[Paper Review] Filament-motor protein system under loading: instability and limit cycle oscillations

Amir Shee, Subhadip Ghosh+1 more|arXiv (Cornell University)|Dec 17, 2020

stochastic dynamics and bifurcation35 references2 citations

TL;DR

This study investigates the dynamics of a rigid filament driven by motor proteins under external loading, showing that constant loading induces instability and detachment, while elastic loading triggers stable limit cycle oscillations via a supercritical Hopf bifurcation. The critical number of motors for oscillations depends on load stiffness, offering a potential cellular mechanosensing mechanism.

ABSTRACT

We consider the dynamics of a rigid filament in a motor protein assay under external loading. The motor proteins are modeled as active harmonic linkers with tail ends immobilized on a substrate. Their heads attach to the filament stochastically to extend along it, resulting in a force on the filament, before detaching. The rate of extension and detachment are load dependent. Here we formulate and characterize the governing dynamics in the mean field approximation using linear stability analysis, and direct numerical simulations of the motor proteins and filament. Under constant loading, the system shows transition from a stable configuration to instability towards detachment of the filament from motor proteins. Under elastic loading, we find emergence of stable limit cycle oscillations via a supercritical Hopf bifurcation with change in activity and the number of motor proteins. Numerical simulations of the system for large number of motor proteins show good agreement with the mean field predictions.

Motivation & Objective

• To understand how external loading influences the stability and dynamics of a filament driven by active motor proteins in a gliding assay.
• To investigate the transition from stable equilibrium to instability and spontaneous oscillations under constant and elastic loading.
• To determine the role of motor protein activity (e.g., ATP concentration) and number in triggering limit cycle oscillations.
• To validate mean-field predictions with stochastic simulations of individual motor proteins.
• To explore the potential of such systems as cellular mechanosensors by linking oscillation onset to load stiffness.

Proposed method

• Modeling motor proteins as active harmonic linkers with load-dependent extension and detachment rates.
• Using mean-field approximation to derive deterministic equations of motion for the filament and bound motor proteins.
• Performing linear stability analysis to identify regions of instability and predict the onset of oscillations.
• Applying Fokker-Planck formalism to derive the mean number of bound motors and the probability distribution of attachment states.
• Conducting direct numerical simulations of the full stochastic model with individual motors to compare with mean-field predictions.
• Analyzing the system's behavior via bifurcation analysis, particularly identifying a supercritical Hopf bifurcation as the mechanism for oscillation onset.

Experimental results

Research questions

• RQ1 How does constant external loading affect the stability of a filament in a motor protein assay?
• RQ2 Under what conditions does elastic loading induce stable limit cycle oscillations in the filament-motor system?
• RQ3 How does the critical number of motor proteins required for oscillations depend on the stiffness of the elastic load?
• RQ4 To what extent do mean-field predictions of oscillation onset match results from stochastic simulations of individual motors?
• RQ5 Can the system's oscillatory behavior serve as a mechanism for sensing mechanical properties of the extracellular matrix?

Key findings

• Under constant loading, the system undergoes a transition from a stable equilibrium to instability, leading to filament detachment.
• With elastic loading, stable limit cycle oscillations emerge via a supercritical Hopf bifurcation when the number of motor proteins exceeds a critical threshold.
• The critical number of motors required for oscillations increases with decreasing load stiffness, indicating a tunable mechanosensing mechanism.
• Numerical simulations of the full stochastic model show excellent agreement with mean-field predictions, confirming the robustness of the Hopf bifurcation boundary.
• Deep in the oscillatory regime, the dynamics exhibit characteristics of relaxation oscillators, with oscillation amplitudes of ~4 nm and frequencies near 0.5 Hz for kinesin-microtubule systems.
• The model's parameter values correspond to biologically relevant systems (e.g., kinesin on microtubules), enabling direct experimental validation.