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Cellular cardiac metabolism: Mechanistic modeling approach
ELSEVIER
Published by Elsevier Science on behalf of IFAC
IFAC PUBLICATIONS www.elsevier.com/locale/ifac
CELLULAR CARDIAC METABOLISM: MECHANISTIC MODELING APPROACH
Gerald M. Saidel,· Lufang ZhOU,1 William C. Stanley3, Marco E. Cabrera l ,2,3
Departments oflBiomedical Engineering. 2Pediatrics. 3Physiology & Biophysics. and Center for Modeling Integrated Metabolic Systems Case Western Reserve University and Rainbow Babies & Children 's Hospital Cleveland. Ohio 44106. USA.
Abstract: A mechanistic model of cardiac cellular metabolism is presented as an example of a top-down systems approach to physiologically oriented cellular metabolism. This model focuses on intennediary metabolism with the inclusion of essential chemical species and pathways distinguishing between metabolic reactions in the cytosol and mitochondria. It accounts for mass transfer between cytosol and mitochondria and between blood that perfuses the tissue and cytosol. The functional relationship of the metabolic fluxes to chemical species concentrations is approached from a phenomenological perspective for simplicity. Keywords: chemical, model, control systems, nonlinearities, response times
Although various metabolic systems models have been developed for analysis of microbial organisms, these are inadequate for understanding more complex tissue-organ systems. The development of quantitative approaches to describe, analyse, and predict the behaviour of tissue-organ metabolic systems requires integration of biochemical, cellular, and physiological data. The goal of such mechanistic modelling is to provide an understanding of the tissue-organ response to hypoxia, exercise, diet, drugs, etc. Indeed, this is a key objective of a newly created Center for Modeling Integrated Metabolic Systems funded by the National Institute of General Medical Sciences ofNIH.
I. INTRODUCTION Research during the past few decades has led to (a) discovery of the component pathways of metabolic processes and the component reactions in the pathways; (b) characterization of the enzymes that catalyse reactions; and (c) identification of the genes that encode the enzymes. However, accumulated knowledge of the components of a metabolic system at all levels of complexity has not been exploited for a quantitative understanding of factors detennining reaction fluxes in different pathways. Furthennore, a theoretical framework is needed to quantify the rates of production and utilization of metabolites based on the metabolic, redox, and energy states of a tissue. This requires a mechanistically based model whose numerical solution yields simulations of metabolic system dynamics for data analysis and interpretation.
A quantitative systems approach to modelling cellular metabolism in a specific tissue/organ has been developed that integrates metabolite kinetics, control mechanisms, and pathways (Cabrera, et al. I998a,b). Of particular interest at the cellular level
527
are the complex pathways which yield energy to sustain biological work. The fuel requirements of these cellular bioenergetic pathways depend on the metabolic characteristics of the cells of the specific tissue/organ. In particular, the cardiac muscle has the capacity to increase its work and metabolic rate several fold, which requires a tight coupling between pathways of ATP synthesis and energy consuming processes (i.e., ATPases). However, most sites of ATP utilization reside in the cytosol while most ATP synthesis takes place in the mitochondria. Since under most circumstances ATP and ADP cannot be transported across the mitochondrial membrane, model simulations can be used to examine the effect of various pathways of energy transfer between these subcellular compartments on the dynamic responses of key species.
Q
Perfused Tissue
Fig. I: Model components for transport and metabolic processes in the system
The mechanistic model of cardiac metabolism described here is more general than an earlier version (Salem, et al. 2002). In this version, we distinguish between metabolic reactions in the cytosol and mitochondria, include additional chemical species and pathways of importance, and present a more general approach to the functional relationship of the metabolic fluxes and chemical species concentrations.
Note: Q: tissue blood flow; Cij: speciesj concentration in compartment i; Uij: species j utilization rate in compartment i; Pij, species j production rate in compartment i; Mito: mitochondria. See text for detail.
The arterial concentration Ca is considered a known input. In volume Vmof the mitochondria (indicated by subscript m), the species j concentration Cmit) may change similarly:
2. MODEL DEVELOPMENT As shown in Fig. I, we consider the heart as perfused tissue with a representative cell having distinct cytosol and mitochondria. Some chemical species diffuse between blood in the vascular bed and the cytosol: glucose, lactate, pyruvate, fatty acids, glycerol, O 2, and CO 2, Some chemical species exist only in the cytosol: glycogen, triglycerides, glucose 6-phosphate, glyceraldehyde 3-phosphate, 1,3CO 2, biphosphoglycerate. Whereas O 2, phosphocreatine, creatine, inorganic phosphate (Pi), and CoA move freely between the cytosol and mitochondria, other chemical species use specific carrier mechanisms for transport between cytosol and mitochondria: pyruvate, fatty acyl-CoA, NADH, NAD, malate, and citrate. Species limited to the mitochondria are a-ketoglutarate, succinyl-CoA, and succinate. Finally, acetyl-CoA, oxaloacetate, ADP, ATP, NAD, and NADH exist at different concentrations in cytosol and mitochondria, but under most circumstances cannot move between them.
In addition to these dynamic balance equations, stoichiometric equations relate ADP to ATP and NAD to NADH within the cytosol and mitochondria. The metabolic processes can be represented as:
PJ
Uj
-
n
m
k=1
k=1
= LPkjifJ~ - LPjkifJ!t
(3)
where
~.
"
r
- K
X
1r
·l Jl 1 + "Kx+X
c 1- I -L-
KJO
J
(4)
where the Ks are coefficients and X represents a metabolic control ratio (e.g., ADP/ATP, NADHINAD). A particular ratio is included in the flux expression only if it is directly coupled to the corresponding reaction.
A mass balance model describes the concentration dynamics of a chemical species j that occur by transport and metabolic processes. In volume Ve of the cytosol (indicated by subscript c), the species j concentration Cq(t) may change because of metabolic production and utilization Pej , Uej, and mass transfer across the cell and mitochondrial membranes. These transport processes are determined by perfusion Q, rate coefficient a.i ' and partition coefficients crj and A{
de, V. -:: =P,.j -l{, + Q
Q
Vascular Bed
3. MODEL JUSTIFICATION The model representation above needs justification and elaboration. First, although we have included many chemical species, we have omitted or lumped together many more. We chose those species because they (I) are known to be important in myocardial metabolism, (2) are measurable or potentially measurable, (3) involve unique energy
(I)
528
transfer processes, (4) are control points in the metabolic pathways, or (5) are potentially important sites for excitation or inhibition with drugs. Second, we relate metabolic fluxes to species concentration to the maximum extent on molecular reaction mechanisms to avoid concentrations with irrational empirical powers. As a practical matter, all reactions are primarily irreversible with different forward and backward mechanisms. We assume that any product inhibition occurs naturally by through the various feedback processes.
their effect on the dynamic responses of key substrates.
ACKNOWLEDGEMENT This work was supported by a grant (GM-66309) from the National Institute of General Medical Sciences, NIH.
REFERENCES 4. SIMULATION STRATEGY AND METHODS
Cabrera, M.E., G.M. Saidel and S.e. Kalhan (l998a). Modeling metabolic dynamics from cellular processes to organ and whole body responses. Prog. Biophys. Molec. Bioi. 69, 539-557. Cabrera, M.E., G.M. Saidel and S.C. Kalhan (1998b). Role of O 2 in regulation of lactate dynamics during hypoxia: mathematical model and analysis. Ann Biomed Eng, 26,1-27. Cabrera, M.E., G.M. Saidel and S.e. Kalhan (1999). Lactate metabolism during exercise: analysis by an integrative systems model. Am. 1. Physiol. 277, RI522-RI536. Salem, J.E., G.M. Saidel, W.e. Stanley, and M.E. Cabrera (2002. Mechanistic model of myocardial metabolism under normal and ischemic conditions. Ann. Biomed Eng. 30,
One of the objectives of our model is to simulate the concentration and flux changes of chemical species that are key to quantifying myocardial metabolism under normal and ischemic conditions. Using parameter values based on our data and the literature, we solve the model equations numerically using a sparse-stiff integrator. In the initial stage of our simulations, we determine the consistency of baseline model parameters under normal, steadystate, resting conditions by comparing the simulated species concentrations to those obtained by experimental measurements (Salem, et al. 2002). Then we simulate the response to ischemia produced by a reduction in coronary blood flow. When the model does not exhibit behaviour according to experimental data, we modify model parameters to achieve consistency.
pp. 202-216, 2002.
5. MODEL RATIONALE AND CHALLENGES While we have shown that this modeling approach has successfully addressed biological questions with simpler metabolic models (Cabrera, at al. 1999, Salem et al. 2002), several aspects remain unresolved, such as the slower than expected response time of several chemical species. This may be corrected in this new model that (I) distinguishes cytosol and mitochondria and (2) includes additional important chemical species that participate in the regulation of energy metabolism (e.g., PJ Among the modeling challenges is the need to account for the transfer of energy between sites of ATP synthesis and sites of ATP utilization. While the creatine kinase reaction PCr + ADP + H+ ~ Cr + ATP appears to occur in both the mitochondria and the cytosol. ADP and ATP are not transported across the mitochondrial membrane. Assuming dynamic equilibrium between cytosolic and mitochondrial PCr, Cr, and P" our preliminary hypothesis is that energy derived from mitochondrial ATP synthesis is transported to the cytosol via PCr, while the products of the CK and ATPase reactions (i.e., Cr and Pi) in the cytosol are transported to the mitochondria. Results of model simulations can be used to examine this and other hypotheses explaining the creatine kinase fluxes and pathways of energy transfer in the subcellular compartments of the beating heart and
529
Published by Elsevier Science on behalf of IFAC
IFAC PUBLICATIONS www.elsevier.com/locale/ifac
CELLULAR CARDIAC METABOLISM: MECHANISTIC MODELING APPROACH
Gerald M. Saidel,· Lufang ZhOU,1 William C. Stanley3, Marco E. Cabrera l ,2,3
Departments oflBiomedical Engineering. 2Pediatrics. 3Physiology & Biophysics. and Center for Modeling Integrated Metabolic Systems Case Western Reserve University and Rainbow Babies & Children 's Hospital Cleveland. Ohio 44106. USA.
Abstract: A mechanistic model of cardiac cellular metabolism is presented as an example of a top-down systems approach to physiologically oriented cellular metabolism. This model focuses on intennediary metabolism with the inclusion of essential chemical species and pathways distinguishing between metabolic reactions in the cytosol and mitochondria. It accounts for mass transfer between cytosol and mitochondria and between blood that perfuses the tissue and cytosol. The functional relationship of the metabolic fluxes to chemical species concentrations is approached from a phenomenological perspective for simplicity. Keywords: chemical, model, control systems, nonlinearities, response times
Although various metabolic systems models have been developed for analysis of microbial organisms, these are inadequate for understanding more complex tissue-organ systems. The development of quantitative approaches to describe, analyse, and predict the behaviour of tissue-organ metabolic systems requires integration of biochemical, cellular, and physiological data. The goal of such mechanistic modelling is to provide an understanding of the tissue-organ response to hypoxia, exercise, diet, drugs, etc. Indeed, this is a key objective of a newly created Center for Modeling Integrated Metabolic Systems funded by the National Institute of General Medical Sciences ofNIH.
I. INTRODUCTION Research during the past few decades has led to (a) discovery of the component pathways of metabolic processes and the component reactions in the pathways; (b) characterization of the enzymes that catalyse reactions; and (c) identification of the genes that encode the enzymes. However, accumulated knowledge of the components of a metabolic system at all levels of complexity has not been exploited for a quantitative understanding of factors detennining reaction fluxes in different pathways. Furthennore, a theoretical framework is needed to quantify the rates of production and utilization of metabolites based on the metabolic, redox, and energy states of a tissue. This requires a mechanistically based model whose numerical solution yields simulations of metabolic system dynamics for data analysis and interpretation.
A quantitative systems approach to modelling cellular metabolism in a specific tissue/organ has been developed that integrates metabolite kinetics, control mechanisms, and pathways (Cabrera, et al. I998a,b). Of particular interest at the cellular level
527
are the complex pathways which yield energy to sustain biological work. The fuel requirements of these cellular bioenergetic pathways depend on the metabolic characteristics of the cells of the specific tissue/organ. In particular, the cardiac muscle has the capacity to increase its work and metabolic rate several fold, which requires a tight coupling between pathways of ATP synthesis and energy consuming processes (i.e., ATPases). However, most sites of ATP utilization reside in the cytosol while most ATP synthesis takes place in the mitochondria. Since under most circumstances ATP and ADP cannot be transported across the mitochondrial membrane, model simulations can be used to examine the effect of various pathways of energy transfer between these subcellular compartments on the dynamic responses of key species.
Q
Perfused Tissue
Fig. I: Model components for transport and metabolic processes in the system
The mechanistic model of cardiac metabolism described here is more general than an earlier version (Salem, et al. 2002). In this version, we distinguish between metabolic reactions in the cytosol and mitochondria, include additional chemical species and pathways of importance, and present a more general approach to the functional relationship of the metabolic fluxes and chemical species concentrations.
Note: Q: tissue blood flow; Cij: speciesj concentration in compartment i; Uij: species j utilization rate in compartment i; Pij, species j production rate in compartment i; Mito: mitochondria. See text for detail.
The arterial concentration Ca is considered a known input. In volume Vmof the mitochondria (indicated by subscript m), the species j concentration Cmit) may change similarly:
2. MODEL DEVELOPMENT As shown in Fig. I, we consider the heart as perfused tissue with a representative cell having distinct cytosol and mitochondria. Some chemical species diffuse between blood in the vascular bed and the cytosol: glucose, lactate, pyruvate, fatty acids, glycerol, O 2, and CO 2, Some chemical species exist only in the cytosol: glycogen, triglycerides, glucose 6-phosphate, glyceraldehyde 3-phosphate, 1,3CO 2, biphosphoglycerate. Whereas O 2, phosphocreatine, creatine, inorganic phosphate (Pi), and CoA move freely between the cytosol and mitochondria, other chemical species use specific carrier mechanisms for transport between cytosol and mitochondria: pyruvate, fatty acyl-CoA, NADH, NAD, malate, and citrate. Species limited to the mitochondria are a-ketoglutarate, succinyl-CoA, and succinate. Finally, acetyl-CoA, oxaloacetate, ADP, ATP, NAD, and NADH exist at different concentrations in cytosol and mitochondria, but under most circumstances cannot move between them.
In addition to these dynamic balance equations, stoichiometric equations relate ADP to ATP and NAD to NADH within the cytosol and mitochondria. The metabolic processes can be represented as:
PJ
Uj
-
n
m
k=1
k=1
= LPkjifJ~ - LPjkifJ!t
(3)
where
~.
"
r
- K
X
1r
·l Jl 1 + "Kx+X
c 1- I -L-
KJO
J
(4)
where the Ks are coefficients and X represents a metabolic control ratio (e.g., ADP/ATP, NADHINAD). A particular ratio is included in the flux expression only if it is directly coupled to the corresponding reaction.
A mass balance model describes the concentration dynamics of a chemical species j that occur by transport and metabolic processes. In volume Ve of the cytosol (indicated by subscript c), the species j concentration Cq(t) may change because of metabolic production and utilization Pej , Uej, and mass transfer across the cell and mitochondrial membranes. These transport processes are determined by perfusion Q, rate coefficient a.i ' and partition coefficients crj and A{
de, V. -:: =P,.j -l{, + Q
Q
Vascular Bed
3. MODEL JUSTIFICATION The model representation above needs justification and elaboration. First, although we have included many chemical species, we have omitted or lumped together many more. We chose those species because they (I) are known to be important in myocardial metabolism, (2) are measurable or potentially measurable, (3) involve unique energy
(I)
528
transfer processes, (4) are control points in the metabolic pathways, or (5) are potentially important sites for excitation or inhibition with drugs. Second, we relate metabolic fluxes to species concentration to the maximum extent on molecular reaction mechanisms to avoid concentrations with irrational empirical powers. As a practical matter, all reactions are primarily irreversible with different forward and backward mechanisms. We assume that any product inhibition occurs naturally by through the various feedback processes.
their effect on the dynamic responses of key substrates.
ACKNOWLEDGEMENT This work was supported by a grant (GM-66309) from the National Institute of General Medical Sciences, NIH.
REFERENCES 4. SIMULATION STRATEGY AND METHODS
Cabrera, M.E., G.M. Saidel and S.e. Kalhan (l998a). Modeling metabolic dynamics from cellular processes to organ and whole body responses. Prog. Biophys. Molec. Bioi. 69, 539-557. Cabrera, M.E., G.M. Saidel and S.C. Kalhan (1998b). Role of O 2 in regulation of lactate dynamics during hypoxia: mathematical model and analysis. Ann Biomed Eng, 26,1-27. Cabrera, M.E., G.M. Saidel and S.e. Kalhan (1999). Lactate metabolism during exercise: analysis by an integrative systems model. Am. 1. Physiol. 277, RI522-RI536. Salem, J.E., G.M. Saidel, W.e. Stanley, and M.E. Cabrera (2002. Mechanistic model of myocardial metabolism under normal and ischemic conditions. Ann. Biomed Eng. 30,
One of the objectives of our model is to simulate the concentration and flux changes of chemical species that are key to quantifying myocardial metabolism under normal and ischemic conditions. Using parameter values based on our data and the literature, we solve the model equations numerically using a sparse-stiff integrator. In the initial stage of our simulations, we determine the consistency of baseline model parameters under normal, steadystate, resting conditions by comparing the simulated species concentrations to those obtained by experimental measurements (Salem, et al. 2002). Then we simulate the response to ischemia produced by a reduction in coronary blood flow. When the model does not exhibit behaviour according to experimental data, we modify model parameters to achieve consistency.
pp. 202-216, 2002.
5. MODEL RATIONALE AND CHALLENGES While we have shown that this modeling approach has successfully addressed biological questions with simpler metabolic models (Cabrera, at al. 1999, Salem et al. 2002), several aspects remain unresolved, such as the slower than expected response time of several chemical species. This may be corrected in this new model that (I) distinguishes cytosol and mitochondria and (2) includes additional important chemical species that participate in the regulation of energy metabolism (e.g., PJ Among the modeling challenges is the need to account for the transfer of energy between sites of ATP synthesis and sites of ATP utilization. While the creatine kinase reaction PCr + ADP + H+ ~ Cr + ATP appears to occur in both the mitochondria and the cytosol. ADP and ATP are not transported across the mitochondrial membrane. Assuming dynamic equilibrium between cytosolic and mitochondrial PCr, Cr, and P" our preliminary hypothesis is that energy derived from mitochondrial ATP synthesis is transported to the cytosol via PCr, while the products of the CK and ATPase reactions (i.e., Cr and Pi) in the cytosol are transported to the mitochondria. Results of model simulations can be used to examine this and other hypotheses explaining the creatine kinase fluxes and pathways of energy transfer in the subcellular compartments of the beating heart and
529