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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

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Inomzhon Mirzaev¹, Jeremy Gunawardena

¹Applied Mathematics Graduate Program, University of Colorado, Boulder, CO, USA, mirzaev@colorado.edu.

Bulletin of Mathematical Biology

|September 11, 2013

Summary

This summary is machine-generated.

This study extends a linear framework for biochemical systems using graph Laplacians. It reveals steady-state dynamics, multistability in gene regulation, and an equivalence with Markov processes for stochastic mechanisms.

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Area of Science:

Biochemical Systems Analysis
Systems Biology
Mathematical Biology

Background:

Previous work introduced a linear framework for time-scale separation in biochemical systems using graph Laplacians.
This framework has successfully derived central results in molecular biology, including enzyme kinetics and gene regulation at equilibrium.

Purpose of the Study:

To lay out the mathematical foundations for accommodating nonequilibrium mechanisms in eukaryotic gene regulation.
To explore the dynamics and steady-state behavior of the linear framework.
To investigate the relationship between deterministic dynamics and stochastic processes.

Main Methods:

Utilized a linear framework based on labelled, directed graphs and associated linear differential equations (dx/dt = L(G) ∙ x).
Analyzed the steady-state properties of the system for any graph and initial condition.
Investigated dynamics in graphs that are not strongly connected, relevant to gene regulation.
Established an equivalence between deterministic Laplacian dynamics and master equations of continuous-time Markov processes.

Main Results:

Demonstrated that the dynamics always reach a steady state, which can be algorithmically calculated.
Showed that non-strongly connected graphs can exhibit flexible behavior resembling multistability, particularly in gene regulation.
Revealed an equivalence between deterministic Laplacian dynamics and continuous-time Markov processes, enabling stochastic, single-molecule mechanism treatment.

Conclusions:

The linear framework provides a robust mathematical foundation for analyzing complex biochemical systems, including nonequilibrium processes.
The framework can capture emergent behaviors like multistability in gene regulation.
The established equivalence with Markov processes allows for the rigorous inclusion of stochasticity and single-molecule dynamics.