Biophys J, September 2002, p. 1368-1373, Vol. 83, No. 3

New and Corrected Simulations of Synaptic Facilitation

Victor Matveev,^* Arthur Sherman,^* and Robert S. Zucker

 ^*Mathematical Research Branch, National Institute of Diabetes and Digestive Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892 and  Molecular and Cell Biology Department, Neurobiology Division, University of California, Berkeley, California 94720 USA

Tang et al. (2000) demonstrated that, at the crayfish neuromuscular junction, both the accumulation and the decay properties of short-term synaptic facilitation (STF) are strongly affected by the addition of a fast high-affinity Ca²⁺ buffer, suggesting a role of residual free Ca²⁺ in the induction of STF. The authors proposed that the experimental results can be explained by a secretion model with two Ca²⁺ binding sites, a secretory site mediating exocytosis and located close to the Ca²⁺ channel (~10-20 nm), and a high-affinity facilitation site located further away (~80-100 nm) from the channel. Here we report that the data presented in Figs. 3, C and D, 6, and 7 of the original article, showing numerical solutions to the Ca²⁺ diffusion equations, are qualitatively inaccurate, because of misstated parameter values and, to a lesser extent, numerical algorithm errors. Therefore, some of the conclusions stated by Tang et al. concerning the proposed model require reexamination. In this letter we show that most of the predictions of the model hold, after an appropriate change of parameter values. Namely, the model correctly predicts the magnitude of STF, and the reduction of STF magnitude and acceleration of its decay in the presence of a fast high-affinity exogenous Ca²⁺ buffer, such as Fura-2. The fast ("F1") and slow ("F2") decay components of STF are also successfully reproduced, if an additional assumption is made that the endogenous Ca²⁺ buffers are immobile. However, our simulations predict that the slower F2 decay component is completely abolished in the presence of Fura-2, contrary to experimental results of Tang et al. (2000) (Fig. 3, A and B, in the original paper). We found that this remaining disagreement can be resolved if one assumes that the diffusion in the synaptic bouton is restricted, so that 1) in a 200-nm layer around the active zone, Fura-2 is immobilized and the diffusion coefficient of Ca²⁺ is reduced fivefold, due to a high degree of tortuosity; and 2) in the rest of the terminal, the diffusion coefficient of Fura-2 is reduced 100-fold (presumably because of binding to various cytosolic compounds). Moreover, contrary to the statement by Tang et al. that their model fails to accurately describe the accumulation time course of STF, we show that the model modifications that we propose lead to a supralinear growth of STF, in agreement with experiment (Fig. 2, A and D).

Facilitation magnitude: role of fixed buffers in STF

An attempt to reproduce the simulation results presented in Figs. 3, C and D, 6, and 7 of Tang et al., 2000, using a newly developed computer program, CalC (see below), revealed significant discrepancies. To resolve this disagreement, we analyzed the computer code used to generate Ca²⁺ traces in Fig. 7 B of the original article (developed by Thomas Schlumpberger), and found that the parameter values used in the simulations were different from the values quoted in the paper. The differences between the two parameter sets are summarized in Table 1. Fig. 1 demonstrates the difference in [Ca²⁺] profiles at the STF site computed using the quoted and actual parameter values (thick and thin lines, respectively), and compares them to the original Fig. 7 B of Tang et al. (dashed line). One can see that, even when using the same parameter values, there remains a significant quantitative disagreement between our simulations and the original figure (compare the dashed and thick curves in Fig. 1). We found that this remaining discrepancy is a result of an algorithm error in the original code, in the realization of the elementary compartment size doubling (see subsection "Implementing differential equations" of the Materials and Methods section of Tang et al., 2000, page 2736).

View larger version (5K): [in this window] [in a new window]   FIGURE 1   Ca2+ concentration at the facilitation site, in response to a 5-pulse stimulation train. Thin line: simulation results for parameter values quoted in Tang et al., 2000; thick line: simulation results for the parameter set used to generate Fig. 7 B of Tang et al., 2000 (see Table 1); dashed line: the control (100% external Ca2+) curve from Fig. 7 B of Tang et al., 2000. The discrepancy between the thick and dashed lines is due to the algorithm errors in the original work.

In a model with a fast STF site (included to account for rapid reduction of STF by photolabile Ca²⁺ buffers; Kamiya and Zucker, 1994), the magnitude of STF is determined by the size of residual [Ca²⁺] relative to the peak [Ca²⁺] achieved during an AP. Fig. 1 shows that, for the quoted parameter values, Ca²⁺ concentration at the STF site is characterized by large transients, which are significantly greater than the residual [Ca²⁺], resulting in small facilitation magnitude (numerical result not shown). STF magnitude computed using the actual parameters, although still below the experimental value (compare Fig. 2, A and B), is significantly larger. Our simulations show that the critical parameters responsible for this difference are the total concentration and the Ca²⁺ binding rate of the fixed buffer, both of which are higher for the actual parameter set. The fixed buffer is critical in maintaining large STF: unlike mobile buffers, which carry Ca²⁺ away from the synaptic terminal, decreasing residual [Ca²⁺] and thereby reducing the STF (unless they are saturated; see below), fixed buffers prevent Ca²⁺ from diffusing away. This slows down the Ca²⁺ signal and prolongs its action near the secretory and facilitation machinery, which in turn leads to greater STF (Sala and Hernández-Cruz, 1990; Nowycky and Pinter, 1993; Neher, 1998; Kits et al., 1999). We note that this mechanism is distinct from the buffer saturation mechanism put forward by E. Neher (Neher, 1998; Rozov et al., 2001) as a potential source of STF. In the latter situation, the increase in response results from the increase in Ca²⁺ transients, caused by reduced buffering capacity associated with buffer saturation, whereas in our simulations the response growth is caused predominantly by the increase in residual [Ca²⁺].

View larger version (21K): [in this window] [in a new window]   FIGURE 2   Accumulation time course of synaptic facilitation elicited by a 5-pulse train, with and without the addition of Fura-2. (A) Experimentally measured facilitation, from Tang et al., 2000, Fig. 2 C (control curve, 100% external Ca2+). (B) Simulation results for the actual parameter set from Table 1. (C) Simulation without mobile buffer (modified parameter set: see text): notice higher STF magnitude. (D) Simulation without mobile buffer, with tortuosity effects included (see text); notice the supralinear shape of the curve, similar to A. In B and C, Ca2+ binding scheme and Fura-2 properties (affinity = 360 nM, DF2 = 118  µm2/s, [Fura-2]total = 400 µM, unbinding rate = 96.7 s1) are the same as in Tang et al., 2000. In D, the Ca2+ affinity of the STF site was changed from 3 to 9 µM; Fura-2 is immobile in a 200-nm layer around the active zone; in the rest of the terminal DF2 = 1.18 µm2/s.

STF decay time course: excluding mobile buffers

The main disagreement between our results and the simulation results reported in the original paper concerns the decay time course of STF: although Fig. 3, C and D, of Tang et al. reveals both the F1 and F2 decay components of STF, we were not able to reproduce the slower F2 component, neither using the quoted parameter values (data not shown; STF is very small in this case), nor the actual values (in the latter case, STF decayed with time constants of 5 ms and 21 ms; see Fig. 3 B and Table 2). This is to be expected, because the time scales of Ca²⁺ processes implemented in the model are either much faster (Ca²⁺ unbinding from buffers, Ca²⁺ unbinding from the STF sensor, buffered diffusion of Ca²⁺) or much slower (Ca²⁺ extrusion by membrane pumps) than the time scale of hundreds of milliseconds characterizing the F2 decay component. The only way to prolong STF decay within the framework of the current model is to slow the buffered diffusion of calcium by assuming that the endogenous Ca²⁺ buffers are predominantly present in fixed form. To this end, we repeated our simulations with a new ("modified") parameter set, which is similar to the actual parameter set in Table 1, with the exception that the mobile buffer is excluded. Furthermore, we placed the secretory site 10 nm away, and the STF site at a distance of 100 nm away from the nearest Ca²⁺ channel. Simulations with this modified set of parameters revealed higher STF magnitude (Fig. 2 C), in better agreement with experiment (Fig. 2 A), and a slower STF decay (Fig. 3 C), with F1 and F2 decay time constants of 28 ms and 135.5 ms, respectively (Table 2). However, as soon as Fura-2 is included in the model, the F2 component isn't just reduced and accelerated, as seen experimentally (Fig. 3 A), but is completely removed (Fig. 3 C; Table 2). This agrees with our earlier remarks about the opposing actions of mobile and fixed buffers on STF. We were not able to obtain the F2 component with Fura-2 present without additional modifications to the model; we believe that the results in the original Fig. 3, C and D of Tang et al. are due to an algorithm or implementation error that we have not been able to identify.

View larger version (23K): [in this window] [in a new window]   FIGURE 3   Decay time course of synaptic facilitation elicited by a 5-pulse train, as a function of time interval between the 4th and 5th pulses, with and without the addition of Fura-2. (A) Experimentally measured facilitation, from Tang et al., 2000, Fig. 3 B. (B) Simulation results for the actual set of parameters from Table 1. (C) Simulation without mobile buffer ("modified" parameter set: see text). (D) Simulation without mobile buffer, with tortuosity effects included. Continuous curves drawn through the data points correspond to exponential fits to the simulation results; the corresponding time constants and amplitudes are listed in Table 2 (for C and D, the corresponding entries are shown in boldface). Parameters are the same as in Fig. 2.

STF decay with Fura-2: implementing tortuosity

To retain the slow decay component of STF in the presence of Fura-2, we have to assume that some additional mechanisms are impeding diffusion in the synaptic bouton. To test this possibility, we incorporated tortuosity into the model by reducing the diffusion coefficients of both Fura-2 and Ca²⁺ in a 200-nm-wide layer around the active zone, assuming that diffusion is obstructed in this region by vesicles and various synaptic proteins. As an alternative, we investigated the effect of slowing the diffusion of Fura-2 alone in the entire volume, simulating the effects of Fura-2 binding to various cytosolic compounds (such an effect has been observed in muscle cells by Konishi et al., 1988 and Baylor and Hollingworth, 1988; see also Kits et al., 1999). We found that either condition by itself is not sufficient to achieve the desired behavior (see, respectively, Table 2, Simulations with tortuosity, row Fura-2, and Modified parameter set, row slow Fura-2), and that both effects have to be included to account for experimental results. The best agreement with experiment was achieved with the following changes to the modified parameter set defined earlier: 1) in a 200-nm layer around the active zone, Fura-2 is immobilized and the diffusion coefficient of Ca²⁺ is reduced fivefold; 2) in the rest of the terminal, the diffusion coefficient of Fura-2 is reduced 100-fold; 3) the facilitation site is moved to 180 nm away from the plasma membrane; and 4) the Ca²⁺ affinity of the STF site is changed from 3  µM to 9  µM.

The corresponding results are shown in Figs. 2 D and 3 D; the decay components are given in Table 2, Simulations with tortuosity, row slower Fura-2. In an attempt to use the most realistic parameters, in these simulations we also reduced the on-rate of the fixed buffer to 0.05  µM¹ ms¹ and the off-rate to 0.8 ms¹. This additional change had only a negligible effect on results. Note that the quoted fixed buffer rate constants of Tang et al. (2000) are slower than the actual rate constants. The quoted values were based on measurements of Xu et al. (1997) for buffer on-rates in bovine chromaffin cells. The ionic strength of crayfish cytoplasm is twice that of vertebrates, and so the on-rate is likely to be even slower, probably 0.05 µM¹ ms¹. To retain the measured buffer ratio of ~500 (Delaney and Tank, 1994), we leave K_D at 16 µM and total buffer concentration at 8 mM, and reduce the off-rate to 0.8 ms¹.

There is an additional reason to use slower buffer kinetics. Flash photolysis of the photolabile calcium chelator DM-nitrophen produces a rapid phase of transmitter release that reflects the presence of a "calcium spike," caused by the rapid release of calcium and its slower (2 ms) rebinding to the unphotolyzed DM-nitrophen (Landò and Zucker, 1994). Simulations of the effects of the native buffer on the "calcium spike," using recently described kinetic constants for DM-nitrophen binding to magnesium, which is responsible for the slow rebinding of calcium (Ayer and Zucker, 1999), show that a buffer on-rate of 0.05 µM¹ ms¹ is most consistent with the experimental results of Landò and Zucker (1994) (R. S. Zucker, unpublished observations).

Supralinear growth of STF

In the original article, the authors stated that their model failed to reproduce the experimentally observed accumulation time course of STF (compare Fig. 2, A and B). However, our results in Fig. 2 D show that the changes listed above, apart from explaining the decay behavior of STF, also enable the model to successfully reproduce the supralinear growth of STF. We found that tortuosity is not necessary for this effect, and that supralinear growth of STF can also be achieved with the modified parameter set, if the distance between the STF site and Ca²⁺ channel is >250 nm (simulations not shown). We found that the presence of fixed Ca²⁺ buffers is crucial for this property. The reason for this is that the fixed buffer effectively adds intermediate Ca²⁺ binding stages to the facilitation process, because the Ca²⁺ ions would have to undergo multiple buffer binding and unbinding steps on their way to the STF site. It has been noted previously by Balnave and Gage (1977) that adding intermediate Ca²⁺ binding steps to a facilitation model is sufficient to reproduce the supralinear STF growth. In general, increasing the number of steps in a kinetic process makes its transient behavior progressively more supralinear in time. Tortuosity potentiates this effect by increasing the effective time it takes a Ca²⁺ ion to diffuse to the STF site, so the supralinearity can be achieved for smaller separations between the STF site and the Ca²⁺ channel. This is desirable because the STF site is likely to be located close to the release machinery, where it can have an immediate effect on the release process. Therefore, including tortuosity allows the model to explain the supralinear growth of STF for more physiologically realistic distances between the STF site and the Ca²⁺ channel.

	CONCLUSION

TOP CONCLUSION REFERENCES

For a suitable choice of parameters, and assuming that endogenous Ca²⁺ buffers are present predominantly in fixed form, the model proposed by Tang et al. (2000) agrees with the observed accumulation and decay properties of STF at the crayfish neuromuscular junction, but predicts stronger reduction of STF by fast high-affinity exogenous Ca²⁺ buffers than seen experimentally. To explain the properties of STF in the presence of Fura-2, we have to impose an additional assumption that the diffusion in the vicinity of the active zone is significantly retarded due to tortuosity effects, and that Fura-2 is also strongly slowed in the rest of the terminal. We also moved the facilitation site to 180 nm from the nearest Ca²⁺ channel. Given the highly constricted space surrounding a docked vesicle, filled with a variety of proteins and cytoskeletal elements, a diffusion distance of 180 nm is not only within the active zone, but may well be within the complex of proteins involved with vesicle exocytosis (Tang et al., 2000). In addition, we decreased the affinity of the facilitation site to 9 µM, less than the affinity of a high-affinity site estimated by Ravin et al. (1997) but consistent with the observations of Delaney and Tank (1994). With these modifications the model also reproduces the experimentally observed supralinear growth of STF. Our analysis of the model reveals that the fixed endogenous Ca²⁺ buffers play a crucial role in the STF process.

Numerical simulations

All new numerical results presented here were performed using the CalC (Calcium Calculator) computer software, freely available from the URL (http://mrb.niddk.nih.gov/matveev). CalC currently runs on Linux, SGI, and Windows/Intel platforms, and is driven by a user-friendly script. We ensured that our program is error-free by checking it against simple, exactly solvable problems, and by reproducing some of the modeling results found in the literature. Furthermore, we verified that our results agreed with the output of the original code in the absence of the compartment size doubling. We also checked the convergence of results when spatial and temporal resolutions were increased. Simulation parameters were chosen to maintain a numerical accuracy of ~5%. To render the results reported here easily reproducible, and to provide a detailed description of the simulations, the corresponding commented script files will be made available at the web site given above.

	ACKNOWLEDGMENTS

We thank Dr. Jim Winslow for careful reading of this manuscript. This work was supported in part by National Institutes of Health Grant NS15114 (to R.S.Z.).

	FOOTNOTES

Address reprint requests to Victor Matveev, 9190 Rockville Pike, Suite 350, Bethesda, MD 20892. Tel.: 301-496-9644; Fax: 301-402-0535; E-mail: matveev{at}nih.gov.

Submitted February 22, 2002, and accepted for publication May 7, 2002.

	REFERENCES

TOP CONCLUSION REFERENCES

• Ayer, R. K., Jr., and R. S. Zucker. 1999. Magnesium binding to DM-nitrophen and its effect on the photorelease of calcium. Biophys. J. 77:3384-3393[Abstract/Free Full Text].
• Balnave, R. J., and P. W. Gage. 1977. Facilitation of transmitter secretion from toad motor nerve terminals during brief trains of action potentials. J. Physiol. 266:435-451[Abstract/Free Full Text].
• Baylor, S. M., and S. Hollingworth. 1988. Fura-2 calcium transients in frog skeletal muscle fibres. J. Physiol. 403:151-192[Abstract/Free Full Text].
• Delaney, K. R., and D. W. Tank. 1994. A quantitative measurement of the dependence of short-term synaptic enhancement on presynaptic residual calcium. J. Neurosci. 14:5885-5902[Abstract].
• Kamiya, H., and R. S. Zucker. 1994. Residual calcium and short-term synaptic plasticity. Nature. 371:603-606[Medline].
• Kits, K. S., T. A. de Vlieger, B. W. Kooi, and H. D. Mansvelder. 1999. Diffusion barriers limit the effect of mobile calcium buffers on exocytosis of large dense cored vesicles. Biophys. J. 76:1693-1705[Abstract/Free Full Text].
• Konishi, M., A. Olson, S. Hollingworth, and S. M. Baylor. 1988. Myoplasmic binding of fura-2 investigated by steady-state fluorescence and absorbance measurements. Biophys. J. 54:1089-1104[Abstract/Free Full Text].
• Landò, L., and R. S. Zucker. 1994. Ca²⁺ cooperativity in neurosecretion measured using photolabile Ca²⁺ chelators. J. Neurophysiol. 72:825-830[Abstract/Free Full Text].
• Neher, E. 1998. Usefulness and limitations of linear approximations to the understanding of Ca⁺⁺ signals. Cell Calcium. 24:345-357[Medline].
• Nowycky, M. C., and M. J. Pinter. 1993. Time courses of calcium and calcium-bound buffers following calcium influx in a model cell. Biophys. J. 64:77-91[Abstract/Free Full Text].
• Ravin, R., M. E. Spira, H. Parnas, and I. Parnas. 1997. Simultaneous measurement of intracellular Ca²⁺ and asynchronous transmitter release from the same crayfish bouton. J. Physiol. 501:251-262[Abstract/Free Full Text].
• Rozov, A., N. Burnashev, B. Sakmann, and E. Neher. 2001. Transmitter release modulation by intracellular Ca²⁺ buffers in facilitating and depressing nerve terminals of pyramidal cells in layer 2/3 of the rat neocortex indicates a target cell-specific difference in presynaptic calcium dynamics. J. Physiol. 531:807-826[Abstract/Free Full Text].
• Sala, F., and A. Hernández-Cruz. 1990. Calcium diffusion modeling in a spherical neuron. Relevance of buffering properties. Biophys. J. 57:313-324[Abstract/Free Full Text].
• Schlumpberger, T. 1999. Computational modeling of calcium action in synaptic facilitation. Ph.D. Dissertation University of California, Berkeley, California.
• Xu, T., M. Naraghi, H. Kang, and E. Neher. 1997. Kinetic studies of Ca²⁺ binding and Ca²⁺ clearance in the cytosol of adrenal chromaffin cells. Biophys. J. 73:532-545[Abstract/Free Full Text].

Biophys J, September 2002, p. 1368-1373, Vol. 83, No. 3
© 2002 by the Biophysical Society   0006-3495/02/09/1368/06  $2.00

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