Rate of DNA synthesis in mammalian cells irradiated with ultravolet light: A model based on the variations in the rate of movement of the replication fork and in the number of active replicons

J. theor. Biol. (1983) 100,359-372

Rate of DNA Synthesis in Mammalian Cells Irradiated with Ultravolet Light: A Model Based on the Variations in the Rate of Movement of the Replication Fork and in the Number of Active Replicons ROGERIO

Department

MENEGHINI

AND ALBERTO

C. DE MELLO

FILHO

of Biochemistry,

Institute of Chemistry, University Sa”o Paulo, CP 20780, Go Paula, Brazil

of

(Received 27 January 1981, and in final form 8 April 1982) A model is proposed to describethe rate of DNA synthesisobserved under certain conditionsin UV irradiated mammaliancells.It is assumed that shortly after irradiation the rate of DNA synthesisdrops mainly as a consequenceof the drop in the rate of movement of the replication fork. This in turn, is due to a pauseat the dimer for a limited length of time. Later on, a recovery in the rate of DNA synthesisoccurs,and it is proposedthat one of the parameterscontributing to that is an increase in the numberof active replicons.This simplemodelenablesone to predict variations in the rate of DNA synthesisas a function both of UV dose and of time after irradiation. Introduction

The study of DNA replication in ultraviolet (UV) irradiated mammalian cells is of importance since the biological consequences of UV light, e.g. cell killing and mutation, are triggered during DNA replication on lesioncontaining templates (Meneghini, 1981; Meneghini, Menck & Schumacher, 1981). In an asynchronous cell population, the overall rate of DNA synthesis is the outcome of three main parameters, namely, the rate of fork movement, the number of active replicons per cell and the percentage of cells in S phase. Upon irradiation, a fourth element, the introduction of the lesion into the DNA, may in principle alter the three parameters above. Moreover, different cells may differ in each of these responses to UV. It would seem unwise to model the effect of UV light on the overall rate of DNA synthesis in very general terms. However, if the study is restricted to relatively low doses, the rate of fork movement between the lesions does not seem to be affected (Povirk & Painter, 1976) and the percentage of cells in S phase is not significantly altered (Moustacchi, Ehmann & 359

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@ 1983AcademicPressInc. (London)Ltd.

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Friedberg, 1979; Lehmann, Kirk-Bell & Stevens, 1979; Dahle, Griffiths & Carpenter, 1980). Hence, the overall rate of DNA synthesis becomes a function of the new number of active replicons and of whatever occurs when the fork reaches the lesion: a halt, a pause or a free bypass. An attempt to model the DNA synthesis response to UV radiation under these conditions seems worthwhile. With this aim, two major assumptions were made: first that the rate of movement of the fork between dimers is normal (Povirk & Painter, 1976), whereas, at the dimer, the fork is arrested for a certain length of time; second, that the number of active replicons increases slowly after irradiation for reasons to be discussed. These two assumptions enable one to develop an expression which gives a full account of the changes in the overall rate of DNA synthesis, both as a function of dose of irradiation and of time. This latter dependence bears on the known phenomenon of recovery of DNA synthesis after UV (Meyn, Kasschau & Hewitt, 1977; Moustacchi et al., 1979; Lehmann et al., 1979). Many factors are likely to be involved in the recovery and the present model attempts to consider only one additional parameter which has so far been overlooked but that, nevertheless, may significantly contribute to this process. Theory (A)

THE

PAUSE

OF THE

THE

REPLICATION

PYRIMIDINE

MACHINERY

AT

DIMERS

In order to develop the model, it is assumed that the frequency of initiation on replicons is constant throughout the S phase. This assumption, although probably not correct for an individual cell, reflects the average situation in an asynchronous cell population. To calculate the frequency of initiation of replicons, i, and the number of active replicons per cell, N, it is necessary to define: R, the rate of fork movement; n, the total number of replicons per cell; r, the length of time corresponding to S phase; L, the size of a replicon. The frequency of initiation of replicons is given by i = n/7.

(1)

At the beginning of the S phase, the number of active replicons will increase according to the rate i, until an equilibrium is reached, when the frequency of termination equals the frequency of initiation of replicons. This steadystate can only be achieved after a minimal period of time, corresponding to the synthesis of a whole replicon. This period of time is thus dependent on the size of the replicon and the rate of the fork movement and is

RATE

OF

DNA

SYNTHESIS

IN

UV-IRRADIATED

CELLS

361

therefore equal to L/2R. The rate of fork movement in this expression is multiplied by two on account of the bidirectional replication which occurs in individual replicons (Huberman & Riggs, 1968). The number of active replicons at the steady-state, iV1, can then be calculated by the expression NI = iLj2R.

(2)

These parameters are defined for a normal, u&radiated cell. If the cells are irradiated with UV-light, lesions are introduced into the genome. It is assumed that pyrimidine dimers are the only important UV lesions producing effects on DNA synthesis, which is a good approximation, as judged by results from photoreactivation experiments (Lehmann & Stevens, 1975; Rosenstein & Setlow, 1980). The frequency of pyrimidine dimers appearing in the DNA is given by Fd, where F is the dose of irradiation and d corresponds to the dimer frequency per unit of DNA length at unitary dose. Suppose that the replication fork pauses at the dimer for a time p and that the interdimer region is replicated at the normal rate R. This latter assumption is supported by recent results (Povirk & Painter, 1976). In this new situation the fork will have a discontinuous movement but the overall situation will be as if a reduced rate of the fork movement, r, had been attained. In order to calculate the new rate of fork movement T, let us calculate the time necessary for the fork to cover an arbitrary length of DNA S, irradiated with a dose F. There will be two components of time, one corresponding to the time necessary for the fork to cover all the interdimer regions which is S/R and the other corresponding to the time the fork remains arrested at the dimers, which is pSFd, where SFd is the total number of dimers in the length S of DNA. Therefore the total time taken by a fork to cover the length S is given by S/R +pSFd. This is equivalent to S/r and hence the rate of fork movement r is given by R r=l+pFdR’

(B)

THE

INCREASE

IN AFTER

THE

NUMBER UV

OF

ACTIVE

REPLICONS

IRRADIATION

In order to develop the model further, we assume, as a first approximation, that the frequency of initiation, i, and the size of the replicon, L, are not modified upon UV irradiation. Because the rate of fork movement decreases upon UV irradiation (equation (3)), the frequency of termination of replicons will temporarily drop. As a consequence, the number of active

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replicons tends to increase until a new steady-state value is attained, when the frequencies of initiation and termination become equal again. It must be pointed out that, according to this view, a replicon arrested at the dimer for a certain length of time is, by definition, an active replicon. The difference is that it is now travelling at a new rate, r (equation (3)). The number of replicons, corresponding to the newly achieved steady-state, Nz, can be calculated by substituting r for R in equation (2): Nz = iL/2r. (C)

KINETICS

OF

INHIBITION

OF

DNA

SYNTHESIS

Both the change of rate of fork movement from R to r, relative to the pre-existing replicons, Ni, and the increase N in the number of active replicons are time dependent. Let us first consider this time dependence for the change of rate of fork movement of the pre-existing N1 replicons. After irradiation the average frequency of dimers is Fd. A given replication fork, when travelling the length Rt (t being the time) will have the probability of not reaching a dimer equal to eFRtFd. To draw this conclusion it was assumed that the dimer distribution follows a Poisson distribution, where RtFd is the average number of dimers contained in the length Rt. Therefore, at a time t after irradiation, part of the original replicons, corresponding to (iL/2R)

is still replicating

epRdFt

at the rate R. The remaining (iL/2R)(l

(4) replicons,

-epRdFt)

(5)

have already reached a dimer and as a consequence are replicating at the new rate r. These two expressions would be sufficient to describe the kinetic curves for inhibition of DNA synthesis by UV irradiation were it not for the increase in the number of replicons described in section (B). To develop a function describing the increase of replicons as a function of time, dN/dt, it must be considered that this increment will be at all times equal to the difference between the frequency of initiation, i, and the frequency of termination, f: dN/dt = i -f. (6) In this expression i is a constant, as assumed in the model, but f is a function of time; at all times f will be equal to the number of active replicons divided by the time taken to replicate a single replicon: f = (iL/2R) epRdFI+ (iL/2R)(l -eeRdFf)+ L/2R L/2r

N L/2r’

(7)

RATE

OF

DNA

SYNTHESIS

IN

UV-IRRADIATED

363

CELLS

According to equations (4) and (5) the first two terms of equation (7) represent the original active replicons, iL/ZR, which are synthesizing DNA at rates R and r, respectively, divided by the time they take to replicate a single replicon. The last term represents the new replicons N which have been added to the original ones, as a consequence of the difference between the frequency of initiation and termination, divided by the time taken to replicate a single replicon. In this case, as a first approximation, all additional replicons N are considered to have been replicating at a rate r since their initiation. Substituting the value off in equation (6) one gets:

dN dt=l-

.

(iL/2R) ebRdFt +(iL/ZR)(l-eeRdF’)+ L/2R L/2r

This expression can be transformed

N L/2r

1 (8) .

into:

dN x=A-A

(9)

epEt--CN

whereA=i(l-r/R);B=RFdandC=2r/L. A solution for the integral of equation (9) is given by:

N,AC

- A C-B

e-B’ + K e-”

where K is a constant of integration (Appendix A). To calculate the value of K, it is considered that at time t = 0 the additional number of replicons N = 0. Thus one gets: K = AB/(C* -BC) which substituted in equation (10) gives:

N,Ac

A C-Be

--Bt

AB +C”-BC

-ct

e

or, replacing the values of A, B and C: &-

2r -LRFd ewRFd* +

RFdL2 4r2-2rRFdL

-2rt/L

e

’

(11)

For large values of t, N tends to the value iL(l -r/R)/2r, which is equal to the difference between the number of replicons in the final steady-state, Nz = iL/2r, and the number of replicons in the initial steady-state, N1 = iL/2R (see section (B)). In order to derive the kinetic curves for inhibition of DNA synthesis (Fig. 1) it is necessary to recall that the overall rate of semiconservative synthesis, 1, is proportional to the rate of fork movement and to the number

364

R. MENEGHINI . i

AND

FILHO

I-O

.Z2

(a)

E 0.5

b)

8

% s ZF .-Lz

A. DE MELLO

(c) 0 lE?z

2

4 Time

6

(hours)

FIG. 1. Rate of DNA synthesis at various times after UV irradiation. The curves correspond to plots of equation (12). The initial value of I when t = 0 was normalized to one. The values of the different parameters in equation (12) were: d (dimer frequency per unit of DNA length at unitary dose) = 2 x lo-* dimers dalton-’ m2 J-r; R (normal rate of fork movement) = 1.2 x lo6 dalton min-r; i (frequency of initiation of replicons) = 43 replicons/min; L (average replicon size) = 128 x lo6 daltons. These values are all for V79 cells. The value of r is calculated by the equation (3) for each dose F of UV irradiation. In equation (3) the time p which the fork pauses at the dimer was considered to be 20 min. The curves were drawn by a model HP-85 Hewlett-Packard computer, for three different doses: (a) 2; (b) 5 and (c) 10 J/m’.

of active replicons.

Taking into account equations (4) and (5) as well one has:

I = (iL/2R)e

-"*'R +(Z/2R)(l

-edRdF’)r +Nr

(12)

where the different values of N as a function of t are given by equation (11). To plot I as a function of t (Fig. 1) and dose (Fig. 2) the parameters from Chinese hamster V79 cells were considered. The value of R was considered to be O-6 k/min or 1.2 x lo6 daltons/min (double stranded

F (J/m’)

2. Rate of DNA synthesis after various doses of UV. Equation (12) was plotted by calculating I for different values of F. In this case time r was fixed as 30 min. The curves were drawn for three different values of p: from top to bottom, 10, 20 and 30min. The values of the remaining parameters were the same as in Fig. 1. The curves were drawn by a model HP-85 Hewlett-Packard computer. The points represent experimental data of DNA synthesis determined by pulse-labeling for 30 min V79 cells with H-thymidine, beginning 30 minutes after irradiation in the way previously described (Cordeiro-Stone et al., 1979). Each point represents the mean of three independent determinations. FIG.

RATE

OF

DNA

SYNTHESIS

IN

UV-IRRADIATED

CELLS

365

DNA) as determined by Dahle et&. (1980). The value of d was determined in this laboratory as 2 x lOPa dimers daltons-’ m2 J-r (Menck & Meneghini, 1982) and is in agreement with other literature data. The value of r is calculated by equation (3) for each dose F and time of pause at the dimer, p, considered. The value of i, according to equation (1) is 15 600/360 or 43. The total number of replicons 15 600 was determined by taking the diploid genome size 2 x 1012 and dividing by the average replicon size, L, which in V79 cells is 128 x lo6 (Dahle et al., 1980). The S phase period for V79 cells corresponds to 360 minutes (our unpublished observations). To plot equation (12) in Fig. 1, the initial value of 1 for t = 0, which is iL/2, was normalized to one and the remaining values were referred to it. The value of p, the length of time which the fork spends halted at the dimer was, in this case, considered to be 20 minutes for the reasons described in section (D). It can be seen that the replication rate decreases rapidly after irradiation. Both the time to reach the minimum and the value of the minimum are dose dependent. During this initial phase the main factor contributing to the change in replication rate is the progressive halting of the replication forks at the dimers, as described by the two first terms of equation (12). By the time the replication rate reaches the minimum, the majority of the replicons will be travelling at a rate r. However, additional replicons N have been added to the original ones since the irradiation, as indicated by the third term of equation (12). When the replication rate reaches a minimum, this continuous increase in the number of active replicons determines a progressive recovery in the rate of replication. The minimum value of I is reached asymptotically as it is the new steady-state number of active replicons, iL/2r. The drop in the rate of fork movement is thus compensated by an increase in the number of active replicons, since the values of 1 in equation (12) for t = 0 and t = 00 are both iL/2. The overall result is, thus, the recovery of control levels of DNA synthesis, occurring in the absence of excision of dimers (Meyn et al., 1977; Dahle et al., 1980). Experimental curves similar to the theoretical ones of Fig. 1 were obtained by Dahle et al. (1980). (D)

REPLICATION

RATE

AFTER

VARIOUS

DOSES

OF

UV

In order to determine the effect of dose F on replication rate, I, one must fix time t in equation (12). As can be seen in Fig. 1, the rate of DNA replication is continuously changing with time; however, for a dose range up to 20 J/m2, minimal values of I are attained between 30 and 60 min and do not change considerably in this interval. In Fig. 2, plots of I versus F were drawn fixing the time as 30 min. Three different values of p, the

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length of time which the replication fork pauses at the dimer, were used (lo,20 and 30 minutes) to determine the one which allows the best fit with experimental data. The experimental values of DNA synthesis were obtained by pulse labeling V79 cells with 3H-thymidine from 30 to 60 minutes after irradiation. As they represent semiconservative synthesis, they are proportional both to the rate of fork movement and to the number of active replicons. It can be seen that the experimental data fit the theoretical curve rather well if p is considered to be 20 minutes. It is clear that the model predicts the rate of DNA synthesis at a given instant whereas the experimental values were determined over a 30 minute time span. Curves like the ones in Fig. 2 were also drawn fixing the time t as 60 minutes (not shown) and although they differ slightly ‘from the ones in which the time t is fixed as 30 minutes, the best fitting is still given by the curve in which p is considered to be 20 minutes. This is why, p was made equal to 20 minutes, in order to plot replication rate as a function of time in Fig. 1. Discussion

DNA replication in mammalian cells is a complicated process which is under many influences and controls. When lesions are introduced in the genome, many of the components of this process may be affected. It seems unlikely that simplified models would give full account of the behavior of the replication machinery upon infliction of damages to the template. However, we considered it worthwhile to derive a relatively simple model, based on a few reasonable assumptions, to point out possible influences on DNA synthesis of parameters that have so far been overlooked. The model turned out to predict rather well the inhibition of DNA synthesis by UV light both as a function of time and of dose. The basic assumptions are that dimers act as temporary blocks for the replication fork and that the frequency of initiation, i, is not modified upon irradiation. The net effect of this temporary block is a decrease in the rate of fork movement. This, in turn, brings about a decrease in the rate of termination of replicons and, since the rate of initiation is not modified, the number of active replicons tends to increase. Hence two parameters influence the overall rate of DNA synthesis after UV irradiation: a progressive decrease in the rate of fork movement of the original active replicons as they encounter dimers and a progressive increase in the number of active replicons. The influence of each of these parameters on DNA synthesis is given by equation (12), where the rate of DNA synthesis, 1, is a function of time, t, and UV dose, F. The kinetic curves generated by this equation

RATE

OF

DNA

SYNTHESIS

IN

UV-IRRADIATED

CELLS

367

(Fig. 1) show a rapid decrease in the rate of replication with time after irradiation, when the two first terms of equation (12), representing the change in the rate of fork movement of the pre-existing replicons, play the most important role. As time goes by, the third term of equation (12), representing the increase in the number of active replicons, determines a slow recovery in the rate of DNA replication. The model predicts that long times after irradiation the increase in the number of active replicons will exactly compensate for the decrease in the rate of fork movement; therefore, a full recovery of DNA synthesis in the absence of dimer excision may occur. The curves generated by equation (12) resemble those experimentally obtained for V79 cells by Dahle et al. (1980). However their curves seem to show a lag prior to the gradual recovery, although this is not perfectly clear due to the scattering of their experimental data. Nevertheless, a possible explanation for this putative lag is that initiation of new replicons is partially inhibited during a limited period of time after irradiation, as suggested by recent observations (Kauffman, Cleaver & Painter, 1980). Because frequency of termination is also partially inhibited (this model), the number of active replicons would stay steady, giving rise to the lag. After that, the normal frequency of initiation would be resumed (Kauffman et al., 1980) and the process of increase in the number of active replicons as described by the model, would start with some delay. Equation (12) also describes the effect of UV dose on the rate of DNA replication, at a fixed time t (Fig. 2). When short times are considered (30-60 mins) DNA synthesis is at its minimum (Fig. l), determined basically by the encounter of the pre-existing replicons with the dimers. This causes the replication fork to pause for a period of time p before resuming replication, and the net result is a reduced rate of fork movement r for these replicons. However, it should be pointed out that according to equation (12), the replication rate never reaches the value given by equation (3) for when all replicons are moving at a rate r. This is because this value would be reached only asymptotically, and the increase in the number of replicons, given by the third term of equation (12), opposes this tendency. In Fig. 2, the value of p which generates the curve that best fits the experimental data is 20 minutes. This value is also suggested by experiments with V79 cells, which showed that the pyrimidine dimers may act as absolute blocks for DNA synthesis only for lo-15 minutes (van Zeeland & Filon, 1980). It is implicit in this calculation that all dimers will act as temporary blocks for DNA synthesis. However, alternative situations could be considered in which only some of the dimers would constitute obstacles to replication. Thus, it is possible that only dimers on the template for the leading strands

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(5’-3’) interrupt the fork (Meneghini, 1976; Cordeiro-Stone, Schumacher & Meneghini, 1979). In this case, half the dimers present in the genome should be considered in equation (3) and consequently, the length of time p would have to be twice as long in order to generate the same curve. Similarly, Park & Cleaver (1979) have proposed that each dimer has a probability, P, of blocking replication. Another model to explain doseresponse curves is that of Edenberg (1976), according to which the relative DNA synthetic rate is equal to the average length between dimers, into which origins of initiation fall, divided by the average replicon size. According to the symbols defined here, the relative DNA synthetic rate should be equal to l/F& instead of the one given by equation (12) of the present model. It can be seen that in Edenberg’s model, the relative rate of DNA synthesis is dependent on the replicon size and, in terms of explaining the dose-response curve, the dimer is considered an absolute block for replication. However, recent data obtained by DNA fiber autoradiography have clearly shown that the fork reduces its rate but does not stop at a dimercontaining template (Dahle etal., 1980). In addition, the experimental data of Fig. 2 do not fit the curve generated by using Edenberg’s equation, regardlesss of the value of d employed. Equation (12) seems to provide a better account of the dose-response curve in terms of inhibition of DNA synthesis. It should be said that the assumption of a pause of the replication fork at the dimer does not imply any specific mode of resumption of synthesis, either the direct bypass or the formation of gaps opposite the lesions being conceivable. This same assumption has been previously considered to derive a mathematical model for DNA synthesis in irradiated cells (Klimek & Vanicek, 1970). The V79 Chinese hamster cells were chosen to illustrate the applicability of the model because they have been frequently used to determine the parameters of DNA replication and because they are a typical example of cells that recover from DNA synthesis inhibition produced by UV, independently of excision of dimers (Meyn et al., 1977; Dahle et al., 1980). However, published dose-response curves for HeLa cells fit equally well the theoretical curves produced by equation (12). Certainly, factors other than accumulation of replicons may play a role in the recovery of DNA synthesis. In human cells proficient in excision repair there is a progressive drop in the frequency of blocks for replication and this is an additional factor in the recovery of the DNA synthesis rate (Moustacchi et al., 1979; Lehmann et al., 1979). Human cells from patients with the disease xeroderma pigmentosum seem to be unable to recover from inhibition of DNA synthesis by UV (Lehmann et al., 1979). Because these cells are defective in excision of dimers one is lead to conclude that in human cells

RATE

OFDNA

SYNTHESIS

IN UV-IRRADIATED

CELLS

369

recovery is absolutely dependent on dimer excision. However, human cells from patients with Cockayne syndrome are normal in excision of dimers and nevertheless are unable to recover from DNA synthesis by UV (Lehmann et al., 1979). If the increase in the number of active replicons plays a significant role in the recovery from DNA synthesis in human cells, then the abnormal recovery in xeroderma pigmentosum and Cockayne syndrome cells may lie on a defective increase in the number of active replicons in these cells. Possible causes for that could be (i) a persistent inhibition of initiation of new replicons after UV irradiation in these cells; (ii) some other effect which could counterbalance the normal tendency of an increase in the number of active replicons, as for instance, a shortening in the average size of replicons in irradiated xeroderma pigmentosum and Cockayne syndrome cells. In the case exemplified here, that of Chinese hamster cells, excision repair of dimers is not an active process (Meyn et al., 1977) and most of the lesions will still be present several hours after irradiation. In this case, it is possible that upon irradiation the replication machinery becomes more capable of replicating the lesion-containing DNA. This would constitute an additional factor for recovery. Although this process is still ill understood, it may also be contributing for the recovery phenomenon in a way not predicted by the present model. In summary, we have addressed ourselves solely to the possibility of accumulation of active replicons as an explanation for the phenomenon of recovery in the rate of DNA synthesis. Other parameters are likely to participate in the process, like excision repair and an induced recovery in the rate of fork movement. The knowledge of the participation of each one, separately, may render it possible a complete quantitative description of the recovery phenomenon. Dr R. Ivan Schumacher’sassistance in computer analysisis greatly appreciated. This work wassupportedby grantsfrom FAPESP, CNPq and FINEP.

REFERENCES CORDEIRO-STONE,M.,SCHUMACHER,R.I.&MENEGHINI,R.(~~~~). Biophys.J.27,287. DAHLE,D.,GRIFFITHS,T.D.&CARPENTER,J. G.(1980). Photochem.Photobiol. 32,157. DONIGER,J. (1978).J. mol. Biol. 120,433. EDENBERG,H. (1976).Biophys.J. 16,849. HUBERMAN,J. A. &RIGGS, A. D. (1968).J. mol.Biol. 32,327. KAUFFMAN,W.K.,CLEAVER,J.E.&PAINTER,R.B. (1980)Biochim.biophys.Acta 680, 191.

KL~MEK, M. & VANICEK, J. (1970).Math.Biosci. LEHMANN,A.R.,KIRK-BELL,S.&MAYNE,L.(~~~~). LEHMANN, A. R. & STEVENS, S. (1975). Biochim.

9,

165.

biophys.

CancerRes.39,4237. Acta 402, 179.

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MENCK,C.F.M. & MENEGHINI, R.(1982).Photochem.Phofobiol.35,507. MENEGHINI, R. (1976). Biochim. biophys. Acta 425,419. MENEGHINI, R. (1981). TrendsBiochem. 6,214. MENEGHINI,R.,MENCK,C.F.&SCHUMACHER,R.I. (1981).Quart.Rev.Biophys. 14,381. MEYN,R.,KASSCHAU,M.R.& H~w11~,R.(1977). Mutat. Res.44, 129. MOUSTACCHI,E.,EHMANN,U.K. &FRIEDBERG,E.C. (1979). Mutat. Res.62,159. PARK,S.D.& CLEAVER,J.E.(~~~~). Proc. naf. Acad.Sci. U.S.A.76.3927. PovIRK,L.F. &PAINTER, R. B.(19j6).Biophys.J. 16,883. ' RosENsTEIN,B.S.&SETLOW,R.(~~~O). Biophys..L 31,195. VAN ZEELAND, A. A. & FILON, A. R. (1980). Abstracts from the VIIIInternational Congress of Photobiology, p. 278. APPENDIX dN x=A-A

A eFB’-CN.

This is a first order linear differential the form.

function,

which can be written in

$+P(x)y =Q(x) or T+

CN = A -A

eeBt.

Let us look for a solution N(t), in the form of a product of two functions in t: N = u(t) x v(t). Deriving:

dN dv -$=u-&+vdt and placing back in equation

du

(Al), one gets:

u$+v$+CN=A-A

u$+v$+Cuv u($+Cv)

eeBr

=A-A +$$=A-A

eeB’ eeBf.

Since the only requirement for v and u is that their product be the solution N(t) we may select the function v such as to satisfy: dv z+cv

=o.

RATE

Therefore:

OF

DNA

SYNTHESIS

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dv/dt = -Cv and dv/v = C dt and integrating:

Ink=-

J

Cdt,

or: where K1 is a constant of integration. Since it is sufficient to obtain a solution for v (t) different to zero we may take KI = 1 or, v(t) = e-ICdt where j C dt is any “primitive” and v(t) # 0. Knowing that (du/dt) + Cv = 0 and substituting the value of v(t) in equation (A2) one gets: du A -A epBt z=

e-JCdt

*

Therefore:

J

A -A

u(t) =

K being a constant of integration.

(I =

,-J-t

dt

+K

e

However, since N(t) = uv :

N = e-JCdt N

emSt

-jCdt

A

A -A e-Bt -JCdt dt+K e --v-~-Bt&+K

>

e-jCdr,

J

Since e+ dt =

e

-Cf

J

A -A epBt dt = -ct e

J

(A ect -A

ecCpBjr) dt

=A Jeczdt-JAe’C-B”dt =--A ecf C

A e’c-B’t (C-B)

*

Therefore

N-A --tK Ae-B’ C

C-B

e-”

which is equation (10). When t = 0 N = 0, thus K = AB/C* N=A A-- AepBf+ C C-B

AB C*-CBe

-ct ’

- CB therefore:

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R. MENEGHINI

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APPENDIX

(a) Equation

MELLO

FILHO

B

A A (10) is: N =---e C C-B

-Bf

AB +C2-BC

-a

e

(b) ~=$($-&e-Br+c2~Bc (c) (5) which is the derivative of the solution equation (9) can be written in the form: (d) :=A-A

(equation

e-B*-C(~-&-e~Bt+C2~BCe~c’).

where N has been replaced by the solution, epBf+Ee CA

(4 Thus: $=-A dN

dN-

-Bt

i.e., equation (10).

ABC -C2-BC

-ct e

--

(0 ordt=e k) dt-

(10)). Now, the initial

CA -A(C-B) C-B

e-Bt

ABC -C2-BC

-ct e

and therefore (h) 5

=&

epBf - $!Ec

ewC’

which is exactly the derivative of the solution (dN/dt),,,

derived above.