Extinction of metastable stochastic populations

Abstract

We investigate the phenomenon of extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n = 0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n = 0. In scenario B there is an intermediate repelling point n = n1 between the attracting point n = 0 and another attracting point n = n2 in the vicinity of which the metastable population resides. The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasi-stationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasi-stationary master equation at small n and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multi-step processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.

Stochastic Tunneling and Metastable States During the Somatic Evolution of Cancer

“…The value of t is crucial to determining the value of p 1 . When calculating the mean fixation time, t, we circumvented this known problem by considering a socalled boundary-layer approach (Assaf and Meerson 2010). The boundary-layer solutions (dashed lines in Figure 5; for details of the calculation see Appendix B) show better agreement with simulation results close to n 1 ¼ 0 than the QSD obtained from the WKB ansatz (solid lines).…”

“…We parameterize the system in terms of n 1 , such that n 1 ¼ 0 is the absorbing state (all cells of type 2) and n 2 ¼ N 2 n 1 . The analysis now closely follows the work of Assaf and Meerson (2010), specifically their scenario A. The outcome of the analysis are expressions for the action Sðx 1 Þ, S 1 ðx 1 Þ (determined up to an additive constant), and the normalization constant C. With this we find an explicit expression for the mean escape time, t, from the metastable state.…”

“…where SðxÞ $ Oð1Þ is known as the action, and exp½2S 1 ðxÞ (S 1 $ Oð1Þ) is the so-called amplitude (Assaf and Meerson 2010). We have introduced C as a normalization constant.…”

“…To find the QSD, we follow e.g., Assaf and Meerson (2010) and expand the QSME (5) in powers of N 21 . Further analysis can be carried out if the QSME has only one variable.…”

Modeling the effect of transcriptional noise on switching in gene networks in a genetic bistable switch

“…The WKB ansatz [15] is given as P m,n ≃exp[−S(m,n)], where S is the action. This ansatz has been used extensively to study population switches between metastable states [19][20][21]. Then, by considering stationary distributions Ṗm;n ¼ Qm;n ¼ 0 and employing the WKB ansatz they arrived at the stationary Hamiltonian Jacobi equation, H ¼ H m; ∂S ∂m ; n; ∂S ∂n À Á ¼ 0.…”