Energetics of Inhibition: Insights with a Computational Model of the Human GABAergic Neuron

Abstract

We investigate metabolic interactions between astrocytes and GABAergic neurons at steady states corresponding to different activity levels using a six-compartment model and a new methodology based on Bayesian statistics. Many questions about the energetics of inhibition are still waiting for definite answers, including the role of glutamine and lactate effluxed by astrocytes as precursors for γ-aminobutyric acid (GABA), and whether metabolic coupling applies to the inhibitory neurotransmitter GABA. Our identification and quantification of metabolic pathways describing the interaction between GABAergic neurons and astrocytes in connection with the release of GABA makes a contribution to this important problem. Lactate released by astrocytes and its neuronal uptake is found to be coupled with neuronal activity, unlike glucose consumption, suggesting that in astrocytes, the metabolism of GABA does not require increased glycolytic activity. Negligible glutamine trafficking between the two cell types at steady state questions glutamine as a precursor of GABA, not excluding glutamine cycling as a transient dynamic phenomenon, or a prominent role of GABA reuptake. Redox balance is proposed as an explanation for elevated oxidative phosphorylation and adenosine triphosphate hydrolysis in astrocytes, decoupled from energy requirements.

Keywords

Bayesian inference flux balance analysis GABA–glutamine cycle GABA synthesis glutamine transfer multicompartment model

Introduction

GABAergic neurons are characterized by a prominent expression of the enzyme glutamate decarboxylase, which converts glutamate to γ-aminobutyric acid (GABA), the most abundant inhibitory neurotransmitter in the mammalian central nervous system.

The source of glutamate precursors required for GABA synthesis has been extensively investigated in vivo and in vitro for more than 30 years, following a pioneering work by Van den Berg and Garfinkel (1971), who, using a multicompartment mathematical model, identified glutamine as a possible precursor of glutamate and proposed cycling of GABA and glutamine between astrocytes and neurons. The interest in GABA precursors was motivated by the lack of pyruvate carboxylase (PC) in neurons, which implied that the loss of GABA released during neurotransmission should be compensated by the flow of a precursor in the opposite direction (Bak et al, 2006). Exogeneous glutamine, formed in astrocytes through the astrocyte-specific enzyme glutamine synthetase, is the most prominent candidate for GABA precursor. In the context of energetics of excitation, glutamine is generally accepted as a metabolic precursor for glutamate synthesis in glutamatergic neurons in which its influx from astrocytes has been experimentally identified and quantified (Sibson et al, 1997; Shen et al, 1999). In GABAergic neurons, in which the machinery for synthesis of GABA from glutamine is more complex and may vary across the different populations of GABAergic neurons (Waagepetersen et al, 1998), the role of glutamine is not yet fully understood. The metabolism of GABA has been investigated with ¹³C NMR (carbon nuclear magnetic resonance) spectroscopy in vitro (Waagepetersen et al, 1998) and in vivo (Patel et al, 2001), but owing to the difficulty in measuring the relative contributions of exogenous glutamate and glutamine to GABA synthesis, quantifications of the GABA-glutamine cycling flux found in the literature differ significantly. Recently, Yang et al (2007) found this flux to be one order of magnitude smaller than previously estimated (Van den Berg and Garfinkel, 1971; Patel et al, 2005). The still largely unknown metabolic interaction between astrocytes and GABAergic neurons adds to the challenge of understanding GABA metabolism and synthesis. However, this is not the only mystery of the energetics of inhibition, because the ability of GABAergic neurons to incorporate carbon into neuroactive amino acids is still far from completely understood (Waagepetersen et al, 1998).

This study, based on a novel six-compartment mathematical model of the astrocyte–GABAergic neuron cellular complex and a new statistical methodology to perform flux balance analysis, investigates the roles of glutamine produced by astrocytes as a precursor for GABA synthesis. Moreover, it sheds some light on how the astrocyte–neuron complex adjusts the metabolic flux balance under different levels of inhibitory activity, and on the roles of glucose and lactate as sources of energy and the importance of redox balance. Scarcity of experimental data for GABA metabolism in humans and sometimes conflicting results, while emphasizing the importance of computational models in understanding the energetics of inhibition, also complicate the model validation against measured data.

Materials and methods

In this section, we briefly describe the biochemical pathways and the six-compartment computational model used to investigate metabolic interactions between astrocytes and GABAergic neurons at steady state under different neuronal activity levels, and the computational and statistical methodology to produce and analyze its output.

Biochemical Pathways

Following the modeling paradigm of Occhipinti et al (2009), the astrocyte–GABAergic neuron cellular complex, the metabolic pathways of which are shown in Figure 1, is assumed to be spatially homogeneous over a region of interest, where both cells are morphologically and functionally uniform. The six compartments accounted for by the model include the cytosol and mitochondria in astrocytes, the cytosol and mitochondria in GABAergic neurons, the extracellular space (ECS), and blood.

Figure 1

Schematics of the biochemical pathways included in the six-compartment model of the astrocyte–GABAergic neuron cellular complex. The cross and box markers on some of the arrows identify the malate/α-ketoglutarate and glutamate/aspartate antiporter pairs of the malate-aspartate shuttle. The shaded areas around the cells indicate the extracellular space and blood domains. ACoA, acetyl-coenzyme A; ADP, adenosine diphosphate; AKG, α-ketoglutarate; Asp, aspartate; ATP, adenosine triphosphate; BPG, biphosphoglycerate; CIT, citrate; CO₂, carbon dioxide; CR, creatine; FUM, fumarate; GABA, γ-aminobutyric acid; GA3P, glyceraldeyde 3-phosphate; GLC, glucose; Gln, glutamine; GLU, glutamate; GLY, glycogen; G6P, glucose 6-phosphate; LAC, lactate; MAL, malate; OAA, oxaloacetate; O₂, oxygen; NADH, nicotinamide adenine dinucleotide (reduced form); NAD⁺, nicotinamide adenine dinucleotide; PCR, phosphocreatine; PYR, pyruvate; SCoA, succinyl-CoA; SSA, succinic semialdehyde; SUC, succinate.

The model equips both astrocytes and GABAergic neurons with complete glycolytic pathways in the cytosol, and tricarboxylic acid (TCA) cycle and oxidative phosphorylation in the mitochondria. A complete malate-aspartate shuttle (MAS) is included in both cells (Dennis and Clark, 1978), as are some anaplerotic reactions, including the malic enzyme (ME) in the mitochondria (Hassel and Brathe, 2000; Hassel, 2001), whereas PC is modeled in the mitochondria of astrocytes only (Yu et al, 1983). The main differences from our previous model for the astrocyte–glutamatergic neuron complex concern GABA transmission and synthesis and the inclusion of separate blood and ECS compartments. Although the fate of GABA, after its release from presynaptic neurons and subsequent interaction with receptors on the postsynaptic neurons, may include returning to the presynaptic neurons or being taken up by astrocytes, our model assumes that all released GABA is taken up by astrocytes, in which it is catabolized to the TCA cycle intermediate succinate through GABA transaminase and succinate semialdehyde dehydrogenase (Bak et al, 2006), adding two new biochemical reactions. The inclusion of the return path, although potentially important for neurotransmitter cycling, is outside the range of steady-state analysis, in which only the net flux of GABA is of interest. During transamination of GABA, α-ketoglutarate can be converted to glutamate, and then to glutamine by the cytosolic astrocyte-specific enzyme glutamine synthetase (Martinez-Hernandez et al, 1976).

The important role of exogenous glutamine, replenishing the precursor pool for GABA biosynthesis, has previously been suggested as a corollary of the inability of GABAergic neurons to synthesize GABA from glucose, because of the lack of the anaplerotic enzyme PC (Bak et al, 2006). The GABA pool is believed to be replenished mostly from astrocyte-released precursors, most prominently glutamine which, upon being taken up by neurons, is further synthesized into glutamate by phosphate-activated glutaminase on the outer surface of the inner mitochondrial membrane (Kvamme et al, 2000). The present model includes pathways for cycling of GABA from neurons to astrocytes and of glutamine from astrocytes to neurons, referred to as the GABA–glutamine cycle.

GABA is oxidized to succinate by the sequential actions of mitochondrial GABA transaminase and succinate semialdehyde dehydrogenase (Berkich et al, 2007), a portion of the metabolic pathway known as GABA shunt, which bypasses two TCA cycle reactions (α-ketoglutarate-dehydrogenase and succinyl-CoA synthase). Although GABA synthesis is confined to GABAergic neurons, GABA degradation can occur in both astrocytes and neurons. Therefore, our model includes glutamate decarboxylase only in the mitochondria of GABAergic neurons, whereas GABA transaminase and succinate semialdehyde dehydrogenase in the mitochondria of both cell types. The bidirectional reaction glutamate dehydrogenase, which has an important role in brain cellular metabolism (Waagepetersen et al, 2003), has been included in the mitochondria of both cells.

Transports of glucose and oxygen are modeled as influxes and those of carbon dioxide as effluxes in astrocytes and neurons, whereas exchanges of lactate between cytosolic domains and ECS, and between ECS and blood are bidirectional, allowing both cells to produce or use lactate according to their metabolic needs.

The Computational Model

We present a brief overview of the computational model used to perform steady-state analysis; motivation and further details can be found in the study by Calvetti and Somersalo (2006) (also see Heino et al, 2007, 2009; Occhipinti et al, 2007, 2009).

In our model, on the basis of stoichiometric properties of a six-compartment metabolic system, rates of change in concentrations of biochemical species are described in terms of mass balances through a system of ordinary differential equations. As the concentration of a metabolite can change over time because of the convective action of blood through capillaries, biochemical reactions, and exchanges with other compartments, a prototype equation is of the form:

V \frac{d C}{d t} = \underset{Reaction}{\underset{︸}{S Φ}} + \underset{Transport}{\underset{︸}{M J}} + \underset{Convection}{\underset{︸}{Q K}},

(1)

with the understanding that not all terms in the right-hand side need be present.

In equation (1), Vis the volume of the compartment, Φ is a vector containing the intracompartmental reaction fluxes and S = [s_ij] denotes the stoichiometric matrix, the entry s_ij of which indicates how many moles of species i are produced (s_ij>0) or consumed (s_ij<0) in the jth biochemical reaction Φ_j. When the species does not participate in the jth reaction, s_ij=0. In the second term, J is the vector of transport rates and M= [m_ij] a matrix the entries of which indicate whether the ith species are transported into (m_ij = 1) a compartment, out of it (m_ij= −1) at the rate J_j, or not exchanged (m_ij = 0). In the convection term, present in the blood compartment only, the matrix Q indicates which biochemical species are affected by convection, which is modeled as K=CBF (C_a–C_v), where CBF is the cerebral blood flow and C_a–C_v the vector of arterial-venous concentration differences of glucose (GLC), lactate (LAC), oxygen (O₂), and carbon dioxide (CO₂) in blood.

Motivated by the assumption that mitochondrial adenosine triphosphate (ATP) is partly consumed in the neuron to maintain the flux of GABA into the synaptic gap, we add a fictitious intermediate subcompartment in the neuronal cytosol, representing the GABA vesicles, and attach an energetic cost to the transport of mitochondrial GABA to this subcompartment by the stoichiometry GABA_m + β × ATP → GABA_v + β × ADP + β × P_i, where GABA_m and GABA_v represent the mitochondrial and vesicular states, and β is the energetic cost per unit of GABA effluxed in the synaptic cleft. We use the value β = 0.21 × 38 = 7.98, in agreement with experimental findings that 21% of the 38 moles of ATP produced per mole of glucose is required to sustain the entire activation machinery associated with the GABA–glutamine cycle (Hyder et al, 2006; Gjedde et al, 2002).

As we are interested in steady-state configurations of the astrocyte–GABAergic neuron cellular complex, we assume that the metabolic system is at or near equilibrium with constant, or near constant, reaction fluxes and transport rates.

The Steady-State Model

At steady state, the derivatives of concentrations vanish, thus reducing the mass balances (1) to a system of linear equations of the form:

A u = b,

(2)

where A = [S M], u = [Φ J] and the vector b = [0 r] has zeroes in correspondence to cytosolic and mitochondrial mass balances, and arterial-venous concentration differences in correspondence to the mass balances in blood domain, i.e., r = CBF (C_a–C_v). The linear system (2) is the steady-state computational model. If the venous concentration of a species is ill-determined, the system can be modified so that r contains only the arterial concentration, whereas the venous concentration becomes an unknown and is added as a component to the vector u; for more details, see Heino et al (2009).

Bayesian Flux Balance Analysis

Traditionally, flux balance analysis seeks to find reaction fluxes and transport rates supporting a steady state. Typically, the linear system (2) admits infinitely many solutions: those that are physiologically meaningless may be ruled out by introducing bound constraints (Schellenberger and Palsson, 2009; Heino et al, 2007). Expressing these constraints in the form Cu≥c, where C is a matrix and c is a constant vector, we restate the flux balance condition as:

A u = b, subject to C u \geq c .

(3)

Different approaches have been proposed to solve steady-state problems related to metabolic systems (Schellenberger and Palsson, 2009). We recast it as a statistical inference problem, modeling all unknowns as random variables. In the Bayesian framework, the nondeterministic nature of the variables reflects our lack of precise knowledge of their values, and to each outcome, a probability is assigned to reflect how credible the outcome is in the light of the model, data, and possible independent additional information. The model and data determine the likelihood, whereas any additional information determines the prior. Thus, any vector u, that is far from representing a steady state, although possible, has an extremely low probability, whereas to any of the possible steady states, a high probability is assigned. The probability combining both sources of information is the posterior probability, which, according to Bayes' formula, is the normalized product of prior and likelihood.

To set up the likelihood, we replace equation (3) with its stochastic extension as:

A u = b + e,

where the random vector e, referred to as noise, accounts for uncertainties in data, pathway network model, and steady-state conditions. Assuming that e is normally distributed with zero mean and Γ diagonal covariance matrix, we can write the likelihood density as:

π (b | u) \propto exp (- \frac{1}{2} {(b - A u)}^{T} Γ^{- 1} (b - A u)),

where the nonzero entries of Γ express our confidence in the system being at steady state or the variances of input blood concentrations.

The prior probability density of u, π_prior (u), an important component in the Bayesian framework, expresses our beliefs about the unknown u and is defined to take on a positive value if all bound constraints are satisfied and to vanish otherwise. In addition, any additional information regarding the values of selected fluxes and transports can be imported as priors into the analysis.

The solution of the Bayesian flux balance analysis is the posterior density of u which, according to Bayes' formula, is proportional to the product of the prior density and the likelihood:

π_{post} (u) = π (u | b) \propto π_{prior} (u) π (b | u) .

(4)

Following the procedure proposed in the study by Calvetti and Somersalo (2006), and further developed in the studies by Heino et al (2007) and Occhipinti et al (2007), to explore the posterior probability density, we generate a representative sample u¹,u²,…,u^N using Markov Chain Monte Carlo sampling described in the study by Occhipinti et al (2009). An efficient public domain computational tool, Metabolica, to facilitate the sampling is available (Heino et al, 2009) and was used for all our computed examples. In plain words, the algorithm randomly draws the vectors u^k so that realizations with high probability appear frequently, whereas vectors with low probability, while possible, do not appear often. In particular, the majority of the vectors u^k represent steady or nearly steady states, the variability being mostly attributed to the fact that different steady states satisfying the input up to a small error are possible.

Results

The experimental data consist of arterial-venous concentration differences for glucose, oxygen, and lactate in humans, and of arterial concentration of CO₂. The values reported in the study by Gibbs et al (1942), C_a,GLC–C_v,GLC = 0.54 mmol/L, C_a,O₂–C_v,O₂ = 2.99 mmol/L, and C_a,LAC–C_v,LAC = −0.18 mmol/L are referred to in the sequel as reference blood concentrations. To compensate for the artificial setup of the model, which only considers astrocytes and GABAergic neurons, thus ignoring glucose and oxygen taken up by glutamatergic neurons, we assume that only a given fraction of the substrates is available for the GABAergic neuron–astrocyte complex. Moreover, to avoid biasing the cerebral metabolic rate of CO₂ and, consequently, the TCA cycle activity, we consider venous concentration of CO₂ as one of the unknowns to be determined by the analysis, giving only the arterial concentration of CO₂ as input. In the likelihood covariance matrix, entries corresponding to the standard deviation (s.d.) values of the steady-state condition are set at 0.005 μmol/g per min, whereas the s.d. values of measured arterial-venous concentration differences are 10% of the specified value, leaving enough flexibility to adjust the cerebral metabolic rate (CMR) values near, but not necessarily precisely at the value predicted by the input blood values. These noise covariances give equal credibility for every substance to be off exact steady state. The CBF value is set at 0.5 mL/g per min. Reaction fluxes and transport rates are expressed in μmol/g per min. The bidirectional reactions lactate dehydrogenase (LDH), ME, glutamate dehydrogenase, and the bidirectional transport of lactate from the ECS to the cytosol in both cell types are estimated as net fluxes that can take on positive or negative values. To ensure that the reaction fluxes and transport rates estimated by our methodology are physiologically meaningful and to avoid exploring unfeasible regions in parameter space, we impose loose upper bounds for absolute values of all unknowns and nonnegativity of reaction fluxes and transport rates with known preferred direction at steady state. We simulate different levels of neuronal activity by introducing a Gaussian prior for the efflux of GABA from neurons to the ECS. The mean value of the prior density, following Hyder et al (2006) and Shulman et al (2004), is J_GABA,N→ECS = 0.13 μmol/g per min at high activity and J_GABA,N→ECS = 0.04 μmol/g per min at low activity, with s.d. 10% of the mean value. Initially, the following computational experiments were performed:

(1)

High- and low-inhibitory activation state, 50% of the reference blood concentrations of glucose and oxygen available for the GABAergic neuron—astrocyte complex;

(2)

high- and low-inhibitory activation states, 25% of the reference blood concentrations available.

For each experiment, a sample of 100,000 points was collected by generating a sample of 120,000 and discarding the first 20,000. The mixing properties of the samples were assessed by calculating the autocorrelation functions for each component, as described in the studies by Heino et al (2007) and Occhipinti et al (2009). A summary of the mean predicted values and s.d. of the main reaction fluxes and transport rates in astrocytes and neurons in the various experiments is reported in Tables 1 and 2. Before discussing the results in more detail, we point out that for each reaction flux and transport rate, the s.d. values are computed as the square root of the mean squared deviation from the mean value. This method of averaging each component separately hides the possible strong correlations between fluxes and transports, which need to be analyzed separately. Consequently, a flux vector with each component separately within the interval defined by the mean and s.d. is not necessarily physiologically meaningful; the components need to be correlated as predicted by the full probability distribution.

Table 1

Estimated reaction fluxes and transport rates in astrocyte for the six different computational experiments

Astrocyte	High	High+Gln	Low	Low+Gln	Reduced high	Reduced low
J _ECS→c,GLC	0.075 ± 0.043	0.077 ± 0.043	0.073 ± 0.041	0.073 ± 0.041	0.042 ± 0.025	0.036 ± 0.021
Φ_PYR→LAC	0.125 ± 0.087	0.126 ± 0.087	0.066 ± 0.093	0.057 ± 0.094	0.091 ± 0.051	0.047 ± 0.046
Φ_ATP→ADP+Pi	3.494 ± 0.460	3.526 ± 0.489	2.752 ± 0.818	2.852 ± 0.873	1.789 ± 0.246	1.554 ± 0.332
MAS_c	0.028 ± 0.019	0.031 ± 0.020	0.083 ± 0.045	0.091 ± 0.048	0.014 ± 0.008	0.032 ± 0.019
MAS_m	0.025 ± 0.019	0.025 ± 0.019	0.080 ± 0.045	0.084 ± 0.048	0.011 ± 0.008	0.029 ± 0.019
J _c→m,PYR	0.027 ± 0.019	0.030 ± 0.020	0.083 ± 0.045	0.091 ± 0.048	0.010 ± 0.008	0.031 ± 0.019
J _m→c,ATP	3.347 ± 0.452	3.383 ± 0.478	2.608 ± 0.814	2.718 ± 0.869	1.691 ± 0.242	1.481 ± 0.332
Φ_GLU→Gln	0.006 ± 0.004	0.013 ± 0.004	0.006 ± 0.004	0.015 ± 0.005	0.006 ± 0.004	0.006 ± 0.004
J _ECS→c,GABA	0.111 ± 0.013	0.112 ± 0.013	0.040 ± 0.006	0.040 ± 0.008	0.075 ± 0.010	0.038 ± 0.006
J _c→ECS,Gln	0.007 ± 0.004	0.020 ± 0.006	0.007 ± 0.004	0.022 ± 0.005	0.007 ± 0.004	0.007 ± 0.004
Φ_GABA→SSA	0.110 ± 0.013	0.110 ± 0.013	0.040 ± 0.010	0.040 ± 0.010	0.071 ± 0.011	0.038 ± 0.010
Φ_PYR→ACoA	0.131 ± 0.021	0.134 ± 0.023	0.125 ± 0.044	0.133 ± 0.047	0.062 ± 0.012	0.065 ± 0.018
Φ_PYR→OAA	0.078 ± 0.050	0.079 ± 0.050	0.102 ± 0.077	0.106 ± 0.080	0.039 ± 0.024	0.048 ± 0.033
Φ_MAL→PYR	0.181 ± 0.053	0.184 ± 0.053	0.143 ± 0.078	0.147 ± 0.081	0.093 ± 0.027	0.082 ± 0.035
Φ_MAL→OAA	0.080 ± 0.050	0.081 ± 0.052	0.103 ± 0.078	0.112 ± 0.085	0.038 ± 0.024	0.048 ± 0.033
Φ_GLU→AKG	0.110 ± 0.015	0.109 ± 0.015	0.040 ± 0.012	0.041 ± 0.012	0.066 ± 0.011	0.030 ± 0.011
ΦO₂→H₂O	0.550 ± 0.073	0.556 ± 0.077	0.431 ± 0.134	0.450 ± 0.143	0.280 ± 0.040	0.245 ± 0.054

Posterior mean values ± s.d. predicted by the steady-state computational model of the human astrocyte–GABAergic cellular complex in the five different experiments, at low and high neuronal activity, with and without prior biasing on the glutamine transport from astrocytes to neurons. This table summarizes the model predictions for the reaction fluxes and transport rates in astrocytes at 50% (first four columns) and 25% (last two columns referred as ‘Reduced’) of the reference blood concentrations. The first and third columns summarize the results at high and low neuronal activity, respectively, without prior biasing on glutamine trafficking, whereas the second and fourth columns summarize the model predictions at high and low neuronal activity with prior on the glutamine transport. The last two columns summarize the results for high and low neuronal activity for the simulations obtained using 25% of the reference blood concentrations. The units are in μmol per gram tissue per minute.

Table 2

Estimated reaction fluxes and transport rates in GABAergic neuron for the six different computational experiments

Neuron	High	High+Gln	Low	Low+Gln	Reduced high	Reduced low
J _ECS→c,GLC	0.077 ± 0.043	0.075 ± 0.043	0.073 ± 0.041	0.073 ± 0.041	0.043 ± 0.025	0.039 ± 0.021
Φ_LAC→PYR	0.083 ± 0.088	0.083 ± 0.088	0.008 ± 0.094	∼0	0.012 ± 0.052	0.006 ± 0.048
Φ_ATP→ADP+Pi	0.853 ± 0.393	0.815 ± 0.405	1.807 ± 0.820	1.710 ± 0.812	0.300 ± 0.164	0.643 ± 0.313
MAS_c	0.238 ± 0.032	0.234 ± 0.032	0.155 ± 0.050	0.147 ± 0.049	0.114 ± 0.022	0.086 ± 0.027
MAS_m	0.242 ± 0.031	0.244 ± 0.031	0.158 ± 0.050	0.157 ± 0.048	0.120 ± 0.022	0.090 ± 0.027
J _c→m,PYR	0.239 ± 0.031	0.235 ± 0.031	0.156 ± 0.049	0.147 ± 0.049	0.117 ± 0.021	0.087 ± 0.026
J _m→c,ATP	0.700 ± 0.378	0.662 ± 0.390	1.660 ± 0.816	1.561 ± 0.807	0.195 ± 0.156	0.562 ± 0.308
Φ_Gln→GLU	0.006 ± 0.004	0.018 ± 0.006	0.006 ± 0.004	0.020 ± 0.005	0.006 ± 0.004	0.006 ± 0.004
J _c→ECS,GABA	0.113 ± 0.012	0.112 ± 0.013	0.039 ± 0.006	0.040 ± 0.006	0.077 ± 0.010	0.038 ± 0.006
J _ECS→c,Gln	0.007 ± 0.004	0.027 ± 0.006	0.007 ± 0.004	0.029 ± 0.003	0.007 ± 0.004	0.007 ± 0.004
Φ_GABA→SSA	0.018 ± 0.015	0.018 ± 0.016	0.046 ± 0.036	0.045 ± 0.036	0.009 ± 0.007	0.017 ± 0.014
Φ_GLU→GABA	0.130 ± 0.017	0.134 ± 0.022	0.084 ± 0.036	0.083 ± 0.036	0.084 ± 0.011	0.054 ± 0.015
Φ_PYR→ACoA	0.138 ± 0.021	0.134 ± 0.022	0.123 ± 0.045	0.114 ± 0.044	0.073 ± 0.011	0.061 ± 0.018
Φ_PYR→MAL	0.102 ± 0.018	0.102 ± 0.018	0.033 ± 0.016	0.033 ± 0.016	0.046 ± 0.015	0.027 ± 0.015
Φ_AKG→GLU	0.110 ± 0.014	0.110 ± 0.014	0.037 ± 0.009	0.038 ± 0.009	0.069 ± 0.011	0.035 ± 0.009
ΦO₂→H₂O	0.260 ± 0.059	0.255 ± 0.060	0.320 ± 0.132	0.304 ± 0.131	0.134 ± 0.025	0.141 ± 0.050

ACoA, acetyl-coenzyme A; AKG, α-ketoglutarate; ADP, adenosine diphosphate; ATP, adenosine triphosphate; Pi, inorganic phosphate; c, cytosol; CO₂, carbon dioxide; ECS, extracellular space; m, mitochondria; GABA, γ-aminobutyric acid; GLC, glucose; Gln, glutamine; GLU, glutamate; LAC, lactate; MAL, malate; MASc and MASm, malate-aspartate shuttle reactions in the cytosol and in mitochondria, respectively; O₂, oxygen; PYR, pyruvate; SSA, succinic semialdehyde. Posterior mean values ± s.d. predicted by the steady-state computational model of the human astrocyte–GABAergic cellular complex in all five different experiments, at low and high neuronal activity, with and without prior biasing on the glutamine transport from astrocytes to neurons. This table summarizes the model predictions for the reaction fluxes and transport rates in GABAergic neuron. The units are in μmol per gram tissue per minute.

The results of the second set of experiments, summarized in the last two columns of the tables, show that at 25% of the reference blood concentrations, the system is unable to produce the requested amount of GABA, suggesting that the reduction in the substrate availability is too strong.

In our Bayesian steady-state analysis at 50% of the reference blood concentrations, the glutamine flux from astrocytes to neurons settles at a negligible level (0.007 ± 0.004 μmol/g per min) regardless of the neuronal activity level. Therefore, we conclude that our model does not support spontaneous steady-state trafficking of glutamine from astrocytes to neurons at both activity levels. This does not necessarily exclude glutamine trafficking between the two cells; it could be a dynamic phenomenon, active only during a transient phase, not sustainable at steady state. To get more insights into the implication of the putative glutamine–GABA cycle, we repeated the experiments introducing a Gaussian prior for the glutamine uptake with mean and variance equal to the GABA efflux from neurons. As this configuration does not necessarily correspond to a steady state, strict flux balance may not be found, and the posterior mean is a compromise between the imposed prior for the inhibitory activity and the steady-state condition, giving an indication of the nature of the transient states during which glutamine shuttling might occur.

The schematics in Figure 2 summarizes this discussion at high neuronal activity with a prior favoring glutamine trafficking (second column from the left in Tables 1 and 2). Our results show that even with a prior for the glutamine shuttling, the computed mean flux values which are 0.027 μmol/g per min at high activity versus an imposed prior expectation of 0.13 μmol/g per min, and 0.007 μmol/g per min at low activity versus an imposed prior expectation of 0.04 μmol/g per min, remain remarkably below the prior mean, indicating that the low glutamine transport in this model is a stable feature of the steady state.

Figure 2

Summary of preferred biochemical pathways. (A) Schematics of the preferred biochemical pathways in astrocytes and GABAergic neurons at high neuronal activity with the substrate availability of 50% of the reference values and priors favoring glutamine transport and GABA release at expected rates of 0.13 μmol/g per min. The transports subject to prior biasing are marked by the symbol . The numbers by the arrows are the posterior conditional mean values in μmol/g per min of the reaction and transport rates estimated from the MCMC sample generated with our computational model. The glycolytic flux values refer to the pyruvate production rates. It must be noted that the rate of glutamine transport from astrocytes to neurons averages ∼0.02 μmol/g per min, which is only 20% of the prior mean. (B) Schematics of the preferred biochemical pathways in astrocytes and GABAergic neurons at high neuronal activity with the substrate availability of 50% of the reference values and a prior suppressing the activity of malic enzyme in neurons. The GABAergic neuron spontanously uptakes glutamine at a level comparable with that reached when using a prior; see (panel A), but it is unable to maintain GABA efflux at the prior mean rate of 0.13 μmol/g per min. GABA, γ-aminobutyric acid; MCMC, Markov Chain Monte Carlo.

Without prior on the glutamine transport, the deviation from steady state, measured as the difference Au–b, is small throughout, ranging between 10⁻⁴ and 10⁻⁷ in the cellular domains. With a prior for the glutamine–GABA cycling, the system shifts slightly off from the strict steady state, the deviation increasing to orders of magnitude between 10⁻² and 10⁻⁵. Interestingly, the steady-state condition is violated more conspicuously by the mass balances of aspartate, glutamine, and glutamate, with discrepancies ∼ −0.05 μmol/g per min in the astrocytic cytosol and ∼0.05 μmol/g per min in the neuronal cytosol, pointing to a slight depletion of these neurotransmitters in the former and corresponding accumulation in the latter. Figure 3A summarizes in a schematic manner the roles of glutamine, glutamate, α-ketoglutarate, and MAS in the synthesis of GABA at high neuronal activity with a prior favoring glutamine trafficking, highlighting the complexity of the process. Figure 3B shows the carbon pathways related to neurotransmitter cycling, with the width of the arrows proportional to the activity and pathways strongly positively correlated in the collected ensemble, marked by stars.

Figure 3

Details of GABA synthesis and trafficking. (A) A detail of the metabolic pathway showing the reactions in the neuron related to the GABA synthesis. The reported values correspond to the experiment summarized in Figure 2A, in which the substrate availability is 50% of the reference value for the tissue, the activity level is high, and glutamine transport is favored by a prior. (B) A summary of metabolic pathways comprising the fate of carbon skeletons related to GABA synthesis and release. The width of the arrows is proportional to the value of the corresponding flux; reactions and transports strongly positively correlated over the collected ensemble are identified by stars. GABA, γ-aminobutyric acid.

In all simulations, the rate of MAS matches the rate at which pyruvate is transferred from the cytosol to the mitochondria. Malic enzyme activity in neurons, glutamate dehydrogenase in both cells, GABA efflux from neurons, and astrocytic GABA shunt settle at approximately the same level in all experiments at 50% of the reference blood concentrations, although the level may vary between experiments.

Regardless of neuronal activity levels and whether glutamine uptake is enforced, glucose uptake and glycolytic activity remain essentially constant in both cells, and with equal glucose partitioning between the cells. We found the average of the ensemble of glycolytic pyruvate production to be in the range of 0.14 to 0.15 μmol/g per min for each cell with 50% glucose availability, and roughly one-half when glucose availability is halved.

The ensembles of LDH flux values in both cells are characterized by high variability, as indicated by the large s.d., with an offset toward pyruvate production in neurons and lactate production in astrocytes. The average LDH flux in neurons is proportional to the activation level, decreasing by a factor of 10, from 0.083 to 0.008 μmol/g per min when passing from high to low neuronal activity. The a priori biasing toward glutamine uptake does not affect the LDH flux in either cell at high activity, whereas decreasing it slightly at low activity. The pyruvate dehydrogenase flux in astrocytes is supported by glycolysis and ME, the latter always showing a preferred direction toward pyruvate production with the average flux value decreasing from 0.181 to 0.143 μmol/g per min when passing from high to low activity. The efflux of carbon skeletons from the TCA cycle by ME is replenished through pyruvate dehydrogenase and PC, the latter always active in astrocytes with flux values averaging ∼0.08 to 0.1 μmol/g per min and exhibiting relatively large variability. Pyruvate carboxylation activity in astrocytes is strongly positively correlated with ME toward pyruvate production and negatively correlated with the malate dehydrogenase flux in the TCA cycle, as clearly visible in the scatter plots shown in Figure 4. Our model also supports ME activity in neurons in the direction of malate production, approximately at the same level at which the TCA cycle intermediate α-ketoglutarate is converted to glutamate, and ∼75% at high activity and 25% at low activity of the pyruvate dehydrogenase flux. The coupling of ME reaction flux with neuronal activity is more pronounced in neurons than in astrocytes, regardless of whether we introduce a prior biasing toward glutamine uptake. Malic enzyme activity in neurons is proportional to glutamate dehydrogenase flux and to GABA release, as confirmed by the positive correlation suggested by the scatter plots displayed in Figure 5.

Figure 4

Scatter plots of astrocytic pyruvate carboxylation versus malic enzyme toward pyruvate (top), and malate dehydrogenase (bottom), with high-activity level (left column) and low-activity level (right column). The prior positivity constraints are clearly visible in these plots. The scatter plots indicate that although the s.d. of a single flux may be large, any combination of values within the s.d. error bars is not an acceptable state, because at the steady state, the flux values depend on each other.

Figure 5

Scatter plots of the malic enzyme in neurons toward malate versus glutamate dehydrogenase (left), and GABA efflux (right) with high and low activity. It must be observed that high neuronal activity shifts the cloud of points, whereas the mutual correlation pattern remains similar at high- and low-activity levels, indicating that malic enzyme is coupled with the neuronal activity level. GABA, γ-aminobutyric acid.

Our findings support higher oxidative activity in astrocytes than in neurons, in which the average oxidative phosphorylation flux is about one-half at high and two-thirds at low activity, and unaffected by biasing of glutamine cycling, unlike that found in the study by Occhipinti et al (2009) for the astrocyte–glutamatergic neuron cellular complex. The diminished oxidative activity in GABAergic neurons compared with the findings for the glutamatergic neurons is in line with the different response of postsynaptic neurons to the release of the two neurotransmitters. Unlike glutamate, the release of GABA in the synaptic cleft inhibits the passage of action potentials, and consequently there is no additional energetic cost for the postsynaptic neurons. The transfer of ATP and ADP (adenosine diphosphate) between the cytosol and the mitochondria is significantly higher in astrocytes than in neurons both at low and high activity, with only a slight increase in astrocytes, and a slight decrease in neurons when biasing the system toward glutamine uptake. In all cases, the statistical spread is rather wide.

Conversely, the activity of MAS, which closely matches the transport of pyruvate from the cytosol to the mitochondria, is approximately a factor of 10 higher in neurons than in astrocytes. Malate-aspartate shuttle flux values decrease with increasing activity level in astrocytes, while increase in neurons. In both cells, the rate at which pyruvate enters the mitochondria is the sum of net contributions of glycolytic and LDH fluxes, the latter oriented toward pyruvate production in neurons and toward lactate production in astrocytes.

Although our experiments suggest that at steady state, GABAergic neurons do not significantly uptake glutamine released by astrocytes, this does not necessarily imply that glutamine is never transported from the ECS into GABAergic neurons, but rather pointing to either significant GABA reuptake by neurons, as suggested by several authors (Bak et al, 2006; Yang et al, 2007), or to a transient nature of this transport, the investigation of which will require a fully kinetic model. Even when biasing the system toward neuronal uptake of glutamine through a prior, the GABA–glutamine cycle does not reach a sustainable equilibrium. In fact, at high activity, even with a prior favoring neuronal glutamine uptake, the glutaminase reaction flux in neurons remains 0.009 μmol/g per min lower than the glutamine influx, pointing either to a transient accumulation of glutamine in neurons or, alternatively, a glutamine leak not explicitly included in the model. This result is in line with the previously pointed offsets from zero in the rates of change of glutamine.

Neuronal glutamine uptake does not affect glycolytic activity in either cell; it minimally changes the rate of LDH in astrocytes at low activity and, by and large, does not change any of the mitochondrial reaction rates, with the obvious exception of the glutamate dehydrogenase flux in astrocytes at high activity.

The fluxes through the mitochondrial aspartate aminotransferase and the glutamate/aspartate antiporter pairs are also unaffected by the prior on glutamine uptake, and mismatches in the remaining constituents of the MAS are equal to the discrepancy in the GABA–glutamine cycle.

The overall routing of carbon skeletons in the two cells can be summarized by following the fate of pyruvate, which under the conditions of our simulations is the principal oxidative substrate. The cytosolic pyruvate pool in GABAergic neurons, supplied mostly by glycolysis, except for a small contribution from LDH, upon entering the mitochondria becomes a substrate for the TCA cycle as either ACoA or as malate, with a contribution from pyruvate dehydrogenase about one and a half times that of ME at high activity and four times at low activity. When suppressing ME activity using a prior, the pyruvate dehydrogenase flux remains unchanged but the TCA cycle activity decreases by one-third; see Figure 2B. The synthesis and release of GABA are also reduced by one-third in the absence of ME in neurons, failing to reach the target flux expressed by the prior. The reduced rate of uptake of pyruvate by the neuronal mitochondria results in the average reversal of the preferred direction of the LDH reaction, with a small release of lactate in the ECS.

Approximately 80% at high activity, and half at low activity, of pyruvate produced by glycolysis in astrocytes is reduced to lactate and effluxed into the ECS at high activity, the remaining portion being transported into the mitochondria. Mitochondrial pyruvate turnover in astrocytes is more complex than in neurons, because of pyruvate recycling resulting from concomitant activation of PC and ME, the preferred direction of which in astrocytes is from malate to pyruvate. In our model, ME is active in both cells, with opposite preferred directions, so that although ME depletes the mitochondrial pyruvate pool in GABAergic neurons, it replenishes it in astrocytes. Pyruvate recycling maintains carbon balance: at high activity on average ME flux (0.181) plus pyruvate transport from the cytosol (0.027) matches pyruvate depletion through pyruvate dehydrogenase (0.131) and PC (0.078).

Discussion

Our analysis suggests that, under steady-state conditions, the transfer of glutamine from astrocytes to GABAergic neurons is negligible. Moreover, when biasing the steady state toward neuronal glutamine uptake by introducing a Gaussian prior with positive mean for the rate of glutamine transport, the system is unable to sustain a strict steady state, as indicated by some imbalance of the masses, indicative of accumulation, depletion, or leaks of substrates. This finding seems to suggest that glutamine transport, if it occurs, is a transient dynamic phenomenon in the astrocyte–GABAergic neuron cellular complex. A plausible explanation for the GABA synthesis would be that neurons reuptake a significant fraction of GABA, as suggested by Yang et al (2007). In the present model, GABA reuptake would decrease the net efflux of GABA, meaning that the simulation labeled in this study as low activity could in fact represent high activity with significant reuptake. The high oxidative activity seems to support this interpretation, but its further investigation requires a kinetic model. Whereas extensive studies of the glutamate–glutamine cycle (Shen et al, 1999) and measurements of glutamine flux from astrocytes to neurons in glutamatergic transmission (Sibson et al, 1997; Su et al, 1997) have been reported, in GABAergic neurotransmission, the difficulty in separating relative contributions of exogenous glutamate and glutamine to GABA synthesis has resulted in large differences in the quantification of the GABA-glutamine flux: a very small glutamine flux from astrocytes to GABAergic neurons, recently reported by Yang et al (2007), emphasized a lower contribution of glutamine to GABA synthesis than previously hypothesized, which is qualitatively in line with our findings.

The present model comprises the astrocyte and GABAergic neuron only, and the omission of the glutamatergic neuron compartment is being taken into account only by reducing the availability of substrates. However, to understand the interplay between the different cell types, it is imperative to develop a three-cell model accommodating both excitatory and inhibitory activities. Not accounting for the role of glutamatergic neurons makes it difficult to compare the results of our study with contributions which include excitatory neurons. The suggestion of our analysis that the stoichiometric equilibrium of the astrocyte–GABAergic neuron complex does not support stronger glutamine trafficking between the cells is not in conflict with findings (Sonnewald et al, 1993; Waagepetersen et al, 2001; Leke et al, 2008) that the astrocyte synthesizes significant amounts of glutamine. Glutamine synthesis under steady state may remain low simply because a flux balance analysis containing only the GABAergic neuron does not allow any other fate for glutamine, unlike in vivo and in vitro experiments, in which a strict mass balance between these two cells cannot be forced. Preliminary tests with a model of a three-cell complex comprising astrocyte, glutamatergic, and GABAergic neurons point toward more active glutamine synthesis, and more complex metabolic interactions among the three cell types.

Our model supports a stricter steady state in the absence of a prior facilitating glutamine transfer; otherwise, the offset from steady state for cytosolic mass balances of aspartate, glutamate, and glutamine in both cells when favoring glutamine uptake is of the order of the leak or depletion/accumulation predicted in the GABA–glutamine cycle. This may be an indication that the present model could be missing details regarding the mass balance of amino groups. Answering this question requires a model addressing carbon and amino groups balance and will be addressed in forthcoming works.

In addition to its metabolic coupling with the TCA cycle activity, how GABA–glutamine cycling affects glycolytic activity and the lactate trafficking between the two cells is also of interest. Our model predictions support the lack of metabolic coupling between GABA uptake by astrocytes and stimulation of astrocytic glycolytic activity, in accordance with the experimental results reported by Chatton et al (2003). Our model also suggests that the release of lactate by astrocytes and their neuronal uptake is coupled with activity level, but not with glucose consumption, regardless of the neuronal uptake of glutamine. The preferred direction of the LDH flux is, on average, skewed toward lactate production in astrocytes and pyruvate production in neurons, and proportional to GABA efflux. Suppressing ME in neuron changes, on average, the favorite direction of the LDH flux in neurons, with the result that both cell types release lactate in the ECS.

The rate at which pyruvate enters the TCA cycle in astrocytes is less pronounced at high activity, suggesting that astrocyte uses GABA as a source of energy during activity.

The present model supports a relatively high PC activity in astrocytes. To further investigate the role of PC, we performed an additional flux balance analysis including a strong Gaussian prior biasing the PC to be around a significantly lower value 0.02 μmol/g per min than what was found based on mere stoichiometry. Equilibrium with PC close to the target value, which was easily found, showed that changes to the overall flux balance were limited to a small decrease in the rate of oxidative phosphorylation in astrocytes in response to reduction in the need for ATP production, and a small increase in ME flux to keep redox balance.

Our results consistently support equal glycolytic activity in GABAergic neurons and astrocytes, and the increased demands on neurons at high activity are met by the uptake of astrocyte-released lactate, in accordance with experimental findings reported in the study by Waagepetersen et al (1998). In view of the large flux of ATP hydrolysis in astrocytes, it is natural to speculate whether glycolytic activity is indeed in response to energetic needs. A closer analysis of the partitioning of the different fluxes points to redox balance as the main reason. The high activity of the TCA cycle in astrocytes required to sustain the cycling of GABA increases the pool of reduced form of nicotinamide adenine dinucleotide (NADH), and an elevated oxidative phosphorylation activity required to maintain cytosolic redox in balance increases the ATP pool the use of which in the cytosol is not specified by the present model. The decoupling of oxidative phosphorylation and proton gradient, and the generation of heat instead of production of ATP for unspecified purposes in astrocytes is a possibility that should be investigated. Whether the level of ATP hydrolysis may decrease in a model which also comprises glutamatergic neurons will be the topic of future investigations. The fate of the significant efflux of lactate in the ECS, if not effluxed further into blood, is not explained by the present model, but can be hypothesized to supply the carbon skeleton required by nearby glutamatergic neurons. Additional testing with an extended model will be required to address this issue.

Footnotes

The authors declare no conflict of intrest.

References

Bak

Schousboe

Waagepetersen

(2006) The glutamate/GABA-glutamine cycle: aspects of transport, neurotransmitter homeostasis and ammonia transfer. J Neurochem 98:641–53

Berkich

Ola

Cole

Sweatt

Hutson

(2007) Mitochondrial transport proteins of the brain. J Neurosci Res 85:3367–77

Calvetti

Somersalo

(2006) Large scale statistical parameter estimation in complex systems with an application to metabolic model. Multiscale Model Simul 5:1333–66

Chatton

Pellerin

Magistretti

(2003) GABA uptake into astrocytes is not associated with significant metabolic cost: implications for brain imaging of inhibitory transmission. Proc Natl Acad Sci USA 100:12456–61

Dennis

Clark

(1978) The regulation of glutamate metabolism by tricarboxylic acid-cycle activity in rat brain mitochondria. Biochem J 172:155–62

Gibbs

Lennox

Nims

Gibbs

(1942) Arterial and cerebral venous blood arterial-venous differences in man. J Biol Chem 144:325–32

Gjedde

Marrett

Vafaee

(2002) Oxidative and nonoxidative metabolism of excited neurons and astrocytes. J Cereb Blood Flow Metab 22:1–14

Hassel

(2001) Carboxylation and anaplerosis in neurons and glia. Mol Neurobiol 22:21–40

Hassel

Brathe

(2000) Neuronal pyruvate carboxylation supports formation of transmitter glutamate. J Neurosci 20:1342–7

10.

Heino

Calvetti

Somersalo

(2009) Metabolica: a statistical research tool for analyzing metabolic networks. Comput Methods Programs Biomed 97:151–167

11.

Heino

Tunyan

Calvetti

Somersalo

(2007) Bayesian flux balance analysis applied to a skeletal muscle metabolic model. J Theor Biol 248:91–110

12.

Hyder

Patel

Gjedde

Rothman

Behar

Shulman

(2006) Neuronal-glial glucose oxidation and glutamatergic-GABAergic function. J Cereb Blood Flow Metab 26:865–77

13.

Leke

Bak

Schousboe

Waagepetersen

(2008) Demonstration of neuron-glia transfer of precursors for GABA biosynthesis in a co-culture system of dissociated mouse cerebral cortex. Neurochem Res 33:2629–35

14.

Kvamme

Roberg

Torgner

(2000) Phosphate-activated glutaminase and mitochondrial glutamine transport in the brain. Neurochem Res 25:1407–19

15.

Martinez-Hernandez

Bell

Norenberg

(1976) Glutamine synthetase: glial localization in brain. Science 195:1356–8

16.

Occhipinti

Puchowicz

LaManna

Somersalo

Calvetti

(2007) Statistical analysis of metabolic pathways of brain metabolism at steady state. Ann Biomed Eng 6:886–902

17.

Occhipinti

Somersalo

Calvetti

(2009) Astrocytes as the glucose shunt for glutamatergic neurons at high activity: an in silico study. J Neurophysiol 101:2528–38

18.

Patel

de Graaf

Mason

Rothman

GDL

Shulman

Behar

(2005) The contribution of GABA to glutamate/glutamine cycling and energy metabolism in the rat cortex in vivo. Proc Natl Acad Sci USA 102:5588–93

19.

Patel

Rothman

Cline

Behar

(2001) Glutamine is the major precursor for GABA synthesis in rat neocortex in vivo following acute GABA-transaminase inhibition. Brain Res 919:207–20

20.

Schellenberger

Palsson

(2009) Use of randomized sampling for analysis of metabolic networks. J Biol Chem 284:5457–61

21.

Shen

Petersen

Behar

Brown

Nixon

Mason

Petroff

OAC

Shulman

Rothman

(1999) Determination of the rate of the glutamate/glutamine cycle in the human brain by in vivo ¹³C NMR. Proc Natl Acad Sci USA 96:8235–40

22.

Shulman

Rothman

Behar

Hyder

(2004) Energetic basis of brain activity: implications for neuroimaging. Trends Neurosci 27:489–95

23.

Sibson

Dhankhar

Mason

Behar

Rothman

Shulman

(1997) In vivo ¹³C NMR measurements of cerebral glutamine synthesis as evidence for glutamate-glutamine cycling. Proc Natl Acad Sci USA 94:2699–704

24.

Sonnewald

Westergaard

Schousboe

Svendsen

Unsgård

Petersen

(1993) Direct demonstration by [13C]NMR spectroscopy that glutamine from astrocytes is a precursor for GABA synthesis in neurons. Neurochem Int 22:19–29

25.

Campbell

Oxender

(1997) Glutamine transport in cerebellar granule cells in culture. Brain Res 751:69–78

26.

Van den Berg

Garfinkel

(1971) A simulation study of brain compartments. Metabolism of glutamate and related substances in mouse brain. Biochem J 123:211–8

27.

Waagepetersen

Bakken

Larsson

Sonnewald

Schousboe

(1998) Metabolism of lactate in cultured GABAergic neurons studied by ¹³C nuclear magnetic resonance spectroscopy. J Cereb Blood Flow Metab 18:109–17

28.

Waagepetersen

Schousboe

Sonnewald

(2001) Elucidation of the quantitative significance of pyruvate carboxylation in cultured cerebellar neurons and astrocytes. J Neurosci Res 66:763–70

29.

Waagepetersen

Sonnewald

Schousboe

(2003) Compartmentation of glutamine, glutamate, and GABA metabolism in neurons and astrocytes: functional implications. Neuroscientist 5:398–403

30.

Yang

Bacher

Shen

(2007) Quantification of cortical GABA-glutamine cycling rate using in vivo magnetic resonance signal of [2-¹³C]GABA derived from glia-specific substrate [2-¹³C]acetate. Neurochem Int 50:371–8

31.

Drejer

Hertz

Schousboe

(1983) Pyruvate carboxylase activity in primary cultures of astrocytes and neurons. J Neurochem 41:1484–7

Energetics of Inhibition: Insights with a Computational Model of the Human GABAergic Neuron–Astrocyte Cellular Complex

Abstract

Keywords

Introduction

Materials and methods

Biochemical Pathways

The Computational Model

The Steady-State Model

Bayesian Flux Balance Analysis

Results

Discussion

Footnotes

References