A Quantitative Model of Neuronal Calcium SignalingSteve Cox, Jane Hartsfield, and Brad Peercy

Rice University, Department of Computational and Applied Mathematics

IntroductionIntroductionA 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 Ca2+ enter through open the former, causing a local depolarization which integrates with others and generates an action potential. Backpropagation of the action potential opens Ca2+ channels. The mGluR bound to a G-protein releases the alpha subunit which combines with PLC, PIP2, and Ca2+ to generate IP3. Coactivation of IP3 receptors by IP3 and Ca2+ induces large Ca2+ efflux from the endoplasmic reticulum (ER).

ApproachApproach

1. Following Nakamura et al. we assume IP3 concentration is uniformly

distributed and elicit BPAPs. Due to disparate timescales of voltage and calcium dynamics, initiation of a Ca2+ wave requires heterogeneity. We hypothesize a variation of IP3 receptor density within the dendrites as a mechanism to generate Ca2+ waves.

2. In reality IP3 concentration is not spatially uniform. We consider a small

region of elevated IP3 distal to the cell body for uniformly distributed IP3 receptors. We describe the solutions for various levels of IP3 and calculate the wave speed of Ca2+ when it exists.

3. Model the cascade of IP3 generation in conjunction with the Ca2+ dynamics

from a local stimulus of glutamate.

Models and ResultsModels and Results

Simplest V Model: Reduced Hodgkin-Huxley

I(x,t) = stimulus at x = cell body

Simplest IP3R Model: Li & Rinzel

h = IP3R gating variable

R = IP3R density factor

Voltage and Calcium Response to Stimulus at the Cell Body.

1. IP3R Density Variation:

Figure 4. IP3R density as a function of distance from the soma.

Results of IP3R 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-[Cai] plot and space parametrized on the phase portrait

in the flip book.

2. IP3 Concentration Variation:

Figure 5. Constant IP3 concentration as a function of space.

When the amount of IP3 is varied as shown in the figure above, a calcium wave may propagate through the cell depending on the values of pbase and pstim. By varying the value of pbase and pstim, 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 IP3 is increased, the calcium wave speed also increases.

.

Figure 8. Phase Portrait for Varying IP3 Concentration. The blue nullclines represent the uncoupled dynamics found in the elevated IP3 region, while the red nullclines represent the uncoupled dynamics found in the basal IP3 region. (Notice that the near vertical nullclines differ due the IP3 dependence of the hinfinity function as opposed to the effect of IP3 receptor variation on the vertical nullcline.)

3. G-protein cascade resulting in IP3 Production:

k1

mGluR + [glu] B k2 [glu] = glutamate concentration

B B = bound mGluR k3

Gq G k4 G = active G-protein subunit

[Cai]+ G + PLC k5 k6

PIP2 IP3 decay

k6/k5 pbase

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; IP3 is generated from PIP2 by the activated G-protein together with [Ca] and PLC.

Figure 9. Cascade of IP3 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 PIP2,Cai, and PLC, which itself is calcium dependent, to create IP3.

Figure 10. Calcium Wave from Transient Glutamate Stimulus.

ConclusionConclusion• Increasing density of IP3 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 IP3 concentration may generate

– only a local response,– simultaneous calcium increase throughout the cell,– or traveling calcium waves.

• Similarly, local elevation of IP3 surrounded by a sufficiently high basal IP3 level admits traveling calcium waves with wave speed increasing as a function of the basal IP3 level.

• Given the ability for each mechanism to generate calcium waves, the interplay between IP3 receptor density and IP3 concentration levels resulting from metabotropic glutamate receptor activation requires further investigation.

ReferencesReferences1. R Bianchi, SR Young and RKS Wong, Group I mGluR activation causes voltage-dependent and -independent calcium rises in hippocampal pyramidal cells. J. Neurophysiol. 81:2903-2913, 1999.

2. S Zhou and WN Ross, Threshold conditions for synaptically evoking calcium waves in hippocampal pyramidal neurons. J. Neurophysiol. 87:1799-1804, 2002.

3. MJ Berridge, Neuronal calcium signaling, Neuron 21:13-26, 1998.

4. T Nakamura, K Nakamura, N Lasser-Ross, J Barbara, VM Sandler, Inositol 1,4,5-trisphosphate (IP3)-mediated Ca2+ release evoked by metabotropic agonists and backpropagating action potentials in hippocampal CA1 pyramidal neurons. J. Neurosci. 20:8365-8376, 2000.

5. Y Li and J Rinzel, Equations for InsP3 receptor-mediated [Ca2+]I oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J. Theor. Biol. 166:461-473, 1994.

* This work was supported by NSF grant #DMS-0240058.

Dv 2e-2 mm2/ms Diffusion constant for axial voltage

VNa 50 mV Reversal potential for sodium

VK -77 mV Reversal potential for potassium

VL -54.4 mV Reversal potential for leak channel

Dc 2.23e-7 mm2/ms Diffusivity constant for calcium in cytosol

r1 0.00111 /ms Maximum IP3R flow rate

r2 0.00002 /ms Leakage rate of calcium from ER

r3 0.0009 M/ms Maximum ER pump rate

kp 0.1 M2 Dissociation constant for ER pump

γ 0.0007 Rate of Ca influx with voltage change

cavg 2 M Total Ca inside the cell

ν 0.185 Ratio of vol. of ER to vol. of cytosol

K1 0.13 M Dissociation constant of IP3 from IP3R

K5 0.082 M Dissociation constant of Ca from IP3R activation site

J16

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Bt = k1(1− B)[glu] − k2B

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Gα t =k 3(1− Gα )B − k4Gα

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[IP3]t = Dp[IP3]xx − k6[IP3] + k5 Gα [Cai]PLC([Cai]) + ε( )

PLC([Cai]) =[Cai]

[Cai] + 0.1μM

Dp 2.8e-7 mm2/ms Diffusion constant for IP3

pbase 0.35 M Basal level of IP3

k1 0.01 /ms/ [glu] binding rate to mGluR

k2 0.0005 /ms [glu] unbinding rate from mGluR

k3 0.0001 /ms/ Activation rate of G-protein

k4 0.0002 /ms Inactivation rate of G-protein

k5 0.01 /ms/ Production rate of IP3

k6 0.00001 /ms Decay rate of IP3

PIP2 1 Amt of PIP2 in cell

glu 0.001 Rate of decay of [glu]

Mag 0.00001 M Amt [glu] applied to stimulus region

c 0.185 Basal amt of Ca in cytosol

x1 0.75 mm Location of beginning of glu stimulus

x2 0.8125 mm Location of end of glu stimulus

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Vt = DVVxx - gNam∞3 (V) (1- n)(V - VNa ) - gKn4 (V - VK ) - gL(V - VL) +I(x,t)

nt =n∞(V ) − n

τ n (V )

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[Cai]t = Dc[Cai]xx + (Rr1x1103 + r2)([Cas] −[Cai]) −

r3[Cai]2

[Cai]2 + K p

2+ γV+

ht =(h∞([Cai];[IP3]) − h)

τ h ([Cai];[IP3])

x110 =[IP3][Ca i]

([IP3] +K1)([Ca i] +K5)h

cavg = [Cas] + [Ca i]

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. Cai Elevation in Response to BPAPs.

Figure 6. Solution Set. If pbase is sufficiently small, for all pstim only small perturbations from rest exist. If pbase and pstim are sufficiently high, calcium waves are initiated as shown in the phase portrait Figure 8. If pbase and pstim 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 pstim depending only on the concentration of IP3 into which the wave is traveling. The wave speed is monotonically increasing as pbase increases.

Figure from Nakamura [4]. Backpropagating action potentials evoke Ca2+ 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 [Ca2+]i (ΔF/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 [Ca2+]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.