Computer simulation of movement-generating cross-bridges.

A stochastic computational method was developed to study properties of cross-bridge models for muscle contraction, by following the time history of individual cross-bridge model of Andrew Huxley (1957) and a modified two-state model with more realistic behavior during steady stretching are used as examples. The method can readily compute steady-state force during shortening and stretching and force-transients following rapid changes in length. Computations of velocity with a steady load and of velocity transients are more sensitive to the randomness inherent in the stochastic method.     

The first three minutes: smooth muscle contraction, cytoskeletal events, and soft glasses.

Smooth muscle exhibits biophysical characteristics and physiological behaviors that are not readily explained by present paradigms of cytoskeletal and cross-bridge mechanics. There is increasing evidence that contractile activation of the smooth muscle cell involves an array of cytoskeletal processes that extend beyond cross-bridge cycling and the sliding of thick and thin filaments. We review here the evidence suggesting that the biophysical and mechanical properties of the smooth muscle cell reflect the integrated interactions of an array of highly dynamic cytoskeletal processes that both react to and transform the dynamics of cross-bridge interactions over the course of the contraction cycle. The activation of the smooth muscle cell is proposed to trigger dynamic remodeling of the actin filament lattice within cellular microdomains in response to local mechanical and pharmacological events, enabling the cell to adapt to its external environment. As the contraction progresses, the cytoskeletal lattice stabilizes, solidifies, and forms a rigid structure well suited for transmission of tension generated by the interaction of myosin and actin. The integrated molecular transitions that occur within the contractile cycle are interpreted in the context of microscale agitation mechanisms and resulting remodeling events within the intracellular microenvironment. Such an interpretation suggests that the cytoskeleton may behave as a glassy substance whose mechanical function is governed by an effective temperature.     

Axial and radial forces of cross-bridges depend on lattice spacing

Nearly all mechanochemical models of the cross-bridge treat myosin as a simple linear spring arranged parallel to the contractile filaments. These single-spring models cannot account for the radial force that muscle generates (orthogonal to the long axis of the myofilaments) or the effects of changes in filament lattice spacing. We describe a more complex myosin cross-bridge model that uses multiple springs to replicate myosin's force-generating power stroke and account for the effects of lattice spacing and radial force. The four springs which comprise this model (the 4sXB) correspond to the mechanically relevant portions of myosin's structure. As occurs in vivo, the 4sXB's state-transition kinetics and force-production dynamics vary with lattice spacing. Additionally, we describe a simpler two-spring cross-bridge (2sXB) model which produces results similar to those of the 4sXB model. Unlike the 4sXB model, the 2sXB model requires no iterative techniques, making it more computationally efficient. The rate at which both multi-spring cross-bridges bind and generate force decreases as lattice spacing grows. The axial force generated by each cross-bridge as it undergoes a power stroke increases as lattice spacing grows. The radial force that a cross-bridge produces as it undergoes a power stroke varies from expansive to compressive as lattice spacing increases. Importantly, these results mirror those for intact, contracting muscle force production.     

Muscle activation and contraction: constitutive relations based directly on cross-bridge kinetics

This paper reviews recent work aimed at deriving tractable constitutive relations for skeletal muscle from biophysical cross-bridge theories. Discussion is focused on a model proposed previously by the first author (the Distribution-Moment Model), which emphasizes the important role of the moments of the actin-myosin bond-distribution function. The theory leads to a relatively simple third order state variable model for contraction dynamics in which the state variables are the three lowest order moments of the bond-distribution function; further, these three moments have simple macroscopic interpretations as muscle stiffness, force, and elastic energy. New results are presented on the formulation of a compatible model for excitation-contraction coupling, and this model requires the introduction of only one more state variable--the free calcium concentration.     

Modeling left ventricular dynamics with characteristic deformation modes

A computationally efficient method is described for simulating the dynamics of the left ventricle (LV) in three dimensions. LV motion is represented as a combination of a limited number of deformation modes, chosen to represent observed cardiac motions while conserving volume in the LV wall. The contribution of each mode to wall motion is determined by a corresponding time-dependent deformation variable. The principle of virtual work is applied to these deformation variables, yielding a system of ordinary differential equations for LV dynamics, including effects of muscle fiber orientations, active and passive stresses, and surface tractions. Passive stress is governed by a transversely isotropic elastic model. Active stress acts in the fiber direction and incorporates length–tension and force–velocity properties of cardiac muscle. Preload and afterload are represented by lumped vascular models. The variational equations and their numerical solutions are verified by comparison to analytic solutions of the strong form equations. Deformation modes are constructed using Fourier series with an arbitrary number of terms. Greater numbers of deformation modes increase deformable model resolution but at increased computational cost. Simulations of normal LV motion throughout the cardiac cycle are presented using models with 8, 23, or 46 deformation modes. Aggregate quantities that describe LV function vary little as the number of deformation modes is increased. Spatial distributions of stress and strain change as more deformation modes are included, but overall patterns are conserved. This approach yields three-dimensional simulations of the cardiac cycle on a clinically relevant time-scale.

     

Fading memory model for airway smooth muscle dynamic response

This work presents the application of a fading memory model to describe the behavior of contracted airway smooth muscle (ASM) for two biophysical cases: finite duration length steps and longitudinal sinusoidal oscillations. The model parameters were initially determined from literature data on transient step length change response and subsequently the model was applied to the two cases. Results were compared with previously published experimental data on ASM oscillations. The model confirms a trend observed in the experimental data which shows that: (i) the value of tissue length change is the most important factor to determine the degree of cross-bridge detachment and (ii) a strong correlation exists between increasing frequency and declining stiffness until a certain frequency (∼25 Hz) beyond which frequency dependence is negligible. Although the model was not intended to simulate biophysical events individually, the data could be explained by cross-bridge cycling rates. As the frequency increases, cross-bridge reattachment becomes less likely, until no further cross-bridge attachment is possible.     

Form birefringence of muscle.

We investigate the sensitivity of measurements of muscle birefringence to cross-bridge dynamics in the resting, active, and rigor states. The theory of form birefringence is reviewed, and an optical model is constructed for the form birefringence of muscle. Values for the parameters in the model are selected or deduced from the literature. As an illustration of the use of the model, plausible distributions for the orientations of cross-bridges in the resting, active, and rigor states are constructed using a model for cross-bridge dynamics suggested by Huxley and Kress (1985). The general magnitude of the predictions of our model is comparable with that of published measurements of muscle birefringence. However, the precise values of the predicted birefringence for the resting, active, and rigor states are sensitive to the assumed orientations of cross-bridges. We also investigate the dependence of muscle birefringence on sarcomere length and on disorder in the orientation of the myofilament array. We conclude that measurements of muscle birefringence can play a useful role in distinguishing between proposed models of cross-bridge dynamics.     

The characteristics method applied to the study of muscle dynamics

In two-state sliding filament models of muscle contraction a partial differential equation must be solved to find the cross-bridge distribution functionn(x, t). In this paper the analytical form of this function is obtained by integration along the characteristic line and special cases are presented in which the explicit expression forn(x, t) can be completely determined. These analytical solutions provide a direct mathematical connection between the microscopic contraction parameters contained in the kinetic theories and macroscopic muscle dynamics and are thus used to investigate what parameters influence the transient contractile tension in typical experimental conditions. The results of this investigation are consistent with relevant aspects of muscle physiology.     

Nonlinear Actomyosin Elasticity in Muscle?

Cyclic interactions between myosin II motor domains and actin filaments that are powered by turnover of ATP underlie muscle contraction and have key roles in motility of nonmuscle cells. The elastic characteristics of actin-myosin cross-bridges are central in the force-generating process, and disturbances in these properties may lead to disease. Although the prevailing paradigm is that the cross-bridge elasticity is linear (Hookean), recent single-molecule studies suggest otherwise. Despite convincing evidence for substantial nonlinearity of the cross-bridge elasticity in the single-molecule work, this finding has had limited influence on muscle physiology and physiology of other ordered cellular actin-myosin ensembles. Here, we use a biophysical modeling approach to close the gap between single molecules and physiology. The model is used for analysis of available experimental results in the light of possible nonlinearity of the cross-bridge elasticity. We consider results obtained both under rigor conditions (in the absence of ATP) and during active muscle contraction. Our results suggest that a wide range of experimental findings from mechanical experiments on muscle cells are consistent with nonlinear actin-myosin elasticity similar to that previously found in single molecules. Indeed, the introduction of nonlinear cross-bridge elasticity into the model improves the reproduction of key experimental results and eliminates the need for force dependence of the ATP-induced detachment rate, consistent with observations in other single-molecule studies. The findings have significant implications for the understanding of key features of actin-myosin-based production of force and motion in living cells, particularly in muscle, and for the interpretation of experimental results that rely on stiffness measurements on cells or myofibrils.     

Distributed Representations for Actin-Myosin Interaction in the Oscillatory Contraction of Muscle

In this paper we suggest and test a specific hypothesis relating the attachment-detachment cycle of cross bridges between actin (I) and myosin (A) filaments to the measured length-tension dynamics of active insect fibrillar flight muscle. It is first shown that if local A-filament strain perturbs the rate constants in the cross-bridge cycle appropriately, then exponentially delayed tension changes can follow imposed changes of length; the latter phenomenon is sufficient for the work-producing property of fibrillar muscle, as measured with small-signal forcing of length and at low Ca²⁺ concentration, and possibly for related effects described recently in frog striated muscle. It is not clear a priori that the above explanation of work production by fibrillar muscle will remain tenable when the viscoelastic complexity of the heterogeneous sarcomere is taken into account. However, White's (1967) recent mechanical and electron microscope study of the passive dynamics of glycerinated fibrillar muscle has produced a model of the distributed viscoeleastic structure sufficiently explicit that alternative schemes for cross-bridge force generation in this muscle can now be tested more critically than previously. Therefore, we derive and solve third-order partial-differential equations which relate local interfilament shear forces associated with the perturbed cross-bridge cycles to the over-all length-tension dynamics of an idealized sarcomere. We then show (a) that the starting hypothesis can account approximately for the small-signal dynamics of glycerinated muscle in the work-producing state over two decades of frequency and (b) that the rate constants for cross-bridge formation and breakage, restricted solely by fitting of the model to the mechanical data, determine a cycling rate of cross bridges in the model compatible with recent measurements of ATP hydrolysis rate vs. stretch in this muscle. Finally, the formulation is extended tentatively to the large-signal nonlinear case, and shown to compare favorably with previous suggestions for the origin of the work-producing dynamics of fibrillar flight muscle.     

Modelling concentric contraction of muscle using an improved cross-bridge model.

Despite its overwhelming acceptance in muscle research, the cross-bridge theory does not account for all phenomena observed during muscular contractions. A phenomenon which has received much attention in the biomechanics literature, but has evaded convincing explanation and is not accounted for in the formulation of the classic cross-bridge theory, is the persistent aftereffects of muscular length changes on force production. For example, following muscle shortening, the isometric force of a muscle is depressed for a long time period ( > 5 s) compared to the corresponding isometric force following no length change. In the present study, the classic cross-bridge model was modified in two ways in an attempt to account for the force depressions following muscle shortening. First, the steady-state force depressions following shortening were described by a single scalar variable: the work performed by the muscle during shortening; and second, the dynamic, history-dependent cross-bridge properties were described using a fading memory function. The proposed model was developed and tested for shortening of the cat soleus at constant speeds ranging from 4 to 32 mm/s, for shortening at changing speeds, and for shortening of different magnitudes ranging from 2 to 10 mm. The history-dependent forces during shortening and the steady-state force depressions following shortening were well captured with the modified cross-bridge model. The present model contains two mathematically simple adaptations to the classic cross-bridge model, and is the first such model to account for the long-lasting force depressions following muscle shortening using a single scalar variable.     

Cross-bridge cycling theories cannot explain high-speed lengthening behavior in frog muscle.

The Huxley 1957 model of cross-bridge cycling accounts for the shortening force-velocity curve of striated muscle with great precision. For forced lengthening, however, the model diverges from experimental results. This paper examines whether it is possible to bring the model into better agreement with experiments, and if so what must be assumed about the mechanical capabilities of cross-bridges. Of particular interest is how introduction of a maximum allowable cross-bridge strain, as has been suggested by some experiments, affects the predictions of the model. Because some differences in the models are apparent only at high stretch velocities, we acquired new force-velocity data to permit a comparison with experiment. Using whole, isolated frog sartorius muscles at 2 degrees C, we stretched active muscle at speeds up to and exceeding 2 Vmax. Force during stretch was always greater than the peak isometric level, even during the fastest stretches, and was approximately independent of velocity for stretches faster than 0.5 Vmax. Although certain modifications to the model brought it into closer correspondence with the experiments, the accompanying requirements on cross-bridge extensibility were unreasonable. We suggest (both in this paper and the one that follows) that sarcomere inhomogeneities, which have been implicated in such phenomena as "tension creep" and "permanent extra tension," may also play an important role in determining the basic force-velocity characteristics of muscle.     

Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations

Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush–Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.     

A Distribution-Moment Approximation for Coupled Dynamics of the Airway Wall and Airway Smooth Muscle

Asthma is fundamentally a disease of airway constriction. Due to a variety of experimental challenges, the dynamics of airways are poorly understood. Of specific interest is the narrowing of the airway due to forces produced by the airway smooth muscle wrapped around each airway. The interaction between the muscle and the airway wall is crucial for the airway constriction that occurs during an asthma attack. Although cross-bridge theory is a well-studied representation of complex smooth muscle dynamics, and these dynamics can be coupled to the airway wall, this comes at significant computational cost—even for isolated airways. Because many phenomena of interest in pulmonary physiology cannot be adequately understood by studying isolated airways, this presents a significant limitation. We present a distribution-moment approximation of this coupled system and study the validity of the approximation throughout the physiological range. We show that the distribution-moment approximation is valid in most conditions, and we explore the region of breakdown. These results show that in many situations, the distribution-moment approximation is a viable option that provides an orders-of-magnitude reduction in computational complexity; not only is this valuable for isolated airway studies, but it moreover offers the prospect that rich ASM dynamics might be incorporated into interacting airway models where previously this was precluded by computational cost.     

Suppression of muscle contraction by vanadate. Mechanical and ligand binding studies on glycerol-extracted rabbit fibers

The suppression of tension development by orthovanadate (Vi) was studied in mechanical experiments and by measuring the binding of radioactive Vi and nucleotides to glycerol-extracted rabbit muscle fibers. During active contractions, Vi bound to the cross-bridges and suppressed tension with an apparent second-order rate constant of 1.34 X 10(3) M-1s-1. The half-saturation concentration for tension suppression was 94 microM Vi. The incubation of fibers in Vi relaxing or rigor solutions prior to initiation of active contractions had little effect on the initial rise of active tension. The addition of adenosine diphosphate (ADP) and Vi to fibers in rigor did not cause relaxation. Suppression of tension only developed during cross-bridge cycling. After slow relaxation from rigor in 1 mM Vi and low (50 microM) MgATP concentration (0 Ca2+), radioactive Vi and ADP were trapped within the fiber. This finding indicated the formation of a stable myosin X ADP X Vi complex, as has been reported in biochemical experiments with isolated myosin. Vi and ADP trapped within the fibers were released only by subsequent cross-bridge attachment. Vi and ADP were preferentially trapped under conditions of cross-bridge cycling in the presence of ATP rather than in relaxed fibers or in rigor with ADP. These results indicate that in the normal cross-bridge cycle, inorganic phosphate (Pi) is released from actomyosin before ADP. The resulting actomyosin X ADP intermediate can bind Vi and Pi. This intermediate probably supports force. Vi behaves as a close analogue of Pi in muscle fibers, as it does with isolated actomyosin.