An Ordered Line Integral Method for Computing the Quasi-potential in the case of Variable Anisotropic Diffusion

TL;DR

This paper extends the Ordered Line Integral Method to efficiently compute the quasi-potential in 2D stochastic differential equations with variable anisotropic diffusion, aiding biological transition modeling.

Contribution

It introduces an extended OLIM solver for anisotropic, position-dependent diffusion in 2D SDEs, with analysis of error dependence and application to biological models.

Findings

The solver accurately computes quasi-potential for anisotropic diffusion.

Error depends on mesh size, update factor, and anisotropy degree.

Application to Lambda Phage model demonstrates biological relevance.

Abstract

Nongradient stochastic differential equations (SDEs) with position-dependent and anisotropic diffusion are often used in biological modeling. The quasi-potential is a crucial function in the Large Deviation Theory that allows one to estimate transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths. Unfortunately, the quasi-potential can rarely be found analytically. It is defined as the solution to a certain action minimization problem. In this work, the recently introduced Ordered Line Integral Method (OLIM) is extended for computing the quasi-potential for 2D SDEs with anisotropic and position-dependent diffusion scaled by a small parameter on a regular rectangular mesh. The presented solver employs the dynamical programming principle. At each step, a local action minimization problem is solved using straight line path segments and the midpoint quadrature rule. The solver is tested on two examples where analytic formulas for the quasi-potential are available. The dependence of the computational error on the mesh size, the update factor K (a key parameter of OLIMs), as well as the degree and the orientation of anisotropy is established. The effect of anisotropy on the quasi-potential and the maximum likelihood paths is demonstrated on the Maier-Stein model. The proposed solver is applied to find the quasi-potential and the maximum likelihood transition paths in a model of the genetic switch in Lambda Phage between the lysogenic state where the phage reproduces inside the infected cell without killing it, and the lytic state where the phage destroys the infected cell.