Department of Physics and Institute of Molecular Biophysics, Tallahassee, Florida 32306, USA

Abstract

Background

The diffusion and reaction of the transmitter acetylcholine in neuromuscular junctions and the diffusion and binding of Ca^{2+ }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.

Result

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 Ca^{2+ }in a dyadic cleft. Our model is similar to those previously solved numerically and our results are also qualitatively similar.

Conclusion

The analytical solution provides a unique benchmark for testing numerical methods and potentially provides a new avenue for modeling biochemical transport.

1. Background

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 ^{2+ }in the dyadic cleft of a ventricular myocyte ^{2+ }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 ^{+ }to flow in and generating an action potential along the muscle fiber. Finally excess acetylcholine molecules around the post-synaptic membrane are broken down by acetylcholinesterase to prevent continued activation of acetylcholine receptors.

Illustration of a neuromuscular junction

**Illustration of a neuromuscular junction**. Presented are the key players in the diffusion and reaction of the neurotransmitter, acetylcholine (ACh).

A related system is a dyadic cleft, which spans the gap between the cell membrane in a transverse tubule and the membrane of a sarcoplasmic reticulum. As Figure ^{2+ }can enter the cell through L-type Ca^{2+ }channels on the cell membrane in response to the arrival of an action potential. The ions then diffuse to reach and activate ryaonodine receptors in the membrane of the sarcoplasmic reticulum. The activated ryaonodine receptors release Ca^{2+ }from the sarcoplasmic reticulum, which ultimately lead to muscle contraction.

Illustration of a dyadic cleft

**Illustration of a dyadic cleft**. Presented are the key players in the diffusion and binding of Ca^{2+}.

Here we propose a simple but not unrealistic model for the diffusion and reaction of acetylcholine in a neuromuscular junction. The model also applies to the diffusion and binding of Ca^{2+ }in a dyadic cleft. We are able to find an analytical solution for this model. For convenience we describe our model in the language of neuromuscular junction. As shown in Figure

Our model for both a neuromuscular junction and a dyadic cleft

**Our model for both a neuromuscular junction and a dyadic cleft**. Panel A: The pre-cleft membrane (top) contains a periodic array of disks for the influx of ligands; the post-cleft membrane (bottom) contains a periodic array of absorbing disks representing receptors. Panel B: The dimensions of a unit cell.

2. Methods

We set up a coordinate system such that the _{z}
_{x }
_{y}

We place the origin of the Cartesian coordinate system at the center of the pre-synaptic face, with the **r **and at time **r**,

where

The boundary condition at

where ^{2 }+ ^{2})^{1/2 }is the distance to the _{z }

We solve the problem in Lapace space. For a function

The solution appropriate for the periodic boundary conditions in the

where

The coefficients _{lm}
_{lm}

To make use of the boundary condition of Eq. (3), we need the 2-dimensional cosine transform of a function in _{x}
_{y}

where _{0 }= 1/2 and _{l }

Applying the boundary condition of Eq. (3), we find

For the boundary condition of Eq. (4) at _{z}

where the quantity

Equation (11) leads to

With Eqs. (10) and (13), we solve for coefficients _{lm}
_{lm}

Inserting Eq. (6) in Eq. (12) and using Eqs. (14) and (9), we find

Finally the total flux through the sink on the post-synaptic face is

The flux accumulated over all times,

is of interest. In our model,

which is independent of

The analytical solution derived above can be implemented on any function

Its Laplace transform is

Our model then contains two parameters related to time: _{0}. For two sets of these parameters, e.g., _{1 }and _{01 }in one and _{2 }and _{02 }in the other, it can be shown that the corresponding response functions satisfy

We now briefly describe the details of our implementation of the analytical solution. To calculate the _{lm }
_{lm }

3. Results

We now present some illustrative results. The parameters of our model are as follows: _{x }
_{y }
_{z }
_{0 }varied from 1 to 10 ms; and ^{5 }nm^{2}/ms.

Our single absorbing disk is used to model ligand binding to multiple receptors. Increasing _{rec}, of receptors per unit cell. We expect _{rec }when _{rec }is small; the increase in _{rec }(see Discussion). Figure _{rec}). As _{rec}) increases, ^{2}
_{0}. Note also that all our

Effect of the size of the absorbing disk on the response function

**Effect of the size of the absorbing disk on the response function**. The values of _{0 }= 1 ms and ^{5 }nm^{2}/ms.

Figure _{0 }so that, effectively, the total number of neurotransmitters entering the synaptic cleft is fixed. It is clear that, as _{0 }increases (i.e., as the speed of neurotransmitter release decreases), the response function rises and decays more slowly. This behavior has previously been specifically modeled by Stiles et al.

Effect of the speed of ligand release on the response function

**Effect of the speed of ligand release on the response function**. Scaling of _{0 }is to ensure that the same number of neurotransmitters is released for all the curves with different _{0 }values, which are shown in the figure in ms. ^{5 }nm^{2}/ms.

There is some uncertainty on the diffusion constant of acetylcholine in neuromuscular junctions. In previous models ^{5 }nm^{2}/ms. In Figure ^{5 }nm^{2}/ms. As expected, when neurotransmitter diffusion slows down, the response function is also delayed. In our model, decreasing _{0 }[see Eq. (19)].

Effect of the ligand diffusion constant on the response function

**Effect of the ligand diffusion constant on the response function**. The values of ^{5 }nm^{2}/ms are shown in the figure. For all curves, _{0 }= 1 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 Ca^{2+ }in a dyadic cleft. Our results are qualitatively similar to those obtained previously from models solved numerically

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

Our model has room for increasing the level of realism and still allows for analytical solution. For example, we modeled ligand binding to receptors as absorbing. The binding can be modeled as partially absorbing if binding does not occur at every ligand-receptor encounter. The partial absorption condition takes the form

where the reactivity

One can thus parameterize

Another simplification of our model is that a single absorbing disk is used to represent the receptors. We accounted for the presence of multiple receptors per unit cell by increasing the radius _{rec }receptors. For the single absorbing disk we have _{rec }receptors is modeled as an absorbing disk with a small radius _{0}, then _{rec}
_{0 }when _{rec }is small _{rec}
_{0 }for small _{rec}. As _{rec }increases, the rate constant _{rec }becomes large

where _{x}L_{y }
_{rec }= _{rec}
_{0}
^{2}/_{0 }is given by

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 Ca^{2+ }in the dyadic cleft. Reversible binding 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

Authors' contributions

JLB implemented the analytical solution, did the calculations, and prepared the figures. HXZ derived the analytical solution and wrote the paper. Both authors read and approved the final manuscript.

Acknowledgements

This work was supported in part by Grant GM58187 from the National Institutes of Health.