Neuromuscular Junction Physiology
Mathematical Modeling:
Khaliq et al. created a 3D model to explain the diffusion and reaction of Ach in the neuromuscular junction in 2011. The model describes the reactions of Ach with the AChR and acetylcholinesterase (the proteolytic enzyme that breaks down the Ach) allowing for its activity for only a certain amount of time. The diffusion is also related to the time required. These models are bsed on nonlinear partial differential equations (PDEs) that can be coupled and can be solved via computer. Khaliq et al. modeled the pre and post-synaptic clefts as circular spaces, creating a 3-D cylindrical space between the two.[4]
As such, the diffusion of Ach based on concentration can be modeled using cylindrical coordinates as described by the equation below in Figure 2. The (A) values relate to the Ach molar concentrations. (AR) and (AR2) represent the AchR with single and double Ach attachment respectively. The k values describe the forward and backwards reactions (denoted with (-) precursor). The model uses the surfaces of the cylinder formed between synapses to form boundary conditions (shown in Figure 3). At these surfaces, they assume the diffusion and reaction equation equals zero. The group utilized analytical software to solve this model for the NMJ. For more information and derivation click on this link to the article.[4] Khaliq article.
Another recent model for the NMJ, proposed by Liu et al. focuses on the use of a reaction and diffusion system much like the previous model. They propose a model that not only incorporates the reaction and diffusion components, but also includes to use of a more accurate depiction or geometry of the synaptic cleft. The numerical methods allow for the concentration of Ach to be computed, solving the diffusion/reaction equation. Figure 4 and 5 show the geometry used in the new model and a depiction of the Ach concentrations found by this model.[5]
Figure 1: This figure from Feher, depicts the function of the neuromuscular junction.
The above videos give good explanations on how the neuromuscular junction works as supplemental studies to the written portion of this site.
Function:
The neuromuscular junction (NMJ)connects the nerve impulse to the muscle membrane. As shown in the diagram below, the action potential from the nerve activates voltage-gated channels, allowing an influx of calcium. This calcium binds with proteins allowing the vesicle to fuse with the membrane leading to the release of acetylcholine (ACh), a neurotransmitter, across the synapse to bind to the ACh receptors. Once enough of these receptors are activated, this will lead to end plate potential or postsynaptic potential through opening of ion channels. This is what leads to muscle fiber function and contraction. Figure 1 below (taken from Feher) gives a good depiction of these steps.[2]
Figure 1 stops at step 5 with the sodium influx and the end plate potential. The potential travels along the sarcolemma through T-tubules and ultimately causes muscle function. The potential leads to the activation of calcium channels in the muscle cells and the intracellular increase of calcium through its release from the sarcoplasmic reticulum. The increased calcium can then bind troponin C to open the ports blocked by the tropomyosin to allow actin and myosin binding. This and ATP allow for the cross bridge cycle to cause contraction of the muscle.[3]
Figure 4: NMJ geometry proposed in the model by Liu et al.
Figure 5: NMJ Ach concentrations from the model by Liu et al.
