Continuum simulations of acetylcholine diffusion with reaction-determined boundaries in neuromuscular junction models

The reaction-diffusion system of the neuromuscular junction has been modeled in 3D using the finite element package FEtk. The numerical solution of the dynamics of acetylcholine with the detailed reaction processes of acetylcholinesterases and nicotinic acetylcholine receptors has been discussed with the reaction-determined boundary conditions. The simulation results describe the detailed acetylcholine hydrolysis process, and reveal the time-dependent interconversion of the closed and open states of the acetylcholine receptors as well as the percentages of unliganded/monoliganded/diliganded states during the neuro-transmission. The finite element method has demonstrated its flexibility and robustness in modeling large biological systems.     

A new 3D mass diffusion�Creaction model in the neuromuscular junction

A three-dimensional model of the reaction-diffusion processes of a neurotransmitter and its ligand receptor in a disk shaped volume is proposed which represents the transmission process of acetylcholine in the synaptic cleft in the neuromuscular junction. The behavior of the reaction-diffusion system is described by a three-dimensional diffusion equation with nonlinear reaction terms due to the rate processes of acetylcholine with the receptor. A new stable and accurate numerical method is used to solve the equations with Neumann boundaries in cylindrical coordinates. The simulation analysis agrees with experimental measurements of end-plate current, and agrees well with the results of the conformational state of the acetylcholine receptor as a function of time and acetylcholine concentration of earlier investigations with a smaller error compared to experiments. Asymmetric emission of acetylcholine in the synaptic cleft and the subsequent effects on open receptor population is simulated. Sensitivity of the open receptor dynamics to the changes in the diffusion parameters and neuromuscular junction volume is investigated. The effects of anisotropic diffusion and non-symmetric emission of transmitter at the presynaptic membrane is simulated.     

Tetrameric mouse acetylcholinesterase: continuum diffusion rate calculations by solving the steady-state Smoluchowski equation using finite element methods

The tetramer is the most important form for acetylcholinesterase in physiological conditions, i.e., in the neuromuscular junction and the nervous system. It is important to study the diffusion of acetylcholine to the active sites of the tetrameric enzyme to understand the overall signal transduction process in these cellular components. Crystallographic studies revealed two different forms of tetramers, suggesting a flexible tetramer model for acetylcholinesterase. Using a recently developed finite element solver for the steady-state Smoluchowski equation, we have calculated the reaction rate for three mouse acetylcholinesterase tetramers using these two crystal structures and an intermediate structure as templates. Our results show that the reaction rates differ for different individual active sites in the compact tetramer crystal structure, and the rates are similar for different individual active sites in the other crystal structure and the intermediate structure. In the limit of zero salt, the reaction rates per active site for the tetramers are the same as that for the monomer, whereas at higher ionic strength, the rates per active site for the tetramers are approximately 67%-75% of the rate for the monomer. By analyzing the effect of electrostatic forces on ACh diffusion, we find that electrostatic forces play an even more important role for the tetramers than for the monomer. This study also shows that the finite element solver is well suited for solving the diffusion problem within complicated geometries.     

Finite element simulations of acetylcholine diffusion in neuromuscular junctions

A robust infrastructure for solving time-dependent diffusion using the finite element package FEtk has been developed to simulate synaptic transmission in a neuromuscular junction with realistic postsynaptic folds. Simplified rectilinear synapse models serve as benchmarks in initial numerical studies of how variations in geometry and kinetics relate to endplate currents associated with fast-twitch, slow-twitch, and dystrophic muscles. The flexibility and scalability of FEtk affords increasingly realistic and complex models that can be formed in concert with expanding experimental understanding from electron microscopy. Ultimately, such models may provide useful insight on the functional implications of controlled changes in processes, suggesting therapies for neuromuscular diseases.     

Synaptic transmission at visualized sympathetic boutons: stochastic interaction between acetylcholine and its receptors.

Excitatory postsynaptic currents (EPSCs) were recorded with loose patch electrodes placed over visualized boutons on the surface of rat pelvic ganglion cells. At 34 degrees C the time to peak of the EPSC was about 0.7 ms, and a single exponential described the declining phase with a time constant of about 4.0 ms; these times were not correlated with changes in the amplitude of the EPSC. The amplitude-frequency histogram of the EPSC at individual boutons was well described by a single Gaussian-distribution that possessed a variance similar to that of the electrical noise. Nonstationary fluctuation analysis of the EPSCs at a bouton indicated that about 120 ACh receptor channels were available beneath boutons for interaction with a quantum of ACh. The characteristics of these EPSCs were compared with the results of Monte Carlo simulations of the quantal release of 9000 acetylcholine (ACh) molecules onto receptor patches of density 1400 microns-2 and 0.41 micron diameter, using a kinetic scheme of interaction between ACh and the receptors similar to that observed at the neuromuscular junction. The simulated EPSC generated in this way had temporal characteristics similar to those of the experimental EPSC when either the diffusion of the ACh is slowed or allowance is made for a finite period of transmitter release from the bouton. The amplitude of the simulated EPSC then exhibited stochastic fluctuations similar to those of the experimental EPSC.     

Cytoplasmic actin in postsynaptic structures at the neuromuscular junction

We used an antibody prepared against Aplysia (mollusc) body-wall actin that specifically reacts with certain forms of cytoplasmic actin in mammalian cells to probe for the presence of actin at the neuromuscular junction. Immunocytochemical studies showed that actin or an actinlike molecule is concentrated at neuromuscular junctions of normal and denervated adult rat muscle fibers. Actin is present at the neuromuscular junctions of fibers of developing diaphragm muscles as early as embryonic day 18, well before postsynaptic folds are formed. These results suggest that cytoplasmic actin may play a role in the clustering or stabilization of acetylcholine receptors at the neuromuscular junction.     

The rising phase of the miniature endplate current at the frog neuromuscular junction

(1) The rising phase of minature endplate currets was recorded at the frog's neuromuscular junction using both the two electrode voltage clamp and a single external electrode, or Strickholm, voltage clamp. (2) The $\text{Q}$(10) of the miniature endplate current rising phase was 2.3 in a variety of solutions selected to alter presynaptic behavior. (3) Increasing the solution's viscosity by an amount sufficient to slow the diffusion coefficient of acetylcholine by a third has no effect on the duration of the rising or the decay phase. This solution does seem to further slow the miniature endplate current decay phase, but not the rising phase, after inhibition of the acetylcholinesterase. (4) As the membrane potential is made more positive, the miniature endplate current rising phase is prolonged, with an $\text{e-}\text{fold}$ slowing per 170 mV change. (5) It is concluded that neither presynaptic nor subsynaptic events determine the rising phase of miniature endplate currents at the frog neuromuscular junction. Rather, the limiting step occurs within the membrane and is most likely a change in the binding constant of the receptor for the acetylcholine molecule.     

Agrin-like molecules at synaptic sites in normal, denervated, and damaged skeletal muscles

Several lines of evidence have led to the hypothesis that agrin, a protein extracted from the electric organ of Torpedo, is similar to the molecules in the synaptic cleft basal lamina at the neuromuscular junction that direct the formation of acetylcholine receptor and acetylcholinesterase aggregates on regenerating myofibers. One such finding is that monoclonal antibodies against agrin stain molecules concentrated in the synaptic cleft of neuromuscular junctions in rays. In the studies described here we made additional monoclonal antibodies against agrin and used them to extend our knowledge of agrin-like molecules at the neuromuscular junction. We found that anti-agrin antibodies intensely stained the synaptic cleft of frog and chicken as well as that of rays, that denervation of frog muscle resulted in a reduction in staining at the neuromuscular junction, and that the synaptic basal lamina in frog could be stained weeks after degeneration of all cellular components of the neuromuscular junction. We also describe anti-agrin staining in nonjunctional regions of muscle. We conclude the following: (a) agrin-like molecules are likely to be common to all vertebrate neuromuscular junctions; (b) the long-term maintenance of such molecules at the junction is nerve dependent; (c) the molecules are, indeed, a component of the synaptic basal lamina; and (d) they, like the molecules that direct the formation of receptor and esterase aggregates on regenerating myofibers, remain associated with the synaptic basal lamina after muscle damage.     

Quantitative simulation of endplate currents at neuromuscular junctions based on the reaction of acetylcholine with acetylcholine receptor and acetylcholinesterase.

Two kinetic models are introduced which predict amplitudes and time-courses of endplate currents and miniature endplate currents at neuromuscular junctions, at both normal and acetylcholinesterase-inhibited endplates. Appropriate differential rate equations reflecting interactions of acetylcholine with acetylcholine receptor and with esterase, diffusion of acetylcholine both within and from the synaptic cleft, and cooperativity between receptor site occupancy and ion channel opening are solved. Acetylcholine release into the cleft is assumed to be instantaneous. The simpler homogeneous reaction space model accurately predicts decay phase time constants are inaccurate. The two-reaction space model predicts amplitudes and time constants within a factor of two of those observed experimentally. The simulations indicate that the amplitudes and time-courses are primarily determined by the chemical reaction rates that characterize acetylcholine interactions with receptor and esterase and that these interactions occur under nonequilibrium conditions. Approximately 50% of the total ion channels in the initial reaction space are predicted to be opened at the peak endplate current. The cooperative opening of ion channels by acetylcholine requires that acetylcholine be introduced into the cleft in discrete, concentrated elements. Virtually all the open channels are confined to the initial reaction space, although acetylcholine-bound receptor sites can be much more widely distributed.     

Counting bungarotoxin binding sites of nicotinic acetylcholine receptors in mammalian cells with high signal/noise ratios

Nicotinic acetylcholine receptors are some of the most studied synaptic proteins; however, many questions remain that can only be answered using single molecule approaches. Here we report our results from single α7 and neuromuscular junction type nicotinic acetylcholine receptors in mammalian cell membranes. By labeling the receptors with fluorophore-labeled bungarotoxin, we can image individual receptors and count the number of bungarotoxin-binding sites in receptors expressed in HEK 293 cells. Our results indicate that there are two bungarotoxin-binding sites in neuromuscular junction receptors, as expected, and five in α7 receptors, clarifying previous uncertainty. This demonstrates a valuable technique for counting subunits in membrane-bound proteins at the single molecule level, with nonspecialized optics and with higher signal/noise ratios than previous fluorescent protein-based techniques.     

Aggregates of acetylcholine receptors are associated with plaques of a basal lamina heparan sulfate proteoglycan on the surface of skeletal muscle fibers

Hybridoma techniques have been used to generate monoclonal antibodies to an antigen concentrated in the basal lamina at the Xenopus laevis neuromuscular junction. The antibodies selectively precipitate a high molecular weight heparan sulfate proteoglycan from conditioned medium of muscle cultures grown in the presence of [35S]methionine or [35S]sulfate. Electron microscope autoradiography of adult X. laevis muscle fibers exposed to 125I-labeled antibody confirms that the antigen is localized within the basal lamina of skeletal muscle fibers and is concentrated at least fivefold within the specialized basal lamina at the neuromuscular junction. Fluorescence immunocytochemical experiments suggest that a similar proteoglycan is also present in other basement membranes, including those associated with blood vessels, myelinated axons, nerve sheath, and notochord. During development in culture, the surface of embryonic muscle cells displays a conspicuously non-uniform distribution of this basal lamina proteoglycan, consisting of large areas with a low antigen site-density and a variety of discrete plaques and fibrils. Clusters of acetylcholine receptors that form on muscle cells cultured without nerve are invariably associated with adjacent, congruent plaques containing basal lamina proteoglycan. This is also true for clusters of junctional receptors formed during synaptogenesis in vitro. This correlation indicates that the spatial organization of receptor and proteoglycan is coordinately regulated, and suggests that interactions between these two species may contribute to the localization of acetylcholine receptors at the neuromuscular junction.     

Phosphorylation and cytoskeletal anchoring of the acetylcholine receptor by Src class protein-tyrosine kinases. Activation by rapsyn.

Src class protein-tyrosine kinases bind to and phosphorylate the nicotinic acetylcholine receptor of skeletal muscle. This study provided evidence for the functional importance of Src kinases in regulating the nicotinic acetylcholine receptor at the neuromuscular junction. Three Src class kinases, Fyn, Fyk, and Src, each formed a complex with the endplate-specific cytoskeletal protein rapsyn. In addition, cellular phosphorylation by each kinase was stimulated by rapsyn in heterologous transfected cells. Several lines of evidence supported rapsyn as a substrate for Src kinases. Most importantly, rapsyn regulation of Fyn, Fyk, and Src resulted in phosphorylation of the nicotinic acetylcholine receptor beta and delta subunits and anchoring of the receptor to the cytoskeleton. Both nicotinic acetylcholine receptor phosphorylation and cytoskeletal anchoring were blocked by the Src kinase-selective inhibitor herbimycin A. Rapsyn alone also induced a modest increase in nicotinic acetylcholine receptor phosphorylation and cytoskeletal translocation. However, inhibition by herbimycin A and a catalytically inactive dominant negative Src demonstrated that the effects of rapsyn were mediated by endogenous Src kinases. These data support the importance of Src class kinases for stabilization of the nicotinic acetylcholine receptor at the endplate during synaptic differentiation at the neuromuscular junction.     

Concurrent musculoskeletal dynamics and finite element analysis predicts altered gait patterns to reduce foot tissue loading

Current computational methods for simulating locomotion have primarily used muscle-driven multibody dynamics, in which neuromuscular control is optimized. Such simulations generally represent joints and soft tissue as simple kinematic or elastic elements for computational efficiency. These assumptions limit application in studies such as ligament injury or osteoarthritis, where local tissue loading must be predicted. Conversely, tissue can be simulated using the finite element method with assumed or measured boundary conditions, but this does not represent the effects of whole body dynamics and neuromuscular control. Coupling the two domains would overcome these limitations and allow prediction of movement strategies guided by tissue stresses. Here we demonstrate this concept in a gait simulation where a musculoskeletal model is coupled to a finite element representation of the foot. Predictive simulations incorporated peak plantar tissue deformation into the objective of the movement optimization, as well as terms to track normative gait data and minimize fatigue. Two optimizations were performed, first without the strain minimization term and second with the term. Convergence to realistic gait patterns was achieved with the second optimization realizing a 44% reduction in peak tissue strain energy density. The study demonstrated that it is possible to alter computationally predicted neuromuscular control to minimize tissue strain while including desired kinematic and muscular behavior. Future work should include experimental validation before application of the methodology to patient care.     

Congenital myasthenic syndromes: spotlight on genetic defects of neuromuscular transmission

The neuromuscular junction (NMJ) is a complex structure that efficiently communicates the electrical impulse from the motor neuron to the skeletal muscle to induce muscle contraction. Genetic and autoimmune disorders known to compromise neuromuscular transmission are providing further insights into the complexities of NMJ function. Congenital myasthenic syndromes (CMSs) are a genetically and phenotypically heterogeneous group of rare hereditary disorders affecting neuromuscular transmission. The understanding of the molecular basis of the different types of CMSs has evolved rapidly in recent years. Mutations were first identified in the subunits of the nicotinic acetylcholine receptor (AChR), but now mutations in ten different genes - encoding post-, pre- or synaptic proteins - are known to cause CMSs. Pathogenic mechanisms leading to an impaired neuromuscular transmission modify AChRs or endplate structure or lead to decreased acetylcholine synthesis and release. However, the genetic background of many CMS forms is still unresolved. A precise molecular classification of CMS type is of paramount importance for the diagnosis, counselling and therapy of a patient, as different drugs may be beneficial or deleterious depending on the molecular background of the particular CMS.     

Evaluation of characteristic parameters for the neurotransmitter release mechanisms at the neuromuscular junction

The process of neurotransmitter release at the neuromuscular junction needs to be represented appropriately in modeling of the synaptic chemical transmission as a reaction-diffusion system. The release mechanisms of the expanding pore and the acceleration are analyzed by the computer simulation with respect to the effects of the characteristic parameters in the mechanisms on spontaneous generation of the miniature endplate current (MEPC), leading to the following evaluation. In the expanding pore mechanism the expanding rate of the pore more than 10 nm ms(-1) and the diffusion coefficient of acetylcholine in the synaptic cleft (D(c)) of about 1.0 x 10(-6) cm2 s(-1) yield the maximum amplitude, the rise time and the decay time constant of the MEPC in agreement with the empirical data. In the active release mechanism the 10-fold acceleration of the natural diffusion and a similar value of D(c) are required to suit for the empirical MEPC.     

Molecular targets for autoimmune and genetic disorders of neuromuscular transmission.

The neuromuscular junction is the target of a variety of autoimmune, neurotoxic and genetic disorders, most of which result in muscle weakness. Most of the diseases, and many neurotoxins, target the ion channels that are essential for neuromuscular transmission. Myasthenia gravis is an acquired autoimmune disease caused in the majority of patients by antibodies to the acetylcholine receptor, a ligand-gated ion channel. The antibodies lead to loss of acetylcholine receptor, reduced efficiency of neuromuscular transmission and muscle weakness and fatigue. Placental transfer of these antibodies in women with myasthenia can cause fetal or neonatal weakness and occasionally severe deformities. Lambert Eaton myasthenic syndrome and acquired neuromyotonia are caused by antibodies to voltage-gated calcium or potassium channels, respectively. In the rare acquired neuromyotonia, reduced repolarization of the nerve terminal leads to spontaneous and repetitive muscle activity. In each of these disorders, the antibodies are detected by immunoprecipitation of the relevant ion channel labelled with radioactive neurotoxins. Genetic disorders of neuromuscular transmission are due mainly to mutations in the genes for the acetylcholine receptor. These conditions show recessive or dominant inheritance and result in either loss of receptors or altered kinetics of acetylcholine receptor channel properties. Study of these conditions has greatly increased our understanding of synaptic function and of disease aetiology.     

A proposed model for the synthesis,storage and release of acetylcholine at the neuromuscular junction

Sythesis, storage and release of acetylcholine at the neuromuscular junction are still area of speculation and controversy. This paper seeks to introduce a lumped parameter mathematical model which may be used to describe and quantify some of the various processes involved. The construction of the model is fully discussed, compared with previous models, and its shortcomings noted.     

The mechanism of agrin-induced acetylcholine receptor aggregation.

Agrin, a protein isolated from the synapse-rich electric organ of Torpedo californica, induces the formation of specializations on myotubes in culture which resemble the post-synaptic apparatus at the vertebrate skeletal neuromuscular junction. For example, the specializations contain aggregates of acetylcholine receptors and acetylcholinesterase. This report summarizes the evidence that the formation of the post-synaptic apparatus at developing and regenerating neuromuscular junctions is triggered by the release of agrin from motor axon terminals and describes results of recent experiments which suggest that agrin-induced tyrosine phosphorylation of the beta subunit of the acetylcholine receptor may play a role in receptor aggregation.