Motivated by the need for an analytical tool that can be used routinely to analyze data collected from isolated, detergent-skinned cardiac muscle fibers, we developed a mathematical model for representing the force response to step changes in muscle length (i.e., quick stretch and release). Our proposed model is reasonably simple, consisting of only five parameters representing: (1) the rate constant by which length change–induced distortion of elastic elements is dissipated; (2) the stiffness of the muscle fiber; (3) the amplitude of length-mediated recruitment of stiffness elements; (4) the rate constant by which this length-mediated recruitment takes place; and (5) the magnitude of the nonlinear interaction term by which distortion of elastic elements affects the number of recruited stiffness elements. Fitting this model to a family of force recordings representing responses to eight amplitudes of step length change (±2.0% baseline muscle length in 0.5% increments) enabled four things: (1) reproduction of all the identifiable features seen in a family of force responses to both positive and negative length changes; (2) close fitting of all records from the whole family of these responses with very little residual error; (3) estimation of all five model parameters with a great degree of certainty; and (4) importantly, ready discrimination between cardiac muscle fibers with different contractile regulatory proteins but showing only subtly different contractile function. We recommend this mathematical model as an analytic tool for routine use in studies of cardiac muscle fiber contractile function. Such model-based analysis gives novel insight to the contractile behavior of cardiac muscle fibers, and it is useful for characterizing the mechanistic effects that alterations of cardiac contractile proteins have on cardiac contractile function.

. Interpreting cardiac muscle force-length dynamics using a novel functional model. Am J Physiol Heart Circ Physiol 286: H1535-H1545, 2004; 10.1152/ ajpheart.01029.2003.-To describe the dynamics of constantly activated cardiac muscle, we propose that length affects force via both recruitment and distortion of myosin cross bridges. This hypothesis was quantitatively tested for descriptive and explanative validity. Skinned cardiac muscle fibers from animals expressing primarily ␣-myosin heavy chain (MHC) (mouse, rat) or ␤-MHC (rabbit, ferret) were activated with solutions from pCa 6.1 to 4.3. Activated fibers were subjected to small-amplitude length perturbations [⌬L(t)] rich in frequency content between 0.1 and 40 Hz. In descriptive validation tests, the model was fit to the ensuing force response [⌬F(t)] in the time domain. In fits to 118 records, the model successfully accounted for most of the measured variation in ⌬F(t) (R 2 range, 0.997-0.736; median, 0.981). When some residual variations in ⌬F(t) were not accounted for by the model (as at low activation), there was very little coherence (Ͻ0.5) between these residual force variations and the applied ⌬L(t) input function, indicating that something other than ⌬L(t) was causing the measured variation in ⌬F(t). With one exception, model parameters were estimated with standard errors on the order of 1% or less. Thus parameters of the recruitment component of the model could be uniquely separated from parameters of the distortion component of the model and parameters estimated from any given fiber could be considered unique to that fiber. In explanative validation tests, we found that recruitment and distortion parameters were positively correlated with independent assessments of the physiological entity they were assumed to represent. The recruitment distortion model was judged to be valid from both descriptive and explanative perspectives and is, therefore, a useful construct for describing and explaining dynamic force-length relationships in constantly activated cardiac muscle. muscle stiffness; cross-bridge recruitment; cross-bridge distortion; model validation; mouse; rat; ferret; rabbit MUSCLE LENGTH modulates cardiac muscle force development, and the resultant force-length relationship (FLR) is basic to the Frank-Starling mechanism of the heart. In general, however, muscle FLRs extend beyond the typical isometric twitching conditions under which force-length data are commonly collected to also include the force response to any length change during contraction. Thus descriptors of muscle FLR should also depict the dynamic force response to length changes that occur during contraction. One often-studied aspect of the dynamic FLR is the force response to sinusoidal length change in a constantly activated muscle fiber. Under these dynamic conditions, the amplitude and phase of the force response depends on both the frequency and amplitude of the sinusoidal length change. Dividing the steady-state sinusoidal force response by the sinusoidal length change yields...

The rate of muscle force redevelopment after release-restretch protocols has previously been interpreted using a simple two-state cross-bridge cycling model with rate constants for transitions between non-force-bearing and force-bearing states, f, and between force-bearing and non-force-bearing states, g. Changes in the rate constant of force redevelopment, as with varying levels of Ca2+ activation, have traditionally been attributed to Ca(2+)-dependent f. The current work adds to this original model a state of unactivated, noncycling cross-bridges. The resulting differential equation for activated, force-bearing cross-bridges, Ncf, was Ncf = -[g+f(K/(K + 1))] Ncf+f(K/(K + 1))NT, where K is an equilibrium constant defining the distribution between cycling and noncycling cross-bridges and NT is the total number of cross-bridges. Cooperativity by which force-bearing cross-bridges participate in their own activation was introduced by making K depend on Ncf. Model results demonstrated that such cooperativity, which tends to enhance force generation at low levels of Ca2+ activation, has a counter-intuitive effect of slowing force redevelopment. These dynamic effects of cooperativity are most pronounced at low Ca2+ activation. As Ca2+ activation increases, the cooperative effects become less important to the dynamics of force redevelopment and, at the highest levels of Ca2+ activation, the dynamics of force redevelopment reflect factors other than cooperative mechanisms. These results expand on earlier interpretations of Ca2+ dependence of force redevelopment; rather than Ca(2+)-dependent f, Ca(2+)-dependent force redevelopment arises from changing expressions of cooperativity between force-bearing cross-bridges and activation.

The heterogenic nature of troponin T (TnT) isoforms in fast skeletal and cardiac muscle suggests important functional differences. Dynamic features of rat cardiac TnT (cTnT) and rat fast skeletal TnT (fsTnT) reconstituted cardiac muscle preparations were captured by fitting the force response of small amplitude (0.5%) muscle length changes to the recruitment-distortion model. The recruitment of force-bearing cross-bridges (XBs) by increases in muscle length was favored by cTnT. The recruitment magnitude was approximately 1.5 times greater for cTnT- than for fsTnT-reconstituted muscle fibers. The speed of length-mediated XB recruitment (b) in cTnT-reconstituted muscle fiber was 0.50-0.57 times as fast as fsTnT-reconstituted muscle fibers (3.05 vs. 5.32 s(-1) at sarcomere length, SL, of 1.9 microm and 4.16 vs. 8.36 s(-1) at SL of 2.2 microm). Due to slowing of b in cTnT-reconstituted muscle fibers, the frequency of minimum stiffness (f(min)) was shifted to lower frequencies of muscle length changes (at SL of 1.9 microm, 0.64 Hz, and 1.16 Hz for cTnT- and fsTnT-reconstituted muscle fibers, respectively; at SL of 2.2 microm, 0.79 Hz, and 1.11 Hz for cTnT- and fsTnT-reconstituted muscle fibers, respectively). Our model simulation of the data implicates TnT as a participant in the process by which SL- and XB-regulatory unit cooperative interactions activate thin filaments. Our data suggest that the amino-acid sequence differences in cTnT may confer a heart-specific regulatory role. cTnT may participate in tuning the heart muscle by decreasing the speed of XB recruitment so that the heart beats at a rate commensurate with f(min).

Cooperativity in contractile behavior of myofilament systems almost assuredly arises because of interactions between neighboring sites. These interactions may be of different kinds. Tropomyosin thin-filament regulatory units may have neighbors in steric blocking positions (off) or steric permissive positions (on). The position of these neighbors influence the tendency for the regulatory unit to assume the on or off state. Likewise, the tendency of a myosin cross-bridge to achieve a force-bearing state may be influenced by whether neighboring cross-bridges are in force-bearing states. Also, a cross-bridge in the force-bearing state may influence the tendency of a regulatory unit to enter the on state. We used a mathematical model to examine the influence of each of these three kinds of neighbor interactions on the steady-state force-pCa relation and on the dynamic force redevelopment process. Each neighbor interaction was unique in its effects on maximal Ca(2+)-activated force, position, and symmetry of the force-pCa curve and on the Hill coefficient. Also, each neighbor interaction had a distinctive effect on the time course of force development as assessed by its rate coefficient, k(dev). These diverse effects suggest that variations in all three kinds of nearest-neighbor interactions may be responsible for a wide variety of currently unexplained observations of myofilament contractile behavior.