A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Abstract

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of degree higher than n. For analysis purpose, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function.In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at-least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximation of the actual moment dynamics.A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.

A data-integrated method for analyzing stochastic biochemical networks

“…18 Previous work has attempted to truncate the system of coupled moment equations by assuming a priori functional forms for approximate moment closure. [18][19][20] While this generates a finite system of equations that can be solved analytically or numerically, the validity of these assumptions might not be a priori warranted.…”

“…In this fashion, a relationship between the N + 1 moments and the first N moments is derived, and the system of moment equations is closed at the Nth moment. For example, one such moment closure approximation known as separable derivative matching 20 approximates the N + 1th moment as a polynomial function of the first N moments. This approach matches time derivatives between the approximate closed system and the exact non-closed system at the initial time t 0 and the given initial conditions.…”

“…This allows the exponents (which remain constant over the simulation) in the polynomial function to be uniquely determined, and the solution turns out to be consistent with the underlying distribution P(x, t) being log-normal. 20 For those biological systems which do not exhibit log-normal behavior, the separable polynomial function maybe too restrictive.…”

“…In fact, ignoring the contribution of the third central moment results in sustained oscillations even in the average trajectory. Furthermore, accounting for the third (un-centered) moment using the separable derivative matching technique 20 (n = 2, i.e., third moment closure approximation (log-normal assumption)) also yields inaccurate results by showing average trajectories with overly damped oscillations (Figures 3(b) and 3(c), moment equations (SDM, n = 2)). This is likely due to the true distribution deviating from log-normal.…”

A rigorous framework for multiscale simulation of stochastic cellular networks

“…Our process of sampling from the distribution P u knowing only its moments ͑Appendix D͒ has not been optimized, and we are currently investigating more general methods. We are also investigating the applicability of alternative moment truncation approximations based on log-normal closure methods 26 while still truncating at the covariances. The bistable and oscillator examples show many regions of parameter space where the RRA algorithm is more efficient than, but has comparable accuracy, to the SSA.…”