How Xenopus laevis embryos replicate reliably: Investigating the random-completion problem

DNA synthesis in Xenopus frog embryos initiates stochastically in time at many sites (origins) along the chromosome. Stochastic initiation implies fluctuations in the time to complete and may lead to cell death if replication takes longer than the cell cycle time $(\approx 25\min )$. Surprisingly, although the typical replication time is about $20\min$, in vivo experiments show that replication fails to complete only about 1 in 300 times. How is replication timing accurately controlled despite the stochasticity? Biologists have proposed two solutions to this “random-completion problem.” The first solution uses randomly located origins but increases their rate of initiation as $S$ phase proceeds, while the second uses regularly spaced origins. In this paper, we investigate the random-completion problem using a type of model first developed to describe the kinetics of first-order phase transitions. Using methods from the field of extreme-value statistics, we derive the distribution of replication-completion times for a finite genome. We then argue that the biologists’ first solution to the problem is not only consistent with experiment but also nearly optimizes the use of replicative proteins. We also show that spatial regularity in origin placement does not alter significantly the distribution of replication times and, thus, is not needed for the control of replication timing.

Figure 1

Schematic of the DNA replication model. A horizontal slice in the figure represents the state of the genome at a fixed time. The lighter (darker) gray represents unreplicated (replicated) regions. Open circles denote initiated origins, while filled circles denote coalescences. The dark dotted line cuts across the last coalescence, which marks the completion of replication. The slope of the lines connecting the adjacent open and filled circles gives the inverse of the fork velocity.

Figure 2

(Color online) (a) End-time distribution with fixed mode $t^{*}=38\min$. Markers are the results of the Monte Carlo simulations. Each distribution is estimated from 3000 end times. The “$\delta$ function” corresponds to initiating all potential origins simultaneously at $t=0\min$. The $n=0,1,2$ cases correspond to constant, linearly increasing, and quadratically increasing initiation rates, respectively. Solid lines are Gumbel distributions with $t^{*}$ and $\beta$ calculated according to Eqs. (7, 8). There are no fit parameters. (b) Initiation distribution ${\varphi }_i\left(t\right)$ for $n=0,1,2$. Parameter values correspond to those in (a). Error bars are smaller than marker size. Solid lines are calculated from Eq. (3). Again, there are no fit parameters.

Figure 3

(a) Schematic diagram of the simple model described in the text. Open circles represent normal proliferating cells, while filled circles are abnormal cells. The numbers indicate the round of cleavage. Once a cell fails to replicate properly, all its progeny will be abnormal. (b) Cumulative distribution of the number of dead cells at gastrulation (between cleavages 10 and 11) generated using Monte Carlo simulation. The distribution satisfies the constraint that 33% of the embryos have five or fewer abnormal cells. The inset shows the convergence of the gradient search to $F=(3.73\pm 0.01)\times {10}^{-3}$. The average and standard deviation of the mean are computed over the last 40 values.

Figure 4

Density of simultaneously active replication forks throughout $S$ phase, $n_f\left(t\right)$. The dotted curve corresponds to the in vitro fork usage while the solid curve is the rescaled fork usage that satisfies the constraints $t^{**}=25\min$ and $F=0.00373$. The rescaled $n_f\left(t\right)$ is generated using $I_{\mathit{\text{vivo}}}(t∕t_{\mathit{\text{vivo}}}^{*})\sim 2I_{\mathit{\text{vitro}}}(t∕t_{\mathit{\text{vitro}}}^{*})$ and $v=1.030\mathrm{kb}∕\min$.

Figure 5

(Color online) (a) Replication end-time distribution with $t^{**}$ fixed to be $25\min$ and $F=0.00373$. Similar to Fig. 2a, the width decreases with an increase in the exponent $n$. (b) Typical maximum number of simultaneously active forks. The curve is obtained by extracting the maximum value of $n_f\left(t\right)$ for different exponents $n$.

Figure 6

(Color online) Results of a numerical search for optimal initiation functions under various constraints. The label “vivo” corresponds to the in vivo case; “optimal” corresponds to optimization of the maximum fork density with no constraint [Eq. (17)]; “$F_{\mathit{\text{vivo}}}$” corresponds to optimization with the constraint that the failure rate be equal the $F_{\mathit{\text{vivo}}}$ extracted in Sec. 3A; “$F_{\mathit{\text{vivo}}}+g\left(0\right)$” corresponds to optimization with the constraint of $F_{\mathit{\text{vivo}}}$ and the constraint that $g\left(0\right)=0$. (a) Finite difference stochastic approximation search. The markers show the search for the case of minimizing the $\max \left\{n_f\right\}$ with no constraint and no boundary condition. The horizontal lines are the maximum fork density for different search conditions. (b) Initiation rate $I\left(t\right)$. The $I_{\mathit{\text{vivo}}}$ shown is in the bilinear form [48]. (c) Fork density $n_f\left(t\right)$. Line types correspond to the same cases as in (b).

Figure 7

(Color online) Schematic diagram of licensing on a lattice genome. (a) Realization of replication using random-licensing ($d_l=0$ case). The gray (white) area represents replicated (unreplicated) domains. Circles denote initiations. (b) Origins are forced to their nearest lattice sites (marked by vertical lines at multiples of $d_l=200\mathrm{kb}$), while initiation times remain the same. (c) The result of the shift in origin positions. Open markers represent “phantom origins” that do not contribute to the replication; filled markers denote the actual origins. Alternatively, a filled marker can be viewed as the origin that initiated in a cluster potential origins. Going from $d_l=0\mathrm{kb}$ in (a) to $200\mathrm{kb}$ in (c), the average initiation time decreased from about $22\min$ to about $10\min$.

Figure 8

(Color online) (a) Distribution of spacings between initiated origins, ${\rho }_i\left(s\right)$, for the $d_l=0$ and $2\mathrm{kb}$ cases ($2\mathrm{kb}$ is chosen to mimic the minimal spacing between origins reported in [29]). The initiation rate and fork velocity are those obtained in Sec. 3B. The mean of the continuous distribution ($d_l=0\mathrm{kb}$ case) is marked $d_{\mathrm{inter}}$ and is $\approx 6.5\mathrm{kb}$. The mode of the discrete distribution ($d_l=2\mathrm{kb}$ case) is marked by ⋆. The probability $P$ at the mode (0.23 in this case) is defined to be the periodicity, a measure of ordering in the system. (b) Average interorigin spacing $d_{\mathrm{avg}}$ as a function of $d_l$. There is a gradual transition from regime I to regime II. In regime I $(d_l⩽d_{\mathrm{inter}})$, $d_{\mathrm{avg}}$ is asymptotically independent of $d_l$ for $d_l\to 0$. In regime II $(d_l>d_{\mathrm{inter}})$, $d_{\mathrm{av}}$ is asymptotically linearly proportional to $d_l$. Inset shows the periodicity $P$ as a function of $d_l$.

Figure 9

(Color online) Evolution of the end-time distribution with increasing spatial ordering due to increasing $d_l$. Each horizontal profile is an end-time distribution. In regime I, the end-time distribution does not change appreciably; in regime II, the mode shifts to the right. The ordering threshold is at $d_l=d_{\mathrm{inter}}\approx 6.5\mathrm{kb}$. The dashed line shows the $d_l=2\mathrm{kb}$ end-time distribution, which corresponds to the lateral inhibition ordering observed experimentally [29].