Deciphering DNA replication dynamics in eukaryotic cell populations in relation with their averaged chromatin conformations

We propose a non-local model of DNA replication that takes into account the observed uncertainty on the position and time of replication initiation in eukaryote cell populations. By picturing replication initiation as a two-state system and considering all possible transition configurations, and by taking into account the chromatin’s fractal dimension, we derive an analytical expression for the rate of replication initiation. This model predicts with no free parameter the temporal profiles of initiation rate, replication fork density and fraction of replicated DNA, in quantitative agreement with corresponding experimental data from both S. cerevisiae and human cells and provides a quantitative estimate of initiation site redundancy. This study shows that, to a large extent, the program that regulates the dynamics of eukaryotic DNA replication is a collective phenomenon that emerges from the stochastic nature of replication origins initiation.

scattering. Therefore, it is difficult to assign the observed scattered intensity to a particular scattering center. One way to overcome this difficulty is to assume that all existing scattering centers participate in the scattering of the incoming beam, hence the scattering effect of the medium is not anymore localized on a single scatterer but is now delocalized over the whole medium 1 .
In the same manner, consider a cell population. The set of potential replication origins (position and time of firing) are not identical from one cell to another. Therefore, similar to the scattered intensity of the incoming beam, the observation of firing of n replication origins at a time t does not represent the firing of the same potential replication origins in all cells but corresponds to the firing of a subset of replication origins present in the cell population. In that sense, one cannot assign the firing probability to a particular potential origin (localized) but one must delocalize it over a subset of potential origins.
As an example, let consider a scalar wave (a firing process in the case of replication), we aim at calculating the scattered intensity (observed number of fired origins) at a position x (at an instant t) and by averaging over the whole volume to get the average intensity scattered by the medium (whole genome averaged firing probability). Suppose we have strong scatterers (fired replication origins), in the limit of point scatterers, the scalar wave evolves in a potential V (r) = where −u is the bare scattering strength (intrinsic firing strength) and X i are the positions of the scatterers randomly distributed in the sample (positions of potential origins). In the bulk of the sample, far from boundaries, one can only detect the intensity of diffuse-scattered beam whose amplitude depends on the realization of disorder (location and orientation of scatterers). It is impossible to calculate in an accurate manner all scattering interactions in the sample.
However, under some approximations, one can reduce the number of interactions to consider, and calculate in a satisfactory manner the intensity of the scattered beam. To define these approximations in relation with replication process, we explicitely take into account the following experimental evidences: i) during replication process an origin cannot fire more than once during a single S phase 2 , and ii) the number of fired origins is smaller than the number of potential origins 3 . The first observation can be interpreted as the fact that a scatterer is only visited once (first order Born approximation 4 ). The second observation can be considered as the fact that the density of strong scatterers is small and leads to the hypothesis that the intrinsic scattering strength of a scatterer can be replaced by an effective (screened) scattering strength that takes into account all possible scattering paths between scatterers (Ladder approximation 5 ). Under these hypotheses, the diffuse intensity at a point x can be written as: where n is the density of scatters, σ sc is the scattering cross-section, G (x − x ) is the dressed Green's function and ψ in (x) is the incoming wave.
The Green's function of the random medium G (x − x ) (also called the dressed propagator), can be obtained by solving the Dyson equation 6 where G 0 is the bare propagator, that is to say, the propagator in the medium without scatterer, and Σ is the self-energy describing the renormalization of single scattering center due to the interaction with the surrounding many-scatterers system.
Having introduced all necessary entities to the calculation of a diffuse intensity, let us now discuss the analogy with DNA replication. As mentioned earlier, we are dealing with DNA replication process in a cell population. Eq. (Sup.1) now represents the total number I (t) of fired origins at instant t in a cell and per genome length. As replication process is independent from one cell to an other, we assume that |ψ (t) where M is the number of cells and δ (t) represents the fact that all cells in the volume are in the S phase. As there is not really an extra-cellular incoming signal that induces the replication process and that could be quantified, we set the scattering crosssection in Eq. (Sup.1) to σ sc = 1. Under these assumptions, Eq. (Sup.1), reduces to from an inactive to an active state is equal to 1. By analogy, this situation corresponds to a medium without scatterer, hence G 0 = 1. Therefore, following Dyson's Eq. (Sup.2), the amplitude of the dressed Green's function is |G (t)| 2 = 1 + Σ + Σ 2 + · · · , where by analogy Σ represents the ability of a potential origin to fire. As previously discussed, the dressed Green's function is defined over the total number of states in the system. Along our definition of G 0 , O total corresponds to the number of strong firing replication origins and the other (m 0 − O total ) replication origins represent the medium. Therefore, a potential origin is either a strong firing replication origin or a replication origin from the medium. As the firing ability ψ of a potential origin is the same for a strong firing origin and an origin from the medium, we express the self-energy as where the factor 1 2 appears because in our analogy each firing event has been counted twice. In order to take into account all possible firing configurations, we expand the upper limit of the summation to infinity and approximate ρ to To derive explicitly Eq. (3), we first differentiate Eq. (Sup.5): m 0 represents the firing probability of an isolated origin, we use Eq. (1) to derive the following evolution equation: where k (t) = m 0 k (t). To get a more compact form for the evolution equation of origin firing probability, we use the following change of variable φ (t) = 1 , where a = 1 C .
3 Is it necessary to consider the origin firing process as delocalised?
Historically the KJMA theory was developed by assuming that origins fire independently of each other, and that they only interact through traveling replication forks (passive replication) 7 . This theoretical framework was later modified to incorporate firing correlation among replication origins 8 ; however, this a posteriori modification required an a priori knowledge of the nature and the range of correlations. As in this methodology the firing probability could be assigned to a single replication origin, it could be considered as a localized theory or a state theory (as was first mentioned by Kolmogrove himself 9 ). The advantage of such a model is that after having defined in an explicit manner the correlation pattern among origins, one could extract from experimental data the local firing probability 10  Using, as initial conditions, the fact that the probability of origin firing at the start of S phase is ψ (t = 0) = ρ (t = 0) = 0, we solve analytically Eqs. (Sup.7) and (Sup.8) and find that: (Sup. 10) Note that by using the expression of ψ (t) (Eq. (Sup.9)) in Eq. (Sup.5), we obtain after some elementary algebra the same expression of ρ (t) (Eq. (Sup.10)) as the one obtained by solving Eq. (Sup.8). As measured experimentally, at the end of S phase (t = t end ) all O total origins have fired. Therefore, the probability of origin firing at the end of S phase is ψ (t = t end ) = Thus at t = t end , Eqs. (Sup.9) and (Sup.10) respectively reduce to: (Sup. 12) By remembering that t end , d f and d w are measurable and therefore finite quantities, a close inspection of expressions (Sup.11) and (Sup.12) shows that while the value of k 0 is finite in Eq. (Sup.12), this is not the case for Eq. (Sup.11) where k 0 should be equal to infinity, meaning that the firing of replication origins is highly efficient which is contradictory to experimental observations 11 . This is the demonstration that expression (Sup.9) for the probability of origin firing does not match with experimental observations, and more generally that the evolution Eq. (Sup.7) does not describe correctly the process of origin firing. Therefore, along the line defined in this work, to describe correctly the replication origin firing process one needs to delocalize this process over the whole potential replication origins distributed along the genome.
As the physical nature of a phenomenon is independent of the picture that is used to describe it, the results obtained here should also be valid for any picture that attempts to describe the replication process. Indeed, as discussed in previous sections, in the simplest form of KJMA model, an initiation event neither impedes nor favors origin initiation at another locus (localization of initiation). However, despite this hypothesis, Baker et al 10,12 have shown that the propagation of replication forks from fired replication origins creates an apparent correlation between firing time and efficiency of two distant fired replication origins due to passive replication. In that sense, in the KJMA model, the propagation of replication forks extends the effect of origin initiation at a particular locus to other distant loci thereby giving to this model a non-localized character. But, in contrast to our non-local modeling of DNA replication based on some analogy with scattering in inhomogeneous media, the KJMA model is based on a state theory where to understand the structure of local (I (x, t)) and/or whole genome averaged (I (t)) initiation function, it is necessary to assume a particular mechanism for the existence of correlations between the firing of replication origins 8 .
Along the same lines of the approach we have used here to study the replication process, Gauthier and Bechhoefer 13 managed to reproduce the genome averaged rate of initiation I (t) in early Xenopus embryos by assuming (i) that during the initial stage of S phase the firing process is reaction limited, while at the end of S phase it is diffusion limited and (ii) because of the fractal nature of the chromatin, the initiator factor undergoes a sub-diffusive dynamics. Indeed, the fact that the authors have assumed that the search process is influenced by the geometry of the chromatin amounts to introduce a memory in the dynamic of firing process. Thus the firing process at a particular locus will necessary depend on the other sites visited by a particular initiator factor consistent with a delocalized picture of firing process 14,15 .

How sensitive is the model to the variation of parameter values?
To address this question we calculated Facs, f DN A (t)), I (t) and N f (t) profiles for both S.cerevisiae