Algebraic formulas for first-passage times of Markov processes in the linear framework

Area of Science:

• Biochemistry
• Systems Biology
• Mathematical Biology

Background:

• The linear framework analyzes biochemical systems using directed graphs.
• It models systems as Markov processes, with the master equation as a linear differential equation.
• The Matrix-Tree theorem provides algebraic access to steady-state probabilities.

Purpose of the Study:

• To extend the linear framework from steady-state analysis to the transient regime.
• To develop algebraic methods for analyzing transient dynamics in biochemical systems.
• To broaden the applicability of the linear framework to new biological problems.

Main Methods:

• Utilizing directed graphs with labelled edges to represent biochemical systems.
• Applying the All-Minors Matrix-Tree theorem to the Laplacian matrix of the system graph.
• Expressing moments of first-passage time distributions and splitting probabilities as rational algebraic functions.

Main Results:

• The study successfully extends the linear framework to the transient regime.
• Moments of conditional first-passage time distributions are expressed as rational algebraic functions of transition rates.
• Splitting probabilities are also derived as rational algebraic functions of transition rates.

Conclusions:

• The extended linear framework provides algebraic solutions for transient biochemical system dynamics.
• This advancement allows for rigorous theoretical analysis of problems previously requiring approximations or simulations.
• The approach enhances the scope of the linear framework, enabling new biological insights.