Pro. Nat. Aad. Si. USA Vol. 73, No. 2, pp. 415-419, February 1976 Biohemistry Studies on nulei aid reassoiation kinetis: Empirial equations desribing DNA reassoiation* (renaturation/nuleation inhibition/single strand ends) RoY J. BRITTENt AND ERI H. DAVIDSON Division of Biology, alifornia Institute of Tehnology, Pasadena, alif. 91125 ontributed by Roy J. Britten, November 26,1975 ABSTRAT The rate of appearane of duplex DNA renaturation, measured with single strand speifi nulease, deviates signifiantly from a seond order reation. Measurements reported in paper I of this series indiate an inhibition in the rate of reassoiation of single strand tails on partially reassoiated moleules by a fator of at least two. Equations are derived that desribe the observed form of reassoiation kinetis as measured with hydroxyapatite and with single strand speifi nulease. The free parameter that desribes the extent of inhibition of nuleation with single strand tails in these equations has been evaluated by least squares methods and agrees with the experimentally measured value. The reassoiation of DNA is now an important tool for the examination of the organization of the genome and for studying its evolution. The kinetis of the reassoiation proess are neededfor the evaluation of the frequeny of ourrene of repeated sequenes and for the design of measurements omparing DNA sequenes from different soures. However, this reation is not fully understood, and the atual form of the kinetis has only been approximately evaluated (1, 2). This paper examines the effet of random shearing of DNA fragments on the kinetis of formation of strand pairs that ontain duplex regions [assayed by hydroxyapatite (HAP)] and on the fration of total nuleotides (NT) that are paired (assayed by single strand speifi nulease, or approximately by optial hyperhromiity). If all DNA fragments terminate at the same points in the sequene and have the same length and if there are no internal repetitions, then eah effetive nuleation leads to omplete pairing of the fragments. Reassoiation then takes the form of a seond order reation however it is assayed: Abbreviations: HAP, hydroxyapatite; NT, nuleotides; ot, mol of nuleotides per liter X se. * This is paper II in a series. Paper I is ref. 5. t Also Staff Member, arnegie Institution of Washington. /o = (1 + KOt)-1 [1 Where o is the DNA onentration (mol of NT/liter) and is onentration in fragments without duplex regions. However, in most measurements the DNA has been sheared and eah fragment terminates at a random plae in the sequene. As a result, most nuleations lead to partial pairing of the fragments and single strand regions remain unpaired. Britten and Kohne (1) showed nevertheless that the kinetis of reassoiation assayed by HAP follow Eq. 1 to a very lose approximation. In an earlier analysis (1, 3), a simple approximation was used to aount for the redution in yield of duplex in later ollisions. It was assumed that effetive nuleation rate is proportional to the square of the onentration of unpaired NT and that the yield of duplex per nuleation is redued as the reation proeeds in proportion to the fration of unpaired NT remaining. The resulting approximate analysis yields Eq. 1 for the fration of fragments without duplex regions (HAP assay) and yields the following equation for the fration of NT unpaired: S/o = (1 + Kot)-n [2] where S is the onentration of NT remaining single stranded and the other symbols are as in Eq. 1. K has the same numerial value as in Eq. 1, and the value of n resulting from this derivation was.5. Morrow (4) and Smith et al. (5), in paper I of this series, measured the kinetis with whih the fration of NT remaining single stranded (S/o) hanges in randomly sheared DNA, using the single strand speifi nulease Si. Their results fit losely the form of Eq. 2. Morrow's data yielded a best value of n =.44, while the best value for the data of Smith et al. was n =.45. It seems lear.from the agreement of these results that n has been determined with some auray. Smith et al. also measured the kinetis with whih the fration of totally single stranded fragments (/ o) hanges, by HAP assay, using portions of the same samples studied by nulease. Their measurements showed that the best value of K in Eq. 1 is equal to the value of K in Eq. 2 for the SI nulease measurements. Thus we have two simple equations that adequately express the kinetis of DNA reassoiation and are suitable for the evaluation and interpretation of measurements. However these expressions provide little insight into the mehanisms responsible. In the urrent work we have taken into aount the variation in length of the free single strands and of the single strands remaining on fragments that already ontain duplex regions. These we term "partiles", sine, as the result of suessive ollisions, suh partially duplexed renaturation produts may grow to very large size. The variation in single strand length during the reation was estimated using a omputer program that models the reassoiation reation by a "Monte arlo" method. These length hanges, of ourse, retard the reassoiation rate at later times. We have shown earlier that, probably due to some form of steri interferene, the per NT rate of reation of the partile single strands is inhibited. Here we show that this inhibition is required to explain quantitatively the observed forms of both HAP and Si nulease kinetis. omparison of observed and omputer simulated reation kinetis A omputer simulation of the reassoiation reation has been used to alulate the expeted or ideal reation kinetis. By summing up the fration of simulated fragments that ontain "duplex" regions, the ideal rate of appearane of HAP binding is alulated. A similar alulation in whih the 415
416 Biohemistry: Britten and Davidson w.9 :e x.8.7.-.6.5 '.4 z.3.2 ll.1 I ).1.1 1. 1.. 1. 1. ot/ot 1/2 Q).8 in a).7 -, I'..6 - \~~~~~~~~~~.5 \~~~~~~~~~~~ =SI.4 z.3 ).2 FIG. 1. omparison of alulated and observed reassoiation kinetis. The solid lines desribing the reassoiation of Esherihia oli DNA, as measured by S1 nulease resistane and by HAP binding, are reprodued from Fig. 1 of Smith et al. (5). The alulated urves are shown by dashed lines. These are the HAP binding urve (inner urve) and the alulated S1 nulease resistane urve (outer urve). The alulation was arried out by a omputer program whih simulates the reassoiation reation using a realisti fragment size distribution as desribed in the text. The omputer alulations presented are the average of eight runs of the simulation program, all losely agreeing. The rate of the alulated urves has been hosen so that the alulated and measured hydroxyapatite urves ross at half-reation. When Eq. 2 is used to fit the omputer generated urves, the alulated HAP binding urve has n = 1.85 and the alulated S1 nulease urve has n =.55..1 Pro. Nat. Aad. Si. USA 73 (1976) "duplex" ontent is summed provides the ideal rate of appearane of S1 nulease resistane. The alulated kinetis are ompared to the observed kinetis in Fig. 1. The reassoiation reation was modeled in the following. way. Two arrays of 5 elements were established representing the omplementary strands of DNA. The array is divided into fragments using a random number generator aording to a Gaussian distribution of lengths with the average and standard deviation speified for eah run. For the results utilized in this paper, the average was 2 elements long and the standard deviation was 1. The distribution was trunated so that no lengths below 2 or above 4 were inluded. This distribution mathed fairly losely the length distribution of sheared fragments determined by eletron mirosopy (6). The array of 5 elements was assumed to repeat every 31 elements, in order to inrease the effetive rate of "nuleation" to a pratial value. The ratio of the repeat length to the average fragment length was hosen so that the fragments terminated randomly within the repeat length. During a simulation run two elements are seleted at random, and if either is paired, this is taken as an unprodutive ollision and the try is repeated. When two unpaired elements are seleted and the two single stranded regions are "zippered" as far as their unpaired regions overlap; that is, eah of the inluded elements is marked as paired. The number of elements paired is sored for "duplex formation," and the total length of the fragments is sored for "HAP binding" if they did not previously have paired regions. Where inhibition of partile single strand ends is to be inluded, a hosen fration of the "nuleations" in suh regions is not sored. Eah try is onsidered a ollision and the number of trys is the measure of the "ot." When the simulated rate is alulated so that the probability of "nuleation" is proportional to the square root of the "fragment" length rather than to the "fragment" length itself, there is little hange in the result Fig. 1 shows that the measured kinetis of the reation are of a different form from the alulated kinetis. The rate of the alulated urves has been hosen so that the alulated and measured hydroxyapatite urves ross at half-reation. Given the same initial rate, the alulated reation would proeed more rapidly than the observed seond order form. Thus when Eq. 2 is used to fit the simulated HAP binding urve, the value obtained for n is 1.85 rather than 1., as in Eq. 1. Similarly, when Eq. 2 is used to fit the simulated SI nulease urve, the value obtained for n is.55, rather than the measured.45. We now onsider the impliations of the disagreement between the ideal and observed kineti alulations. As pointed out above, the rate of disappearane of free single strands depends not only on the reation between free single strands, but also on the reation between free single strands and the single stranded regions on partiles. A orret desription of this proess would take the form d/dt = -K2 - K(S - ) [3] where the meaning of K and are as above and (S-) is the onentration of single stranded NT on partiles. As a first approximation the same rate onstant, K, is used for both terms of the expression. Eq. 3 is of ourse a non-seond order form, and it predits that measurements of single stranded DNA onentration made with HAP should display faster than seond order kinetis, given the same initial rates. The effet of fragment length variation is to blur this distintion somewhat by slowing down the reation as it approahes termination. This is beause longer fragments tend to reat first so that the mean free fragment length, and hene the reation rate, dereases. However a realisti fragment length distribution is already inluded in the simulated reations illustrated in Fig. 1. The onlusion is that the reation as measured with HAP has a rate whih in the seond half of the reation is slower than the predition even when the effet of fragment length distribution is inluded. This result strongly implies the existene of some form of inhibition that retards the later phase of the reation. Suh an inhibition has also been diretly measured by Smith et al. (5). Similar onsiderations apply to the omparison of alulated and observed S1 nulease kinetis. Here again it is evident that some inhibition exists whih has the effet of retarding the atual appearane of S1 nulease resistane. The alulated uninhibited ase begins at the expeted rate, with the n of Eq. 2 at.55. This is equal to our best estimate of the extent of overlap (a) of two random sheared single strands as disussed in Smith et al. (5). The simulated reation ontinues throughout in aordane with Eq. 2 with n =.55. The observed reation begins at this rate sine the measured value of a is.55. Soon, however, the overall rate delines, due to some form of inhibition, and the best fit value of n for the whole reation is.45. We onlude that an inhibition affeting the latter part of the reassoiation reation is required in order to explain the experimental measurements of both HAP and S1 nulease kinetis. Neither the apparent seond order form desribed by Eq. 1 for HAP kinetis nor the use of Eq. 2 with n =.45 for S1 nulease
Biohemistry: Britten and Davidson measurements an be quantitatively understood without onsidering suh an inhibition. Equations desribing reassoiation in more detail We now desribe differential equations for the disappearane of free single strands (HAP kinetis) and the disappearane of single stranded NT (SI nulease kinetis). These equations inlude appropriate terms for the reation of free single strands with partile single strands and for the reation of partile single strands with eah other. They permit a more detailed analysis of the nature of the apparent inhibition than has previously been possible. The parameters used in the following expressions are the same as those defined earlier in this paper. In addition, there are terms for fragment length, partile inhibition, and nuleation rate onstant. The symbols applied are: o, the total DNA NT onentration; L, the average free single strand fragment length at any time in the reation; LR, the average length of single strands on partiles at any time in the reation; K, the nuleation rate for eah NT, not varying with fragment length (but varying aording to the omplexity); E, a partile inhibition fator expressed as a fration of the uninhibited rate; a, the average overlap between two free single strands of average length L or between two partile single strands of average length LR, expressed as fration of single strand length; and ar, the average overlap between free single strands of average length L and partile single strands of average length LR expressed as a fration of LR. In onstruting these equations we use the following assumptions: (i) The observed rate of nuleation varies with fragment length as K,,L1/2 (2). (ii) In reations between roughly equal onentrations of longer and shorter fragments the observed rate of nuleation varies with the length of the shorter partiipant (7). While this assumption may not be orret, it is in aord with the best urrent measurements on whih we rely in the following. Thus the observed nuleation rate in ollisions between free single strands and partile single strands is onsidered to vary as K,,LR-1/2 sine generally LR < L. (iii) The yield per free single strand in a suessful ollision between free single strands is L NT withdrawn from, and is al NT withdrawn from S. Similarly, in nuleations between a free single strand and a partile single strand the amount of NT removed from S per strand is arlr. The yield for a partile-partile ollision is alr, and so forth. For the disappearane of free single strands (HAP kinetis) we write d = K-Ll/22L - L(S - )L [4] This is equivalent to Eq. 3, exept that the effet of length on rate and yield are here expliit, and a fator is provided for partile inhibition. In ontrast to Kn of Eq. 4, the Ks of Eqs. 1, 2, and 3 all inlude the effets of yield and fragment length. In Eq. 4 the rate of disappearane of free single strands is taken to be proportional to L1/2 (2). Similarly, for the disappearane of free single strands (Si nulease kinetis): ds= 2aL - L (S - OaRLR - EK -[ LR1/2(S - )kxlr [5] -.9.8 _ D.7.6.5-.4.3.2- o LL_ Pro. Nat. Aad. Si. USA 73 (1976) 417.1 1. 1. 1. Ot/ O 1/2 FIG. 2. alulated hange in length of free single strands (LI Lo) and single strand regions in partiles (LILR) with ot. The ordinate is expressed as the single strand lengths relative to the starting fragment length, Lo. The alulations were arried out with the omputer simulation of the reassoiation reation desribed in the text. The absissa is expressed in terms of ot normalized to a simulated HAP binding rate of 1. liter mol-' se-1. The funtions used to generate the lines plotted are Eq. 6 for LR/Lo (dashed line) and Eq. 7 for L/Lo (solid line), least squares fitted to the results of the omputer simulation. Eqs. 4 and 5 state the expeted strand length effets on nuleation rate, and the expeted per ollision yield for eah portion of the reation. We assume that the value of E is the same for partile-partile as for partile-free reations and that E is independent of L. The integration of these differential equations would provide solutions suitable for the quantitative examination of the rate of duplex formation and HAP binding. In order to integrate these equations, we must evaluate L and LR as a funtion of t or ot. We follow an approximation method whih, though it may lak elegane, is suffiiently aurate for the study of DNA reassoiation kinetis. On the right-hand side of the differential equations we express and S approximately in terms of ot using Eqs. 1 and 2. In the next setion we desribe the evaluation of the variation of L and LR with ot and the method for the solution of Eqs. 4 and 5. Approximate solution of the differential equations Fig. 2 shows a alulation of partile single strand length (LR) as a funtion of ot obtained with the "Monte arlo" reassoiation program and the distribution of fragment lengths desribed above. The graph shows the average length of single strand regions in partiles (LR) as a fration of Lo, the starting single strand length. Thus, in Fig. 2 LR/Lo is plotted against the "ot" of the simulated reation. The urve shown in Fig. 2 an be fit reasonably well by an expression of the form (LR/LO) = (1 - a)(1 + Kot)-Y [6] where y is about.24. The value of L, the free single strand length, also hanges as the reation proeeds, due to the fat that longer fragments tend to reat more rapidly, always leaving a distribution of free single strands of shorter mean length in the yet
418 Biohemistry: Britten and Davidson unreated lass. An analysis similar to that in Fig. 2 shows that a reasonable form to desribe the hange in L as a funtion of ot is (L/Lo) = (1 + Kot-Y [7] where y is evaluated at about.34 for the fragment length distribution we have studied. The observed disappearane of free single strands is fit almost perfetly by the simple seond order expression, Eq. 1. Exept for the first few perent of the reation, the observed SI nulease kinetis are desribed adequately by Eq. 2. In making these substitutions we insert approximate solutions for and S. When integrated we obtain more aurate funtions for and S whih furthermore inlude expliit length orretions. Thus all the variables in Eqs. 4 and 5 an be related to ot by expressions of the form (1 + Kot)-Y. For, y = 1. (Eq. 1); for S, y =.45 (Eq. 2); for L,y =.34 (Eq. 7); and for LR, y.24 (Eq. 6). The only remaining problem is the evaluation of ar. As will be desribed elsewhere (Britten and Davidson, unpublished), ar/a an be shown analytially to be equal to [1 + (a/2)] for all ases where L is muh longer than LR. Using these approximations, Eqs. 4 and 5 may now be integrated either algebraially or by numerial proedures. When integrated, Eqs. 4 and 5 yield the following expressions, where m = y/2 in Eq. 6 or about.12, and x = y/2 in Eq. 7 or about.17. For HAP reassoiation kinetis -xx[ - 1 exp x E[V(l+m-n-2x) - 1] (1 + m - n - 2xXl - a)112 + (E[ -2x)(1- ])1/2] [8] where V = 1 + Kot. Here, as earlier, K is the same as in Eq. 1, i.e., the observed seond order rate onstant for the HAP reation. For S1 nulease reassoiation kinetis S (V(n-x-1) - 1) o AP (n X - 1) + E - 1 - a)1/2(1 - V-r) m E(1 - a)y/2(vlnm - 1) (1 - n - m) + E'-;(l ma) 12(V(.-l) - 1) (n - m - l) We note that all of the onstants in Eqs. 8 and 9 exept E, the partile inhibition, have been evaluated from other evidene or alulations and are not free. Evaluation of the partile inhibition, E Eqs. 8 and 9 were evaluated at a series of ot's to determine the best value of E. As a hek, Eqs. 4 and 5 were numerially integrated as well, with almost idential results. In Fig. 3 the data of Fig. 1 of Smith et al. (5) are replotted with the best least squares fits of Eq. 9. The form of Eq. 9 an be seen to fit the data exellently. The value of a was taken as.55, and K is normalized to 1. for ease of alulation. The best value of E is then.6. That is, slightly less than a 2-fold partile inhibition fator is required. The main signifiant fea- [9] 1..~.9.8 @.7 _.6 o.2 3 z.2.3 a.. Pro. Nat. Aad. Si. USA 73 (1976).1 1. 1. 1. Ot/Ot 1/2 1. 1,. FIG. 3. S1 nulease reassoiation data fit with Eq. 9. Data are transribed diretly from Fig. 1 of Smith et al. (5). The solid line is alulated with Eq. 9. By least squares analysis the best fit value of E was.6. Other parameters were K = 1., n =.45, m =.12, and x =.17, as desribed in the text. ture of the alulation illustrated in Fig. 3 is that it shows that the "length effet" built into Eqs. 8 and 9 ombined with the "partile inhibition" suffies to explain the observed S1 nulease kinetis. It is possible that the length effet is slightly different from that we have alulated. New evidene shows that the dimensions of the longer strand may have to be taken into aount in onsidering reations between longer and shorter strands. As noted above we have followed Wetmur (7) in alulating the nuleation rate as a funtion of the shorter strand length. The net effet would be small, however, resulting in a minor inrease in the partile inhibition. That is, there would be a slight redution in the value of E if the assumption that rate depends solely on the length of the smaller reating partiipant is wrong. We find that a slightly larger inhibition is also needed to explain the observed HAP kinetis, with a best value of E around.2, but less disrimination is possible than with the S1 nulease kinetis. Furthermore the variation in root mean square error in fitting the alulated urves to the data with hange in E is gradual. The most onservative onlusion is that an inhibition of 1.8- to 4-fold (i.e., E of.25-.6) is required in order to provide a quantitative interpretation of the observed kinetis. We now summarize the evidene for the existene of the partile inhibition fator. If the value of E is set at 1. (i.e., no inhibition) a less satisfatory fit of Eq. 9 to the data is obtained, and the root mean square error rises to about 1.7 times its value when E =.6. In addition experiments of Smith et al. (5) indiate a partile inhibition fator in the range.5-.6, in exellent agreement with the quantitative treatment just desribed. However, both the numerial interpretation of these experiments and the alulation shown in Fig. 3 rely in part on the length redution funtion illustrated in Fig. 2 and formalized in Eq. 6. If for some reason length redution for partile single strands follows a muh different ourse than the omputer simulation predits, the value of the partile inhibition fator would hange. However we see no reason why the length redution alulation should not be realisti. Our belief is that the partile inhibition probably aounts for about half of the total observed retardation. That is, if a alulation similar to those resulting in Eqs. 8 and 9 and Fig. 3 is arried out without any length
Biohemistry: Britten and Davidson redution terms, the value found for E is about.25, a little less than one-half that observed when the expeted length redution is taken into aount. The general onlusion is that whatever its exat auses, a progressive redution of the partile-single strand nuleation rate ours. This rate redution is the underlying reason that kinetis follow the familiar forms desribed by Eqs. 1 and 2. We thank Drs. Norman Davidson, Mihael Smith, and Lynn Klotz for their ritial reviews of a draft of this paper. This work was supported by NIH Grants HD-5753 and GM-2927 and by NSF Grant GB-33441. Pro. Nat. Aad. Si. USA 73 (1976) 419 1. Britten, R. J. & Kohne, D. E. (1967) arnegie Inst. Wash. Year Book 65, 78-16. 2. Wetmur, J. G. & Davidson, N. (1968) J. Mol. Biol. 31,349-37. 3. Britten, R. J., Graham, D. E. & Neufeld, B. R. (1974) in Methods in Enzymology, eds. Grossman, L. & Moldave, K. (Aademi Press, New York), Vol. 29, part E, pp. 363-418. 4. Morrow, J. (1974) Ph.D. Thesis, Stanford University. 5. Smith, M. J., Britten, R. J. & Davidson, E. H. (1975) Pro. Nat. Aad. Si. USA. 72, 485-489. 6. hamberlin, M. E., Britten, R. J. & Davidson, E. H. (1975) J. Mol. Biol. 96,317-333. 7. Wetmur, J. G. (1971) Biopolymers 1, 61-613.