A general two-cycle network model of molecular motors

Molecular motors are single macromolecules that generate forces at the piconewton range and nanometer scale. They convert chemical energy into mechanical work by moving along filamentous structures. In this paper, we study the velocity of two-head molecular motors in the framework of a mechanochemical network theory. The network model, a generalization of the recently work of Liepelt and Lipowsky (PRL 98, 258102 (2007)), is based on the discrete mechanochemical states of a molecular motor with multiple cycles. By generalizing the mathematical method developed by Fisher and Kolomeisky for single cycle motor (PNAS(2001) 98(14) P7748-7753), we are able to obtain an explicit formula for the velocity of a molecular motor.

💡 Research Summary

The paper presents a comprehensive theoretical framework for describing the motion of two‑head molecular motors using a generalized mechanochemical network that incorporates multiple reaction cycles. Building on the earlier two‑cycle model of Liepelt and Lipowsky (PRL 98, 258102, 2007) and the single‑cycle formalism of Fisher and Kolomeisky (PNAS 98, 7748‑7753, 2001), the authors derive an explicit analytical expression for the average velocity of a motor that can accommodate any number of intertwined forward and backward cycles.

The authors first discretize the motor’s conformational and chemical states into a set of nodes and represent allowed transitions as directed edges with rates (k_i^{\pm}). Each edge follows an Arrhenius‑type dependence on the external load (F) and on the chemical potential difference (\Delta\mu) (primarily set by ATP, ADP, and Pi concentrations). By arranging the transition‑rate matrix into block‑diagonal form, they separate the contributions of individual cycles and apply a graph‑theoretic Laplace expansion to compute the steady‑state probability currents around each closed loop. The net current (J) is the weighted sum of the cycle currents, and the motor’s mean velocity follows from (V = d,J), where (d) is the mechanical step size.

The central result is the velocity formula

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