The colocalization transition of homologous chromosomes at meiosis

Meiosis is the specialized cell division required in sexual reproduction. During its early stages, in the mother cell nucleus, homologous chromosomes recognize each other and colocalize in a crucial step that remains one of the most mysterious of meiosis. Starting from recent discoveries on the system molecular components and interactions, we discuss a statistical mechanics model of chromosome early pairing. Binding molecules mediate long-distance interaction of special DNA recognition sequences and, if their concentration exceeds a critical threshold, they induce a spontaneous colocalization transition of chromosomes, otherwise independently diffusing.

Figure 1

(Color online) Main panel shows the evolution of the square distance between “chromosome recognition sequences,” $d^2\left(t\right)$ (expressed as a fraction of the system size, $L$), as a function of time, $t$ (in ${\text{log}}_{10}$ scale), for the given values of the concentration of molecular mediators, $c$ (here the molecule-chromosome and molecule-molecule affinities are, respectively, $E_X=1.2kT$ and $E_0=0$). Chromosomes are initially at a distance $L$, i.e., $d^2\left(0\right)=100\%$. At long times $d^2\left(t\right)$ plateaus to its equilibrium value approximately exponentially (superimposed fit), which for $c=0.3\%$ corresponds to the expected average square distance between independent randomly diffusing directed strings, $\sim 40\%$; for $c=2.5\%$, instead, $d^2\left(t\right)$ collapse to almost zero, signaling that the two chromosomes have colocalized. Inset shows (in a range of values of $E_X$ and $E_0$) that the characteristic time to approach equilibrium, $\tau$ (derived from exponential fits), increases as a function of $c$. The superimposed fits are power laws.

Figure 2

(Color online) Left panel The equilibrium average chromosome square distance, $d^2$, from Fig. 1 ($E_X=1.2kT$, $E_0=0$) is shown as a function of $c$ (in ${\text{log}}_{10}$ scale): for $c<c_{\text{tr}}\simeq 0.7\%$, $d^2$ approaches values as big as the system size and chromosomes are randomly and independently diffusing (horizontal dotted line gives the independent diffusion value $d^2\sim 40\%L^2$); for $c>c_{\text{tr}}$, $d^2$ rapidly decays to zero, showing that they have colocalized. Around $c_{\text{tr}}$, there is a narrow crossover regime where chromosomes are only transiently colocalizing. Right panels show pictures of typical equilibrium configurations, in the “independent” (upper panel, $c=0.3\%$) and “colocalized’ phase” (lower panel, $c=2.5\%$). Molecular mediators are pictured as orange floating particles.

Figure 3

(Color online) The system phase diagram is plotted in the $\left(E_X,c\right)$ plane (the $c$ axis is in ${\text{log}}_{10}$ scale). The transition line, $c_{\text{tr}}\left(E_X\right)$, delimiting the phase where chromosomes move independently from the phase where they colocalize, is marked by filled circles in the case $E_0=0$. For comparison, the transition line is also plotted in the case $E_0=1.2kT$ (empty circles). Fits superimposed to the two transition lines are inverse power laws (see text).