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Conceived and designed the experiments: AG HL KM OH. Performed the experiments: AG HL KM. Analyzed the data: AG HL KM OH. Wrote the paper: AG OH.

Current address: Marie Curie Research Institute, The Chart, Oxted, Surrey, United Kingdom

The authors have declared that no competing interests exist.

Eukaryotic cells seem unable to monitor replication completion during normal S phase, yet must ensure a reliable replication completion time. This is an acute problem in early

Using both previous and novel DNA combing data we have confirmed that

This work provides a refined temporal profile of replication initiation rates and a robust, dynamic model that quantitatively explains replication origin usage during early embryonic S phase. These results have significant implications for the organisation of replication origins in higher eukaryotes.

Eukaryotic cells must ensure the complete duplication of their genome in a predictable time

The replication completion problem is particularly crucial in early embryos of the frog ^{−1} ^{5} initiation events can be more than 20 kb from its neighbours (and even less if origins fire late in S phase). However, replication origins lack any sequence specificity in ^{−20/10} = 0.135. Even with a mean spacing of 5 kb the probability would still be 0.018, a figure incompatible with replication completion in <20 min. This paradox is known as the “random completion” or “random gap” problem

The problem might be solved if new origins could be continuously laid down on unreplicated DNA during S phase. However, this idea conflicts with what is known about origin regulation

Two models have been proposed to solve the random completion problem. In the “regular spacing” model ^{5} consecutive origins could be lethal. The “origin redundancy” model

Given the complexity of this problem, the agreement of any model with the data must be assessed by quantitative analysis. The formal analogy between DNA replication and one-dimensional crystal nucleation (initiation), growth (elongation) and coalescence (termination), has allowed a mathematical analysis of the extensive datasets generated by DNA combing

The open circles are the data points and the two dashed lines are linear fits presented in Figure 10 b in

In this article we have generated novel DNA combing data that allowed us to refine the temporal profile of

Our model genome consists of a lattice of ^{6}×100 bp blocs. Each bloc is represented by a 0 (0-block) if unreplicated or a 1 (1-block) if replicated. At the start of the computation, all blocks have a value of 0. Each 0-block is competent to initiate replication (i.e., potential origins are in vast excess). At each round of computation (equivalent to 10 seconds), a fraction of the 0-blocks initiate replication and are converted to 1-blocks. Replication forks are defined by the boundary between a 0- and a 1-block. At each round of computation, forks move by one block. Replication fork velocity (^{−1} (nt.s^{−1}). Converging forks stop when they merge. Blocks with value 1 cannot rereplicate. Replication is finished when all blocks have a value of 1.

We envision here a simple model of initiation governed by the encounter of a particle with a 0-block to convert it to a 1-block with probability _{T}(t)_{F}(t)_{B}(t)

We assume that the chromosome is initially surrounded by _{0}_{u}(t)*P(t)_{u}(t)_{t}_{t}_{u}(t)*P(t)_{u}(t)*P(t)_{F}(t)_{t}_{F}(t)_{t}/L_{u}(t)_{F}(t)

In the above model, the total number of particles _{T}(t)_{T}

In stationary scenarios, _{T}_{T}_{T}^{4}. If ^{−3} kb^{−1} s^{−1}), then _{T}

Open circles are numerical simulation data points. (A) Particle recycling scenario: _{T}^{4};. ^{−3} kb^{−1} s^{−1}. (B) Particle abundance scenario: _{T}^{5}; ^{−4} kb^{−1} s^{−1}. Blue and red curves represent the simulated replicated fraction and the fork density, respectively.

In the “abundance” scenario, _{T}_{T}^{5} and ^{−4} kb^{−1} s^{−1}. For most of the time _{T}

In this scenario, _{T}_{T}_{0}_{0}_{0}^{−1} and ^{−3} kb^{−1} s^{−1}, a good fit to the increasing part of the data is obtained (^{2}^{−8}, calculated using a statistical weighting of each data point as described ^{2}^{−8}) or with the previously proposed quadratic function _{2}t^{2}_{2}^{−10} kb^{−1} s^{−3}; ^{2}^{−7}) _{T}_{F}(t)

The solid black line is the best fit to the increasing part of the data using a Levenberg-Marquardt algorithm coupled with a dynamic Monte Carlo method (_{0}^{−1} and ^{−4} kb^{−1} s^{−1}; ^{2}^{−8}). Blue and red curves represent the simulated replicated fraction and the fork density, respectively.

In this scenario, _{T}_{0}_{0}_{0}^{−3}+10^{−4}_{T}^{4} (^{−4}+10^{−5}_{T}^{5} (

Open circles are numerical simulation data points. (A) Limiting particles scenario: ^{−3}+10^{−4}_{T}^{4}. (B) Abundant particles scenario: ^{−4}+10^{−5}_{T}^{5}. Blue and red curves represent the simulated replicated fraction and the fork density, respectively.

In the previous scenario, the chosen variation of _{B}(t)_{B}(t)_{B}(t))_{0}_{1}_{B}(t)/N_{c}_{min}_{0}_{B}_{max}_{0}_{1}_{C}_{C}^{3}) _{0}^{−3} kb^{−1} s^{−1}, _{1}^{−3} kb^{−1} s^{−1}, _{T}^{4} and _{C}^{3} (

Open circles are numerical simulation data points. (A) Limiting particles scenario: _{0}^{−3} kb^{−1} s^{−1}, _{1}^{−3} kb^{−1} s^{−1}, _{T}^{4} and _{C}^{3}. (B) Abundant particles scenario: _{0}^{−4} kb^{−1} s^{−1}, _{1}^{−3} kb^{−1} s^{−1}, _{T}^{5} and _{C}^{3}. Blue and red curves represent the simulated replicated fraction and the fork density, respectively.

Let us now explore the case where particles are abundant: _{0}^{−4} kb^{−1} s^{−1}, _{1}^{−3} kb^{−1} s^{−1}, _{T}^{5} and _{C}^{3} (

The increasing particle availability scenario on the one hand and the fork-dependent affinity scenarios on the other hand, have complementary strengths and weaknesses. The former better explains the increasing part of the data, the latter the decreasing part. In this section, we explore whether combining them can improve the fit to the data. This was indeed the case. The solid line on _{C}_{C}^{3}) and adjusting the four free parameters (_{0}^{−4} kb^{−1} s^{−1}, _{1}^{−3} kb^{−1} s^{−1}, ^{−1} and _{0}^{2}^{−9}). The fit shows an excellent agreement with both the increasing and decreasing part of the data. One might wonder whether the decrease in

The solid black line is the best fit to the increasing part of the data using a Levenberg-Marquardt algorithm coupled with a dynamic Monte Carlo method (_{0}^{−4} kb^{−1} s^{−1}, _{1}^{−3} kb^{−1} s^{−1}, ^{−1}and _{0}^{2}^{−9}). Blue and red curves represent the simulated replicated fraction and the fork density, respectively.

We note that the model makes two predictions concerning the end of S phase. First, the previously neglected decreasing part of the data is biologically significant. Second, the very end of S phase is marked by a lack of initiations.

To check these predictions and the robustness of the model, we analyzed a novel set of experimental data from which we directly extracted

(A) Comparison of the novel data points (open circles, see text) with a rescaled fit to the previous data set (

_{B}(t)_{B}(t)_{B}(t)_{F}(t)_{B}(t)_{0}^{−4} kb^{−1} s^{−1}, _{1}^{−3} kb^{−1} s^{−1}, ^{−1} and _{0}_{C}^{3}) to the experimental data without further fit. A good agreement of the model with the experimental results is observed. Overall, these results provide good evidence for the robustness of the model.

(A) Comparison of the simulated fork density profile (solid red curve) and simulated

In this paper we provide a refined determination and analysis of the time-dependent frequency of replication initiation

The bimolecular interaction of a trans-acting factor (particle) with an origin gives rise to initiation with a probability _{T}_{0}_{f}

One prediction of this model is that a limiting component of replication forks accumulates during S phase. Biochemical experiments in _{F}(t)≪L_{u}(t)*P(t)

A second prediction of the model is that the rate of origin firing is tied to the density of replication forks. It has been previously proposed that _{B}(t)

Our model does not explain the occurence of synchronous origin clusters

Another limitation of our model is that it postulates a homogeneous intranuclear concentration of a limiting “particle” whereas the intranuclear concentration of many replication factors shows inhomogeneities known as replication foci

A further limitation of this study is that all the models explored are based on recycling of a replication-fork factor. It is certainly conceivable that other mechanisms independent of recycling could fit the data equally well. However, the simplicity of our model, and the robustness of the fit to the data leads us to hope that the model captures some of the logic of S phase. The model makes sense regarding the necessarily limited amount of resources required to assemble replication forks and the desired autonomy and robustness of the mechanisms that control orderly progression through S phase and timely completion of DNA replication. The general nature of the proposed model pushes us to explore, in a future work, whether these findings can be generalized to other eukaryotes. It will also be interesting to see whether this model predicts the pattern of origin firing following perturbation of replication fork progression and can be modified to incorporate the effects of checkpoint mechanisms influencing origin firing.

All simulations were performed using a dynamic Monte-Carlo method using a Matlab platform. The source code can be obtained upon request.

The data shown on

This work and previous modelling of DNA combing data

We thank John Bechhoefer for providing the