Edited by: Gennady Cymbalyuk, Georgia State University, United States
Reviewed by: Victor Matveev, New Jersey Institute of Technology, United States; Pragya Goel, Harvard Medical School, United States; Xiaomin Xing, The University of Iowa, United States
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
The relation of form and function, namely the impact of the synaptic anatomy on calcium dynamics in the presynaptic bouton, is a major challenge of present (computational) neuroscience at a cellular level. The Drosophila larval neuromuscular junction (NMJ) is a simple model system, which allows studying basic effects in a rather simple way. This synapse harbors several special structures. In particular, in opposite to standard vertebrate synapses, the presynaptic boutons are rather large, and they have several presynaptic zones. In these zones, different types of anatomical structures are present. Some of the zones bear a so-called T-bar, a particular anatomical structure. The geometric form of the T-bar resembles the shape of the letter “T” or a table with one leg. When an action potential arises, calcium influx is triggered. The probability of vesicle docking and neurotransmitter release is superlinearly proportional to the concentration of calcium close to the vesicular release site. It is tempting to assume that the T-bar causes some sort of calcium accumulation and hence triggers a higher release probability and thus enhances neurotransmitter exocytosis. In order to study this influence in a quantitative manner, we constructed a typical T-bar geometry and compared the calcium concentration close to the active zones (AZs). We compared the case of synapses with and without T-bars. Indeed, we found a substantial influence of the T-bar structure on the presynaptic calcium concentrations close to the AZs, indicating that this anatomical structure increases vesicle release probability. Therefore, our study reveals how the T-bar zone implies a strong relation between form and function. Our study answers the question of experimental studies (namely “Wichmann and Sigrist, Journal of neurogenetics 2010”) concerning the sense of the anatomical structure of the T-bar.
Understanding the plasticity of neuronal connections, neuronal activity, and the relation of form and function of neuronal anatomical structures is a major challenge of present day neuroscience.
Of special importance are the relation of calcium dynamics, anatomical shapes, and release probabilities of synaptic vesicles at AZs of neuronal synapses. For an extended overview we refer to e.g., Naraghi and Neher,
Here, we repeat in brief the major elements relevant as the basis for our study:
When an axon of a nerve is stimulated with an action potential, the elevated voltage causes calcium influx at the voltage-gated calcium channels of the AZs. This means that calcium is transported actively from the synaptic cleft into the presynaptic boutons. The calcium that enters the presynaptic boutons creates a calcium microdomain close to the release zone. Calcium microdomains trigger the fusion of presynaptic vesicles with the membrane
In more detail: after the calcium has entered the presynaptic zone, it experiences several processes that reduce its concentration again rather soon: It diffuses away inside the presynaptic boutons, it gets buffered with the aid of specific buffers such as calbindin, and calcium pumps pump it again into the synaptic cleft. Therefore, the calcium microdomain will be decreased soon again. Anyhow, the small but strong calcium influx has an impact upon the vesicles, especially for those who belong to the readily releasable pool, which presumably is located close to the AZ. Vesicle release probability is related to calcium concentration in presynaptic AZs. Higher calcium concentrations in the calcium microdomains, which arise due to action potential stimulation enhance vesicle release probability. At each action potential, the calcium concentration peak triggers vesicle release (Parsons et al.,
In the case of many synapses, the presynaptic region close to the AZ can be considered to be more or less homogeneous besides the vesicles, which are located there.
However, many synapses show distinct morphological features such that the presynaptic AZ bears specific anatomic structures, which might have an influence upon calcium dynamics and profile (Chou et al.,
Hence, distinct morphological structures and features are important for the function of several synapse types of neuronal transmission.
Indeed, some synapses harbor a special structure called a T-bar (cf. e.g., the review Wichmann and Sigrist,
Thus, a prominent example of a geometric non-homogeneous anatomical structure within an AZ is the T-bar structure of the Drosophila neuromuscular junction (NMJ) (Jan and Jan,
Indeed, the Drosophila larval NMJ is a relatively simple model system of synaptic plasticity, as it is an NMJ with only glutamatergic synapses. It harbors only excitatory effects, and no inhibitory ones, so that action potential input and evoked postsynaptic potentials (EPSPs) can be directly related. Therefore, this NMJ often serves as the basis for studying the biophysical interplay, which appears in the presynaptic vesicle cycle, the calcium dynamics, and the postsynaptic glutamate receptors. Concerning literature in this context, we refer to e.g., Allbritton et al.,
In the review, Wichmann and Sigrist (
More recent experimental studies (Graf et al.,
All these studies unveiled important aspects of the T-bar such as size, its impact upon calcium dynamics and vesicle dynamics, as already summarized by Wichmann and Sigrist (
In this study, we consider the dynamics of calcium, which underlies vesicle dynamics. In detail, we model and simulate the calcium current dynamics and calcium microdomain concentration of one AZ under action potential stimulation of the NMJ. We consider the relation of form and function, which appear at single AZs of the presynaptic boutons of the Drosophila larval NMJ.
To unveil the role of the T-bar at the active zone, we need to compare the different AZ types of the presynaptic boutons of the Drosophila larval NMJ. In detail, at the presynaptic boutons, there are AZs, which in part bear a T-bar, while others do not bear a T-bar. Thus, the major aim of this study will be to unveil the relation between the T-bar anatomical obstacle, its presence and absence, and the impact of this specific structure upon the calcium concentration profile within the AZ, close to the membrane. Our aim is to compare the calcium dynamics for AZs with T-bar with AZs without T-bar.
To perform the comparison of AZs with T-bar and AZs without T-bar, in this study, we generate
In order to investigate the interplay of the shape of the AZ and the distribution of the calcium channels, we developed a typical 3D geometry of a synapse with a T-bar and compared the calcium concentrations close to the AZ for the case of geometry with a T-bar and clustered channels with an AZ without T-bar, but still clustered channels, and an active zone without T-bar and without clustering of the calcium channels.
We simulated calcium influx under repetitive action potential stimulation and compared the calcium concentrations for the different cases in a fully 3D spatio-temporal resolved manner. This indicates that we calculated the calcium concentrations within the AZ at single points of the 3D space to reveal exactly the calcium distribution not only averaged over some compartments but fully resolved in 3D space and time. We evaluate the calcium concentrations as if we would have a high resolution microscope, but just
The study we present here is a purely theoretical study, without any earlier unpublished experimental data input, but using a significant amount of biological model parameters taken from the literature. The major questions of our study are to unveil the influence of the T-bar anatomic obstacle and the clustering structure of the calcium channels upon the calcium microdomain in the presynaptic AZ. Our
This means that the T-bar obstacle has a substantial impact upon calcium microdomain concentration and shape, and this indicates that the T-bar anatomical structure likely has an important role to enhance vesicle release probability, as the concentration of calcium at the vesicular release sites is a major trigger of synaptic activity. Various biophysical models and simulations deal with the dynamics of this calcium in the context of synaptic activity, such as Naraghi and Neher (
The T-bar AZ likely is an optimal object to study the relation of form and function and their interplay within synaptic processes such as calcium dynamics and vesicle release probability.
Indeed, there exist several studies which considered the influence of an obstacle for calcium dependent vesicle exocystosis (cf. e.g., Kits et al.,
Hence, there is ongoing modeling and simulation research to unravel the role of calcium at release sites, and it is promising to apply modeling and simulation techniques to the calcium dynamics processes appearing at the Drosophila NMJ synapses.
In Section 2, Materials and Models, we introduce the geometries representing typical anatomical scenarios for the synapses under consideration. The meshes created for these geometries will serve as the basis for the simulations. In particular, we introduce the mathematical model of calcium diffusion, reaction, and influx we use to describe calcium dynamics at the T-bar zone of the Drosophila NMJ larval synaptic AZ. In Section 3, Results, we present the simulation results of calcium influx and evaluation of calcium concentrations at the different geometric scenarios for the presynaptic AZ of the Drosophila larval NMJ boutons. In particular, we present the calcium concentration profiles in a spatially resolved manner and show how the calcium concentration behaves in the different scenarios. We can clearly relate the T-bar geometric obstacle with a quantitatively clearly enhanced calcium concentration at the AZ. In Section 4, Discussion, we interpret the simulation results and relate them to unravel the relation of form and function of the T-bar at the presynaptic AZ of the Drosophila larval NMJ boutons. In Section 5, Conclusion, we put our study in the broader context of present scientific research in the field of the interplay of calcium and vesicle dynamics at synaptic zones.
To motivate the construction of the anatomy we use as the basis for our simulations, we re-consider the basic process of calcium influx, calcium microdomains, and vesicles in the AZ, cf.
Aim of this figure: Display the biological process under consideration, and display the realization of the geometry of the anatomic structures for the three different anatomic structures under consideration.
We made a mathematical model describing calcium dynamics in three dimensions in space plus in time, and simulated the influx of calcium caused by repetitive action potential stimulation. In particular, we modeled the intracellular calcium concentration by means of a diffusion-reaction model.
As N-type calcium channels are the major voltage-gated calcium channel type at synaptic terminals (Weber et al.,
Ultrastructural analysis of electron microscope images of the AZs at the presynaptic zones of the presynaptic boutons of the Drosophila NMJ revealed anatomical structures resembling “T-shaped structures” (Shaw and Meinertzhagen,
To facilitate the naming of “roof” and “base” for readers, which have their origin more in the simulation community, we will use the more figurative words “table socket” and “table desk” in a synonymous way.
Using our mesh creation tool ProMesh4 (Reiter,
The calcium channels are distributed in a continuous manner allover the 2D influx zone
As the calcium concentration far away from the AZ can be assumed to be small, as there is no influx, we may consider a region big enough around the T-bar AZ for our computations as the region of interest, where the simulations have to be performed, as the rest of the inner presynaptic zone will have negligible influence on the calcium concentrations, so we do not need to consider complete presynaptic boutons, in contrast to our former study about the relation of form and function of the presynaptic boutons (Knodel et al.,
In the case of the T-bar, the T-bar space is left blank in order to account for the obstacle of the T-bar, in the other case, the full space is meshed, in the case of the T-bar, we displayed a cut through the coarse form of the volume mesh, (cf.
Aim of this figure: Display the computational domain in order to explain the geometric Finite Element mesh basis of the simulation. Computational domain displayed opened by a cut plane to allow insight into the region where the T-bar AZ is located, and to explain the subdomains which are important to understand the regions where computations are evaluated in this study.
The computational domain is denoted as
We model calcium influx and ion distribution by means of a diffusion-reaction 3D PDE model. Note that the use of a continuum approach to calculate calcium ion distribution has to be interpreted as a probability to find an ion in a certain location, similar to e.g., the Schrödinger equation in quantum mechanics to find an electron. In principle, the number of calcium (Ca2+) ions under the T-bar table is comparably few. The use of a continuum model, in fact, resembles the averaging of a huge number of computations performed e.g., with random walk algorithms. If we would perform a single computation with e.g., random walk models, the results would have to be averaged anyhow at the end. The use of a continuum model in the sense of a distribution probability enables to homogenize the model and to perform much fewer computations to arrive at the same biophysical result. These homogenization techniques are well-known since Einstein's derivation of the diffusion equation based on random walk models (Einstein,
Several more recent studies serve as practical examples showing that Einstein's homogenization approach for the derivation of the diffusion equation based on random walk scenarios applies also to the case of random walk based calcium dynamics, such that the diffusion-reaction equation approach even is justified in the presence of bi-molecular binding interactions (Hake and Lines,
Our mathematical model consists of two concentrations, namely calcium and a calcium buffer. As in general, calbindin is the major buffer of calcium within presynaptic regions, we assume that the buffer substance is calbindin (Nadkarni et al.,
for the concentrations of calcium [
The kinetic reactions follow the law of mass action (LaMA) (Neher,
Parameters of the partial differential equation (PDE) simulations Equations (1), (2).
200 | [μm2 s−1] | Allbritton et al., |
|
30 | [μm2 s−1] | Luby-Phelps et al., |
|
0 - 44 | [s−1 (μM)−1] | Nadkarni et al., |
|
0 - 36 | [s−1] | Nadkarni et al., |
|
50 | [nM] | Schneggenburger and Neher, |
|
[ |
40 | [μM] | Neher, |
1.5 | [mM] | Egelman and Montague, |
|
8·10−24 | [mol/s] | Graupner, |
|
60 | [nM] | Elwess et al., |
|
2 | Graupner, |
||
2.5·10−21 | [mol/s] | Graupner, |
|
1.8 | [μM] | Graupner, |
|
1 | Graupner, |
Furthermore, we use the notations
That is, we assume constant extracellular calcium concentration
As the initial condition for the concentrations inside the computational domain, we choose constant values all over the computational domain, which are based on the values given in
the initial values of free calcium and free buffer are computed as Neher (
At the sides and the top of the computational domain, i.e., at
as we assume the bouton to be sufficiently large so that concerning short-term plasticity, the VGCC caused calcium influx will not have a substantial influx upon the total calcium concentration inside the bouton but just for the calcium microdomains at the AZ.
At the ground of the cuboid, we impose Neumann flux conditions.
Except for the case of channel zone
At the surface where the VGCCs are located, i.e., at the channel zones
Written in a closed form, the Neumann flux boundary conditions read:
with the membrane current given per surface unit
The membrane current/flux is the superposition of the voltage-gated calcium (VGCC) influx current/flux
Note that the membrane streams are multiplied with their corresponding densities per unit area. For example, the density of the VGCC channels ρ
To describe the VGCCs of N-type calcium channels, we use the model published in Borg-Graham (
The buffers, pumps, and leakage are incorporated technically as described by Graupner (
The dynamics of ionic fluxes of VGCC in the case of a neuron membrane entering (11) is described in the following way (published in Borg-Graham
Thus, we have a product of a (or several) gating functions
The current for a single channel, i.e., the flux of calcium can be described as by means of the Goldman-Hodgkin-Katz equation (GHK) (Jack et al.,
is given in [A] (Ampère), where
The gating functions may be a product of several functions of type, usually of the form
The powers of
The
which is a Hodgkin-Huxley-like model, adapted for calcium and other ion types by Borg-Graham (
We assume that the VGCCs are distributed continuously over the surface
In both cases, we assume 6 VGCC calcium channels of N-type (Borg-Graham,
The channel surface
The channel surface
Note that in the case of not clustered channels, the overall surface covered by the channels is larger, such that the total number of channels is the same in all cases.
Despite the fact that for our standard parameter setup, we assume 6 VGCCs per AZ for the main results, we also performed a sensitivity analysis using other numbers of VGCCs per AZ: Since the channel number is only an estimation and has not been measured experimentally to our knowledge, we investigate the sensitivity of our results with respect to the channel number. We probe for a wide range of other numbers of VGCCs per (AZ), ranging from only 1 up to 200, as Nadkarni et al. (
Nevertheless, we assume that the number of VGCCs at the Drosophila larval NMJ will be much lower than 200 per AZ, and presumably is in the region between 1 up to maximally 10. Our results reported below show results for the variation of the VGCCs in the range of Nadkarni et al. (
For the PMCA and the NCX current (Graupner,
with the values given in
Note that following (Graupner,
For the density of the PMCA pumps ρ
The leak current density and the leak current were adjusted such that at rest, inner calcium concentration is at equilibrium in the absence of action potentials.
In this study, we considered as well the case with nonzero buffering constant
The major studies, we perform with the case with the buffer reaction parameters from the literature, as this case reflects the “worst-case” scenario for concentrations to develop a difference between the geometries - if more calcium gets bound, the diffusion barrier of the T-bar anatomical structure likely will have less influence on the calcium accumulation at the active zone compared to the case without calcium reduction due to bindings to the buffer.
The parameters used in this study and their respective literature basis are reported in
We impose an action potential at the synaptic membrane “felt” by the calcium channels. The action potential has a typical form and is displayed by (
Among the simulations, we compared the arising calcium concentrations over time, and depending on the distance from the center of the AZs, we created circles around the T-bar socket with different heights, to evaluate in later steps the calcium concentrations for specific radii (distance from the T-bar socket) and heights (measured from the ground of the AZ, i.e., the membrane).
where
where we choose Δ
is the vector to the single circles, and it is given by means of cylinder coordinates. This means that in the first and second directions, we have spherical coordinates in two dimensions, whereas in the third coordinate, we have a simple value. The spherical coordinates describe a circle in two dimensions with a given radius around the center, which is given by the third component vector. The different values of
Aim of this figure: Display the regions where the calcium concentrations will be evaluated which arise within our simulations. The regions of evaluations are circles around the T-bar “socket”. The circles are placed such that they cover the region below the T-bar “roof” in a rather dense way, as well in height computed from the membrane base, and also in distance from the T-bar “socket”. In the Results, Section 3, we evaluate the averaged concentrations of each circle, as evaluations at only single points might not be so favorable, even though we also check that our results are independent of the computational grid. Constructing circles around the T-bar socket to measure calcium at equidistant radii and heights from the T-bar socket. Note the thin magenta colored lines around the table leg, which mark the evaluation lines. Circles constructed based upon (18).
The PDEs are discretized by means of a vertex-centered finite volume scheme for unstructured grids and solved by means of a Newton solver. The arising linear equations are solved with the aid of a BiCGSTAB solver preconditioned with a geometric multigrid GMG.
Our simulations are performed with the simulation framework UG4 (Heppner et al.,
Note that at the first glance, one could get the impression that in Equation (13), the value of intracellular calcium
However, indeed, this does not hold true for that type of the vertex centered finite volume methods we use, as we apply a weak formulation-based approach (Bey,
Within the weak solution-based vertex-centered finite volume approach, the solutions are not simply computed element by element, but moreover in a continuous manner. In particular, the starting point of weak methods based on finite volume or finite element approaches is
To ensure numerical robustness, numerical grid convergence tests can be applied, and we perform this procedure to ensure numerical grid convergence of our results.
The number of degrees of freedom (DoFs) for the geometries and the corresponding refinement level of the 3D tetrahedral volume mesh are displayed in
Degree of freedom (DoF) number for the different geometric configuration volume meshes, for the different levels of grid refinement.
0 | 690 | 542 |
1 | 4,778 | 3,842 |
2 | 35,018 | 28,890 |
3 | 266,802 | 223,954 |
4 | 2,079,970 | 1,763,362 |
Several years ago, there were attempts to model the interplay of calcium and vesicle dynamics based on 3D PDEs, which were solved by simplifying the equations for enabling analytic solutions by Fogelson and Zucker (
All these examples, algorithm developments and program codes are caused by the need to understand the relation of form and function of e.g., synaptic processes at a cellular level. Of course, the list of articles and programs in the field is not intended to be complete, as here, we do not write a review but just want to give typical examples of the field.
Of course, we do not claim that our simulation framework is superior to the others used in the field. Nevertheless, we believe our framework helps to consider additional research topics in a new light and complements the standard methods with stimulating techniques.
In order to facilitate qualitative understanding of the mathematically quite complex model scenario described in the Materials and Model Section 2 also for experimental scientists which are not familiar with mathematically complex formulae, but with complex experimental setups, we repeat in brief the major aspects of our simulations before evaluating in detail the simulation results.
We applied repetitive action potential stimulation to the membrane of the AZ. This action potential stimulation was realized technically by imposing a given time dependent voltage, which was applied to the calcium N-type channels located at the membrane of the AZ. We simulated the influx of calcium into the active zone by means of VGCC calcium N-type channels and the intracellular calcium microdomain concentration dynamics by means of a diffusion-reaction model. We considered the case of three different geometrical scenarios (note that in this context, the geometrical and anatomical scenarios can be considered as synonyms):
Active zone with T-bar and calcium channels clustered around the T-bar socket, directly at the presynaptic AZ membrane.
Active zone without T-bar, but still clustered calcium channels around an “imaginative” T-bar socket.
Active zone without a T-bar, and the calcium channel number is the same as in the two other scenarios, but the channels are distributed over a bigger surface, such that this example realizes the case of not clustered calcium channels.
Simulation of calcium and buffer concentration under repetitive action potential stimulation for the three different anatomical cases. Screenshots are taken at the first calcium microdomain peak. Full simulation movies are available as
This figure displays the simulation movie screenshot at peak of calcium influx in a zoomed version of the calcium concentration for the three different anatomic scenarios, i.e., it is a magnified version of
The 3D simulation movies which we supply can be viewed as some sort of “
Once an action potential arrives, in all three scenarios, calcium enters the presynaptic bouton close to the AZ. Inside the presynaptic bouton, the calcium concentration moves by means of diffusion and reacts with the buffer. (Note that we also display the free buffer concentration, but in our textual description, we focus upon calcium concentration).
In all three anatomic cases, at each action potential, a calcium microdomain appears close to the channels in the presynaptic boutons.
In all three cases, once when calcium enters, free buffer binds to calcium but has only a limited effect upon the microdomain shape. The free buffer amount decreases once calcium enters and the shape of the buffer concentration mirrors the calcium concentration in an inverse way, as the free buffer decreases when calcium increases.
As calcium influx only appears at the peaks of the action potentials, after when the voltage decreases again, the calcium microdomain disappears quite fast in all three anatomical cases. This means that the calcium in the major part diffuses away, whereas a small part of the calcium gets pumped out again.
As the boutons are considered to be quite big, also the buffer is replenished rather fast by means of diffusion from the other regions of the presynaptic bouton.
So before when the next action potential arrives, the presynaptic calcium and buffer concentrations are already again at effectively the same level as at the very beginning, before when stimulation started.
When the next action potential arrives, the calcium microdomain effect happens again as in the former case.
When comparing the three geometric scenarios using the same scale for the simulations, we see that in the case of the presence of the T-bar, more calcium accumulates close to the channels than in the two cases without T-bar. This is obviously due to the T-bar “table desk” obstacle, which slows down the spread of calcium inside the presynaptic bouton. It is obvious that the T-bar has a major effect on the calcium microdomain peak concentration. Furthermore, comparing the two cases without a T-bar, but with and without clustered channels, we observe that the channel clustering still causes a higher calcium microdomain peak concentration compared to the case when the channels are more widely spread. Also, the channel clustering itself still has a substantial impact.
In conclusion, the combination of T-bar and channeled clusters causes a much higher calcium microdomain concentration compared to the case without T-bar, and even more to the case where also no clustering applies.
This first qualitative evaluation of our simulations already shows a substantial influence of the T-bar structure and even of the clustering structure upon the shape and concentration intensity of the calcium microdomain at each action potential.
This observation is possible already based upon simple evaluations by the view, but it remains to quantify this comparison by means of quantitative measures, which we display in the next sections.
In order to quantify the differences in the calcium microdomain concentrations, we perform quantitative evaluations of the computed concentrations for the three given anatomical cases.
If we evaluate the temporal evolution of the calcium concentrations at fixed places close to the T-bar (or the corresponding locations without T-bar), i.e., close to the membrane of the AZ, but below the (virtual) T-bar “socket,” averaged for a circle with fixed radius and fixed height, we get profiles over time such as displayed in
This figure displays quantitative evaluations of the temporal dynamics of calcium concentrations at fixed spatial locations. We display the concentration value of calcium for a given height averaged over a circle with a given radius. The height and the radius are based upon the circles as described in Section 2 and displayed in
The evaluation displayed in
We see clearly that the case with T-bar has a substantially higher peak than the case without T-bar. In particular, the case without a T-bar and no clustering of channels have a much smaller peak.
As we intend to study the behavior for different radii and heights,
Nevertheless, it is obvious that this strategy of evaluation over time is not sufficient, and more elaborated calcium profile evaluations have to be considered to understand the spatial profile of calcium dynamics.
To investigate the calcium microdomain concentration profile more in detail, we evaluated the concentrations for a fixed time point, at the peak value, and display the variation of the calcium concentration for a fixed height but varying radius, i.e., distance from the (imaginative in case of no T-bar) T-bar socket center.
If not noted otherwise, the evaluations are performed at the first peak marked in
Spatial profiles of calcium concentrations at a fixed time for the three different anatomic cases. The evaluations of the concentrations are performed at the first peak which is marked with a red arrow in
In the case of clustered channels, with and without T-bar, for each given height, when varying the radius, we observe that the highest value is reached close to the (virtual) socket, whereas we see a drop-down of the concentration when going farther away from the socket. When comparing the clustered channel cases with and without T-bar, we see that close to the membrane, for small height, the values in the case with T-bar are about 25% higher than in the case without T-bar, when we are close to the T-bar socket. The farther we move away from the membrane and closer to the (virtual in case of no T-bar) T-bar “roof,” which means that the height of the evaluation circle becomes larger, the greater the difference gets. Below the T-bar “roof,” the concentration in the case of T-bar anatomy is two times bigger compared to the case where no T-bar is present. When going to higher radii, the differences decrease, as the “blocking” effect of the “roof” has fewer influences, as the sites are open also in the case with T-bar, where the calcium can diffuse away more easily. The clustered channels cause that close to the center, the concentrations are the highest, even without T-bar.
At all spatial locations, the not clustered channel case without T-bar shows concentrations strongly below the case of clustered channels, with or without T-bar. The differences are in the size of magnitude of a factor of about 10, despite the fact that we consider the same number of channels.
The difference in calcium concentration is impressive for the three cases. The variation of the height reveals that the farther away we are from the membrane, i.e., from the channels, the more important becomes the influence of the T-bar “roof.” We see a strong influence on the possibility for calcium to diffuse in the case of the T-bar obstacle present, in the sense that the T-bar indeed acts as a diffusion obstacle.
There are two major sources of different local calcium concentrations: Close to the membrane, the major difference arises from the channel location structure (clustered - not clustered), but the T-bar has an additional non-negligible influence on the peak concentration. Far away from the membrane, close to, i.e., below the T-bar roof (virtual “T-bar” roof in case of no T-bar), the impact of the T-bar is even more important than close to the membrane: Even in the case of clustered channels, there is a strong difference of local calcium concentration between the case with and without T-bar.
In order to test if the frequency of action potential stimulation has an influence upon the relation of the calcium microdomain sizes at the peaks of calcium influx, we varied the frequency of the action potential stimulation. Our question was if maybe at different frequencies, after e.g., several action potentials, the relation of the peak sizes changes for the three different anatomic cases which we consider. To investigate this question, we performed the action potential stimulations with several realistic frequencies: 20 Hz, 40 Hz (i.e., the case already considered before), 80 Hz, and 100 Hz.
Temporal evolution of the concentration overtime at a fixed spatial location. Fixed height and fixed radius averaged values for different frequencies concentration evaluation around the (imaginative in the case without T-bar) T-bar socket center at radius
Note that we do not observe calcium accumulation over several action potentials, as the local calcium concentration at the AZ is cleared before the next action potential arrives, also for higher frequencies.
For further quantitative comparisons, we repeated the evaluation method already applied above: We evaluated the concentration profile for a given height, varying the radius at a fixed time point. To perform the comparisons, we evaluated the different curves at the same time point for two different representative frequencies, namely 20 Hz and 100 Hz.
In
Comparison for the calcium concentration under variation of radius and height at the fixed time point for different stimulation frequencies, i.e., evaluation of the spatial profile of calcium concentrations at peak time after several action potentials, for the same time point. Note the red arrows in
We see that the curves show the same behavior as we have seen before in the case of 40 Hz, (we recall
As in the case of the first peak of the 40 Hz stimulation considered before, we observe:
For a height close to the membrane, the difference of the concentrations in the case with and without T-bar, but clustered channels is approximately 25%. However, in the case of a height close to below the T-bar “roof,” which only is virtual in case of no T-bar, the difference is about 50%, i.e., the concentration in the case of the presence of a T-bar is about two times larger compared to the case without T-bar. For values of the radius farther away from the T-bar “socket” center, the values get more close.
In the case of not clustered channels and no T-bar, the values are much smaller for all considered cases, about an estimated factor of about 10 compared to the case of a T-bar and clustered channels, for all considered radii, and for all considered heights. In conclusion, the T-bar in combination with clustered channels is strongly superior to the case of no T-bar and clustered channels, and the case of no T-bar and not clustered channels even cannot compete with the other two cases.
So the calcium concentration behavior remains similar to the already discussed cases, also for ongoing action potentials, not only for the first peak. As well under variation of the frequency, at the peak moments, there are no major differences in the relations of the concentration relations compared to the first peak case we considered before.
With ongoing firing rate, we see a small synaptic depression caused by the calcium N-type channel behavior, that causes a rather small decrease in the peak size, which however has only a very minor influence on the quantitative results and no influence on the qualitative results for the comparison of the three geometry types. This small reduction of the calcium influx is due to VGCC gating depression effects.
To test for the influence of the buffer mechanism, the pumps, and the leakage, we performed the simulations switching off these effects. When comparing the three different anatomic cases, the results are very similar to the case with these effects, (cf.
Compare the case of calcium microdomain concentrations in the presence of buffers and pumps with the case of the absence of buffers and pumps for all three geometric scenarios. Left column: Case without buffers and pumps, right column: standard case with buffers and pumps (to facilitate direct comparison, we repeat parts of
Even qualitatively, the calcium microdomain shapes show effectively the same behavior as in the case with buffer, pumps, and leakage. There is no major difference between the case of a large forward buffering constant and active pumps, and the case without buffering and without pumps.
Having varied different “software” parameters, we vary next to a “hardware” parameter: We vary the number of channels per AZ to test if the results we derived so far might depend upon the number of the channels at the AZ. We probe for if different channel numbers might cause a change in the relation of the concentration of the three geometric cases, the AZ with T-bar, the AZ without T-bar but clustered channels, and the case without T-bar and without clustered channels.
For the VGCC number per AZ, we varied in a wide range (Nadkarni et al.,
Sensitivity of results under variation of VGCC number, rest of parameters standard setup. Left column
We again show variations of the concentrations for the three cases over time for a fixed radius and a given height. As well, we evaluate the variation of the concentrations for a fixed time point at different heights, considering the variation of the radius with respect to the center of the (virtual) T-bar center. As the relations are the same as in all cases so far considered, we omit to repeat these results in an extended manner in the text, as they remain the same for all numbers of VGCCs. Whereas the scale changes, as more influx means more concentration, the relations do not change when comparing the three anatomic cases. The variation of the number of VGCCs has no major impact upon the relative influence of the T-bar upon calcium concentration compared to the case, where the T-bar is not present.
Quantitatively, an increase in VGCC numbers has a strong effect, but in a similar way for all three geometric cases: in all cases, we arrive at the result that the T-bar (clustered channels) has higher calcium microdomain concentrations, then follows clustered channels without T-bar, and at the end, lowest calcium microdomain concentration, the case of not clustered channels without T-bar. Indeed, the number of VGCCs is nearly proportional to the peak concentration of calcium.
The statement that the channel number is not important is due to the fact that we always compare the three geometric cases with each other for the
For the sake of completeness, we also display a comparison of channel number variation results with the aid of a 3D plot, (cf.
Displaying the calcium concentration profiles under variation of the number of calcium channels with the aid of 3D plots, where the number of channels represents the third dimension. Standard parameter set (besides channel number). In the upper row, we consider the time evolution of the calcium concentration at fixed spatial locations, in the lower row, we consider the spatial profile of the calcium concentration at the peak time and vary the radius of the evaluation for fixed height. In the left column, we consider a linear scale for the calcium concentrations, on the right column, we consider logarithmic scales for the calcium concentrations. VGCC number is always displayed with a logarithmic scale.
Also, we compare the concentration profile for the averaged calcium concentration below the (virtual) table, i.e., within the computational subdomain
As we perform numerical computations based upon finite volume discretizations, one could ask if the results are numerically robust, or if they might, e.g., depend upon the computational grid we use or upon the grid refinement level. In order to be able to exclude that our results show grid dependent artifacts, we performed the analysis of our results by means of numerical grid convergence studies. For diffusion-reaction equations, results should become more and more stable with increasing grid refinement levels. To do so, we study the behavior of our results for the standard parameter set under grid level refinement. We study the case of the evaluation at a fixed spatial location over time, and we study the behavior in the case of the variation of the radius for a given peak and fixed height. For the latter case, we also compute the relative differences between the different levels for each given anatomical setup, and we see excellent numerical grid convergence. The differences to higher levels are below one per mill already for the standard refinement level 2 compared to levels 3 and 4. Therefore, we can conclude that our results are not dependent on our grid nor the grid resolution, and thus, our numerical results are highly reliable.
In detail: Numerical grid convergence tests.
The aim of this figure is to demonstrate that the results we compute by means of vertex-centered finite volume methods are independent of the computational mesh, which is validated if they are independent of the grid refinement level. This means that we compare the results for making the grid finer and finer. If one can show this numerical grid convergence, one can trust the results. Here, we display results for the standard parameter set for different grid refinement levels: Standard spatial refinement level 2 as used for the results presented in the other figures, and spatial refinement levels 3 and 4. The first two rows display such a refinement test. The first row shows results for evaluations of calcium concentrations over time for a fixed height and radius, whereas the second row shows results for fixed time at the first peak, evaluated for a fixed height and varying radius. In the last row, we display the relative differences for the three different anatomic scenarios for the spatial profile. In detail:
In this study, we created a mathematical model describing calcium influx into the presynaptic active zone of the Drosophila NMJ for three different anatomical cases, namely for a geometry with T-bar, for a geometry without T-bar and clustered calcium channels, and for a geometry without T-bar and without clustered channels, i.e., with more broadly distributed channels.
Our
This means that our results enable us to determine in a quantitative manner the efficacy of the T-bar anatomical structure upon presynaptic calcium concentration. In this study, we computed the calcium concentration in a fully spatio-temporal resolved manner (3 spatial dimensions plus one temporal one) for the Drosophila larval NMJ presynaptic AZ for those scenarios which are realized in biology, and we computed quantitatively the differences of the calcium concentrations at each point of the AZ.
Interestingly, the maximum values of the intracellular calcium microdomains close to the membrane/below the T-bar (as computed by us) are consistent with the values reported by Schneggenburger and Neher (
All simulations for all variation of parameters such as stimulation frequency or VGCC number have revealed that the calcium microdomain concentrations deploy the highest peak values in case of the presence of the T-bar diffusion obstacle, whereas in the case of an absence of the T-bar, the concentrations are substantially reduced, in particular, if the channels are not clustered. Given the experimentally observed fact that the T-bar comes in combination with clustered channels, and that without the T-bar, the channels are not clustered, the biological sense of the T-bar is clearly explained such that it is a major player to enhance calcium concentration strongly.
The influence of the buffer is of minor importance for our results, as it has no impact on the qualitative results, and only a nearly negligible influence on the quantitative results. It seems that the calcium diffuses away from the point of influx and the AZ before the buffer can grab the major part upon the influx of the calcium to react with it in the AZ. This could mean that in the context of calcium microdomains at the AZ of the NMJ with and without T-bar, calcium diffusion and influx have a more significant effect on the concentration and shape of the calcium microdomain.
This observation in the context of calcium dynamics is consistent with the findings of Sneyd and Tsaneva-Atanasova (
The influence of the number of VGCCs is quantitative, but not of qualitative order. We also assume that for more than 20 VGCCs, the calcium microdomain peak concentrations reach values, which do not seem to be very realistic anymore, which we use as a further hint to assume that the VGCC number at the NMJ AZs are located in the lower region of the values reported by Nadkarni et al. (
Our simulation results consider e.g.,
Indeed, there are a lot of mechanisms described in the literature that account for calcium channel inactivation in many cases, but also activation in some circumstances (Lee et al.,
If the VGCC depression would be stronger due to e.g., the presence of proteins that tend to block the channels, the reduction of the calcium influx would be stronger, such that at each following action potential, the peak of the influx would be smaller.
However, very likely, this would not affect the validity of the results derived here, because this influx reduction would affect all three anatomical cases we discussed in this study in a similar manner. After all parameter studies we performed, the relation of the peak concentrations for the case with and without T-bar remained similar. In particular, we have demonstrated this property by means of varying the VGCC number, which resulted in a large difference in scale of the total influx, where however the relations of total peak concentrations between the different anatomical cases remained effectively the same. Therefore, we can conclude that also with stronger reduced calcium influx, our statement that the T-bar obstacle enhances calcium concentration remains unchanged also in case of stronger calcium-dependent VGCC inactivation.
Finally, calcium-dependent channel inactivation effects happen at a significantly smaller scale than the effects considered within our study. We are using the homogenized model developed by Borg-Graham (
Note that the effect of small peak reduction with ongoing firing rate due to synaptic channel depression is not the synaptic plasticity we mean in this study, and we think this small effect has nothing to do with synaptic plasticity.
Indeed, the application of the terminus “synaptic plasticity” might be considered as ambiguous, but is getting clear based on our definition we mean in this context:
When talking about plasticity effects of the T-bar, we mean that the presence (after growth) of this obstacle has a substantial impact upon calcium microdomain concentrations, i.e., the synaptic plasticity we mean refers to the difference of the presence and absence of the T-bar.
Synaptic plasticity in our context means to compare the synaptic behavior of the three different geometric types, i.e., to evaluate the influence on synaptic effects in the case with and without T-bar and in the case of clustered and not clustered channels. In this sense, the T-bar causes the plasticity effect to enhance calcium concentration.
However, we do not model the growth of the T-bar itself but compare the synaptic behavior for the three cases. In this sense, the T-bar has a measurable effect upon synaptic plasticity, but not at the time scale at which we compute, but when comparing the result of a synapse with the grown T-bar with a synapse where no T-bar is present.
A major question of Wichmann and Sigrist (
In this sense, we have revealed why and in which sense the T-bar is a “plasticity module,” as stated in the review by Wichmann and Sigrist (
Note that in this approach, we consider comparably short intervals of action potential stimulation and calcium influx, as the major question of our study is to relate the anatomical structure of the T-bar to the calcium concentration, comparing the case of AZs with and without T-bar. Our study does not consider long-term comparisons, as they would ask for additional effects such as major changes in calcium influx, which might differ for the three cases. However, in such a case, it would not be possible anymore to study the question of the influence of the anatomical structure of the T-bar in a way independent of additional effects.
The T-bar has a significant influence on the intracellular calcium concentrations close to the AZ and hence has a substantial influence upon the release probability of mature vesicles if those are present close to the AZ.
If the vesicle is located farther away from the membrane, but “below” the T-bar, the T-bar obstacle substantially increases the calcium concentration (even more than close to the membrane) and, thus, establishes a substantially enhanced release probability also for vesicles not directly located at the membrane - even compared to the case of clustered channels but absent T-bar. Thus, clustering of channels alone enhances release probability close to the membrane, but the T-bar obstacle makes a major difference in particular for vesicles not directly located at the membrane, but “below” the “roof” of the obstacle of the T-bar.
As the T-bar causes higher calcium concentration, it causes also vesicles not directly located at the membrane to have a strongly enhanced release probability compared even to the case of clustered channels, but no T-bar. The difference for the case with and without T-bar might decide if a vesicle either will be exocytosed or not.
Indeed, our statement that the T-bar AZ enhances calcium concentration and vesicle release probability fits very well with the experimentally observed fact that synapses with several T-bars show permanent enhanced vesicle release, as stated by Wichmann and Sigrist (
Furthermore, the enhanced calcium concentration, in the case of a T-bar, as shown by our simulations, explains why high release probability synapses are synapses that harbor T-bars, but low-release probabilities are those synapses without T-bars, as also assumed by Wichmann and Sigrist (
Our study explains in a biophysical manner
These biophysical well-founded conclusions also allow us to understand why the BRP mutants, where the calcium channel clustering is disturbed and which do not have a T-bar deploy reduced calcium influx and low release probability (Goel et al.,
As experimental studies have shown (Graf et al.,
In particular, our framework and even the data presented in this study could be used to compute release probabilities for vesicles based on calcium concentrations, depending on the geometric context, i.e., to relate the form and function of synapse structures. Such evaluations could be done based on our given data set by means of combining them with available calcium release probability models. Indeed, we plan such applications of our data and our framework for future studies, also in the interplay with the vesicles.
In the middle run, it would be an interesting task for future studies to combine calcium simulations as performed in this study with vesicle dynamics computations. This combination (incorporating also interesting information such as given in Gonzalez-Bellido et al.,
Such a model combining calcium concentration dynamics and vesicle dynamics might facilitate understanding in more detail how and why the release probability also depends upon the spatial distance between the docking vesicles, and the impact upon endocytosis efficiency. While at the present stage, such questions are beyond the scope of the present study; however, a future model and simulation approach might also help to get further insight into the different basic mechanisms between tonic and phasic synapses (Atwood et al.,
In a former study, we investigated the relation of form and function of the different bouton types of the Drosophila larval NMJ, which appear in this context (Knodel et al.,
In detail, the basic question of the former article was to understand why the NMJ harbors two different bouton types, namely the small 1s boutons and the big 1b boutons (Johansen et al.,
The restriction to 1s and 1b boutons was due to consideration of experiments performed at abdominal muscles 6 and 7, which are exclusively innervated by 1s and 1b boutons. Bouton 1s and 1b have glutamatergic synapses. Among other muscle segments, namely muscles 12 and 13, there exists an additional bouton type called type II. Type II bouton harbors aminergic synapses. Type 1s boutons and type II boutons show phasic, while type 1b show tonic behavior.
Assuming a diffusion process describing vesicle movement within the 3D boutons, we developed a mathematical model of spatio-temporal resolved vesicle dynamics and applied this model to different sized boutons which aimed for 1s and 1b boutons with different numbers of synapses as realized typically in the NMJ. The model approach of our former study simulated vesicle dynamics within complete single presynaptic boutons. A major element of the evaluations was to apply sensitivity analysis upon bouton size, vesicle release probability and active zone number. These studies allowed us to unveil that high vesicle release probabilities allow for strong vesicle release and thus high EPSPs quasi independent of the bouton size at the beginning of the action potential stimulation. However, our investigations showed as well that the high support in case of high release probability drops down rapidly independent of the bouton size. Moreover, we showed that large boutons allow for long-term support of vesicle release quasi independent of the release probability due to the interplay of release probability and available vesicles. Therefore, we predicted that the NMJ is constructed such that the small 1s boutons are equipped with high release probability sites, and the large 1b boutons are equipped with low release probability sites. The experimental data, which were a complementary part of our study (Knodel et al.,
Whereas, our former study proposed that the 1s and 1b boutons have different release probability patterns leading to different plasticity patterns (long-term low level EPSPs for the 1b bouton due to low release probability combined with huge size of the ready-releasable-pool vs. short-term high level EPSPs for the 1s boutons due to high release probability leading to fast depletion of the ready-for-releasable pool, which, however, would be similar also in a bigger bouton, so a small bouton suffices for strong short-term EPSPs), we did not speculate on the question of the reason of these different patterns.
Independent experimental studies by Kurdyak et al. (
In particular, it is tempting to ask for the reason for the different calcium profiles. One possible solution might be that different anatomical structures at the AZs would be responsible for different calcium microdomain concentrations leading to different release probabilities, which coincides with the before already described challenge to understand the role of the T-bar in the AZ and its putative impact upon calcium concentration profiles. Indeed, in this study, we investigated the relation of form and function at a slightly smaller scale compared to our former study of vesicle dynamics at the scale of complete boutons. Moreover, instead of considering vesicle dynamics, we considered the dynamics of calcium, which underlies vesicle dynamics. A challenging question is if it might be possible to combine the results of both model simulation studies, this study and the former study (Knodel et al.,
Therefore, it would be tempting to speculate if the difference in release probability between 1s and 1b boutons might have its reason in enhanced numbers of T-bars in the case of 1s and type II boutons such that the enhanced number of T-bars might be responsible for enhanced calcium dynamics in 1s and type II boutons compared to 1b boutons.
Indeed (Xing and Wu,
Despite the fact that (He et al.,
Therefore, the results of our study open a perspective if maybe the relation of AZs with and without T-bar might be different between the different bouton types of the NMJ.
However, we are quite careful with the speculation of some sort of relation of T-bar number and bouton type. Explicitly, we name our idea an hypothesis but not any kind of result of our study.
It is very important to keep into account that in this study, we always compared the T-bar anatomy with the case of no T-bar always using the same parameters concerning other effects. However, when comparing 1s, 1b, and type II boutons, presumably, this method cannot be applied. Presumably, the surrounding parameters are different for different bouton types, as reported e.g., in Xing and Wu (
Besides the fact that this article demonstrated that the hypothesis of our former study (Knodel et al.,
However, only in case, a statistical counting of T-bars in the different bouton types would reveal that the high release probability boutons have more T-bars, then the results of our study would imply that these T-bars likely are a major factor of the different firing patterns of the different bouton types.
So while our study is able to unveil the sense of the T-bar in an AZ compared to the case that the T-bar is absent and, thus, gives important insight into the biological reason of the T-bar growth, we have to restrict ourselves: With those experimental data known to us so far combined with the results of this study, it is not possible to give any kind of well-founded answer to the question why 1s boutons and type II boutons show higher release probability than 1b boutons. While we dare the hypothesis that maybe the relative number of T-bars might be enhanced in phasic boutons (1s, type II) of the NMJ compared to the 1b case, we do not claim that this hypothesis is safe at all. The conditions and parameters might vary strongly between the different bouton types.
An additional possibility would be further to speculate if the T-bars in case of the 1s boutons and type II boutons are bigger compared to the 1b case, but also this assumption is highly speculative and not any kind of result of this study. We are not aware of any kind of relevant experimental data in this direction.
Exploiting the results of the simulations of our model, in this study, we elaborated the relation of form and function and the biological sense of the T-bar anatomical structure at the Drosophila neuromuscular junction presynaptic AZ, and this study gives concise answers to at least a part of the major questions asked in the review (Wichmann and Sigrist,
Indeed, our study gives a very simple answer to the question asked in the title of the review (Wichmann and Sigrist,
Interestingly, we found this answer by means of a purely
Our investigations in this study motivate experimental research to count and compare the AZs with and without T-bar to compare the relations for the different bouton types of the NMJ in order to evaluate if maybe the portion of AZs with and without T-bar might influence if a bouton type is a tonic or phasic, but this speculation is not a result of this study and cannot give a definitive answer to this question at this stage.
As detailed investigations have shown that the T-bar shape is more complex than just a simple “umbrella” (Wichmann and Sigrist,
We intend to apply our framework and our data to study in more detail the relation between the form of synapses and vesicle release probability caused by different calcium concentration profiles in different geometric scenarios, also in the case of other synapse types.
The review of Wichmann and Sigrist (
In conclusion, our framework and methods complement given research published by other groups, and our methods and results might stimulate further research for the relation of form and function at synapses, concerning calcium dynamics and their influence on vesicle release probabilities, and help develop a detailed understanding of the relation of form and function for neuronal processes.
The original contributions presented in the study are included in the article/
MK and GW developed the model. MK and RD determined the model parameters based on the literature. MK performed the simulations and performed the evaluation of the results with the support of RD. MK, RD, and GW wrote the manuscript. All authors contributed to the article and approved the submitted version.
This work has been supported by the German Ministry of Economics and Technology (BMWi) in the project HYMNE (02E11809B).
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
MK thanks Anton Vadimovich Chizhov of the Ioffe Institute (Russian Academy of Sciences), Sankt Petersburg, Russian Federation, for providing data for validation of the implementation of the VGCC calcium influx model, for various other stimulating discussions on the subject, for very helpful hints concerning literature study, data evaluation, and writing of this paper as well as for his very kind proof reading the paper. Further, MK thanks Sebastian Reiter (GCSC Frankfurt) for his kind support in the development of the geometric setup and meshing of the computational grid, Michael Heisig (GCSC Frankfurt) for his very kind proof reading of the manuscript, and the High-Performance Computing Center Stuttgart (HLRS) for the supplied computing time at the Apollo Hawk supercomputer, where the computations of this project were performed. The authors wish to express their sincere thanks to the Referees for their thorough and critical reviews of our work.
The Supplementary Material for this article can be found online at:
The supplemental material displays simulation movies of calcium dynamics influx into the presynaptic AZ of the Drosophila larval NMJ, for the case of anatomy with T-bar, and anatomy without T-bar but clustered channels, and anatomy without T-bar and channels not clustered. Buffer concentration simulation movies are attached as well for the three geometric cases. Simulations fully spatio-temporally resolved, 3D in space plus time. The simulation domain opened by means of a cut plane. Simulations displayed for the case where the major part of the computational domain (as opened by means of a cut plane) is shown, and also for the case where the AZ around the T-bar (and corresponding regions for the other two geometric scenarios without T-bar, i.e., with and without clustered channels) are zoomed. Furthermore, a supplemental file displays several figures to which we refer in the text of the article itself as additional information and a list of abbreviations. For the details of the movie type (anatomical structure, perspective, concentration type), we refer to the table of section 1 “SUPPLEMENTAL MOVIE DESCRIPTION” of the supplementary text file.
1Note that we use the words “flux” and “current” like a synonym, as mathematically, we have a flux, whereas biophysically, this corresponds to a current.