Nonlinear myofilament regulatory processes affect frequency-dependent muscle fiber stiffness

Analysis of metabolite and strain effects on cardiac cross-bridge dynamics using model linearisation techniques

Multi-scale models of cardiac energetics are becoming crucial in better understanding the prevalent chronic diseases operating at the intersection of metabolic and cardiovascular dysfunction. Computationally efficient models of cardiac cross-bridge kinetics that are sensitive to changes in metabolite concentrations are necessary to simulate the effects of disease-induced changes in cellular metabolic state on cardiac mechanics across disparate spatial scales. While these models do currently exist, deeper analysis of how the modelling of metabolite effects and the assignment of strain dependence within the cross-bridge cycle affect the properties of the model is required. In this study, model linearisation techniques were used to simulate and interrogate the complex modulus of an ODE-based model of cross-bridge kinetics. Active complex moduli were measured from permeabilised rat cardiac trabeculae under five different metabolite conditions with varying ATP and Pi concentrations. Sensitivity to metabolites was incorporated into an existing three-state cross-bridge model using either a direct dependence or a rapid equilibrium approach. Combining the two metabolite binding methods with all possible locations of strain dependence within the cross-bridge cycle produced 64 permutations of the cross-bridge model. Using linear model analysis, these models were systematically explored to determine the effects of metabolite binding and their interaction with strain dependence on the frequency response of cardiac muscle. The results showed that the experimentally observed effects of ATP and Pi concentrations on the cardiac complex modulus could be attributed to their regulation of cross-bridge detachment rates. Analysis of the cross-bridge models revealed a mechanistic basis for the biochemical schemes which place Pi release following cross-bridge formation and ATP binding prior to cross-bridge detachment. In addition, placing strain dependence on the reverse rate of the cross-bridge power stroke produced the model which most closely matched the experimental data. From these analyses, a well-justified metabolite-sensitive model of rat cardiac cross-bridge kinetics is presented which is suitable for parameterisation with other data sets and integration with multi-scale cardiac models.

Skeletal muscle: Modeling the mechanical behavior by taking the hierarchical microstructure into account

Skeletal muscles ensure the mobility of mammals and are complex natural fiber-matrix-composites with a hierarchical microstructure. In this work, we analyze the muscle's mechanical behavior on the level of fascicles and muscle fibers. We introduce continuum mechanics hyperelastic material models for the connective tissue endomysium and the embedded muscle fibers. The coupled electrical, chemical and mechanical processes taking place in activated contracting muscle fibers are captured including the temporal change of the activation level and the spatial propagation of the activation potential in fibers. In our model, we investigate the material behavior of fascicle, fiber and endomysium in the fiber direction and examine interactions between muscle fiber and endomysium by considering the temporal and spatial change of muscle fiber activation. In addition, a loading case of normal and shear forces is applied to analyze the fiber lifting force and the lifting height of unipennate muscles with different pennation angles. Moreover, the development of local stresses and strains in fibers and endomysium for different strains are studied. The simulation results allow to identify regions in high risk of damage. Optimal arrangements of unipennate muscle microstructure are found for either very small or very large pennation angles.

Multiscale modeling of the neuromuscular system: Coupling neurophysiology and skeletal muscle mechanics

Mathematical models and computer simulations have the great potential to substantially increase our understanding of the biophysical behavior of the neuromuscular system. This, however, requires detailed multiscale, and multiphysics models. Once validated, such models allow systematic in silico investigations that are not necessarily feasible within experiments and, therefore, have the ability to provide valuable insights into the complex interrelations within the healthy system and for pathological conditions. Most of the existing models focus on individual parts of the neuromuscular system and do not consider the neuromuscular system as an integrated physiological system. Hence, the aim of this advanced review is to facilitate the prospective development of detailed biophysical models of the entire neuromuscular system. For this purpose, this review is subdivided into three parts. The first part introduces the key anatomical and physiological aspects of the healthy neuromuscular system necessary for modeling the neuromuscular system. The second part provides an overview on state-of-the-art modeling approaches representing all major components of the neuromuscular system on different time and length scales. Within the last part, a specific multiscale neuromuscular system model is introduced. The integrated system model combines existing models of the motor neuron pool, of the sensory system and of a multiscale model describing the mechanical behavior of skeletal muscles. Since many sub-models are based on strictly biophysical modeling approaches, it closely represents the underlying physiological system and thus could be employed as starting point for further improvements and future developments. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

A new myofilament contraction model with ATP consumption for ventricular cell model

A new contraction model of cardiac muscle was developed by combining previously described biochemical and biophysical models. The biochemical component of the new contraction model represents events in the presence of Ca2+-crossbridge attachment and power stroke following inorganic phosphate release, detachment evoked by the replacement of ADP by ATP, ATP hydrolysis, and recovery stroke. The biophysical component focuses on Ca2+ activation and force (F b) development assuming an equivalent crossbridge. The new model faithfully incorporates the major characteristics of the biochemical and biophysical models, such as F b activation by transient Ca2+ ([Ca2+]-F b), [Ca2+]-ATP hydrolysis relations, sarcomere length-F b, and F b recovery after jumps in length under the isometric mode and upon sarcomere shortening after a rapid release of mechanical load under the isotonic mode together with the load-velocity relationship. ATP consumption was obtained for all responses. When incorporated in a ventricular cell model, the contraction model was found to share approximately 60% of the total ATP usage in the cell model.

Rat cardiac troponin T mutation (F72L)-mediated impact on thin filament cooperativity is divergently modulated by α- and β-myosin heavy chain isoforms

The primary causal link between disparate effects of human hypertrophic cardiomyopathy (HCM)-related mutations in troponin T (TnT) and α- and β-myosin heavy chain (MHC) isoforms on cardiac contractile phenotype remains poorly understood. Given the divergent impact of α- and β-MHC on the NH2-terminal extension (44-73 residues) of TnT, we tested if the effects of the HCM-linked mutation (TnTF70L) were differentially altered by α- and β-MHC. We hypothesized that the emergence of divergent thin filament cooperativity would lead to contrasting effects of TnTF70L on contractile function in the presence of α- and β-MHC. The rat TnT analog of the human F70L mutation (TnTF72L) or the wild-type rat TnT (TnTWT) was reconstituted into demembranated muscle fibers from normal (α-MHC) and propylthiouracil-treated (β-MHC) rat hearts to measure steady-state and dynamic contractile function. TnTF72L-mediated effects on tension, myofilament Ca(2+) sensitivity, myofilament cooperativity, rate constants of cross-bridge (XB) recruitment dynamics, and force redevelopment were divergently modulated by α- and β-MHC. TnTF72L increased the rate of XB distortion dynamics by 49% in α-MHC fibers but had no effect in β-MHC fibers; these observations suggest that TnTF72L augmented XB detachment kinetics in α-MHC, but not β-MHC, fibers. TnTF72L increased the negative impact of strained XBs on the force-bearing XBs by 39% in α-MHC fibers but had no effect in β-MHC fibers. Therefore, TnTF72L leads to contractile changes that are linked to dilated cardiomyopathy in the presence of α-MHC. On the other hand, TnTF72L leads to contractile changes that are linked to HCM in the presence of β-MHC.

A Spatially Detailed Model of Isometric Contraction Based on Competitive Binding of Troponin I Explains Cooperative Interactions between Tropomyosin and Crossbridges

Biophysical models of cardiac tension development provide a succinct representation of our understanding of force generation in the heart. The link between protein kinetics and interactions that gives rise to high cooperativity is not yet fully explained from experiments or previous biophysical models. We propose a biophysical ODE-based representation of cross-bridge (XB), tropomyosin and troponin within a contractile regulatory unit (RU) to investigate the mechanisms behind cooperative activation, as well as the role of cooperativity in dynamic tension generation across different species. The model includes cooperative interactions between regulatory units (RU-RU), between crossbridges (XB-XB), as well more complex interactions between crossbridges and regulatory units (XB-RU interactions). For the steady-state force-calcium relationship, our framework predicts that: (1) XB-RU effects are key in shifting the half-activation value of the force-calcium relationship towards lower [Ca²⁺], but have only small effects on cooperativity. (2) XB-XB effects approximately double the duty ratio of myosin, but do not significantly affect cooperativity. (3) RU-RU effects derived from the long-range action of tropomyosin are a major factor in cooperative activation, with each additional unblocked RU increasing the rate of additional RU’s unblocking. (4) Myosin affinity for short (1–4 RU) unblocked stretches of actin of is very low, and the resulting suppression of force at low [Ca²⁺] is a major contributor in the biphasic force-calcium relationship. We also reproduce isometric tension development across mouse, rat and human at physiological temperature and pacing rate, and conclude that species differences require only changes in myosin affinity and troponin I/troponin C affinity. Furthermore, we show that the calcium dependence of the rate of tension redevelopment k_tr is explained by transient blocking of RU’s by a temporary decrease in XB-RU effects.

Force generation in cardiac muscle cells is driven by changes in calcium concentration. Relatively small changes in the calcium concentration over the course of a heart beat lead to the large changes in force required to fully contract and relax the heart. This is known as ‘cooperative activation’, and involves a complex interaction of several proteins involved in contraction. Current computer models which reproduce force generation often do not represent these processes explicitly, and stochastic approaches that do tend to require large amounts of computational power to solve, which limit the range of investigations in which they can be used. We have created an new computational model that captures the underlying physiological processes in more detail, and is more efficient than stochastic approaches, while still being able to run a large range of simulations. The model is able to explain the biological processes leading to the cooperative activation of muscle. In addition, the model reproduces how this cooperative activation translates to normal muscle function to generate force from changes in calcium across three different species.

A multiscale chemo-electro-mechanical skeletal muscle model to analyze muscle contraction and force generation for different muscle fiber arrangements

The presented chemo-electro-mechanical skeletal muscle model relies on a continuum-mechanical formulation describing the muscle's deformation and force generation on the macroscopic muscle level. Unlike other three-dimensional models, the description of the activation-induced behavior of the mechanical model is entirely based on chemo-electro-mechanical principles on the microscopic sarcomere level. Yet, the multiscale model reproduces key characteristics of skeletal muscles such as experimental force-length and force-velocity data on the macroscopic whole muscle level. The paper presents the methodological approaches required to obtain such a multiscale model, and demonstrates the feasibility of using such a model to analyze differences in the mechanical behavior of parallel-fibered muscles, in which the muscle fibers either span the entire length of the fascicles or terminate intrafascicularly. The presented results reveal that muscles, in which the fibers span the entire length of the fascicles, show lower peak forces, more dispersed twitches and fusion of twitches at lower stimulation frequencies. In detail, the model predicted twitch rise times of 38.2 and 17.2 ms for a 12 cm long muscle, in which the fibers span the entire length of the fascicles and with twelve fiber compartments in series, respectively. Further, the twelve-compartment model predicted peak twitch forces that were 19% higher than in the single-compartment model. The analysis of sarcomere lengths during fixed-end single twitch contractions at optimal length predicts rather small sarcomere length changes. The observed lengths range from 75 to 111% of the optimal sarcomere length, which corresponds to a region with maximum filament overlap. This result suggests that stability issues resulting from activation-induced stretches of non-activated sarcomeres are unlikely in muscles with passive forces appearing at short muscle length.

Length-dependent changes in contractile dynamics are blunted due to cardiac myosin binding protein-C ablation

Enhanced cardiac contractile function with increased sarcomere length (SL) is, in part, mediated by a decrease in the radial distance between myosin heads and actin. The radial disposition of myosin heads relative to actin is modulated by cardiac myosin binding protein-C (cMyBP-C), suggesting that cMyBP-C contributes to the length-dependent activation (LDA) in the myocardium. However, the precise roles of cMyBP-C in modulating cardiac LDA are unclear. To determine the impact of cMyBP-C on LDA, we measured isometric force, myofilament Ca²⁺-sensitivity (pCa₅₀) and length-dependent changes in kinetic parameters of cross-bridge (XB) relaxation (k_rel), and recruitment (k_df) due to rapid stretch, as well as the rate of force redevelopment (k_tr) in response to a large slack-restretch maneuver in skinned ventricular multicellular preparations isolated from the hearts of wild-type (WT) and cMyBP-C knockout (KO) mice, at SL's 1.9 μm or 2.1 μm. Our results show that maximal force was not significantly different between KO and WT preparations but length-dependent increase in pCa₅₀ was attenuated in the KO preparations. pCa₅₀ was not significantly different between WT and KO preparations at long SL (5.82 ± 0.02 in WT vs. 5.87 ± 0.02 in KO), whereas pCa₅₀ was significantly different between WT and KO preparations at short SL (5.71 ± 0.02 in WT vs. 5.80 ± 0.01 in KO; p < 0.05). The k_tr, measured at half-maximal Ca²⁺-activation, was significantly accelerated at short SL in WT preparations (8.74 ± 0.56 s⁻¹ at 1.9 μm vs. 5.71 ± 0.40 s⁻¹ at 2.1 μm, p < 0.05). Furthermore, k_rel and k_df were accelerated by 32% and 50%, respectively at short SL in WT preparations. In contrast, k_tr was not altered by changes in SL in KO preparations (8.03 ± 0.54 s⁻¹ at 1.9 μm vs. 8.90 ± 0.37 s⁻¹ at 2.1 μm). Similarly, KO preparations did not exhibit length-dependent changes in k_rel and k_df. Collectively, our data implicate cMyBP-C as an important regulator of LDA via its impact on dynamic XB behavior due to changes in SL.

Modeling the chemoelectromechanical behavior of skeletal muscle using the parallel open-source software library OpenCMISS

An extensible, flexible, multiscale, and multiphysics model for nonisometric skeletal muscle behavior is presented. The skeletal muscle chemoelectromechanical model is based on a bottom-up approach modeling the entire excitation-contraction pathway by strongly coupling a detailed biophysical model of a half-sarcomere to the propagation of action potentials along skeletal muscle fibers and linking cellular parameters to a transversely isotropic continuum-mechanical constitutive equation describing the overall mechanical behavior of skeletal muscle tissue. Since the multiscale model exhibits separable time scales, a special emphasis is placed on employing computationally efficient staggered solution schemes. Further, the implementation builds on the open-source software library OpenCMISS and uses state-of-the-art parallelization techniques taking advantage of the unique anatomical fiber architecture of skeletal muscles. OpenCMISS utilizes standardized data structures for geometrical aspects (FieldML) and cellular models (CellML). Both standards are designed to allow for a maximum flexibility, reproducibility, and extensibility. The results demonstrate the model's capability of simulating different aspects of nonisometric muscle contraction and efficiently simulating the chemoelectromechanical behavior in complex skeletal muscles such as the tibialis anterior muscle.

The tropomyosin binding region of cardiac troponin T modulates crossbridge recruitment dynamics in rat cardiac muscle fibers

The cardiac muscle comprises dynamically interacting components that use allosteric/cooperative mechanisms to yield unique heart-specific properties. An essential protein in this allosteric/cooperative mechanism is cardiac muscle troponin T (cTnT), the central region (CR) and the T2 region of which differ significantly from those of fast skeletal muscle troponin T (fsTnT). To understand the biological significance of such sequence heterogeneity, we replaced the T1 or T2 domain of rat cTnT (RcT1 or RcT2) with its counterpart from rat fsTnT (RfsT1or RfsT2) to generate RfsT1-RcT2 and RcT1-RfsT2 recombinant proteins. In addition to contractile function measurements, dynamic features of RfsT1-RcT2- and RcT1-RfsT2-reconstituted rat cardiac muscle fibers were captured by fitting the recruitment-distortion model to the force response of small-amplitude (0.5%) muscle length changes. RfsT1-RcT2 fibers showed a 40% decrease in tension and a 44% decrease in ATPase activity, but RcT1-RfsT2 fibers were unaffected. The magnitude of length-mediated increase in crossbridge (XB) recruitment (E0) decreased by ~33% and the speed of XB recruitment (b) increased by ~100% in RfsT1-RcT2 fibers. Our data suggest the following: (1) the CR of cTnT modulates XB recruitment dynamics; (2) the N-terminal end region of cTnT has a synergistic effect on the ability of the CR to modulate XB recruitment dynamics; (3) the T2 region is important for tuning the Ca(2+) regulation of cardiac thin filaments. The combined effects of CR-tropomyosin interactions and the modulating effect of the N-terminal end of cTnT on CR-tropomyosin interactions may lead to the emergence of a unique property that tunes contractile dynamics to heart rates.

Cardiac myosin isoforms exhibit differential rates of MgADP release and MgATP binding detected by myocardial viscoelasticity

We measured myosin crossbridge detachment rate and the rates of MgADP release and MgATP binding in mouse and rat myocardial strips bearing one of the two cardiac myosin heavy chain (MyHC) isoforms. Mice and rats were fed an iodine-deficient, propylthiouracil diet resulting in ~100% expression of β-MyHC in the ventricles. Ventricles of control animals expressed ~100% α-MyHC. Chemically-skinned myocardial strips prepared from papillary muscle were subjected to sinusoidal length perturbation analysis at maximum calcium activation pCa 4.8 and 17°C. Frequency characteristics of myocardial viscoelasticity were used to calculate crossbridge detachment rate over 0.01 to 5mM [MgATP]. The rate of MgADP release, equivalent to the asymptotic value of crossbridge detachment rate at high MgATP, was highest in mouse α-MyHC (111.4±6.2s(-1)) followed by rat α-MyHC (65.0±7.3s(-1)), mouse β-MyHC (24.3±1.8s(-1)) and rat β-MyHC (15.5±0.8s(-1)). The rate of MgATP binding was highest in mouse α-MyHC (325±32 mM(-1) s(-1)) then mouse β-MyHC (152±23 mM(-1) s(-1)), rat α-MyHC (108±10 mM(-1) s(-1)) and rat β-MyHC (55±6 mM(-1) s(-1)). Because the events of MgADP release and MgATP binding occur in a post power-stroke state of the myosin crossbridge, we infer that MgATP release and MgATP binding must be regulated by isoform- and species-specific structural differences located outside the nucleotide binding pocket, which is identical in sequence for these four myosins. We postulate that differences in the stiffness profile of the entire myosin molecule, including the thick filament and the myosin-actin interface, are primarily responsible for determining the strain on the nucleotide binding pocket and the subsequent differences in the rates of nucleotide release and binding observed among the four myosins examined here.

An analysis of deformation-dependent electromechanical coupling in the mouse heart

To investigate the effects of the coupling between excitation and contraction on whole-organ function, we have developed a novel biophysically based multiscale electromechanical model of the murine heart. Through comparison with a comprehensive in vivo experimental data set, we show good agreement with pressure and volume measurements at both physiological temperatures and physiological pacing frequencies. This whole-organ model was used to investigate the effects of material and haemodynamic properties introduced at the tissue level, as well as emergent function of our novel cell contraction model. Through a comprehensive sensitivity analysis at both the cellular and whole organ level, we demonstrate the sensitivity of the model's results to its parameters and the constraining effect of experimental data. These results demonstrate the fundamental importance of length- and velocity-dependent feedback to the cellular scale for whole-organ function, and we show that a strong velocity dependence of tension is essential for explaining the differences between measured single cell tension and whole-organ pressure transients.

Cardiomyopathy-Related Mutations in Cardiac Troponin C, L29Q and G159D, Have Divergent Effects on Rat Cardiac Myofiber Contractile Dynamics

Previous studies of cardiomyopathy-related mutations in cardiac troponin C (cTnC)-L29Q and G159D-have shown diverse findings. The link between such mutant effects and their divergent impact on cardiac phenotypes has remained elusive due to lack of studies on contractile dynamics. We hypothesized that a cTnC mutant-induced change in the thin filament will affect global myofilament mechanodynamics because of the interactions of thin filament kinetics with both Ca(2+) binding and crossbridge (XB) cycling kinetics. We measured pCa-tension relationship and contractile dynamics in detergent-skinned rat cardiac papillary muscle fibers reconstituted with the recombinant wild-type rat cTnC (cTnC(WT)), cTnC(L29Q), and cTnC(G159D) mutants. cTnC(L29Q) fibers demonstrated a significant decrease in Ca(2+) sensitivity, but cTnC(G159D) fibers did not. Both mutants had no effect on Ca(2+)-activated maximal tension. The rate of XB recruitment dynamics increased in cTnC(L29Q) (26%) and cTnC(G159D) (25%) fibers. The rate of XB distortion dynamics increased in cTnC(G159D) fibers (15%). Thus, the cTnC(L29Q) mutant modulates the equilibrium between the non-cycling and cycling pool of XB by affecting the on/off kinetics of the regulatory units (Tropomyosin-Troponin); whereas, the cTnC(G159D) mutant increases XB cycling rate. Different effects on contractile dynamics may offer clue regarding how cTnC(L29Q) and cTnC(G159D) cause divergent effects on cardiac phenotypes.

Cardiac electromechanical models: from cell to organ

The heart is a multiphysics and multiscale system that has driven the development of the most sophisticated mathematical models at the frontiers of computational physiology and medicine. This review focuses on electromechanical (EM) models of the heart from the molecular level of myofilaments to anatomical models of the organ. Because of the coupling in terms of function and emergent behaviors at each level of biological hierarchy, separation of behaviors at a given scale is difficult. Here, a separation is drawn at the cell level so that the first half addresses subcellular/single-cell models and the second half addresses organ models. At the subcellular level, myofilament models represent actin–myosin interaction and Ca-based activation. The discussion of specific models emphasizes the roles of cooperative mechanisms and sarcomere length dependence of contraction force, considered to be the cellular basis of the Frank–Starling law. A model of electrophysiology and Ca handling can be coupled to a myofilament model to produce an EM cell model, and representative examples are summarized to provide an overview of the progression of the field. The second half of the review covers organ-level models that require solution of the electrical component as a reaction–diffusion system and the mechanical component, in which active tension generated by the myocytes produces deformation of the organ as described by the equations of continuum mechanics. As outlined in the review, different organ-level models have chosen to use different ionic and myofilament models depending on the specific application; this choice has been largely dictated by compromises between model complexity and computational tractability. The review also addresses application areas of EM models such as cardiac resynchronization therapy and the role of mechano-electric coupling in arrhythmias and defibrillation.

The role of thin filament cooperativity in cardiac length-dependent calcium activation

Length-dependent activation (LDA) is a prominent feature of cardiac muscle characterized by decreases in the Ca(2+) levels required to generate force (i.e., increases in Ca(2+) sensitivity) when muscle is stretched. Previous studies have concluded that LDA originates from the increased ability of (strong) cross-bridges to attach when muscle is lengthened, which in turn enhances Ca(2+) binding to the troponin C (TnC) subunit of the troponin complex. However, our results demonstrate that inhibition of strong cross-bridge attachment with blebbistatin had no effect on the length-dependent modulation of Ca(2+) sensitivity (i.e., EC(50)) or Ca(2+) cooperativity, suggesting that LDA originates upstream of cross-bridge attachment. To test whether LDA arises from length dependence of thin-filament activation, we replaced native cTnC with a mutant cTnC (DM-TnC) that is incapable of binding Ca(2+). Although progressive replacement of native cTnC with DM-TnC caused an expected monotonic decrease in the maximal force (F(max)), DM-TnC incorporation induced much larger increases in EC(50) and decreases in Ca(2+) cooperativity at short lengths than at long lengths. These findings support the conclusion that LDA arises primarily from the influence of length on the modulation of the Ca(2+) cooperativity arising from interaction between adjacent troponin-tropomyosin complexes on the thin filament.

A metabolite-sensitive, thermodynamically constrained model of cardiac cross-bridge cycling: implications for force development during ischemia

We present a metabolically regulated model of cardiac active force generation with which we investigate the effects of ischemia on maximum force production. Our model, based on a model of cross-bridge kinetics that was developed by others, reproduces many of the observed effects of MgATP, MgADP, Pi, and H(+) on force development while retaining the force/length/Ca(2+) properties of the original model. We introduce three new parameters to account for the competitive binding of H(+) to the Ca(2+) binding site on troponin C and the binding of MgADP within the cross-bridge cycle. These parameters, along with the Pi and H(+) regulatory steps within the cross-bridge cycle, were constrained using data from the literature and validated using a range of metabolic and sinusoidal length perturbation protocols. The placement of the MgADP binding step between two strongly-bound and force-generating states leads to the emergence of an unexpected effect on the force-MgADP curve, where the trend of the relationship (positive or negative) depends on the concentrations of the other metabolites and [H(+)]. The model is used to investigate the sensitivity of maximum force production to changes in metabolite concentrations during the development of ischemia.

Myofilament length dependent activation

The Frank-Starling law of the heart describes the interrelationship between end-diastolic volume and cardiac ejection volume, a regulatory system that operates on a beat-to-beat basis. The main cellular mechanism that underlies this phenomenon is an increase in the responsiveness of cardiac myofilaments to activating Ca(2+) ions at a longer sarcomere length, commonly referred to as myofilament length-dependent activation. This review focuses on what molecular mechanisms may underlie myofilament length dependency. Specifically, the roles of inter-filament spacing, thick and thin filament based regulation, as well as sarcomeric regulatory proteins are discussed. Although the "Frank-Starling law of the heart" constitutes a fundamental cardiac property that has been appreciated for well over a century, it is still not known in muscle how the contractile apparatus transduces the information concerning sarcomere length to modulate ventricular pressure development.

Simulation of steady state and transient cardiac muscle response experiments with a Huxley-based contraction model

A cardiac muscle model is presented with the purpose of representing a wide range of mechanical experiments at constant and transient Ca(2+) concentration. Modifications of a previous model were: weak and power attached crossbridge states, a troponin system involving three consecutive regulatory troponin-tropomyosin units acting together in Ca(2+) kinetics and detachment constants depending on crossbridge length. This model improved cooperativity (Hill coefficient close to 4) and the force-velocity relationship, and incorporated the representation of the four phases of muscle response to length and force steps, isotonic shortening and isosarcometric contractions, preserving previous satisfactory results. Moreover, experimentally reported effects, such as length dependence on Ca(2+) affinity, the decreased cooperativity at higher Ca(2+) concentrations, temperature effects on the stiffness-frequency relationship and the isometric internal shortening due to series elasticity, were obtained. In conclusion, the model is more comprehensive than a previous version because it is able to represent a wider variety of steady state experiments, the mechanical variables in twitches can be adequately related to intracellular Ca(2+), and all the simulations were performed with the same set of parameters.

Force transients and minimum cross-bridge models in muscular contraction

Two- and three-state cross-bridge models are considered and examined with respect to their ability to predict three distinct phases of the force transients that occur in response to step change in muscle fiber length. Particular attention is paid to satisfying the Le Châtelier-Brown Principle. This analysis shows that the two-state model can account for phases 1 and 2 of a force transient, but is barely adequate to account for phase 3 (delayed force) unless a stretch results in a sudden increase in the number of cross-bridges in the detached state. The three-state model (A-->B-->C-->A) makes it possible to account for all three phases if we assume that the A-->B transition is fast (corresponding to phase 2), the B-->A transition is of intermediate speed (corresponding to phase 3), and the C-->A transition is slow; in such a scenario, states A and C can support or generate force (high force states) but state B cannot (detached, or low-force state). This model involves at least one ratchet mechanism. In this model, force can be generated by either of two transitions: B-->A or B-->C. To determine which of these is the major force-generating step that consumes ATP and transduces energy, we examine the effects of ATP, ADP, and phosphate (Pi) on force transients. In doing so, we demonstrate that the fast transition (phase 2) is associated with the nucleotide-binding step, and that the intermediate-speed transition (phase 3) is associated with the Pi-release step. To account for all the effects of ligands, it is necessary to expand the three-state model into a six-state model that includes three ligand-bound states. The slowest phase of a force transient (phase 4) cannot be explained by any of the models described unless an additional mechanism is introduced. Here we suggest a role of series compliance to account for this phase, and propose a model that correlates the slowest step of the cross-bridge cycle (transition C-->A) to: phase 4 of step analysis, the rate constant k(tr) of the quick-release and restretch experiment, and the rate constant k(act) for force development time course following Ca(2+) activation.

Multi-scale computational modelling in biology and physiology

Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.

A mathematical model of fatigue in skeletal muscle force contraction

The ability for muscle to repeatedly generate force is limited by fatigue. The cellular mechanisms behind muscle fatigue are complex and potentially include breakdown at many points along the excitation-contraction pathway. In this paper we construct a mathematical model of the skeletal muscle excitation-contraction pathway based on the cellular biochemical events that link excitation to contraction. The model includes descriptions of membrane voltage, calcium cycling and crossbridge dynamics and was parameterised and validated using the response characteristics of mouse skeletal muscle to a range of electrical stimuli. This model was used to uncover the complexities of skeletal muscle fatigue. We also parameterised our model to describe force kinetics in fast and slow twitch fibre types, which have a number of biochemical and biophysical differences. How these differences interact to generate different force/fatigue responses in fast- and slow- twitch fibres is not well understood and we used our modelling approach to bring new insights to this relationship.

Dynamic left ventricular elastance: a model for integrating cardiac muscle contraction into ventricular pressure-volume relationships

To integrate myocardial contractile processes into left ventricular (LV) function, a mathematical model was built. Muscle fiber force was set equal to the product of stiffness and elastic distortion of stiffness elements, i.e., force-bearing cross bridges (XB). Stiffness dynamics arose from recruitment of XB according to the kinetics of myofilament activation and fiber-length changes. Elastic distortion dynamics arose from XB cycling and the rate-of-change of fiber length. Muscle fiber stiffness and distortion dynamics were transformed into LV chamber elastance and volumetric distortion dynamics. LV pressure equaled the product of chamber elastance and volumetric distortion, just as muscle-fiber force equaled the product of muscle-fiber stiffness and lineal elastic distortion. Model validation was in terms of its ability to reproduce cycle-time-dependent LV pressure response, DeltaP(t), to incremental step-like volume changes, DeltaV, in the isolated rat heart. All DeltaP(t), regardless of the time in the cycle at which DeltaP(t) was elicited, consisted of three phases: phase 1, concurrent with the leading edge of DeltaV; phase 2, a brief transient recovery from phase 1; and phase 3, sustained for the duration of systole. Each phase varied with the time in the cycle at which DeltaP(t) was elicited. When the model was fit to the data, cooperative activation was required to sustain systole for longer periods than was possible with Ca(2+) activation alone. The model successfully reproduced all major features of the measured DeltaP(t) responses, and thus serves as a credible indicator of the role of underlying contractile processes in LV function.

Interaction between myosin heavy chain and troponin isoforms modulate cardiac myofiber contractile dynamics

Coordinated expression of species-specific myosin heavy chain (MHC) and troponin (Tn) isoforms may bring about a dynamic complementarity to match muscle contraction speed with species-specific heart rates. Contractile system function and dynamic force-length measurements were made in muscle fibers from mouse and rat hearts and in muscle fibers after reconstitution with either recombinant homologous Tn or orthologous Tn. The rate constants of length-mediated cross-bridge (XB) recruitment (b) and tension redevelopment (k(tr)) of mouse fibers were significantly faster than those of rat fibers. Both the tension cost (ATPase/tension) and rate constant of length-mediated XB distortion (c) were higher in the mouse than in the rat. Thus the mouse fiber was faster in all dynamic and functional aspects than the rat fiber. Mouse Tn significantly increased b and k(tr) in rat fibers; conversely, rat Tn significantly decreased b and k(tr) in mouse fibers. Thus the length-mediated recruitment of force-bearing XB occurs much more rapidly in the presence of mouse Tn than in the presence of rat Tn, demonstrating that the speed of XB recruitment is regulated by Tn. There was a significant interaction between Tn and MHC such that changes in either Tn or MHC affected the speed of XB recruitment. Our data demonstrate that the dynamics of myocardial contraction are different in the mouse and rat hearts because of sequence heterogeneity in MHC and Tn. At the myofilament level, coordinated expression of complementary regulatory contractile proteins produces a functional dynamic phenotype that allows the cardiovascular systems to function effectively at different heart rates.

Mechanism of the Frank–Starling law—A simulation study with a novel cardiac muscle contraction model that includes titin and troponin I

A stretch-induced increase of active tension is one of the most important properties of the heart, known as the Frank-Starling law. Although a variation of myofilament Ca(2+) sensitivity with sarcomere length (SL) change was found to be involved, the underlying molecular mechanisms are not fully clarified. Some recent experimental studies indicate that a reduction of the lattice spacing between thin and thick filaments, through the increase of passive tension caused by the sarcomeric protein titin with an increase in SL within the physiological range, promotes formation of force-generating crossbridges (Xbs). However, the mechanism by which the Xb concentration determines the degree of cooperativity for a given SL has so far evaded experimental elucidation. In this simulation study, a novel, rather simple molecular-based cardiac contraction model, appropriate for integration into a ventricular cell model, was designed, being the first model to introduce experimental data on titin-based radial tension to account for the SL-dependent modulation of the interfilament lattice spacing and to include a conformational change of troponin I (TnI). Simulation results for the isometric twitch contraction time course, the length-tension and the force-[Ca(2+)] relationships are comparable to experimental data. A complete potential Frank-Starling mechanism was analyzed by this simulation study. The SL-dependent modulation of the myosin binding rate through titin's passive tension determines the Xb concentration which then alters the degree of positive cooperativity affecting the rate of the TnI conformation change and causing the Hill coefficient to be SL-dependent.

Troponin T modulates sarcomere length-dependent recruitment of cross-bridges in cardiac muscle

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).

Oscillations in molecular motor assemblies

Autonomous oscillations in biological systems may have a biochemical origin or result from an interplay between force-generating and visco-elastic elements. In molecular motor assemblies the force-generating elements are molecular engines and the visco-elastic elements are stiff cytoskeletal polymers. The physical mechanism leading to oscillations depends on the particular architecture of the assembly. Existing models can be grouped into two distinct categories: systems with a delayed force activation and anomalous force-velocity relations. We discuss these systems within phase plane analysis known from the theory of dynamic systems and by adopting methods from control theory, the Nyquist stability criterion.

The sarcomeric control of energy conversion

The Frank-Starling Law, Fenn Effect, and Suga's suggestions of cardiac muscle constant contractile efficiency establish the dependence of cardiac mechanics and energetics on the loading conditions. Consistent with these observations, this review suggests that the sarcomere control of contraction consists of two dominant feedbacks: (1) a cooperativity mechanism (positive feedback), whereby the number of force-generating cross-bridges (XBs) determines the affinity of calcium binding to the troponin regulatory protein; and (2) a mechanical (negative) feedback, whereby the filament shortening velocity affects the rate of XB turnover from the force to the non-force generating conformation. The study explains the roles of these feedbacks in providing the adaptive control of energy consumption by the loading conditions and validates the dependence of the cooperativity mechanism on the number of strong XBs. The cooperativity mechanism regulates XB recruitment. It explains the cardiac force-length calcium relationship, the related Frank-Starling Law of the heart, and the adaptive control of new XB recruitment and the associated adenosine triphosphate (ATP) consumption. The mechanical feedback explains the force-velocity relationship and the constant and high-contractile efficiency. These mechanisms were validated by testing the force responses to large amplitude (100 nm/sarcomere) sarcomere length (SL) oscillations, in intact tetanized trabeculae (utilizing 30 microM cyclopiazonic). The force responses to large-length oscillations lag behind the imposed oscillations at low extracellular calcium concentration ([Ca(2+)](0)) and slow frequencies (<4 Hz, 25 degrees C), yielding counterclockwise hystereses in the force-length plane. The force was higher during shortening than during lengthening. The area within these hystereses corresponds to the external work generated from new XB recruitment during each oscillation, and it is determined by the delay in the force response. Characterization of the delayed response and its dependence on the SL, force, and calcium allows identification of the regulation of XB recruitment. The direct dependence of the phase on force indicates that XB recruitment is determined directly by the force (i.e., the number of strong XBs) and indirectly by SL or calcium. The suggested feedbacks determine cardiac energetics: 1) the constant and high contractile efficiency is an intrinsic property of the single XB, due to the mechanical feedback; and 2) the XBs are the myocyte sensors that modulate XB recruitment in response to length and load changes through the cooperativity mechanism.

Dynamic myocardial contractile parameters from left ventricular pressure-volume measurements

A new dynamic model of left ventricular (LV) pressure-volume relationships in beating heart was developed by mathematically linking chamber pressure-volume dynamics with cardiac muscle force-length dynamics. The dynamic LV model accounted for >80% of the measured variation in pressure caused by small-amplitude volume perturbation in an otherwise isovolumically beating, isolated rat heart. The dynamic LV model produced good fits to pressure responses to volume perturbations, but there existed some systematic features in the residual errors of the fits. The issue was whether these residual errors would be damaging to an application where the dynamic LV model was used with LV pressure and volume measurements to estimate myocardial contractile parameters. Good agreement among myocardial parameters responsible for response magnitude was found between those derived by geometric transformations of parameters of the dynamic LV model estimated in beating heart and those found by direct measurement in constantly activated, isolated muscle fibers. Good agreement was also found among myocardial kinetic parameters estimated in each of the two preparations. Thus the small systematic residual errors from fitting the LV model to the dynamic pressure-volume measurements do not interfere with use of the dynamic LV model to estimate contractile parameters of myocardium. Dynamic contractile behavior of cardiac muscle can now be obtained from a beating heart by judicious application of the dynamic LV model to information-rich pressure and volume signals. This provides for the first time a bridge between the dynamics of cardiac muscle function and the dynamics of heart function and allows a beating heart to be used in studies where the relevance of myofilament contractile behavior to cardiovascular system function may be investigated.

Analysis of hystereses in force length and force calcium relations

Analysis of the hystereses in the force-length relationship at constant Ca(2+) concentration and in the force-calcium relationship at constant sarcomere length (SL) provides insight into the mechanisms that control cross-bridge (XB) recruitment. The hystereses are related here to two mechanisms that regulate the number of strong XBs: the cooperativity, whereby the number of strong XBs determines calcium affinity, and the mechanical feedback, whereby the shortening velocity determines the duration for which the XBs are in the strong state. The study simulates the phenomena and defines the role of these feedbacks. The model that couples calcium kinetics with XB cycling was built on Simulink software (Matlab). Counterclockwise (CCW) hysteresis, wherein the force response lags behind the SL oscillations, at a constant calcium level, is obtained in the force-length plane when neglecting the mechanical feedback and accounting only for the cooperativity mechanism. Conversely, the force response precedes the SL oscillations, yielding a clockwise (CW) hysteresis when only the mechanical feedback is allowed to exist. In agreement with experimental observations, either CW or CCW hysteresis is obtained when both feedbacks coexist: CCW hystereses are obtained at low frequencies (<3 Hz), and the direction is reversed to CW at higher frequencies (>3 Hz). The cooperativity dominates at low frequencies and allows the muscle to adapt XB recruitment to slow changes in the loading conditions. The changeover frequency from CCW to CW hysteresis defines the velocity limit above which the muscle absorbs rather than generates energy. The hysteresis in the force-calcium relation is conveniently explained by the same cooperativity mechanism. We propose that a single cooperativity mechanism that depends on the number of strong XBs can explain the hystereses in the force-length as well as in the force-calcium relationships.

Approaches to modeling crossbridges and calcium-dependent activation in cardiac muscle

While the primary function of the heart is a pump, ironically, the development of myofilament models that predict developed force have generally lagged behind the modeling of the electrophysiological and Ca2+-handling aspects of heart cells. A major impediment is that the basic events in force generating actin-myosin interactions are still not well understood and quantified despite advanced techniques that can probe molecular levels events and identify numerous energetic states. As a result, the modeler must decide how to best abstract the many identified states into useful models with an essential tradeoff in the level of complexity. Namely, complex models map more directly to biophysical states but experimental data does not yet exist to well constrain the rate constants and parameters. In contrast, parameters can be better constrained in simpler, lumped models, but the simplicity may preclude versatility and extensibility to other applications. Other controversies exist as to why the activation of the actin-myosin is so steeply dependent on activator Ca2+. More specifically steady-state force-[Ca2+] (F-Ca) relationships are similar to Hill functions, presumably as the result of cooperative interactions between neighboring crossbridges and/or regulatory proteins. We postulate that mathematical models must contain explicit representation of nearest-neighbor cooperative interactions to reproduce F-Ca relationships similar to experimental measures, whereas spatially compressing, mean-field approximation used in most models cannot. Finally, a related controversy is why F-Ca relationships show increased Ca2+ sensitivity as sarcomere length (SL) increases. We propose a model that suggests that the length-dependent effects can result from an interaction of explicit nearest-neighbor cooperative mechanisms and the number of recruitable crossbridges as a function of SL.

Interpreting cardiac muscle force-length dynamics using a novel functional model

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 alpha-myosin heavy chain (MHC) (mouse, rat) or beta-MHC (rabbit, ferret) were activated with solutions from pCa 6.1 to 4.3. Activated fibers were subjected to small-amplitude length perturbations [deltaL(t)] rich in frequency content between 0.1 and 40 Hz. In descriptive validation tests, the model was fit to the ensuing force response [deltaF(t)] in the time domain. In fits to 118 records, the model successfully accounted for most of the measured variation in deltaF(t) (R(2) range, 0.997-0.736; median, 0.981). When some residual variations in deltaF(t) were not accounted for by the model (as at low activation), there was very little coherence (<0.5) between these residual force variations and the applied deltaL(t) input function, indicating that something other than deltaL(t) was causing the measured variation in deltaF(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.

Hystereses in the force-length relation and regulation of cross-bridge recruitment in tetanized rat trabeculae

Various mechanisms have been suggested to explain cardiac force-length Ca2+relations. The existence of a cooperativity mechanism, whereby cross-bridge (XB) recruitment is affected by the number of active XBs, suggests that the force response to length oscillations should lag length oscillations. Consequently, the oscillatory force response should be larger during shortening than during lengthening. To test this prediction, force responses to large-sarcomere length (SL) oscillations (36.7 ± 16.0 nm) at different SLs ( n = 6) and frequencies ( n = 7) were studied in intact tetanized trabeculae dissected from rat right ventricle ( n = 13). Stable tetani were obtained by utilizing 30 μM cyclopiazonic acid in Krebs-Henseleit solution containing 6 mM extracellular Ca2+at 25°C. SL was measured by laser diffraction techniques (Dalsa). Force was measured by silicone strain gauge. Instantaneous dynamic stiffness during large oscillations was measured by superimposing additional fast (50 or 200 Hz) and small-amplitude (2.25 ± 0.25 nm) oscillations. The force responses lagged the SL oscillations at slow frequencies (112 ± 41 ms at 1 Hz), and counterclockwise hystereses were obtained in the force-length plane: the force was higher during shortening than during lengthening. The delay in the force response decreased as the frequency of the SL oscillation was increased. Clockwise hysteresis, where the force preceded the SL, was obtained at frequencies >4 Hz. Similar hysteresis characteristics were obtained in the force-SL and stiffness-SL planes. Maximal lag was observed at the shortest SL, and the delay decreased with sarcomere elongation: 131.1 ± 31.7 ms at 1.78 ± 0.03 μm vs. 14.7 ± 18.5 ms at 1.99 ± 0.015 μm. The results establish the ability of cardiac fiber to adapt XB recruitment to changes in prevailing loading conditions. This study supports the stipulated existence of a cooperativity mechanism that regulates XB recruitment and highlights an additional method to characterize regulation of the force-length relation.

Cardiac myofilaments: mechanics and regulation

The mechanical properties of the cardiac myofilament are an important determinant of pump function of the heart. This report is focused on the regulation of myofilament function in cardiac muscle. Calcium ions form the trigger that induces activation of the thin filament which, in turn, allows for cross-bridge formation, ATP hydrolysis, and force development. The structure and protein-protein interactions of the cardiac sarcomere that are responsible for these processes will be reviewed. The molecular mechanism that underlies myofilament activation is incompletely understood. Recent experimental approaches have been employed to unravel the mechanism and regulation of myofilament mechanics and energetics by activator calcium and sarcomere length, as well as contractile protein phosphorylation mediated by protein kinase A. Central to these studies is the question whether such factors impact on muscle function simply by altering thin filament activation state, or whether modulation of cross-bridge cycling also plays a part in the responses of muscle to these stimuli.