A biologically inspired two-species exclusion model: effects of RNA polymerase motor traffic on simultaneous DNA replication
Soumendu Ghosh, Bhavya Mishra, Shubhadeep Patra, Andreas Schadschneider, Debashish Chowdhury
AA biologically inspired two-species exclusion model:effects of RNA polymerase motor traffic on simultaneous DNA replication
Soumendu Ghosh, Bhavya Mishra, Shubhadeep Patra, Andreas Schadschneider, and Debashish Chowdhury Department of Physics, Indian Institute of Technology Kanpur, 208016, India ISERC, Visva-Bharati, Shantiniketan 731235, India Institute for Theoretical Physics, University of Cologne, K¨oln, Germany
We introduce a two-species exclusion model to describe the key features of the conflict betweenthe RNA polymerase (RNAP) motor traffic, engaged in the transcription of a segment of DNA,concomitant with the progress of two DNA replication forks on the same DNA segment. One ofthe species of particles ( P ) represents RNAP motors while the other ( R ) represents replicationforks. Motivated by the biological phenomena that this model is intended to capture, a maximumof only two R particles are allowed to enter the lattice from two opposite ends whereas the unre-stricted number of P particles constitute a totally asymmetric simple exclusion process (TASEP)in a segment in the middle of the lattice. Consequently, the lattice consists of three segments; theencounters of the P particles with the R particles are confined within the middle segment (segment2) whereas only the R particles can occupy the sites in the segments 1 and 3. The model capturesthree distinct pathways for resolving the co-directional as well as head-collision between the P and R particles. Using Monte Carlo simulations and heuristic analytical arguments that combine exactresults for the TASEP with mean-field approximations, we predict the possible outcomes of the con-flict between the traffic of RNAP motors ( P particles engaged in transcription) and the replicationforks ( R particles). The outcomes, of course, depend on the dynamical phase of the TASEP of P particles. In principle, the model can be adapted to the experimental conditions to account for thedata quantitatively. I. INTRODUCTION
The totally asymmetric simple exclusion process(TASEP) [1–4] was originally introduced as a simplifiedmodel describing the kinetics of protein synthesis [5, 6].Since then it has found many more applications to bi-ological systems, especially to situations [7–26], wherethe kinetics is dominated by the traffic-like collective mo-tion of molecular motors (for reviews, see [27–31]). Ge-netic message encoded chemically in the sequence of themonomeric subunits of DNA is transcribed into an RNAmolecule by a molecular motor called RNA polymerase(RNAP). In each of its step on the DNA track a RNAPmotor elongates the nascent RNA molecule by a singlesubunit using the same DNA strand as the correspondingtemplate [32]. TASEP-based models have also been de-veloped for the traffic-like collective movements of RNAPmotors on the same segment of DNA while each RNAPsynthesizes a distinct copy of the same RNA [33–38].A segment of DNA can undergo multiple rounds oftranscription during the lifetime of a cell. In contrast,each DNA molecule is replicated once, and only once,just before the cell divides into two daughter cells [32].A molecular machine called DNA polymerase (DNAP)is a key component of a replisome which is a multi-machine macromolecular complex that replicates DNA.As the replisomes unzip a duplex DNA and replicate thetwo exposed strands, Y-shaped junctions called replica-tion forks, are formed. The progress of replication canbe described in terms of the movement of two replicationforks; replication of a segment of DNA is completed whentwo replication forks, approaching each other from oppo-site ends of the segment, collide head-on [32]. Theoreti- cal models for the “nucleation” of replication competentreplication forks and growth of the replicated domains ofthe DNA have been reported in the past [39–45] (see also[46, 47] for reviews).Interestingly, transcription and replication can occursimultaneously on the same segment of DNA. How-ever, typically, at a time only one of the two DNAstrands of the DNA undergoes transcription by a traf-fic of RNAPs while both the strands are simultaneouslyreplicated by distinct replisomes. Obviously, head-oncollisions between a replication fork and RNAP mo-tors is possible. Moreover, since the rate of replica-tion is 10-20 times faster than that of transcription, areplication fork can catch up with a RNAP from be-hind thereby causing co-directional collision. Both typesof collisions can have disastrous consequences [48], un-less the transcription-replication conflict is resolved suf-ficiently rapidly to ensure maintenance of genomic sta-bility. Nature has adopted multiple mechanisms of res-olution of such conflict [49–52]. However, to our knowl-edge, no quantitative theoretical model of transcription-replication conflict and their resolution has been reportedso far.Here we propose a TASEP-based minimal model thatcaptures the essential aspects of RNAP traffic on a seg-ment of DNA concomitant with the progress of DNAreplication forks from the two ends of the same DNAsegment. The kinetics of the model incorporates all theknown natural mechanisms of resolution of conflicts be-tween DNA replication and transcription. This formu-lation, as explained in the next section, leads to a two-species exclusion process on a 3-segment lattice in onedimension which also includes “Langmuir kinetics”, i.e. a r X i v : . [ q - b i o . S C ] J a n attachment and detachment of particles in the bulk [53].One of the two species of particles represents RNAP mo-tors all of which move co-directionally, i.e., say, from leftto right. In contrast, only two particles of the secondspecies, each representing a DNA replication fork, ap-proach each other from opposite ends of the same track,i.e., one from the left and the other from the right. Be-cause of the decrease of the separation between the tworeplication forks with the passage of time, the spatial re-gion of conflict between the two species of particles alsokeeps shrinking. Thus, the model of the two-species ex-clusion process developed here is highly non-trivial.By a combination of analytical arguments and com-puter simulations, we investigate the effect of the two pro-cesses, i.e., transcription and replication, on each other.More specifically, we indicate (a) the trends of variationof the mean time for completion of replication and (ii)the statistics of the successful and unsuccessful replica-tion events, in the different phases of the RNAP traffic[2–4]. II. MODEL
The schematic diagram of the model is shown in Fig. 1.For simplicity, motion of both species of particles are as-sumed to occur along a single common track representedby a one dimensional lattice of total length L , where, L is the total number of equispaced sites on the lattice.The lattice consists of three segments: sites i = 1 to i = L − i = L to i = L (segment 2),and site i = L + 1 to i = L (segment 3).One of the two species of particles, labelled by P , rep-resent the RNAP motors; all the P particles can move,by convention, only from left to right, i.e., from i to i + 1.There is no restriction on the number of P particles thatcan populate the lattice, except the limits arising natu-rally from the rates of entry, exit and forward hoppingthat are described below. In contrast, not more thantwo particles of the second species, labelled R and rep-resenting the replication forks, can ever enter the latticeirrespective of the kinetic rates, i.e., probabilities per unittime of the various kinetic processes that are describedbelow. One of the R particles, denoted by R (cid:96) moves fromleft to right ( i to i + 1) whereas the other, denoted by R r moves from right to left ( i + 1 to i ) on the lattice.The R (cid:96) particle can enter the lattice only at i = 1with the probability γ per unit time. Similarly, the par-ticle R r can enter the lattice only at i = L with theprobability δ per unit time. After entry, the particles R (cid:96) and R r can hop to the next site in their respective pre-determined directions of motion with the rates B and B , respectively (see Fig. 1). Both these particles cancontinue hopping, obeying the exclusion principles andrules of resolution of encounter with P particles as de-scribed below, till they encounter each other head-on attwo nearest-neighbor sites on the lattice indicating com-pletion of replication. Unlike the R particles, all the P particles can enterthe lattice only at the site i = L with the attachmentrate (i.e., probability per unit time) α q provided thatsite is not already occupied by any other particle of ei-ther species. Once entered, a P particle can hop forwardto the next site with the rate q if, and only if, the targetsite is not already occupied by any other P or R parti-cle. A P particle can continue forward hopping, obeyingthe exclusion principle and the rules of resolution of en-counter with R particles, till it reaches the site i = L from where it can exit with the rate β q .Thus, the division of lattice into three segments isbased on the scenario that the lattice sites in the seg-ments 1 and 3 can be occupied exclusively by only the R particles whereas the sites in middle region (i.e. seg-ment 2) can get populated by both the P and R particles.However, in the segment 2 the P particles encounter the R (cid:96) particle co-directionally and R r particle head-on. Thefinal encounter between the two R particles, when theymeet each other at two nearest-neighbour sites, is head-on.Next we described the kinetics of both types of parti-cles in the segment 2 which capture the mutual exclusionof the RNAP motors as well as the rules of resolution ofthe conflicts between transcription and replication. Mu-tual exclusion is captured by the simple rule that no sitecan be occupied simultaneously by more than one parti-cle irrespective of the species to which it belongs. Thethree possible outcomes of the encounter between a P particle at the site i and a R (cid:96) particle at site i − R r particle at the site i + 1 are as follows:(a) The R particle can bypass the P particle with therates p co and p contra , in the cases of co-directional andcontra-directional encounter respectively, without dis-lodging the latter from the lattice and, therefore, boththe particles can continue hopping in their respective nat-ural direction of movement after the encounter.(b) The R particle can knock the P particle out of thetrack, with the rate D irrespective of the direction (co- orcontra-directional) encounter and it resumes its hoppingafter the P particle is swept out of its way thereby abort-ing the transcription by that P particle prematurely.(c) Upon encounter with P particles, a R particle doesnot necessarily always win. In such situations, occasion-ally, the R particle detaches from the lattice with a prob-ability C per unit time irrespective of the direction ofencounter; this scenario captures the possible collapse ofthe replication fork that can causes genome instability.Once the replication fork collapses, the victorious P par-ticle(s) resume their onward journey on the lattice.If the R particles, entered from the sites i = 1 and i = L , eventually meet each other on a pair of nearest-neighbour sites of the lattice, thereby indicating comple-tion of the replication of the entire stretch of DNA from i = 1 to i = L , we identify it as a successful event oftype 1 (from now onwards we referred to it to as sr1).On the other hand, if one of the R particle stalls or col-lapses at any site in between L and L while the other FIG. 1. A schematic diagram of the model. The whole latticeis divided into three segments { , , . . . , L − } , { L , . . . , L } and { L , . . . , L } . A R particle (green arrow) can enter, eitherfrom the first site of segment 1 (i.e. i = 1) with the probability γ per unit time or from the last site of segment 3 (i.e. i = L ), with the probability δ per unit time. A R particle thatenters through i = 1 is allowed to hop from left to right (i.e. i → i + 1) with rate B , if the target site is empty. But, if a R particle enters through i = L it is allowed to hop only fromright to left (i.e. i → i − B . Both the R particlescontinue their motion until they meet each other, at a pair ofnearest neighbour sites. However, inside segment 2, a new P particle (yellow circle) can attach only at i = L , with rate α q , only if this site is empty. Once attached, a P particlecan hop forward only from left to right (i.e. i → i + 1) by asingle site in each step, with rate q , provided the target site isempty. Normally, a P particle would continue its hopping tillit reaches the the site L from where it detaches with rate β q .Thus, the lattice sites in the segments 1 and 3 can be occupiedby only the R particles, whereas a mixed population of R and P particles can exist in the segment 2. p co D DC C (a) (b)(c) (d)(e) (f) p contra FIG. 2. Schematic representation of interference between the R particles and the P particles. In (a) and (b) R particle canpass the P particle, with rates p co and p contra . In (c) and (d)the R particle can knock the P particle out of the track, withrate D . In (e) and (f) P particle can block the progress of the R particle thereby causing its eventual collapse, with rate C . continues hopping until it reaches a nearest neighbor ofthat particular site of stall or collapse, it also indicatessuccessful completion of replication and, therefore, iden-tified as a successful event of type 2 (from now onwards,referred to as sr2). But if both the R particles are stalled(i.e., replication fork collapsed) before completely cover-ing the entire lattice together by hopping between thesites 1 and L , then the process is identified as unsuccess-ful event (usr). Dividing the sum total of the times taken by all the sr1 and sr2 events by the total number of allsuch events we obtain the mean hopping time of a R par-ticle ( τ ), which is the mean time required for successfulcompletion of replication of the DNA of length L (in theunits of “base pairs”). III. RESULTS
Although our model captures just a few key aspectsof the biological processes involved in the transcription-replication conflict, the proposed model is already toocomplex to allow a rigorous analytical treatment. Wetherefore rely mainly on Monte Carlo (MC) simulations.However, in certain limiting situations, the computersimulations are complemented by an approximate ana-lytical theory, that draws heavily on the known exactresults for TASEP with single species of particles. Thetransparent arguments of the analytical derivations pro-vide some insight into the underlying physical processes.However, the analytical derivation is based primarily onheuristic arguments some steps of which are essentiallyequivalent to mean-field approximations. Therefore, theaccuracy of our heuristic analytical arguments have beenchecked by comparison with the corresponding data ob-tained from the MC simulations.In the MC simulations we adopted random sequentialupdating to investigate the effects of traffic of P particleson the kinetics of R particles, i.e, the effects of ongoingtranscription on replication. The data collected duringthe simulations are averaged over 10000 realizations eachstarting from a fresh initial configuration. We convert therates into probabilities by using the conversion formula p k = k dt where, k is an arbitrary rate constant and dt is an infinitesimally small time interval; the typicalnumerical value of dt used in our simulations is dt =0 . s . Unless stated explicitly otherwise, the numericalvalues of the relevant parameters used in the simulationsare L = 2000, L = 500, L = 1500, B = B = 300 s − , β q = 1000 s − and q = 30 s − . A. Effects of steady traffic of RNA polymersaseson replication time
The time needed for a successful completion of replica-tion (now onwards, referred to as “replication time”) isidentified as the time taken by the two R particles to meethead-on, starting from their simultaneous entry into thelattice through i = 1 and i = L . In order to measure thereplication time in the steady traffic of P particles in theMC simulations, we first switch on the entry of only the P particles (i.e., transcription) through i = L . The two R particles are allowed to enter simultaneously, through i = L and i = L only after the flux of the P attains itsconstant value in the non-equilibrium steady-state of theTASEP. Once the R particles enter the segment 2 andstart encountering the P particles, the rate of replicationbegins to get affected adversely.We first present the derivation of the analytical resultsbefore comparing with the corresponding data obtainedfrom MC simulation. Suppose n denotes the number ofparticles in the segment 2 of the lattice. In the limit n (cid:29)
1, the effects of a single R particle on the flow ofthe P particles is expected to be negligibly small so thatthe movement of the P particles can be approximatedwell by a purely single-species TASEP in the segment 2.Under this assumption, the flux J P of the P particlescorresponding to the number density ρ inside segment 2is given by the standard formula (see, for example, [54]) J P = qρ (1 − ρ ) . (1)Since, because of the open boundaries, n fluctuates withtime even in the steady state, the number density ρ = n/ ( L − L ), also fluctuates with time. The effectivevelocity of the P particles corresponding to the flux (1)in segment 2 is given by, v P = J P ρ = q (1 − ρ ) . (2)First we explore the parameter regime where β is solarge that at sufficiently low values of α the P particleswould be in the low-density (LD) phase of the TASEP(in the “initiation”-limited regime in the terminology oftranscription). With the increase of α the system wouldmake a transition to the maximal current (MC) phaseof TASEP (“elongation-limited” regime of transcription).For the analytical derivation, we assume the followingsimplified situations:(a) None of the R particles collapse (i.e. C = 0) uponencounter with P particles,(b) None of the R particles can detach from the latticeprematurely (i.e. D = 0),(c) In the absence of any hindrance, the rate of replicationby both the forks are identical, i.e., the symmetric case: B = B = B .Since the time intervals between the entry of the P particles at i = L is quite long, the number of P particlesencountered co-directionally by R (cid:96) and that head-on by R r would be almost identical under the conditions (a)-(c) above, the most-probable location for the head-onmeet of the two oppositely moving R particles is expectedto be the midpoint of the segment 2 (i.e. at i, i + 1 ≈ L/ , L/ ± ≈ L/ α q q → α , β q q → β. (3)as the rescaled initiation and termination rates, respec-tively, of a P particle. Using the well known results forthe flux and density profile of TASEP under open bound-ary conditions [55, 56], we get expressions for J and v , in all three possible phases: In the low density (LD)phase, J P = qα (1 − α ) , v P = q (1 − α ) , (4)in the high density (HD) phase, J P = qβ (1 − β ) , v P = q (1 − β ) , (5)and in the maximal current (MC) phase, J P = q , v P = q . (6)Next, we define the effective velocity of a R particle insidesegment 2. For R (cid:96) v R (cid:96) = (cid:40) B (1 − ρ ) if no P particle in front p co (1 − ρ ) if P particle is in front (7)whereas for R r v R r = (cid:40) B (1 − ρ ) if no P particle in front p contra (1 − ρ ) if P particle is in front (8)Therefore, the relative velocities v r with which a R particle approaches a leading P particle, are v R (cid:96) − v P and v R r + v P for co-directional and contra-directional en-counter, respectively. The average separation d betweenthe P particles, i.e., distance headway between the suc-cessive particles, in the segment 2 is d = 1 ρ . (9)From expressions (2), (7) and (8), the average inter-action times τ co and τ contra , between a R particle and a P particle, during co-directional and contra-directionalencounter, are given by, τ co = dv R (cid:96) − v P = 12 ρ (cid:34) B (1 − ρ ) − q (1 − ρ ) + 1 p co (1 − ρ ) − q (1 − ρ ) (cid:35) , (10)and τ contra = dv R r + v P = 12 ρ (cid:34) B (1 − ρ ) + q (1 − ρ ) + 1 p co (1 − ρ ) + q (1 − ρ ) (cid:35) . (11)where we have arrived at the expression for τ co assumingit to be an average of the contributions from the twosituations mentioned in (7). Similarly the expression for τ contra is also the average of the two contributions fromthe alternative cases mentioned in (8).Next, we define N co and N contra , as the total numberof encounters that a R particle can suffer inside the seg-ment 2. We derive approximate expressions for N co and N contra . When the conditions (a)-(d) are satisfied, N co and N contra are given by the expressions N co = N contra ≈ ρL . (12)where the factor L/ R particles has to traverse a distance of L/ R particleinside segment 2, i.e. τ int , as a product of total numberof interactions and average encounter time. In the steadystate, τ int is given by τ int = N co τ co + N contra τ contra = ρL τ co + τ contra ) . (13)Further, we calculate the total replication time τ as asummation of replication times inside segment 1 and 3and replication time τ int inside segment 2, τ = τ int L B = ρL τ co + τ contra ) + L B , (14)where the extra factor of 1 / τ on α q arisesin (14) from the use of the result ρ ( α ) = α for the LDphase of P particles where the relation between α and α q is given by (3).In Fig. 3, we show the variation of replication time τ with the rate of transcription initiation (i.e., entry rate α q of the P particles), for a constant transcription termina-tion rate (exit rate of P particles) β q . With the increaseof α q , τ increases, and eventually saturates, above a criti-cal value of α q . This behavior is qualitatively reproducedby the heuristic analytical arguments. The latter, how-ever, tends to slightly overestimate the mean replicationtime.The steady state density profiles of the P particle areplotted in the inset of Fig. 3 for few different values of α q .The trend of variation of the profiles with α q is consistentwith the transition from the LD phase to MC phase ofthe TASEP of the P particles. For all those values of α q ,for which system is in LD phase, particle density ρ in-creases with increase of α q . Therefore, the total numberof encounters that a R particle can have inside segment2 also increases, which results the increase in τ . Above acritical value of α q , the TASEP in the segment 2 makesa transition to the MC phase where the number density ρ of the P particles and, hence, τ , becomes independentof α q .In Fig. 3 we have plotted τ against α q for two distinctcases. In the first p co = p contra = 30 s − (i.e., the rates of ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲ ▲ ▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲ p contra < p co p contra = p co p contra < p co ▲ p contra = p co α q = α q = α q = ����������� �������� ρ ��� � �� �������������� α � ( � - � ) τ ( � ) FIG. 3. Variation in average hopping time ( τ ) is plottedwith α q , for two different sets of values of p co and p contra .In first case, p co = p contra = 30 s − and in second case p co = 30 s − , p contra = 20 s − . Our theoretical predictions,based on heuristic analytical arguments, are drawn by con-tinuous curves and numerical data obtained from our MC-simulations are shown by discrete points. In inset, we plotthe density profile of P particles along the lattice, for threedifferent values of α q . The density profile data have been ob-tained only from MC simulations. The other relevant modelparameters are β q = 1000 s − , q = 30 s − , C = D = 0 s − . passing is same irrespective of the direction of encounter).But in the second case rates of passing are asymmetric,i.e., p contra < p co , where p co = 30 s − and p contra = 20s − . The lower value of τ in the latter case shows thateven if one of the passing rates decreases, it leads to alowering of the time needed for completion of replicationbecause a R particle has to pause for longer duration.Next, based on similar heuristic mean-field-type argu-ments, we derive analytical expressions for the averagereplication time in the opposite limit where α is suffi-ciently high. In this parameter regime, at sufficientlysmall values of β , the system is in the high density(HD) phase of TASEP (“termination”-limited regime oftranscription), but makes a transition to the MC phasewith the increase of β . For the analytical arguments,we assume the same special scenario (a)-(c) above, i.e., C = 0 = D and B = B = B .In this case, because of the high value of α the parti-cle R r is expected to suffer large number of encounterswith P particles all of which approach it head-on. Evenif it succeeds entering the segment 2 through i = L andmove ahead at a slow pace by passing oncoming P parti-cles, new P particles continue to make fresh entries intothis segment through i = L . Thus, the number of parti-cles to be bypassed by R r keep increasing as time passestill R r exits the segment 2 through i = L . In contrast,the particle R (cid:96) encounters far fewer P particles because,after it enters the segment 2, the new entrant P parti-cles would be falling behind it and even some of thosein front would make their exit from i = L before R (cid:96) catches up co-directionally from behind. Therefore, wemake the simplifying assumption (perhaps, slight over-simplification) that the particle R r remains stalled at i = L + 1 and replication is completed only when R (cid:96) reached i = L .The average number of P particles within the segmentfrom L to L is ( L − L ) ρ where ρ is the average numberdensity of the R particles in this segment. The averagespatial gap between the P particles, as given by eq. (9),is 1 /ρ and the number of gaps to be covered by a R particle is ( L − L ) ρ . Therefore, the total time spentby the R particle in exchanging its position with the co-directionally moving P particles is τ exch = ( L − L ) ρ/p co . (15)The effective velocity of a R particle in the segment be-tween L and L is B − q . The total time spent by the R particle in covering all the gaps by forward hopping is τ hop = ( L − L ) ρ ρ B − q ) = ( L − L ) / ( B − q ) . (16)The time taken by the R particle to reach L from i = 1is τ arr = L /B. (17)Thus, finally, the total time taken to complete replicationis τ = ( L − L ) ρp co + ( L − L )( B − q ) + L B . (18) β q = β q = β q = ������������ �������� ρ ��� � �� �������������� β � ( � - � ) τ ( � ) FIG. 4. The replication time τ is plotted as a function of β q . Our theoretical predictions, based on heuristic analyticalarguments, are drawn by continuous curves and numericaldata obtained from our MC-simulations are shown by discretepoints. In inset, we plot the density profile of P particlesalong the lattice, for three different values of β q . The densityprofile data have been obtained only from MC simulations.The other relevant model parameters are α q = 100 s − , q =30 s − , p co = p contra = 30 s − . In Fig. 4, we show the variation of replication time τ with the rate of transcription termination (i.e., exit rate β q of the P particles), for a constant transcription ini-tiation rate (entry rate of P particles) α q . Note that the dependence of τ on α q arises in (18) from the useof the result ρ ( β ) = 1 − β for the HD phase of P par-ticles where the relation between β and β q is given by(3). With the increase of β q , τ decreases, and eventuallysaturates, above a critical value of β q . This behavior isqualitatively reproduced by the heuristic analytical argu-ments which slightly overestimate the mean replicationtime. The steady state density profiles of the P particleare plotted in the inset of Fig. 4 for few different valuesof β q . The trend of variation of the profiles with β q isconsistent with the transition from the HD phase to MCphase of the TASEP of the P particles. For all thosevalues of β q , for which system is in HD phase, particledensity ρ decreases with increase of β q . Therefore, thetotal number of encounters that a R particle can have in-side segment 2 also decreases, which results the decreasein τ . Above a critical value of β q , the TASEP in thesegment 2 makes a transition to the MC phase wherethe number density ρ of the P particles and, hence, τ ,becomes independent of β q .In the absence of collapse of the replication fork ( C =0) and premature detachment of RNAP ( D = 0), thereplication time is essentially decided by the density ofthe RNAP motors (i.e., R particles). Since one or two R particles make hardly any noticeable perturbation of thedensity that is prescribed by the exact theory for a pureTASEP of P particles, the expressions (14) and (18) forthe replication time τ are in excellent agreement with thecorresponding data obtained from MC simulation of themodel. B. Histograms of the Number of Successful andUnsuccessful Replication Events
In this subsection we show the effects of transcriptionon replication. For this purpose, we calculate how thedistributions of the three processes, namely, sr1, sr2 andusr are affected by the encounter of R particles with the P particles. • Special case of C (cid:54) = 0 and D = 0Nonzero C gives rise to two other alternative scenarios.If only one of the R particles collapses and the otherdoes not replication is completed via the alternative routethat we defined as sr2. Similarly, collapse of both the R particles leads to nonzero probability of usr. Notethat any increase in the probabilities of sr2 or usr, orboth, cause reduction in the probability of sr1 because P sr1 + P sr2 + P usr = 1.Suppose, on the average, the total number of P parti-cles in the segment between i = L and i = L is N . If N events of passing in the segment between i = L and i = L is required for completion of replication withoutsuffering collapse of either of the two replication forks, (a) . . . . . P r ob a b ilit y o f e v e n t s α q (s -1 ) P sr1 P sr2 P usr (b) . . . . . P r ob a b ilit y o f e v e n t s β q (s -1 ) P sr1 P sr2 P usr FIG. 5. Distribution of sr1, sr2 and usr in the special case C (cid:54) =0, D = 0 is plotted for five different values of (a) α q , for fixed β q = 1000 s − and (b) β q , for fixed α q = 1000 s − . The dataused for the bar plots have been obtained by MC-simulations.Lines have been obtained from the analytical expressions (19),(20) and (21). Dotted line corresponds to P sr1 , dashed linecorresponds to P sr2 and continuous line corresponds to P usr .The other relevant parameters used in this figure are L =2000, q = 30 s − , C = 0 . s − , D = 0 and p co = p contra = p = 20 s − . then the probability of sr1 is P sr1 = (cid:18) pp + C (cid:19) N (19)For sr2, one of the forks ( R particles) has to collapse whilethe remaining stretch of the segment between i = L to i = L is covered by the surviving fork. One of theforks may collapse after passing n number of P particles;the probability of its occurrence is [ C/ ( p + C )][ p/ ( p + C )] n ; the probabilty that the surviving fork passes theother remaining P particles is [ p/ ( p + C )] N − n . Thus, the (a) . . . . . P r ob a b ilit y o f e v e n t s α q (s -1 ) P sr1 P sr2 P usr N α q (s -1 ) (b) . . . . . P r ob a b ilit y o f e v e n t s β q (s -1 ) P sr1 P sr2 P usr
100 200 300 400 0.1 1 10 100 N β q (s -1 ) FIG. 6. Distribution of sr1, sr2 and usr in the general case C (cid:54) = 0, D (cid:54) = 0 is plotted for five different values of (a) α q ,for fixed β q = 1000 s − and (b) β q , for fixed α q = 1000 s − .In inset we plot the variation in N (i.e. average number of P particle detached from the track during their encounterwith R particles) with (a) α q and (b) β q . These data havebeen obtained only from MC-simulations. The other relevantparameters used in this figure are L = 2000, q = 30 s − , C = 0 . s − , D = 10 s − and p co = p contra = p = 20 s − . probability of sr2 is P sr2 = N − (cid:88) n =0 (cid:20)(cid:18) Cp + C (cid:19)(cid:18) pp + C (cid:19) n (cid:21)(cid:18) pp + C (cid:19) N − n = N (cid:18) Cp + C (cid:19)(cid:18) pp + C (cid:19) N (20)Exploiting normalization, we get the probability for usr P usr = 1 − P sr1 − P sr2 . (21)Note that N = ρ ( L − L ) is the average number of P particles in the interaction segment between i = L and i = L . In the LD regime of P particles ρ = α = α q /q .In Fig. 5, we plot the distributions of ’sr1’, ’sr2’ and‘usr’ as histograms for (a) five distinct values of α q anda constant value of β q , and (b) five distinct values of β q and a constant value of α q . The analytic approximations(19)–(21) reproduce qualitatively the behavior observedin the MC simulations. For a given sufficiently high valueof β q , segment 2 is in the LD phase at small values of α q .In this regime the number of eventual collapse of a R par-ticle during its encounters with P particles is negligiblysmall. Therefore, for these small values of α q , number ofevents of the type ’sr2’ and ‘usr’ are low and, hence theprobability of sr1 is very weakly affected (see Fig. 5(a)).As α q increases further, the number of eventual collapseincreases because of the increasing number of encoun-ters with P particles which is reflected in the significantincrease in ’sr2’ and ‘usr’ in Fig. 5(a). Increase in theprobabilities of sr2 and usr results in the correspondingdecrease in the probability of sr1 because of the normal-ization of the probabilities mentioned above. Numberof the events ’sr1’, ’sr2’ and ‘usr’ attain their respectivesaturation values as α q increases above the critical valuewhere the transition from LD phase to MC phase takesplace (see Fig. 5(a)).Similarly, for a sufficiently high value of α q , withincreasing β q the P particles exhibit a transition fromthe HD phase to the MC phase. Consequently, thedecrease in the frequency of encounter of the P particleswith the R particles. The likelihood of collapse of boththe R particles in any run decreases as indicated bythe increase of the probability of sr2. The concomitantincrease of the probabiity of sr is also shown in Fig. 5(b). • Special case of C (cid:54) = 0 and D (cid:54) = 0Now, we consider the general case of our model allow-ing for the possibilities that C (cid:54) = 0 and D (cid:54) = 0. As wehave already done in the restricted case of C (cid:54) = 0, D = 0,we characterize the effect of nonzero C and D also interms of the statistics of ’sr1’, ’sr2’ and ’usr’.In Fig. 6(a) and (b), we plot the distributions of theevents ’sr1’, ’sr2’ and ’usr’ as histograms for (a) five dif-ferent values of α q at a constant high value of β q and(b) five different values of β q at a constant high value of α q . The trends of variation of these three probabilitiesare explained by the transition to the MC phase from (a)LD phase and (b) HD phase.In the inset of Fig. 6, we display the effects of repli-cation on transcription. We show the variation in thenumber N of P particles that detach from the latticewhen they encounter a R particle, for a given rate α q .The trend of variation and the physical reason for thistrend is also well explained by the transition from LDphase to MC phase. C. Distribution of Detachments of P Particles
In MC simulations, we measure the time intervalsbetween two consecutive P detachment events as δt , δt . . . δt n , if n +1 detachments takes place in a single MCsimulation run. Since this is a stochastic process thesetime intervals δt , δt . . . δt n are, in general, differentfrom each other. We compute the number of consecutive P detachment events corresponding to a given interval δt , i.e. if two time intervals are identical ( δt i = δt j = δt ),then, number of consecutive P detachment events withtime interval δt is 2. We repeat the procedure over 10000MC simulation runs to calculate total number of consec-utive P detachment events with given interval δt , andthen we divide this number with number of MC simula-tion runs, i.e. 10000, to calculate the average number N α of consecutive P detachments within the time interval δt for a fixed rate α q . N α δ t (s)-4-2 0 2 0 0.1 0.2 0.3 l n N α δ t (s) FIG. 7. Distribution of N α is plotted with δt for a constant α q = 100 s − . In inset we plot N α with δt on a semi-log axisto show the exponential fall of N α with δt . These data havebeen obtained only by MC-simulations. The other relevantparameters used in this figure are C = 0 . s − , D = 10 s − , p co = 20 s − and p contra = 20 s − . In Fig. 7 we show the variation in N α with δt andwe find that N α falls exponentially as the time intervalbetween two consecutive P detachment events increases.To confirm the exponential behavior, in the inset we showthe variation in N α on a semi-log axis with δt . IV. SUMMARY AND CONCLUSION
In this paper we have developed the first minimalmodel that captures the key kinetic rules for the reso-lution of conflict between transcription and concomitantreplication of the same stretch of DNA. This model hasbeen formulated in terms of a two-species exclusion pro-cess where one species of particles (denoted by P ) rep-resents the RNA polymerase motors while the two mem-bers of the other species (denoted by R ) represent thetwo replication forks.In contrast to all the multi-species exclusion models re-ported so far, the allowed populations of the two speciesare quite different in our model. A maximum of only two R particles are allowed to enter the lattice; impositionof this restriction on the number of R particles is mo-tivated by the fact that none of the segments of DNAshould be replicated more than once during the life timeof a cell. In sharp contrast, the number of P particles isnot restricted except for the control of their populationby the rate constants for their entry, exit and hopping.This choice is consistent with the fact that the multiplerounds of transcription of the same segment of DNA isnot only possible but resulting synthesis of multiple iden-tical transcripts is also desirable for the proper biologicalfunction of the cell. Moreover, all the P particles moveco-directionally, from left to right whereas one of the R particles (namely, R (cid:96) ) moves from left to right while theother R particle (namely R r ) approaches it head-on fromthe opposite end. Another distinct feature of this modelis that the lattice consists of three segments; the encoun-ters of RNAP motors ( P particles) with the replicationfork ( R particles) are confined within the middle segment(segment 2) whereas only the R particles can occupy thesites in the segments 1 and 3.By a combination of analytical treatment, based onheuristic arguments, and Monte Carlo simulations wehave analyzed the effects of the RNA polymerase mo-tor traffic on the DNA replication and vice versa. 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