Multiphysics model of a rat ventricular myocyte: A voltage-clamp study
© Krishna et al; licensee BioMed Central Ltd. 2012
Received: 23 May 2012
Accepted: 27 July 2012
Published: 21 November 2012
The objective of this study is to develop a comprehensive model of the electromechanical behavior of the rat ventricular myocyte to investigate the various factors influencing its contractile response.
Here, we couple a model of C a2 + dynamics described in our previous work, with a well-known model of contractile mechanics developed by Rice, Wang, Bers and de Tombe to develop a composite multiphysics model of excitation-contraction coupling. This comprehensive cell model is studied under voltage clamp (VC) conditions, since it allows to focus our study on the elaborate C a2 + signaling system that controls the contractile mechanism.
We examine the role of various factors influencing cellular contractile response. In particular, direct factors such as the amount of activator C a2 + available to trigger contraction and the type of mechanical load applied (resulting in isosarcometric, isometric or unloaded contraction) are investigated. We also study the impact of temperature (22 to 38°C) on myofilament contractile response. The critical role of myofilament C a2 + sensitivity in modulating developed force is likewise studied, as is the indirect coupling of intracellular contractile mechanism with the plasma membrane via the N a+ /C a2 + exchanger (NCX). Finally, we demonstrate a key linear relationship between the rate of contraction and relaxation, which is shown here to be intrinsically coupled over the full range of physiological perturbations.
Extensive testing of the composite model elucidates the importance of various direct and indirect modulatory influences on cellular twitch response with wide agreement with measured data on all accounts. Thus, the model provides mechanistic insights into whole-cell responses to a wide variety of testing approaches used in studies of cardiac myofilament contractility that have appeared in the literature over the past several decades.
Cardiac muscle contraction is a result of a transient increase in myoplasmic C a2 + concentration C a2 + myo . Sarcolemmal (SL) membrane depolarization triggers C a2 + influx via dihydropyridine (DHP)-sensitive L-type C a2 + channels. Following diffusion across a small sub-membrane dyadic space, this influx activates ryanodine receptors (RyRs) controlling ryanodine-sensitive C a2 + release channels in the junctional portion of the sarcoplasmic reticulum (jSR). Fabiato and Fabiato  named the process calcium-induced calcium release (CICR). C a2 + subsequently diffuses from the dyadic space into the myoplasm. Ultimately, myoplasmic C a2 + concentration C a2 + myo is returned to resting levels by combination of: (a) C a2 + buffering in the dyadic space and myoplasm; (b) sequestration of C a2 + y by sarcoplasmic/endoplasmic reticulum C a2 + -ATPase (SERCA)-type calcium pumps lining the longitudinal portion of the sarcoplasmic reticulum (LSR); and (c) C a2 + extrusion from the myoplasm by N a+ /C a2 + exchangers and C a2 + -ATPase pumps on the sarcolemmal membrane.
C a2 + is an extremely important and highly versatile second messenger in cardiac cells, which plays a crucial role not only in excitation-contraction (E-C) coupling but also in excitation-transcription coupling . Various inter-connected C a2 + signalling pathways help preserve the integrity of the cellular C a2 + system despite any disturbances (e.g., changes in stimulation frequency or inotropic state). A key role for the dyadic C a2 + release system is the formation of an adequate myoplasmic C a2 + transient that can serve as an input driving signal for the intracellular contractile machinery (the myofilaments). The resultant contractile response is conditioned by a number of additional factors that include the mechanical load; sarcomere equilibrium length; myofilament C a2 + sensitivity; and the temperature. Although it is well-known that the contractile response of a cell is a function of the stimulation frequency (its force-frequency response (FFR)), this study is limited to an investigation at 5 Hz (unless otherwise specified), a physiologically relevant heart rate for a rat ventricular myocyte.
Our objective was to develop an integrated model of the rat ventricular cell under voltage clamp conditions, which includes the description of various C a2 + signalling pathways in the dyadic space, the myoplasmic medium and the sarcoplasmic reticulum (Figure 1, adopted from Krishna et al. ) as well as a comprehensive coupled mechanical system describing the contractile machinery responsible for force generation.
Electrochemical description of C a 2 + sub-system
Q 10 values used to model temperature variation
Mechanical description of myofilament contractile system
We employ our coupled multiphysics model describing the electrochemical as well as the mechanical subsystems to study cellular contraction emphasizing the various modulatory influences that are at play. We begin with a detailed analysis of different types of twitch response to better understand the influence of various factors such as sarcomere length, peak C a2 + myo and the stiffness of the contractile element in cell shortening, followed by a comparative study of these twitch responses. The negative feedback of cellular contraction on the myoplasmic C a2 + -transient [19, 20] is also investigated. We then perform an idealized virtual experiment similar to that carried out in an experimental study  to uncover the regulation of cell contraction by N a+ /C a2 + exchange, and in the process identifying the role of the SERCA pump in facilitating this effect. We then model the effects of temperature  on cardiac contractile response. This is followed by a study identifying the role of myofilament C a2 + sensitivity as a key factor influencing the degree of cell shortening. In particular, the effect of β-adrenergic agonist isoproterenol, which is known to cause a decrease in myofilament C a2 + sensitivity , is investigated. Hence, we have developed an integrated multiphysics model of rat ventricular cell electromechanics and now seek to study its response to various tests prescribed by elements of the virtual protocol above. In doing so, we hope to identify and clarify the role played by key factors involved in modulation of the cell’s contractile response.
From our modeling standpoint, the dyadic coupling unit (DCU) as defined by Krishna et al.  is a fundamental element involved in the mechanism of CICR. This previous study described the control features of this unit, as well as its interaction with the SERCA pump and free sarcolemmal pumps and exchangers to achieve a homeostatic regulation of myoplasmic C a2 + concentration. We now extend our voltage clamp studies to address the subject of force generation following CICR, starting with the classical twitch responses below.
A twitch response, which is a brief contractile response of a cardiac cell elicited by dynamically changing activator C a2 + , is a commonly used experimental characterization. Following are the three types of recordings commonly used to quantify force generation in isolated cardiac cells: (1) isosarcometric contraction which is experimentally obtained by incorporating feedback sarcomere length (SL) control using laser diffraction techniques [23–25]; (2) unloaded contraction where the cell is allowed to contract freely; and (3) isometric contraction where overall muscle length is fixed, but sarcomere length is not controlled, allowing considerable internal shortening as a result of compliant end connections (series elastic element). All the twitch studies are carried out at 22.5°C  to be consistent with the data cited. Under all three loading conditions, the cell is subjected to conditioning train of steady state (100 cycles) voltage clamp pulses at 5 Hz followed by a 0.8 s rest interval (allowing decay to a zero force resting state) which is subsequently followed by a single test twitch for which a transient lasting 1 s is obtained. The voltage clamp protocol used is a 50 ms step pulse to 10 mv from a holding potential of -40 mv. In this study, we report a normalized force with a maximum value of 1 possible under optimal conditions such as high C a2 + myo , isosarcometric loading (SL = 2.3 μ m) and low temperature allowing maximum overlap of thick and thin filaments.
The steady state force-C a 2 + (F-Ca) relationship shown in Figure 3A-i exhibits a leftward shift and an increase in developed maximum plateau force as SL is clamped at increasing lengths. This leftward shift results from an increase in myofilament C a 2 + sensitivity as SL is increased. Figure 3A-ii shows the temporal course of normalized force as SL is changed in steps from 1.8 to 2.3 μ m. The waveshape of standard C a 2 + -transient is overlaid in dotted lines in this figure. Although an increase in SL (traces marked + to ∗) does not cause a large variation in the time to peak force (TTP), it does result in an increase in peak force magnitude and twitch duration as the result of an increase in myofilament C a 2 + sensitivity. These characteristics show a strong correspondence with measured data from rat ventricular myocytes tested at similar (∼ 22.5°C) temperatures [14, 24, 26]. The correlation coefficient of the speed of contraction and relaxation has been experimentally observed  to be very close (> 0.98). The inset in Figure 3A-ii is a plot of the rate of relaxation (reciprocal of time taken for 50% sarcomere relaxation (RT50)) versus rate of contraction (reciprocal of time taken for peak sarcomere contraction (TTP)) for increasing SL. This linear relationship highlights contraction-relaxation coupling, and represents a key intrinsic property of the contractile myofilaments . Figure 3A-iii shows the phase plots of self normalized force versus the instantaneous C a 2 + concentration in the cytosol for increasing SL (traces marked + to ∗) overlayed with two steady state F-Ca relationships corresponding to SL = 1.8 μ m (+) and SL = 2.3 μ m (∗). The assessment of dynamic and steady-state C a 2 + relationships allows better analysis of the phase-plane loops of force versus C a 2 + . The active twitch curve is related to the steady-state values to determine, at what isochrone the dynamic force-C a 2 + value equals that obtained in the steady-state relationship. This point of intersection of the steady state F-Ca trace and the corresponding phase plot gives the contraction-relaxation coupling point (CRCP, marked as ∘) from initiation of stimulation . Time is implicit on the phase trajectory and at time instants prior to reaching the critical coupling point for a particular trajectory, C a 2 + myo exceeds the value of C a 2 + predicted by the steady state F-Ca relationship. This excess favors continued sarcomere contraction. At later time points beyond the CRCP, the developed force is greater than that predicted by the steady state curve, which favors myofilament relaxation.
Increasing background C a 2 + myo causes a leftward shift in steady state F-SL relationship as shown in Figure 3B-i. The increase in maximal plateau force with increase in background C a 2 + myo is observed to be less prominent at higher levels of activator C a 2 + in the myoplasm. In Figure 3B-ii the activator C a 2 + is varied by modulating the peak of the C a 2 + myo transient by adjusting the voltage clamp pulse duration (an increase in pulse duration from 5 ms to 50 ms increased peak C a 2 + myo from 0.5 to 1.1 μ M respectively). This protocol allows for the peak of the transient to be changed without a significant change in the duration of the transient (Krishna et al.  ; Figure 4). The traces correspond to increasing peak values from 0.5 (∙) to 1.1 (∗) μ M. Although similar to case with increasing SL, increasing activator C a 2 + results in a relatively non-linear increase in peak force generated. As shown in Figure 3A-i, the steady state F-Ca relationship is characterized by a Hill function as experimentally observed . The time-to-peak force (TTP) remains relatively unaffected by the amount of activator C a 2 + causing the twitch response. Inset in Figure 3B-ii shows the dependence of TD50 (time taken from 50% activation to 50% relaxation) on peak C a 2 + myo indicating an increase in twitch duration with increasing levels of activator C a 2 + . Figure 3B-iii shows the phase plots of self normalized force versus the instantaneous C a 2 + concentration in the myoplasm for increasing peak C a 2 + myo (traces marked ∙ to ∗) overlayed with a steady state F-Ca relationships corresponding to SL = 2.3 μ m (∗). The contraction-relaxation coupling point (∘) traverses along the F-Ca relationship to increasing values of C a 2 + and force with increasing peak C a 2 + myo .
In the case of an isosarcometric contraction the net force comprises only the active component due to the tension generated by the sarcomere trying to contract. Phase ‘a’ in Figure 7B-i is indicative of the delay in the contractile response when compared to the [C a2 + ] myo transient. However, soon after the [C a2 + ] myo reaches its peak (∙), the sarcomere begins to contract, resulting in a gradual increase in force, achieving a maximum (■) as seen in phase ‘b’ of Figure 7B-i. As active force increases, a decline in level of activator C a2 + in the cytosol ultimately causes a recovery to the minimum contractile state (▸), as shown in phase ‘c’ of Figure 7B-i.
Total force generated by an unloaded cell during contraction against its internal restoring force is a combination of the active component attributed to tension generating action of cycling crossbridges and the passive component attributed to titin and other cytoskeletal elements. Passive force generated has a negative contribution to the net force for SL values lower than the equilibrium length. Hence, the competition between the active and passive components of force gives the trace in Figure 7B-ii its characteristics. During phase ‘a’ in Figure 7B-ii, as the C a2 + level in the cytosol increases towards its peak value (∙), the sarcomere attains maximum contraction velocity (▾), which is followed by a steep increase in net force as shown in phase ‘b’ of Figure 7B-ii. However, decreasing sarcomere length increases the negative contribution from the passive component of force resulting in the first transient decline shown in phase ‘c’ of Figure 7B-ii. The increased C a2 + binding to troponin in response to the rise in activator C a2 + in the cytosol enhances the active component of force, allowing the cell to reach peak net contractile force (■). The fast decline in sarcomere length which causes a rapid increase in the passive component of force results in the second transient decline as shown in phase ‘d’ of Figure 7B-ii. However, a subsequent decrease in cell shortening soon allows an increase in net force lasting for a short duration (while the SL reaches its minimum (♦)). The declining C a2 + level in the cytosol, an outcome of SR uptake, then causes a gradual recovery to the resting state (▸).
As shown in Figure 7A, for the same activator C a2 + , although the total muscle force generated during isometric contraction exhibits a triphasic response similar to an isosarcometric contraction, its own unique characteristics are a delayed time to peak (an increase from 84.5 ms to 119 ms) and relatively smaller magnitude. As shown in phase ‘a’ of Figure 7B-iii, the increase in C a2 + concentration is not reflected in a fast mechanical response. After [C a2 + ] myo reaches its peak (∙), the tension in the sarcomere begins to build up (phase ‘b’ of Figure 7B-iii) although at a slower rate (compare isometric and isosarcometric traces in Figure 7A), causing a delayed time to peak (84 ms and 120 ms in isosarcometric and isometric cases, respectively) due to the presence of the series elastic element which facilitates slow internal shortening. The sarcomere achieves peak contraction (♦) when the total muscle force reaches its maximum (■), following which the cell recovers back to the minimum contractile state (▸) as shown in phase ‘c’, Figure 7B-iii. During isometric contraction (KSE = 2) the afterload (due to the series elastic element) tracks the active component of force generated due to the tension developed in the sarcomere while the passive component of force (attributed to titin and other cytoskeletal elements) is small in magnitude owing to a much smaller degree of sarcomere contraction achieved when compared to the unloaded case (compare trace marked ∗ in Figure 5B and the trace for KSE=2 in Figure 5D).
Effect of contraction on the [C a 2 + ] myo transient
Regulation of isometric cell shortening by N a + / C a 2 + exchange
Effect of temperature on contractile performance
Although skinned rat ventricular preparations have been used extensively in studies of the cardiac contractile process, data from such preparations violates the assumptions of our whole cell electromechanical model. The integrity of the plasma membrane components and intricate C a2 + -regulatory system are compromised to some degree, regardless of the skinning technique used. Therefore, we have chosen to consider only data from rat ventricular myocytes or ultra-thin rat ventricular trabeculae to help validate the model.
The decrease in integrated I Ca,L and faster SR uptake with increasing temperature together cause a decline in peak and duration of the C a2 + -transient as shown in Figure 10B. This agrees with the experimental findings by Janssen et al. (; Figure 1) on thin rat ventricular trabeculae. RT50,C corresponding to the C a2 + -transient (time taken for 50% decline in C a2 + myo ) decreased from 22.8 ms at 22°C to 18.5 ms at 38°C (inset in Figure 10B). Myofilament C a2 + sensitivity increases with an increase in temperature as a result of a temperature dependent enhancement in crossbridge cycling rate [34–36]. Traces for the steady state F-Ca relationship in Figure 10C show a temperature dependent increase in myofilament C a2 + sensitivity with no significant change in maximum plateau force. An increase in temperature from 22°C (+) to 38°C (∗) results in a decrease in EC50 from 0.55 μ M to 0.44 μ M (inset in Figure 10C).
As experimentally observed , a decrease in peak and duration of the C a2 + myo -transient with increasing temperature results in a corresponding overall decrease in peak developed contractile force and twitch duration. Our simulations show changes in I Ca,L and C a2 + myo in Figures 9A and B, and corresponding changes in developed force in Figure 10D with increasing temperature. As the temperature is increased from 22°C to 30°C, traces for normalized force in Figure 10D show a small increase in peak amplitude despite a decline in the amount of activator C a2 + responsible for contraction due to a temperature dependent increase in myofilament C a2 + sensitivity (Figure 10C). This increase in peak developed force at low, non-physiological temperatures is confirmed by Janssen et al. (; Figure 1). However, as the temperature is increased further from 30°C to 38°C, further decreases in peak C a2 + myo cause strong decreases in peak developed force. Figure 10D shows that an increase in temperature was accompanied by a decrease in both time to peak (TTP) as well as RT50,F (time taken for 50% decline in force). The inset in Figure 10D shows the linear relationship [27, 37] between rate of relaxation versus rate of contraction for increasing temperature (reciprocal of RT50,Fversus reciprocal of TTP).
Figure 10E shows phase plots of normalized force versus instantaneous C a2 + concentration in the myoplasm for increasing temperatures (22°C (+) to 38°C (∗) in steps of 4°C) overlayed with two steady state F-Ca relationships corresponding to 22°C (+) and 38°C (∗). The contraction-relaxation coupling point (∘) moves to decreasing values of [C a2 + ] and force with increasing temperature. Phase-plane analysis of normalized force versus instantaneous C a2 + concentration in the myoplasm reveals that, as the temperature is increased from 22°C to 38°C, the relaxation phase moves to the right towards the corresponding steady-state F-Ca relationship. This suggests that, with increase in temperature there is a departure from cross-bridge kinetics being the rate-limiting step in cardiac relaxation to a more C a2 + -driven mechanism. As evident from Figure 10D, the inset in Figure 10E shows the temperature dependence of peak force developed indicating a moderate increase at low (< 30°C) temperatures with a strong decline at temperatures above 30°C. Traces for SL shortening in Figure 10F corresponding to [C a2 + ] myo transients in panel B, indicate an overall decrease in sarcomere shortening with increasing temperature which agrees with the trend in developed force in panel D. Comparison of insets in Figure 10E and F shows the correlation between peak force developed and the corresponding delta change in SL. The effect of change in temperature on myofilament contractility is a two-stage response with feedback. Firstly, the [C a2 + ] myo transient is temperature sensitive owing largely to the temperature dependence of the trigger current I Ca,L . Secondly, the cellular contractile machinery is highly temperature sensitive due the strong temperature dependence of rate kinetics involved in the formation of crossbridges. In addition, the process of crossbridge formation is known to have a small feedback effect on the [C a2 + ] myo transient as seen in Figure 8. As the temperature is increased from 22°C to 30°C, the temperature sensitivity of crossbridge kinetics predominates the temperature dependence of [C a2 + ] myo transient in determining the contractile response. However, for temperatures from 30°C to 38°C the opposite holds true.
Role of myofilament C a 2 + sensitivity
Figure 11B shows traces of normalized force captured at steady state in response to a [C a2 + ] myo transient (overlaid) in the control case as well as with modified myofilament C a2 + sensitivity. A 30% decrease (from control value) in MCS causes a much larger change in peak isometric force generated than a 30% increase, highlighting the nonlinear response. Figure 11B shows that an increase in MCS causes a faster onset of the upstroke in force response and a delay in the start of recovery as a result of enhancement in C a2 + binding to troponin. This delay in recovery manifests in an increase in time to peak from 119.5 ms (∙) in the control case to 124 ms (▴). Similarly, a decrease in MCS produces an opposite effect resulting in a decrease in time to peak (112 ms (▾)). The two distinct slopes during the upstroke in force response in Figure 11B are a result of an initial contribution of the strongly bound pre-rotated state (XB_PreR in Figure 2) followed by the effect of increase in strongly bound post-rotated state (XB_PostR in Figure 2). As shown in the inset in Figure 11B, an increase in MCS causes an increase in TD50 (time taken from 50% activation to 50% relaxation).
Phase plots of normalized force versus the instantaneous C a2 + concentration in the cytosol are shown in Figure 11C. As observed experimentally , a decrease in myofilament C a2 + sensitivity causes the gradient (units of Normalized force/μ M) of the trajectory during the relaxation phase of the twitch contraction (marked by thick arrows in Figure 11C) to decrease from 12.4 in the control case to 7.0. A similar but opposite effect was observed in the rate of relaxation with a corresponding increase in MCS. A delayed onset of the upstroke in force response as a result of a decrease in MCS (Figure 11B) causes a distinct loop at high C a2 + concentrations (trace marked ▾ in Figure 11C). The traces for sarcomere length in Figure 11D reflect the changes in force developed, showing an increased degree of shortening with an enhancement in MCS. Peak shortening velocity also increases with an increase in myofilament C a2 + sensitivity as shown in Figure 11E.
Effect of Isoproterenol: The β-adrenergic agonist isoproterenol (ISO) is known to cause a decrease in myofilament C a2 + sensitivity as a result of PKA-mediated phosphorylation of troponin I at Ser23/Ser24 [15, 17]. However, the increase in amplitude of myoplasmic C a2 + transient more than compensates for the decrease in C a2 + sensitivity in order to facilitate the inotropic effect of β-adrenergic stimulation. Here, we adopt a 1 Hz stimulation protocol used by Roof et al. (; Figure 1) to study the effect of 1 μ M isoproterenol on isometric contraction and compare it with the effect of increasing extracellular C a2 + concentration (C a2 + o ) from 1 mM (control) to 3 mM. As shown in Figure 4A, administration of 1 μ M isoproterenol or an increase in C a2 + o to 3 mM causes a substantial (3-fold) enhancement in peak myoplasmic C a2 + myo as observed experimentally (; Figure 1). The isoproterenol-dependent effect is a result of PKA mediated dose-dependent increase (23%) in peak I Ca,L current together with an increase (17%) in the maximal uptake rate of the SERCA pump when compared to the control case (Appendix, Eqns. 57). In the isoproterenol, the significant increase in peak C a2 + myo causes an increase in the strength of isometric tension developed (Figure 4B) despite a decrease in myofilament C a2 + sensitivity. Increase in C a2 + o to 3 mM causes a similar effect of increasing isometric tension due to an increase in activator C a2 + in the cytosol. The increase in peak C a2 + myo is a result of an increase in SR C a2 + content due to enhanced C a2 + entry via I Ca,L assisted by impaired N a+ /C a2 + exchange due to elevated C a2 + o . However, compared to β-adrenergic stimulation, the lack of a decrease in MCS results in a moderately larger developed force. Figure 4C shows the phase plots of self normalized force versus the instantaneous C a2 + concentration in the cytosol. As expected, either the presence of isoproterenol or an increase in C a2 + o causes rightward extension (due to increase in peak C a2 + myo ) of the phase plot and increases the area enclosed by it due to elevated myoplasmic C a2 + level combined with an increase in tension developed. In the presence of isoproterenol, the increase in isometric tension developed occurs despite a decrease in myofilament C a2 + sensitivity (Figure 4C) which manifests as an increase (0.65 (Figure 4C) which manifests as an increase (0.65 μ M to 0.85 μ M) in EC50, the average C a2 + myo at 50% of maximal developed force.
Myofilament dynamics have been captured by various representations ranging from the highly simplified models to complex empirical [41, 42] and biophysical models . While simplified models tend to use an explicit parabolic tension profile , the empirical models use predefined expressions to specify the average force developed by the cross bridges, based on experimental observations of isolated muscle contraction under different loading conditions. On the contrary, biophysical models of cardiac myofilament dynamics include descriptions of cross-bridge cycling and their elastic properties. An extensive review of various myofilament models in the literature is given by Trayanova and Rice . We have developed a composite multiphysics model of excitation-contraction coupling in the rat ventricular myocyte based on a mechanistic electrochemical model of calcium-induced calcium-release (CICR)  and a detailed mechanochemical model of cooperative activation and crossbridge cycling . After integrating these component models, the resultant multiphysics model of cardiac electromechanics is used to examine the mechanisms regulating myofilament contractility in an isolated rat ventricular myocyte under a voltage clamp protocol.
In particular, we have studied the role of different modulatory factors in influencing the various types of twitch response (isosarcometric, unloaded and isometric) elicited by an isolated rat ventricular myocyte. The dependence of isosarcometric contraction on the amount of activator C a2 + available in the myoplasm to trigger contraction and the sarcomere length which modulates myofilament C a2 + sensitivity is demonstrated (Figure 3). Unloaded cell contraction is investigated to understand the influence of C a2 + myo transient on the degree of sarcomere contraction achieved by an unrestrained cell highlighting the enhanced shortening velocity and rate of recovery with increasing peak C a2 + myo (shorter time to peak and faster rate of relaxation in the inset in Figure 11B). In agreement with Rice et al.  we demonstrate that in isometric contraction (Figure 6), there can be significant internal shortening of the sarcomere as the result of compliant end connections (low KSE values).
C a2 + released as a result of CICR is known to act as an actuator, triggering myofilament contraction by binding to the low affinity regulatory sites on troponin C which act as the sensor, the C a2 + affinity of which is a function of dynamically changing sarcomere length as well as the fraction of strongly bound crossbridges. This hence facilitates not only a feedforward but also a moderate feedback interaction between the [C a2 + ] myo transient and the myofilament contractile mechanism. This feedback effect is seen in Figure 8C where the presence of isometric contraction results in a small decrease in peak C a2 + transient, a result of the C a2 + buffering action of troponin.
Although it is well known that myoplasmic C a2 + concentration has a direct influence on beat to beat cell shortening, other indirect modulatory influences exist. Of particular interest is the indirect coupling of the cellular plasma membrane with the myofilament contractile mechanism. A decrease in N a+ /C a2 + exchanger function in extruding C a2 + from the cytosol results in an increased relative role for the SERCA pump, augmenting SR C a2 + content. This in turn results in enhanced release increasing the availability of post-release activator C a2 + in the cytosol, thus translating into greater degree of contraction (Figure 9). This completes a control loop that allows modulation of NCX activity to force a readjustment in myofilament contractility.
Experimental conditions such as temperature strongly influence cardiac myofilament contractility. An increase in temperature results in an increase in sensitivity of the myofilaments to myoplasmic C a2 +. As the temperature is increased from 22°C to 30°C, increasing myofilament C a2 + sensitivity (Figure 10C) causes a moderate increase in peak force despite a decline in peak C a2 + myo . However, a further increase to body temperature results in a steep decline in force developed (Figure 10D). Hence, an increase in temperature from 22°C to 38°C which results in a decline in peak C a2 + myo , causes an overall decrease in force developed. This translates into an overall decrease in the degree of cell shortening with increasing temperature (Figure 10F). This temperature dependent behavior is not captured by the model proposed by Rice et al.  where an increase in temperature causes an increase in peak force developed as a result of the large Q10 values used (Qf app , Qh f , Qh b and Qg xb , Table 1, ).
Modulation of myofilament C a2 + sensitivity (MCS) as a result of a change in Troponin I (TnI) phosphorylation by PKA has been implicated in heart failure . Here we study the role of MCS in modulating myofilament contractile response (Figure 11) an aspect overlooked in recent modeling studies including Rice et al. . In particular, we model the effect of isoproterenol, a β-adrenergic agonist, which is known to cause a decline in myofilament C a2 + sensitivity as a result of protein kinase A (PKA) mediated phosphorylation of troponin I at Ser23/Ser24 [15, 16]. Such a decline in C a2 + sensitivity aids myofilament relaxation in the presence of increased levels of activator C a2 + . Figure 4 shows that an increase in amplitude of the myoplasmic C a2 + transient (a cumulative effect of enhancement in trigger current and increase in uptake rate of the SERCA pump) is more than adequate to compensate for the decrease in C a2 + sensitivity, thus facilitating the desired effect of β-adrenergic stimulation, namely an increase in contractile force generated.
Our model of a rat ventricular myocyte is limited to C a 2 + related channels, exchanger and pumps (I Ca,L , I NaCa , I PMCA , I ryr and SERCA pump), while lacking exclusive N a + or K + related channels and transporters (described in Krishna et al. ). In our voltage clamp protocol, external solution in the bath is modeled as normal Tyrode with C s + substituted for K + . We have represented I NaCs in our model (described in Krishna et al. ) by the expression for N a + /K + pump formulated by Lindblad et al.  replacing K + ion concentrations with the C s + ion concentration. While ensuring whole cell N a + ion balance, the peak N a + /C s + pump current is modified to be one-sixth to account for the decreased potency of the cation C s + in activating the pump. The voltage-dependence of I NaCs is adopted from the data on N a + /K + pump from Hansen et al. . The role of mitochondrial C a 2 + uniporter is not modeled in this study due to its negligible (< 1%) role in C a 2 + transport from the cytosol . This model is aimed at mimicking voltage clamp conditions where channels other than calcium are blocked, and it cannot be used to study any action potential-induced C a 2 + transient-frequency relationship. However, its focus on the C a 2 + dynamics allows one to comprehend more clearly the important role of C a 2 + signalling pathways and feedback control systems in maintaining whole cell homeostasis over a prolonged period of time.
The cooperative activation of the thin filament and the strain-dependent transitions of the crossbridge cycle have been approximately modeled as non-spatial, state-variables. However, this simplification is valid as these transitions are inherently local phenomena and the model reproduces a wide range of steady state and dynamic responses in cardiac muscle . Although a CaM-dependent pathway is reported  to be responsible in modulation of myofibrillar contractility implying a possible CaM mediated role for Ca-dependent kinases or phosphatases in regulating myofilament contractility (particularly in frequency dependent acceleration of relaxation), further studies are required to clarify the molecular mechanisms involved.
The temperature dependence of passive force attributed to titin and other cytoskeletal elements is not modeled in this study and the assumption of constant stiffness for the series elastic element (KSE) does not account for temperature dependent effects. This is an area where the model can be expanded but additional measured data on the temperature sensitivity of these elements is necessary. Regardless, our model provides reasonable approximations to the temperature dependence of developed force in intact thin rat ventricular trabeculae .
We have developed a composite mathematical model for cardiac electromechanics which includes a detailed description of C a2 + dynamics under voltage clamp conditions in the rat ventricular myocyte, based on experimental data [4, 13]. We have investigated the role of different factors including the myoplasmic C a2 + myo transient and the sarcomeric length in influencing various types of twitch responses obtained under different loading conditions (including isosarcometric, isometric and unloaded conditions). Various control loops influencing cell shortening have been explored. In particular, the bidirectional interaction of the C a2 + transient with the myofilament contractile mechanism and the importance of indirect SR mediated interaction of the sarcolemma with the contractile machinery is highlighted by showing the regulation of isometric contraction by the degree of NCX activity. The effect of temperature on cell shortening is investigated identifying the differential sensitivity of the C a2 + myo transient and the myofilament contractile mechanism. The important role of myofilament C a2 + sensitivity in force generation is studied with particular emphasis on the effect of β-adrenergic stimulation on cardiac contractile response. In agreement with Janssen , we also demonstrate a key linear relationship between the rate of contraction and relaxation, which is shown here to be intrinsically coupled over the full range of physiological perturbations (including temperature, sarcomeric length, activator C a2 + , and isoproterenol; e.g. see insets in Figures 10D, 3A-ii).
This study demonstrates that the model has long-term stability in regulating myoplasmic C a2 + , as shown in the 18-sec duration experiments at a physiological rate of stimulation (conditioning train of voltage clamp pulses at 5 Hz) shown in Figure 9. This long term C a2 + balance under physiological conditions is crucial in facilitating implementation of this model in large scale simulations such as frequency dependent studies analyzing cellular force-frequency response. Our study also provides mechanistic insights into whole-cell responses to a wide variety of testing approaches used in studies of cardiac myofilament contractility that have appeared in the literature over the past two decades (Figures 3,4,5,6,7,8,9,10 and 11). Thus, the model serves as a platform for the predictive modeling of VC investigations of cardiac electromechanics pertaining to the rat ventricular myocyte in a number of areas. These are fundamental issues that would benefit from a better mechanistic understanding of the cardiac contractile mechanism in the rat ventricluar myocyte. This study is aimed at providing an initial step towards this goal.
Equations governing electro-mechanics modified (from Rice et al. ) in the model
= 4.5408 μ L s−1(Krishna et al. ).
calcium ion concentration
myoplasmic Ca2+ concentration
extracellular Ca2+ concentration
contraction-relaxation coupling point
dyadic coupling unit
half maximal effective concentration
force versus Ca2+
force frequency response
force versus sarcomere length
L: L-type Ca2+ current
sodium calcium exchanger current
plasma membrane Ca2+ ATPase pump current
junctional portion of the sarcoplasmic reticulum
stiffness coefficient of the non-contractile series elastic element
longitudinal portion of the sarcoplasmic reticulum
long lasting type
myofilament Ca2+ sensitivity
ordinary differential equation
pico amps per pico farad
protein kinase A
time required for 50% sarcomere relaxation
C: time required for 50% decline in Ca2+-transient
F: time required decline in force response
I: time required for 50% ICa,L inactivation
sarcoplasmic reticulum Ca2+ ATPase
time taken from 50% activation to 50% relaxation
fraction of unphosphorylated Troponin I
time required to attain peak value.
This work was supported by a research grant from Methodist Hospital Research Institute. The authors would like to thank Liang Sun for his early contributions to this work.
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