An integrative dynamic model of brain energy metabolism using in vivo neurochemical measurements

Abstract

An integrative, systems approach to the modelling of brain energy metabolism is presented. Mechanisms such as glutamate cycling between neurons and astrocytes and glycogen storage in astrocytes have been implemented. A unique feature of the model is its calibration using in vivo data of brain glucose and lactate from freely moving rats under various stimuli. The model has been used to perform simulated perturbation experiments that show that glycogen breakdown in astrocytes is significantly activated during sensory (tail pinch) stimulation. This mechanism provides an additional input of energy substrate during high consumption phases. By way of validation, data from the perfusion of 50 µM propranolol in the rat brain was compared with the model outputs. Propranolol affects the glucose dynamics during stimulation, and this was accurately reproduced in the model by a reduction in the glycogen breakdown in astrocytes. The model’s predictive capacity was verified by using data from a sensory stimulation (restraint) that was not used for model calibration. Finally, a sensitivity analysis was conducted on the model parameters, this showed that the control of energy metabolism and transport processes are critical in the metabolic behaviour of cerebral tissue.

Appendices

Appendix A: Mass balances and kinetic equations of the brain energy metabolism model

This Appendix presents the mass balances, kinetic equations and parameters of the model.

The metabolic model is presented as a group of interacting subsystems that describe specific cellular functions, while performing critical interactions with other subsystems. It is our belief that the decomposition of the model will allow a better systems understanding of the problem of simulating brain physiology. The subsystems considered in the modelling are described in the following Appendixes:

• (A1) Compartments and exchange systems (GLC, LAC, O2, CO2)

• (A2) Central energy metabolism (glycolysis + mitochondrial oxidation)

• (A3) Neuronal stimulation system and glutamate cycling for astrocytes-neurons coordination

• (A4) Glycogen storage (astrocytes only)

Each of these cellular subsystems will be described in detail (differential equations and kinetic parameters). Appendix A.5 then presents an overview of the complete model. A complete set of model parameters is presented in section A.6. Regulation mechanisms and changes in behaviour in the model were described using a ‘switch function’ (f(t,δ,a)) which is described in section A.7.

1.1 A.1 Compartments and exchange systems

The exchange systems account for the transfer fluxes between the cerebral compartments Fig. 6. The model considers 4 compartments:

• Capillary (V_c) = 0.55% of total volume

• Extracellular space (V_e) = 20% of total volume

• Neuronal volume (V_n) = 45% of total volume

• Astrocytic, or glial volume (V_g) = 25% of total volume

Fig. 6

Exchange systems

The exchange fluxes between these compartments consider the compartment’s volumetric ratios in order to have consistent mass balances (for example R_cg = V_c/V_g = 0.055/0.25 for exchanges between capillaries and astrocytes). Each exchange flux is defined with a given compartment as reference volume, so no volumetric adjustment is required in these specific cases. The following hypotheses are made to describe the transfer of substrate and by-products of cerebral metabolism:

• Transfer of gaseous species (O2 and CO2) between cellular compartments and capillaries is assumed to be direct. O2 transfer is described with the mechanisms used in Aubert and Costalat (2005).

• GLC and LAC exchanges between the compartments are regulated by transporters that are described by facilitated diffusion equations.

• CBF regulation was built upon works by Aubert and Costalat (2005), but the regulation of blood flow in this study is described using a generic sigmoid switch (see Appendix A.6).

• No hypotheses are made as regards the ANLS. Lactate transfer is described by facilitated diffusion and transfer from astrocytes to neurons occurs only if the gradients are favourable. Since two mechanisms excrete lactate from the cerebral tissue (v^ec _LAC and v^gc _LAC), there is no ‘structural bias’ in the model to force lactate transfer from astrocytes to neurons Tables 1 and 2.

  Table 1 Variables for exchange and transport systems

  Table 2 Kinetic equations for exchange and transport systems

1.2 A.2 Central energy metabolism

Figure A-2 presents the reactions that are considered for central energy metabolism. In this work, we consider the processing of GLC through glycolysis, with mitochondrial oxidation of PYR to be the major pathways of central energy metabolism. The action of ATPases and phosphocreatine (PCr) buffering, as well as the action of lactate dehydrogenase (vLDH) are considered. These reactions occur in both neurons and astrocytes Fig. 7.

• Glycolysis is simplified and represented by 5 reactions but these still accounts for ATP/ADP and NAD/NADH balances and regulation (discussed in the text).

• Glycolytic reactions include the same reactions as were presented in works by Aubert and Costalat (2005), with the difference that the conversion of GLC to GAP is now described in three reactions instead of one. This allows a better description of regulatory phenomenon, especially as regards to glycogen storage (astrocytes, see Appendix A.4), and GLC consumption dynamics by the hexokinase (vHK) a reaction highly inhibited by its product, G6P.

• A constant amount of energetic shuttles (ATP+ADP+AMP = ANP) is assumed. Distribution between ATP, ADP and AMP varies depending on energetic requirements, kinetic reactions and adenylate kinase equilibrium. Equilibrium description from Heinrich and Schuster (1996) was used for adenylate kinase.

• The same hypothesis of a constant sum is made for cofactors NADH + NAD = NADHtot as well as for phosphocreatine buffer (PCr+Cr = PCrtot).

• Inhibition of mitochondrial activity at high ATP/ADP ratio is considered (see Table A.2.2, equations 6 and 15).

• The stoichiometry and overall mass balances of the reactions of central energy metabolism are the same in neurons and astrocytes, with the same regulation mechanisms. However, the kinetic parameters are different, which allows describing the different metabolic behaviour of neurons and astrocytes at rest or during active periods.

• No hypotheses or fluxes are imposed a priori as regards to the ANLS. The model simply describes the kinetic behaviour of key enzymatic reactions. The Lactate dehydrogenase enzyme, which catalyses the conversion between PYR and LAC, is usually assumed to favour the forward reaction (PYR → LAC). This is seen in the parameters (see Table A.6) as the forward rate constant is much higher than the reverse rate constant (kⁿ _LDH,f > kⁿ _LDH,r). However, if the conditions are favourable (NADH/NAD and PYR/LAC ratios), the reaction could produce PYR from LAC Tables 3 and 4.

  Table 3 Central energy metabolism

  Table 4 Kinetic equations for central energy metabolism

Fig. 7

Central energy metabolism

1.3 A.3 Neuronal stimulation and glutamate cycling for astrocytes-neurons coordination

The stimulatory subsystem is built upon works by Aubert and Costalat (2005) to describe neuronal stimulation (vn stim). A critical addition to the stimulation system is the glutamate cycling proposed by Pellerin and Magistretti (1994) for neuronal-astrocytic coordination Fig. 8. The following mechanisms and hypotheses are modelled:

• A sodium inflow (vⁿstim described by vⁿ1 and vⁿ2 in Table A.3.2) stimulates neuronal ATPase. This inflow is considered to be an ‘input’ in the sense that it can be modulated to get a better description of physiological stimulations (strength and duration).

• Upon neuronal stimulation, GLU is released by neurons and taken up by astrocytes (veg

• GLU) with the co-transport of 3 sodium ions.

• The conversion of GLU to glutamine (GLN) in astrocytes for its ‘non-stimulatory’ transfer to neurons is modelled as one reaction (vgn GLU).

• This simplified model (neglecting GLN) retains the critical feature of the GLU cycling: a coordinated activation of ATPases in astrocytes upon neuronal activation.

• The stoichiometry for ATP consumption in the cycle is respected, as 2 molecules of ATP are required in astrocytes for each molecule of GLU circulating in the cycle (one for Na pumping and one for GLU conversion and transfer).

Fig. 8

Stimulation and astrocytes-neurons coordination

Considering the dynamics of GLU cycling allows describing cerebral activity with a physiological approach, instead of using two distinct stimulations for neurons and astrocytes, as was the case in Aubert and Costalat (2005). The equations for the stimulation system are presented in Tables 4 and 5.

Table 5 Stimulation related variables

1.4 A.4 Glycogen storage system

Figure A-4 presents the mechanisms considered for glycogen storage in astrocytes and its link to central energy metabolism through G6P. The reactions for GLY storage are considered only for astrocytes, as neurons don’t accumulate GLY. Glycogen levels in astrocytes can vary depending on the energetic requirements of the tissue and time of the day. Levels of glycogen up to 4.2 mMol are reported for astrocytes (Brown and Ransom, 2007; units adjusted for consistency). The pool of glycogen in astrocytes is thus not negligible and could potentially sustain cerebral activity for a few minutes. Qualitatively, it would seem that under resting conditions, glycogen storage (up to 4.2 mMol) would be favoured, as glucose is available from the extracellular space and glycolytic requirements are lower. During high activity periods (or hypoglycemia), glycogen breakdown can be initiated by neurotransmitters (noradrenaline) to induce glycogenolysis. However, as was recently reported in a special issue of Glia (Vol. 55, No.12, 2007) the role of astrocytic glycogen in brain energy metabolism is much more complex than that if a simple energy reserve. The following hypotheses and mechanisms are considered in the glycogen storage system Fig. 9:

• Glycogen synthesis from G6P is described in one reaction, the glycogen synthase (vg GYS) and breakdown is described also in one reaction, the glycogen phosphorylase (vg PYG).

• Michaelis-Menten kinetics are used to describe both reactions

• GLY accumulation will be favoured when the entering flux of GLC exceeds the glycolytic requirements (i.e. at rest). However, GLY cannot be accumulated at concentrations higher than 4.2 mM.

• Upon neuronal activation, a ‘signal’ (noradrenaline) is assumed to induce GLY breakdown. This is modelled as a fractional increase of vg PYG for a certain time following the stimulation. This signal, as was the case for the stimulatory signal, can vary depending on the stimulation. This signal is, however, assumed to start after neuronal stimulation onset (delay).

• Inhibition of vg HK by its product, G6P, is considered in the model (see equation 10 in Table A.2.2). Thus, if GLY breakdown increase the G6P pool, an inhibition of vg

Fig. 9

Glycogen storage system

HK can occur, although the overall glycolytic flux would be maintained through GLY consumption. This would allow astrocytes to switch from ‘pure’ GLC usage to a mix of GLC and GLY usage.

The model we present here for glycogen storage considers the dynamic, regulatory response of the glycogen pool and its integration in brain energy metabolism. The last hypothesis is critical as regards the role of GLY in energy metabolism. Switching from ‘pure’ glucose usage to a mix of glycogen and glucose usage during high activity periods would have a critical effect on the ‘energy balance’ in astrocytes. Glycogenolysis has a higher energetic yield (initiating glycolytic flux with G6P instead of GLC temporarily saves one molecule of ATP). Thus, during low demand periods, the glycogen pool can be replenished, as the glycolytic requirements are lower and this would later constitute a reserve of substrate with a higher energetic yield for high demand periods. The glycogen pool must thus be viewed not only as a substrate reserve, but also as a dynamic energy reserve. Using the glycogenolytic potential in astrocytes (even for normal cerebral activity) would allow astrocytes to buffer their energy budget between high and low demand periods. Moreover, glycogen dynamics are important for the ANLS theory. As reviewed by Pellerin et al. (2007) the GLY ‘dynamic usage’ hypothesis is not contradictory with the ANLS. Transferring the ’energetic potential’ of astrocytic GLY to neurons could be done through LAC shuttling. But the fact that astrocytes can switch to GLY usage during high demand period also potentially leaves more GLC for neurons to consume. Thus, a precise quantification of glycogen dynamics and its integration in models for brain energy metabolism is going to be crucial for future developments (Pellerin et al. 2007) Table 5, 6 and 7.

Table 6 Kinetic equations for stimulation related fluxes

Table 7 Glycogen storage system

1.5 A.5 The complete model

Figure 10 (next page) presents the complete model used in this study.

Fig. 10

Presents the complete model used in this study

1.6 A.6 Model parameters

Table A.6.1 presents model kinetic parameters. These parameters were found through model calibration (see section on model calibration) with in vivo data on energy metabolism (see article) and brain physiology data (typical fluxes and ratios, see Appendix B). Table A.6.2 presents the physical constants and known concentrations that are used in the model. Table A.6.3 presents the ‘input’ parameters that are used to define the effect of neuronal stimulation.

1.7 A.7 Switch function

The present model needs to describe physiological changes in behaviour (changes in CBF during stimulation etc.) which are not well described by ‘pure’ on/off switches. Sigmoid behaviour is much more common with biological systems. The following function can be used to describe a sigmoid switch:

$$ f\left( {t,\delta, a} \right) = \frac{1}{{1 + e^{{ - a \cdot \left( {t - \delta } \right)}} }} $$

where ‘t’ is the time, ‘δ’ is the time of the ‘event’ inducing switch in behaviour and ‘α’ is the sharpness of the change or the ‘slope’ during the change in behaviour. Figure 11 presents profiles of that function.

Fig. 11

Switch function behaviour

Thus, by combining different forms of that function, it is possible to describe ‘smooth’ or ‘sharp’ changes in biological behaviour without having to use binary switching. This approach allows a better description of biological behaviours and reduces discontinuities, which is important in numerical solvers for differential equations. The switch can also be used with concentrations (instead of time) to represent metabolic regulation (i.e. maximum accumulation levels, inhibitions etc.) Tables 8, 9, 10.

Table 8 Model parameters

Table 9 Physical constants and known concentrations

Table 10 Stimulation parameters

Appendix B: Model calibration with fluxes and ratios from literature

An important consideration in building a realistic model for brain energy metabolism is the overall balancing of substrates consumption. Typical rates for GLC and O2 consumption at rest and during stimulation are presented in Figure 12.

Fig. 12

Simulated fluxes for GLC oxidative metabolism and O2/GLC ratio during resting and neuronal stimulation (5 min tail pinch starting at t = 1000s).

The glutamate cycling, induced by neuronal stimulation, is characterized by the circulation of GLU in astrocytes (νgn GLU, as described, for example, in Hyder et al. 2006). The profile for νgn GLU produced by the model (Figure B-1a) is within the ranges reported in the literature (see Hyder et al. 2006). The most relevant glycolytic rate here is assumed to be the PFK rate (Figure B-1b), as it accounts for substrate coming from both GLC and GLY (see Appendix A.4 and Figure A-5 for further details). Steady-state consumption fluxes for GLC and O2 are within the ranges reported in the literature for overall GLC and O2 consumption by the brain. Using the glycolytic rates (vn PFK + vg PFK) at rest and during stimulations, our model shows overall GLC consumption rates in the range of 90 to 105g per day (Figure B-1f). The O2 consumption rate and LAC excretion by the tissue would also be in ranges described in the literature since the simulated O2 to GLC ratio (5.75 at rest and going down during stimulation, see Figure B-1d) is also in the range of reported values. The rate of GLC consumption (Figure B-1f) is only marginally smaller at rest (90g·d-1) than during neuronal activation (100-105g·d-1 depending on stimulation parameters). This is in accordance with the literature, as a high baseline rate for GLC metabolism in the brain is the general consensus (Magistretti 2006 and references therein). Thus the model calibration, as regards the overall GLC, O2 and LAC rates (at rest and during stimulation) produced a physiological behaviour representative of the cerebral environment.

The phenomenon of predominant GLC consumption in astrocytes, with further oxidation in neurons (ν^g _,PFK > vⁿ _PFK while vⁿ _mito > v ^g_mito , as was the case in the Hyder et al. framework and in many ANLS supporting arguments) is not observed in the simulations. In our model, GLC consumption happens mostly in neurons (at rest: v ⁿ_PFK ≈ 6 μM·s⁻¹ and ν ^g_,PFK ≈ 4.5 μM·s⁻¹, Figure B-1b). During stimulation, however, glycolytic rate in astroyctes can be slightly higher than in neurons (depending on the strength of the stimulus).

PYR oxidation is much more pronounced in neurons (v ⁿ_mito ≈ 13 μM·s⁻¹ and v ^g_mito ≈ 6 μM·s⁻¹, see Figure B-1c) and this induces a slight shuttling of lactate from astrocytes to neurons (Figure B-1e, vn LDH is negative). However, glycolysis remains the major source of PYR for mitochondrial oxidation. In our model, LAC shuttling accounts only for ≈7% of the energetic requirements of resting neurons. This proportion rises to ≈12–15% during stimulations periods, which allows increases in energy demand in neurons be met by a sharp increase in mitochondrial rate rather than an increase in glycolytic flux. However, as regards the ANLS, our simulation results are not in agreement with a ‘full’ lactate shuttle (i.e. all the LAC produced in astrocytes being transferred to neurons for oxidation). This would lead to an O2 to GLC ratio of 6 (with no variations during stimulation), which is clearly not the case in both our simulations and in many reports from the literature.