A solvable model for the diffusion and reaction of neurotransmitters in a synaptic junction

Here we present an analytical solution to a model for the diffusion and reaction of acetylcholine in a neuromuscular junction and for the diffusion and binding of Ca2+ in a dyadic cleft. Our model is similar to those previously solved numerically and our results are also qualitatively similar.The analytical solution provides a unique benchmark for testing numerical methods and potentially provides a new avenue for modeling biochemical transport.In intercellular and intracellular spaces, passive transport of biomolecules is a common phenomenon. Because such processes are difficult to probe directly by experiments, numerical modeling is increasingly used to gain insight. Two processes that have been extensively modeled are the diffusion and reaction of the transmitter acetylcholine in a neuromuscular junction [1-6] and the diffusion and binding of Ca2+ in the dyadic cleft of a ventricular myocyte [7,8]. In contrast to previous numerical approaches, here we present an analytical solution of a model for the diffusion and reaction of acetylcholine in a synaptic cleft (or Ca2+ in a dyadic cleft). Our model is similar to those previously solved numerically; hence our analytical solution potentially provides a new avenue for modeling biochemical transport. More importantly, an analytical solution provides a unique benchmark for testing numerical methods. Such a solution has been lacking up to now; the present work fills this gap.Neuromuscular junction refers to the cleft between a motor neuron and a muscle fiber. As illustrated in Figure 1, the neuronal signal for muscle contraction is mediated by acetylcholine. These neurotransmitter molecules are initially inside vesicles located in the pre-synaptic axon terminal. When an action potential reaches the axon terminal, the vesicles release acetylcholine molecules into the synaptic cleft. These molecules then diffuse toward the post-synaptic membrane and bind to acetylcholine receptors in the membrane. Acetylcholine binding ac