Pis'ma v ZhETF, vol. 102, iss. 1, pp. 60-66 © 2015 July 10
On the scattering of DNA replication completion times
E. Z. Meilikhov+*1\ R. M. Farzetdinova+
+ National Research Centre "Kurchatov Institute", 123182 Moscow, Russia * Moscow Institute of Physics and Technology, 141707 Dolgoprudny, Russia
Submitted 16 April 2015 Resubmitted 18 May 2015
Stochasticity of Eukaryotes' DNA replication should not lead to large fluctuations of replication times which could result in mitotic catastrophes. Fundamental problem that cells face is how to be ensured that entire genome is replicated on time. We develop analytic approach of calculating DNA replication times, that being simplified and approximate, leads, nevertheless, to results practically coincident with those which have been obtained by some sophisticated methods. In the framework of that model we consider replication times' scattering and discuss the influence of repair stopping on kinetics of DNA replication. Our main explicit formulae for DNA replication time tr <x VlnJV (N is the total number of DNA base pairs) is of general character and explains basic features of DNA replication kinetics.
DOI: 10.7868/S0370274X15130123
The dream of every cell is to become two cells [1]. This dream is realized by the process of cell division, in which a cell duplicates its contents in order to provide sufficient materials for both daughter cells. Before every cell division, DNA replication must be carried out. In Eukaryotes, the replication starts in multiple origins being activated in diverse time and space points during S phase, when DNA replication proceeds [2]. DNA synthesis occurs in two opposite directions on replication forks and terminates when two converging forks meet. Complete genome replication in Eukaryotic cells should come to end in a definite time [3, 4]. Loss of the only origin or stopping a single fork may result in that DNA does not succeed to replicate before mitosis, making inviable daughter cells.
In the course of lengthy genome replication, numerous potential origins arise stochastically [5]. Nevertheless, that stochasticity does not lead to large fluctuations of replication times which could, otherwise, result in frequent mitotic catastrophes. This means that, despite the random character of arising and activating origins, duration of S phase has solid timetable. That is the essence of the so-called random-completion paradox: stochasticity leads to the exponential distribution of interorigin gaps, whereby the probability of large gaps (and, thus, unduly long replication times) is too high [6, 7].
To solve the random completion problem two distinct models have been proposed [4]:
^e-mail: meilikhov@yandex.ru
1) the model of regular spacings [8] with origins being positioned at regular (not random) intervals. The lack of this model is that accidental failure of just one or two out of consecutive origins could be lethal;
2) the origin redundancy model [3, 9] with randomly spaced origins being much more abundant than actual activated replication centers whose number increases as S phase progresses to allow rapid completion of unrepli-cated gaps.
Though experiments [10] favor the second model, much needs to be done for developing quantitative description of the replication process. For this purpose the formal analogy between DNA replication and one-dimensional crystal growth could be employed [11]. In this model, the replication process is defined by two base parameters - the replication fork velocity u (assumed to be constant through S phase), and the time-dependent rate I(t) of origin activation assumed to be spatially homogeneous.
In the framework of that model, a temporal profile I(t) extracted from the data occurred to be growing through S phase [12]. A formal study based on the above-mentioned analogy demonstrates that initiating all origins at the beginning of S phase leads to a broad completion time distribution, whereas a growing dependency I(t) narrows this distribution [12]. Therefore, both experimental and theoretical work support the origin redundancy model with a non-constant I(t) (though the molecular mechanism that underlies the observed changes in I(t) remained unknown).
A fundamental control problem that cells face is how to be ensured that every last part of the genome is replicated on time [3]. To answer this question, ideas of condensed-matter physics have been used in [13], where stochastic Ivolmogorov model [14] has been employed to describe the kinetics of replication. In that solidification model, the kinetics results from three simultaneous processes: (i) nucleation of solid domains, (ii) growth of domains, and (iii) domain coalescence, which occurs when two expanding domains merge. In the simplest form of the model, solid domains nucleate anywhere in the liquid, with equal probability for all locations.
These features of Ivolmogorov model can be adapted to DNA replication: (i) DNA replication starts at multiple activated origins, where replication forks are created, (ii) DNA synthesis propagates bidirectionally from each activated origin, and (iii) DNA synthesis stops when two replication forks meet. For cells to achieve an acceptable distribution of replication completion times, the initiation rate I(t) should increase during replication [13], in agreement with extracted values of I(t) from experimental data.
Though being new and perspective, the analytical study [13] of replication kinetics is rather complicated. Here we present some simple approach which seems to be more transparent and intuitive, and delivers some simple expressions for scattering replication times.
In [15] another problem has been considered - how do defects of replication slow the duplication process and, thereby, result in increasing replication times. However, they could not obtain analytic expressions for the replication kinetics and completion times, and the analysis has been based on the numerical solution of kinetic equations. We will show that Ivolmogorov model (appropriately generalized) allows to obtain some simple analytic relations in that CclSG, clS well.
In the present paper, we analyze replication kinetics of DNA basing on the Ivolmogorov approach [14]. Replication begins around randomly arising special centers (origins) after their activation (firing). The process is analogous to the melt crystallization under decreasing temperature, when solid state nuclei arise randomly in the bulk, grow gradually, and eventually forms a complete solid state due to coalescence.
Ivolmogorov model includes two physical parameters: the time-varying rate I(t) of generating new phase nuclei (in our case, the rate of arising activated centers of replication - origins) and constant velocity u of those nuclei growth (in our case, the movement velocity of the boundary between the replicated and non-replicated DNA parts). Below, for brevity we call non-replicated
DNA part as phase 0, while the replicated one - as the phase 1.
The Ivolmogorov result for the disappearing phase 0
qo(t) = exp
-2uj
(1)
wherefrom it follows that the fraction </i(f) = 1 — q0(t) of the new phase varies by the law
çi (i) = 1 - exp
E
-2u J
(2)
If the rate of generating active replication centers is constant (I(t) = an, we will call such a source as a-source), then
(/i(i) = l-expH2/r02], (3)
where rn = (mo)-1^2-
One of the relevant experimental results is shown in the inset of Fig. 1 where the dynamics of genome repli-
a,
S
i/Vn
l/N 1/2N
'min ^av
Fig. 1. Average (1), slow (2), and fast (3) replication kinetics for non-replicated DNA fraction qo(t). In the inset - experimental dependence qo(t) (points) and the theoretical curve calculated with Eq. (3)
cation in the baker's yeast Saccharomyces cerevisiae is presented [16]. Points are experimental data, the solid curve is calculated one, corresponding to rn ~ 30 min.
Formally, the phase transition finishes at the moment when the fraction qi(t) of new phase turns to be unit. From this viewpoint, the phase transition in the infinite medium is never ending process because the fraction (/o = l— qi of the old phase never becomes equal to unit exactly (see Fig. 1). In the finite system consisting of a large but finite number N 1 of molecules (in
our case, base pairs) the transition (replication) is completed at the moment when the fraction of old phase turns out to be less than 1/N, that corresponds to one non-replicated DNA base pair only (curve 1 in Fig. 1). That condition defines the replication time
roVlnJV.
t*
(4)
However, due to stochasticity of the process that time fluctuates from one cell to another, so Eq. (4) could be considered as definition of some average ("standard") value of time required for the total replication. One could estimate typical scattering of replication completion times from following reasons. Due to fluctuations, relaxation dependencies of the type (3) are not identical for different cells, and near entering q\(t) to the plateau (where 1 — q\ 1) they disperse by the value on the order of mean-square deviation 1/y/N. As the formal reason of that scattering one could consider variations of the effective values of the parameter ro .
As it has been mentioned, the typical replication time is defined by the relation qi(t*) = 1/N, or (cf. (3))
exp(-i*2/t02) = 1/N.
(5)
Another replication curve, displaced due to the fluctuation shift ro —> r, goes beneath (curve 2 in Fig. 1), and for it
exp(-t*2/t2) = 1/VN. (6)
That curve corresponds to later completion of replication at the time moment tmax > t*, defined by the condition
exp(-tmax2/r2) = l/W. (7)
From (5)-(7), it follows
^max
(8)
i.e., upward scattering of replication times reaches At « « imax - t* = (V2 - l)i* « OAt*. That is provided by twofold fluctuation decrease of the parameter ro (ro —> t = to/2), associated with the process stochasticity and possible random variations of the generation rate «o and replication velo
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