Nucleation and growth in one dimension. II. Application to DNA replication kinetics

Inspired by recent experiments on DNA replication, we apply a one-dimensional nucleation-and-growth model to DNA-replication kinetics, focusing on how to extract the time-dependent nucleation rate $I\left(t\right)$ and growth speed $v$ from data. We discuss generic experimental problems: namely, spatial inhomogeneity, measurement noise, and finite-size effects. After evaluating how each of these affects the measurements of $I\left(t\right)$ and $v$, we give guidelines for the design of experiments. These ideas are then discussed in the context of the DNA-replication experiments.

Figure 1

Mapping DNA replication onto the one-dimensional KJMA model.

Figure 2

Parameter extraction from an almost ideal data set. (a) Inferred nucleation rate vs time. (b) Velocity vs time. (c) Average domain sizes vs time. (d) Island fraction vs time; theory and extracted $f\left(t\right)$ overlap. In (c), $\ell *$ is the minimum average eye-to-eye spacing and sets the basic length scale. In (d), $t*$ is the time at which 50% of the genome has replicated. It sets the basic time scale.

Figure 3

(Color online). Inversion results in the presence of asynchrony and finite-size effects. (a) $I∕2v$ vs $2vt$. The arrows indicate where $f=0.8$ in $f$ vs $t$ curves in (d) for three different molecule sizes: ${10}^4$ (unchopped), 1000 and 250 (chopped). (b) $\rho (f,t_i)$ for six time points 60, 80, 100, 120, 140, 160 (from left to right). The circles are simulation data; the solid lines are from Eq. (7), using the extracted parameters in Table . (c) Optimization results for the starting-time distribution $\varphi \left(\tau \right)$. The solid line is a Gaussian fit. (d) $f$ vs $2vt$ for ${\ell }_c=250$ and ${\ell }_c=1000$. The solid line is the unchopped case (size ${10}^4$). (e) Average domain sizes vs $f$. The open circles are for the unchopped case, while the dotted and dashed curves correspond to ${\ell }_c=1000$ and 250. (f) Plot of $\log {\chi }^2$ $\left[\rho (f,t_i)\right]$ (arbitrary units) vs $v$ for size ${10}^4$. The complete fit results are shown in Table . See also text.

Figure 4

(Color online). Rescaled graphs for finite-size effects.

Figure 5

The finite-size effects and changes in the basic time and length scales. Shown are two different initiation rates $I\left(t\right)={10}^{-5}t$ and $I\left(t\right)=0.001$. The vertical line is where the average number of domains per molecule is 10. The $y$ axis has been normalized relative to the initiation rate for an infinite system $(\beta \to \infty )$.

Figure 6

(Color online). The effect of coarse graining. (a) $f$ vs $2vt$. From left to right, $\Delta x*=0,1,5$. (b) $I∕2v$ vs $2vt$. From top to bottom, the coarse-graining factor $\Delta x*=0$ (no coarse graining), 1 (comparable to optical resolution), and 5. (c) Average domain sizes vs $f$. The open circles are for no coarse graining, while the dashed lines are for $\Delta x*=1$ and 5 (dotted and dashed lines, respectively). (d)–(f) Rescaled graphs.