Multiscale modelling of birth-death processes

Area of Science:

• Computational Biology
• Mathematical Biology
• Systems Biology

Background:

• Biological systems display multiscale dynamics with both high and low copy number species.
• Hybrid modeling approaches balance computational efficiency and accuracy for these systems.
• Threshold-based methods like Jump-Switch-Flow (JSF) simulate species differently based on copy number, but threshold selection is often empirical.

Purpose of the Study:

• To develop a principled method for selecting the threshold parameter $\Omega$ in threshold-based hybrid simulations.
• To provide a computable, method-agnostic error bound for extinction probability in hybrid models.
• To enable users to specify an error tolerance and obtain an optimal $\Omega$.

Main Methods:

• Formalized JSF as a piecewise-deterministic Markov process.
• Derived backward equations for extinction probability under exact and hybrid dynamics.
• Analyzed dynamics near extinction boundaries, reducing them to time-inhomogeneous linear birth-death processes.
• Developed a rigorous error decomposition and a heuristic based on solving a scalar Riccati equation.

Main Results:

• Introduced a computable, method-agnostic error bound for threshold-based hybrid simulations.
• Provided an explicit rule for selecting the threshold $\Omega$ based on desired error tolerance for extinction probability.
• Demonstrated that the heuristic reliably upper-bounds empirical error in extinction probability using Monte Carlo simulations on a Lotka-Volterra model.

Conclusions:

• The developed framework offers a principled approach to threshold selection in hybrid simulations, moving beyond trial-and-error.
• The method is applicable to any threshold-based hybrid simulation scheme, not limited to JSF.
• This work significantly improves the accuracy and efficiency of modeling biological systems with multiscale dynamics.