Random walk analysis of restricted metabolite diffusion in skeletal myofibril systems

The purpose of this work was the development of a basal mathematical model for the diffusion of low-molecular metabolites in a skeletal muscle cell. A three-dimension diffusion of low-molecular particles was simulated by a Monte-Carlo method (random walks of diffusing molecules). The model takes into account the following structural elements: (i) a regular lattice of actin and myosin filaments inside a myofibril; (ii) the membranes of sarcoplasmic reticulum and mitochondria surrounding the myofibrils; (iii) a set of myofibrils inside a skeletal muscle cell. We simulated diffusion of particles in the bulk of intracellular water phase and their reflections from the rigid surfaces of intracellular structures. The model allowed to calculate the apparent coefficients of particle diffusion in the axial and radial directions, Dparallel(app) and Dperpendicular(app), respectively. In accordance with experimental data from literature, the coefficient Dparallel(app) was independent of time. The coefficient of radial diffusion Dperpendicular(app) decreased with time to steady state values similar to that determined by the NMR diffusion spectroscopy methods. The interactions of diffusing particles with thin and thick filaments of myofibrils could explain the decrease in the Dperpendicular(app) value by a factor of 20%. The collisions of particles with myofilaments began to reveal themselves as a gradual decrease in the Dperpendicular(app) value at early stages of diffusion (t1/2 approximately equal to 0.05 microsec). The contribution of particle reflections from the membranes of sarcoplasmic reticulum and mitochondria to the retardation of the radial diffusion was about of 20-30%, depending on porosity of a membranous shield around the myofibril. For conventional sizes of a membranous shield (diameter 2 microm), the interactions of particles with the shield caused a decrease in the Dperpendicular(app) value with a half-time t1/2 approximately equal to 0.5 msec. This time is essentially lower by a factor about of 100 than that found in published NMR measurements. When we considered diffusion of particles inside a cell compartment confined to impermeable membranous shield, the reflection of particles from this shield led the drastic decrease in the radial diffusion coefficient (Dperpendicular(app) --> porportional to when t --> porportional to). This pattern of the Dperpendicular(app)(t) time-course might be expected in the NMR measurements on skeletal muscle tissue where a sarcolemma represents an impermeable shield for ATP and PCr molecules.