In 2005, Furusawa et al. reported log-normal distributions in protein concentration in bacteria, and showed how these log-normal distributions might arise from Brownian dynamics. Not all cell characteristics arise from Brownian dynamics (i.e., multiplicative stochastic dynamics with a linear noise term) or exhibit log-normal distributions. For example, cell size distributions appear to arise from Langevin dynamics (i.e., multiplicative stochastic dynamics with a sublinear noise term) and produce distributions that remain stationary in time. See this work, for example. But based on Furusawa et al., it seems reasonable to predict that the cellular concentration of a protein like ATP synthase might obey Brownian dynamics.
As pointed out recently in a paper from Ole Peters and Alexander Adamou, Brownian dynamics have some interesting statistical properties. In particular, ergodicity is broken because the log-normal distributions that arise from Brownian dynamics are not stationary. Rather, over time the size of fluctuations (the variance) increases as the square-root of time. This asymmetry in time can be rather subtle because it's a difference in variance that grows at a sublinear rate. If the variance is large relative to the mean, it can take a lot of measurements (and often a long time) to be sure you're seeing a difference in variance and rule out the possibility that your distribution is stationary.
But when Brownian dynamics are present, there are potentially some interesting consequences. For example, as Peters and Adamou suggest, if we model "cooperation" between two entities (like intracellular organelles?) as the sharing and mixing of what would otherwise be two independent populations (or concentrations of a chemical?), we see a damping or smoothing in the fluctuations (the variance). As time goes on, the cooperating populations grow faster in concentration than wholly independent populations thanks to that damping effect. See Figure 2:
Abstracted away from the details, we have Brownian dynamics and an opportunity for "cooperation" (i.e., mixing or sharing) leading over time to faster growth rates (e.g., in a concentration of proteins or chemicals) through a cooperative damping of fluctuations in concentrations of the same proteins or chemicals. Kind of nice to see how simple assumptions like these could lead to evolutionary dynamics.