Nonequilibrium Statistical Mechanics of Dividing Cell Populations

We present and study a model for the nonequilibrium statistical mechanics of protein distributions in a proliferating cell population. Our model describes how the total protein variation is shaped by two processes: variation in protein production internal to the cells and variation in division and inheritance at the population level. It enables us to assess the contribution of each of these components separately. We find that, even if production is deterministic, cell division can generate a large variation in protein distribution. In this limit we solve exactly a special case and draw an analogy between protein distribution along cell generations and stress distribution in layers of granular material. At the other limit of extremely noisy protein production, we find that the population structure restrains variation and that the details of division do not affect the tail of the distribution.

Figure 1

Coefficient of variation in protein distribution vs division parameters. $a$ represents the variation at cell division, and $f$ is the underlying asymmetry of this division, with $f=1/2$ corresponding to symmetric division.

Figure 2

Population distribution with constant protein production. (a) Generating function $G(s)$ for a uniform division distribution $\eta (q)=1$, computed exactly as a sum over cumulants (○), with asymptotic approximation $G(s)\sim c/x$ (line; $c=\mathrm{const}$). (b) Monte Carlo simulation of the distribution function $P(x)$ (○), compared to the asymptotic approximation equation (9) (line). Also depicted (□) is the distribution for a division fraction with a smaller standard deviation (square of width $a=0.8$ around $f=\frac{1}{2}$).

Figure 3

(a) Topology of the inheritance in a population of dividing cells compared to (b) the $q$ model for stress transmission in granular packings.

Figure 4

Protein distributions for highly variable production: the analytical solution (line) and Monte Carlo simulation (○) for a uniform division function; the narrower division function (□, square of width $a=0.8$ around $f=\frac{1}{2}$, same as in Fig. 2); and the model with synchronous uniform division and an exponentially distributed production (+).