…even though it’s really tough.
And studying the full spatial behavior of stochastic processes (including evolutionary theory, in its many guises), especially when interaction and fitness are relative to a complex network of contacts or relationships, is hard. Usually so hard, that we don’t have analytic models for the full behavior of sets of stochastic processes operating on complex networks, or interacting in complex ways. We resort to simulation since the models we can solve are very simple, and few. And we seek guidance for the “average” behavior — the nonspatial global behavior of a model — in mean-field approximations. We temporarily ignore fluctuations, write deterministic mean-field equations for the dynamics, analyze those, and then add fluctuations back, in the form of simple white noise. We take the deterministic mean-field equations and derive pair-approximations or moment closures, and analyze at least the summary statistics for correlations between classes or traits we’re tracking, since we can’t analyze much else spatially. We reduce complex epidemic diffusion models to percolation problems. But mostly, we simulate.
And we do this, and let the mathematicians like Thomas Liggett and Rick Durrett and their students attempt to extend the realm of analytic solutions, because it turns out dynamical systems act differently when interactions are local than when interactions are global. It’s universal, wherever we look. Prisoner’s dilemma models give different answers concerning optimal strategies when the model is well-mixed, and when individuals can only interact with local “neighbors.” Cooperation flourishes when interaction (and dependency) is local, and fade when groups get large and impersonal. Epidemics spread through populations differently when everyone is in mutual contact, and when individuals have sparse and clustered social connections, encountering only subsets of the community. Ideas and cultural information flow differently through populations, depending upon structure. Genes are subject to different selective forces, again depending upon structure.
And it turns out, that even at the level of the molecular chemistry of proteins within our cells. This week Koishi Takahashi and colleagues discovered that when you compare detailed molecular models, between standard mean-field chemical rate equations and a full spatially explicit molecular simulation, that spatiotemporal correlations between molecules matter. A protein kinase cascade, called MAPK, is widespread in eukaryotic organisms, and it turns out that spatial association between enzyme binding sites changes the rate and dynamics of the whole process, which explains why enzymes might release ADP slowly to dampen and stabilize the cascade against the instabilities caused by spatial correlations introducing rate variations.
The details don’t really matter here, but the point is, even down to the molecular level, the dynamics of biological and evolutionary processes are inherently spatiotemporal. Mean-field models are useful for theory building and giving us null models against which the behavior of spatiotemporal models can be compared, and of course there are empirical situations which can be well approximated by mean-field dynamics. But it turns out that mean-field analysis doesn’t tell us the full picture; it’s not just a matter of not having good predictions about spatial distributions in empirical cases, although that’s important. The real point is that mean-field models simply cannot explain some of the phenomena we see in the real world.
And that makes the study of complex spatiotemporal processes worthwhile, despite the fact that it’s hard, and that we lack the ability to even formulate the full dynamics of such systems in many cases. And despite the fact that we have to get good at simulating complex systems and analyzing the behavior of computational models.
