Reconstructing DNA replication kinetics from small DNA fragments

In higher organisms, DNA replicates simultaneously from many origins. Recent in vitro experiments have yielded large amounts of data on the state of replication of DNA fragments. From measurements of the time dependence of the average size of replicated and nonreplicated domains, one can estimate the rate of initiation of DNA replication origins, as well as the average rate at which DNA bases are copied. One problem in making such estimates is that, in the experiments, the DNA is broken up into small fragments, whose finite size can bias downward the measured averages. Here, we present a systematic way of accounting for this bias by deriving theoretical relationships between the original domain-length distributions and fragment-domain length distributions. We also derive unbiased average-domain-length estimators that yield accurate results, even in cases where the replicated (or nonreplicated) domains are larger than the average DNA fragment. Then we apply these estimators to previously obtained experimental data to extract improved estimates of replication kinetics parameters.

Figure 1

Definitions of domain types. (a) An infinitely long segment of DNA consists of eyes (replicated regions) and holes (nonreplicated regions). (b) On a finite fragment, there are interior, exterior, and oversized domains for both the eyes (i) and holes (h). Shaded vertical wedges denote cuts in the DNA molecule.

Figure 2

The free length over which an interior domain of size $X$ can be moved within a fragment of length $L$ without being chopped is $L-X$, implying a free fraction of $1-x$.

Figure 3

An illustration of the duality between domains and fragments. (a) Oversized domain; (b) interior domain. (a) and (b) may be derived from each other by interchanging fragments with domains. Shaded vertical wedges denote places where the molecule is cut.

Figure 4

An example of the transformation of an original uniform distribution of domains when cut into uniformly distributed fragments. Interior, edge, and oversized frequency distributions are shown. Monte Carlo simulations, with a total number of domains $N_t={10}^4$, are overlaid. There are no adjustable parameters.

Figure 5

Performance of different estimators, based on an exponential distribution of domain sizes $\rho \left(x\right)=(1∕\mu )\exp (-x∕\mu )$, with average $\mu$. All lengths are scaled by $L$, the (unique) fragment size. Unbiased, interior, and interior-edge estimators are shown, with Monte-Carlo simulation data overlaid. Points from the two unbiased estimators are indistinguishable. To guarantee a 10% bias, the interior estimator requires $x<0.08$, while the interior-edge estimator requires only that $x<0.4$.

Figure 6

(a) Distribution of lengths of combed DNA fragments. The average length is $102\mathrm{kb}$, with a standard deviation of $75\mathrm{kb}$. There are 1142 fragments in the dataset. (b) Distribution of the replication fraction of the DNA fragments.

Figure 7

Histograms of interior and edge hole domains, for replication fraction $f$ between 0 and 0.2. Solid lines are fits to exponential distributions, excluding the first point (see the discussion in the text).

Figure 8

Ratio of average edge domain length to average interior domain length as a function of the replication fraction. The dashed lines show the expected limits, 1.5 for small $f$, or large domains, and 1 for large $f$, or small domains.

Figure 9

Average hole (a) and eye (b) lengths as a function of replication fraction $f$. Open circles are computed with the interior estimator, filled squares with the interior-edge estimator. The solid curve is a fit based on the theory of Herrick et al. [10].

Figure 10

Extracted initiation rate $I\left(t\right)$ as a function of time $t$ using data estimated from the interior estimator (a) and interior-edge estimator (b).