Mathematical model of compartmentalized energy transfer:: Its use for analysis and interpretation of 31P-NMR studies of isolated heart of creatine kinase deficient mice

A mathematical model of the compartmentalized energy transfer in cardiac cells is described and used for interpretation of novel experimental data obtained by using phosphorus NMR for determination of the energy fluxes in the isolated hearts of transgenic mice with knocked out creatine kinase isoenzymes. These experiments were designed to study the meaning and importance of compartmentation of creatine kinase isoenzymes in the cells in vivo. The model was constructed to describe quantitatively the processes of energy production, transfer, utilization, and feedback between these processes. It describes the production of ATP in mitochondrial matrix space by ATP synthase, use of this ATP for phosphocreatine production in the mitochondrial creatine kinase reaction coupled to the adenine nucleotide translocation, diffusional, exchange of metabolites in the cytoplasmic space, and use of phosphocreatine for resynthesis of ATP in the myoplasmic creatine kinase reaction. It accounts also for the recently discovered phenomenon of restricted diffusion of adenine nucleotides through mitochondrial outer membrane porin pores (VDAC). Practically all parameters of the model were determined experimentally. The analysis of energy fluxes between different cellular compartments shows that in all cellular compartments of working heart cells the creatine kinase reaction is far from equilibrium in the systolic phase of the contraction cycle and approaches equilibrium only in cytoplasm and only in the end-diastolic phase of the contraction cycle. Experimental determination of the relationship between energy fluxes by a(31)P-NMR saturation transfer method and workload in isolated and perfused heart of transgenic mice deficient in MM isoenzyme of the creatine kinase, MM -/- showed that in the hearts from wild mice, containing all creatine kinase isoenzymes, the energy fluxes determined increased 3-4 times with elevation of the workload. By contrast, in the hearts in which only the mitochondrial creatine kinase was active, the energy fluxes became practically independent of the workload in spite of the preservation of 26% of normal creatine kinase activity. These results cannot be explained on the basis of the conventional near-equilibrium theory of creatine kinase in the cells, which excludes any difference between creatine kinase isoenzymes. However, these apparently paradoxical experimental results are quantitatively described by a mathematical model of the compartmentalized energy transfer based on the steady state kinetics of coupled creatine kinase reactions, compartmentation of creatine kinase isoenzymes in the cells, and the kinetics of ATP production and utilization reactions. The use of this model shows that: (1) in the wild type heart cells a major part of energy is transported out of mitochondria via phosphocreatine, which is used for complete regeneration of ATP locally in the myofibrils - this is the quantitative estimate for PCr pathway; (2) however, in the absence of MM-creatine kinase in the myofibrils in transgenic mice the contraction results in a very rapid rise of ADP in cytoplasmic space, that reverses the mitochondrial creatine kinase reaction in the direction of ATP production In this way, because of increasing concentrations of cytoplasmic ADP, mitochondrial creatine kinase is switched off functionally due to the absence of its counterpart in PCr pathway, MM-creatine kinase. This may explain why the creatine kinase flux becomes practically independent from the workload in the hearts of transgenic mouse without MM-CK. Thus, the analysis of the results of studies of hearts of creatine kinase-deficient transgenic mice, based on the use of a mathematical model of compartmentalized energy transfer, show that in the PCr pathway of intracellular energy transport two isoenzymes of creatine kinase always function in a coordinated manner out of equilibrium, in the steady state, and disturbances in functioning of one of them inevitably result in the disturbances of the other component of the PCr pathway. In the latter case, energy is transferred from mitochondria to myofibrils by alternative metabolic pathways, probably involving adenylate kinase or other systems. The merits of several mathematical models of muscle energy metabolism are also discussed.