J Mol Cell Cardio124,
Theoretical
97-104 (1992)
Model and Computer Simulation of lkitatio~ontraction Coupling of Mammalian Cardiac Muscle Anushka
P. Michailova
and Velin
Z. Spassov
Central Laboratory of Biophysics, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria (Received 8 October 1990, accepted in revisedform 16 September 1991) A. P. MICHAILOVA AND V. 2. SPASSOV. Theoretical Model and Computer Simulation of Excitation-contraction Coupling of Mammalian Cardiac Muscle. Journal ofMolccularnnd Cellular Cardioloo (1992) 24, 97-104. A mathematical model is developed to investigate the kinetics of electrical, mechanical and molecular processes in mammalian cardiac muscle. Isometric contractions at different muscle length and frequency of stimulation in response to a rhythmically applied clamp pulse or artificial action potential are simulated. Numerical results show that concentration of Ca2+ ions, bound to Ca2 + -specific sites on protein troponin C, could be a regulatory factor in actin-myosin interactions and subsequent production of force in Huxley’s mathematical approach for the sliding mechanism. The behavior of the model is compared to that of living cardiac muscle. KEY WORDS: Excitation-contraction matical approach for the sliding
coupling; Mammalian cardiac mechanism; Regulatory proteins.
Introduction The impressive progress in the field of molecular biology of isolated muscle proteins (Chapman, 1979; Holroyde et al., 1980), experimental data for the kinetics of Ca*+ transients obtained with the photoprotein aequorin and the recordings of action potential and tension in cardiac muscle cells (Morgan and Blinks, 1982; Morgan et al., 1984; Gwathmey et al., 1987, 1988) offer new physiological and pharmacological ideas and opportunities for
muscle; Computer
model;
Huxley’s
mathe-
deeper understanding of excitation-contraction coupling. A number of mathematical models also have been suggested to investigate the complex relationship between electrical, molecular and mechanical processes in cardiac muscle cells (Wong, 1971, 1981; Kaufmann et al., 1974; Shumakov et al., 1978; Robertson et al., 1981; Adler et al., 1985; Regirer et al., 1986; Backx et al., 1989). These models are able to explain, confirm or refute some of the physiological and
Abbrmiafions: Notations of the parameters and constants used throughout the study: S sarcolemma; SP sarcoplasm; SR sarcoplasmic reticulum; CSR cisternal sarcoplasmic reticulum; LSR longitudinal sarcoplasmic reticulum; MF myofibril; AC active center on actin filament; TNC troponin C; TN1 troponin I; TNT troponin T; TM tropomyosin; PE parallel elastic element; SE series elastic element; CE contractile element; 1 time after stimulus; T,, time from onset of contraction to peak isometric muscle tension; Xc,, Ca*+ concentration in CSR; XLsR Ca*’ concentration in LSR; Xsp Ca2* concentration in SP; Xsp normalized Xsp; pcSR initial Ca2+ concentration in f+ CSR; X&R initial Ca’+ concentration in LSR; psp initial Ca m SP; y(t) concentration of Ca’+ ions, bound to Ca’+-specific type sites on TNC; 7” normalized -y(t); ki velocity constants (i = 1,7); k;f,, k&CCa’+ on- and off-rate velocity constants; trop free troponin concentration; I,, voltage dependent slow inward Ca2’ current; E(f) action potential or clamp pulse; E, reversal membrane potential; d(E(t)), f(E(f)) the time courses of activation and inactivation; d,, f, Pisometric muscle tension; Pn normalized P; steady-state values of d(E(f)) and@(i)); gca maximum Ca*+ conductance; PCE tension in CE; PSE tension in SE; PL preIoad; k, constant characterizing SE; PPE tensions in PE; PO, kp constants characterizing PE; L initial muscle length; L, resting muscle length at zero preload; L,, muscle length at which the developed peak tension is maximum; AL extension of series elastic element; R empirical constant; k, stiffness of crossbridge; n(u,t) distribution of actin-myosin bonds; A, Blimits at which the value oFn(u,l) is significant; I distance of actin from the equilibrium position of myosin; h largest displacement at which cross-bridge can become attached to an actin-filament;j, g velocity constants;fi(l), g,(L) length-dependent functions; VP plateau potential; V, resting potential; V, overshoot; V,,, membrane voltage; T duration of action potential or clamp pulse; V, potential when f = T; a. b. c constants. Please
physics,
address
Bulgarian
0022-2828/92/010097
all correspondence
Academy
to: Division of Mathematical Modeling in Biology, Central of Sciences, Acad. G. Bonchev str., block 21, 1113 Sofia, Bulgaria.
+ 9 $03.00/O
Laboratory
0 1992 Academic
of Bio-
Press Limited
A. P. Miehailova
98
and V. Z. Spassov
pharmacological hypotheses, and simulate different contractile events. The main purpose of this study was to confirm theoretically the idea that the concentration of Ca*+ ions, bound to Ca*+-specific sites on troponin C (TNC*), could be an important regulatory factor in actin-myosin interactions and subsequent production of force in Huxley’s mathematical approach for the sliding mechanism (Huxley, 1957; Robertson et al., 1981).
Materials
SP
and Methods
Basic physiological postulates of the model
MF
In this model it is suggested that upon depolarization, Ca2+ current not only raises the sarcoplasmic Ca* + concentration, but also induces the release of Ca2+ from cisternal sarcoplasmic reticulum (CSR), whose rate of release is a function of an action potential [E(t)]. These two main sources of Ca*+ increase sarcoplasmic Ca*+ concentration. Free sarcoplasmic Ca*+ binds to Ca2+-specific sites on TNC, causing a conformational change in the structure of the protein. This signal is transmitted across actin filament through troponin I (TNI) to troponin T (TNT) and tropomyosin (TM), inducing the formation of actomyosin crossbridges and the generation of force in muscle fiber. The uptake and recycling of Ca2+ to CSR is accomplished by longitudinal sarcoplasmic reticulum (LSR) and permits consequent relaxation (Backx et al., 1989; Holroyde et al., 1980; Robertson et al., 1981; Wong, 1981) (Fig. 1). The analysis by Robertson et al. (1981) suggests that large rapid changes in the amount of Ca2 + bound to Ca* + -specific sites of TNC are induced by each transient increase in free sarcoplasmic Ca*+ concentration. Thus, they concluded that only sites of Ca* + -specific type can act as rapid Ca* + -regulatory sites in muscle. In contrast, in their paper it is shown that large changes in Ca*+ occupancy of Ca2+ -Mg2 + type sites on TNC can occur with repeated stimulation. Thus, they assumed that Ca* + content in these sites is a measure of the intensity and frequency of recent muscle activity. In the framework of our model, the binding and exchange of Ca*+ and Mg*+ with *See
‘Nomenclature’
for the notations
of the parameters
FIGURE 1. Schematic diagram of excitation-contraction model. S, sarcolemma; SP, sarcoplasm; CSR, cisternal sarcoplasmic reticulum; LSR, longitudinal sarcoplasmic reticulum; MF, myofibril; AC, active center on actin filament. See text and ‘Nomenclature’ for details.
Ca* + -Mg2+ regulatory sites on TNC, calmodulin, parvalbumin and myosin, the binding and exchange of Ca* + with Ca* + -specific regulatory sites on calmodulin and intracellular free Mg2+ are ignored, because, according to the results of Robertson et al. (1981), these metal binding type sites play an indirect role in Ca* + regulation of contraction (Fig. 1). Most of the studies on cardiac excitation-contraction coupling deal mainly with Ca*+ sources and influx (Julian, 1969; Wong, 1971, 1981; Kaufmann et al., 1974; Stein and Wong, 1974; Shumakov et al., 1978; Robertson et al., 1981; Regirer et al., 1986; Stein et al., 1988; Backx et al., 1989). In our model Na+-Ca*+ exchange mechanism, which is thought to be responsible for extrusion of Ca*+ ions out of the cell, is also ignored. We use the assumptions of Wong (1981) for Ca*+ efflux (Fig. 1). Kaufmann (1978), Chapman (1979) and Wong (1981) suppose that the contribution of cardiac mitochondria to sarcoplasmic Ca2+ concentrations (less than 2OOpM), and passive diffusion across external membrane could be negligible. On these grounds, we did not include mitochondria and passive diffusion in the framework of our study (Fig. 1). and
constants
used
throughout
the study.
Modeling
Cardiac
Jhitation-Contraction
Mathematical formulation Wong (1971) modified the activation function for skeletal muscle of Julian (1969) in Huxley’s mathematical approach for the sliding mechanism, having in view slower onset of active state and longer duration of E(t) in mammalian cardiac cells. Stain and Wong (1974), using first or second order kinetics for binding and dissociation of Ca2+ to the protein troponin, showed that this activation function for skeletal muscle could be derived from more realistic assumptions about Ca2+ kinetics. Numerical analysis of Robertson et al. (1981) suggested that only sites of Ca’+-specific type on TNC can act as rapid Ca*+-regulatory sites in cardiac muscle cells, but in their study it is not shown how the time course of Ca2+ binding to TNC correlates with force development. To simulate force, developed by a contractile element in Hill’s model, Shumakov et al. (1978) assumed that Ca2+-troponin concentration is directly proportional to contractile force. Wong (1981) assumed that the activation function for cardiac muscle in Huxley’s mathematical approach for the sliding mechanism could be the concentration of free sarcoplasmic Ca2+. In our study we suppose that the activation function (y(t)) in Huxley’s for mathematical approach the sliding mechanism (Eqn 14) could be the concentration of Ca2+ ions bound to Ca2+-specific sites on TNC, deriving y(t) from second order kinetics (Eqn 4). We describe the kinetics of Ca2+ ions in mammalian cardiac muscle cells (Eqns l-5)* (Fig. 1).
99
Coupling
Details for the kinetics of Ca2+ ions in CSR, LSR, sarcoplasm (SP) and a voltage-dependent slow inward Ca*+ current (Zca) (Eqns l-3, 5) are described by Wong (1981). In our study, mechanical properties of living cardiac muscle cells are included by Hill’s three-component model (Fig. 2). We used the assumptions of Wong (1971) for isometric muscle tension (p), series and parallel elastic tensions (PsE, PPE), (Eqns 6-9). Contractile tension (Z’& is obtained, using Huxley’s mathematical approach for the sliding mechanism (Wong, 1971) (Eqns 10-15).
P
FIGURE 2. Hill’s three-component model: lel elastic element; SE, series elastic element; tractile element.
PE, paralCE, con-
P=PSE+PpE
dY,,,ldt = - k,da, (Xs, - X“SP)XCSR + k& VLSR - PLSR)-k5d=&XR -k6(X~~~ -x&R,>
(6)
PCE=PSE
(7)
PPE =~,(ex~(k~(~-M-l)
(8)
PSE= PL(exp(k$. AL)-1)
(9)
PCE= AsBk,.n(u,t).u du
(10)
(1)
6n(u,t)/& =Al-n)-gn
(11)
where:
dXLsR/dt=(kz+k~~)(Xs~-X~~) -~&@LsR -pLSR)
C-3
u = Nh ~spldt=k,~c,+k,dmtXsp-X~P)XcSR - & +~&)VSP -X8,>
- dr(W
Xsp - k&y(t)
(4)
Zca= gca.d@(t)) .AE(t)). (E(t)-Ed
(5)
dyldt = k&(trop-y(t))
‘See ‘Nomenclature’
for the notations
AcucO
(3)
of the parameters
0
Theoretical
97-104 (1992)
Model and Computer Simulation of lkitatio~ontraction Coupling of Mammalian Cardiac Muscle Anushka
P. Michailova
and Velin
Z. Spassov
Central Laboratory of Biophysics, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria (Received 8 October 1990, accepted in revisedform 16 September 1991) A. P. MICHAILOVA AND V. 2. SPASSOV. Theoretical Model and Computer Simulation of Excitation-contraction Coupling of Mammalian Cardiac Muscle. Journal ofMolccularnnd Cellular Cardioloo (1992) 24, 97-104. A mathematical model is developed to investigate the kinetics of electrical, mechanical and molecular processes in mammalian cardiac muscle. Isometric contractions at different muscle length and frequency of stimulation in response to a rhythmically applied clamp pulse or artificial action potential are simulated. Numerical results show that concentration of Ca2+ ions, bound to Ca2 + -specific sites on protein troponin C, could be a regulatory factor in actin-myosin interactions and subsequent production of force in Huxley’s mathematical approach for the sliding mechanism. The behavior of the model is compared to that of living cardiac muscle. KEY WORDS: Excitation-contraction matical approach for the sliding
coupling; Mammalian cardiac mechanism; Regulatory proteins.
Introduction The impressive progress in the field of molecular biology of isolated muscle proteins (Chapman, 1979; Holroyde et al., 1980), experimental data for the kinetics of Ca*+ transients obtained with the photoprotein aequorin and the recordings of action potential and tension in cardiac muscle cells (Morgan and Blinks, 1982; Morgan et al., 1984; Gwathmey et al., 1987, 1988) offer new physiological and pharmacological ideas and opportunities for
muscle; Computer
model;
Huxley’s
mathe-
deeper understanding of excitation-contraction coupling. A number of mathematical models also have been suggested to investigate the complex relationship between electrical, molecular and mechanical processes in cardiac muscle cells (Wong, 1971, 1981; Kaufmann et al., 1974; Shumakov et al., 1978; Robertson et al., 1981; Adler et al., 1985; Regirer et al., 1986; Backx et al., 1989). These models are able to explain, confirm or refute some of the physiological and
Abbrmiafions: Notations of the parameters and constants used throughout the study: S sarcolemma; SP sarcoplasm; SR sarcoplasmic reticulum; CSR cisternal sarcoplasmic reticulum; LSR longitudinal sarcoplasmic reticulum; MF myofibril; AC active center on actin filament; TNC troponin C; TN1 troponin I; TNT troponin T; TM tropomyosin; PE parallel elastic element; SE series elastic element; CE contractile element; 1 time after stimulus; T,, time from onset of contraction to peak isometric muscle tension; Xc,, Ca*+ concentration in CSR; XLsR Ca*’ concentration in LSR; Xsp Ca2* concentration in SP; Xsp normalized Xsp; pcSR initial Ca2+ concentration in f+ CSR; X&R initial Ca’+ concentration in LSR; psp initial Ca m SP; y(t) concentration of Ca’+ ions, bound to Ca’+-specific type sites on TNC; 7” normalized -y(t); ki velocity constants (i = 1,7); k;f,, k&CCa’+ on- and off-rate velocity constants; trop free troponin concentration; I,, voltage dependent slow inward Ca2’ current; E(f) action potential or clamp pulse; E, reversal membrane potential; d(E(t)), f(E(f)) the time courses of activation and inactivation; d,, f, Pisometric muscle tension; Pn normalized P; steady-state values of d(E(f)) and@(i)); gca maximum Ca*+ conductance; PCE tension in CE; PSE tension in SE; PL preIoad; k, constant characterizing SE; PPE tensions in PE; PO, kp constants characterizing PE; L initial muscle length; L, resting muscle length at zero preload; L,, muscle length at which the developed peak tension is maximum; AL extension of series elastic element; R empirical constant; k, stiffness of crossbridge; n(u,t) distribution of actin-myosin bonds; A, Blimits at which the value oFn(u,l) is significant; I distance of actin from the equilibrium position of myosin; h largest displacement at which cross-bridge can become attached to an actin-filament;j, g velocity constants;fi(l), g,(L) length-dependent functions; VP plateau potential; V, resting potential; V, overshoot; V,,, membrane voltage; T duration of action potential or clamp pulse; V, potential when f = T; a. b. c constants. Please
physics,
address
Bulgarian
0022-2828/92/010097
all correspondence
Academy
to: Division of Mathematical Modeling in Biology, Central of Sciences, Acad. G. Bonchev str., block 21, 1113 Sofia, Bulgaria.
+ 9 $03.00/O
Laboratory
0 1992 Academic
of Bio-
Press Limited
A. P. Miehailova
98
and V. Z. Spassov
pharmacological hypotheses, and simulate different contractile events. The main purpose of this study was to confirm theoretically the idea that the concentration of Ca*+ ions, bound to Ca*+-specific sites on troponin C (TNC*), could be an important regulatory factor in actin-myosin interactions and subsequent production of force in Huxley’s mathematical approach for the sliding mechanism (Huxley, 1957; Robertson et al., 1981).
Materials
SP
and Methods
Basic physiological postulates of the model
MF
In this model it is suggested that upon depolarization, Ca2+ current not only raises the sarcoplasmic Ca* + concentration, but also induces the release of Ca2+ from cisternal sarcoplasmic reticulum (CSR), whose rate of release is a function of an action potential [E(t)]. These two main sources of Ca*+ increase sarcoplasmic Ca*+ concentration. Free sarcoplasmic Ca*+ binds to Ca2+-specific sites on TNC, causing a conformational change in the structure of the protein. This signal is transmitted across actin filament through troponin I (TNI) to troponin T (TNT) and tropomyosin (TM), inducing the formation of actomyosin crossbridges and the generation of force in muscle fiber. The uptake and recycling of Ca2+ to CSR is accomplished by longitudinal sarcoplasmic reticulum (LSR) and permits consequent relaxation (Backx et al., 1989; Holroyde et al., 1980; Robertson et al., 1981; Wong, 1981) (Fig. 1). The analysis by Robertson et al. (1981) suggests that large rapid changes in the amount of Ca2 + bound to Ca* + -specific sites of TNC are induced by each transient increase in free sarcoplasmic Ca*+ concentration. Thus, they concluded that only sites of Ca* + -specific type can act as rapid Ca* + -regulatory sites in muscle. In contrast, in their paper it is shown that large changes in Ca*+ occupancy of Ca2+ -Mg2 + type sites on TNC can occur with repeated stimulation. Thus, they assumed that Ca* + content in these sites is a measure of the intensity and frequency of recent muscle activity. In the framework of our model, the binding and exchange of Ca*+ and Mg*+ with *See
‘Nomenclature’
for the notations
of the parameters
FIGURE 1. Schematic diagram of excitation-contraction model. S, sarcolemma; SP, sarcoplasm; CSR, cisternal sarcoplasmic reticulum; LSR, longitudinal sarcoplasmic reticulum; MF, myofibril; AC, active center on actin filament. See text and ‘Nomenclature’ for details.
Ca* + -Mg2+ regulatory sites on TNC, calmodulin, parvalbumin and myosin, the binding and exchange of Ca* + with Ca* + -specific regulatory sites on calmodulin and intracellular free Mg2+ are ignored, because, according to the results of Robertson et al. (1981), these metal binding type sites play an indirect role in Ca* + regulation of contraction (Fig. 1). Most of the studies on cardiac excitation-contraction coupling deal mainly with Ca*+ sources and influx (Julian, 1969; Wong, 1971, 1981; Kaufmann et al., 1974; Stein and Wong, 1974; Shumakov et al., 1978; Robertson et al., 1981; Regirer et al., 1986; Stein et al., 1988; Backx et al., 1989). In our model Na+-Ca*+ exchange mechanism, which is thought to be responsible for extrusion of Ca*+ ions out of the cell, is also ignored. We use the assumptions of Wong (1981) for Ca*+ efflux (Fig. 1). Kaufmann (1978), Chapman (1979) and Wong (1981) suppose that the contribution of cardiac mitochondria to sarcoplasmic Ca2+ concentrations (less than 2OOpM), and passive diffusion across external membrane could be negligible. On these grounds, we did not include mitochondria and passive diffusion in the framework of our study (Fig. 1). and
constants
used
throughout
the study.
Modeling
Cardiac
Jhitation-Contraction
Mathematical formulation Wong (1971) modified the activation function for skeletal muscle of Julian (1969) in Huxley’s mathematical approach for the sliding mechanism, having in view slower onset of active state and longer duration of E(t) in mammalian cardiac cells. Stain and Wong (1974), using first or second order kinetics for binding and dissociation of Ca2+ to the protein troponin, showed that this activation function for skeletal muscle could be derived from more realistic assumptions about Ca2+ kinetics. Numerical analysis of Robertson et al. (1981) suggested that only sites of Ca’+-specific type on TNC can act as rapid Ca*+-regulatory sites in cardiac muscle cells, but in their study it is not shown how the time course of Ca2+ binding to TNC correlates with force development. To simulate force, developed by a contractile element in Hill’s model, Shumakov et al. (1978) assumed that Ca2+-troponin concentration is directly proportional to contractile force. Wong (1981) assumed that the activation function for cardiac muscle in Huxley’s mathematical approach for the sliding mechanism could be the concentration of free sarcoplasmic Ca2+. In our study we suppose that the activation function (y(t)) in Huxley’s for mathematical approach the sliding mechanism (Eqn 14) could be the concentration of Ca2+ ions bound to Ca2+-specific sites on TNC, deriving y(t) from second order kinetics (Eqn 4). We describe the kinetics of Ca2+ ions in mammalian cardiac muscle cells (Eqns l-5)* (Fig. 1).
99
Coupling
Details for the kinetics of Ca2+ ions in CSR, LSR, sarcoplasm (SP) and a voltage-dependent slow inward Ca*+ current (Zca) (Eqns l-3, 5) are described by Wong (1981). In our study, mechanical properties of living cardiac muscle cells are included by Hill’s three-component model (Fig. 2). We used the assumptions of Wong (1971) for isometric muscle tension (p), series and parallel elastic tensions (PsE, PPE), (Eqns 6-9). Contractile tension (Z’& is obtained, using Huxley’s mathematical approach for the sliding mechanism (Wong, 1971) (Eqns 10-15).
P
FIGURE 2. Hill’s three-component model: lel elastic element; SE, series elastic element; tractile element.
PE, paralCE, con-
P=PSE+PpE
dY,,,ldt = - k,da, (Xs, - X“SP)XCSR + k& VLSR - PLSR)-k5d=&XR -k6(X~~~ -x&R,>
(6)
PCE=PSE
(7)
PPE =~,(ex~(k~(~-M-l)
(8)
PSE= PL(exp(k$. AL)-1)
(9)
PCE= AsBk,.n(u,t).u du
(10)
(1)
6n(u,t)/& =Al-n)-gn
(11)
where:
dXLsR/dt=(kz+k~~)(Xs~-X~~) -~&@LsR -pLSR)
C-3
u = Nh ~spldt=k,~c,+k,dmtXsp-X~P)XcSR - & +~&)VSP -X8,>
- dr(W
Xsp - k&y(t)
(4)
Zca= gca.d@(t)) .AE(t)). (E(t)-Ed
(5)
dyldt = k&(trop-y(t))
‘See ‘Nomenclature’
for the notations
AcucO
(3)
of the parameters
0
