Immigration-induced phase transition in a regulated multispecies birth-death process

Abstract

Power-law-distributed species counts or clone counts arise in many biological settings such as multispecies cell populations, population genetics, and ecology. This empirical observation that the number of species c k represented by k individuals scales as negative powers of k is also supported by a series of theoretical birth-death-immigration (BDI) models that consistently predict many low-population species, a few intermediate-population species, and very high-population species. However, we show how a simple global population-dependent regulation in a neutral BDI model destroys the power law distributions. Simulation of the regulated BDI model shows a high probability of observing a high-population species that dominates the total population. Further analysis reveals that the origin of this breakdown is associated with the failure of a mean-field approximation for the expected species abundance distribution. We find an accurate estimate for the expected distribution c k by mapping the problem to a lower-dimensional Moran process, allowing us to also straightforwardly calculate the covariances c k c ℓ . Finally, we exploit the concepts associated with energy landscapes to explain the failure of the mean-field assumption by identifying a phase transition in the quasi-steady-state species counts triggered by a decreasing immigration rate.

Dynamics of T cell receptor distributions following acute thymic atrophy and resumption

“…A recent investigation into the fully stochastic model indicates that the true c k differs from the c k derived using the mean-field assumption (Eqs. 2-4 and 5) only for very large k ≈ N [52]. Thus, our analyses may be inaccurate only if a single large clone dominates the whole population.…”

How heterogeneous thymic output and homeostatic proliferation shape naive T cell receptor clone abundance distributions

“…When the total T cell immigration rate is extremely small, the above constraint is violated and a single large clone arises due to competitive exclusion [34,43,44]. In this case, an accurate approximation for the steady-state clone abundance c k can be obtained using a variation of the two-species Moran model [34]. Thus, except under such extreme scenarios, the parameters associated with the human adaptive immune system satisfy the condition for the mean-field approximation to be accurate.…”

“…However, the mean field approximation to large-k clone counts is not accurate. For very large clone sizes, the total population constraint forces c k to rapidly approach zero for k N * , a feature that is not accurately reflected in the mean-field approximation [34]. Finally, large clones likely include memory T cells that have been produced after antigen stimulation of specific clones.…”

“…7 because this limit allowed relatively easy derivations of analytical results. This was also the strategy for previous modeling work [4,6,7,34,35]. However, the per clone immigration and proliferation times may be on the order of months or years, a time scale over which thymic output can change significantly as an individual ages [33,37,47,48] and as thymic output diminishes [33,37].…”

“…We now use the results for the mean steady-state population N * to find the clone counts averaged over all realizations of the underlying stochastic process. The mean-field equations for the dynamics of these mean clone counts in the neutral model were derived in [34,35] and are reviewed in Appendix A. In the neutral model, we assume that all effective clones Q are associated with the same rates α and r so that the mean field evolution equation for c k (α, r) is [34,35]…”