A solvable model for the diffusion and reaction of neurotransmitters in a synaptic junction
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The diffusion and reaction of the transmitter acetylcholine in neuromuscular junctions and the diffusion and binding of Ca2+ in the dyadic clefts of ventricular myocytes have been extensively modeled by Monte Carlo simulations and by finite-difference and finite-element solutions. However, an analytical solution that can serve as a benchmark for testing these numerical methods has been lacking.
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.
KeywordsAcetylcholine Sarcoplasmic Reticulum Synaptic Vesicle Neuromuscular Junction Synaptic Cleft
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, 2, 3, 4, 5, 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.
We set up a coordinate system such that the x and y axes are parallel to the pre- and post-synaptic membranes. The synaptic junction has depth L z . The junction is periodic in the x and y directions, with periodicities of L x and L y , respectively. In each "unit cell", a synaptic vesicle bursts at time t = 0, releasing the neurotransmitters into the cleft. We model the release of the neurotransmitters as a transient flux, u(t), that is confined to a circular opening with radius R. We place the synaptic vesicle at the center of the pre-synaptic face of the unit cell. After diffusing to the post-synaptic membrane, neurotransmitters are absorbed by a circular disk in each unit cell, with radius a. We place this "sink" also at the center of the post-synaptic face of the unit cell. The exact shapes and locations of the pre-synaptic opening and the post-synaptic sink are not essential for the analytical solution of our model. The quantity of interest is the total flux, J(t), through the post-synaptic face of each unit cell.
The coefficients α lm (s) and β lm (s) are to be determined from boundary conditions.
which is independent of a. Equation (17b) is simply a consequence of ligand conservation, i.e., the total number of ligands released from the synaptic vesicle is the same as the total number of ligands absorbed by the receptors.
We now briefly describe the details of our implementation of the analytical solution. To calculate the q lm and p lm coefficients of Eqs. (8) and (9), we first carried out the integration over y analytically. The remaining integration over x was done numerically using the Gauss-Legendre quadrature with two points. The summations over l and m in Eq. (15) were truncated at l = m = 40. The Laplace transform of was inverted by the Stehfest algorithm . A Fortran90 code for the implementation is available upon request.
We now present some illustrative results. The parameters of our model are as follows: L x = L y = 500 nm; L z = 50 nm; R = 20 nm; a varied from 2.5 to 40 nm; t0 varied from 1 to 10 ms; and D varied in (0.4-4) × 105 nm2/ms.
4. Discussion and Conclusion
We have presented an analytical solution to a model for the diffusion and reaction of acetylcholine in a neuromuscular junction. The model also applies to the diffusion and binding of Ca2+ in a dyadic cleft. Our results are qualitatively similar to those obtained previously from models solved numerically [1, 2, 3, 4, 5, 6, 7].
Perhaps the greatest value of our analytical solution is that it provides a benchmark for testing numerical methods. Diffusion and reaction of ligands in intercellular and intracellular spaces have been modeled either on a particle description or a concentration description. The former type of models have been solved by Monte Carlo simulations [1, 7], while the latter type of models have been solved by either finite-difference [2, 6] or finite-element [3, 4, 5] methods. The two types of models have been shown to give equivalent results . The level of realism of our model approaches those of the models solved numerically; hence our analytical solution will be able to serve as a good benchmark for the numerical methods.
One can thus parameterize a and κ by matching k with experimental data for the ligand-receptor binding rate constant.
We have modeled ligand-receptor binding as irreversible. This is somewhat justified for modeling the neuromuscular junction, in which acetylcholinesterase can break down acetylcholine molecules newly released from the receptors. No such mechanism is present for Ca2+ in the dyadic cleft. Reversible binding can be treated by appropriate boundary conditions  on the post-synaptic face. Another important detail is that both acetylcholine and ryanodine receptors have multiple binding sites for their ligands so that there are multiple ligand-occupation states for the receptors. Again, these can be treated by appropriate boundary conditions.
The geometries of some of the models previously solved numerically are more sophisticated than that of our model. In particular, secondary folds of the neuromuscular junction has been included in some of the previous models [1, 3, 4, 5]. A formalism for treating ligand binding to a site buried in a narrow tunnel has been developed  and may be adopted for treating the narrow secondary folds in the neuromuscular junction. However, analytical solution requires idealized geometries; the kind of realistic geometries drawn from electron microscopy that can be handled by a finite-element method [4, 5] is beyond the reach of analytical solution. Nevertheless, with all the new ingredients outlined above, analytical solution will potentially provide a new avenue for modeling biochemical transport.
This work was supported in part by Grant GM58187 from the National Institutes of Health.
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