the duplex on hydroxyapatite. The other half was put - 86 - Figure 7: Melt of twice renatured long repeated DNA fragments The long repeat DNA fraction isolated and used denatured and incubated in Plate I was to Cot 5 at 60 °c in 0.12 M PB a second time. The twice renatured material was also used for Plate IB. This sample, a sample of the original long repeated fraction (incubated to Cot 5 once during isolation) and native DNA were melted in a soectrophotometer equipped with a thermal cuvette. (□) long repeated DNA fragments twice renatured to long repeated DNA fragments renatured once to Cot 5. (◊) Cot 5. (X) native DNA. - 87 - ti ---u 0.9 ~ ◊ 01 ◊ ◊ 0 co ◊ --0) ◊ 0 ◊ <.O C\J ~ 0 <.O C\J X ◊ # ◊ ~ <r 0.8 / ◊ ◊ ◊ ◊ ◊ ◊ 0.7----------'------'-------L-----'----......I 40 50 60 70 80 90 100 TEMPERATURE °C - 88 - Figure 8: Renaturation of unsheared long repeated DNA sequences A long 3H labeled Materials and Methods. DNA fraction was isolated as described The DNA was alkaline denatured and incubated to the Equivalent Cot values shown in 0.3 M NaCl 0.01 M Pipes pH 6.7. samples were ul/mg), then divided. The The One half was treated with S-1 nuclease (50 made 0. 12 M PB and hydroxyapatite. in the duplex other half of the fractions separated on sample was not treated with nuclease but made 0. 12 M PB and duplex containing strands measured on hydroxyapatite. (■) Fraction DNA bound to hydroxyapatite after S-1 nuclease. Fraction DNA containing duplex regions (binding without nuclease). (0) - 89 - FRACTION DNA BOUND TO HAP . .0 .00) .0N 0 .0 0 o~---r--~---,---r--,----,r----r--,-----,r--__,.._ 0 0 0 • 0 0 b ,,,0 C ~ ro fTl:.... z •• --i () 0 ("'+ 0 • b 0 0 0 b L..J.--'--..L-----J....__-.1.._...1..-__.....____,__.._--...1..___ _ - 90 directly over hydroxyapatite and the duplex containing fraction with any single strand tails bound and eluted. without nuclease are shown digested and undigested molecules are single The renaturation curves with and in Figure 8. samples show stranded. The differences between the 50% or more of the hybrid This may be due to internal short sequences or random overlap of fragments which are shorter parent sequences. than their These problems are discussed further below. Discussion The measurements we have made on the repetition frequency and complexity of long repetitive DNA in the rat show little difference between the long repeated sequences and repetition repeats in whole DNA. frequency of long sequences is quite close to the repetition frequency of repeats in whole DNA and the complexity of the class long size exactly the complexity expected for a fraction of sequences purified from whole DNA. If long and short repeated DNA sequences share the is The same sequences, the 7.0% of the DNA isolated as long repeated DNA would have 16% of rat DNA complexity and hybridize two fold more slowly. This experiment is not very sensitive, however, and inconsistency with the evidence from the other experiments may be more apparent than real. The least squares fit provides the interpretation that the data are consistent with long repeats being a pure fraction. The presence of two components and the variable recovery of the slow component provide a lot of flexibility in the interpretation. not be fully recovered The slow repeat component will by hvbridization to Cot 5, and its presence allows the fittin~ pro~ram to adjust the fraction of the slow component to fill in renaturation by some hig;h complex_ity component. - 91 The question of sequences shared by long and short repeats is directly addressed by cross- hybridization of short repetitives driven by long repetitive sequences. interpretation is Here the result is different; the simplest that these two classes share sequences. surprising, many of the short sequences must be derived This is not from randomly sheared long sequences which formed short complements with other sheared long sequences. Measuring the interspersion period of sequences in long repeated DNA is a more accurate probe for short/long sequence overlap. Our preparation of long repeats hybridizes with short sequences which are interspersed repeated sequences in whole DNA. like all At the short interspersion period (2500 nucleotides), 42% of the DNA is bound when hybridized to Cot 0.5. The lower Cot was used to adjust for the ten fold lower complexity of the repetitive driver. that 42%/57%=74% of repeated 2500 nucleotides hybridizin~ sequences hybridizing preparation. The result suggests sequences interspersed with a period of by Cot 5 in by Cot 50 can rat and 42%/73%=58% of be driven by our long repeat The best fit data from this experiment suggest that some sequences in long repeated DNA are interspersed with a longer period. The self-reaction of long repeat result. The initial self- reaction of long repeat fragments indicates that more than 50% of the stranded. Later in sequences presents a similar structures containing duplex are single the reaction there are no single stranded regions and the structures formed are branched . - 92 All of these results suggest that long repeated DNA contain internal sequences which hybridize with interspersed throughout the genome. problems with fragments may short sequences Unfortunately there are substantial each of the experiments. The cross-reaction and interspersion experiments are sensitive to cross shorts with longs and longs with shorts. contamination of And the self-reaction result may be artifactual due to the unknown length of the internal elements of the long repeat sequence. The kinetic complexity and cross reaction experiments are extremely sensitive to contamination of short repeated DNA with long repeated sequences found in short fragments. Random shear in the initial preparation of the DNA or mechanical shear of the repeat duplexes on hydroxyapatite generate short sequences experiments (Davidson et. al., 1973b; et. al., 1977b) long ones. Goldberg et. suggest many of the different from long repeats. from short Melting al., 1975; repeat Pearson sequences are Short sequences exhibit more mismatch than long sequences. But some short sequences must be derived sequences and the melt of the short repetitive fragments shows a high precision component (Pearson et. from long al., 1977b). It is difficult to estimate how many of the short sequences are derived from long sequences without knowing the in vivo length of long sequences in the genome . than the 4000 length of the 1500-2000 We do not know that length. nucleotide fragments long duplexes excluded nucleotides, but these may used from It may be longer in these experiments . agarose A-50 The is about include molecules which were mechanically sheared during hydroxyapatite fractionation. - 93 It is possible to put an upper limit on the amount of contamination by placing a lower limit on the number of short sequences interspersed throughout the genome. At Cot 5, 57% of 2500 nucleotide contain a short repeated DNA sequence. DNA fragments If the short repeated sequences are 250 nucleotides long, 250/2500 = 10% of the bound DNA is repetitive so 5.7% of the DNA must contain short reoetitive sequences hybridizing by Cot 5. At this Cot, .16 of the genome is in 0.057/0.16=0.356 of duplex sequences must be short. true duolex so We find 0.5 to 0.6 of duplex DNA included on agarose A-50, so 15% to 25% of that DNA or 15%/50%=30% to 25%/60%=42% of the short duplexes may be derived from long sequences. longer than period. This is an upper limit. Many short sequences are much 250 nucleotides and others may be interspersed at a longer If 33% of short sequences are derived short sequences would experiment. drive all long from long sequences, sequences in a cross reaction Conversely, 30%-40% of the short fragments would be driven by long sequences. In addition, the long sequences. This could sequences may be contaminated by short account for more of the long/short cross-hybridization reaction and the aoparent short interspersion period of sequences in long DNA. The hybridization curves of Figure 3 indicate that as much as 20% of the long sequences may cont~in (10% reaction at Cot 2900 and 10% unreacted). single copy DNA In that case, 9.6% (16% duplex at Cot 5 * 60% included on A-50) of the 20% single copy or 1.92% could be short repeat contamination. In addition, an equal fraction might be present because of incomplete digestion of single strand tails. This 4% contamination would drive the short DNA sequences at a 25 fold reduced rate '. This reaction, following a conservative 20%-30% cross - 94 reaction due to short contamination by longs might account for the results of figure 4. The problem would be more serious if as much as 10% of the single strand tails were not digested by the nuclease. Such contamination would be difficult to detect by melting experiments, which are the only independent test of long sequence purity. A low melting shoulder on the long repeated DNA melting curve could be due to short repeated DNA contamination or mismatched long repeated sequences. Some of the interspersion of long repeat sequences in whole DNA is due to sequences spaced at lengths longer than nucleotide interspersion period in rat (Pearson et. the average 2500 al., 1977b). If short DNA sequences were spaced at 1000 nucleotides and long repeated sequences were spaced at 4000 nucleotides, a small (4%) amount of short sequence overlap or contamination might explain the results. These data, and data from these kinds of experiments in general, is not accurate enough to determine the fraction of short sequences required to exolain the reaction. caused In addition, much of the by sequences which renature interspersion curve may be well before Cot 5. The 4% contaminant of short repeated sequences in long fragments may be more repetitive that whole repeated DNA. This is likely, more repeated sequences will have better chances for multiple collisions to form concatenate structures which could be nuclease resistant. long If so, the contaminant could account for some of the binding at the short fragment lengths. This experiment does not measure the repetition frequenc y of the sequences driving the sequences generating the interspersion curve . - 95 The concentration/repetition frequency interpretation of much of these data. very small level frequency affects the cause of the affects the the One can never be certain that a of contaminant repeated with higher is not particularly argument hybridization. cross-hybridization and than average This problem interspersion experiments. Interpretation of the self-renaturation of lorn;i; repeated sequences in Figure 8 is dependent on the length of the long sequences in shown the DNA. Figure 9 shows two repeated sequences are possible longer than interpretations. the fragments If the long of DNA renatured (Figure 9A) the second time, 50% of the DNA on a duplexed structure bound to hydroxyapatite will be single stranded (Britten and Davidson, 1976) because of random overlap. On the other hand, the single strands associated with duplexes may be due to more complex branched structures as shown in Figure 98. Electron microscope visualization of the self-reaction products supports structures in Plate IB could arrangements in the reflect different different long repetitive fragments. due to overreaction of the repeated tails to second model. long The multi-ended internal sequence They may also be sequences allowing single strand react with other molecules to build up larger structures with branches. Problems with the long fragment self-reaction experiment are more easily solved. These results are not affected by moderate levels of contamination in the long sequence population as these experiments look at the majority of the mass of the long sequences . experiments measure properties of the most prevalent species. Self- driven - 96 - Figure 9: repeated DNA This figure presents two models for the result of Figure 7: that fragments: renatured Formation of duplexes between long two models long repetitive DNA fra~ents form single strand tails on a second renaturation. length 1 were Part A shows what would happen shorter if the fragment than the repetitive sequence length ft. Part B shows the sequence arrangement necessary to explain single strand structures if the fragment length 1 is ~reater than the repetitive sequence length R. The numbers (1 - 4) are the steps in the preparation of long repeated DNA fragments from randomly sheared ( X marks random cuts) cDNA fragments. 1. Denaturation and renaturation of fragments of length 1. 2. S-1 nuclease digestion of renatured duplexes. a 1a 2 and a 3 for The gaps between example are due to the single strand breaks in the parent duplex. 3. Second denaturation of long repetitive fragments including molecules from other regions of the genome. 4. Second renaturation of long repeated fragments. single strand The extent of regions depends on the number of collisions for the case of Part A while more collisions among Part B structures would more complex molecules with single stranded regions. lead to - 97 - B. A. ----R---- )( I >< >< X X I 4----L--- XI )( )( I X IX \'----- --==~--...:--___,r l o, 02 2 04 03 05 06 05 06 --- a, 02 03 04 3 l ~ o,~ a, 03) 04) 05 ~ (06 ~ 0---' 02 5 o 02 6 l a' o 2 o 3 o 4 o5 ~ ~ o, 4 o2 03 r--==. 05 o6 o6 X l - 98 - Longer DNA can be used to make repetitive duplexes. from longer fragments If duplexes are not longer than long duplex structures from shorter frafs!Tlents, the length of the long repeats has been found. sequence length fragments will not have overlap errors. Hydroxyapatite binding with and Full single strand tails due to without nuclease digestion may conclusively show short internal repeat sequences. These short duplexes may also be sized after S-1 nuclease to find length the of the short sequence element. Electron microscopy can also be used to repeats. look for At early stages of the self renaturation each molecule should only have experienced one collision on average. single internal short stranded If the long repeats are because of random overlap, the hybrid molecules should be linear with only two tails. If internal sequences are hybridizing the molecules should have 3 and 4 tail structures due to single strands on either side of the duplex. distinguish between the two This topological criterion can possibilities and is independent of the fragment length of the strands used. Conclusions The renaturation, cross-hybridization and melting have experiments we presented all suggest that there are no significant differences in complexity or repetition frequency between long and short sequences in the rat. Eden et. al. (1977) repeated DNA have measured the repetition frequency, complexity and sequence overlap between long and short repeated sequences in repetition frequencies of the the sea urchin. long and They found that the short repetitive sea urchin - 99 sequences in whole DNA are similar, with the long sequences repeated slightly less. sequences And they found is about three the kinetic complexity of the short times that of the long DNA sequences, reflecting the fraction of long sequences in the whole sea urchin repetitive sequence population (about 30%). The cross-hybridization results - using long to drive short repetitive sequences - different from the results with rat DNA. was driven at a rate in repetitive sequences the sea urchin are quite The short repetitive tracer 1/10 that of the long repetitive driver. This result suggests that 10% or fewer of the sequences in long repeated DNA fragments can hybridize with· short repeated DNA sequences. contrasts sharply with the results presented for the show any difference rat - The result we cannot between short repetitive tracer hybridization and the self-reaction of the long repetitive driver. We believe there are two explanations which could account conflicting results. First, the higher fraction for the of long repeated sequences in the rat may cause more difficult contamination problems. Second, the internal structure of the long repeated duplexes may be different in the rat. Under virtually identical digestion and we find from 40% fractionation conditions, to more than 50% of rat repeated DNA sequences from 4000 nucleotide fragments are excluded on agarose A-50 while Eden al. (1977) found about fragments were excluded. et . 30% of the sequences from 2000 nucleotide The calculations we presented earlier suggest that as much as 70% of rat repeated DNA may be longer than 1000 nucleotides. Experiments usin~ a wide range of digestion conditions for - 100 sea urchin DNA (Britten et. al., 1977) show that 47% of isolated repeat sequence duplexes are longer than 1000 nucleotides after mild digestion. A much make smaller fraction of true short repeats in the rat genome would isolation of a short repeat contamination much more difficult. fraction free of long Much of the discussion described the technical problems involved in these kinds of experiments; be simple repeat while it may to show two repetitive sequence populations are distinct, it is difficult to unambiguously demonstrate true sequence overlap. Some of the long repeated sequences must multigene families (Galau et. short repeated sequences. al., include the transcribed 1976) and must be distinct from Long repeated DNA sequences also form higher precision hybrids than most of the short repeated DNA (Davidson et. al., 1973b; Goldberg et. et. 1977b) al., al., 1975, Britten et. which are al., 1977; from short sequence probably distinct Pearson hybrids. We believe there is evidence that fragments share sequences in the rat. long and short repeated DNA Possible contaminant artifacts contribute to ambiguity in many of our results, but .able to provide any strong evidence sequence sets are distinct in the rat. we have not been that long and short repetitive The circular structures found in Plate I may be due to tandem arrangement of short sequences in long DNA fragments. of The branched structures must reflect different arrangements sequences interpretation internal to long repeated DNA is consistent with our most long/short sequence sharing: fragments. sensitive tests This for the cross-hybridization, interspersion and unsheared self-renaturation experiments. - 101 If long and short repetitive sequences are shared number of structural and organizational in the questions are raised. rat a Some short repeated sequences may be present as a single sequence within a long repeated sequence or long repeated sequences may be arranged as tandem arrays of short repeats. This interpretation is suggested by the cross-hybrization and interspersion experiments. In addition, some long repeated sequences may be made up of a number of different genome as short repeated sequences which are also found in the interspersed repeat sequences were arranged results of the electron differently microscopy experiments are easily explained. sequences. If the same short in different long sequences, the and long self-renaturation Either structural model could provide the "integrator gene" function suggested by Britten and Davidson (1969) . - 102 - References cited Angerer, R. C., Davidson, E. H. and Britten, R. J. (1975) "DNA sequence organization in the mollusc Aplysia californic1a" Cell 6:29-39 Bonner, J., Garrard, W. Sevall, J. S. T., Gottesfeld, J. and Wilkes, M. M. M. , Holmes, D. S., (1974) "Functional organization of the mammalian genome" Cold Spring Harb. Quant, Biol. and Stafford, D. ( 1974) ~ 38:303-310 Birnstiel, M., Telford, J., Weinberg, E. "Isolation and some properties of the genes coding for the histone proteins" Proc. Acad, J. and Davidson, E. Britten, R. higher ceHs: Sci, H. (1969) "Gene regulation in A theory" Science 165:349-357 Britten, R. acid ~ 71 :2900-2904 N.ab_ J. and Davidson, E. Britten, R. (1976) "Studies on nucleic kinetics: empirical equations describin~ DNA Nat. ~ llS 73:415-419 E. and Neufeld, reassociation reassociation" Proc. H. /\Cad. J., Graham, D. B. R. ( 1974) "Analysis of repeating DNA. sequences by reassociation" Methods in Enzymology (L. Grossman and K. Britten, R. and Davidson, J., Graham, D. E. H. (1977) Moldave, eds.) Vol 29 part E 363-418 E., Eden, F. C., Painchaud, D. M. "Evolutionary divergence and length of repetitive sequences in sea urchin DNA." J..:.. Mol. Evol. in press - 103 Brown, D. D. and Sugimoto, evolution of ribosomal mulleri: Quant. K. and 5S DNAs the evolution of a gene ( 1974) "The structure and in Xenopus laevis and Xenopus Cold Spring flac!h. family" Sm._ filQ.L. 38:501-505 Chamberlin, M. E., Britten, R. "Sequence organization MQL. Biol. in Xenopus J. and Davidson, E. H. ( 1975) studied by electron microscopy"~ 96:317-333 Davidson, E. transcription and H. and Britten, R. J. (1973) "Organization, regulation in the animal genome" Quart. B..~ Biol. 48:565-613 Davidson, E. J. (1973) H., Hough, B. "General R., Amenson, C. S. and Britten, interspersion of repetitive with non-repetitive sequence elements in the DNA of Xenoous" !.L_ MQ.L.. Biol. E., Davidson, E. H., Graham, D. Amenson, S., C. R. Hough, E., Neufeld, B. B. R. R., 77: 1-23 Chamberlin, and Britten, R. J. M. (1974) "Arrangement and characterization of repetitive sequence elements in animal DNAs" Cold Spring Harb. Davis, R. microscope W., Simon, SymQ,... Quant. M. ~ and Davidson, heteroduplex methods 38:295-301 N. Eden, F. E. Graham, D. and Britten, R. J. "Electron for mapping regions of base sequence homology in nucleic acids" In Methods in Enzymology (L. Moldave, eds.) Vol XXI, Part D p. (1971) Grossman and K. 413-428 E., Painchaud, D. M., Davidson, E. H. (1977) "Exploration of long and short repetitive sequence relationships in the sea urchin genome" Nuc. Acids Res. in - 104 press Efstratiadis, A., Crain, W. and Kafatos, F. N.a.t.:.. Acad. A., Chamberlin, M. and Davidson, E. DNA" Chap. MA: H. H. Sci. US 73: 2289-2293 E., Hough, B. R., Britten, R. J. ( 1976) !'Evolution of repetitive and non-repetitive 12 of Molecular EY_olution (F. ,J. Avala, ed.) Sunderland, Sinauer Assoc. pp 200-224 Goldberg, R. B., Crain, W. R., Ruderman, J. Higgins, R. C., Gelfand, and Davidson, E. H. (1975) "DNA sequence organization Barnett, J. Britten, R. in J., Davidson, E. (1976) "DNA sequence organization in the lepidopteran Antherea pernyi" Proc. Galau, G. R., Britten, R. R., J. the genomes of five marine V., R. invertebrates" More, G. P., A., Galau, G. A., Chromosoma (Berl.) 51:225-251 Graham, D. J. E., Neufeld, B. R., Davidson, E. H. and Britten, R. (1974) "Interspersion of repetitive and non-repetitive DNA sequences in the sea urchin genome" Cell 1:127-137 Holmes, D. S. and Bonner, J. (1974) "Sequence composition of rat nuclear deoxyribonucleic acid and high molecular weight ribonucleic acid" Biochemistry 13:841-848 Noll, (1967) H. "Characterization of molecules by constant velocity sedimentation" Nature 215:360-363 Pearson, W. program for R., Davidson, E. least H. and Britten, R. J. (1977a) "A squares an;:ilysis of reassociation and hybridization data" submitted to Nuc. Acids Res. - 105 Pearson, W. R. , Wu, J. R. ~md Bonner, J. ( 1977b) "Analysis of rat repetitive DNA sequences" Thesis California Institute of Technology Schmid, C. W. and Deininger, P. L. (1975) "Sequence organization of the human genome" Cell 6:345-358 Sober, H. (1968) "Deoxyribonucleic acid content A.. per cell of various organisms: Table X Mammals" Handbook of Biochemistry Cleveland: Chemical Rubber Co. p. Studier, F. W. H-58 (1965) "Sedimentation studies on the size and shape of DNA" .J..... MQL. Jil..Qlt.. 11 : 373-390 - 106 - CHAPTER III ABSENCE OF SHORT PERIOD INTERSPERSION OF REPETITIVE AND NON-REPETITIVE SEQUENCES IN THE DNA OF DROSOPHILA MELANOGASTER - 107 Chromosoma (Berl.) 56, 309-326 (1976) CHROMOSOMA © by Springer-Verlag 1976 Absence of Short Period Interspersion of Repetitive and Non-repetitive Sequences in the DNA of Drosophila melanogaster William R. Crain 1 , Francine C. Eden 1 • 2 , William R. Pearson 1 , Eric H. Davidson 1 and Roy J. Britten 1 * 1 2 Division of Biology, California Institute of Technology, Pasadena, California 91125; present address: Department of Zoology, Indiana University, Blovmington, Indiana 47401, U.S.A. Abstract. A sensitive search has been made in Drosophila melanogaster DNA for short repetitive sequences interspersed with single copy sequences. Five kinds of measurements all yield the conclusion that there are few short repetitive sequences in this genome: l) Comparison of the kinetics of reassociation of short (360 nucleotide) and long (1,830 nucleotide) fragments of DNA; ·2) reassociation kinetics of long fragments (2,200 nucleotide) with an excess of short (390 short nucleotide) fragments; 3) measurement of the size ofSl nuclease resistant reassociated repeated sequences; 4) measurement of the hyperchromicity of reassociated repetitive fragments as a function of length; 5) direct assay by kinetics of reassociation of the amount of single copy sequence present on 1,200 nucleotide long fragments which also contain repetitive sequences. Introduction Interspersion of short repetitive sequences with single copy sequences of one to a few thousand nucleotides appears to be a very widespread-even nearly universal-feature of higher animal genomes. DNAs of fifteen organisms widely distributed on the phylogenetic tree have now been examined. In the typical case a majority of the genome consists of short (300 nucleotide pairs) repeated sequences adjacent to longer (1,000-2,000 nucleotide pairs) single copy sequence (recently reviewed by Davidson et al., 1975). Drosophila melanogaster DNA does not share this pattern of sequence organization (Manning et al., 1975) and at this time two other exceptions have been found among the insects (Crain, in preparation; Manning, personal communication). These differences are of conceptual importance since DNA sequence organization probably has significant implications for the functional organization of the genome and for genetic regulation (Britten and Davidson, 1969, 1971; Davidson and Britten, 1973). • Also Staff Member, Carnegie Institution of Washington, Washington. D.C. 20015, U.S.A. - 108 - 310 W.R. Crain et al. Previous studies of the Drosophila genome (Manning et al., 1975) were done principally by electron microscopy of reassociated DNA, while most of the data obtained on more typical genomes have been derived from other methods such as hydroxyapatite assay of reassociation of short and long fragments and SI nuclease resistance. The question obviously arises as to whether the differences between Drosophila DNA and other DNAs are more apparent than real; that is, due to differences in experimental approach. Studies of sequence arrangement in the Xenopus genome by electron microscopy (Chamberlin et al., 1975) have yielded results which are entirely consistent with those obtained by other methods. Xenopus DNA has a typical pattern of short period sequence interspersion and is the type organism, since its DNA was the first to be studied in detail (Davidson et al., 1973). An additional effort to determine sequence organization in the Drosophila genome by several methods other than electron microscopy appeared justified to us. Not only is Drosophila an important experimental species, but conceivably the application of other methods might uncover short repeated sequences missed in the electron microscopic measurements. In addition earlier measurements on Drosophila DNA (Wu et al., 1972) had been differently interpreted. In this paper we report a search for short repeated sequences in the Drosophila genome by a combination of five different methods. Manning et al. (1975) concluded that the number average length of the repeated sequences in Drosophila DNA is about 5,500 nucleotides and that these are probably terminated by single copy sequences. They estimated that there are 2,800 repeats in the genome. From these results we may estimate the quantity of single copy sequence expected to be linked to repeated sequences by a calculation described in the Discussion. If there were a significant number of previously unobserved short interspersed repetitive sequences the measurements reported here would show a larger fraction of single copy DNA linked to repetitive sequences than suggested by Manning et al. The conclusions from our measurements are in substantial agreement with those of Manning et al. (1975). Material and Methods I. DNA Preparalion. Unlabeled DNA was prepared from syncytium stage eggs that had been stored at - 70° C. The eggs were dechorionated by suspension in 50% Chlorox for 2 min at room temperature (2- 3 ml/g tissue). They were then caught on a Nitex screen and washed sequentially with distilled H 2 0, 70% EtOH and H 2 0. The eggs were gently removed from the screen and allowed to settle three times in ice-cold saline-Triton (0.9%NaCI, 0.1 % Triton-JOO; 10 ml/g tissue). The dechorionated eggs (Elgin and Hood, 1973) sink in this mix. The eggs that did not sink were washed again with distilled H 2 0 . The saline-Triton washed eggs were then washed once more with H 2 0 (10 ml/g tissue) and suspended in 0.05 M Tris-maleate buffer, pH 7.4; 0.005 M MgCl 2 (5 ml/g tissue). The eggs were homogenized with two strokes in a Dounce homogenizer and the homogenate filtered through two layers of Miracloth. The material that was trapped on the Miracloth was resuspended in a small volume of Tris-maleate buffer, rehomogenized in the Dounce homogenizer and filtered through Miracloth again. The filtered material was centrifuged at 2,000 rpm for 10 min through a 1/ 3 volume sucrose cushion (0.2 M sucrose in Tris-maleate buffer) to pellet the nuclei. The pellet was resuspended in one-half the original volume of Tris-maleate buffer plus 0.1 % Triton, and centrifuged a second time through a sucrose cushion. The nuclear pellet was resuspended in 0. 1 M Tris. 0.1 M EDT A. pH 8.0 ( I ml buffer to I g starting material) and the nuclei lysed in I% SOS. - 109 DNA Sequence Organization in Drosophila melanugaster 311 The sample was treated with 100 µg/ml predigested pronase (B grade, Calbiochem) for 2 h at room temperature. The mixture was brought to 1.0 M NaCIO 4 and extracted with an equal volume of phenol: IAC (I : I) and [AC (24 : I chloroform:isoamyl alcohol). The DNA was spooled after precipitation with two volumes of 95 % Et OH and redissolved in 0.015 M NaCl, 0.05 M Tris, 0.01 M EDTA , pH 7.6. The DNA was further purified by treatment with 50 µg/ml RNase A (chromatographically pure, Worthington Biochemical Co.) for I h at 37° C. The solution was adjusted to 0. 1 M NaCl and incubated with 200 µg/ml pre-digested pronase at 37° C for 1.5 h. After digestion the mixture was extracted with an equal volume of IAC and reprecipitated with ethanol. The yield of DNA was approximately 300 µg/g eggs. As an additional purification step sheared DNA was bound to hydroxyapatite at 60° C and washed with 0.12 M phosphate buffer followed by elution with 0.4 M phosphate buffer. Tritium labeled Drosophila DNA purified from tissue culture cells was generously provided to us by Ors. Ray White and David Hogness. A 60 ml suspension of cells that had been grown in the presence of 3 1-1-thymidine was centrifuged and the cells washed in medium without serum. The pellet was suspended in 0.5 ml medium without serum and brought to 10.5 ml with 0.5 M EDTA, pH 9.55; 2.5% sarcosyl; followed by incubation at 60° C. After 2 h 11 ml of 0.1 M EDTA containing I mg/ml proteinase K (EM Laboratories) was added and the mixture was incubated at 37° C for 7 h. CsCI was added ( 1.21 g/ml) and the samples were centrifuged at 33,000 rpm for 3 days at 15° C in a Spinco 40 rotor. The fractions containing the tritium counts were pooled and dialyzed extensively against 0. 1 M NaAcetate. The 3 H label was shown to be greater than 97% sensitive to DNase and completely resistant to RNase. All sheared tracer preparations were subjected to one further purification step by binding to and elution from hydroxyapatite. The purified tracer melted with an optical hyperchromicity of 95% of that expected for pure DNA. Its specific activity was determined to be I x 10 5 cpm/µg . 2. Preparation o( DNA Fragments. DNA fragments of desired lengths were obtained by shearing in a Yirtis 60K homogenizer as previously described (Britten et al., 1974). 3. Sizing 11( DNA Fragments. The single strand length of the DNA fragments was determined on isokinetic alkaline sucrose gradients as described by Noll (1967). The parameters were as follows : Vm;, =9.84 ml, C,.,=43% w/v, Cna,k = 16% w/v in 0.1 M NaOH. The gradients were centrifuged at 41,000 rpm for 20- 24 hat 20° C in the Spinco SW41 rotor. The weight average fragment length was determined according to the equations of Studier ( 1965) relative to markers sized by electron microscopy. 4. DNA Rl!a.uociation . Unless otherwise indicated the DNA was reassociated in 0. 12 M phosphate buffer at 60° C or 0.4 M phosphate buffer at 66° C. C 0 t values determined in the higher salt were corrected to "'equivalent " C 0 t by multiplying by 4.9. All buffers and DNA solutions were passed through BioRad Chelex 100 equilibrated in the desired buffer. The fraction of DNA molecules containing double strand regions was assayed by hydroxyapatite chromatography (Britten et al., 1974). The reassociation of DNA was monitored in some experiments by measuring the hypochromicity of denatured DNA samples at 260 nm in 0.12 M phosphate buffer in a water jacketed cuvette in a Beckman ACT A Mark III spectrophotometer. 5. SI Nuclease Digestion and Sizing o( Resistant Rl!gions. Reassociated DNA was adjusted to 0.15 M NaCl, 0.025 M NaAcetate, 0.005 M PIPES, 0.1 mM ZnSO 4 , 0.005 M /l-mercaptoethanol, pH 4.4, and digested for 45 min at 37° C with enough single strand specific SI nuclease to digest all single strand regions (Ando, 1966). The reaction was terminated by addition of phosphate buffer to a final concentration or0.12 M and SI resistant duplex regions were collected on hydroxyapa tite at 60° C. 14 C labeled sea urchin DNA incubated to appropriate C 0 t was included as an internal control in each SI digestion experiment. After elution from hydroxyapatite with 0.4 M phosphate buffer the resistant region s were chromatographed on a BioRad Biogel A-50 column ( 1.5 x 110 cm) in 0. 12 M phosphate buffer. The column was poured around a support of 6 mm glass beads. An exclusion marker of long native DNA and an inclusion marker of 32 P0 3 were used to calibrate the column. - 110 - 312 W.R. Crain et al. Results I. Reassociation Kinetics of Drosophila DNA Figure l shows the reassociation kinetics of two different fragment sizes of Drosophila melanogaster DNA assayed by binding to hydroxyapatite. The curves show that Drosophila melanogaster has a relatively small quantity of middle 0.------.------.------~----~-----, 20 ~ :, 0 40 CD ~ z 0 ~ 60 80 100~---~----~---~----~---~ 0.01 0.1 1.0 10 100 1000 Equivalent C0 t Fig. I. Reassociation kinetics of Drosophila DNA. Drosophila DNA fragments of 360 (e) and 1,830 (o) nucleotides were prepared by mixing unlabeled DNA from syncytium stage eggs with 3 H-labeled DNA from Drosophila tissue culture cells (mass ratios of 1,500/1 and 750/1 respectively). These mixtures were sheared in the Virtis homogenizer (Britten et al. , 1974). Reassociation and assay of the fraction of fragments containing double stranded regions was carried out by binding to hydroxyapatite (see Materials and Methods). The least squares solution of the data yielded components with the following fraction of the genome (Q) and rate constants (k) : For the 360 nucleotide fragments, very highly repetitive sequences and foldback: Q=0. 12, k cannot be clearly defined; middle repetitive sequences, Q=0.108, k=2.I M - 1s - 1; single copy sequences, Q=0.751 , k=0.0121 M - 1s - 1. Least squares analysis on the I 830 nucleotide fragment data was carried out with the rate constant for single copy sequence fixed to a value of 0.0273 M - 1s •• 1. This value was arrived at by adjusting the k for single copy sequence from that measured with 360 nucleotide fragments. Thus k 1.830 = 2 )" =0.0273. The values for the components in the 1,830 nucleotide curve are as follows : k 3 0 0 ( -I •830 360 very highly repetitive and fold back Q =0.15, knot defined ; middle repetitive, Q =0.22, k =0.94 M ·· 1s - 1; single copy, Q=0.64, k=0.0273 M - 1s - 1 - 111 - DNA Sequence Organization in Drosophila me/anogaster 313 repetitive DNA, as previously observed (Laird, 1971; Wu et al., 1972; Schachat and Hogness, 1973; Manning et al., 1975). The kinetic curves are dominated by the single copy DNA reassociation. About 10% of Drosophila DNA binds at very early times to hydroxyapatite. This is probably composed of satellites and inverted repeats (Gall et al., 1971; Peacock et al., 1973; Wilson and Thomas, 1974; Manning et al., 1975; Schmid et al., 1975). We have not investigated this fraction in the present work. The fraction of the DNA fragments containing middle repetitive sequences or only single copy sequences was determined by least squares analysis. These quantities and the most likely rate constant are shown in the legend to Figure 1. The reassociation of the fragments averaging 1,830 nucleotides is somewhat faster than that of the 360 nucleotide fragments. Such a result is expected for the following reasons. First, for those fragments which are purely repetitive ot purely single copy the rate of reassociation is increased approximately in proportion to the square root of the fragment size (Wetmur and Davidson, 1968; Wetmur, 1976). Some of the fragments from interspersed regions contain both repetitive and single copy sequences linked together. These fragments are bound to hydroxyapatite when the repetitive regions are reassociated, and therefore when longer fragments are used a greater fraction of the single copy DNA is bound at a given C\t. This process supplies one of the major methods for the detection and measurement of short period interspersion of repeated and single copy sequences. The rate of reassociation of long and short fragments has been measured for the DNA of a number of species (Davidson et al., 1973; Goldberg et al., 1975; Davidson et al., 1975; Angerer et al., 1975). In each case in which other methods show the presence of short period interspersion the reassociation rate of fragments a few thousand nucleotides long is observed to be strikingly faster than that of fragments a few hundred nucleotides long. However for Drosophila melanogaster DNA the effect of increased fragment length is relatively modest The rate of reassociation of the 1,830 nucleotide fragments is faster than that of the 3,160 nucleotide fragments by a factor which is only slightly larger than the ratio of the square roots of the fragment lengths. This is true for both the repetitive and single copy parts of the reaction. It follows that only a small fraction of the 1,830 nucleotide long fragments are made up of linked repetitive and single copy sequences. We do not yet know whether the rate of reassociation is exactly proportional to the square root of length for fragments as short as 360 nucleotides, and thus a quantitative examination of such measurements is limited in accuracy. The accuracy of this method is not sufficient to detect a small amount of relatively short interspersed repetitive sequences if they exist. 2. Reassociation of Long Labeled Fragments with an Excess of Short Fragments The reassociation of long labeled fragments with an excess of short fragments was the first method used to demonstrate repetitive sequence interspersion (Britten and Smith, 1970) and it has been used for the measurements of the length of the interspersed single copy sequences in the DNA of several species (e.g., - 112 314 W.R. Crain et al. Davidson et al., 1973; Graham et al. , 1974; Angerer et al., 1975; Efstradiatis et al., 1976). Figure 2 shows the kinetics of reassociation of labeled 2,200 nucleotide long Drosophila DNA fragments with excess 390 nucleotide unlabeled fragments. As above the rate of reassociation of the long fragments is affected by their length, so that a small amount of sequence interspersion is difficult to detect. Previous measurements with bacterial DNA fragments (Davidson et al., 1973) indicated that the rate of reassociation of long labeled fragments with an excess of short fragments is directly proportional to the length of the long fragments. The least squares solutions for the rates and quantities of the repetitive and single copy components are shown in the legend to Figure 2. The single copy rate is in fact accelerated by just the ratio of the lengths (2,200/390). Thus the majority of the single copy sequences are not linked to repetitive sequences on fragments of 2,200 average nucleotide length. In Figure I the quantity of the repetitive component appears somewhat larger for the 2,200 nucleotide labeled fragments than for the 360 nucleotide fragments. This is in part due to linkage of single copy sequences to the ends of long repetitive sequences. However the accuracy of these measurements is not sufficient to determine whether the quantity is greater than that expected from the electron microscopic measurements of Manning et al. (1975) (see Discussion). It is possible that short, interspersed repetitive sequences exist but are either very short or very divergent and thus cannot be bound to hydroxyapatite at standard criterion. If this were true it might be possible to detect these less stable sequences by lowering the criterion of reassociation. To test this possibility the incubations and hydroxyapatite assays were done in 0.12 M phosphate buffer at 50° C, 10° Clower than the standard criterion. Under these conditions more divergent or shorter repeats would reassociate and bind to hydroxyapatite than at the 60° C criterion used in the preceding experiments. The lower curve of Figure 2 shows the result. Additional binding occurred at very low C 0 t but no substantial change is observed in the middle repetitive and single copy regions of the kinetics. The implication of the low C 0 t binding is that under these conditions a greater amount of DNA was associated with foldback or satellite sequences. Since in Drosophila (Schmid et al., 1975) and other species the foldback or palindrome (Wilson and Thomas, 1974) sequences are interspersed throughout much of the genome it is likely, though not proved here, that we are seeing the effect of a larger class of such interspersed sequences. There is, however, no sign of an increased amount of middle repetitive sequences. Thus no evidence for unusually divergent or short interspersed middle repetitive sequences can be obtained by lowering the criterion 10° C. 3. The Size of SJ Nuclease Resistant Reassociated Repetitive Sequences Under appropriate conditions (Britten, Graham, Eden, Painchaud and Davidson, in preparation) the single strand specific nuclease SI from Aspergillus (Ando, 1966; Vogt, 1973; Sutton, 1971) is useful for determining the length of repetitive sequences. With a carefully controlled digestion single strands can - 113 - DNA Sequence Organization in Drosophila melanogaster 315 20 "C C 40 ::, 0 ID <( z 0 ct!! 60 80 100..___ _ __.__ _ ____._ _ _ __.__ _ ____.__ _ ____. 0.01 0.1 1.0 10 100 1000 Equivalent C0 t Fig. 2. Reassociation kinetics of 2,200 nucleotide labeled fragments in the presence of an excess of 390 nucleotide DNA fragments. The upper curve (e) shows the binding of the labeled fragments to hydroxyapatite at 60° C in 0.12 M phosphate buffer after incubation under the same conditions. The lower curve ( o) shows the binding to hydroxyapatite at 50° C in 0.12 M phosphate buffer after incubation under these conditions. Least squares analysis was performed on the data at 60° C and 50° C yielding the following components : 60° C. Very highly repetitive and foldback, Q=0.17, k not defined; middle repetitive, Q=0.27, k= 1.66 M - 1 s 1 ; single copy, Q=0.584, k=0.074 M - 1 s - 1 . 50° C. Very highly repetitive and foldback. Q=0.34, knot defined; middle repetitive, Q=0.166, k = 1.06 M - 1 s 1 ; single copy, Q=0.497, k =0.048 M - •s - 1 be effectively removed while the enzyme does not attack duplexes containing even 10% to 20% mismatched base pairs. The following procedure has been developed to measure the length distribution of short repetitive sequences: a) reassociation of approximately 2,000 nucleotide long fragments to a C 0 t at which few single copy sequences are reassociated; b) limited digestion with SI nuclease to cleave single strands from duplexes; c) binding to hydroxyapatite (0.12 M phosphate buffer, 60° C) to separate duplexes from undigested single strands; d) gel filtration analysis of the SI resistant duplexes on agarose A-50 columns. This method has been applied to the DNA of a number of species which possess typical short interspersed repetitive sequences. In every case the resistant duplexes separate into two size classes: a long fragment class, which is excluded from the agarose column; and a short fragment class with a mode 114 316 W.R. Crain et al. 12 10 4 2 40 60 Fraction Fig. 3. Length of SI resistant duplexes formed by renaturing highly repetitive and middle repetitive Drosophila DNA. Highly repetitive. and middle repetitive DNA sequences were prepared from 2,400 nucleotide fragments as described in text and treated with SI nuclease sufficient to remove single strands without digesting mismatched regions. 436 µg of highly repetitive and 600 µg of middle repetitive DNA were digested under the following conditions; 0.15 M NaCl, 0.005 M PIPES, 0.025 M Na Acetate, pH 4.3, 0.1 mM ZnSO 4 , 37° C, 45 min. The resistant duplex fraction was collected on hydroxyapatite and chromatographcd on agarose A-50 in 0.12 M phosphate buffer. Sea urchin DNA (2,000 nucleotides) reassociated to C 0 t 4 was added to the Drosophila DNA before SI digestion as a control. The arrows indicate the positions of exclusion and inclusion markers. • Drosophila satellite DNA (C 0 t 0.05 bound). o Drosophila middle repetitive DNA (C 0 t 0.05- 3.0), • Sea urchin repetitive DNA size of about 300 nucleotide pairs (Goldberg et al., 1975 ; Britten, Graham, Eden, Painchaud and Davidson, in preparation). The fraction of the repeats which fall in the short repetitive sequence class varies from 35-85%, depending on the species (Davidson et al., 1975). In order to achieve greater sensitivity the Drosophila DNA fragments were partially separated into a very rapidly reassociating fraction and a middle repetitive fraction. For this purpose the 2,400 nucleotide long fragments were incubated to C 0 t 0.05 and passed over hydroxyapatite. The bound fraction was eluted with 0.4 M phosphate buffer. The unbound fraction was reincubated to C 0 t 3. Both fractions were digested with S 1 nuclease and the size of the resistant duplex structure measured by gel filtration on agarose A-50. Details of the procedure and the yield of DNA in each fraction are given in Materials and Methods. The size distributions of the SI resistant duplexes are shown in Figure 3 - 115 - 317 DNA Sequence Organization in Drosophila melanogaster along with a control, labeled sea urchin DNA. Identical patterns were obtained for the sea urchin DNA in both runs. The yield of the short fragments of sea urchin DNA is a little less than the amount expected but the SI digestion was clearly effective. The C 0 t 0.05 bound fraction of the Drosophila DNA shows no measurable quantity of short S 1 resistant duplexes. The middle repetitive fraction (C 0 t 0.05 to 3) appears to show a small quantity of short resistant duplexes. About 25% of the DNA has a k •• greater than 0.3, or a size Jess than 1,000 nucleotide pairs. This result is entirely consistent with the pattern of long repetitive sequences measured by the electron micrographic measurements of Manning et al. (1975). 4. The Hyperchromicity of Reassociated Repetitive Fragments When DNA fragments containing both repetitive and single copy sequences are reassociated to low C 0 t the single copy regions remain unpaired. The quantity of these unpaired regions can be measured by collecting all of the fragments containing paired regions on hydroxyapatite and measuring their hyperchromicity. The amount of single copy DNA linked to repetitive sequences on such fragments depends on the sequence interspersion pattern. If there are a large number of short interspersed repeats a large quantity of single copy sequences Table I. Hyperchromicity of middle repetitive DNA sequence after reassociation Preparation of middle repetitive DNA fraction Initial average fragment length (nucleotides) 350 1,600 3,200 First incubation C0 t Fraction bound Fraction to total DNA 3 0.24 0.24 3 0.31 0.31 3 0.32 0.32 0.012 0.41 0.096 0.016 0.55 0.17 0.016 0.56 0.18 0.29 0.44 0.042 0.51 0.30 0.05 0.54 0.22 0.039 12.6 300 0.22 0.95 78.1 15.3 1,000 0.20 0.83 83.5 11.7 1,600 0.19 0.79 84.5 Second incubation C0 t Fraction unbound Fraction of total DNA Third incubation C0 t Fraction bound Fraction of total DNA Hyperchromicity determination Fourth incubation (for spectrophotometric melt) C0t Final fragment size (nucleotides) Hyperchromicity Fraction of native DNA hyperchromicity Tm (° C) - 116 - 318 W.R. Crain et al. will be linked to them, and a low hyperchromicity will be observed. The amount of hyperchromicity decreases as the length of the fragments increases. This effect has been demonstrated in measurements of the hyperchromicity of partially reassociated DNA fragments in several species that display typical interspersion of short repetitive sequences (Davidson et al., 1974; Graham et al., 1974; Goldberg et al., 1975; Angerer et al., 1975). The results for the middle repetitive fraction of Drosophila are listed on the lower part of Table 1. The upper parts of this table show the incubations and hydroxyapatite fractionation steps used in purifying the appropriate middle repetitive fraction of Drosophila DNA. The reassociated fragments which are 300 nucleotides long at the end of the procedure have about 95% of the hyperchromicity of native DNA, indicating that very little single copy sequence is linked to them. Even the longer fragments (I ,000 and 1,660 nucleotides) show hyperchromicities of 83% and 79% of native DNA. These values suggest that some single copy DNA is linked to repeats in fragments of this size and indeed some is expected if the average repetitive sequence is 5,000 nucleotides long (Manning et al., 1975). Table 2. Estimates from reassociation kinetics of the components of Drosophila DNA Component Single copy Middle repetitive Fast Data presented in this paper Quantity Repetition frequency Complexity Rate constant in whole DNA Rate constant if pure 0.74 1.0 6.7 X I0 7 NTP 0.012) a 0.0164 0.09 30 3.14x 10 6 0.35b 3.89 0.14 0.70 I 0.0069 0.12 70 0.5 0.12 Data of Manning et al.• Quantity Repetition frequency Rate constant in whole DNA • In the fit described here the single copy rate constant was fixed to agree with that of the free fit for the 360 nucleotide fragment reassociation curve b For this fit the rate constant for the middle repetitive component in the 1,830 and 2,200 nucleotide curves were corrected to their predicted value for 360 nucleotide fragments and the average rate constant determined. This average middle repetitive rate constant was then fixed in the least squares fit of the 360 nucleotide fragment reassociation curve. The predicted rate constants for 360 nucleotide fragments from the two longer fragment size curves were calculated as follows: For 1,830 nucleotide fragments, where 0.936 M - 1 s- 1 is the rate observed for the middle repetitive sequences in the experiments of Fig. I. 0 112 k=0.936 (~- - ) =0.45 . 1,830 For 2,200 nucleotide fragments driven by 390 nucleotide fragments, where 1.66 M - 1 s - 1 is the rate observed for the middle repetitive sequences in the experiment of Fig. 2. 11 2 390 k = 1.66 ( ) (~~) =0.283 . 2,200 390 The average of these two rate constants is 0.35 M • 1 s - 1 . • Fits of data from Manning et al. ( 1975) for Drosophila DNA fragments of 400 nucleotides length - 117 DNA Sequence Organization in Drosophila me/anogaster 319 5. Kinetic Assay for Single Copy DNA Sequences Linked to Repetitive Sequences The following is probably the most direct and most sensitive of the methods used in this paper to determine the amount of single copy DNA sequences which are linked to repetitive DNA sequences. Long labeled DNA fragments which contain repetitive sequences were selected. These fragments were sheared and reassociated with an excess of total Drosophila DNA fragments. The reassociation kinetics were analyzed for evidence of fragments bearing only single copy sequences. The sensitivity of this experiment is such that a few percent of single copy DNA can be detetced. The first step was to isolate a large quantity of short repetitive DNA fragments to be used to select the long labeled fragments containing repeats. The frequency with which middle repetitive sequence occur in the Drosophila genome is relatively low, about 30 copies of each sequence. This agrees well with the previously reported repetition frequency of the middle repetitive component (Wu et al., 1972; Schachat and Hogness, 1974; Manning et al., 1975). This makes it difficult to separate single copy from repetitive DNA and a single fractionation step does not suffice. Table 3 show the fractionation steps and yield at each stage in comparison with that expected from our best estimates of the repetitive frequency components of Drosophila DNA (Table 2). Twenty mg of Drosophila ONA were sheared to 300 nucleotides and 14% was recovered _ Table 3. Fractionation of Drosophila repetitive DNA Fraction bound to hydroxyapatite 40 20 0.005 5 Fraction unbound 0.47 0.63 Predicted fraction• unbound 0.47 0.54 0.724 0.74 Predicted fraction• bound 0.63 0.73 • The fraction of the DNA bound to hydroxyapatite at the different steps in the fractionation was arrived at by calculating the theoretical fraction reassociated for each component according to the equation C where C is the concentration of totally single stranded fragments, C0 is the total DNA nucleotide concentration, tis time and k is the second order rate constant. The predicted recoveries for the individual components were added to give the predicted total fractions bound or unbound, as shown in columns 4 and 5. The rate constants of each component were adjusted according to the new concentration in the remaining DNA in order to calculate the recovery in the next step. This calculation was made using the components arrived at from these data: 14% zero time binding and highly repetitive DNA; 9% middle repetitive sequences with a rate constant of 0.35 M - 1 s - 1 , and 74% single copy with a rate constant at 0.0121 M- 1 s - 1 . The rate constants were adjusted to their values for 290 nucleotide length fragments and a rate constant of 100 M - 1 s • 1 was estimated for the fast component - 118 - 320 W.R. Crain et al. 0 • 20 • • -o 40 • C :, 0 ID zcl: 0 al! 60 80 1OO-------'------'-------L------'------' 0 .01 0 .1 1.0 10 100 1000 Equivalent C0 t Fig. 4. Reassociation of sheared selected repetitive DNA in the presence of excess whole DNA. Tritium labeled Drosophila DNA (tracer) fragments which had been stripped of zero time binding sequences were reassociated to C 0 t 3.1 in the presence of a 50-fold excess of 290 nucleotide purified Drosophila repetitive DNA (driver). The tracer which had reassociated with the repetitive driver was collected by hydroxyapatite chromatography. The single strand size of the tracer after elution from hydroxyapatite was 1,225 nucleotides. The DNA that bound to hydroxyapatite was recovered and sheared, reducing the tracer length to 375 nucleotides. The sheared material was mixed with 290 nucleotide whole Drosophila DNA in a ratio of I to 5 and reassociated to various C0 t values. The fraction of fragments containing duplexed regions was measured on hydroxyapatite. The curve represents a least squares fit to the data with the rate constant for the single copy component fixed at 0.012 M - 1 s·· 1 according to the following description. The rate of the single copy component for the 375 nucleotide fragment tracer in this curve was arrived at by adjusting the rate for 360 nucleotide fragments as follows: 2 k2oo =k 300 ( 290)" (375) =0.014M - 1 s - 1 . 360 290 This rate constant was further adjusted to allow for the dilution of the single copy sequences when whole DNA driver was mixed with purified repeat driver. Assuming that the repeat driver contained no single copy seq uences the single copy sequences would be diluted by a factor of 0.833. The resulting rate constant for single copy seq uences is thus: 0.833 (0.014) = 0.0l2 M • 's - 1 as the repetitive fraction. This should consist of 34% highly repetitive, 66% middle repetitive and 0.2% single copy DNA, were there no sequence interspersion in this genome. The reassociation kinetics of this fraction were measured in the spectrophotometer by optical hyperchromicity. The results were consistent - 119 DNA Sequence Organization in Drosophila melanogaster 321 with the expected components and the reassociation went to completion (final optical density=0.75 of the initial denatured A260 ). No single copy component could be observed in the reaction kinetics. The second step was to prepare long labeled fragments. 3 H labeled DNA was sheared to 1,800 nucleotide pairs, denatured, incubated to C 0 t 0.005 and passed through hydroxyapatite to remove the zero time binding or foldback fraction. This step was repeated, and a total of 10.2% of the fragments were removed. The remaining fragments were incubated with a 50 fold excess of the short repetitive fragments (described above) to C 0 t 3.2. The sample was then passed over hydroxyapatite in 0.12 M phosphate buffer at 60° C. Twentyseven percent of the long labeled fragments and 79% of the short driver fragments were bound. The bound fragments were eluted at 98° C and the tracer recovered was 1,225 nucleotides long, as assayed by alkaline sucrose sedimentation. These fragments were sheared to 375 nucleotides and incubated with a 5 fold excess of total Drosophila DNA. The kinetics of the reassociation are shown in Figure 4. It is clear that the principal portion of the labeled fragments is made up of repetitive DNA sequences. Least squares analyses were carried out to determine the best estimates of the fraction of the sheared tracer fragments which contained only single copy sequence. For this purpose the rate constant for the single copy component was fixed at 0.012 M- 1s- 1, the rate constant of single copy sequence in the driver DNA. All of the other parameters were allowed to remain free. The solution indicates that 9% of the labeled fragments are single copy sequences. We have also made a least squares analysis with the middle repetitive as well as the single copy rate constants fixed at the values shown in Table 2 (corrected for the additional repetitive sequences present in this mixture). With these restrictions the calculated fraction of fragments bearing only single copy sequence is 14%. Discussion Our purpose here is to calculate the expected amount of single copy sequence linked to repetitive sequences and to examine the quantitative conclusions from the five types of measurements. In this calculation G is the genome size in nucleotide pairs and L is the fragment length. For a particular set of repetitive sequences (indicated by the subscript 1) S 1 is the length of the interspersed single copy sequences, Q 1 the fraction of the genome in these single copy sequences and F 1 the number of interspersed repetitive or single copy sequences in this set. Figure 5 shows for an ideal case the increase with fragment length of the fraction of fragments containing single copy DNA linked to repeated sequences. The slope (for L < S 1) in Figure 5 is dR/dL=Q 1 /S 1 . Since Q 1 =F1 Si/G the slope can be written : (l) This simple relationship is for a particular set of interspersed sequences and we may add up the contribution to the slope of each such set as long as the fragment length is less than the length of the single copy sequences. - 120 322 W.R . Crain et al. --s,--.i R l L Fig. 5. The contribution of one element of a sequence interspersion pattern to hydroxyapatite binding. Shown as a function of fragment length (L) is R, the amount of DNA in fragments which contain one set of repeated sequences spaced by- single copy sequences of length S 1 . The total quantity of single copy DNA interspersed with this set is Q 1 and the quantity of repeated DNA is P 1 . For short fragments little single copy DNA is linked to the repeats and in the limit only the repetitive DNA (P 1) would be included . The amount of single copy DNA increases in proportion to the fragment length until the length of the single copy sequences is reached (plus enough length to recognize the succeeding repeat). The slope of the linear rising portion is Si/Q 1 Thus for L < S the total initial slope of the curve for the fraction of fragments containing single copy DNA linked to repetitive sequence is: (2) where F1 , F 2 etc. are the numbers of single copy (or repetitive) sequence elements in each interspersed set and N is the total number of interspersed sequences in the genome. For L less than S the fraction of the genome which is in single copy sequences linked to repetitive sequences ( Y) follows directly: Y=L(dR/dL)=L N/G. (3) Manning et al. (1975) take the genome size of Drosophila melanogaster to be 1.3 x 10 8 nucleotide pairs (Fristrom and Yund, 1973) and estimate that there are 2,800 interspersed repeated sequences with an average length of 5,500 nucleotides. If we assume that all of these are interspersed with single copy sequences that are more than few thousand nucleotides long, then for a 1,000 nucleotide fragment length: Y= 1,000 x 2,800/1.3 x 10 8 =0.0215. Therefore we except to find about 2% of the genome (2.8 x 106 nucleotide pairs) as single copy sequence linked to fragments 1,000 nucleotides long which contain middle repetitive sequences. From Table 2 the amount of middle repetitive DNA is 9% of the genome or 11. 7 x I 0 6 nucleotide pairs. Therefore 1,000 nucleotide long fragments of middle repetitive DNA would contain 19% (2.8 x 10 6 / - 121 323 DNA Sequence Organization in Drosophila melanogaster Table 4. Fraction of middle repetitive and single copy sequences in Drosophila genome Fragment size 360 nucleotides 1,830 nucleotides 2,200 nucleotides Middle repetitive component Single copy component Fraction Fraction of fragof genome ments that is observed• single copy attached to repeatsb Corrected fraction of genome in middle repetitive sequence• Fraction of fragments• Fraction of genome that is single copy attached to repeatsb Corrected fraction of genome in single copy sequence• 0.091 0.221 0.267 0.083 0.182 0.220 0.739 0.637 0.584 0.008 0.039 0.047 0.747 0.676 0.631 0.008 0.039 0.047 Quantities of components determined from least squares fit of reassociation data at given fragment length (from Figs. I and 2 and Table 2) b Fraction of genome that is single copy sequences attached to repeated sequences at the various fragment lengths. Calculation as described in text Fraction of genome composed of component after correction for fraction of DNA which is single copy linked to repeats at the fragments length used 2.8 x 106 + l l. 7 x 106 ) single copy DNA. We take this to be the best estimate from the electron micrographic measurements of Manning et al. For calculations involving other fragment lengths the quantity of linked single copy DNA is simply proportional to the fragment length as long as the length of typical single copy sequence is not exceeded. Calculations from the Kinetics of Reassociation ( Parts I and 2) Table 4 shows the expected fraction of the single copy DNA calculated as above from the number of middle repetitive sequences (2,800) estimated by Manning et al. (1975). We have subtracted this estimate from the least squares estimates of the fraction of DNA in the middle repetitive kinetic components. If the number of interspersed middle repetitive sequences were significantly greater than 2,800 we would observe, after this subtraction, an increase in the middle repetitive fraction with fragment length. Though small, such an effect is seen in Table 4 and it is this uncertain result which led us to attempt corroboration with other, more sensitive methods. Measurement of the Size of the SJ Nuclease Resistant Fragments The &ize distribution of SI resistant middle repetitive duplexes is shown in Figure 3 as open circles. The great majority of the fragments are excluded from agarose and are therefore greater than 2,000 nucleotides in length. A small fraction, perhaps 15% of the mass of DNA has a kav greater than about 0.4, and thus their length is less than 400 or 500 nucleotide pairs. This represents - 122 - 324 W.R. Crain et al. the best estimate by this method of the fraction of the middle repeats of about 300 nucleotide average length. While this is a simple and direct method, the following calculation shows that it is not a sensitive way to detect a small number of short repetitive sequences. If there were, for example, as many 300 nucleotide repetitive duplexes as all others combined we would expect 300/5,600= 5% of the total DNA in this region of the A-50 chromatogram. We obviously can not rule out such a possibility from this measurement. In fact the data suggest a potentially larger number of short repeats. However there are a variety of artifactual possibilities which could give rise to short fragments, for example shear breakage, and internal nicking by the SI nuclease. We prefer to draw no conclusion from the tail of short fragments shown in Figure 3. In any case it is clear from this measurement that the bulk of the middle repetitive sequences are long. Hyperchromicity of Reassociated Repetitive Fragments The results of fractionating Drosophila DNA fragments of various sizes are shown on Table I along with the hyperchromicity of the purified middle repetitive fractions. The fractionation indicates that only a small increase in single copy DNA linked to middle repetitive DNA occurs with fragment size. The three final fragment lengths are 300, 1,000 and 1,600 nucleotides. The hyperchromicity data imply for these three fragment lengths respectively that 5%, 16.5% and 21 % were made up of linked single copy DNA. The calculation described above, assuming that there are just 2,800 interspersed middle repetitive sequences in the genome yields 5%, 14% and 21 % for the three lengths. This agreement is surprisingly good considering the possible sources of error in these measurements. Direct Kinetic Assay for Single Copy Sequences Linked to Middle Repeats The least squares solutions of the data shown in Figure 4 indicate that 9-14% of the selected 1,225 nucleotide long fragments is made up of linked single copy DNA sequences. The calculation described above yields 17% for this fragment length. This agreement is certainly within error. This measurement is the only one of all of the approaches we have applied to Drosophila DNA which directly implies that single copy DNA is interspersed with the repeats at any length. The other measurements have relied, for example, on the increase with length of the fraction of the genome associated with middle repeats or on the fact that the linked DNA was single stranded. In every case it was reasonable but unproved assumption that the linked DNA is actually single copy sequence. The electron micrographic measurements of Manning et al. are subject to the same proviso. However the least squares solution of the data of Figure 4 includes a component with the kinetics of single copy DNA which is of about the quantity expected if there are 2,800 repetitive sequences interspersed with single copy sequences. 123 DNA Sequence Organization in Drosophila melanogaster 325 Some of the fragments of Drosophila DNA that have been "cloned" and examined by Wensink et al. (1974) contain both single copy and repetitive sequences. In addition all of the cloned middle repetitive sequences so far examined are several thousand nucleotides or more in length. The data shown in Figure 4 give the best direct estimate of the number of places in the Drosophila genome at which repeated sequences are linked to single copy sequences. The actual number of such transitions in the DNA sequence is, of course, just twice the number of interspersed repetitive and single copy sequences. Conclusion Each of the measurements described in this paper gives a result that is consistent, within error, with the conclusion that about 2,800 relatively long middle repetitive sequences are interspersed with single copy sequences in the Drosophila genome. We may ask how many more interspersed repetitive sequences could be added to the 2,800 used in the calculations above without creating an unacceptable deviation from the observed measurements. It seems very unlikely that a doubling of the number would be acceptable. While this would not introduce a problem of interpretation for the kinetics and SI resistance methods, serious deviations would be revealed by the last two methods described. The hyperchromic shift in one set of measurements and the amount of single copy DNA in the other would have to be in error by a factor of two. We do not believe this is at all likely and conclude that a two fold greater number of interspersed repeats is ruled out. Of course smaller deviations are possible. Acknowledgements. We wish to thank Dr. Ray White for his generous gift of tritium labeled Drosophila DNA. This work was supported by grants from the National Institutes of Health (HD-05753 and GM-20927) and from the National Science Foundation (BMS75-07359). W.R.C. holds a postdoctoral fellowship from the National Institutes of Health (GM-55726). F.C.E. held a postdoctoral fellowship from the American Cancer Society (PF-955). W .R .P. is a predoctoral fellow on a National Institutes of Health training grant (GM-00086). References Ando, T. : A nuclease specific for heat-denatured DNA isolated from a product of Aspergillus oryzae. Biochim. biophys. Acta (Arnst.) 114, 158-168 (1966) Angerer, R.C., Davidson, E.H., Britten, R.J.: DNA sequence organization in the mollusc Aplysia californica. Cell 6, 29-39 (1975) Britten, R.J ., Davidson, E.H.: Gene regulation for higher cells: a theory. Science 165, 349-357 (1969) Britten, R.J., Davidson, E.H.: Repetitive and nonrepetitive DNA sequences and a speculation on the origins of evolutionary novelty. Quart. Rev. Biol. 46, 111-138 (1971) Britten, R.J ., Graham, D.E., Neufeld, B.R.: Analysis of repeating DNA sequences by reassociation. In : Methods in enzymology (L. Grossman and K. Moldave, eds.), Vol. 29, Part E, pp. 363-418. New York : Academic Press 1974 Britten, R.J. , Smith, J.: A bovine genome. Carnegie Inst . Wash. Yearb. 68, 378-386 (1970) 124 326 W.R. Crain et al. Chamberlin, M.E., Britten, R.J. , Davidson, E.H .: Sequence organization in Xenopus DNA studied by the electron microscope. J. molec. Biol. 96, 317-333 (1975) Davidson, E.H., Britten, R.J.: Organization, transcription and regulation in the animal genome. Quart. Rev. Biol. 48, 565-{;)3 (1973) Davidson, E.H., Galau, G.A ., Angerer, R.C., Britten, R.J.: Comparative aspects of DNA sequence organization in metazoa. Chromosoma (Berl.) 51, 253-259 (1975) Davidson, E.H., Graham, D.E., Neufeld, B.R., Chamberlin, M.E., Amenson, C.S., Hough, B.R., Britten, R.J. : Arrangement and characterization of repetitive sequence elements in animal DNAs. Cold Spr. Harb. Symp. quant. Biol. 38, 295-301 (1974) Davidson, E.H ., Hough, B.R., Amenson, C.S., Britten, R.J.: General interspersion of repetitive with hon-repetitive sequence elements in the DNA of Xenopus. J. molec. Biol. 77, 1-23 (1973) Efstradiatis, A. , Crain, W.R., Britten, R.J., Davidson, E.H ., Kafatos, F. : DNA sequence organization in the lepidopteran Antherea pernyi. Proc. nat. Acad. Sci. (Wash.) (in press, 1976) Elgin, S.C.R., Hood, L.E.: Chromosomal proteins of Drosophila embryos. Biochemistry 12, 49844991 (1973) Fristrom, J.W., Yund, M.A.: Genetic programming for development in Drosophila. Crit..~ Biochem. 1, 537- 570 (1973) Gall, J.G. , Cohen, E.H., Polan, M.L.: Repetitive DNA sequences in Drosophila. Chromosoma (Berl.) 33, 319-344 (1971) Goldberg, R.B., Crain, W.R., Ruderman, J.V., Moore, G .P., Barnett, T.R., Higgins, R.C., Gelfand, R.A., Galau, G.A ., Britten, R.J., Davidson, E.H. : DNA sequence organization in the genomes of five marine invertebrates. Chromosoma (Berl.) 51, 225-251 (1975) Graham, D.E. , Neufeld, B.R., Davidson, E.H., Britten, R.J .: Interspersion of repetitive and nonrepetitive DNA sequences in the sea urchin genome. Cell 1, 127-137 (1974) Laird, C.D.: Chromatid structure: relationship between DNA content and nucleotide sequence diversity. Chromosoma (Berl.) 32, 378-406 (1971) Manning, J.E., Schmid, C.W., Davidson, N. : Interspersion of repetitive and non-repetitive DNA sequences in the Drosophila melanogaster genome. Cell 4, 141-155 (1975) Noll, H.: Characterization of macromolecules by constant velocity sedimentation. Nature (Lond.) 215, 360-363 (1967) Peacock, W.J., Brutlag, D., Goldring, E., Appels, R. , Hinton, C.W., Lindsley, D.L. : The organization of highly repeated DNA sequences in Drosophila melanogaster chromosomes. Cold Spr. Harb. Symp. quant. Biol. 38, 405- 416 (1973) Schachat, F.H., Hogness, D.S. : Repetitive sequences in isolated Thomas circles from Drosophila melanogaster. Cold Spr. Harb. Symp. quant. Biol. 38m 371 - 381 (1974) Schmid, C. W., Manning, J.E., Davidson, N.: Inverted repeat sequences in the Drosophila genome. Cell 5, 159- 172 (1975) Studier, F.W.: Sedimentation studies of the size and shape of DNA. J. molec. Biol. 11, 373-390 (1965) Sutton, W.D.: A crude nuclease preparation suitable for use in DNA reassociation experiments. Biochim. biophys. Acta (Arnst.) 240, 522- 531 (1971) Vogt, V. M.: Purification and further properties of single-strand-specific nuclease from Aspergillus oryzae. Europ. J. Biochem. 33, 192- 200 (1973) Wensink, P.C., Finnegan, D.J ., Donelson, J.E., Hogness, D.S.: A system for mapping DNA sequences in the chromosomes of Drosophila melanogaster. Cell 3, 315-325 (1974) Wetmur, J.G .: Hybridization and renaturation kinetics of nucleic acids. Ann. Rev. biophys. Bioeng. 5 (in press, I976) Wetmur, J.G., Davidson, N.: Kinetics of renaturation of DNA . J. molec. Biol. 31, 349- 370 (1968) Wilson, D.A., Thomas, C.A., Jr.: Palindromes in chromosomes. J. molec. Biol. 84, 115-144 (1974) Wu, J.-R., Hurn, J., Bonner, J.: Size and distribution of the repetitive segments of the Drosophila genome. J. molec. Biol. 64, 211-219 (1972) Received March 2, 1976 / Accepted March 22, 1976 by J.G. Gall Ready for press March 30, 1976 - 125 - CHAPTER IV KINETIC DETERMINATION OF THE GENOME SIZE OF THE PEA - 126 Introduction In the past few years, substantial data on the organization of repeated and single copy sequences in animal DNAs have been accumulated. There have been relatively few studies on DNA sequence organization in plants (Walbot and Dure, 1976; Smith et. evidence that plant DNAs may contain sequences. large al., 1976). fractions There is of repetitive Studies on the cotton genome (Walbot and Dure, 1976) suggest that the organization of the large repetitive fraction may be different from the typical 300 nucleotide short repeat/1000-2000 nucleotide long single copy spacer pattern seen Davidson et. al., 1975). in most animals. (for review, see Walbot and Dure (1975) have reported that 40% of the cotton genome is repeated and that the reoeated sequences average 1250 nucleotides in length while the interspersed single copy sequences are 2000 nucleotides long. We have measured the fraction of repetitive and single copy DNA in the pea genome. Our measurements on the single copy rate of reassociation suggest that the size of the pea genome smaller than previous reports Hoff and Sparrow, 1963; the DNA sequences is substantially (McLeisch and Sunderland, 1961; Birnsteil et. al., 1964). More Van't than 70% of in the pea are repeated about 500 fold and the rate constant for the single copy reassociation is .00215, corresponding to a genome size of .459 pg. according to our best estimates. - 127 - Materials and methods Isolation of crude chromatin from pea embryos 100 lb of Alaskan peas (Pisum sativum) were germinated for 3 days. 23 lb of pea embryos were isolated using a system for large scale preparation of pea embryo tissue. Embryos were stored at -20 °c. For preparation of crude chromatin, 4 lb of frozen homogenized in embryos were a 1 gal Waring blender in 0.25 M sucrose 50 mt~ Tris pH 8.0 1 mM MgC1 2 and centrifuged at 5000g in a Sorvall GSA rotor min. The crude nuclear pellet was for separated from a starch pellet, resuspended in grinding buffer and centrifuged at 10,000g for The nuclear pellet was 30 again separated from the resuspended in 50 mM Tris pH 8 and centrifuged at 15 min. starch pellet, 10,000g for 20 min twice. Isolation of DNA The crude chromatin pellet was suspended in 0.1 M Tris 0.1 pH 8.5 and sodium dodecyl sulfate added to 1%. The viscous mixture was brought to 37 °c and incubated with 50 ug/ml Pronase B) until the DNA went into solution. (Calbiochem grade The DNA was extracted twice with a mixture of phenol:chloroform:isoamyl alcohol twice M EDTA with chloroform:isoamyl alcohol (25:24:1) (24:1). and extracted The DNA was then precipitated at room temperature in 2.5 volumes of 95% ethanol allowed to dissolve overnight in 0.05 M Tris 0.01 M EDTA pH 8.5. and - 128 Once in solution, the DNA was digested with RNase A (pre-incubated 10 min at 95 °c) at 50 ug/ml for 90 min at 37 °c and digested with 200 ug/ml Pronase (pre-incubated 45 min at 37 °c) for 90 min at 37 °c. DNA was then The extracted with chloroform:isoamyl alcohol (24:1) three times and reprecipitated with ethanol. The DNA was redissolved and reprecipitated with ethanol. The DNA was sheared to 350 nucleotides by three french pressure cell at 50,000 psi. passes through a and passed over Chelex-100 (BioRad) before renaturation. Preparation of 1251 labeled DNA 350 nucleotide pea DNA was incubated to Cot 400 stranded fraction (15%) isolated on hydroxyapatite. and the single This material was concentrated and labeled with 125 1 using a modified Commerford (1971) procedure (Holmes and Bonner, 1974) . Renaturation of DNA DNA fragments were denatured at 100 °c and renatured in 0.12 M sodium phosphate buffer (PB) at 60 °c and 70 °c and in 0.48 M PB at 71 0 c and 81 °c. Samples were then diluted to 0.03 M PB and passed over hydroxyapatite at 60 °c to check for degradation . than 10% of the DNA not binding to hydroxyapatite discarded. The in Samples with more 0.03 M PB were single strand fraction was then eluted with 0.12 M PB and the double strand fraction denatured at 100 °c and eluted with 0.12 M PB. - 129 Optical renaturation was carried out in a Gilford 2400 spectrophotometer equipped with a water jacketed sample compartment. Samples were denatured, loaded decrease into the sample compartment and in hyperchromitcity monitored. The data in Figure 2A were measured in O. 12 M PB at 60 °c in 10 mm and 1 mm path in 0.41 M PB at 71 °c in 10 mm cuvettes. and the length cuvettes The data in Figure 2B was measured at 66 °c in 10 nm and 1 rmn cuvettes. Hybridization of 1251 labeled DNA was carried out in 0.12 M PB at 60 °c and in 0.48 M PB at 65 °c. After renaturation, samples were diluted to 0. 12 M PB and passed over hydroxyapatite in 0.12 M PB. double strand fraction was eluted with 0.48 M PB. the driver DNA was followed optically; The The renaturation of the fraction of the 1251 tracer in hybrid was measured by TCA precipitation of the hydroxyapatite fractions. Fits of second order components to the data were calculated using a non-linear least squares computer program (Pearson et. al., 1977). Results To determine the fraction of single copy and the single copy rate constant we have made three types of measurements. We have renatured pea DNA at 60 °c and 70 °c assaying the fraction duplex by hydroxapatite (HAP) binding. We have optically renatured pea DNA at 60 °c and 66 °c to determine the true fraction duplex. And we have driven an isolated labeled slowly renaturing fraction to look for a small fraction highly complex component. Renaturations were carried out at two criterion - 130 temperatures to look for apparent repetitive fractions caused by poorly matched sequences. Both hydroxyapatite and ootical Cot curves were used to discover if an apparent increased repetitive fraction might be due to repeated sequences interspersed with short single copy spacers. The hydroxyapatite Cot curves Figures 1A and 1B show the renaturation of 350 nucleotide long pea DNA fragments at 60 °c and 70 °c. The 60 °c (Tm-25°c) incubations were carried out at 60 °c in .12 M sodium phosphate buffer (PB, and at 71 °c for .41 M PB. .18 M Na+) The 70 °c (Tm-15°c) curve was carried out at 70° for .12 M PB and 81 °c for .41 M PB. The higher temperatures were used to correct for the higher melting temperature of DNA in .41 M salt. These high temperatures may account for some of the scatter in the data and the final fraction unreacted at high Cot's. Tables IA. and IB present the results of a variety of least squares fits to the data. The tables start with the results of an unconstrained two component fit. This is the "best fit" possible for the data. second fits show the results when the The single copy rate from the renaturation at one temperature is forced on the Cot curve at the other temperature . difference This calculation gives another measure of the between the temperatures. 70 °c Cot curves repeated These slow comoonent rates at the real different fits suggest the difference between the 60 °c and is not significant. The apparent shift of some sequences to the slow component increases the rate of the slow component and may be due separated by two decades. to the difficulty of resolving components - 131 - Figure 1: Renaturation of pea DNA measured by hydroxyapatite binding Denatured 350 nucleotide pea DNA was renatured in 0.12 M PB and 0.41 M PB at 60 °c and 71°C (Tm-25 °c) or 70 °c and 81 °c (Tm-15 °c) respectively . hydroxyapatite The fraction single chromatography. stranded was determined by Equivalent Cot is Cot the (moles/liter-sec) times salt concentration factors of 1.0 for 0. 12 M PB incubations and 5.0 for 0.41 M PB incubations (Britten et. The solid lines plotted show the best least squares data. al., 1974). fit to the The dashed line shows the contribution of a component with a rate of 0.0004 liter-sec/mole. A. Renaturation at Tm-25 °c. B. Renaturation at Tm-15 °c. - 132 - 0.001 0.01 0.1 10 100 1000 10,000 100,000 0.0 A 0.2 B 0.8 1.0 0.001 0.01 0.1 I 10 I 00 1000 10,000 100,000 EQUIVALENT C0t - 133 - Table I Renaturation of 350 nucleotide pea DNA fragments at 60 °c and 70 °c Renaturation measured by hydroxyapatite binding A. Unconstrained fits: (no parameters fixed) 1. 60 °c (T m-25°C) reaction Goodness of fit: 3.7% Component Fraction Rate Cot 1/2 Repetition Frequency 1 0. 114 > 50 0.02 >20,000 2 0.583 (+0.046) 0.399 (±_0.128) 2.51 138 3 0. 160 (±_0.043) 0.00288 (±_0.00315) 347 Final fraction unreacted: 2. 70 oc 0.143 1 ±0.029 (T m-15°C) reaction Goodness of fit: 4.0% 1 0. 1840 > 50 0.02 >20,000 2 0.411 (±0.043) 0.452 ( +O. 17 4) 2.21 148 3 0.220 (±0.044) 0.00305 (±0.00202) 3279 1 Final fraction unreacted: 0.185 ±0.015 - 134 B. Constrained fit: 1. 60 OC single copy rate from reaction at one temperature used to fit reaction at the other temperature (Tm-25°c) reaction Goodness of fit: 3.5% 1 0. 113 > 50 0.02 >20,000 2 0.581 (±0,034) 0 . 403 (+0.108) 2.48 132 3 0 . 161 (±0,039) 0.0Q.3Q2 328 1 Final fraction unreacted: 1 . 70 oC 0 . 145 ±0,018 (Tm-15°C) reaction Goodness of fit: 3.9% 1 0. 184 > 50 0.02 >20,000 2 0.414 (±0,029) 0 . 443 ( ±0. 130) 2 . 25 154 3 0.217 (±0,031) 347 1 Q.~(20281.i Final fraction unreacted: 0.185 ±0,013 - 135 C. Constrained fit: 1. 60 °c three fitted components single component with rate 0.0004 copy (Tm-25°c) reaction Goodness of fit : 3.1% 0.0650 > 50 0.02 >125 , 000 2 0.282 (±0 . 108) 2 . 74 ( ±2 . 79) 0.365 6850 3 0.428 (±0 . 111 ) 0. 100 (±0.057) 10 250 4 0. 116 (±0 . 050) 2500 1 Q...._Q.Q.Q.!!_Q_ Final fraction unreacted: 2 . 10 oC 0.109 ±0 . 033 (Tm-15°C) reaction Goodness of fit : 4. 1% 1 2 3 0 .179 > 50 0 . 02 >125,000 0.382 0 . 547 l±0.29b) 1.tl3 136 '( (±0.0 '(0) 0. 172 l:t0 . 0'73) 0.0103 l±0 . 01bb) 0 . 09b (±O.Otl4) ~Q.Q.Q.40 Final fraction unreacted : 9'( 25000 o. 1'71 ±0.022 2b 1 - 13b A third set of fits is included to test for a small fraction of sequences hybridizing at the rate of 0.0004 liter-sec/mole with a Cot 1/2 of 2500. These fits indicate our measurements cannot exclude possibility of a very slowly renaturing fraction. the The dashed lines in Figures 1A and 1B show how this hypothetical component would renature. From both of these measurements it appears that about 60% of pea DNA is repetitive with a Cot 1/2 of 1 and another 25% is less repetitive, with a Cot 1/2 of 350 corresponding to a genome size of pg. There .35 is not a substantial difference between the renaturation at 60 °c and at 70 °c. The optical Cot curves Hydroxyapatite fractionation measures containing complexes. the formation of duplex Strands containing duplexes longer than 20-40 nucleotides are retained on HAP while a large fraction of the retained strands may be single stranded. Repeated sequences 300 nucleotides long separated by single cooy sequences of equal length would cause amount of repetitive HAP twice the actual amount of binding to be the repetitive sequences. Optical renaturation - measuring the fraction duplex by renaturing the DNA in a spectrophotometer - measures the true duplex fraction in the DNA. Single strand ends of duplexed molecules contribute no more to the hyperchomicity measurement than unduplexed single strand molecules. - 137 - Figure 2: Optical renaturation of pea DNA 350 nucleotide pea DNA fragments were denatured and renatured in a water jacketed spectrophotometer chamber. Samples were incubated in 0.12 MPB at 0.41 M PB at bO 0 c and 71 °c (Tm-25 °c) in 0.12 MPB at 0 c. bb Fraction DNA reacted is the fraction of hyperchomicity lost after renaturation. Equivalent Cot was calculated as in Figure 1. The solid lines plotted show the results of the best least squares fits to the data. The dashed lines show the contribution of a component renaturing with a rate of 0.0002 liter-sec/mole. A. Renaturation at Tm-25 °c . B. Renaturation at Tm-19 °c. - 138 - 10 100 1000 10,000 100,000 • 10 100 1000 10,000 100,000 0.2 0.4 0.6 0 w ~ u 0.8 <( w n:: <( 1.0 0 0.0 z z 0 ~ 0.2 LL 0.4 u <( n:: 0.6 0.8 1.0 0.001 0.01 0.1 I EQUIVALENT C0 t - 139 - Table II Renaturation of pea DNA fragments at 60 °c and 6b 0 c Renaturation measured optically A. Unconstrained fit: (no parameters fixed) 1. 60 °c (T m-25 °c) reaction Goodness of fit: 2.6% Fraction Rate Cot 1/2 repetition frequency 0.261 ( ±0. HS 1) 6.70 (±12.9) 0. 15 3~00. 2 0.414 (±0.051) 0. 167 (±0.046) 6.0 95. 3 0.295 (±0.016) 0.00176 (±0,00040) 56~.2 1. Component Final fraction unreacted: 2. bb 0 0.122 ±0.017 c (Tm-19 °c) reaction Goodness of fit: 0.5% 1 0.071 2 0.423 (±0.0112) 3 0.246 (±0,010) 4 0.214 ( ±0. 0. 15) 0.01 >100,000 0.926 1030. 0.0467 (±0,0075) 21 . 41 45. 0.00104 (t0.00031) 9b 1. 5 1. >100 Final fraction unreacted: 0.04b ±0.022 - 140 B. Constrained fit: (single copy rate fixed at 0.0002 liter-sec/mole) 1. 60 oc (Tm-25 oC) reaction Goodness of fit: 2.7% 1 0.067 >10 2 0. 465 C+o.017) 0.323 C±0.046) 3. 1 O 1600. 3 0.229 (+0.016) 0.00551 C±o.00163) 181. 5 27.6 4 0.219 C±o.016) 0.0002 Final fraction unreacted: <O. 1 5000. >5000. 1. 0.020a 2. 66 oc (Tm-19 oC) reaction Goodness of fit: 0.8% 1 0.087 <0.01 >100. >50,000. 2 0.482 (+0.007) 0.752 (+0.042) 1.33 3760. 3 0.259 C±o.007) 0.0134 C±o.0014) 75. 66.7 4 0. 152 C±o.005) 0.0002 5000. 1. Final fraction unreacted: 0 . 020a Footnotes to Table II a These values were fixed to allow the program to do the fit. - 141 Figures 2A and 2B display the renaturation of pea DNA at 60 °c and 66 °c in the spectrophotometer. These curves show the same large repetitive fraction seen in the HAP Cot curves. about 45%. data. for In these curves it is Again a number of multicomponent curves were fit to the Tables IIA and IIB show the results of those fits. each set of data is the best two component fit. three component fit. The first fit We also include a The third fit forces the slow component rate one renaturation temperature on the data of the other. from The final curve forces a slow component appropriate for a 2.5 pg genome size. A comparison of these data with the HAP cot curves indicates that perhaps 30% of the 60% repetitive fraction from the HAP curve may be due to single strand tails. The rate constants for the two components are not significantly different. Hybridization of labeled slowly renaturing DNA Each of the renaturation curves we have run suggest there are only two major components in the pea genome. The slow component reanneals quite rapidly, about 20 times faster than that expected from the 5 pg. genome size et. predicted from a nuclear DNA content of 9.7 pg (Birnsteil al., 1964). Because of the poor termination data and small fraction of the DNA in the complex component it is difficult to rule out a minor component of with a Cot 1/2 of 2500. We have increased the sensitivity of our measurement more than four fold by isolating a slowly renaturing fraction of the genome, labeling it in vitro with 125 r and driving that tracer with whole pea DNA. Pea DNA was renatured to Cot 400 and the single strand fraction separated on - 142 - Hybridization of 1251 labeled single copy pea DNA Figure 3: A fraction of pea DNA (15%) enriched in single copy sequences was isolated and labeled as described in the text. was driven by a 5000 fold excess equivalent and This labeled fraction of unfractionated pea DNA to the Cots shown. The renaturation of the driver DNA (not shown) 1251 single copy tracer was measured hydroxyapatite. The solid after fractionation on line shows the best least squares fit with the components described in Table III. ( ■ ) fraction of 1251 DNA in duplex. - 1~3 - FRACTION DNA BOUND TO HYDROXYAPATITE 0 p p 0 . .0 0 (X> CJ) ~ N 0 .00 0 9 0 ri, oc < )> r- ri,o z ~ () 0 -0 -o 0 0 0 0 b 0 0 0 0 g L..J_ 0 _J__..J...._...L--L----L----'---....&....-...___ _ __ - 144 - Table III Hybridization of 125 1 labeled Slowly renaturing pea DNA Goodness of fit: 3.6% Component Fraction Rate Cot 1/2 repetition frequency 1 0. 109 (±0.051) 0.4b3 (±0.972) 2 . .1 b 215. 2 0.580 (±0.053) 0.00215 (±0.00082) 4b5. 1. Final fraction unreacted: 0 . 263 ±0.044 - 145 HAP. This material was labeled with 1251 using Holmes and Bonner, the Commerford (1971, 1974) procedure and driven by whole unlabeled DNA. Fifteen percent of the DNA was isolated and labeled for this experiment so a 10% single copy fraction in whole DNA should appear as more than 50% of the DNA after the purification. The hybridization curve in Figure 3 does not indicate any component with a Cot 1/2 greater than 500. Discussion These measurements are all in good agreement as to the fraction of the genome in the repetitive component. The HAP Cot curves suggest that about b0% of the DNA is repetitive while the optical Cot curves indicate that slightly less than 45% is repetitive. Both curves show the same two component The fast is centered at Cot 1, the slow component around Cot 400. The hybridization of the isolated kinetic components. slow component DNA driven by The conclusion that about 45% or more of the DNA sequences in the unfractionated pea DNA shows the same slow component. pea genome are repeated is a strong one. Changes in the incubation temperature which should melt poorly matched duplexes do not alter renaturation curve. The large fraction repetitive is really due to duplexed DNA, not to single strand tails or closely spaced duplexes. are the We therefore confident that the fraction of repeated sequences we have determined is correct. - 146 There is also good agreement on renaturing complex component - about the Cot 450. 1/2 of the slowly If this slow component represents the single copy DNA of the pea the genome size is about pg. .46 Unfortunately it is difficult to be certain that the final fraction unreacted does not contain a more slowly renaturing component. Ten percent of the DNA could easily be renaturing but hidden in the final unreacted fraction of either HAP Cot curve or the optical Cot curves. We believe it unlikely that a more slowly reannealing component is to be found in the pea genome. The hybridization experiment displayed in Figure 3 sets an upper limit on the fraction of very slow component below 5% of the pea genome. We believe that the true size of the pea genome is 5 x 108 base pairs. This is in contrast to the 5 x 109 base pairs/genome based on the chemical determination of DNA in nuclei isolated from stem tips (Birnsteil Sunderland, 1961; and Torrey, 16X polyploid. measuring et. al., 1964) or root tips Van't Hoff and Soarrow, 1963). pea seedling (McLeisch and We now know (Libbenga 1973) that the majority of stem cells of the pea are 4X to Walbot and Dure (1975) made a similar discovery on the kinetics of reassociation of cotton DNA. We believe that the complexity measured from the rate of renaturation of single copy DNA (Britten and Kohne, 1968; for the genome size. Laird, 1971) provides a more accurate value - 147 - References cited Birnsteil, M. L., Chipchase, M. "On the chemistry and Biophys. Acta. I. H. and Flamm, W. G. (19b4) organization of nucleolar proteins" Biochem. 87:111-122 J., Graham, Britten, R. D. E. and Neufeld, B. R. (1974) "Analysis of repeating DNA sequences by reassociation" Methods .in Enzymology (L. Grossman and K. J. Britten, R. Moldave, eds.) Vol 29 part E 3b3-41~ and Kohne, D. E. (19b8) "Repeated sequences in DNA" Science 161:529-540 Commerford, S. L. ( 1971) "Iodination of nucleic acids in vitro" Biochemistry 10:1993-2000 Davidson, E. J. H., Galau, G. A., Angerer, R. C. and Britten, R. (1975) "Comparative aspects of DNA sequence organization in metazoa" Chromosoma (Berl.) 51:253-259 Holmes, D. S. and Bonner, J. (1974) "Sequence composition of rat nuclear deoxyribonucleic acid and high molecular weight ribonucleic acid" Biochemistry 13:841-~4~ Laird, C. D. DNA content and 32:378-406 (1971) "Chromatid structure: relationship between . nucleotide sequence diversity" Chromosoma (Berl.) - 148 Libbenga, K. R. and Torrey, J. G. (1973) "Hormone induced endoreduplication prior to mitosis in cultured pea root cortex cells" American Journal of Botany b0:293-299 McLeisch, J. and Sunderland, N. (19b1) "Measurements on desoxyribosenucleicacid (ONA) in higher plants by Feulgen photometry and chemical methods"~ !&ll_ Res. Pearson, W. program for R., Davidson, E. H. and Britten, R. by (1977) "A D. B., Acids Res. Rimpau, J. and "Interspersion of different repeated revealed J. least squares analysis of reassociation and hybridization data" submitted to Nuc. Smith, 24:527-540 Flavell, B. sequences in interspecies DNA/DNA hybridization" ( 197b) the wheat genome NJJ.Q..... ~ ~ 10: 2811-2825 Van't Hoff, J. and Sparrow, A. H. (19b3) "A relationship between DNA content, nuclear volume and minimum mitotic cycle time" Proc. Acad. Nat, SQ.h i.USl 49:897-902 Walbot, V. cotton seed and Dure, L. S. (197b) "Developmental biochemistry of embryogenesis and germination. the cotton genome" J..... Mol. Biol, 101 :503-53b VII. Characterization of - 149 - A.PPENDIX A A PROGRA.M FOR LE~ST SQUARES ANALYSIS OF REASSOCIATION AND HYBRIDIZATION DA.TA - 150 Abstract A computer program is described for the rapid calculation of least squares solutions for data fitted to different functions normally used in reassociation and hybridization kinetic measurements. for The equations the fraction not reacted as a function of Cot follow: order, (1+kCot)- 1; First order, exp(-kCot); second approximate fraction ( 1+kCot)-· 44 ; and a function describing the pairing of tracer when rate constant for the of DNA sequence tracer (k) is variable order, remaining (1+kCot)-n; single stranded, the distinct from the driver rate Several may be used for most of these functional forms. components The standard deviations of the individual parameters at the solutions are calculated. - 151 Introduction The quantitative examination of reassociation and hybridization kinetic measurements has become increasingly imoortant as sophistication of the measurements has grown. a general In this paper we describe computer program which can conveniently apoly the variety of functions now used to interpret kinetic measurements. use the We have chosen to a non-linear least squares method so that the solutions give equal weight to all assumptions of the need individual measurements and be made no preliminary about the initial or terminal values of the reaction. Least squares computer programs have been applied to the problem of resolving repetitive and single copy kinetic components in DNA renaturation experiments carried out on many organisms and Britten, 1973; Davidson et . al., 1975a). (e.g. Davidson They are also used to determine rate constants in RNA excess hybridization reactions Galau et . al . , 1975). The (e.g. use of cDNA probes to determine the complexity of RNA populations by kinetic rather measurements Ryffel and McCarthy, 1975) (e . g. Bishop et. al., 1974; than saturation also relies heavily on the resolution of abundance classes by accurate determination of their rate constants . The five functions listed in Table 1 are used for of the following kinds of measurements . (FINGER) describes accurately the form assayed The second order equation of DNA reassociation kinetics by hydroxyapatite chromatography (Britten Wetmur and Davidson, 19b~) though for fairly complex and Davidson, 1976; the examination Smith et . al. , 1975). and Kohne, 1966; reasons (Britten The pseudo-first- order modified second order assay of hybridization NUFORM EXCESS fiexp(-kiCot) WHTCMP 3 4 5 fi(1 + kiCot)-0.44 fiexolkil 1-(kdCot) 1-nJ/Lkrl(1-n)J} describes rate of tracer reaction wnen tracer rate constant k differs from driver rate constant~ first order function: f(1 + kCot)-n WHATOR 2 function: S-1 nuclease for RNA excess experiments variable form reaction: to determine values of n when the aooarent order of the reaction is unknown second order reaction: DNA renaturation by hydroxyapatite chromatography fi(1 + kiCot)- 1 FINGER measured Use Form N::l.o.'Tle Number TABLE 1 Functional forms used by the program ........ I f's) V1 - 153 equation (EXCESS) applies when the nucleic remains unpaired as in RNA driven acid driving the reactions. The reaction third function (WHATOR) has a form which can be varied by changing the value of the exponent, and is useful when the apparent order of the reaction is not known or there is a need to components. test the heterogeneity of the reacting When the exponent n=1, it reduces to second order form. When high quality single component hydroxyapatite measurements are analyzed, the least squares solution yields n=1.0 (Smith et. As n becomes large this equation 1975). values of first order. approaches first al., order and n as high as 10 yield a form which is indistinguishable from The next functional form (WHTCMP) reassociation when the is applicable to DNA reaction is assayed by S-1 nuclease (Smith et. al., 1975) and expresses the fraction of the length of the DNA sequence present which remains sin~le stranded. The last equation (NUFORM) applies when the rate of reassociation of tracer with the driver and the rate of driver renaturation itself differ. available is assumed to follow the equation n=.44 is usually used and The amount of driver (1+kCot)-n. A value of gives a good approximation to the actual capacity of the remaining single stranded regions to tracer molecules (Davidson et. reassociate with al., 1975b). The program: Use The program presented in timesharing computer. It Appendix III is in use on a PDP-10 provides for interactive input but has been designed for ease of conversion to a Batch processing system. run is presented A typical in Appendix I, with user responses underlined. Batch environment the underlined inputs would be submitted on cards. In a - 154 Upon execution in a timesharing environment, the program types a list of options available: 1START,2ADD,3GUESS,5PLOT,6CPLOT,~LPLOT,9STOP Options 1 and 2 are for input, option 3 is used to and options 5, 6 and~ are used for output. find the solution, Option 9 terminates the program. Input options: (1START,2ADD) 1START is used at the beginning of the program to read in a new se~ of data. The program asks for a data file name which should contain input data consisting of: 1. Title of the data file 2. C/Co (the fraction of DNA or RNA remaining single stranded), and the value of the Cot for each data point 3. 3.0 in the C/Co pcsition to terminate input The form of these data is described more fully in the program listing . Option 2 permits addition of a new file to a pre-existing set of data. This program handles up to 200 data points. The fitting option: The user is information: requested (3GUESS) to provide the following three lines • of - 155 1. FUNCTION(-1HELP),ITERATIONS,RMS QUIT,DELMX QUIT,(NIT) 2. parameter estimates 3. index of parameters to be fit First line: FUNCTION may be given the following values (see Table I): -1. (HELP) prints out the names and numbers of the functions available. OSAME,1FINGER,2WHATOR,3NUFORM,4EXCESS,5WHTCMP 0. (SAME) The previously used function, parameter values and fixed parameters are saved to make a new approximation 1. set of second order components (FINGER) <C!Co> = F + [r f.(1+k.Cot)- 1 r<=5 . ]_ ]_ 1=/ 2. variable exponent form to examine the order or heterogeneity of a reaction (WHATOR) <C!Co> = F + f(1+kCot)-n 3. tracer reaction with driver of different rate constant (NUFORM) ,.. <C!Co> = F + [ f.explk.L 1-(1+kdCot) 1-nJ/lkd(1-n)J} r<=2 j:-1 4. ]_ ]_ set of first order components (EXCESS) r <C!Co> = F + [. r<=5 . f.exp(-k.Rot) ]_ ]_ I::. I 5. set of components for reassociation assayed strand specific enzyme (WHTCMP) <C/Co> = F + r<=5 t,., with a single 44 f.(1+k.Cot)-· l l. - 156 In the equations <C!Co> is the calculated fraction remains the assay involved. single stranded according · to of the DNA which f is the fraction which is not reacted at the termination of the reaction of the components described. fi is the fraction in an individual component and f and k are these parameters for a single .ki is the rate constant. component case. kd is the rate constant for self-reassociation of the n is the variable value of the exponent where required. driver DNA. ITERATIONS is the number of iterations to be used function to sufficient. the data. fit the guesses 5 iterations are If the fit has not converged after 20 iterations, orobably not converge. PROBLEMS. For reasonable to it will These situations are discussed in the section on To calculate the error estimates for an arbitrary set of parameters without iterating, 0 ITERATIONS should be typed. RMS QUIT and DELMX QUIT provide additional criteria for terminating the data fitting process. The process will terminate either when the difference between the RMS and the current and less than RMS QUIT or when previous is zero, the is the DELMX value (the largest fractional change in any parameter) is less than DELMX QUIT. QUIT iterations If RMS QUIT or DELMX process will terminate when either the RMS or the parameters are not changing. stragtegy used in the (NIT) program. It is a value for a corrective is set to 10 in the program by default, and in general should not be used. Second line: The initial parameter guesses are to be specified by the user the following order: in F, f 1 , k 1 , f2, k2, f3, k3, f4, k4, f5, k5 for functions - 157 1(FINGER), 4(EXCESS) and 5(WHTCMP). F, f, k, n. For function 2(WHATOR) the order is For function 3(NUFORM) the order is F, f1,k1, f2,k2, kd, n. Third line: The index or order number of the parameters which are to be at their initial values is specified by the user. fixed Any parameter or number of parameters may be fixed. After these specifications are supplied the performed. The RMS, DELMX (the maximum fractional parameter) and the parameter values are until printed after iterations are change in any each iteration the ITERATIONS are finished or one of the termination conditions specified by RMS QUIT or DELMX QUIT is satisfied. When the iterations stop, the parameter correlation matrix and criteria for two the error of the fit, the RMS and the GOODNESS OF FIT are printed, as well as the NUMBER OF data POINTS and the number of DEGREES OF FREEDOM. The number of DEGREES OF FREEDOM is the NUMBER OF POINTS less the number of variable GOODNESS OF (not fixed) FIT values are calculated parameters. The RMS and from the sum of the squared errors. SUMSQ = I (C/Co - <C/Co>) 2 for each of the data points. of the C/Co is the fraction unreacted at the Cot data point and <C/Co> is the expected value calculated from the function used to fit the data. RMS= The RMS value printed then is: SUMSQ/p where .~ is the number of data points. The GOODNESS OF FIT value uses the DEGREES OF FREEDOM instead of the number of points: - 15~ GOODNESS OF FIT= ~SUMSQ/ ( p-v) where y is the number of variable parameters in the fit. Output options: (5PLOT,6CPLOT,~LPLOT) Options 5 and~ print an alphanumeric plot of the solution and data on the teletype or line printer respectively. file to be written which can be used for the Option b causes a plotting by another program (such as one which uses an X-Y plotter). For example, the following sequence of responses could be used to fit a second order function to the data in the file named FILE and have the result plotted on the line printer (see Appendix I). 1. 1 ;start 2. FILE ;name of a PDP-10 data file ;(this statement would probably be deleted ; in a batch system) 3, 1 ;type out the data 4. 3 ;(GUESS) do the fit 5. 1 10 0. 0 . 005 ;fit a second order function (FINGER) ;use 10 ITERATIONS ;terminate if RMS is not changin~ ;terminate if parameters are changing ; by 0.5% or less b. 0.05 0.25 10.0 0.65 0.001 ;parameter estimates 7. 5 ;the rate constant of the second ; component is fixed ;plot the fit on line printer 9. 9 ;stop the pro~ram - 159 A typical run using the program is presented in Appendix I. It shows a number of the problems encountered when fitting hybridization data which are discussed in the next section. discussion of the program. Appendix II structure and mathematical techniques used in the It is not necessary to read Appendix II to use the material presents a is directed the program; to those who wish to modify the program or apply it to a different problem. Appendix III presents a highly annotated listing of the We have made efforts to provide a program that will run on different computers but some statements (particularly input statements) to program. be changed when the program is used on other systems. may have The program listing describes in some detail how these changes might be carried out. Analysis of hybridization data The program strategy This non-linear least squares program is designed to converge on a solution yielding parameters which minimize the least squares deviation of the function from a set of data. whether the There are two main concerns: final solution is biased by the input parameter estimates, and whether further iterations will improve the solution. independent of input A solution is parameter estimates and insensitive to further iterations if it has converged. Convergence is indicated by the amount of change in the RMS from one iteration to the next and by the change in the DELMX parameter. technique used Because of the by this program, Taylor's the rapidly in the neighborhood of a solution. series approximation parameter values converge very Although this neighborhood - 160 may be difficult to find for the first few iterations (the algorithm's strategy uses a "gradient" technique to find the neighborhood), once found, successive iterations will improve the parameter values by one to two significant figures with each iteration. reflected 10- to 100- fold in a iterations. changes This rapid decrease very little. This improvement is decrease in DELMX with successive in DELMX may occur while the RMS Attempts to improve the solution beyond convergence may cause the message CORRECTIVE ITERATIONS EXCEEDED to be printed. Possible problems This program rapidly converges to a unique solution from a wide range of parameter estimates when the data provide adequate constraints. New parameter estimates are always better (in the least than the previous values but the parameters may become negative during the process. to squares sense) become component Inadequate termination data may allow the FINAL parameter negative while the FRACTION par~~eter increases without bound. unreassociated (presumably near zero) Fixing the for the slowest FINAL fraction or close scrutiny of the DELMX values to find regions of local convergence may solve the oroblem. Often when the series of iterations do not immediately converge the program hesitates in a region of local convergence. When pressed to improve the RMS the program may go off in a direction (such as negative parameters) where termination is impossible. 1.0 to 5.0, then decreases to 0.02 to 0.1 largest as parameter change is less than 2-10%). Usually DELMX starts at the fit converges (the DELMX may then jump by a factor of 50 to 200 and start on another (possibly unterminated) path. - 161 The small DELMX value indicates a provide good values for examined. the To determine local low RMS region which might parameters but they must be carefully the quality of the parameter values, plot the solution. Occasionally the progra~ returns negative rate constants components of the reaction. program to remove that component This is usually due to an attempt by the from the solution usually shows why this was done . do not show in the plot should for solution. Plotting the Single erroneous points that be searched for and perhaps a new solution should be attempted with fewer components. Parameter fixing Data from other measurements may supply fixed values for parameters in a solution. For example, chemical measurements of genome size may be used to establish the rate constant of the single copy component. "mini-cot" experiments (Britten et. al., 1974) often provide the most reliable determinations of repetitive and which may be genome used as fixed Slave single copy rate constants values to calculate the fraction of the associated with the different rate components in a total reassociation curve. Fixing the single copy rate constant at a value determined from the genome size or minicot analysis may be particularly important for DNA which contains a small fraction repeated 2-20 fold. class is virtually indistinguishable This low repetition from single copy DNA but can increase the apparent single copy rate constant by a factor of two. - 162 - Parameter fixin~ also provides information about the variety of similar least squares solutions which describe the data equally well. graph of RMS vs. informative, A a set of fixed values of a parameter can be particularly if it turns out to be a very shallow curve near the minimum. Parameter statistics Parameter standard deviations and correlation coefficients provide information about the range of parameter values which may adequately describe the data. These numbers, calculated by the program along with the RMS and DELMX, indicate the significance of the parameter values obtained at the least squares solution. The parameter standard deviation and correlation coefficient calculation is similar to standard deviation and correlation coefficient calculations in linear regression analysis. deviations and inverted. The calculation assumes To obtain the standard correlation coefficients the data covariance matrix is that the Taylor's series linear approximation is accurate in the neighborhood of the solution (indicated by a low DELMX at convergence). parameter standard To examine the usefulness of the deviations and correlation coefficients, artificial test data were generated. Some examples of solutions using the second order (FINGER) function are shown in Table 2. The data set for each of these analyses Gaussian random number generator a l=• a for the final unreacted quantity as follows: C/Co = GAUS(F,FSD) + [ was calculated using 1 f.(1+k-Cot)1 1 a 0.196 0.017 0.198 0.019 0.212 0.015 0.208 0.017 0.180 0.015 0.200 0.007 O.d40 0. 35 0.298 0.017 O.dOl 0. 32 0.274 0.040 0.609 0. 38 0.2~~ 0.0 f2 O.d28 0. 38 0.297 0.038 0.106 0.02 5.82 2.47 10.2 4.2 9.28 1 .81 1. 18.4 16:g r~g 10.00 k2 0.0403 0.0379 0.0382 0.0392 0.0466 0.0350 RMS For this examole, Eis 0.2 O. 17 0.600 0.14g O.d1~ 0. 3_ 0.05 0.0819 0.600 0.0397 0. 41 0. 110 o.cto2 0.039 0. 35 0.05~6 0.02 9 0.106 0.048 o. 1000 0.300 k1 Noise factor equal to FSD described in the text. 200 40 40 40 40 40 f1 Parameter values at solution 0.200 0.300 0.040a Number of points Final Parameters used to generate data Solution 1 Error Solution 2 Error Solution 3 Error 4 Solution Error Solution 5 Error Solution sum of Error trials above Value Error Trial TABLE g_ EFrECT OF GAUSSIAN NOISE ON PARAMETER VALUES AND ERROR ESTIMATES Successive trials with random noise ....... O' w - 1b4 - (GAUS is a computer subroutine which generates a series of values following a Gaussian distribution with average E and standard deviation FSD.) This method of generating variation in the "data" is a good analogue of actual fluctuations from measurement to measurement, such as the binding of DNA to hydroxyapatite . The fluctuations ob.served in Table 2 for the various solutions and the standard deviations shown are probably representative of what would happen in measurements which showed repetitions of actual a comparable RMS. It is clear that the standard deviations of the rate constants are much larger than those of the component quantities. When analyses of this sort are carried out with components only a factor of ten apart in rate constant (instead of the factor of 100 for the example illustrated in Table 2) the rate constant fluctations are very large. It can be seen from Table 2 that there is a reasonable quantitative relationship between the fluctuations from set to set and the calculated standard deviations. This table exhibits one of the characteristic problems of fitting reassociation kinetic data and indicates the insight into the accuracy of individual parameter estimates provided by the standard deviation calculations. two kinetic components are It is clear that where present, even a factor of one hundred apart, very accurate data are needed to obtain estimates of rate constants with moderate accuracy. - 165 Discussion We have described a powerful and flexible squares analysis of the program for kinetics of reassociation. the least This program has been used for the analysis of an extensive series of measurements (Galau et. al., 1974; Davidson Britten and Davidson, 1976; analysis et . al., 1975b; Galau et. al., Smith et. 1976). al., 1975; Least squares is necessary for reproducible and clear interpretation of such measurements. We interpretation , refer while to in these papers for this discussion we examples of use and focus on over-interpretation or misinterpretation of the solutions. analytical tools of this problems of As powerful sort are developed it becomes very easy to simply accept the "output" and lose touch with its meaning. It is important to strategy provides biological that any successful least squares parameter values which "fit the data better" in the "least squares sense" . or remember The solution does not guarantee either physical reality. Components may be used to fit small peculiarities in the data and have little physical meaning. usually evident from Parameter values may also be the fluke particular importance. of a a plot of the fit. This is set of data and have little absolute The parameter standard deviations calculated by the program provide confidence limits for the oarameter values given the data . Where it is known from other measurements that a DNA homogeneous fraction is or where a single copy fraction has been purified the least squares analysis orovides an excellent means for evaluating its rate constant and permits a more accurate calculation of its complexity . In general it is not known that components are homogeneous and the often - 166 values of parameters derived from the least squares solutions may be averages of a set of unresolved components. In such a case the solution • makes an excellent model of the set of repetitive sequences which may be used in a variety of calculations even though the components are important issue not known. Where an true potential heterogeneity of a component other tools must ' individual rests on the be used. For example fractionation of double from single standed DNA could be carried out at the midpoint of its reassociation and the rate of reassociation of the two fractions carefully compared. Another consideration is as important as questions of parameter meaning: solution uniqueness. If a parameter can change over a wide range without affecting the quality of the fit measured by the RMS or GOODNESS OF suspect. FIT, conclusions based on a specific parameter value are In many cases, this problem will parameter standard deviation. be indicated by a high More quantitative insight into the significance of a particular parameter value can be gained by plotting the RMS or GOODNESS OF FIT criterion as a function of the best fit using different fixed values for the parameter in question. the best in the GOODNESS OF FIT is less than 10% over this be If the variation range of rate no conclusions can be based on the 0.05 value which would be different for the other values. to other solutions should with the rate constant fixed at 0.15 and 0.015. constants, if fit of the data gives a rate constant of 0.05 for a slow component with a GOODNESS OF FIT of 0.02, found For example, 100% with variations If the GOODNESS OF FIT changes by 50- in the parameter value the value is probably uniquely determined by the data. parameter value and While the method of fixing one varyin~ the others to find the best least squares - 167 - solution can be used for any parameter, when measuring rate it is particularly important constant parameter values. Data with two rate constants differing by a factor of 10-30 can usually be described by a . very wide range of rate constants . In summary, once a fully convergent least squares solution has been established for interpretation: a set of data we must consider four 1. the measurement There may be systematic errors in such as DNA degradation or unknown fragment size. 2. issues of The individual set of data may be atypical, particularly if only a few measurements are available and the standard deviations are then a poor measure of the possible error. Deviations in reassociation kinetic measurements are typically not due to random statistical variables such as sampling from a population or radioactive decay, but are more likely due to variations in assay procedures such as hydroxyapatite batch or temperature or volume of samples. deviations. 3. This weakens the significance of the standard The components may not represent true individual rate components but be averages of unresolved sets of components. least 4. The squares minimum in the region of the solution may be shallow. In that case, the set of solution parameter values may not uniquely fit the data; other parameter sets with substantially different values might provide equally valid interpretations of the data. In many cases such problems can be shown quantitative significance. to be of minor Least squares solutions such as those generated by this program represent the best oresently known approach to the interpretation of reassociation and hybridization kinetics. - 168 - References Cited Bishop, J. 0., Morton, J. . (1974) G., Roshbash, M. "Three abundance classes and Richardson, M. in HeLa cell messenger RNA" Nature 250:199-204 Britten, R. J. and Kohne, D. E. (1968) "Repeated sequences E. and Neufeld, in DNA", Science 161:529-540 Britten, R. J . , Graham, D. R. B. "Analysis of repeating DNA sequences by hybridization" in Methods in. Enzymology L. Grossman and K. Moldave, eds. Vol. 29 Part E pp. 363-4 rn Britten, R. acid J. and Davidson, E. H. , (1976) "Studies on nucleic reassociation kinetics: Empirical equations describing DNA reassociation" Proc. Nfil.... Acad. Sci. Davidson, E. H. transcription and and Britten, US. 73:415-419 R. J. , ( 1973) "Organization, regulation in the animal genome" Quart. Rev . Biol, 48:565-613 Davidson, E. J., (1975) H., Galau, G. A.., Angerer, R. C. and Britten, R. "Comparative aspects of DNA organization in metazoa" Chromosoma 51 : 253-259 Davidson, E. ( 1975) H., Hough, B. "Structural R. , Klein, W. genes adjacent sequences" .Q.e.ll 4 :217- 238 to H. and Britten, R. interspersed J. repetitive DNA. - 169 Galau, G. A., Britten, R. and Davidson, J. E. H., ( 1974) "A measurement of the sequence complexity of polysomal messenger RNA in sea urchin embryos" Cell 2:9-20 R. Galau, G. A., Klein, W. H., Davis, M. J. Davidson, H. and E. M., Wold, B. J. Britten, (197b) "Structural gene sets active in embryos and adult tissues of the sea urchin" Cell 7:487-505 Laird, C. D. {1971) "Chromatid structure: Relationships between DNA content and nucleotide sequence complexity" Chromosoma 32:378-406 Marquardt, D. (1963) W. estimation of non-linear "An algorithm for parameters" J..:.. Soc. least Indust. squares Aru2.L.. Math. 11:431-441 Martin, R. S., Peters, G., and Williamson, J. H. "Symmetric decomposition of a positive definate matrix" Algebra by J. H. Wilkinson and C. Reinsch Vol. AUTOMATIC COMPUTATION F. L. Olver, H. Samelson and Rutishauser, K. Springer-Verlag pp. S. E. in Linear of illlliJlaQQK FOR Householder, F. Statel, ed. W. J. New York: 9-30 Morrow, J. (1974) Ph. Ryffel, G. u. cytoplasmic Bauer, A. II (1971) D. Thesis, Stanford University and McCarthy, B. J. ( 1975) "Complexity of RNA in different mouse tissues measured by hybridization of polyadenylated RNA to complementary DNA" Biochemistry 14:1379-1384 Smith, M. J., Britten, R. J. and Davidson, E. H. (1975) "Studies on nucleic acid reassociation kinetics: Reactivity of single stranded DNA tails in DNA-DNA renaturation" Proc. Nat. 72:4805-4809 AQ&L.. ~ US - 170 - APPENDIX l This is a sample run of the program with user underlined responses .RU NNNBAT 1START,2ADD,3GUESS,5PLOT,6CPLOT,8LPLOT,9STOP _1 TYPE RED DATA FILENAME RTDNA 106 PT RAT COT CURVE TO LIST DATA TY 1 start by getting data name of file on disk do not· list datc:t 1START,2ADD,3GUESS,5PLOT,6CPLOT,8LPLOT,9STOP do a fit FUNCTION (-1HELP), ITERATIONS, RMS QUIT, DELMX QUIT, (NIT) .3.... :::.L check the names of the functions OSAME,1FINGER,2WHATOR,3NUFORM,4EXCESS,5WHTCMP FUNCTION (-1HELP), ITERATIONS, RMS QUIT, DELMX QUIT, (NIT) 1 I 10 I • 0001 I • 05 FINAL,FRACT(1),K(1),FRACT(2),K(2),FRACT(3),K(3) .05 .17 2 •. 08 .01 .65 .0005 initial parameter guesses TYPE INDEX OF PARAMETERS TO BE FIXED. no fixed parameters ORIGINAL RMS= 0.0447680 GOODNESS OF FIT 0.0463237 RMS DELMX PARAMETERS 0.0304963 1.7717584 0.393E-01 0.171E+OO 0.722E+OO 0.979E-01 0.985E-02 O.b02E+OO 0.455E-03 0.0298000 0.2591739 0.383E-01 O. 174E+OO 0.974E+OO 0.100E+OO 0.8b7E-02 0.598E+OO 0.445E-03 0.0297149 0.0705356 - 171 0.374E-01 0.176E+OO 0.105E+01 0.103E+OO 0.817E-02 0.594E+OO 0.438E-03 iterations stop because RMS is changing by less than RMS QUIT=0.0001 covariance matrix I 2 4 3 5 6 7 1 0. 100E+01 2-0.251E+OO 0.100E+01 3 0.175E+00-0.612E+OO 0. 100E+01 4-0.699E+OO 0.312E+00-0.180E+OO 0.100E+01 5 0.559E+00-0.715E+OO 0.555E+00-0.825E+OO 0. 100E+01 6 0.353E+00-0.526E+OO 0.385E+00-0.876E+OO 0.901E+OO 0.100E+01 7 0.861E+00-0.406E+OO 0.289E+00-0.941E+OO 0.820E+OO 0.759E+OO 0.100E+01 106 PT RAT COT CURVE RMS= 0.0297149 GOODNESS OF FIT 0.0307474 106 POINTS 99 DEG OF FREEDOM .. P( 1) ...•.. P(2) ...... P(3) ...... P(4) ...... P(5) ...... P(6) ...... P(7) .. . parameters are printed in the same order they were input. FINAL,FRACT(1),K(1), ... the first line is the parameter values 0.374E-01 0.176E+OO 0.105E+01 0. 103E+OO 0.817E-02 0.594E+OO 0.438E-03 0.325E-01 0.198E-01 0.451E+OO 0.695E-01 0.116E-01 0.609E-01 0.140E-03 the second line is the parameter errors 1START,2ADD,3GUESS,5PLOT,6CPLOT,8LPLOT,9STOP Do another fit to illustrate OSAME function and a local minimum without convergence FUNCTION (-1HELP), ITERATIONS, RMS QUIT, DELMX QUIT, (NIT) J_ 0,10,0.,.01 OSAME, the fit continues from where it left off ORIGINAL RMS= 0.0297149 GOODNESS OF FIT 0.0307474 RMS DELMX PARAMETERS 0.0296630 0.0514820 0.366E-01 0.177E+OO 0.106E+01 0.107E+OO 0.777E-02 0.591E+OO 0.431E-03 0.0296151 0.0407157 - 172 0.358E-01 0.177E+OO 0.106E+01 0.110E+OO 0.746E-02 0.589E+OO 0.425E-03 0.0295696 0.0350217 Note the change in DELMX after 0.349E-01 0.177E+OO 0.106E+01 0.114E+OO 0.721E-02 0.586E+OO 0.419E-03 0.0295148 0.6947887 this point. A local minimum was found, 0.264E-01 0.179E+OO 0.103E+01 0. 143E+OO 0.425E-02 0.563E+OO 0.367E-03 0.0290103 0.4464171 going on caused a shift to a path 0.182E-01 0.175E+OO 0.112E+01 0.166E+OO 0.531E-02 0.553E+OO 0.331E-03 0.0288781 1.4516340 leading to a negative parameter 0.744E-02 0.177E+OO 0.106E+01 0.189E+OO 0.382E-02 0.538E+OO 0.295E-03 0.0286363 6.6211100 -0. 132E-02 0.175E+OO 0.112E+01 0.208E+OO 0.402E-02 0.530E+OO 0.268E-03 0.0285198 0.8757134 -0.106E-01 0.176E+OO 0.109E+01 0.225E+OO 0.351E-02 0.521E+OO 0.244E-03 0.0284246 0.4295531 -0.187E-01 0.176E+OO 0 : 110E+01 0.240E+OO 0.337E-02 0.515E+OO 0.225E-03 0.0283610 0.2825794 now the fit will not converge -0.260E-01 0.176E+OO 0.110E+01 0.252E+OO 0.317E-02 0.510E+OO 0.209E-03 I 1 2 3 4 5 6 7 1 0. 100E+01 2-0.177E+OO 0.100E+01 3 0.132E+00-0.380E+OO 0.100E+01 4-0.722E+OO 0.345E+00-0.251E+OO 0.100E+01 5 0.535E+00-0.607E+OO 0.490E+00-0.885E+OO 0.100E+01 6 0.967E-01-0.455E+OO 0.358E+00-0.747E+OO 0.828E+OO 0.100E+01 7 0.900E+00-0.305E+OO 0.233E+00-0.937E+OO 0.774E+OO 0.498E+OO 0.100E+01 106 PT RAT COT CURVE RMS= 0.0283610 GOODNESS OF FIT 0.0293465 106 POINTS 99 DEG OF FREEDOM .. P( 1 ) ... . .. P (2) ...... P ( 3) ..... . P( 4) ...... P( 5) ...... P(6) ...... P(7) .. . -0.260E-01 0.176E+OO 0.110E+01 0.252E+OO 0.317E-02 0.510E+OO 0.209E-03 0.565E-01 0.135E-01 0.383E+OO 0.829E-01 0.176E-02 0.635E-01 0.101E-03 1START,2ADD,3GUESS,5PLOT,6CPLOT,8LPLOT,9STOP a fit to show parameter fixing FUNCTION (-1HELP) , ITERATIONS, RMS QUIT, DELMX QUIT, (NIT) 3._ 0,15,0. o. FINAL,FRACT(1),K(1),FRACT(2),K(2),FRACT(3),K(3) - 173 .05 .17 2 .• 08 .01 .65 .00035 k(3) is set to rate calculated from genome size TYPE INDEX OF PARAMETERS TO BE FIXED. L ORIGINAL RMS= 0.0608586 GOODNESS OF FIT 0. 0626578 RMS DELMX hold parameter 7 constant. note that with this constraint the fit converges PARAMETERS 0.0438545 7.8186761 0.997E-02 0.176E+OO 0.496E+OO 0. 134E+OO 0.113E-02 0.589E+OO 0.350E-03 0.03400~8 0.6862643 0.318E-01 0.186E+OO 0.772E+OO 0 . 108E+OO 0.315E-02 0.583E+OO 0.350E-03 0.0293703 0.5452864 0 . 235E-01 0.170E+OO 0.111E+01 0.149E+OO 0.692E-02 0.570E+OO 0.350E-03 0.0292464 0.6662786 0.228E-01 0.178E+OO 0.105E+01 0.158E+OO 0.415E-02 0.553E+OO 0.350E-03 0.0290227 0.2421386 0 . 301E-01 0.175E+OO 0. 112E+01 0.171E+OO 0.468E-02 0.536E+OO 0.350E-03 0.0290199 o. 1418406 0,334E-01 0.177E+OO 0. 107E+01 0. 176E+OO 0.410E-02 0.525E+OO 0.350E-03 ' 0.0290172 0.0786546 0.320E-01 O. 176E+OO 0.111E+01 0.174E+OO 0.445E-02 0. 530E+OO 0 . 350E-03 0.0290166 0.0502960 0.330E-01 0.177E+OO 0.109E+01 0. 176E+OO 0.424E-02 0.526E+OO 0.350E-03 0.0290163 0.0301044 0.325E-01 O. 176E+OO 0 . 110E+01 0.175E+OO 0.437E-02 0.529E+OO 0 . 350E-03 0.0290163 0.0187524 0.32~E-01 O. 177E+OO 0. 109E+01 0.1 75E+OO 0.429E-02 0.527E+OO 0.350E-03 0.0290162 0.0113841 0.326E-01 O. 176E+OO 0.110E+01 0.175E+OO 0.434E-02 0.528E+OO 0.350E-03 0 . 0290162 0.0070176 0 . 327E-01 0.176E+OO 0 .1 09E+01 0.175E+OO 0.431E-02 0.528E+OO 0.350E-03 0.0290162 0 . 0042845 0.327E-01 0. 176E+OO 0.110E+01 0. 175E+OO 0.433E-02 0.528E+OO 0.350E-03 0.0290162 0.0026309 0.327E-01 0. 176E+OO 0. 109E+01 0.175E+OO 0 . 431E-02 0.528E+OO 0.350E-03 0.0290162 0.0016100 - 174 - 0.327E-01 0.176E+OO 0.110E+01 0.175E+OO 0.432E-02 0.528E+OO 0.350E-03 I 1 2 3 4 5 6 1 0. 100E+01 2 0.216E+OO 0.100E+01 3-0.163E+00-0.413E+OO 0.100E+01 4 0.773E+OO 0.131E+00-0.468E-01 0.100E+01 5-0.547E+00-0.654E+OO 0.529E+00-0.664E+OO 0.100E+01 6-0.900E+00-0.379E+OO 0.291E+00-0.918E+OO 0.807E+OO 0.100E+01 106 PT RAT COT CURVE RMS= 0.0290162 GOODNESS OF FIT 0.0298740 106 POINTS 100 DEG OF FREEDOM .. P( 1) ...... P(2) ...... P(3) ...... P(4) ...... P(5) ...... P(6) ...... P(7) .. . 0.327E-01 0. 176E+OO 0.110E+01 0.175E+OO 0.432E-02 0.528E+OO 0.350E-03 0.192E-01 0.142E-01 0.396E+OO 0.300E-01 0.223E-02 0.509E-01 O.OOOE+OO the covariance matrix does not include parameter 7 and the parameter 7 error is 0.000 because it is fixed 1START,2ADD,3GUESS,5PLOT,6CPLOT,8LPLOT,9STOP 1- error estimates without iteration FUNCTION (-1HELP), ITERATIONS, RMS QUIT, DELMX QUIT, (NIT) .LQ O iterations gives error estimates FINAL,FRACT(1),K(1),FRACT(2),K(2),FRACT(3),K(3) .0327 .176 1,10 .175 .00432 .528 ,00035 TYPE INDEX OF PARAMETERS TO BE FIXED. check the error without the parameter fixed I 1 2 3 4 5 6 7 1 0. 100E+01 2-0.201E+OO 0.100E+01 3 0.145E+00-0.460E+OO 0.100E+01 4-0.735E+OO 0.361E+00-0.254E+OO O. 100E+01 5 0.579E+00-0.627E+OO 0.496E+00-0.900E+OO 0.100E+01 6 0.465E+00-0.484E+OO 0.368E+00-0.933E+OO 0.937E+OO 0.100E+01 7 0.877E+00-0.341E+OO 0.252E+00-0.959E+OO 0.828E+OO 0.81bE+OO 0.100E+01 106 PT RAT COT CURVE - 175 RMS= 0.0290170 GOODNESS OF FIT 0.0300253 106 POINTS 99 DEG OF FREEDOM . . P( 1) . .. . . . P(2) . ..... P(3) ...... P(4) .. .. .. P(5) .... .. P(6) . .... . P(7) .. . 0.327E- 01 O. 176E+OO 0. 110E+01 0. 175E+OO 0 . 432E-02 0 .528E+OO 0. 350E-03 0 . 401E- 01 0. 152E- 01 0 . 414E+OO 0. 106E+OO 0. 399E-02 0.883E-01 0. 160E-03 1START ,2ADD,3GUESS , 5PLOT,6CPLOT,8LPLOT,9STOP 9-. STOP - 176 - Appendix II The program: Structure and Method Appendix III provides an annotated listing of the main program is divided (1START,2GUESS), (5PLOT,8LPLOT) into least squares program. The four blocks corresponding to input fitting (3GUESS), plot and data output for future plotting (bCPLOT). output The input section scans the input until a 3.0 is found in the C/Co value and sorts the data using the subroutine SORTI. subroutine NPLMLT. for the then Plotting is done with the To plot the data on the line printer the unit number output statements is changed and the program goes to the 5PLOT section. In the 3GUESS option, the parameter input of the FUNCTION line and guesses are handled by the PARMIO subroutine. The number of parameters to be fit is counted from the input data by COUNT, the of parameters to routine, REFIN, Once the parameters are specified, the main is called with one argument, DIFSUB, used to specify which of the the subroutines FINDIF, WHTDIF, FRMDIF, WHADIF should be index be fixed is read in and an array of parameters to be varied is generated by MAP. fitting the used EXCDIF or to provide the actual derivatives and function values for the fitting routine. The algorithm used in REFIN was developed by Marquardt non-linear least squares problems. (1963) The algorithm first attempts to find improved parameter values using Taylor's expansion around the region the old parameter estimates . for of If the new parameters improve the fit they - 177 are used for the subsequent iteration, vector is rotated in the until a new parameter set (SYMDET and SYMSOL), if not, the direction of the decreasing error gradient better fits manipulate the the data. Two subroutines, symmetric partial derivative matrices used to solve for the parameter changes. adapted the calculated and terminate, inverted correlation matrix. control These from ALGOL versions published by Martin et. iterations parameter change the partial a~. programs were (1971). derivative matrix When is again by SYMINV to provide the error estimates and WCOVAR writes out the correlation matrix and is returned to the main program, where the parameters and error estimates are printed. REFIN is particularly efficient in storage and manipulation of the partial derivative matrices. Only one new matrix with dimensions (number of variable oarameters) 2 is stored. This results in si~nificant savings of space over methods which store two square partial derivative matrices, one for the matrix and one for its inverse, and also a matrix storing the partial derivative for the jth parameter at the ith data point with dimensions (number of parameters* number This latter matrix of data ~rows with the number of data points and thus with the quality of the fit, requiring more storage for better data. this points). While is a minor consideration for nucleic acid experiments where more than 7 parameters and 50 points are rarely fit, it becomes important for t.he analysis of other data (particularly protein bands on polyacrylamide gels fit by Gaussian functions) where as many as 100 parameters may be fit to 1000 data points. - 17~ - Parameter fixing uses parameters to an array, IMAP, to map the indices of be varied into the partial derivative matrices. derivatives at each data point are calculated for all Partial parameters, whether they are varied or not, but when the partial derivative matrices are built up, only the parameters with indices in the used. holding any number or This provides a versatile method for combination of parameters constant during the fit . particularly useful array IMAP Parameter fixing in careful analyses of the are is significance of particular sets of parameter values . Fitting a nu~ber of functions decreases the computational efficiency of this progra~ by requirin~ a subroutine call for each data point during the generation of the partial derivative matrix. efficient A. more program fitting one function can be written putting the loop from the appropriate DIFSUB subroutine into REFIN. Analytic derivatives are used for four of the five Empirical derivatives used for functions fit . the NUFORM function are calculated by successive evaluation of the function at one parameter value and a value 1% different. It is important to note that this partial differentiating routine can be used to fit any other function as well, simply by substituting the new function for the subroutine FORM. This program, and the REFIN fitting subroutine in provide required. particular, can a versatile tool whenever non-linear least squares analysis is The algorithm used is efficient, the internal storage required is compact and the ability to fix any combination of parameters provides unique insight into the physical meaning of the solution. - 179 APPENDIX III C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C (C) COPYRIGHT 1977 WILLIAM PEARSON THIS VERSION SUPPORTS FIXED PARAMETERS, ERROR ESTIMATES AND ERRORS FROM BATCH. IT HAS BEEN TESTED FOR CORRECT ERROR ESTIMATES AND PARTIAL DERIVATIVES FOR FINGER,WHATOR, WHTCMP AND EXCESS. SUBROUTINES CALLED REFIN,PARMIO,MAP ,RRJE,COUNT,PLMLT,SORTI,FORM,WCOVAR PROBLEMS WITH THIS PROGRAM MAY BE ANSWERED BY BILL PEARSON KERCHOFF MARINE LAB 101 DAHLIA CORONA DEL MAR, CA. 92625 PH. (714) 675-2159 VARIABLE USAGE: ARRAYS COT(200) YY(200) NMAX:200 COT VALUES FRACTION SINGLE STRANDED MAXIMUM NUMBER OF DATA POINTS P(11) 0(11) PARAMETER VALUES PARAMETER ERRORS, PARAMETER CHANGE VECTOR FOR 4 FINGR MAXIMUM NUMBER OF PARAMETERS NPMAX:11 IA(11) ARRAY OF INDICES FOR PARAMETERS HELD CONSTANT MAPPING OF INDICES OF PARAMETERS ALLOWED TO CHANGE ARRAY OF VALUES TO BE PLOTTED IMAP(11) ROUT( 18) COMMON /IO/ COMMON BLOCK FOR INPUT/OUTPUT DEVICE mJMBERS LRD INPUT DEVICE; 5 FOR TTY TIMEHARING, PROBABLY ALSO 5 FOR CDR IN BATCH MODE OUTPUT DEVICE; 5 FOR TTY TIMESHARING , PROBABLY b FOR LPT TN BATCH STORAGE DEVICE; 20 FOR DSK ON PDP-10 DATA IS STORED IN DSK FILES. OPEN SETS UP DSK FILES POR READS AND WRITES C C LWD C C C LSD - 180 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C PROBABLY 5 IN A BATCH SYSTEM. NO OUTPUT TO LSD IS NECESSARY UNLESS AN OPTION 6 SYSTEM IS SET UP :EXTERNAL SUBROUTINES THIS PROGRAM WILL FIT 5 DIFFERENT FUNCIONS ENCOUNTERED ED IN DNA-DNA AND DNA-RNA HYBRIDIZATION EXPERIMENTS. THE DIFFERENT FUNCTIONS ARE SPECIFIED BY THE VALUE OF IWHAT=1 .. 5 AND PARTIAL DERIVATIVES AND ERRORS ARE EVALUATED BY THE EXTERNAL SUBROUTINES FINDIF, WHADIF, FRMDIF, EXCDIF·, WHTDIF. THE FUNCTIONS ARE: !WHAT EXTERNAL FUNCTION 1 FINDIF (FINGER) MULTICOMPONENT 2ND ORDER REASSOCIATION AS MEASURED ON HYDROXYLAPATITE. F(X,P)=P(1)+P(2I)/(1.P(2I+1)*X) FOR I=1 TO NC COMPONENTS 2 WHADIF (WHATOR) FUNCTION USED TO STUDY KINETCS OF DNA REASSOCIATION MEASURED BY NUCLEASE DIGESTION. F(X,P)=P(1)+P(2)/(1.+P(3)*X)**P(4) 3 FRMDIF (NUFORM) FUNCTION WHICH DESCRIBES THE HYBRIDIZATION OF UP TO 2 DRIVEN COMPONENTS WITH RATES P(3),P(5) DRIVEN BY A VAST EXCESS OF A COMPONENTS HYBRIDIZING WITH RATE P(b). P(2,4) ARE THE FRACTIONS OF THE COMPONENTS DRIVEN. P(7) IS THE FACTOR ANALOGOUS TO P(4) IN WHATOR. P(1):THE FINAL AMOUNT UNREACTED. F(X,P)=P(1)+P(2I)*EXP(P(2I+1)*( 1.-(1+P(6)*X)**(1-P(7)))/(P(6)*(1.-P(7)))) 4 EXCDIF (EXCESS) 1ST ORDER REASSOCIATION (RNA EXCESS) F(X,P)=P(1)+P(2I)*EXP(-P(2I+1)*X) 5 WHTDIF (WHTCMP) MULTICOMPONENT REASSOCIATION MEASURED BY NUCLEASE F(X,P)=P(1)+P(2I)/(1.+P(2I+1)*X)**(0.44) F(X,P) IS EACH CASE IS THE FRACTION SINGLE STRANDED AT COT X. THIS PROGRAM CAN FIT OR PLOT UP TO 5 COMPONENTS FOR FINGER, EXCESS OR WHTCOMP AS THE NUMBER OF PARAMETERS IN THOSE CASES IS NP=2*NC+1 AND NPMAX:11. - 181 C C C NOTE, STATEMENTS FLAGGED: *-------------------c ARE SUBROUTINES OR OPTIONS POSSIBLY SPECIFIC FOR THE PDP-10 C C C AND MAY NOT WORK ON OTHER SYSTEMS DIMENSION COT(200),YY(200),P(11),D(11),IMAP(11),IA(11),ROUT(18) DOUBLE PRECISION CNAME,TITLE COMMON /IO/LRD,LWD,LSD EXTERNAL FINDIF,WHADIF,FRMDIF,EXCDIF,WHTDIF DATA NMAX,NPMAX/200,11/ C C C C COMCON IS THE CONSTANT RELATING RATE CONSTANT TO NUCLEOTIDES COMPLEXITY DATA COMCON/1015000./ C LRD=5 LWD=5 LSD=20 C C C C C LWDSAV IS USED TO SAVE THE WRITE DEVICE WHEN IS IS CHANGED IS OPTION 8. THIS WAY ONLY THE ABOVE THREE STATEMENTS NEED BE CHANGED TO CHANGE IO DEVICES LWDSAV=LWD C C C LPTD=3 IS TO USE THE LINE PRINTER IN OPTION 8 LPTD=3 IWHAT=1 C C C C MAIN SWITCH POINT. AFTER ANY OPTION IS DONE, RETURN HERE TO CHOOSE NEW OPTION 10 WRITE(LWD,20) 20 FORMAT('O 1START,2ADD,3GUESS,5PLOT,6CPLOT,8LPLOT,9STOP'/) READ(LRD,30) ICHO *-------------------c 30 FORMAT(I2) 30 FORMAT(I) GOTO (100,105,200,10,500,600,10,800,900),ICHO GOTO 10 C C C C C C C INPUT OPTION GET A DATA FILE NAME OPEN FILE FOR READING READ TITLE READ FRACT SS, COT AND COUNT NUMBER OF POINTS UNTIL FRACT SS=3. - 182 C C C C C C C C C THE FORM OF THE INPUT DATA FILE SHOULD BE FIRST CA.RD: SECOND CARD: NEXT CARDS: A.NY 54 CHA.R TITLE, 1ST CHAR BLA.NK FRACT. SINGLE STRANDED,COT. SAME LA.ST CARD: 3.0 100 KEND=O 105 WRITE(LWD,110) 110 FORMAT(' TYPE RED DA.TA. FILENAME'/) REA.D(LRD,115) TITLE 115 FORMAT(A10) *-------------------OPEN(UNIT=LSD,A.CCESS='SEQIN' ,FILE:TITLE) READ(LSD,120) 120 FORMA.T(54H------------------------------------------------------) WRITE(LWD, 120) 125 KEND=KEND+1 READ(LSD,130) YY(KEND),COT(KEND) *-------------------C 130 FORMAT(1X,F7.5,E11.4) 130 FORMAT(2F) IF (YY(KEND).GE.3.0) GOTO 135 IF (KEND.GT.NMAX) GOTO 160 GOTO 125 135 KEND=KEND-1 C C C SORT DATA IN INCREASING ORDER BY COT. NECESSARY FOR PLOTTING C CALL SORTI(COT,KEND,YY) WRITE(LWD, 140) 140 FORMAT(' TO LIST DA.TA. TY 1'/) READ(LRD,145) ICHO *------------------c ~45 FORMAT(I2) 145 FORMAT(I) IF (ICHO.NE.1) GOTO 10 WRITE(LWD, 155) 155 FORMAT(8X,'COT' ,2X,'FRACT SS'//) WRITE(LWD,150) (COT(I),YY(I),1=1,KEND) 150 FORMAT(5X,F10.3,Fb.3) GOTO 10 C C TOO MUCH DATA, ONLY REA.DIN FIRST NMAX POINTS C 160 WRITE(LWD,165) 165 FORMAT(' MAX NUMBER OF POINTS EXCEEDED'/' SOME DATA LOST'/) GOTO 135 C - 183 C C C C GUESS OPTION FOR PARTIAL DERIVATIVE LEAST SQUARES FIT INPUT IWHAT, PARAMETERS AND CALCULATE NP(NO. OF PARAM). AND NC(NO. OF COMPONENTS). 200 CONTINUE 210 CALL PARMIO(P,NP,NC,NPMAX,IWHAT,NINT,RMSQU,DLMXQU,NIT,&228) WRITE(LWD,220) 220 FORMAT(' TYPE INDEX OF PARAMET~RS TO BE FIXED.'/) READ(LRD,225) (IA(I),I=1,NPMAX) *-------------------C 225 FORMAT(11I2) 225 FORMAT( 11I) C C MAP PARAMETERS TO BE VARIED IN IMAP . C CALL MAP(IA,NP,IMAP,NDIFF) 228 CONTINUE C C C CALL LEAST SQUARES FITTING ROUTINE USING APPROPRIATE PARTIAL DERIVATIVE SUBROUTINE. C GOTO (255,260,265,270,275),IWHAT 255 CALL REFIN(COT,YY,KEND,P,D,NP,NC,IMAP,NDIFF,NIT,O.,FINDIF,DIFFSQ, 1NINT,RMSQU,DLMXQU) GOTO 290 260 CALL REFIN(COT,YY,KEND,P,D,NP,NC,IMAP,NDIFF,NIT,O.,WHADIF,DIFFSQ, 1NINT,RMSQU,DLMXQU) GOTO 290 265 CALL REFIN(COT,YY,KEND,P,D,NP,NC,IMAP,NDIFF,NIT,O.,FRMDIF,DIFFSQ, 1NINT,RMSQU,DLMXQU) GOTO 290 270 CALL REFIN(COT,YY,KEND,P,D,NP,NC,IMAP,NDIFF,NIT,O.,EXCDIF,DIFFSQ, 1NINT,RMSQU,DLMXQU) GOTO 290 275 CALL REFIN(COT,YY,KEND,P,D,NP,NC,IMAP,NDIFF,NIT,O . ,WHTDIF,DIFFSQ, 1NINT,RMSQU,DLMXQU) GOTO 290 290 CONTINUE C C OUTPUT TITLE , PARAMETERS AND PARAMETER ERRORS . C WRITE(LWD,120) IDENOM=KEND-NDIFF IF (IDENOM.LT.1) IDENOM=KEND RMS:SQRT(DIFFSQ/KEND) CHISQ=SQRT(DIFFSQ/IDENOM) WRITE(LWD,235) RMS,CHISQ,KEND,IDENOM 235 FORMAT('ORMS=' ,F13.7,' GOODNESS OF FIT' ,F13 .7/5X,I3,' POINTS' ,5X, 113,' DEG OF FREEDOM'/) • WRITE(LWD,230) 230 FORMAT('O . .. P(1) ...... P(2) .... . . P(3) ...... P(4) .... .. P(5) . ..... P(6) 1.... .. P(7) ... '/) - 184 WRITE(LWD,240) (P(I),I=1,NP) 240 FORMAT(1X, 10E10 . 3,/) WRITE(LWD,240) (D(I),I=1,NP) GOTO 10 C C C C C C C C OPTION 5 PLOT. PLOTS DATA AND APPROPRIATE FUNCTION ON TTY WITH A LOO SCALE. LPST:8 IS FIRST POSITION FOR DATA LPSP=18 IS LAST POSITION FOR DATA II IS BEGINNING X VALUE FOR PLOT IEND IS EXPONENT OF LAST POINT PLOTTED. K IS CURRENT DATA POINT TO BE PLOTTED C 500 CALL PLMLT(X,ROUT,L,1,100.,0 . ) LPST:8 LPSP:18 II=12.*ALOG10(COT(1)) IEND= (ALOG 10( COT(KEND) )-ALOG10( COT( 1)) )"t 12 .+24. K=1 C C ZERO ARRAY OF VALUES TO BE PLOTTED C DO 515 L=1,LPSP 515 ROUT(L)=O. C C FOR EACH PLOT INCREMENT ALONG THE X AXIS C DO 510 J=1,IEND XL=FLOAT(II+J)/12.-1. X:10 .**XL ROUT(1)=0. L=LPST-1 C C IF THERE ARE STILL DATA POINTS TO BE PLOTTED C IF(K.GT.KEND) GOTO 530 C C CHECK TO SEE IF A DATA POINT SHOULD BE PLOTTED C 525 IF (ALOG10(COT(K)).GT.(XL+.04)) GOTO 530 C C STORE THE ERROR IN THE XVALUE OF THE POINT C ROUT(1)=XL-ALOG10(COT(K))+.1 L=L+1 C C C STORE THEY VALUES OF THE POINT AND INCREMENT THE PLOT ARRAY INDEX(L) AND DATA ARRAY INDEX(K) C ROUT(L)=YY(K) K=K+1 IF (L.GE.LPSP) GOTO 530 - 185 IF (K.LE.KEND) GOTO 525 C C C CALCULATE THE COMPONENT AND FUNCTION VALUES AND STORE IN PLOT ARRAY. C 530 CONTINUE GOTO (1540,1550,1560,1570,1580),IWHAT C C FINGER C 1540 SUMC=P(1) DO 1542 I=1,NC JF=2*I JK=JF+1 COMP=P(JF)/(1 .0+X*P(JK)) ROUT(I+2)=COMP 1542 SUMC=SUMC+COMP ROUT(2)=SUMC GOTO 570 C C WHATOR C 1550 ROUT(3)=0. ROUT(5)=0. ROUT(4)=P(2)/(1.0+X*P(3)) COMP=P(2)/(1.0+X*P(3)) **P(4) ROUT(2)=COMP+P(1) GOTO 570 C C NUFORM C 1560 CALL FORM(P,X,ROUT(2)) COMP=P(4) P(4)=1 . 0 CALL FORM(P,X,ROUT(5)) P(4)=COMP GOTO 570 C C EXCESS C 1570 SUMC=P(1) 00 1572 I=1,NC JF=2*I JK=JF+1 XE=X*P(JK) COMP=O . IF (XE.LT.10.) COMP=P(JF) *EXP(-XE) ROUT(I+2)=COMP 1572 SUMC=SUMC+COMP ROUT(2)=SUMC GOTO 570 C - 186 C WHTCMP C 1580 SUMC=P(1) DO 1582 I=1,NC JF=2*I JK=JF+1 COMP=P(JF)/(1.+X*P(JK))**(0.44) ROUT(I+2) =COMP 1582 SUMC=SUMC+COMP ROUT(2)=SUMC GOTO 570 C C PLOT DATA AND COMPONENTS IN ARRAY ROUT C 570 CALL PLMLT(X,ROUT,L,3,100.,0.) 510 CONTimJE C C PUT ON BOTTOM AXIS C CALL PLMLT(X,ROUT,L,2,100.,0.) WRITE(LWD,572) 572 l<'ORMAT( 1HO) WRITE(LWD, 120) WRITE(LWD,235) RMS,CHISQ WRITE(LWD,575) 575 FORMAT('O COMPONENT ... FRACTION ... COMPLEXITY .......... K........... . 1KPURE' I) C C OUTPUT FINAL COMPLEXITIES C DO 580 J= 1 ,NC L=2*J Q=COMCON*P(L)/P(L+1) QQ=P(L+1)/P(L) WRITE(LWD,585) J,P(L),Q,P(L+1),QQ 585 FORMAT(1HO,I8,F10.4,3E15.4) 580 CONTINUE WRITE(LWD,590) P(1) 590 FORMAT('OFINAL=' ,F7.4) LWD=LWDSAV GOTO 10 C C C C C THIS SECTION WRITES DATA FILES FOR PLOTTING BY OTHER PROGRAMS SUCH AS A CALCOMP PLOTTER PROGRAM. IT MAY BE REMOVED ON OTHER SYSTEMS 600 WRITE(LWD,610) 610 FORMAT(' GIVE CALCOMP PLOT DATA FILE NAME'/) READ(LRD,615) CNAMf. 615 FORMAT(A10) *-------------------OPEN(UNIT=LSD,ACCESS='SEQOU' ,FILE=CNAME) - 187 - WRITE(LWD,620) 620 FORMAT(' TYPE TITLE FOR PLOT'/) READ(LRD,625) 625 FORMAT(54H------------------------------------------------------) WRITE(LSD,625) WRITE(LSD,630) KEND,NP,IWHAT 630 FORMAT(3I6) WRITE(LSD,633) (P(I),I=1,NP) 633 FORMAT(5E16.9) WRITE(LSD,635) (COT(I),YY(I),I=1,KEND) 635 FORMAT(5(E11.4,F5.3)) END FILE LSD GOTO 10 C C C C C THIS SECTION CHANGES LWD TEMPORARILY SO THAT THE PLOT OF THE DATA IS WRITEEN OUT TO THE LINE PRINTER INSTEAD OF THE TTY 800 LWDSAV=LWD LWD=LPTD WRITE(LWD,810) 810 FORMAT( 1H1) WRITE(LWD,820) TITLE 820 FORMAT(' FILE NAME= ',A10,//) WRITE(LWD, 120) WRITE(LWD,230) WRITE(LWD,240) (P(I),I=1,NP) WRITE(LWD,240) (D(I),I=1,NP) WRITE(LWD,572) GOTO 500 900 CONTINUE STOP END REFBAT.F4 C VERSION 306 C C C C C C C C C C C C C C C C 20 AUG 1976 THIS VERSION 305 USES A BATCH METHOD AND SCALES THE PARTIAL DERIVATIVE MATRIX AND USES THE COSINE CORRECTION DONE CORRECTLY THIS VERSION SUPPORTS FIXED PARAMETERS, CALCULATES PARAMETER ERRORS CORRECTLY, AND ALLOWS FOR TYPING OF A PARAMETER CORRELATION MATRIX. THIS IS A GENERAL LEAST SQUARES FITTING PROGRAM USING A MODIFIED DAMP LEAST SQUARES ALGORITHM DESIGNED BY MARQUARDT. J. SOC. IND. APPL. MATH. (1965) 11:431-441 ARGUMENTS: C XP ( 1) C YP(1) ARRAY OF DATA X VALUES ARRAY OF DATA Y VALUES - 1!j!j - C N NUMBER OF DATA VALUES P(1) D(1) NP PARA.METER VAUES PARAMETER ERRORS NUMBER OF PARAMETERSS IMAP ND INDEX MAPPING OF PARAMETERS TO BE VARIED NUMBER OF PARAMETERS VARIED NIT NUMBER OF CORRECTIVE ITERATIONS ALLOWED C C C C C C C C C C C C C C C FIN RMS CHANGE CONSIDERED SIGNIFICANT, SHOULD BE O. SUBDIF SUBROUTINE PROVIDING PARTIAL DERIVATIVES A.tID FUNCTION VALUES. DIFFSQ ERROR OF THE FIT C C C NINT NUMBER OF ITERATIONS, IF LEO ERCAL=.TRUE. AND NO ITERATIONS RMSQU QUIT CRITERIA FOR RMS C C C C DLMXQU QUIT CRITERIA FOR DELTA MAX C C C C C THIS SUBROUTINE FORMS THE MATRICES PP AND PY AND THEN USES THEM TO SOLVE THE SET OF EQUATIONS C C PP*DELTA=PY C C AND THEN CHANGES THE PARAMETER VALUES BY DELTA, I.E. C C P(IMAP(I))=DELTA(I) C C C C C IMAP IS USED TO HOLD PARAMETERS CONSTANT. A CONSTANT PARAMETER INDEX IS NOT INCLUDED IN IMAP(I) SO THAT PARAMETER WILL NOT BE CHANGED. A MAPPING MUST BE USED FOR CORRECT CALCULATION OF DELTA. C C C C THE RMS OF THE NEW PARAMETER VALUES IS CALCULATED. IF THE FIT IS IMPROVED THE ALGORITHM STARTS OVER. IF NOT, THE DIAGONAL OF PP IS INCREASED AND NEW DELTAS CALCULATED. C C PP(I,J)=PP(I,J)+PART(IMAP(I))*PART(IMAP(J)) C C SUMMED OVER ALL THE PARTIAL DERIVATIVES OF ALL THE DATA POINTS. C C PY(I)=PY(I)+(F(X,P)-Y)*PART(IMAP(I))) C C C SUMMED OVER ALL THE DATA. - 189 C C C C C THE MATRIX PP IS SYMMETRIC SO ONLY THE UPPER RIGHT HAND TRIANGLE IS STORED AND THE LOWER LEFT HAND TRIANGLE AND THE ARRAY DIA IS USED TO STORE THE CHOLESKY DECOMPOSITION OF THE MATRIX. THE INVERSE OF THE MATRIX IS LATER STORED THERE FOR THE ERROR CALCULATION. C C C C THE MATRIX HANDLING ROUTINES WHICH DO THIS ARE SYMDET,SYMSOL (WHICH SOLVES THE LINEAR EQUATIONS USING PY). AND SYMINV WHICH INVERTS THE MATRIX. C SUBROUTINE REFIN(XP,YP,N,P,D,NP,NC,IMAP,ND,NIT,FIN,SUBDIF,DIFFSQ, 1NINT,RMSQU,DLMXQU) DIMENSION XP(1),YP(1),P(1),D(1),IMAP(1) DIMENSION PP(11,11),PY(11),DIA(11),PART(11),0P(11),DELTA(11) LOGICAL ERCAL COMMON/IO/LRD,LWD,LSD C C C C C C ITNO HOLDS NUMBER OF REAL ITERATIONS ITC HOLDS THE NUMBER OF CORRECTIVE ITERATIONS. XLAM IS THE CORRECTION FACTOR ERCAL IS A FLAG FOR DOING THE ERROR CALCULATION GAMCRT IS THE CRITERION FOR USING THE COSINE CORRECTION METHOD C GAMCRT:3. 14159265/4. ERCAL= (NINT.LE.O) ITNO=O DENOM=N-ND IF (DENOM.GE.1) GOTO 14 WRITE(LWD,12) 12 FORMAT(' SOLUTION IS UNDERDETERMINED'/) DENOM:N 14 IF (ERCAL) GOTO 40 C CALCULATE ORIGINAL ERRORS C ITC=O XLAM=.01 DIFFSQ=O. 0010I=1,N CALL SUBDIF(XP(I),Y,P,PART,NP,NC,SUMC,2) DIFF=YP(I)-SUMC 10 DIFFSQ=DIFFSQ+DIFF*DIFF C C STORE IN RMSO FOR TERMINATION CALCULATION C RMSO=SQRT(DIFFSQ/N) CHISQ:SQRT(DIFFSQ/DENOM) WRITE(LWD,15) RMSO,CHISQ 15 FORMAT(' ORIGINAL HMS=',F13.7,/' GOODNESS OF FIT',F13.7/) C WRITE OUT FORMAT OF RMS AND PARAMS C WRITE(LWD,20) 20 FORMAT( '0' ,5X, 'RMS' ,5X, 'DELMX' ,5X, 'PARAMETERS') - 190 C C MAIN LOOP: CALCULATE PY,PP MATRICES C 40 ITNO=ITN0+1 DO 45 J:1,NP 45 OP(J)=P(J) DIFFSQ=O. DO 50 I=1,ND PY(I)=O. DO 50 J=1,ND 50 PP(I,J)=O. DO 100 I=1,N X=XP(I) Y=YP(I) CALL SUBDIF(X,Y,P,PART,NP,NC,SUMC,1) DIFF=Y-SUMC DO 80 J=1,ND PY(J)=PY(J)+DIFF*PART(IMAP(J)) DO 80 K=J,ND 80 PP(J,K)=PP(J,K)+PART(IMAP(J))*PART(IMAP(K)) 100 DIFFSQ=DIFFSQ+DIFF*DIFF C C IF ERCAL THEN DO NOT CHANGE PARAMETERS C IF(ERCAL) GOTO 245 C C SCALE THE MATRIX C DO 105 I=1,ND D(I)=SQRT(PP(I,I)) XTEM=O. IF (D(I).NE. 0.0) XTEM=PY(I)/D(I) PY(I)=XTEM DO 105 J= 1,I XTEM=O. IF ((D(I).NE. 0.0).AND.(D(J).NE. 0.0)) XTEM=PP(J,I)/(D(I)*D(J)) PP(J,I)=XTEM 105 CONTINUE C C FLAM=O. XKGAM=1.0 C C CORRECTIVE LOOP C 110 FLAM=XLAM-FLAM C C CORRECT PP MATRIX C DO 120 I= 1 , ND 120 PP(I,I)=FLAM+PP(I,I) C - 191 C DO DECOMPOSITION AND SOLVE EQUATIONS FOR DELTA C CALL SYMDET(PP,ND,DIA,DET,&210) CALL SYMSOL(PP,ND,DIA,PY,DELTA) C C FORM NEW PARAMETERS C C C SCALE THE DELTAS FIRST C DO 123 J:1,ND XTEM=O.O IF (D(J).NE.0.0) XTEM=DELTA(J)/D(J) 123 DELTA(J)=XTEM C C 125 DO 130 J:1,ND 130 P(IMAP(J))=P(IMAP(J))+XKGAM*DELTA(J) C C CALCULATE NEW CHI SQUARE C DIFFN=O. 00150I=1,N CALL SUBDIF(XP(I),Y,P,PART,NP,NC,SUMC,2) DIFF=YP(I)-SUMC 150 DIFFN=DIFFN+DIFF*DIFF C C IF FIT IS BETTER, GOTO 160 ELSE GOTO 170 C IF (DIFFSQ-DIFFN-FIN) 170,160,160 C C C C C FIT IS WORSE , INCREMENT CORRECTION COUNTER REPLASE P(I) WITH OLD PARAMETERS CHANGE CORRECTION FACTOR LOOP AROUND C 170 ITC=ITC+ 1 DO 173 J=1,NP 173 P(J)=OP(J) C C 175 IF (ITC.GT.NIT) GOTO 180 C C CALCULATE VALUES FOR COSINE CORRECTION C GAMNUM=O. PYSQ=O. DELSQ=O. DO 178 I=1,ND PYTEM=PY(I) DELTE"1=DELTA(I) GAMNUM=GAMNUM+PYTEM*DELTEM - 192 PYSQ=PYSQ+PYTEM*PYTEM 178 DELSQ=DELSQ+DELTEM*DELTEM GAMNUM=GAMNUM/(SQRT(PYSQ)*SQRT(DELSQ)) IF (GAMNUM .GT. 1.0) GAMNUM=1 .0 GAMNUM=ACOS(GAMNUM) IF (GAMNUM.LT.GAMCRT) GOTO 176 XLAM=10.*XLAM GOTO 110 176 CONTINUE C WRITE(LWD,179) GAMNUM C 179 FORMAT( 'OUSING COSINE CRITERION, GAMMA=' ,E10.3) XKGAM=XKGAM/2. GOTO 125 180 WRITE(LWD,185) 185 FORMAT(' CORRECTIVE ITERATIONS EXCEEDED'//) GOTO 230 C C C C C FIT IS BETTER. RESETS CORRECTION COUNTER DECREASE CORRECTION FACTOR DISPLAY ERROR AND DECIDE TO GO ON C 160 ITC=O XLAM=XLAM/10. C C C CALCULATE THE MAXIMUM RELATIVE CHANGE IN PARAMETERS FOR DISPLAY C DELMX=O. DO 163 I=1,ND IF (P(IMAP(I)).NE.0.0) XTEM=ABS(DELTA(I)/P(IMAP(I))) 163 IF (XTEM.GT.DELMX) DELMX=XTEM RMS=SQRT(DIFFN/N) 164 WRITE(LWD,165) RMS,DELMX,(P(I),I=1,NP) 165 FORMAT('0',2F10.7/1X,11E10.3) C C TERMINATION TEST C IF ((RMSO-RMS).LE.RMSQU) GOTO 230 IF (DELMX.LE.DLMXQU) GOTO 230 IF(ITNO.GE.NINT) GOTO 230 C C UPDATE OLD RMS C RMSO=RMS GOTO 40 210 WRITE(LWD,215) 215 FORMAT(' EQUATIONS SINGULAR'/) C C GO BACK ONE ITERATION AND QUIT C 220 DO 225 I=1,NP - 193 225 P(I)=OP(I) C C DO THE ERROR CALCULATION FOR THE PARAMETERS C 230 ERCAL=.TRUE. GOTO 40 245 DO 240 I=1,NP 240 D(I)=O.O CALL SYMDET(PP,ND,DIA,DET,&200) CALL SYMINV(PP,ND,DIA) CHISQ=SQRT(DIFFSQ/DENOM) DO 250 I= 1 ,ND 250 D(IMAP(I))=SQRT(DIA(I))*CHISQ C C REQUEST IF CORRELATION MATRIX SHOULD BE DISPLAYED. C CALL WCOVAR(PP,DIA,IMAP,ND) 200 RETURN END C C C C C C C C C C C C C C C C C cc C C C SYMSUB.F4 VERSION 302 13-FEB-1976 THE FOLLOWING SUBROUTINES MANIPULATE SYMMETRIC MATRICES, FINDING THE DETERMINANT, SOLVING LINEAR EQUATIONS, AND INVERTING THE MATRIX. THE FORTRAN PROGRAMS ARE MODIFICATIONS OF THE ALGOL PROGRAMS PUBLISHED BY R.S. MARTIN, G. PETERS AND J.H. WILKINSON 'SYMMETERIC DECOMPOSITION OF A POSITIVE DEFINATE MATRIX" PP 9-25 IN "LINEAR ALGEBRA", J.H. WILKINSON AND C REINSCH ED. VOL II OF HANDBOOK FOR AUTOMATIC COMPUTATION. NEW YORK: SPRINGER-VERLAG 1971 439 PP. SYMDET FORM CHOLESKY DECOMPOSITION OF SYMMETERIC MATRIX AND PLACE IN LOWER TRIANGLE AND ARRAY P. SUBROUTINE SYMDET(A,N,P,DET,*) DOUBLE PRECISION X,Y,Z,INPROD DI~NSION A(11,11),P(1) DET=1. DO 100 I=1,N DO 100 J=1,I X=A(J,I) IF (I.EQ.J) GOTO 10 GOTO 50 10 K=J-1 15 IF (K.LT.1) GOTO 20 Y=A(I,K) - 194 Z=Y*P(K) A(I,K)=Z X=X-Y*Z K=K-1 GOTO 15 20 DET=DET*X IF (X.EQ.O.) RETURN 1 P(I):1/X GOTO 100 50 K=J-1 60 IF (K.LT.1) GOTO 70 INPROD=A(I,K)*A(J ,K) X=X-INPROD K=K-1 GOTO 60 70 A(I,J)=X 100 CONTINUE RETURN END C C C C C SYMSOL: SOLVE A SET OF LINEAR EQUATIONS USING A LOWER TRIANGULAR CHOLESKY DECOMPOSITION. C SUBROUTINE SYMSOL(A,N,P,B,X) DOUBLE PRECISION Y,INPROD DIMENSION A(11,11),P(1),B(1),X(1) DO 50 I=1,N Y=B(I) K=I-1 10 IF (K.LT.1) GOTO 20 INPROD=A(I,K)*X(K) Y=Y-INPROD K=K-1 GOTO 10 20 X(I)=Y 50 CONTINUE I=N 55 IF (I.LT.1) GOTO 100 Y=X(I)*P(I) K=I+1 60 IF (K.GT . N) GOTO 70 INPROD=A(K,I) *X(K) Y=Y-INPROD K=K+1 GOTO 60 70 X(I)=Y I=I-1 GOTO 55 100 CONTINUE RETURN - 195 END C C C C C SYMINV: INVERT A MATRIX USING A LOWER TRIANGULAR CHOLESKY DECOMPOSITION MATRIX C SUBROUTINE SYMINV(A,N,P) DIMENSION A(11,11),P(1) DOUBLE PRECISION X,Y,Z,INPROD DO 50 1=2,N DO 40 J=2,I J1=J-1 X=-A(I,J1) K=I-1 10 IF (K.LT.J) GOTO 20 INPROD=A(I,K)*A(K,J1) X=X-INPROD K=K-1 GOTO 10 20 A(I,J1)=X 40 CONTINUE 50 CONTINUE DO 100 J= 1 ,N DO 100 I=J,N IF (I.EQ.J) GOTO 70 X=A(I,J) IF (I.EQ.N) GOTO 100 11=1+1 DO 60 K=I1 ,N INPROD=A(K,I)*A(K,J) 60 X=X+INPROD A(I,J)=X GOTO 100 70 X=P(I) IF (I.EQ.N) GOTO 90 11=1+1 DO 80 K=I1 ,N Y=A(K,J) Z=P(K)*Y A(K,J)=Z X=X+Y*Z 80 CONTINUE 90 P(I)=X 100 CONTINUE RETURN END C ERPBAT.F4 VERSION 305 1 JUN 1976 C C C C 31-0CT-1975 WCOVAR VERSION 300 - 196 C C C THIS ROUTINE DISPLAYS THE CORRELATION MATRIX AFTER CALCULATING IT FROM THE COVARIANCE MATRIX PP. C C C C C IT IS DESIGNED FOR A SYMMETRIC COVARIANCE MATRIX STORED IN THE LOWER LEFT TRI!\NGLE OF PP WITH VARIANCES STORED IN DIA . IMAP ASSOCIATES THE MATRIX WITH APPROPRIATE PARAMETER INDICES. C C C C C C SUBROUTINE WCOVAR(PP,DIA,IMAP,ND) DIMENSION PP(11,11),DIA(1),IMAP(1) COMMON /IO/LRD,LWD,LSD WRITE(LWD,10) 10 FORMAT(' TY 1 FOR CORRELATION MATRIX'/) READ(LRD,15) ICHO 15 FORMAT(I) IF (ICHO.NE.1) RETURN DO 25 I=1,ND 25 DIA(I)=SQRT(DIA(I)) C C C USE IFS AND GOTOS TO HANDLE SPECIAL CASES INSTEAD OF USING DO LOOPS C I=1 30 IF (I.GT. ND) GOTO 50 J::1 35 IF (J.GT.I-1) GOTO 40 PP(I,J):PP(I,J)/(DIA(I)•DIA(J)) J=J+1 GOTO 35 40 I=I+1 GOTO 30 50 XDIA=1. WRITE(LWD,60) (IMAP(I),I=1,ND) 60 FORMAT('OI ',11(3X,I3,4X)/) WRITE(LWD,70) IMAP(1),XDIA 70 FORMAT(1X ,I2,11E10.3/) IF (ND.LE.1) GOTO 100 DO 90 I=2,ND I1=I-1 90 WRITE(LWD,70) (IMAP(I),(PP(I,J),J=1,I1),XDIA) 100 WRITE(LWD,110) 110 FORMAT(1X,//) RETURN END C MS2SUB.F4 VERISON 302 19 JAN-1976 C C C CONTAINS MAP, PARMIO MAP FIXED TO FIX PARAMETERS PROPERLY C C C MAP - 197 C C C THIS SUBROUTINE TAKES THE ARRAY IA(NP) OF PARAMETERS TO BE FIXED AND SEARCES IT FOR EACH OF THE NP PARAMETERS TO FORM THE ARRAY IMAP(ND) OF PARAMETERS TO BE VARIED. C C SUBROUTINE MAP(IA,NP,IMAP,ND) DIMENSION IA(1),IMAP(1) LOGICAL TEST ND=NP DO 50 I=1,NP IF (IA(I).EQ.O) GOTO 60 50 ND=ND-1 60 IM=O NF=NP-ND DO 100 I= 1 ,NP TEST= .TRUE. DO 80 J= 1,NF IF (IA(J).NE . I) GOTO 80 TEST=.FALSE. GOTO 90 80 CONTINUE 90 IF (.NOT.TEST) GOTO 100 IM=IM+1 IMAP(IM)=I 100 CONTINUE RETURN END C C C C C C C C C C C C C C C C C PARMIO VERSION 304 28 MAY 1976 MODIFIED FOR BATCH NNNGER SUPPORTS SAME PARAMETERS PARMIO IS THE BASIC PARAMETER INPUT ROUTINE USED BY OPTION 2,GUESS, 4,FINGR AND 7,ERROR . IT DETERMINES WHICH OF THE F.'UNCTIONS WILL BE USED (IWHAT), THE PARAMETER VALUES, THE NUMBER OF PARAMETERS AND THE NUMBER OF COMPONENTS RY CALLING COUNT FOR IWHAT=1,4,5. SUBROUTINE PARMIO(P ,NP,NC,NPMAX,IWHAT,NINT,RMSQU,DLl'v1XQU ,NIT, *) DIMENSION P(1) COMMON /IO/LRD ,LWD,LSD C C C CHOOSE FUNCTION TO BE FIT OR PLOTTED. IF OSAME CHOSEN, JUMP OUT WITH SAME PARAMETERS AND SAME FIXED PARAMETERS . C 5 WRITE(LWD, 10) - 198 10 FORMAT(' FUNCTION (-1HELP), ITERATIONS, RMS QUIT, DELMX QUIT, (NIT 1) I/) READ(LRD,15) ICHO,NINT,RMSQU,DLMXQU,NIT *-------------------------c 15 FORMAT(I2,I3,2E10.2,I5) 15 FORMAT(2I,2F,I) IF (ICHO.LT.O) GOTO 17 GOTO 19 17 WRITE(LWD, 18) 18 FORMAT(' OSAME,1FINGER,2WHATOR,3NUFORM,4EXCESS,5WHTCMP'/) GOTO 5 19 IF (NIT.LE.0) NIT:10 IF (ICHO.EQ.O) RETURN 1 IWHAT=ICHO IF ((IWHAT.LT.1).0R.(IWHAT.GT.5)) GOTO 5 GOTO (20,25,30,20,20),IWHAT 20 WRITE(LWD,21) 21 FORMAT(' FINAL,FRACT(1),K(1),FRACT(2),K(2),FRACT(3),K(3)'/) READ(LRD,22) (P(I),I=1,NPMAX) *-------------------c 22 FORMAT(8E10.3) 22 FORMAT(11F) CALL COUNT(P,NC,NPMAX,2) NP=2*NC+1 GOTO 35 25 WRITE(LWD,26) 26 FORMAT(' FINAL,FRACT,K,SXP'/) READ(LRD,22) (P(I),I:1,4) NP=4 NC=1 GOTO 35 30 WRITE(LWD,31) 31 FORMAT(' FINAL,FRACT,K:TRACER-1,F,K:TRACER-2,DRIVER K,0.44'/) READ(LRD,22) (P(I),I=1,7) NP=7 NC=1 GOTO 35 35 CONTINUE RETURN END C C C C C C C C C C C MISSUB.F4 VERSION 301 CORRECTED FOR 370 FORTRAN 27 OCT-1976 CONTAINS SORTI,COUNT SORT: A BUBBLE SORT SORTING A ANDY USING A. SUBROUTINE SORTI(A,N,Y) - 199 DIMENSION A(1),Y(1) LOGICAL KEY IF (N.LE.1) GOTO 30 N1=N-1 DO 20 J=1,N1 L=N-.J KEY=.TRUE. DO 10 K=1,L IF (A(K).LE.A(K+1)) GOTO 10 ATEM=A(K) A(K)=A(K+1) A(K+1)=ATEM YTEM=Y(K) Y(K)=Y(K+ 1) Y(K+1 )=YTEM KEY=.FALSE. 10 CONTINUE IF (KEY) GOTO 30 20 CONTINUE 30 RETURN END C C C C C C COUNT: COUNTS THE NUMBER OF PARAMETERS. NOTE EITHER A 0. OR NEGATIVE PARAMETER IS CONSIDERED THE END OF THE PARAMETERS. C C SUBROUTINE COUNT(P,NC,NM,NPC) DIMENSION P(1) NC=O DO 10 I=2,NM,NPC IF (P(I)) 30,30,20 20 NC=NC+1 10 CONTINUE 30 RETURN END C PLMLT.F4 VERSION 303 29 OCT-1975 C C C C THIS SUBROUTINE PLOTS A LINE ON A TTY OR LPT PUTTING SYMBOLS OF LOCATIONS SPECIFIED IN THE ARRAY RAND USING P,Q AS SCALE FACTORS. IN GENERAL P:100,Q=O. C C C C C C C Y IS THE ABSCISSA VALUE PLOTTED OUT N IS THE NUMBER OF POINTS TO BE PLO\TTED. NW IS THE WIDTH OF THE PLOT IR IS AN ARRAY OF SYMBOLS PLOTTED. THE NUMBER IN R(I) IS PLOTTED WITH SYMBOL IR(I) LUMP(I) IS THE ACTUAL LINE OF SYMBOLS PLOTTED C SUBROUTINE PLMLT(Y,R,N,LF,P,Q) - 200 C PART OF FINGER.CMD. SAVE. INTEGER B,C DIMENSION LUMP(51),IR(18),R(1) COMMON /IO/LRD,LWD,LSD DATA NW/51/ RNDZ(X)=IFIX(X+SIGN(0.5,X)) DATA IR/ 1H. , 1H-, 1H1, 1H2, 1H3, 1H4, 1H5, 1HO, 1HA, 1HB, 1HC, 1HD, 1HE, 1HF, 11HG, 1HH, 1HI, 1HI/ DATA B/1H / DATA C/1HI/ GO TO (100,100,200),LF C C WRITE THEY AXIS C 100 WRITE(LWD,101) 101 FORMAT(' DATA FIT 1---.8---.9----1') RETURN ECOT 0---. 1---.2---.3---.4---.5---.6---.7 C C BLANK LINE C 200 DO 210 I=1,NW 210 LUMP(I)=B JEND=1 IF (N.LT.1) GOTO 400 DO 220 IPL=1,N C C KP IS THE POSITION OF THE SYMBOL TO BE PLOTTED ON THE LINE C KP=IFIX((R(IPL)*P+Q)/2+0.5)+1 IF ((KP.GT.NW).OR.(KP.LE.O)) GOTO 220 IF (KP.GT.JENO) JEND=KP C C C LUMP BEOMES AN ARRAY OF SYMBOLS TOBE PLOTTED AT POSITION KP BLANK OTHERWISE C LUMP(KP)=IR(IPL) 220 CONTINUE LS=RNDZ(Q) C C PUT THE BASELINE ON THE PLOT LAST SO IT IS NOT OVERWRITTEN C LUMP(LS+1)=C 400 WRITE(LWD,401 )R(N) ,R(2) ,Y, (LUMP(J) ,J=1,JEND) 401 FORMAT (1H ,F5.3,F6.3,E9.2,1X,51A1) RETURN END C C C C C DIFSUB.F4 VERSION 302 19 FEB 1976 - 201 C C C THIS VERSION HAS BEEN EXTENSIVELY TESTED FOR FINGER, EXCESS, WHATOR AND WHTCMP AND SHOULD BE CORRECT FOR THOSE FUNCTIONS. C C TESTED FOR NUFORM AND ERRORS CORRECTED C C C C FOR EACH OF THESE SUBROTINES, IF IFUNC = 2 ONLY THE FUNCTION VALUE IS CALCULATED C C C C THE EMPIRICAL DERIVATIVES USED IN FRMDIF CAN BE USED WITH ANY FUNCTION WITH NO CHANGE IN THIS DIFFERENTIATING SUBROUTINE C C C C FINDIF: FINGER PARTIAL DERIVATIVES, USES ANALYTIC DERIVATIVES C C SUBROUTINE FINDIF(X,Y,P,PART,NP,NC,SUMC,IFUNC) DIMENSION P(1),PART(1) SUMC=P(1) GOTO (5,50),IFUNC 5 PART ( 1) =1. DO 10 J=1,NC JF=2*J JK=JF+1 DF=1.0/(1.0+P(JK)*X) DK=P(JF)*DF SUMC=SUMC+DK PART(JF)=DF 10 PART(JK)=-DF*DK*X RETURN 50 DO 60 J=1,NC JF=2*J JK:JF+1 60 SUMC=SUMC+P(JF)/(1.0+P(JK)*X) RETURN END C C C C C WHTDIF: ANALYTIC PARTIAL DERIVATIVES FOR WHTCMP SUBROUTINE WHTDIF(X,Y,P,PART,NP,NC,SUMC,IFUNC) DIMENSION P(1),PART(1) SUMC=P(1) GOTO (5,50),IFUNC 5 PART ( 1) =1. DO 10 J=1,NC JF=2*J JK:JF+1 - 202 DF=(1.+P(JK)*X)**(-0.44) DK=P(JF)*DF SUMC=SUMC+DK PART(JF)=DF 10 PART(JK)=-0.44*X*P(JF)/(1.+X*P(JK))**(1.44) RETURN 50 DO 60 J=1,NC JF=2*J JK=JF+1 60 SUMC=SUMC+P(JF)*(1.0+P(JK)*X)**(-0.44) RETURN END C C C C EXCDIF: ANALYTIC DERIVATIVES FOR EXCESS C SUBROUTINE EXCDIF(X,Y,P,PART,NP,NC ,SUMC,IFUNC) DIMENSION P(1),PART(1) SUMC=P(1) GOTO (5,50),IFUNC 5 PART ( 1) =1. DO 10 J=1,NC JF=2*J JK=JF+1 DF=EXP(-P(JK)*X) DK=P(JF)*DF SUMC=SUMC+DK PART(JF) =DF 10 PART(JK)=-X*DK RETURN 50 DO 60 J:1,NC JF=2*J JK=JF+1 60 SUMC=SUMC+P(JF)*EXP(-P(JK)*X) RETURN END C C C C WHADIF: ANALYTIC DERIVATIVES FOR WHATOR C SUBROUTINE WHADIF(X,Y,P,PART,NP,NC,SUMC,IFUNC) DIMENSION P(1),PART(1) SUMC=P(1) GOTO (5,50),IFUNC 5 PART ( 1) =1 . DO 10 J=1,NC JK=3*J JF=JK-1 JE=JK+1 DF:1/(1.+P(JK) *X)**P(JE) - 203 DK=P(JF)*DF PART( JF) =DF PART(JK)=-P(JE)*P(JF)*X/(1.+P(JK)*X)**(1.+P(JE)) PART(JE)=-DK*ALOG(P(JF)*(1.+X*P(JK))) 10 SUMC=SUMC+DK RETURN 50 DO 60 J =1 , NC JK=3*J JF=JK-1 JE=JK+1 60 SUMC=SUMC+P(JF)/(1.+P(JK)*X)**P(JE) RETURN END C C C C C FRMDIF: EMPIRICAL DERIVATIVES FOR ANY FUNCTION FORM OF 7 PARAMTERS C SUBROUTINE FRMDIF(X,Y,P,PART,NP,NC,SUMC,IFUNC) DIMENSION P(1),PART(1) CALL FORM(P,X,SUMC) IF (IFUNC.EQ.2) RETURN DEL=.01 DO 10 J= 1 ,NP TEMP=P(J) IF (TEMP.NE.0.0) GOTO 15 PART(J) =0. GOTO 10 15 P(J)=TEMP*(1.+DEL) CALL FORM(P,X,YVAL1) PART(J)=(YVAL1-SUMC)/(DEL*TEMP) P(J)=TEMP 10 CONTINUE RETURN END C NFORM.F4 VERSION 1 18-FEB-1976 C C C C C FORM: CALCULATES THE FUNCTION DECRIBED IN NUFORM. C C C C C F(X,P)=FRACTION SINGLE STRANDED TRACER WITH RATES P(3),P(5) AND FRACTIONS P(2),P(4) DRIVEN BY DRIVER WITH RATE P(6) WITH SINGLE STRAND DRIVER GIVEN BY P(7)=0.44 USUALLY. C SUBROUTINE FORM(P,COT,VAL) REAL P( 1) V=1+P(6)*COT PWR=1-P(7) - 204 VAL 1=P( 1) VAL2=P(3)/P(6)*(1-V**PWR)/PWR VAL2=P(2)*EXP(VAL2) VAL3=P(5)/P(6)*(1.-V**PWR)/PWR VAL3=P(4)*EXP(VAL3) VAL=VAL1+VAL2+VAL3 RETURN END