A two-dimensional compartment model for the reaction-diffusion system of acetylcholine in the synaptic cleft at the neuromuscular junction

Abstract

A minimal compartment model of the reaction-diffusion system (RD system) of a neurotransmitter in a two-dimensional space of axis-symmetrical disc is proposed to represent the chemical transmission process of a quantum of acetylcholine (ACh) in the synaptic cleft at the neuromuscular junction. The behavior of the RD system for ACh is expressed by a two-dimensional diffusion equation with nonlinear reaction terms due to the rate processes for ACh receptor and acetylcholinesterase. The simulation analysis of the RD system reveals that the radial diffusion process of ACh has more distinctive effects on spontaneous generation of the miniature endplate current (MEPC) than the transverse process. The anisotropic diffusion is effective in the RD system since the diffusion coefficient of ACh in the radial direction is evaluated to be about 1.0×10⁻⁶ cm² sec⁻¹ for appropriate characterization of the MEPC, on which the diffusion coefficient in the transverse direction larger than 2.0×10⁻⁶ cm² sec⁻¹ has virtually no effects. The compartment model is thus appropriately constructed to comprise three elements on the transverse coordinate and ten elements on the radial coordinate in the disc with 500 nm of radius and 50 nm of height.

Introduction

In processing of the neuronal signals, the synaptic chemical transmission is an important process, and investigation of the molecular events in the process has led to the neurotransmitter theory (Mathews, 1986). The dynamic behavior of the neurotransmitter in diffusion through the synaptic cleft and action at the synaptic membranes is engaged in a fundamental function for the transmission process. Analysis of such behavior can be performed most appropriately with representation of the transmission process as a reaction-diffusion system (RD system) for the neurotransmitter because the experimental analysis still is practically difficult for the molecular processes in the cleft. Some mathematical models for the dynamic behavior of acetylcholine (ACh), a typical neurotransmitter, in spontaneous generation of the miniature endplate current (MEPC) at the neuromuscular junction have been proposed to analyze the transient process of the synaptic chemical transmission. In the model of Rosenberry (Rosenberry, 1979), the radial diffusion of ACh is treated in the two axis-symmetrical compartments with homogeneity in the transverse direction, while in the model of Thomas (Friboulet and Thomas, 1993) the transverse coordinate is discretized and the radial diffusion undergoes simple efflux of ACh due to the concentration gradient.

In this study, the one-dimensional compartment models of Rosenberry and Thomas are extended to a two-dimensional compartment model for examination of the effects of transverse and radial diffusion of ACh on the transient behavior of the chemical transmission process. The process is hence represented as an RD system, in which the ACh concentration varies with time and position in a two-dimensional space of axis-symmetrical disc of the synaptic cleft, due to transverse and radial diffusion of ACh and its interaction with the ACh receptor (AChR) and acetylcholinesterase (AChE). The variation in concentration of the open channel form of AChR in response to the interaction of AChR with incoming ACh corresponds to transient evolution of the MEPC. The behavior of this RD system of ACh is mathematically expressed by a two-dimensional diffusion equation with nonlinear reaction terms for ACh and a set of nonlinear ordinary differential equations governing the rate processes for AChR and AChE, which are distributed at the spatial points in the disc.

The analysis of temporal behavior of the RD system employs computer simulation, that is, discretization of both the transverse and radial coordinates in the space for the partial differential equation and numerical integration of the governing ordinary differential equations under various conditions (Hayashi and Sakamoto, 1986). Simulation of the response of the RD system to release of a single quantal packet of ACh into the cleft leads to characterization of the transverse and radial diffusion processes of ACh in the chemical transmission with reference to their effect on spontaneous generation of the MEPC. Based on an optimal selection of the subdivision numbers and critical radius for the simulation, a minimal compartment model is proposed which comprises three and ten compartments in the transverse and radial directions, respectively, in a disc with 500 nm of radius and 50 nm of height. Evaluation of the diffusion coefficients in the transverse and radial directions suggests anisotropic diffusion for this two-dimensional compartment model to represent the characteristic behavior of the chemical transmission process.