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J Cereb Blood Flow Metab. Dec 2010; 30(12): 1982–1986.
Published online Aug 11, 2010. doi:  10.1038/jcbfm.2010.132
PMCID: PMC3002878

Comment on recent modeling studies of astrocyte–neuron metabolic interactions

Abstract

Recent years have seen a surge in mathematical modeling of the various aspects of neuron–astrocyte interactions, and the field of brain energy metabolism is no exception in that regard. Despite the advent of biophysical models in the field, the long-lasting debate on the role of lactate in brain energy metabolism is still unresolved. Quite the contrary, it has been ported to the world of differential equations. Here, we summarize the present state of this discussion from the modeler's point of view and bring some crucial points to the attention of the non-mathematically proficient reader.

Keywords: astrocytes, glucose, lactate, mathematical modeling, neuronal–glial interaction

Comment

Let us first summarize the situation. Two biophysical models have attracted some attention. The first one is the model originally introduced by Aubert et al. This model was progressively refined and now exists under different phenotypes (Aubert et al, 2007; Cloutier et al, 2009). The second model is the one originally proposed by Simpson et al (2007), which was recently used in a second installment (Mangia et al, 2009). Both are biophysical models formulated as a series of coupled differential equations and focusing on the transport and processing of metabolites through the main metabolic pathways. Both describe these metabolic pathways in a simplified manner neglecting some regulatory mechanisms and non-critical pathways. Finally, both aim at providing an integrative framework capable of explaining different in vivo experimental measurements. Yet, they differ in a key prediction; the Aubert model and all of its derivatives support the idea of an astrocyte-to-neuron transfer of lactate (ANLS), whereas the Simpson model supports the idea of a neuron-to-astrocyte transfer of lactate (NALS). Surprisingly, they agree on the idea that lactate is transferred from one compartment to the other but disagree on the direction of this transfer. It is appropriate to remind here that modeling studies based on mass-balance approaches all agree with the Aubert model about an astrocyte-to-neuron lactate shuttle (Hyder et al, 2006; Jolivet et al, 2009). More recently, DiNuzzo et al (2010) have tried to unify these two models, and we will discuss this approach later.

Beyond the biological dispute, one should speculate why two mathematical models so similar in structure differ on such an important prediction (Simpson versus Aubert). Let us now look more closely at the arguments developed around the Simpson model and in particular in Mangia et al (2009). In the latter publication, the authors show that their ‘core model' of a neuron-to-astrocyte lactate transfer correctly predicts glucose and lactate in vivo measurements. Their study shows that the ANLS model of an astrocyte-to-neuron lactate transfer is also possible if (1) the astrocytic glucose transport capacity is increased 12-fold compared with the core model and (2) that neurons do not respond to activation with increased glycolysis, and they proceed to argue that these two conditions are not supported by the current literature. It must be noted that the Aubert model also correctly predicts glucose and lactate transients while supporting an astrocyte-to-neuron lactate shuttle (Aubert et al, 2007; Cloutier et al, 2009). Let us first deal with the second argument. It was shown already in 2004 that glutamate induces inhibition of glucose transport in neurons (Porras et al, 2004). It is also known that mature neurons only regulate their glucose utilization weakly compared to young neurons (Patel and Brewer, 2003). More recently, it was shown that neurons are unable to upregulate their glycolysis without diverting glucose out of the pentose phosphate pathway, resulting in oxidative stress and apoptotic death (Herrero-Mendez et al, 2009). Therefore, contrary to what Mangia and colleagues argue, the observation that neurons do not respond to activation with increased glycolysis (or glucose transport) is strongly supported by the current literature and might be logical in the context of oxidative damage containment, even though the exact mechanisms uncoupling oxidative and glycolytic capacities remain to be elucidated. Thus, the only argument left that seemingly speaks against the ANLS model is the extent of the astrocytic glucose transport capacity.

Let us look more specifically at the simulations carried out by Simpson, Mangia, and colleagues and in particular at fluxes j3, −j5, and j6 (Figure 1A). In their framework, j3 describes the uptake of glucose by the astrocytic compartment from the basal lamina, −j5 describes the uptake of glucose by the astrocytic compartment from the interstitium, whereas j6 describes the uptake of glucose by the neuronal compartment. We are interested in measuring how much glucose is entering the astrocyte versus the neuron; hence, in the ratio

equation image

when running the original simulations by Simpson and colleagues, we find that this ratio, the astrocytic fraction of glucose capture, rglc,astro=0.132 at steady state and decreases to 0.018 during activation (Figure 1B; it varies between 0.205 and 0.220 in the more recent installment of the model). This is in stark contrast to findings showing that glucose captured by glial cells represents at least 50% of the total glucose utilization, i.e., rglc,astroobserved>0.5 (Barros et al, 2009; Nehlig et al, 2004), a proportion that increases, and not decreases, with activation (Chuquet et al, 2009; Nehlig et al, 2004; Véga et al, 2003). All modeling studies yielding predictions for this ratio agree with high values for astrocytic glucose utilization, possibly well above 50% (Hyder et al, 2006; Jolivet et al, 2009). Hence, it is no surprise that the Simpson model predicts a transfer of lactate from neurons to astrocytes because the neuronal compartment in their model captures most glucose (1−rglc,astro; hence, 78.0% to 86.8%) and because lactate is produced from glucose. However, this prediction goes against all available literature regarding the compartmentalization of glucose capture between neuronal and glial somata. It must also be noted that no glucose analog uptake was observed in proximal dendrites of Purkinje cells in Barros et al (2009), suggesting that distal dendrites were not loaded either. Conversely, the ANLS model presented in Mangia et al (2009) predicts a fraction of glucose captured by astrocytes above 70% and increasing with activation consistent with the experimental literature (Figure 1B).

Figure 1
The Simpson model. (A) Structure of the Simpson model and definition of fluxes j1 to j18 (adapted from Simpson et al (2007)). (B) The fraction of glucose being transported into astrocytes (rglc,astro) in both the original Simpson model (orange) and in ...

The first conclusion that can be drawn from these results is the following. As two models supporting two opposing hypotheses predict glucose and lactate transients equally well, these data cannot be used as a basis to determine which hypothesis prevails. Although these measurements were invaluable in showing that lactate is being used and later on replenished in the course of activation, they are not sufficient when considered alone to make an educated guess about the direction of the neuron–astrocyte metabolic coupling through lactate. In fact, it is obvious that any model with two compartments (one provider and one user) in which the provider upregulation onset lags behind the user upregulation onset can yield a curve like the lactate transient. Second, the compartmentalization of glucose capture between neurons and astrocytes seems to be the critical factor to determine the direction of the shuttle, because it is established that neurons are responsible for the bulk of oxygen utilization. In other words, the issue rests on the relative distribution and effective utilization of glycolytic and oxidative capacities among different cellular and intracellular compartments. Why the Simpson model fails to predict this compartmentalization according to the experimental literature is not entirely clear, but we will cite a number of points that should be investigated further. First, the model structure extensively relies on one single image of brain tissue, oversimplifying brain tissue three-dimensional organization and its dynamics during activation. Yet, the model is used with metabolite data collected at a macroscopic scale. Most importantly, Simpson and colleagues determined the concentrations of glucose transporters by measuring cytochalasin B binding in cultures, which is in no way a quantitative technique for the in vivo situation. Along the same lines, glucose utilization is complex and under the regulation of potentially different rate-limiting steps, e.g., at a minimum transport and phosphorylation, depending on tissue or conditions (i.e., activation versus resting state), implying that in no way can it be reduced to the sole determination of the amount of transporters (Barros et al, 2007). Fundamentally, it is not justifiable to derive quantitative in vivo properties of transport, particularly when considered as a rate-limiting step, from at best semi-quantitative data of transporter density obtained in cultures.

As mentioned earlier, DiNuzzo et al have recently proposed a unified model using the rate equations from the Aubert model for intracellular reactions and the topology and rate equations from the Simpson model for diffusion and transports. Some parameters were also slightly adapted (DiNuzzo et al, 2010). A look at their Table 1 shows that the astrocytic fraction of glucose capture is ~23% at rest and decreases during activation. Hence, even though some equations from the Aubert model were integrated into their model, it still retains flaws from the Simpson model. Only when they halve the neuronal versus astrocytic glycolytic capacity, this ratio increases to 37% at rest but it still decreases during activation.

Table 1
Summary table

Finally we would like to bring to the reader's attention two issues with the Simpson model that we uncovered while preparing this paper. First, a close look at the rates shows that a constant fraction (1/12) of the glucose entering both the neuronal and astrocytic compartments is diverted from glycolysis to nonoxidative processes. As indicated in Simpson et al (2007), one of those metabolites in the specific case of astrocytes is glycogen. This is included in the following flux equation

equation image

with AstroGlc the glucose content in astrocytes, K1=98.3667 and K2=1.0215 × 10−15 two constants, and GlycolyticRatio an arbitrary function modeling activation (in Mangia et al (2009), j8 is called AstroOther and K2=0.8874 × 10−15). First of all, glycogen does not belong to that flux at steady state. Indeed, although the brain constantly recycles glycogen, there is no permanent accumulation of glycogen over time. Thus, production and degradation rates must cancel out. However, the model also fails to account properly for glycogen degradation during activation. A look at the term ‘AstroGlc/(K2+AstroGlc)' shows that it is essentially constant during the simulation with values for AstroGlc in the range 1 × 10−14 to 2.75 × 10−14 well above the constant K2. Therefore, j8 is essentially constant and positive at rest and transiently increases proportionally with GlycolyticRatio. This is true for both Simpson and Mangia et al (2009) (Figure 1C) but neglects the degradation of glycogen during activation, the end product of which was shown to be lactate. Hence, if the model was correctly accounting for glycogen degradation during activation, j8 should decrease or at least increase only slightly. Of course, degradation of glycogen into lactate contradicts the conclusions of the Simpson model regarding a transfer of lactate from neurons to astrocytes as it would rather favor the opposite direction for such a shuttle. Simpson and colleagues do not provide the identity of other metabolites that could underlie the behavior of j8. Second, the variables of the model are not properly initialized and are still evolving toward their steady-state value when the stimulation is triggered (Figure 1D). Again, this is true for both Simpson et al (2007) and Mangia et al (2009). How far reaching the consequences of these two issues are is unclear, but we believe that these points and those aforementioned should be addressed before any conclusions can be drawn from these modeling studies because at least in the case of glycogen, the overlooked issue seems to directly contradict the main conclusion of these studies.

As we already pointed out, mathematical models are very powerful tools but they are ultimately only partial replicas of the system they model, not the system itself. As a consequence, although mathematical models are very useful as guides, it is always dangerous to draw definitive conclusions based on modeling studies only. One useful guide in choosing a mathematical model over another is to measure the range of observations successfully explained by the different models (such as the explained variance in regressions). The Simpson model successfully explains lactate and glucose transients observed during activation but fails to predict a correct compartmentalization of glucose capture, the non-upregulation of the neuronal glycolysis, and was not compared with any other experimental data set so far (Table 1). Like the Simpson model, the Aubert model successfully explains lactate and glucose transients but also successfully predicts qualitatively—and to some extent quantitatively—the compartmentalization of glucose fluxes and oxygen utilization in vivo, NADH (nicotinamide adenine dinucleotide) and oxygen transients, as well as features of the blood oxygen level-dependent signal. Thus, it seems that the Aubert model remains the best biophysical model of neuron–astrocyte metabolic interactions available.

Notes

The authors declare no conflict of interest.

References

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