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

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