# American Institute of Mathematical Sciences

2014, 11(5): 1027-1043. doi: 10.3934/mbe.2014.11.1027

## A theoretical study of factors influencing calcium-secretion coupling in a presynaptic active zone model

 1 Dept. Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Santander, 39005 2 Area de Química Aplicada, Universidad Autónoma Metropolitana-Azcapotzalco, 02200-México D.F., Mexico 3 Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005-Santander, Spain 4 Instituto de Neurociencias, Centro Mixto Universidad Miguel Hernández-CSIC, Sant Joan d'Alacant, Alicante, Spain

Received  April 2013 Revised  April 2014 Published  June 2014

A theoretical analysis of some of the relevant factors influencing the calcium time course and the strength and timing of release probabilities of vesicles evoked by an action potential in a calyx-type active zone is presented in this paper. In particular, our study focus on the comparison of cooperative vs non-cooperative calcium binding by the release site and the effect of the number of Ca$^{2+}$ binding sites on the calcium sensitivity for release. Regarding the comparison of cooperative and non-cooperative kinetic schemes, our simulations show that quite different results are obtained when considering one or another: a reduction in the release probability of more than a $50\,\%$ is obtained when considering the cooperative kinetic scheme. Also, a delay in the average time for release appears when using this model for the calcium sensor.
Our study also shows that a non-cooperative kinetic binding scheme gives rise to a well defined average calcium level for release assuming that the same kinetic constants are considered for all the sites. Our results also suggest that the central value of the calcium sensitivity for release depends on the number of binding sites $N$ and the dissociation constant $K_{D}$ with a scaling law depending on $N K_{D}$.
Citation: Amparo Gil, Virginia González-Vélez, Javier Segura, Luis Miguel Gutiérrez. A theoretical study of factors influencing calcium-secretion coupling in a presynaptic active zone model. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1027-1043. doi: 10.3934/mbe.2014.11.1027
##### References:
 [1] S. Blanes, F. Casas, J. A. Oteo and J. Ros, The magnus expansion and some of its applications, Physics Reports, 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001. [2] J. H. Bollmann and B. Sakmann, Control of synaptic strength and timing by the release-site $Ca^{2+}$ signal, Nat. Neurosci., 8 (2005), 426-434. [3] J. H. Bollmann, B. Sakmann and J. G. G. Borst, Calcium sensitivity of glutamate release in a calyx-type terminal, Science, 289 (2000), 953-957. doi: 10.1126/science.289.5481.953. [4] J. G. G. Borst and B. Sakmann, Calcium influx and transmitter release in a fast cns synapse, Nature, 383 (1996), 431-434. doi: 10.1038/383431a0. [5] J. G. G. Borst and B. Sakmann, Calcium current during a single action potential in a large presynaptic terminal of the rat brainstem, J. Physiol., 506 (1998), 143-157. doi: 10.1111/j.1469-7793.1998.143bx.x. [6] A. Gil and V. González-Vélez, Exocytotic dynamics and calcium cooperativity effects in the calyx of held synapse: A modelling study, J. Comp. Neurosci., 28 (2010), 65-76. doi: 10.1007/s10827-009-0187-x. [7] A. Gil and J. Segura, Ca3D: A Monte Carlo code to simulate 3D buffered calcium diffusion of ions in sub-membrane domains, Comput. Phys. Commun., 136 (2001), 269-293. [8] A. Gil, J. Segura, J. A. G. Pertusa and B. Soria, Monte Carlo simulation of 3-D buffered $Ca^{2+}$ diffusion in neuroendocrine cells, Biophys. J., 78 (2000), 13-33. [9] Y. Han, P. S. Kaeser, T. C. Südhof and R. Schneggenburger, Rim determines Ca(2+) channel density and vesicle docking at the presynaptic active zone, Neuron., 69 (2011), 304-316. [10] S. Hefft and P. Jonas P, Asynchronous gaba release generates long-lasting inhibition at a hippocampal interneuron-principal neuron synapse, Nat. Neurosci., 8 (2005), 1319-1328. doi: 10.1038/nn1542. [11] O. Kochubey and R. Schneggenburger, Regulation of transmitter release by $Ca^{2+}$ and synaptotagmin: Insights from a large cns synapse, Trends Neurosci., 34 (2011), 237-246. [12] O. Kochubey and R. Schneggenburger, Synaptotagmin increases the dynamic range of synapses by driving $Ca^{2+}$-evoked release and by clamping a near-linear remaining $Ca^{2+}$ sensor, Neuron., 69 (2011), 736-748. [13] X. L. Lou, V. Scheuss and R. Schneggenburger, Allosteric modulation of the presynaptic $Ca^{2+}$ sensor for vesicle fusion, Nature, 435 (2005), 497-501. [14] W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., 7 (1954), 649-673. doi: 10.1002/cpa.3160070404. [15] C. J. Meinrenken, J. G. G. Borst and B. Sakmann, Calcium secretion coupling at calyx of held goverened by nonuniform channel-vesicle topography, J. Neurosci., 22 (2002), 1648-1667. [16] C. J. Meinrenken, J. G. G. Borst and B. Sakmann, Local routes revisited: The space and time dependence of the $Ca^{2+}$ signal for phasic transmitter release at the rat calyx of held, J. Physiol., 547 (2003), 665-689. [17] M. Müller, F. Felmy, B Schwaller and R Schneggenburger, Parvalbumin Is a Mobile Presynaptic $Ca^{2+}$ Buffer in the Calyx of Held that Accelerates the Decay of $Ca^{2+}$ and Short-Term Facilitation, The Journal of Neuroscience, 27 (2007), 2261-2271. [18] K. Sātzler and et al., Three-dimensional reconstruction of a calyx of held and its postsynaptic principal neuron in the medial nucleus of the trapezoid body, J. Neurosci., 22 (2002), 10567-10579. [19] R. Schneggenburger and E. Neher, Intracellular calcium dependence of transmitter release rates at a fast central synapse, Nature, 406 (2000), 889-893. [20] J. Segura, A. Gil and B. Soria, Modeling study of exocytosis in neuroendocrine cells: Influence of the geometrical parameters, Biophys. J., 79 (2000), 1771-1786. doi: 10.1016/S0006-3495(00)76429-0. [21] J. Sun, Z. P. Pang, D. Qin, A. T. Fahim, R. Adachi and T. C. Sudhof, A dual-$Ca^{2+}$-sensor model for neurotransmitter release in a central synapse, Nature, 450 (2007), 676-683. [22] L.-Y. Wang, E. Neher and H. Taschenberger, Synaptic vesicles in mature calyx of held synapses sense higher nanodomain calcium concentrations during action potential-evoked glutamate release, J. Neurosci., 28 (2008), 14450-14458. doi: 10.1523/JNEUROSCI.4245-08.2008.

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##### References:
 [1] S. Blanes, F. Casas, J. A. Oteo and J. Ros, The magnus expansion and some of its applications, Physics Reports, 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001. [2] J. H. Bollmann and B. Sakmann, Control of synaptic strength and timing by the release-site $Ca^{2+}$ signal, Nat. Neurosci., 8 (2005), 426-434. [3] J. H. Bollmann, B. Sakmann and J. G. G. Borst, Calcium sensitivity of glutamate release in a calyx-type terminal, Science, 289 (2000), 953-957. doi: 10.1126/science.289.5481.953. [4] J. G. G. Borst and B. Sakmann, Calcium influx and transmitter release in a fast cns synapse, Nature, 383 (1996), 431-434. doi: 10.1038/383431a0. [5] J. G. G. Borst and B. Sakmann, Calcium current during a single action potential in a large presynaptic terminal of the rat brainstem, J. Physiol., 506 (1998), 143-157. doi: 10.1111/j.1469-7793.1998.143bx.x. [6] A. Gil and V. González-Vélez, Exocytotic dynamics and calcium cooperativity effects in the calyx of held synapse: A modelling study, J. Comp. Neurosci., 28 (2010), 65-76. doi: 10.1007/s10827-009-0187-x. [7] A. Gil and J. Segura, Ca3D: A Monte Carlo code to simulate 3D buffered calcium diffusion of ions in sub-membrane domains, Comput. Phys. Commun., 136 (2001), 269-293. [8] A. Gil, J. Segura, J. A. G. Pertusa and B. Soria, Monte Carlo simulation of 3-D buffered $Ca^{2+}$ diffusion in neuroendocrine cells, Biophys. J., 78 (2000), 13-33. [9] Y. Han, P. S. Kaeser, T. C. Südhof and R. Schneggenburger, Rim determines Ca(2+) channel density and vesicle docking at the presynaptic active zone, Neuron., 69 (2011), 304-316. [10] S. Hefft and P. Jonas P, Asynchronous gaba release generates long-lasting inhibition at a hippocampal interneuron-principal neuron synapse, Nat. Neurosci., 8 (2005), 1319-1328. doi: 10.1038/nn1542. [11] O. Kochubey and R. Schneggenburger, Regulation of transmitter release by $Ca^{2+}$ and synaptotagmin: Insights from a large cns synapse, Trends Neurosci., 34 (2011), 237-246. [12] O. Kochubey and R. Schneggenburger, Synaptotagmin increases the dynamic range of synapses by driving $Ca^{2+}$-evoked release and by clamping a near-linear remaining $Ca^{2+}$ sensor, Neuron., 69 (2011), 736-748. [13] X. L. Lou, V. Scheuss and R. Schneggenburger, Allosteric modulation of the presynaptic $Ca^{2+}$ sensor for vesicle fusion, Nature, 435 (2005), 497-501. [14] W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., 7 (1954), 649-673. doi: 10.1002/cpa.3160070404. [15] C. J. Meinrenken, J. G. G. Borst and B. Sakmann, Calcium secretion coupling at calyx of held goverened by nonuniform channel-vesicle topography, J. Neurosci., 22 (2002), 1648-1667. [16] C. J. Meinrenken, J. G. G. Borst and B. Sakmann, Local routes revisited: The space and time dependence of the $Ca^{2+}$ signal for phasic transmitter release at the rat calyx of held, J. Physiol., 547 (2003), 665-689. [17] M. Müller, F. Felmy, B Schwaller and R Schneggenburger, Parvalbumin Is a Mobile Presynaptic $Ca^{2+}$ Buffer in the Calyx of Held that Accelerates the Decay of $Ca^{2+}$ and Short-Term Facilitation, The Journal of Neuroscience, 27 (2007), 2261-2271. [18] K. Sātzler and et al., Three-dimensional reconstruction of a calyx of held and its postsynaptic principal neuron in the medial nucleus of the trapezoid body, J. Neurosci., 22 (2002), 10567-10579. [19] R. Schneggenburger and E. Neher, Intracellular calcium dependence of transmitter release rates at a fast central synapse, Nature, 406 (2000), 889-893. [20] J. Segura, A. Gil and B. Soria, Modeling study of exocytosis in neuroendocrine cells: Influence of the geometrical parameters, Biophys. J., 79 (2000), 1771-1786. doi: 10.1016/S0006-3495(00)76429-0. [21] J. Sun, Z. P. Pang, D. Qin, A. T. Fahim, R. Adachi and T. C. Sudhof, A dual-$Ca^{2+}$-sensor model for neurotransmitter release in a central synapse, Nature, 450 (2007), 676-683. [22] L.-Y. Wang, E. Neher and H. Taschenberger, Synaptic vesicles in mature calyx of held synapses sense higher nanodomain calcium concentrations during action potential-evoked glutamate release, J. Neurosci., 28 (2008), 14450-14458. doi: 10.1523/JNEUROSCI.4245-08.2008.
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