[Mathematical modelling in physiology].

The article illustrates the method of mathematical modelling in physiology as a unique tool to study physiological processes. A number of demonstrated examples appear as a result of long-term experience in mathematical modelling of electrical and mechanical phenomena in the heart muscle. These examples are presented here to show that the modelling provides insight into mechanisms underlying these phenomena and is capable to predict new ones that were previously unknown. While potentialities of the mathematical modelling are analyzed with regard to the myocardium, they are quite universal to deal with any physiological processes.

[Assessment of the mechanical activity of cardiac isomyosins V1 and V3 by an in vitro motility assay with regulated thin filaments].

A series of experiments in an in vitro motility assay with reconstructed thin filaments has been performed to determine the dependence of the velocity of thin filament movement on the concentration of calcium in solution (in the pCa range from 5 to 8) for rabbit cardiac isomyosins V1 and V3. The "pCa-velocity" curves had the sigmoid form. It was found for each isoform that sliding velocities of regulated thin filaments (at the saturating calcium concentration (pCa 5)) and actin filaments did not differ from each other. The Hill coefficient was 1.04 and 0.75 for isomyosins V1 and V3, respectively. The calcium sensitivity of V3 was found to be higher than that of V1. In the framework of the same method, the relationship between the velocity of thin filament sliding and the concentration of the actin-binding protein a-actinin (analog of the "force-velocity" relationship) has been estimated for each isoform V1 and V3 at the saturating calcium concentration. The results obtained suggest that the calcium regulation of the contractile activity of isomyosins V1 and V3 occurs by different mechanisms.

[Mathematical models for the study of electromechanical and mechanoelectrical coupling in the myocardium].

Mathematical models have been developed to describe interactions of electrical, mechanical and chemical processes in cardiomyocytes. The models simulate wide range of experimental data on excitation-contraction coupling and, more importantly, on mechanoelectric feedback in heart muscle. The model results clearly show that mechano-dependence of intracellular calcium handling due to cooperative effects of contractile proteins activation plays a key role in cardiac mechanoelectric coupling. At the same time, mechanosensitive currents can also contribute to action potential responses to mechanical perturbations. Using this model to study the heterogeneous myocardium we have shown that temporal and functional electromechanical heterogeneity of coupled cardiomyocytes can essentially determine the myocardium contractility. Optimization of the electromechanical function of contractile system emerges from the fine coordination between the activation sequence of cardiomyocytes, their local electromechanical properties and the mechanical interaction during contraction.

[Mechanisms of electromechanical function disturbances in cardiomyocytes overloaded with calcium. The theoretical study].

Arrhythmias and mechanical disturbances are simulated in a mathematical model of cardiomyocyte electromechanical activity. The simulated pattern is similar to that observed for acute heart failure associated with calcium overloading of myocardium cells. Special attention was paid to the calcium overloading resulting from the reduced Na+,K+ pump activity. In the framework of the model, it was shown that mechanical factors could promote arrhythmia initiation when the pump activity reduced. Different approaches to electrical and mechanical function restoration during acute heart failure associated with calcium overloading were suggested and analyzed in the model.

[Modeling of mechanoelectric coupling in cardiomyocytes in norm and pathology].

We developed mathematical models of the electromechanical function of cardiomyocytes and the simplest mechanically heterogeneous myocardial systems, muscle duplexes. By means of these models we studied the contribution of mechanoelectric feedbacks to the contractile activity of the myocardium in norm and pathology. In particular, we simulated and clarified the effects of mechanical conditions on both the form and the duration of the action potential during contractions. From this standpoint different kinds of myocardium mechanical heterogeneity were analyzed. As we have established, the latter can play both a positive and a negative role, depending on the distribution of mechanical nonuniformity and the sequence of activation of heterogeneous myocardium system elements. By means of the same models, we studied the contribution of mechanical factors to the arrhythmogenicity in the case of the cardiomyocyte calcium overload caused by the attenuation of the sodium-potassium pump and outlined the ways for correcting the contractile function in these disturbances.

[The in vitro motility assay to study the calcium-mechanical relationship in skeletal and cardiac muscles].

In a series of experiments on regulated contractile systems (i.e., in vitro mobile systems with reconstructed thin filaments), the velocities of the movement of a thin filament on the surface covered by either rabbit skeletal or rat cardiac myosin at various concentrations of calcium ions in solution (in the pCa range from 4 to 8) were assessed. The corresponding "pCa-velocity" relationships were plotted, which proved to be of the sigmoid form. It was found that, at a saturating calcium concentration (pCa 4), the velocity of regulated thin filaments was 65% higher than for unregulated ones in the case of skeletal myosin and 87% higher than for unregulated thin filaments in the case of cardiac myosin. It was also found that the Hill coefficient was 1.95 and 2.5 for skeletal and cardiac myosins, respectively. The difference in the Hill coefficients for skeletal and cardiac myosins is discussed in terms of the difference in contribution of cooperativity mechanisms of contractile and regulatory proteins in the regulation of contraction in these types of muscles.

[Mathematical model of a generalized calcium buffer in cardiac muscle cells]. / Matematicheskaia model' obobshchennogo kal'tsievogo bufera v kletakh serdechnoi myshtsy.

The problem of reduction of the earlier model for the calcium buffer system of cardiomyocytes was solved. A simpler model of generalized buffer, was obtained, which reproduces the behavior of the initial system in typical situations. To identify the parameters of the equations of the generalized buffer, recursive algorithms of filtration were developed. The model consists of two equations, which describe separately the fast and slow components of the buffer system. It is shown that the error of the generalized buffer model relative to the initial model of the calcium buffer system does not exceed 10% in a wide range of Ca2+ transients. The introduction of the generalized buffer into the model of mechanochemical coupling provides a more realistic description of a decrease in free calcium concentration. The new model adequately describes all mechanical events simulated by the initial model. In terms of the model, the effect of the slow component of the generalized buffer in a single contraction-relaxation cycle was studied. It was shown that the presence of the slow component leads to a marked decrease in the amplitude of contraction and increase in relaxation time.

[The role of nonspecific troponin in kinetics of intracellular calcium in cardiomyocytes]. / Rol' nespetsificheskogo troponina v kinetike vnutrikletochnogo kal'tsiia v kardiomitsitakh.

According to literature data nonspecific troponin TnC2 binds calcium in cardiac myocytes slowly and does not influence calcium transient. Therefore TnC2 often was not taken into consideration in models of the intracellular calcium kinetics. The mathematical model of the intracellular Ca(2+)-binding system we developed includes TnC2 as a ligand. Rate constants accepted for the complexation of Ca2+ by intracellular ligands give good agreement of the theoretical data with observed physiological ones. We compared the results of numerical experiments on this model with similar results obtained on the other model where TnC2 had been ignored. Some essential differences were revealed. In the model without TnC2 the fraction of calcium binding to specific troponin TnC1 always is smaller than the fraction of Ca2+ bound to the other ligands together. In our model TnC1 fraction (it regulates the mechanical response of muscle) is bigger that means more effective Ca(2+)-binding during activation of contractile proteins. The results above show that within the frame of our model the constants taken for Ca(2+)-binding ligands are more substantiate.

[The contribution of the myocardial segmental nonhomogeneity of the left ventricular walls to its contractile and pumping functions]. / Vklad segmentarnoi neodnorodnosti miokarda stenok levogo zheludochka v ego sokratitel'nuiu i nasosnuiu funktsii.

A normalised variation coefficient of the partial systolic fraction served as the measure of nonhomogeneity of the segmentary kinetics of the left ventricle's wall (parameter J). A significant correlation was found between the parameter J and the fraction of the ventricle output (the correlation being negative one) both in normal subjects and in patients with cardiac pathology. The parameter J was also found to be a sensitive index of the heart pumping and contractile functions. The local cardiotopodynamics as expressed via the segmentary nonhomogeneity seems to be able to contribute much into the regulation or modulation of the heart pumping function.