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Simulating stochastic population dynamics: The Linear Noise Approximation can capture non-linear phenomena

Frederick Truman-Williams, Giorgos Minas

TL;DR

A new framework based on centre manifold theory is introduced, a classical concept from non-linear dynamical systems, that enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar non-linear dynamical systems.

Abstract

Population dynamics in fields such as molecular biology, epidemiology, and ecology exhibit highly stochastic and non-linear behavior. In gene regulatory systems in particular, oscillations and multistability are especially common. Despite this, none of the currently available stochastic models for population dynamics are both accurate and computationally efficient for long-term predictions. A prominent model in this field, the Linear Noise Approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation; however, it is only accurate for linear systems and short-time predictions. Other models may achieve greater accuracy across a broader range of systems, but they sacrifice computational efficiency and analytical tractability. This paper demonstrates that, with specific modifications, the LNA can accurately capture non-linear dynamics in population processes. We introduce a new framework based on centre manifold theory, a classical concept from non-linear dynamical systems. This approach enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar non-linear dynamical systems. With these modifications, the LNA can achieve accurate long-term simulations without compromising computational efficiency. We apply our methodology to classes of oscillatory and bi-stable systems, and present multiple examples from molecular population dynamics that demonstrate accurate long-term simulations alongside significant improvements in computational efficiency.

Simulating stochastic population dynamics: The Linear Noise Approximation can capture non-linear phenomena

TL;DR

A new framework based on centre manifold theory is introduced, a classical concept from non-linear dynamical systems, that enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar non-linear dynamical systems.

Abstract

Population dynamics in fields such as molecular biology, epidemiology, and ecology exhibit highly stochastic and non-linear behavior. In gene regulatory systems in particular, oscillations and multistability are especially common. Despite this, none of the currently available stochastic models for population dynamics are both accurate and computationally efficient for long-term predictions. A prominent model in this field, the Linear Noise Approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation; however, it is only accurate for linear systems and short-time predictions. Other models may achieve greater accuracy across a broader range of systems, but they sacrifice computational efficiency and analytical tractability. This paper demonstrates that, with specific modifications, the LNA can accurately capture non-linear dynamics in population processes. We introduce a new framework based on centre manifold theory, a classical concept from non-linear dynamical systems. This approach enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar non-linear dynamical systems. With these modifications, the LNA can achieve accurate long-term simulations without compromising computational efficiency. We apply our methodology to classes of oscillatory and bi-stable systems, and present multiple examples from molecular population dynamics that demonstrate accurate long-term simulations alongside significant improvements in computational efficiency.

Paper Structure

This paper contains 40 sections, 2 theorems, 84 equations, 21 figures, 25 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Population dynamics
  3. Reaction networks
  4. Example (SIR model)
  5. System Size Expansion
  6. Example (SIR model, continued)
  7. The Linear Noise Approximation (LNA)
  8. Example (SIR model, continued):
  9. Accuracy of the LNA
  10. The phase corrected Linear Noise Approximation (pcLNA)
  11. Phase
  12. The pcLNA algorithm
  13. Choosing the phase-defining metric $d$ and set of reference trajectories $\Gamma$
  14. Topological equivalence
  15. Decomposition
  16. ...and 25 more sections

Key Result

Theorem 1

Let $E^c$ denote the eigenspace of $\mathcal{A}$ corresponding to the $n_0+1$ eigenvalues with zero real part. Then there exists a smooth $(n_0+1)$-dimensional invariant manifold $\mathcal{W}^c$, defined locally near $({\bm x}_{\alpha_p},\alpha_p)$ and tangent to $E^c$ at that point. Since $\dot\alp are invariant under bifode2. Consequently, for each $\alpha$ near $\alpha_p$, the center manifold i

Figures (21)

  • Figure 1: A. Phase portraits of solutions of the RRE in \ref{['eq:bifode1']} (black) and a SSA simulation (red) for the system in WILHELM1995. B. Phase portraits of 20 SSA simulations (color) and the equilibria of the RRE in \ref{['eq:bifode1']} (crosses) for the system in Gardner2000. The qualitative changes from left to right figure are due to changes in the bifurcation parameter value.
  • Figure 2: LNA method summary. A. LNA is a stochastic approximation of the stochastic population dynamics that can be exactly simulated using the SSA. LNA uses the numerical solutions of the deterministic RRE and a SDE that can be solved exactly. The SIR population dynamics model is used here as an example. B. Comparison between SSA and LNA on the SIR population dynamics in terms of simulated $99\%$ Confidence Intervals (CI) shows astonishing agreement as the blue lines and shades almost completely overlap with the pink lines and shades
  • Figure 3: General pcLNA algorithm over steps $i-1,i,i+1$. For simplicity we illustrate the case where $j-1=j=j+1$, so the same RRE solution is used over these steps. pcLNA proceeds with standard LNA steps interrupted by phase corrections with ${\bm x}^{(j_i)}(t_i)$ replaced by $G({\bm X}(t_i))={\bm x}^{(j_i)}(s_i)$, and the perturbations $\bm \xi(t_i)$ replaced by $\bm \kappa(s_i)$.
  • Figure 4: The dynamics of a generic normal form of the supercritical Hopf bifurcation. The two-dimensional centre manifold has coordinate ${\bm u}=(u_1,u_2)$, while the third coordinate $v$ satisfies $\dot{v}=-v$. Selected orbits for the three different cases are presented.
  • Figure 5: Phase correction. The phase-correcting map $G({\bm X}_i)$ for a point ${\bm X}_i$ on the stochastic trajectory is computed in three steps. First, the RRE solution and point ${\bm X}_i$ is projected into the centre coordinates of the non-linear Centre manifold. Then, the point, ${\bm u}(s_i)$ lying on the projected RRE trajectory $\{{\bm u}(t):t\geq 0\}$ that minimises the distance to ${\bm U}_i$ is found. Finally, the corresponding point, ${\bm x}(s_i)$, with the same time-point $s_i$ with ${\bm u}(s_i)$ and that lies on the RRE solution $\{{\bm x}(t):t\geq 0\}$ is found. The point ${\bm x}(s_i)$ and the perturbation $\bm \kappa_i$ (instead of ${\bm x}_i$ and $\bm \xi_i$) are used in the next LNA transition. The magnitude of $\bm \kappa_i$ is smaller than or equal to the magnitude of $\bm \xi_i$ on the centre coordinates.
  • ...and 16 more figures

Theorems & Definitions (4)

  • Remark 1: Bistability
  • Definition 1: Topological equivalence
  • Theorem 1: Center Manifold Theorem
  • Theorem 2: Takens