Formulation of Chromatin Mobility as a Function of Nuclear Size during C. elegans Embryogenesis Using Polymer Physics Theories

During early embryogenesis of the nematode, Caenorhabditis elegans, the chromatin motion markedly decreases. Despite its biological implications, the underlying mechanism for this transition was unclear. By combining theory and experiment, we analyze the mean-square displacement (MSD) of the chromatin loci, and demonstrate that MSD-vs-time relationships in various nuclei collapse into a single master curve by normalizing them with the mesh size and the corresponding time scale. This enables us to identify the onset of the entangled dynamics with the size of tube diameter of chromatin polymer in the C. elegans embryo. Our dynamical scaling analysis predicts the transition between unentangled and entangled dynamics of chromatin polymers, the quantitative formula for MSD as a function of nuclear size and timescale, and provides testable hypotheses on chromatin mobility in other cell types and species.

Figure 1

Chromatin mobility correlates with nuclear size. (a) MSD of different nuclei sizes. Different colors represent different sizes of nuclei. Nuclear $\text{radius}=2.0$ (brown), 2.4 (red), 2.8 (orange), 3.2 (yellow), 3.6 (light green), 4.0 (green), 4.4 (light blue), [$\mu \mathrm{m}$]. The solid lines represent fitting to Eq. (1), whereas the dotted lines are fitting to Eq. (1) with $\alpha$ fixed to 0.5. (b) MSD exponent $\alpha$ (blue) and MSD coefficient $A$ (assuming the constant exponent of $\alpha =0.5$, red) of each group. The error bars represent S.D (standard deviation). The blue dotted line indicates $\alpha =0.5$, and the red dotted curve represents a fit $A_{0.5}\sim R^{1.5}$, which are expected from the Rouse dynamics.

Figure 2

Changes in nuclear size induce changes in chromatin motion. (a) Distribution of the radii of nuclei in the eight-cell stage. Control (gray), ima-3 (RNAi) (blue), and C27D9.1 (RNAi) (orange). (b) Correlation between the MSD coefficient and the nuclear radius in the eight-cell stage (filled circle). Blank circles represent the MSD coefficient calculated from all stages as shown in Fig. 1. The error bars represent S.D.

Figure 3

(a) Summary of the relationship among MSD, mesh size $\xi$, tube diameter $a$, time- and nuclear-size dependencies of MSD discussed in the text. Top schematics illustrate the mesh size and the tube diameter in the chromatin solution. (b) Quantitative comparison of the mesh size, tube diameter and MSD in different nuclear size. The mesh size $\xi$ (blue dotted line) is calculated according to Eq. (4) as $\xi =4.2\times {10}^{-3}\times R^3[\mu \mathrm{m}]$. The tube diameter estimated from the experiment (red solid line) as the border of the broken line in Fig. S5 [24] is $a=\xi \times {48}^{1/2}=2.9\times {10}^{-2}\times R^3[\mu \mathrm{m}]$. Black line is ${\mathrm{MSD}}^{1/2}$ at $\tau =20\mathrm{s}$ obtained from the experiment [Fig. 1]. (c) The master curve of MSD constructed by time-nuclear size superposition (double logarithmic scale) by normalizing MSD and $\tau$ by ${\xi }^2$ and ${\tau }_{\xi }$, respectively. The color code is the same as in Fig. 1. The broken line on the plot is obtained by fitting the plot as described in Ref. [24]. The larger spots correspond to the measurements with $\mathrm{S}.\mathrm{E}.\mathrm{M}./\mathrm{MSD}\le 0.06$, whereas the others are shown with the smaller spots.