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Glutamate regulation of calcium and IP3 oscillating and pulsating dynamics in astrocytes

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An Erratum to this article was published on 19 December 2009

An Erratum to this article was published on 19 December 2009

Abstract

Recent years have witnessed an increasing interest in neuron–glia communication. This interest stems from the realization that glia participate in cognitive functions and information processing and are involved in many brain disorders and neurodegenerative diseases. An important process in neuron–glia communications is astrocyte encoding of synaptic information transfer—the modulation of intracellular calcium (Ca2 + ) dynamics in astrocytes in response to synaptic activity. Here, we derive and investigate a concise mathematical model for glutamate-induced astrocytic intracellular Ca2 +  dynamics that captures the essential biochemical features of the regulatory pathway of inositol 1,4,5-trisphosphate (IP3). Starting from the well-known two-variable (intracellular Ca2 +  and inactive IP3 receptors) Li–Rinzel model for calcium-induced calcium release, we incorporate the regulation of IP3 production and phosphorylation. Doing so, we extend it to a three-variable model (which we refer to as the ChI model) that could account for Ca2 +  oscillations with endogenous IP3 metabolism. This ChI model is then further extended into the G-ChI model to include regulation of IP3 production by external glutamate signals. Compared with previous similar models, our three-variable models include a more realistic description of IP3 production and degradation pathways, lumping together their essential nonlinearities within a concise formulation. Using bifurcation analysis and time simulations, we demonstrate the existence of new putative dynamical features. The cross-couplings between IP3 and Ca2 +  pathways endow the system with self-consistent oscillatory properties and favor mixed frequency–amplitude encoding modes over pure amplitude–modulation ones. These and additional results of our model are in general agreement with available experimental data and may have important implications for the role of astrocytes in the synaptic transfer of information.

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Acknowledgements

The authors wish to thank Vladimir Parpura, Giorgio Carmignoto, and Ilyia Bezprozvanny for insightful conversations. V. V. acknowledges the support of the U.S. National Science Foundation I2CAM International Materials Institute Award, Grant DMR-0645461. This research was supported by the Tauber Family Foundation, by the Maguy-Glass Chair in Physics of Complex Systems at Tel Aviv University, by the NSF-sponsored Center for Theoretical Biological Physics (CTBP), grants PHY-0216576 and 0225630, and by the University of California at San Diego.

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Authors and Affiliations

  1. School of Physics and Astronomy, Tel Aviv University, 69978, Ramat Aviv, Israel

    Maurizio De Pittà, Mati Goldberg & Eshel Ben-Jacob

  2. Center for Theoretical Biological Physics, UCSD, La Jolla, CA, 92093-0319, USA

    Vladislav Volman & Eshel Ben-Jacob

  3. Computational Neurobiology Lab, The Salk Institute, La Jolla, CA, 92037, USA

    Vladislav Volman

  4. Project-Team Alchemy, INRIA Saclay, 91893, Orsay, France

    Hugues Berry

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  1. Maurizio De Pittà
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Correspondence to Eshel Ben-Jacob.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s10867-009-9182-8

Electronic supplementary material

Below is the image is a link to a high resolution version

Supplementary Figure 1

The product of two Hill functions (a-b) with sufficiently distant midpoints is equivalent to the Hill function with the largest midpoint (c). Namely: Hill(x, K 1) · Hill(x, K 2) ≈ Hill(x, K 2) where K 1<<K 2. Midpoints are marked by vertical dashed lines; K 1: red; K 2: blue. (GIF 22.5KB)

High resolution image file (EPS 146KB)

Supplementary Figure 2

Hill functions of Hill functions (a-b) can also be approximated by Hill functions. (c-d) Hill (Hill(x, K 2), K 1) = (1+K 1)−1. Hill(x, K 1 K 2(1+K 1)−1). In this case the midpoint of the resulting Hill function depends on the specific values of the midpoints of the original Hill functions considered in the composition of the Hill-of-Hill function. (e-h) Hill(x, K 1 · Hill (x, K 2)) = (x + K 2)/(x + K 1 + K 2) = Hill (x, (K 1 + K 2)) + f(x), where f(x) = K 2/(x + K 1 + K 2). Notably, f(x⟶0) = K 2/(K 1 + K 2) whereas f(x⟶∞) ≈ 0, so that the resulting Hill curve is essentially comprised within the interval [K 2/(K 1 + K 2),1). (GIF 50.6KB)

High resolution image file (EPS 284KB)

Supplementary Figure 3

Bifurcation diagrams for a modified ChI model and prototypical sets of (a-c) AM-encoding and (d-f) FM-encoding L-R parameters. The bifurcation diagrams were computed after introduction into the ChI model of the rate of glutamate-dependent IP3 production, v glu , as a free bifurcation parameter, namely \(\bar I = v_{glu} + v_\sigma (C,I) - v_{3K} (C,I) - v_{5P} (I)\). This figure shows that the ChI model can still display oscillations in presence of an external non-specific bias of IP3 production. This is a first suggestion that the corresponding glutamate-dependent G-ChI model may also display oscillations. The parameters are taken from Table 1. (GIF 44.6KB)

High resolution image file (EPS 191KB)

Appendices

Appendix 1

For the sake of simplicity, we have adopted throughout the text the following notation for the generic Hill function:

$$ {\rm Hill}\!\left( {x^n,K} \right)\equiv \frac{x^n}{x^n+K^n} $$

where n is the Hill coefficient and K is the midpoint of the Hill function, namely the value of x at which Hill\(\left. {\left( {x^n,K} \right)} \right|_{x=K} =1 \mathord{/\, {\vphantom {1 2}} \kern-\nulldelimiterspace} 2\).

It can be shown that the product of two Hill functions can be approximated by the Hill function with the greatest midpoint, when the two midpoints are distant enough from each others, that is:

$$ {\rm Hill}\left( {x^n,K_1 } \right)\cdot {\rm Hill}\!\left( {x^n,K_2 } \right)\approx {\rm Hill}\!\left( {x^n,K_2 } \right) $$

if and only if K 1 < < K 2 (Fig. 1, Online Supplementary Material). Indeed, under such conditions, Hill\(\left( {x^n,K_1 } \right) \cdot {\rm Hill}\!\left( {x^n,K_2 } \right)>\!> 0\) only when x > > K 1, hence

$$\begin{array}{lll} {\rm Hill}\!\left( {x^n,K_1 } \right) \cdot {\rm Hill}\!\left( {x^n,K_2 } \right)&=&\frac{x^{2n}}{x^{2n}+\left( {K_1^n +K_2^n } \right)x^n+K_1^n K_2^n }\\ &\approx& \frac{x^{2n}}{x^{2n}+K_2^n x^n}={\rm Hill}\!\left( {x^n,K_2 } \right). \end{array}$$

This result can be extended to the product of N Hill functions, that is:

$$ \prod\limits_{i=1}^N {{\rm Hill}\!\left( {x^n,K_i } \right)} \approx {\rm Hill}\!\left( {x^n,\max \left( {K_1 ,\ldots ,K_N } \right)} \right) $$

provided that K 1 < < K 2 < <...< < K N .

Notably, the product of Hill function is not the only case in which a functions composed by Hill functions can be approximated by a mere Hill function: other examples are given by functions of the type \({\rm Hill}\!\left( {\left( {{\rm Hill}\!\left( {x^n,K_1 } \right)} \right)^m,K_2 } \right)\) or \({\rm Hill}\!\left( {x^m,K_1 \cdot {\rm Hill}\!\left( {x^n,K_2 } \right)} \right)\) (see Fig. 2 in Online Supplementary Material).

Appendix 2

We seek an expression for [CaMKII*] based on the following kinetic reaction scheme:

$$ \label{eq21} 4{\rm Ca}^{2+}+{\rm CaM}\mathop\rightleftarrows\limits^{k_b}_{k_u} {\rm CaM}^+ $$
(21)
$$ \label{eq22} {\rm KII}+{\rm CaM}^+\mathop\rightleftarrows\limits^{k_1}_{k_{-1}}{\rm CaMKII}\mathop\rightleftarrows\limits^{k_2}_{k_{-2}}{\rm CaMKII}\mbox{*} . $$
(22)

Let us first consider the reaction chain (22). We can assume that the second step is very rapid with respect to the first one [58, 104] so that generation of CaMKII* is in equilibrium with CaMKII consumption, namely:

$$ \label{eq23} \left[ \mbox{CaMKII*} \right]\approx \frac{k_2 }{k_{-2} }\left[ \mbox{CaMKII} \right]. $$
(23)

Then, under the hypothesis of quasisteady state for CaMKII, we can write:

$$ \label{eq24} \frac{d}{dt}\left[ {{\rm CaMKII}} \right]=k_1 \left[ {{\rm KII}} \right][ {{\rm CaM}^+} ]-\left( {k_{-1} +k_2 } \right)\left[ {{\rm CaMKII}} \right]+k_{-2} \left[ {\mbox{CaMKII*}} \right]\approx 0. $$
(24)

It follows that incorporation of (23) into (24) leads to:

$$ \label{eq25} \left[ \mbox{{CaMKII}* } \right]=K_1 K_2 \left[ {{\rm KII}} \right][ {{\rm CaM}^+} ] $$
(25)

where \(K_i ={k_i } \mathord{/{\kern1pt} {\vphantom {{k_i } {k_{-i} }}} \kern-\nulldelimiterspace} {k_{-i} }\). Defining \(\left[ {{\rm KII}} \right]_{\rm T} =\left[ {{\rm KII}} \right]+\left[ {{\rm CaMKII}} \right]+\left[ {\mbox{CaMKII*}} \right]\) as the total kinase II concentration and assuming it constant, we can rewrite (25) as follows:

$$ \label{eq26} \left[ \mbox{{CaMKII}* } \right]=\frac{K_2 \left[ {{\rm KII}} \right]_{\rm T} }{1+K_2 }\frac{[ {{\rm CaM}^+} ]}{[ {{\rm CaM}^+} ]+K_m } $$
(26)

with \(K_m =\left( {K_1 \left( {K_2 +1} \right)} \right)^{-1}\).

The substrate concentration for the enzymatic reaction (22) is provided by reaction (21) according to which:

$$ \label{eq27} [ {{\rm CaM}^+} ]=\left[ {{\rm CaM}} \right]\frac{[ {{\rm Ca}^{2+}} ]^4}{[ {{\rm Ca}^{2+}} ]^4+K_{\rm d} } $$
(27)

with \(K_{\rm d} ={k_u } \mathord{/{\kern1pt} {\vphantom {{k_u } {k_b }}} \kern-\nulldelimiterspace} {k_b }\). Therefore, substituting (27) into (26), we obtain:

$$ \label{eq28} \left[\mbox{{CaMKII}*}\right]=\frac{K_2 \left[ {{\rm KII}} \right]_{\rm T} }{1+K_2 }\left( {1+\frac{K_{\rm m} }{\left[ {{\rm CaM}} \right]}} \right)^{-1}\frac{[ {{\rm Ca}^{2+}} ]^4}{[ {{\rm Ca}^{{\rm 2}+}} ]^4+\frac{K_{\rm m} K_{\rm d} }{K_{\rm m} \,+\, \left[ {{\rm CaM}} \right]}} $$
(28)

so that \(\left[\mbox{{CaMKII}*}\right]\propto {\rm Hill}\left( {[ {{\rm Ca}^{2+}} ]^4,K_{\rm D} } \right)\) with \(K_{\rm D} =\left( {\frac{K_{\rm m} K_{\rm d} }{K_{\rm m} \,+\,\left[ {\rm CaM} \right]}} \right)^{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}\).

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De Pittà, M., Goldberg, M., Volman, V. et al. Glutamate regulation of calcium and IP3 oscillating and pulsating dynamics in astrocytes. J Biol Phys 35, 383–411 (2009). https://doi.org/10.1007/s10867-009-9155-y

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