Changes in glucose uptake rather than lactate shuttle take center stage in subserving neuroenergetics: evidence from mathematical modeling

Abstract

In this paper, we combined several mathematical models of cerebral metabolism and nutrient transport to investigate the energetic significance of metabolite trafficking within the brain parenchyma during a 360-secs activation. Glycolytic and oxidative cellular metabolism were homogeneously modeled between neurons and astrocytes, and the stimulation-induced neuronal versus astrocytic Na⁺ inflow was set to 3:1. These assumptions resemble physiologic conditions and are supported by current literature. Simulations showed that glucose diffusion to the interstitium through basal lamina dominates the provision of the sugar to both neurons and astrocytes, whereas astrocytic endfeet transfer less than 4% of the total glucose supplied to the tissue. Neuronal access to paracellularly diffused glucose prevails even after halving (doubling) the ratio of neuronal versus astrocytic glycolytic (oxidative) metabolism, as well as after reducing the neuronal versus astrocytic Na⁺ inflow to a nonphysiologic value of 1:1. Noticeably, displaced glucose equivalents as intercellularly shuttled lactate account for ∼6% to 7% of total brain glucose uptake, an amount comparable with the concomitant drainage of the monocarboxylate by the bloodstream. Overall, our results suggest that the control of carbon recruitment for neurons and astrocytes is exerted at the level of glucose uptake rather than that of lactate shuttle.

Introduction

Cerebral metabolism is traditionally assumed to be nearly fully aerobic at the organ level, with glucose as the sole energy substrate of the resting as well as the activated brain (Siesjo, 1978; Sokoloff et al, 1977). Although it has multiple metabolic fates, the glucose molecule is primarily catabolized in the brain to yield adenosine triphosphate (ATP), which is the obligatory substrate of most endergonic biochemical reactions. In particular, the hydrolysis of ATP by the Na⁺/K⁺-ATPase for the maintenance and restoration of ionic gradients is thought to establish the coupling between brain electrical activity and metabolism (Roland, 1993). The transient local increase in neural activity after physiologic stimulation raises specifically nonoxidative glucose metabolism, as revealed by the excess tissue uptake of glucose with respect to the oxygen required for its complete oxidation. Moreover, within the activated area, the intraparenchymal lactate production exceeds its use and disposal, as increases in lactate concentration during increased neuronal activity have been reported in several studies using magnetic resonance spectroscopy in the human cortex (reviewed by Mangia et al, 2009a). The critical observation that glutamate can induce glucose uptake and lactate release in cultured astrocytes (Pellerin and Magistretti, 1994), although not observed in many astrocyte preparations (Dienel and Cruz, 2004), provided adequate ground for the astrocyte-neuron lactate shuttle (ANLS) hypothesis, shifting neuronal energetics to rely significantly on lactate derived by glycolysis in astrocytes (Magistretti and Pellerin, 1994, 1999). Although conclusive experimental evidence in situ remains to be obtained, discussions about the plausibility of such a scenario have been afforded without consensus (Pellerin and Magistretti, 2003; Chih and Roberts, 2003).

The understanding of cerebral metabolism requires the knowledge of the underlying metabolic network organization. An apparent simplification of the adult brain metabolism derives from the relative isolation maintained by the blood–brain barrier, which singles out glucose as the exclusive source of the carbon equivalents for energy production under physiologic conditions not associated with fasting (hyperketonemia) or intense exercise (hyperlactatemia) (van Hall et al, 2009). However, the metabolic fate of glucose hinges on the substantial heterogeneity of the cerebral tissue, which is dictated by the cellular ultrastructure as well as by the enzymatic specialization of individual cell types. It should be realized that establishment of the energetic significance of intercellular metabolite trafficking could be compromised by an inaccurate description of the functional role of astrocytes during neurotransmission. Recent estimates for the amount of energy consumed by the processes that generate and transmit neural signals assign to astrocytes only a few percent of the total cerebral energy expenditure (Attwell and Laughlin, 2001; Lennie, 2003). On the contrary, both synaptic and spiking activity can activate astrocytic metabolism (Hertz et al, 2007), although the extent of their specific contribution is difficult to predict on the basis of current knowledge. In this respect, appraising the energy demand of either action potentials or synaptic potentials fails to figure out whether their proportion is comparable (Lennie, 2003) or strikingly unbalanced (Attwell and Laughlin, 2001). These calculations predict that neuronal action and synaptic potentials together consume most of the energy produced by glucose catabolism, whereas, according to these estimates, energy would be needed by astrocytes on account of the sole glutamate recycling. This would mean that neuronal versus astrocytic stimulation, as the ATP consumed for cellular ion homeostasis (Attwell and Laughlin, 2001), is in a ∼20:1 ratio. Yet astrocytes are known to be involved in other activation-related tasks, such as the clearance of extracellular potassium after neuronal action potentials (Hertz et al, 2007). Within astrocytes, the elevation of extracellular K⁺ activates both the low-affinity Na⁺/K⁺-ATPase and the Na⁺/K⁺/2Cl^– cotransporter (which further stimulates the Na⁺/K⁺-ATPase). Thus, if the astrocytic functional involvement is grounded on the sole glutamate cycling, the corresponding energy demand is significantly underestimated (Hertz et al, 2007). Unfortunately, there are conflicting results on the measurement of the Na⁺/K⁺-ATPase exercise in astrocytes compared with neurons (Roland, 1993). However, when assigning half the energy expenditure of action potentials to astrocytes, the aforementioned stimulation ratio is still greater than 3:1. In addition, recent experimental evidence has shown that the energetic cost associated with action potentials is lower than that previously anticipated (Alle et al, 2009), thus indicating that the fraction 3:1 is a low-end estimate.

The cellular metabolic and physiologic processes underlying the activation of the human brain have been recently applied to kinetic modeling, thus formalizing the central biochemical relationship between function and metabolic regulation in the study of cerebral activity (Aubert et al, 2001; Aubert and Costalat, 2002, 2005; Aubert et al (2005, 2007); Pellerin et al, 2007; Aubert and Costalat, 2007). The transport of nutrients to the brain has been further addressed by a refined theoretical account (Simpson et al, 2007; Mangia et al, 2009b), thereby endorsing the various enzyme-catalyzed reactions constituting cell metabolism with the accurate kinetic properties of glucose transporter (GLUT) proteins and monocarboxylate transporters (MCTs). When modeling cerebral metabolism at the cellular level, the distinct enzymatic activity of neurons and astrocytes has a fundamental role. Unfortunately, there is relative unavailability of quantitative information about the details of specific biochemical reactions taking place in different brain cells. As a first approximation, neuronal and astrocytic metabolic capacity and regulation are assumed to be homogeneous between the two cell types, a scenario that is partly confirmed by experimental data (Hertz et al, 2007; Lovatt et al, 2007). Given this assumption, the knowledge of the precise functioning and tissue distribution of cerebral GLUTs and MCTs is crucial in modeling the functional metabolism of neurons and astrocytes (Simpson et al, 2007). Although the reversible Michaelis–Menten kinetics used in previous models (Aubert and Costalat, 2005; Aubert et al (2005, 2007); Aubert and Costalat, 2007) may appear to be appropriate for the description of glucose and lactate transporters, it cannot account for the fact that these carriers are asymmetric and their kinetic characteristics are profoundly different depending on the direction of transport (Simpson et al, 2007). In addition, appraising the model response to changes in the proportion of energy produced by glycolytic and oxidative pathways is all important, inasmuch as this analysis can seldom be based on the sole experimental measurements. In particular, the previous models of compartmentalized cerebral metabolism (Aubert and Costalat, 2005; Aubert et al (2005, 2007); Pellerin et al, 2007; Aubert and Costalat, 2007) assumed that the oxidative capacity of astrocytes was more than four times lower than that of neurons. This assumption has recently been criticized on the basis of several experimental evidences that indicate that oxidative metabolism in astrocytes is at least comparable with their volume fraction (Hertz et al, 2007; Lovatt et al, 2007). Similarly, the sole observation of high levels in key glycolytic enzyme hexokinase in neurons (Chih and Roberts, 2003) could be insufficient to uphold higher neuronal glycolytic capacity in theoretical modeling (Simpson et al, 2007).

In this paper, we investigate the involvement of neurons and astrocytes in the coupling between cerebral electrical activity, hemodynamics, and metabolism by encompassing in a unique biochemical framework the available information about cellular ion exchanges, enzyme-catalyzed reactions, and nutrient delivery and transport systems within the brain tissue. We specifically formulated a model that unifies the models presented by Aubert and Costalat (2002, 2005) and by Simpson et al (2007) and Mangia et al (2009b). This approach integrates the features of the former theoretical accounts, which link neuronal activity to metabolic demands, with the characteristics of the latter, which take into account both the detailed structure of the blood–brain barrier and the properties and the cellular distribution of GLUTs and MCTs in the brain. The goal of this study was to test the outcomes of the simulations under various assumptions related to (1) the activation-derived sodium inflow in astrocytes compared with neurons; (2) the relative neuronal and astrocytic glycolytic and oxidative competence; and (3) the glucose transport capacity of the two cell types. Our aim was to evaluate the impact of these changes while adhering to previously validated kinetic equations and parameter estimates (Aubert and Costalat, 2005; Simpson et al, 2007). Although mathematical equations can only be considered as very simplified translations of the relevant in vivo processes, it is important to remark that the overall results of the theoretical descriptions have revealed a supportive agreement with experimental data, thus establishing a landmark in the modeling of compartmentalized cerebral metabolism (Aubert and Costalat, 2005; Simpson et al, 2007). Taking into consideration the limitations of a theoretical description, the conclusions drawn from the present modeling work are discussed with respect to the possible physiologic mechanisms that are currently believed to have a role in functional brain energy metabolism.

Materials and methods

The present model grounds on the premise that neuronal activation enhances sodium influx into cells and cerebral blood flow (Aubert et al, 2001). We adopted the approach of recasting the neuronal and astrocytic ATP-consuming processes involved in neurotransmission as the gross Na⁺ entry into cells (Attwell and Laughlin, 2001; Aubert and Costalat, 2005). We further assumed that the resting ATP expenditure is determined by basal metabolic activity (i.e., housekeeping) plus the action of Na⁺/K⁺-ATPase after sodium leakage, both in neurons and astrocytes (Aubert et al, 2001). Sodium influx resulting from brain stimulation is described by a time-dependent activation function that includes an alpha-type pulse with a time constant of 2 secs plus a term remaining fixed throughout the entire stimulation interval (Aubert and Costalat, 2002), coherent with the in vivo intracortical electrophysiologic recordings of axonal, synaptic, and dendritic activity showing the adaptation of the aggregate tissue electrical response (Logothetis et al, 2001). The activation-induced increase in intracellular Na⁺ concentration stimulates the Na⁺/K⁺-ATPase, which in turn causes a decrease in the ATP/ADP ratio (Aubert et al, 2001; Aubert and Costalat, 2002, 2005). To maintain this ratio high enough to efficiently hydrolyze ATP, the energy metabolism is increased by glycolysis (Heinrich and Schuster, 1996), oxidative phosphorylation (Gjedde, 1997; Aubert and Costalat, 2002), and buffering of ATP through creatine kinase (Aubert et al, 2001) and adenylate kinase (Rapoport et al, 1976), resulting in a decrease of tissue glucose and phosphocreatine and an accompanying increment of lactate (Aubert and Costalat, 2002, 2005). Blood flow mediates the increase in nutrient transport from the blood to the brain (Buxton et al, 1998), whereby glucose and lactate cross the blood–brain barrier through the endothelium and basal lamina, and exchange among cellular and extracellular compartments by means of facilitative transport or diffusion (Simpson et al, 2007), whereas oxygen simply diffuses from the capillary to the cells (Vafaee and Gjedde, 2000).

On the basis of the theoretical accounts summarized above, we performed kinetic modeling simulations by formulating for each metabolite an ordinary differential equation describing the relevant enzyme-catalyzed reaction/transport process with the corresponding parameter values. We adapted the different mathematical descriptions by constraining model parameters to both the known steady-state metabolite concentrations and the experimental activation-derived concentration transients. Changes in cell stimulation and metabolic/transport capacity were further limited by known metabolic rates of oxygen and glucose for the human brain (Mangia et al, 2009a, 2009b). As a result, only minimal quantitative adjustments were necessary to obtain stationary and dynamic results consistent with previous modeling analyses (Aubert and Costalat, 2005; Simpson et al, 2007). In particular, regulatory kinetic parameters were not altered, whereas adjustments to reaction/transport rates were in the range of 1% to 5% (with the exception of paracellular diffusion, see ‘Results' section), which is well within the actual uncertainty for these values. We conformed to earlier works (Aubert et al, 2001; Aubert and Costalat, 2002, 2005) by adopting the convention that cell outward fluxes are positive and cell inward fluxes are negative, whereas transport rates were expressed per unit of destination compartment volume. Furthermore, throughout the text, we expressed simulated lactate flow rates as fluxes of glucose equivalents. The system of coupled differential equations, which we delineate in the Appendix, was numerically solved using algorithms for stiff problems based on the Rosenbrock formula implemented in MATLAB (version 7.0.4 R14) software (The Mathworks Inc., Natick, MA, USA; http://www.mathworks.com/).

Results

To examine the impact of parameter estimates on resting steady state, we performed local sensitivity analysis of glucose and lactate flow rates as the time derivative of fluxes with respect to each model parameter. Glucose diffusion rate between the basal lamina and interstitium was found to have the largest effect on intercellular metabolite trafficking. Accordingly, steady state of the system—that is, condition consistent with known brain metabolite concentrations—was obtained with the only significant revision to previous models of nutrient transport (Simpson et al, 2007; Mangia et al, 2009a) covering the glucose paracellular diffusion coefficient, which must be decreased 3-fold, from 0.44 μm²/s (Simpson et al, 2007) to approximately 0.14 μm²/s, to avoid metabolic intermediates from piling up within the cerebral tissue. This suggests that a measure of the interstitium tortuosity may underestimate the actual hindrance imposed by diffusion barriers created by the extracellular matrix at the level of the basement membrane (Simpson et al, 2007), which must be traveled to a great extent until the small gaps between the adjacent astrocytic endfeet can be surpassed. Nonetheless, the concerted model supports the conclusion that diffusion is a key component of nutrient delivery to the brain under resting conditions (Simpson et al, 2007). Indeed, basal levels of glucose and lactate are dramatically decreased when the passage of glucose is constrained only through the astrocytic endfoot, thus invalidating the otherwise good correspondence with experimental data of total brain glucose versus serum glucose (Simpson et al, 2007). It is noted that lactate is a smaller molecule with a larger mobility than glucose and that its diffusion is correspondingly greater (Pfeuffer et al, 2000).

To allow model comparison, we have chosen a 360-secs stimulation period to simulate sustained activation (see Aubert and Costalat, 2005). According to the duration of the stimulus, we distinguish a biphasic metabolic response to activation characterized by an early phase followed by a late phase, a result that is in agreement with previous theoretical accounts (Aubert and Costalat, 2002, 2005). The transition of brain metabolism toward delayed behavior reflects the adaptation of neural electrical activity (e.g., neuronal habituation) (Logothetis et al, 2001; Aubert and Costalat, 2005) as well as the reduction of intracellular ATP buffering capacity because of phosphocreatine depletion (Aubert et al, 2001; Aubert and Costalat, 2002), and it is eventually absent for short stimulation periods (Simpson et al, 2007). In particular, energy recovery through kinase-dependent rephosphorylation of adenosine phosphates is preferred over pyruvate oxidation during the early phase of stimulation, whereas recoupling of glucose uptake and oxygen use is found to take place during the late reduction in the activity of the phosphocreatine buffer system. Noticeably, alteration of tissue phosphocreatine concentration at the stimulus onset affects both the magnitude and the peak delay of the metabolic response (data not shown). This indicates that the phasic nature of the metabolic response depends on local energy availability rather than on the stimulation paradigm.

The simulated time courses of metabolite levels during neuronal activation obtained in our simulations were found to be in good agreement with previous findings. In particular, concentration transients of glyceraldehyde 3-phosphate, phosphoenolpyruvate, intracellular oxygen, nicotinamide adenine dinucleotide, phosphocreatine, and ATP are consistent with theoretical outcomes of previous works (for references about the experimental support, see Aubert and Costalat, 2002, 2005). Furthermore, we reproduced experimental data for the uncoupling between the cerebral metabolic rate of oxygen (CMR_O2) and glucose (CMR_Glc), as well as tissue (cellular and extracellular) glucose and lactate concentration transients (Aubert and Costalat, 2005). These findings validate the present concerted model and are not further discussed here. In the following, we focus on the different glucose and lactate trafficking patterns that are compatible with underlying similar changes in tissue metabolite concentrations (Aubert and Costalat, 2005; Mangia et al, 2009b).

In the first simulated scenario, glycolytic and oxidative cellular metabolism is homogeneously modeled between neurons and astrocytes, whereas we considered a significant astrocytic activation by adopting a 3:1 neuronal versus astrocytic stimulation ratio (Figure 1). Simulations show that both neurons and astrocytes take up most of the glucose from the interstitium, which absorbs from the basal lamina more than 25 times the glucose picked up by astrocytic endfeet (Figures 1A and 1B). This suggests that astrocytic endfeet do constitute an impediment rather than a gateway structure for glucose supply to the brain cells, which is consistent with the fact that these processes are not part of the blood–brain barrier (Gjedde and Marrett, 2001). As glucose uptake into astrocytes from the basal lamina is relatively slow, diffusion (Simpson et al, 2007) and not saturable kinetics (Aubert and Costalat, 2005) is required to account for paracellular delivery of glucose to the brain. Our results show that glucose flux is not directed specifically to astrocytes either at rest or during augmented energy demand. Indeed, on activation, the increase of astrocytic glucose uptake from the basal lamina is negligible, and glucose entry into these cells from the interstitium is significantly reduced (Figure 1B). Concomitant glucose consumption in neurons drives the rise in diffusion of the sugar into extracellular space and its subsequent rapid uptake by high capacity neuronal GLUTs (Figure 1A), consistent with previous theoretical outcomes (Simpson et al, 2007). Interestingly, under these conditions, tissue lactate flow transients are entirely due to neurons, which exploit the ability to import extracellular glucose to increase their glycolytic metabolism thereupon exporting lactate to the interstitium (Figure 1C), from where it diffuses out of the brain (Figure 1C) or is taken up by astrocytes (Figure 1D).

Figure 1.

Simulated glucose and lactate carbon equivalent fluxes in response to sustained activation. (A) Glucose flow rates between the neuron or basal lamina and interstitium. (B) Glucose flow rates between the astrocyte and interstitium or basal lamina. Glucose taken up by the brain surpasses the endothelium and reaches the basal lamina, from where it goes either to the interstitium or to the astrocytic endfeet. Interstitium takes up most of the glucose at rest, namely 5.48 μmol/L per s compared with 0.21 μmol/L per s transported from the basal lamina directly into the astrocytes. These values correspond to a basal CMR_Glc of 0.34 μmol/g per min. During stimulation (time interval 0 to 360 secs), routing of the sugar is identified by the increased neuronal and decreased astrocytic glucose uptake from the interstitium. Glucose transport rate to the neurons increases from 4.36 μmol/L per s to more than 7 μmol/L per s in the early phase of stimulation and then settles to 6 μmol/L per s for the rest of the stimulation period. At the same time, glucose transport rate to the astrocytes decreases from 1.12 to 0.75 μmol/L per s on average. In terms of cerebral metabolic rate of glucose, this translates to a mean value during stimulation of 0.41 μmol/g per min (+20%). Note that in spite of the relatively large maximal velocity of glucose transporters (e.g., 0.58 mmol/L per s for neurons), the actual transport rate remains low (in the order of μmol/L per s) because of the high Michaelis constant (K_m) for glucose and the rather small difference in its concentration within compartments. (C) Lactate flow rates (as glucose equivalents) between the neuron or basal lamina and interstitium. (D) Lactate flow rates (as glucose equivalents) between the astrocytes and interstitium or basal lamina. At rest, both neurons and astrocytes release lactate to the interstitium at a rate of 0.21 and 0.02 μmol/L per s, respectively. Excess extracellular lactate resulting from these fluxes is transported out of the brain. Astrocytes also release lactate directly to the basal lamina through the astrocytic endfeet (0.05 μmol/L per s). On stimulation, neuronal lactate release to the interstitium increases to 0.86 μmol/L per s on average and extracellular lactate is taken up by—that is, shuttled to—the astrocytes at a mean rate of 0.40 μmol/L per s. Activation-induced accumulation of tissue lactate results in the rise of clearance of the monocarboxylate to a mean rate of 0.39 μmol/L per s. Note that the values of lactate flow rates have to be doubled to obtain uncorrected fluxes of the monocarboxylate (see Table 1).

To examine the role of cell metabolism on the modulation of lactate exchanges between tissue compartments, we modified the glycolytic and oxidative metabolic capacities of neurons and astrocytes. We adopted conditions favoring the occurrence of ANLS by considering increased glycolysis and decreased oxidative metabolism in astrocytes compared with neurons. Simulations show that halving the ratio of neuronal versus astrocytic glycolytic metabolism induces small changes in glucose routing to cells (Figure 2A), introducing a small astrocyte to neuron lactate shuttle at rest and eventually promoting it during the late phase of stimulation (Figure 2B). Similarly, doubling the neuronal versus astrocytic oxidative metabolism ratio only moderately affects the overall cellular glucose uptake profiles (Figure 2C), yielding a moderate resting lactate flow from astrocytes to neurons and a slightly more pronounced ANLS in the late stimulation period (Figure 2D). These results indicate that the presence of a resting lactate flux from astrocytes to neurons is not per se supportive or contrary to a functional role of ANLS in neurometabolic coupling (Pellerin et al, 2007).

Figure 2.

Simulated metabolite carbon equivalents trafficking after alteration of cell metabolic competence or stimulation. (A, B) Effect of halved neuronal versus astrocytic glycolytic capacity on the glucose and lactate flow rates between the neuron or basal lamina and interstitium and between the astrocytes and interstitium or basal lamina. Glycolysis (maximum velocity of glycolytic chain enzymes) of the neurons and astrocytes is changed accordingly. For example, the reaction rate of the neuronal hexokinase-phosphofructokinase (HK-PFK) system (ν_HK-PFK) is reduced from 0.11 to 0.09/s, whereas astrocytic HK-PFK velocity is increased from 0.06 to 0.10/s (values expressed per unit of intracellular volume). In these conditions, the difference in the basal glucose uptake between the neurons and astrocytes becomes smaller (from 3.24 to 1.68 μmol/L per s). As a result, the neurons take up astrocyte-released lactate at a rate of 0.66 μmol/L per s (resting astrocyte-neuron lactate shuttle, ANLS). However, both the glucose and lactate dynamics during activation is only moderately affected, and lactate flows from the neurons to the astrocytes. (C, D) Effect of doubled neuronal versus astrocytic oxidative capacity on glucose and lactate flow rates between the neuron or basal lamina and interstitium and between the astrocytes and interstitium or basal lamina. Glucose routing is almost unaffected when oxidative metabolism (maximum rate of mitochondrial respiration) (V_max,Mito) of neurons is increased from 18.5 to 21 μmol/L per s and that of astrocytes is concurrently reduced from 10.2 to 5.8 μmol/L per s. However, this affects lactate exchanges both at rest (0.29 μmol/L per s as resting ANLS) and during the late stimulation period, where neurons import lactate derived from astrocytes, although there is no net neuronal lactate uptake on average. (E, F) Effect of halved neuronal versus astrocytic stimulation on glucose and lactate flow rates between the neuron or basal lamina and interstitium and between the astrocytes and interstitium or basal lamina. The amount of Na⁺ influx attributed to stimulation is reduced from 190 to 175 mmol/L in neurons (v₁ⁿ=0.4, v₂ⁿ=16.4), and increased from 65 to 115 mmol/L in astrocytes (v₁^a=0.2, v₂^a=21.7). The load of intracellular sodium results in a different activation of the neuronal and astrocytic Na⁺/K⁺-ATPase, which produces changes in cell glucose uptake and lactate exchange. In particular, activation-induced changes in channeling of glucose to neurons is more restricted to the early phase after stimulation and has a reduced magnitude (from +1.69 to +1.03 μmol/L per s with respect to resting conditions), whereas astrocytic glucose uptake increases from 1.12 to 1.35 μmol/L per s). In these conditions, neurons do not import lactate from the interstitium either at rest or during stimulation. It is noted that tuned changes of metabolic capacity and stimulation parameters (in the range of those used here) can result in minor lactate exchanges both at rest and during activation.

Following the significant astrocytic need for energy, the reduction in glucose uptake by astrocytes (Figures 1B, 2A and 2C) is accounted for by the sharp increase in lactate flow into these cells (Figures 1D, 2B and 2D). To address whether the degree of activation predisposes glucose access into neurons, we halved the neuronal versus astrocytic stimulation, further imposing an early activation level greater in astrocytes (Figures 2E and 2F). This produces a qualitatively similar time course of the neuronal and astrocytic glucose uptake. However, the magnitude of the activation-induced glucose intake by neurons is reduced, whereas the glucose taken up by astrocytes increases during the late phase of activation (Figure 2A). However, in the early phase of stimulation, it is still the energy demand of neurons to push the channeling of glucose in these cells, without a substantial change for its transport into astrocytes (Figure 2E). Furthermore, the initial lactate flux again shows the net neuronal release and astrocytic absorption (Figure 2F), which ceases only during the late phase of the stimulation. However, on average, there is still no net astrocyte to neuron lactate shuttle until the neuronal versus astrocytic activation-induced Na⁺ influx ratio is further reduced to 1:1 (see Table 1). These conditions of high astrocytic activation result in a seemingly unphysiologic glucose depletion in these cells (data not shown), suggesting that astrocytic glucose transport capacity, as presently determined (Simpson et al, 2007), is insufficient to meet an elevated stimulation of astrocytes, compatible with previous modeling results (Mangia et al, 2009b). The inadequate glucose transport capacity of astrocytes under this specific condition may form a biochemical/transport basis for the requirement that glycogen is located in astrocytes, not neurons. It should be realized, however, that the relevant in vivo kinetic properties of GLUT proteins expressed by neurons and astrocytes may be different from those used in this model, which are partly derived by cell cultures (Simpson et al, 2007). Interestingly, simulated glucose and lactate flows survived almost unchanged even after a more than 10-fold reduction of the neuronal versus astrocytic glucose transport capacity (see Table 1), reflecting the main role of intracellular ATP homeostasis in controlling the intercellular exchange of energy substrates. Analysis of enzymatic activity indicates that the apparent insensitivity of glucose routing on maximum transport velocity is a primary consequence of the substantially different affinity between GLUTs (K_m∼10 mmol/L) and hexokinase (K_m=0.05 mmol/L) for glucose. Specifically, hexokinase remains constantly saturated by its substrate under simulated conditions. The fact that glucose phosphorylation, not transport, is the rate-limiting step in the use of the sugar, together with the rapid equilibration of glucose (and lactate) among the different compartments of the brain, suggests that any supposed regulation of metabolite trafficking cannot be accomplished directly at the supply level. This is in agreement with the observation that, given the abundance of GLUTs and MCTs in both neurons and astrocytes and their passive nature (hence the similar neuronal and astrocytic glucose and lactate concentrations), selectivity in a compartmentalized metabolism of these substrates remains difficult to realize (Gjedde, 2001) unless individual cellular demand is exceptionally substrate-specific, a condition that still necessitates a convincing experimental basis (Ames, 2000).

Table 1. Simulated mean glucose and lactate flow ratesa during rest and activation.

Simulation	Neuron interstitiumb		Astrocyte interstitiumb		Interstitium-basal laminab		Astrocyte-basal laminab
	Basal	Stimulated	Basal	Stimulated	Basal	Stimulated	Basal	Stimulated
Glucose
1	−4.36	−6.05	−1.12	−0.75	−5.48	−6.46	−0.21	−0.22
2	−3.54	−5.51	−1.86	−1.39	−5.39	−6.50	−0.25	−0.26
3	−4.31	−5.89	−1.10	−0.93	−5.41	−6.44	−0.22	−0.25
4	−4.36	−5.39	−1.12	−1.35	−5.48	−6.37	−0.21	−0.26
5	−4.26	−5.88	−1.18	−0.77	−5.44	−6.29	−0.15	−0.18
6	−4.36	−4.59	−1.12	−1.96	−5.48	−6.21	−0.21	−0.29
Lactate
1	0.21	0.86	0.02	−0.40	0.23	0.35	0.05	0.04
2	−0.66	0.29	0.82	0.20	0.16	0.33	0.08	0.07
3	−0.29	0.06	0.51	0.41	0.22	0.33	0.07	0.08
4	0.21	0.29	0.02	0.16	0.23	0.32	0.05	0.07
5	0.17	0.77	0.05	−0.34	0.22	0.33	0.05	0.04
6	0.21	−0.48	0.02	0.88	0.23	0.29	0.05	0.10

^aGlucose and lactate flow rates are expressed as glucose equivalents (as in Figures 1 and 2). To obtain uncorrected lactate flow rates the relevant values have to be multiplied by a factor of 2.

^bThe source compartment is indicated on the left within each pair. Positive (negative) values of glucose/lactate flow mean release (uptake) with respect to the source compartment.

Simulation numbers indicate the following conditions: (1) core model (3:1 neuronal versus astrocytic stimulation ratio and homogeneous metabolic competence); (2) halved neuronal versus astrocytic glycolytic capacity ratio; (3) doubled neuronal versus astrocytic oxidative capacity ratio; (4) halved neuronal versus astrocytic stimulation ratio; (5) reduced neuronal versus astrocytic glucose transport capacity ratio (T_max,GLCⁱⁿ from 0.58 to 0.22 mmol/L per s, T_max,GLC^ai from 0.14 to 0.65 mmol/L per s); (6) 1:1 neuronal versus astrocytic stimulation-induced Na⁺ influx ratio (ν₁ⁿ=0.36, ν₂ⁿ=15.7, ν₁^a=0.28, ν₂^a=23.7).

Mean values have been obtained for a 360-secs stimulation period.

Note that flow rates under stimulated condition do not sum to zero because of the changes in intracellular glucose (0.3 to 0.4 mmol/L mean tissue decrease) and lactate (0.2 to 0.3 mmol/L mean tissue increase) concentration in both neurons and astrocytes.

All fluxes are expressed in units μmol/L per s.

We next calculated the basal and stimulated glucose and lactate fluxes resulting from all sets of assumptions (Table 1). It is noted that under all the simulated conditions, the magnitude of the lactate shuttle is much smaller than the concomitant neuronal glucose use. In particular, when metabolism is homogeneous between neurons and astrocytes, neither of the two cell types take up lactate under resting conditions. Even after a significant unbalancing of cellular metabolism, maximum basal carbon equivalents attributed to lactate trafficking represent only approximately 15% of the accompanying glucose taken up by neurons. Moreover, activation substantially alters neuronal glucose flux regardless of any lactate dynamics, whereas late (but not early) astrocytic glucose uptake follows intercellular lactate shuttle to a greater extent. Importantly, neither the metabolic (glycolytic and oxidative) competence nor the degree of stimulation prevented glucose from being diverted to neurons in the early phase of activation, even after significant changes in cell glucose transport capacity. Noticeably, the amount of activation-induced lactate shuttle with respect to the concurrent glucose flow is comparably much smaller in terms of glucose equivalents. Under core model conditions, the fraction of lactate release with respect to glucose uptake at rest is 5% for both neurons and astrocytes; during activation, lactate released by neurons and taken up by astrocytes accounts for 15% and 35%, respectively, of neuronal and astrocytic glucose uptake. Simulations showed that for the 360-secs stimulation period, the proportion of the mean shuttled lactate uptake either by astrocytes (conditions 1 and 5 in Table 1) or by neurons (condition 6 in Table 1) corresponds to approximately 6% to 7% of the total brain glucose uptake during the same interval, a quantity comparable with the transfer of the monocarboxylate to the basal lamina (Table 1). The lactate/glucose carbon flux ratio is further reduced after relatively small departures from the physiologic values of cell metabolism and amount of stimulation (conditions 2 to 4 in Table 1), which may prevent any important resting as well as activation-induced lactate shuttle between neurons and astrocytes (see also Figure 2). Instead, lactate flux through the basal lamina, which largely reflects lactate washout by the bloodstream (data not shown), stays almost independent of metabolic and stimulation charge of the two cell types (Table 1). The simulated efflux of lactate out of the brain and the concomitant increase in the brain lactate level are each found to account for approximately 0.3 mmol/L of the pyruvate not oxidized during the 360 secs stimulation, thus contributing equally to the shift toward glycolysis over oxidation. It should be realized, however, that the net accumulation and disposal of lactate represent only a relatively minor fraction of the pyruvate generated by the flux from glucose, supporting the notion that the majority of the ATP is generated by oxidative metabolism, at rest as well as during activation (Mangia et al, 2009a).

Discussion

In this modeling work, we examined the effects of several determinants for energy use and metabolite trafficking during normal and enhanced cerebral activity, namely cellular transport and metabolic capacity and specialization, and energy-consuming processes related to activation. We theoretically examined the significance of these parameters on the metabolism of the resting and stimulated brain. Remarkably, the model supports the central dogma of cerebral energy metabolism, endorsing the view that glucose is the major neuronal energetic fuel (Bak et al, 2009).

Simulations show that glucose is primarily taken up by neurons through the extracellular space, at rest as well as during activation (Table 1), thereby supporting the importance of glucose paracellular diffusion (Simpson et al, 2007). Moreover, net resting gradient-driven cellular lactate uptake, both neuronal and astrocytic, can occur and is governed solely by intracellular metabolism. Under stimulated conditions, the kinetic properties of GLUTs and MCTs favor glucose routing to neurons (Figures 1A, 1B, 2A, 2C, and 2E) as well as lactate release by these cells (Figures 1C, 2B, 2D, and 2E) (Hertz and Dienel, 2005; Simpson et al, 2007). Theoretical results show that, at least during the early phase of activation, lactate is primarily oxidized in astrocytes (Figures 1D, 2B, 2D and 2E), consistent with the fact that working astrocytes oxidize lactate and other fuel when extracellular potassium levels increase after neuronal action potentials (Hertz et al, 2007). Accordingly, simulated glucose uptake by astrocytes decreases on activation, whereas astrocytic energy production relies on neuron-derived lactate, upholding the available experimental evidences of high oxidative metabolic rates in astrocytes and the increase in astrocytic tricarboxylic acid cycle induced by stimulation (Cruz et al, 2005). This model is compatible with a dependence of lactate transient evolution on neuronal energy demand during activation, and with an astrocytic lactate uptake under such standard circumstances (Simpson et al, 2007).

Alterations of glycolysis and oxidative metabolism influence both the magnitude and direction of the resting lactate shuttle, as well as the late and poststimulus periods, whereas the lactate flux during the early phase of stimulation (Figures 2B and 2D) and the cellular glucose uptake (Figures 2A and 2C) are only minimally affected. A significant increase in the stimulation level of astrocytes with respect to neurons does not prevent neuronal access to paracellularly diffused glucose (Figure 2E), whereas glucose uptake by astrocytes becomes rate-limiting for their glycolytic metabolism. As neuronal glucose uptake decreases toward basal level in the late phase of the stimulation, the intercellular transfer of lactate terminates (Figure 2F). Interestingly, physiologic changes of both the neuronal versus astrocytic stimulation ratio and cell metabolic capacity are compatible with a negligible activation-induced lactate shuttle (see Figure 2 and Table 1). Overall, these results indicate that lactate trafficking attributed to an increase in energy demand balances the availability of carbon skeletons between neurons and astrocytes as an effect of how efficiently glucose can be secured by each individual cell type.

The outcomes of the model are in agreement with the commonly accepted idea that oxygen and glucose enter the cells through the blood, which on the contrary allows drainage of excess lactate from the brain (Gjedde, 1997). Interestingly, although the velocity of lactate clearance by the bloodstream was found to be relatively low, washout of the monocarboxylate remained elevated for the entire stimulation and the poststimulus period, upholding the notion that the brain disperses accumulated lactate (either released from the neurons or astrocytes), rather than exclusively shuttling lactate as an intracerebral energy substrate between different cell types. Mechanisms for regional lactate dispersal may include diffusion to the blood as well as long-range gap junctional transfer of the monocarboxylate through astrocytic syncytium (Dienel and Cruz, 2008).

Under core model conditions, we obtained a lactate transfer from neurons to astrocytes. Extensive evaluation of the consequences of alterations in the properties of nutrient transporter proteins supported this view (Simpson et al, 2007). Only when the stimulation-induced cellular Na⁺ influx is comparable in neurons and astrocytes is a small astrocyte to neuron lactate shuttle obtained (Table 1), whereas attaining a significant ANLS throughout the stimulation requires, in addition to a substantial astrocytic activation, a very low oxidative metabolism in astrocytes and much higher astrocytic glucose transport capacity (Aubert and Costalat, 2005), all assumptions that are not supported by the available experimental evidence (Hertz et al, 2007; Lovatt et al, 2007; Simpson et al, 2007; Mangia et al, 2009b and references therein).

Taken together, these simulations reveal that the cellular metabolic resources and the degree of activation modulate the resting and the stimulus-induced lactate shuttle. In particular, the neuronal versus astrocytic stimulation ratio has the largest modulatory effect on cell glucose uptake and early lactate shuttle, whereas intracellular metabolism has a greater influence on lactate shuttle at rest and during the late stimulation period. More importantly, the glucose transport capacity of neurons and astrocytes is found to be broadly subordinate to cellular energetics—that is, the amount of energy required and the ability to produce it. Unfortunately, accurate estimates of the functional contribution of each cell type to energy consumption represent a presently unknown factor. Brain glucose and lactate kinetics reflect the balance between supply and demand, which ultimately depends on the interplay of energy needs as well as on transport and metabolic competence. On the basis of the results of this study, none of these facets as such pose a substantial constraint on the physiologic response of the brain to activation. Although the model results suggest that glucose availability to neurons is a fundamental requirement, it is evident from simulations that intercellular lactate exchanges may be common appanages of the normal brain functioning. Nevertheless, displaced lactate accounts for less carbon equivalents with respect to glucose use, probably reflecting distinct roles of glucose and lactate trafficking in managing the cellular accessibility to energy substrates. The functional significance of the intercellular lactate shuttling is further reduced by the increase in stimulation-induced lactate spillover to the blood resulting from our simulations. This is compatible with the experimental findings showing during activation a disproportionate increase in glucose compared with oxygen consumption that is not explained by the quantity of lactate accumulation in the brain (Madsen et al, 1999) or total carbohydrate use (Quistorff et al, 2008). Our results are also consistent with the lack of retention of [6-¹⁴C]-glucose metabolites in activated tissue, which is inferred from a number of autoradiographic studies that assayed glucose use with [¹⁴C]-glucose and [¹⁴C]-deoxyglucose (Cruz et al, 2005, and references therein). The large underestimation of metabolic activation by [¹⁴C]-glucose observed in these experiments is compatible with the substantial release of newly synthesized labeled lactate, which is partly accounted for by the present simulation results.

On the basis of this work, we propose that small deviations in the regional transport and metabolic capacity as well as in degree of local stimulation may represent intrinsic factors in the physiology of the brain, which capitalize the substrate availability underlying different metabolite relocations. Besides the large variability within the available experimental data (Mangia et al, 2009a), the regional heterogeneity of the brain in terms of cellular metabolism and activation-induced energetic stress is also evidenced by divergent results after neuronal MCTs inhibition (see Erlichman et al, 2008, and references therein). However, the modulation of substrate availability behind cellular energy demand applies crucially to glucose uptake, though much less to lactate shuttle. Our results show that the early metabolic response to physiologic stimulation coincides with interstitial glucose routing rather than astrocytic lactate shuttling to activated neurons, suggesting that the intercellular lactate flow is an epiphenomenon reflecting the control of glucose channeling to cells. The possible absence of considerable cellular lactate intake during large remodulations of glucose uptake to neurons and astrocytes advances the possibility that the metabolic cooperation between these two cell types is actualized before glucose phosphorylation. Similarly, the existence of physiologic stimulations resulting in small or no local alteration of resting lactate shuttle may introduce a challenge to the attention reserved to this compound as an obligatory cell-to-cell energy substrate, which is consistent with a relatively high individual self-sufficiency in both the glycolysis and oxidative metabolism of neurons and astrocytes (Lovatt et al, 2007). Recent proposals have been put forward about the potential physiologic roles of lactate (Cerdan et al, 2006; Schurr, 2006), as well as about the emerging concern of lactate as a signaling molecule capable of influencing neuronal activity (see Iadecola, 2007). Furthermore, it should be realized that the consequences of ultrastructural characteristics pertaining to subcellular domains (Hertz et al, 2007), or the physiologic shift in the nature of the energy demand toward processes primarily served by glycolysis (Ames, 2000), cannot be ruled out. All these considerations emphasize the inherent limitations of the present theoretical account and further complicate the picture, though they lie in the realm of hypotheses that shape the ongoing detailing of the brain cellular physiology.

In conclusion, the outcomes of this metabolic model support changes in glucose uptake, not lactate shuttle, to take center stage in the usage of energy substrates by neurons and astrocytes. Under the assumptions supported by current literature, the model predicts a lactate shuttle from neurons to astrocytes, which nevertheless is secondary to direct neuronal glucose uptake.

Appendix

Table A1

Table A1. Balance equations and simulated steady-state concentrations for brain metabolites.

Table A2

Table A2. Rate equations for reaction/transport processes.

Table A3

Table A3. Parameter values used in model transport/reaction rate equations.

Definition		Value	Definition		Value
57. Neuronal membrane area divided by intracellular volume	Snm/Vn	90000/cm	105. Affinity constant of endothelial GLUTs for glucose	KeG	10 mmol/L
58. Astrocytic membrane area divided by intracellular volume	Sam/Va	35000/cm	106. Zero-trans entry resistance of endothelial GLUTs for glucose	ReG,oi	1.0
59. Membrane sodium conductance	gNa	0.0039 ms/cm2	107. Zero-trans exit resistance of endothelial GLUTs for glucose	ReG,io	1.0
60. Resting neuronal membrane potential	Vmn	−70 mV	108. Equilibrium exchange resistance of endothelial GLUTs for glucose	ReG,ee	1.0
61. Resting astrocytic membrane potential	Vma	−80 mV	109. Affinity constant of astrocytic GLUTs for glucose	KaG	10 mmol/L
62. Kinetic constant of Na/K ATPase	kpump	0.29 × 10−6 cm L/mmol per s	110. Zero-trans entry resistance of astrocytic endfeet GLUTs for glucose	RbaG,oi	1.0
63. Michaelis constant of Na/K ATPase for ATP	Km,Pump	0.5 mmol/L	111. Zero-trans exit resistance of astrocytic endfeet GLUTs for glucose	RbaG,io	0.73
64. Stimulation-induced neuronal sodium influx rate (pulse term)	ν1n	0.41 mmol/L per s	112. Equilibrium exchange resistance of astrocytic endfeet GLUTs for glucose	RbaG,ee	0.73
65. Stimulation-induced neuronal sodium influx rate (constant term)	ν2n	18 mmol/L per s	113. Zero-trans entry resistance of astrocytic GLUTs for glucose	RaiG,oi	1.0
66. Stimulation-induced astrocytic sodium influx rate (pulse term)	ν1a	0.12 mmol/L per s	114. Zero-trans exit resistance of astrocytic GLUTs for glucose	RaiG,io	1.36
67. Stimulation-induced astrocytic sodium influx rate (constant term)	ν2a	11.7 mmol/L per s	115. Equilibrium exchange resistance of astrocytic GLUTs for glucose	RaiG,ee	1.36
68. Characteristic time of cellular sodium influx	τStimx	2 s	116. Affinity constant of neuronal GLUTs for glucose	KnG	4 mmol/L
69. First-order kinetic constant of neuronal hexokinase-phosphofructokinase	kHKPFKn	0.11/s	117. Zero-trans entry resistance of neuronal GLUTs for glucose	RnG,oi	1.0
70. First-order kinetic constant of astrocytic hexokinase-phosphofructokinase	kHKPFKa	0.06/s	118. Zero-trans exit resistance of neuronal GLUTs for glucose	RnG,io	0.72
71. Inhibition constant of the substrate inhibition of HKPFK by ATP	KI,ATP	1 mmol/L	119. Equilibrium exchange resistance of neuronal GLUTs for glucose	RnG,ee	0.72
72. Cooperativity coefficient of the substrate inhibition of HKPFK by ATP	nH	4	120. Affinity constant of neuronal MCTs for lactate	KnL	2 × 10−5 mmol/L
73. Michaelis constant of hexokinase for glucose	Kg	0.05 mmol/L	121. Zero-trans entry resistance of neuronal MCTs for lactate	RnL,oi	0.54
74. Apparent second-order kinetic constant of neuronal phosphoglycerate kinase	kPGK′n	42.6 L/mmol per s	122. Zero-trans exit resistance of neuronal MCTs for lactate	RnL,io	0.54
75. Apparent second-order kinetic constant of astrocytic phosphoglycerate kinase	kPGK′a	23.6 L/mmol per s	123. Equilibrium exchange resistance of neuronal MCTs for lactate	RnL,ee	0.08
76. Second-order kinetic constant of neuronal pyruvate kinase	kPK	86.7 L/mmol per s	124. Affinity constant for endothelial MCTs for lactate	KeL	1 × 10−4 mmol/L
77. Second-order kinetic constant of astrocytic pyruvate kinase	kPKa	48.2 L/mmol per s	125. Zero-trans entry resistance of endothelial MCTs for lactate	ReL,oi	1.0
78. Second-order forward kinetic constant of neuronal lactate dehydrogenase	kLDHn+	2000 L/mmol per s	126. Zero-trans exit resistance of endothelial MCTs for lactate	ReL,io	0.71
79. Second-order reverse kinetic constant of neuronal lactate dehydrogenase	kLDHn−	44.8 L/mmol per s	127. Equilibrium exchange resistance of endothelial MCTs for lactate	ReL,ee	0.71
80. Second-order forward kinetic constant of astrocytic lactate dehydrogenase	kLDHa+	1110 L/mmol per s	128. Affinity constant for astrocytic MCTs for lactate	KaL	2 × 10−4 mmol/L
81. Second-order reverse kinetic constant of astrocytic lactate dehydrogenase	kLDHn−	25 L/mmol per s	129. Zero-trans entry resistance of astrocytic endfeet MCTs for lactate	RbaL,oi	0.51
82. Maximal neuronal mitochondrial respiration rate	Vmax,Miton	0.018 mmol/L per s	130. Zero-trans exit resistance of astrocytic endfeet MCTs for lactate	RbaL,io	0.51
83. Maximal astrocytic mitochondrial respiration rate	Vmax,Mitoa	0.01 mmol/L per s	131. Equilibrium exchange resistance of astrocytic endfeet MCTs for lactate	RbaL,ee	0.08
84. Michaelis constant for pyruvate uptake by mitochondria	Km,Mito	0.05 mmol/L	132. Zero-trans entry resistance of astrocytic MCTs for lactate	RaiL,oi	0.54
85. Michaelis constant for oxygen uptake by mitochondria	Km,O2	0.001 mmol/L	133. Zero-trans exit resistance of astrocytic MCTs for lactate	RaiL,io	0.54
86. Michaelis constant of mitochondrial respiration for ADP	Km,ADP	0.005 mmol/L	134. Equilibrium exchange resistance of astrocytic MCTs for lactate	RaiL,ee	0.08
87. Second-order forward kinetic constant of neuronal creatine kinase	kCKn+	0.37 L/mmol per s	135. Apparent maximal glucose transport rate between capillary and endothelium	Tmax,GLCce	5.67 mmol/L per s
88. Second-order reverse kinetic constant of neuronal creatine kinase	kCKn−	0.002 L/mmol per s	136. Apparent maximal lactate transport rate between capillary and endothelium	Tmax,LACce	0.26 mmol/L per s
89. Second-order forward kinetic constant of astrocytic creatine kinase	kCKa+	0.2 L/mmol per s	137. Apparent maximal glucose transport rate between endothelium and basal lamina	Tmax,GLCeb	6.41 mmol/L per s
90. Second-order reverse kinetic constant of astrocytic creatine kinase	kCKa−	0.001 L/mmol per s	138. Apparent maximal lactate transport rate between endothelium and basal lamina	Tmax,LACeb	0.30 mmol/L per s
91. Product of capillary permeability by its surface area divided by neuronal volume	PSc/Vn	1.58/s	139. Apparent maximal glucose transport rate between basal lamina and astrocyte	Tmax,GLCba	0.08 mmol/L per s
92. Product of capillary permeability by its surface area divided by astrocytic volume	PSc/Va	0.83/s	140. Apparent maximal lactate transport rate between basal lamina and astrocyte	Tmax,LACba	0.03 mmol/L per s
93. Product of oxygen solubility coefficient by P50	KO2	0.036 mmol/L	141. Apparent maximal glucose transport rate between astrocyte and interstitium	Tmax,GLCai	0.14 mmol/L per s
94. Product of hemoglobin concentration by its oxiphoric power	[Hb]OPHb	8.6 mmol/L	142. Apparent maximal lactate transport rate between astrocyte and interstitium	Tmax,LACai	0.04 mmol/L per s
95. Hill coefficient for hemoglobin	Nh	2.73	143. Apparent maximal glucose transport rate between interstitium and neuron	Tmax,GLCin	0.58 mmol/L per s
96. Blood oxygen concentration	[O2s]	8.34 mmol/L	144. Apparent maximal lactate transport rate between interstitium and neuron	Tmax,LACin	0.07 mmol/L per s
97. Blood glucose concentration	[GLCs]	4.8 mmol/L	145. Glucose diffusion rate between basal-lamina tand interstitium	kDiff,GLCbi	0.023/s
98. Blood lactate concentration	[LACs]	0.3 mmol/L	146. Lactate diffusion rate between basal-lamina and interstitium	kDiff,LACbi	0.046/s
99. Stimulation-induced blood flow increase fraction	αF	0.7	147. Conserved adenylate moiety (ATP+ADP+AMP)	A	2.212 mmol/L
100. Basal cerebral blood flow	CBF0	0.012/s	148. Conserved creatine moiety (Cr+PCr)	C	10 mmol/L
101. Blood flow rise time	t1	5 secs	149. Conserved NAD moiety (NAD++NADH)	N	0.212 mmol/L
102. Faraday's constant	F	96500 C/mol	150. Intracellular sodium concentration	[Nai+]	150 mmol/L
103. Gas constant	R	8.314 J/K per mol	151. Intracellular hydrogen concentration	[H+]	62.5 mmol/L
104. Absolute temperature	T	310 K	152. Equilibrium constant of adenylate kinase	qAK	0.92

^(57−58)Koch (1999); ^(59,152)Gjedde (1997); ^{(60−61,100)}Roland (1993); ^{(62,69−72,76−77)}Heinrich and Schuster (1996); ^(63,148)Erecinska and Silver (1989); ^(64−68)Aubert and Costalat (2002, 2005), Logothetis et al (2001); ^(73,97)Gruetter et al (1992); ^{(74−75,78−81,149)}Joshi and Palsson (1990); ^(82−83)Gjedde (1997), Hertz et al (2007); ^(84−86)Gjedde (1997), Kemp (2000); ^{(87−90,147)}Roth and Weiner (1991); ^(91−92)Kassissia et al (1995), Vafaee and Gjedde (2000); ^(93−96)Vafaee and Gjedde (2000); ^(98,101)Aubert and Costalat (2002); ⁽⁹⁹⁾Buxton et al (1998); ^{(105−146,151)}Simpson et al (2007); ⁽¹⁵⁰⁾McCormick (1998).

All parameter values used in this work have been obtained from previously published models (Aubert and Costalat, 2002, 2005; Simpson et al, 2007). Specifically, kinetic parameters were taken by published in vivo experimental data where available, otherwise they were adjusted such that values of stationary state metabolites agree with the the relevant values for the brain (Table A1). For details on how metabolic model parameters were estimated see Aubert et al (2001) and Aubert and Costalat (2002, 2005). Values of parameters for MCTs and GLUTs have been obtained from Simpson et al (2007) and Mangia et al (2009b).

Note that, because of the asymmetrical character of glucose and lactate carrier proteins, the actual maximum transport rate is T_max/R_oi for inward flux and T_max/R_io for outward flux, where R_oi and R_io are zero-trans entry and zero-trans exit resistance terms of the transporter (Simpson et al, 2007).