A Quantitative Model of Neuronal Calcium Signaling Steve Cox, Jane Hartsfield, and Brad Peercy Rice University, Department of Computational and Applied Mathematics Introduction A number of laboratories, [1,2], have succeeded in eliciting intracellular evanescent calcium waves in pyramidal cells. As rises in intracellular calcium bring about (a) modulation of subsequent action potentials via calcium dependent potassium channels, (b) modulation of conductance values and/or densities of ionotropic glutamate receptors, and (c) activation of transcription factors, there is considerable interest in ascertaining the conditions required for the existence of such waves. Theory and experiment have together identified the fundamental cascade: (1) glutamate binds to metabotropic glutamate receptors (mGluRs), (2) bound mGluRs activate phospholipase C (PLC) to produce Inositol 1,4,5-Trisphosphate (IP3), (3) IP3 binds to IP3 receptors (IP3Rs) on the endoplasmic reticulum (ER) that release calcium, and steps (2) and (3) are both facilitated by moderate levels of calcium and so participate in calcium-induced-calcium-release, the putative [3] mechanism underlying evanescent calcium waves. In this work we construct mathematical models of mGlu and IP3 receptors and integrate these with active models of neuronal and ER membrane and so arrive at a computational platform with which we may investigate (a) the distribution and type of mGluRs and IP3Rs required to support intracellular evanescent calcium waves, (b) the distribution of neuronal calcium channels and the number of back propagating action potentials required to replenish the ER with calcium, (c) and the range of permissible rates of such calcium waves. Experimental Data Targets of Glutamate Figure1: Glutamate Effect at the Synapse. Glutamate targets the receptors, NMDAR, AMPAR, and mGluR. Na + and Ca 2+ enter through open the former, causing a local depolarization which integrates with others and generates an action potential. Backpropagation of the action potential opens Ca 2+ channels. The mGluR bound to a G- protein releases the alpha subunit which combines with PLC, PIP 2, and Ca 2+ to generate IP 3. Coactivation of IP 3 receptors by IP 3 and Ca 2+ induces large Ca 2+ efflux from the endoplasmic reticulum (ER).Approach 1. Following Nakamura et al. we assume IP 3 concentration is uniformly distributed and elicit BPAPs. Due to disparate timescales of voltage and calcium dynamics, initiation of a Ca 2+ wave requires heterogeneity. We hypothesize a variation of IP 3 receptor density within the dendrites as a mechanism to generate Ca 2+ waves. 2. In reality IP 3 concentration is not spatially uniform. We consider a small region of elevated IP 3 distal to the cell body for uniformly distributed IP 3 receptors. We describe the solutions for various levels of IP 3 and calculate the wave speed of Ca 2+ when it exists. 3. Model the cascade of IP 3 generation in conjunction with the Ca 2+ dynamics from a local stimulus of glutamate. Models and Results Simplest V Model: Reduced Hodgkin-Huxley I(x,t) = stimulus at x = cell body Simplest IP 3 R Model: Li & Rinzel h = IP 3 R gating variable R = IP 3 R density factor Voltage and Calcium Response to Stimulus at the Cell Body. 1. IP 3 R Density Variation: Figure 4. IP 3 R density as a function of distance from the soma. Results of IP 3 R Density Variation: - Elevated IP3 receptor density affects the ([Cai],h) phase portrait. - For certain effective calcium stimuli, the high density region is super threshold while the low density region is subthreshold. - However the coupling through calcium diffusion is sufficient to propagate a calcium wave visualized as a space-[Ca i ] plot and space parametrized on the phase portrait in the flip book. 2. IP 3 Concentration Variation: Figure 5. Constant IP 3 concentration as a function of space. When the amount of IP 3 is varied as shown in the figure above, a calcium wave may propagate through the cell depending on the values of p base and p stim. By varying the value of p base and p stim, we see that some combinations result in no waves, some in a simultaneous increase in calcium levels throughout the cell, while a critical relationship exists for those values which produce calcium waves. As the base amount of IP 3 is increased, the calcium wave speed also increases.. Figure 8. Phase Portrait for Varying IP 3 Concentration. The blue nullclines represent the uncoupled dynamics found in the elevated IP 3 region, while the red nullclines represent the uncoupled dynamics found in the basal IP 3 region. (Notice that the near vertical nullclines differ due the IP 3 dependence of the h infinity function as opposed to the effect of IP 3 receptor variation on the vertical nullcline.) 3. G-protein cascade resulting in IP 3 Production: k 1 mGluR + [glu] B k 2 [glu] = glutamate concentration B B = bound mGluR k 3 Gq G  k 4 G  = active G-protein subunit [Ca i ]+ G  + PLC k 5 k 6 PIP 2 IP 3 decay  k 6 /k 5 p base PLC = phospholipase C Parameters Used Application of glutamate initiates a G-protein cascade: mGluRs are bound by [glu]; G-proteins are activated by the bound receptors; IP 3 is generated from PIP 2 by the activated G-protein together with [Ca] and PLC. Figure 9. Cascade of IP 3 Generation. A transient local stimulus of glutamate creates a short term increase in bound mGluRs. This results in a blip of activated G  -protein subunit which unites with PIP 2,Ca i, and PLC, which itself is calcium dependent, to create IP 3. Figure 10. Calcium Wave from Transient Glutamate Stimulus.Conclusion Increasing density of IP 3 receptors distal to the cell body in the apical trunk can account for calcium propagation toward the nucleus as seen in the Nakamura experiments. For a given calcium influx, variation in local IP 3 concentration may generate – only a local response, – simultaneous calcium increase throughout the cell, – or traveling calcium waves. Similarly, local elevation of IP 3 surrounded by a sufficiently high basal IP 3 level admits traveling calcium waves with wave speed increasing as a function of the basal IP 3 level. Given the ability for each mechanism to generate calcium waves, the interplay between IP 3 receptor density and IP 3 concentration levels resulting from metabotropic glutamate receptor activation requires further investigation.References 1. R Bianchi, SR Young and RKS Wong, Group I mGluR activation causes voltage-dependent and -independent calcium rises in hippocampal pyramidal cells. J. Neurophysiol. 81: , S Zhou and WN Ross, Threshold conditions for synaptically evoking calcium waves in hippocampal pyramidal neurons. J. Neurophysiol. 87: , MJ Berridge, Neuronal calcium signaling, Neuron 21:13-26, T Nakamura, K Nakamura, N Lasser-Ross, J Barbara, VM Sandler, Inositol 1,4,5- trisphosphate (IP 3 )-mediated Ca 2+ release evoked by metabotropic agonists and backpropagating action potentials in hippocampal CA1 pyramidal neurons. J. Neurosci. 20: , Y Li and J Rinzel, Equations for InsP 3 receptor-mediated [Ca 2+ ] I oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J. Theor. Biol. 166: , * This work was supported by NSF grant #DMS DvDv 2e-2 mm 2 /msDiffusion constant for axial voltage V Na 50 mVReversal potential for sodium VKVK -77 mVReversal potential for potassium VLVL mVReversal potential for leak channel DcDc 2.23e-7 mm 2 /msDiffusivity constant for calcium in cytosol r1r /ms Maximum IP3R flow rate r2r /ms Leakage rate of calcium from ER r3r  M/ms Maximum ER pump rate kpkp 0.1  M 2 Dissociation constant for ER pump γ0.0007Rate of Ca influx with voltage change c avg 2  M Total Ca inside the cell ν0.185Ratio of vol. of ER to vol. of cytosol K1K  M Dissociation constant of IP 3 from IP 3 R K5K  M Dissociation constant of Ca from IP 3 R activation site J16 DpDp 2.8e-7 mm 2 /msDiffusion constant for IP 3 p base 0.35  M Basal level of IP 3 k1k /ms/  [glu] binding rate to mGluR k2k /ms[glu] unbinding rate from mGluR k3k /ms/  Activation rate of G-protein k4k /msInactivation rate of G-protein k5k /ms/   Production rate of IP 3 k6k /msDecay rate of IP 3 PIP 2 1  Amt of PIP 2 in cell  glu 0.001Rate of decay of [glu] Mag  M Amt [glu] applied to stimulus region c0.185Basal amt of Ca in cytosol x1x mmLocation of beginning of glu stimulus x2x mmLocation of end of glu stimulus IP 3 Generation Cascade Figure 2. Back Propagating Action Potentials. Three action potentials recorded near the cell body from stimulus applied at the cell body. Figure 3. Ca i Elevation in Response to BPAPs. Figure 6. Solution Set. If p base is sufficiently small, for all p stim only small perturbations from rest exist. If p base and p stim are sufficiently high, calcium waves are initiated as shown in the phase portrait Figure 8. If p base and p stim are too high a large excursion in calcium is found everywhere. Figure 7. Calcium Wave Speed. For the region in which calcium waves do exist the speed is calculated. The wave speed is independent of p stim depending only on the concentration of IP 3 into which the wave is traveling. The wave speed is monotonically increasing as p base increases. Figure from Nakamura [4]. Backpropagating action potentials evoke Ca 2+ waves in the presence of t-ACPD. In control ACSF ( first column) a pair of spikes initiated with brief intrasomatic pulses caused rapid, almost simultaneous, increases in [Ca 2+ ] i ( Δ F/F) at all locations in the cell. This increase is shown in two ways. In the top panel the amplitude changes are indicated by a change in color. Positions along the ordinate correspond to the pixels overlaid on the cell image in the fourth column. Positions along the abscissa correspond to the same time scale as in the bottom panel. Below the pseudocolor image the same data are plotted as time-dependent changes for the two regions of interest (red and green boxes) indicated on the image of the cell. In ACSF containing 30 mM t-ACPD, the same pair of spikes caused the same synchronous increases in [Ca 2+ ] i, followed by larger increases at different times in different locations. The pseudocolor image shows that this secondary increase propagated as a wave that initiated at a location ~50  m from the soma. This wave did not propagate into the distal apical dendrites nor into the basal dendrites. Note that the pseudocolor scale has been changed to include the larger-amplitude secondary response. Five spikes (third column) caused a similar secondary response that initiated earlier and more synchronously at different dendritic locations.