A mathematical model for sarcomere mechanics in cross-striated muscles taking into account the stretch and twist of actin filaments

A model of sarcomere mechanics, which takes into account the elongation of the actin and myosin filaments and twisting of the actin filaments during muscle contraction is suggested. The model accounts for the experimentally observed phenomena of the stretch and twist of actin filaments due to strong binding of myosin heads and pulling force. Some model parameters were estimated from published experimental data. The results of modelling suggest that the twist of actin filaments may play an essential role in mechanical responses of contracting muscle fibres to stepwise changes in their length.     

The active state of cross-striated muscle cell sarcomere (an imitation model)]

An index of the sarcomere active state is introduced. It reflects the most important processes of sarcomere contraction, such as calcium diffusion taking into account permeability of calcium channels and its binding with troponin, formation of energy supplies and interaction of sarcomere contractile proteins. The effect of changes of the values of diffusion coefficients and chemical reactions rates was studied theoretically.     

A mechanochemical model of actin filaments

In eukaryotic cells, actin filaments are involved in important processes such as motility, division, cell shape regulation, contractility, and mechanosensation. Actin filaments are polymerized chains of monomers, which themselves undergo a range of chemical events such as ATP hydrolysis, polymerization, and depolymerization. When forces are applied to F-actin, in addition to filament mechanical deformations, the applied force must also influence chemical events in the filament. We develop an intermediate-scale model of actin filaments that combines actin chemistry with filament-level deformations. The model is able to compute mechanical responses of F-actin during bending and stretching. The model also describes the interplay between ATP hydrolysis and filament deformations, including possible force-induced chemical state changes of actin monomers in the filament. The model can also be used to model the action of several actin-associated proteins, and for large-scale simulation of F-actin networks. All together, our model shows that mechanics and chemistry must be considered together to understand cytoskeletal dynamics in living cells.     

Stress-induced alignment of actin filaments and the mechanics of cytogel

Experimental evidence suggests that anisotropic stress induces alignment of intracellular actin filaments. We develop a model for this phenomenon, which includes a parameter reflecting the sensitivity of the microfilament network to changes in the stress field. When applied to a uniform cell sheet at rest, the model predicts that for sufficiently large values of the sensitivity parameter, all the actin filaments will spontaneously align in a single direction. Stress alignment can also be caused by a change in external conditions, and as an example of this we apply our model to the initial response of embryonic epidermis to wounding. Our solutions in this case are able to reflect the actin cable that has been found at the wound edge in recent experiments; the cable consists of microfilaments aligned with stress at the wound boundary of the epithelium. These applications suggest that stress-induced alignment of actin filaments could play a key role in some biological systems. This is the first attempt to include the alignment phenomenon in a mechanical model of cytogel.     

Efficiency of light diffraction by cross-striated muscle fibers under stretch and during isometric contraction.

When light is diffracted by a single frog muscle fiber the intensities I kappa of the different orders kappa (kappa = 1,2,3) strongly depend on the angle between the axis of the incident beam and the fiber axis. Maximum intensity is not obtained with perpendicular incidence (omega = 0 degree) but at angles that can be calculated for each order number and sarcomere length using Bragg's formula. In analogy to techniques developed for x-ray structure analysis of mosaic crystals we have rotated the fiber around an axis perpendicular to the fiber axis and to the incident beam axis within an angular range delta omega = +/- 35 degrees and recorded the light intensities I kappa. Diffraction efficiencies defined as E kappa = integral of I kappa d omega were studied as a function of sarcomere length and during isometric contraction. The sarcomere length dependences of the efficiencies E kappa of the first three orders show characteristic trends. E1 increases with fiber stretch, E2 has a minimum at a sarcomere length near 2.8 micrometers, and E3 has a maximum near 2.5 micrometers. These trends as well as the observed efficiency ratios are in fairly good agreement with predictions by the intensity formula developed for x-ray structure analysis. During isometric contraction, the diffraction efficiencies of the fiber decrease, with the decreases becoming greater the higher the order number. These decreases might be caused by a longitudinal displacement of myofibrils of up to 0.4 micrometers. The efficiency of light diffraction strongly depends on the tonicity of the bathing fluid. Hypertonic (3/2 x normal) solution reduces E1 to less than half, hypotonic (2/3 x normal) solution increases E1 to almost twice the value obtained in normal Ringer's solution.     

Specific interaction of cytochalasins with muscle and platelet actin filaments in vitro

The cytochalasins (CE, CD, CB and H₂CB) inhibit numerous cellular processes which require the interaction of actin with other structural and contractile proteins. In this report we describe the effects of the cytochalasins on the viscosity and morphology of muscle and platelet actin. The cytochalasins decreased the viscosity of F-actin solutions. The effect of H₂CB, CB and CD on F-actin viscosity was maximal at concentrations of 20–50μM and did not increase with time. In contrast, CE caused a progressive decrease in the viscosity of F-actin solutions which was dependent upon the concentration of CE and the duration of incubation of the CE-actin mixture. After two hours of incubation of drug-actin mixtures, the relative effectiveness of the cytochalasins in reducing the viscosity of F-actin was CE > CD > CB = H₂CB. The effects of CD and CE were paralleled by morphologic changes in negatively stained actin filaments. The effects of the cytochalasins on the viscosity and morphology of muscle and platelet actin were the same whether the drugs were added before or after the polymerization of the protein. These studies show that the interaction of the cytochalasins with actin is highly specific. Because the relative potencies of these drugs for affecting motile processes and the relative affinities of the drugs for binding sites within a variety of cells are CE > CD > CB = H₂CB, the effects of cytochalasins on actin described here may contribute to some of the biological effects of the drugs on motile processes.     

A model of inspiratory muscle mechanics

We have previously shown that the costal and crural parts of the diaphragm have different actions on the rib cage (RC) and that the tension developed in one part is not transmitted perfectly to the other. Thus the diaphragm can be modeled pneumatically or electrically as two generators or pumps in series between the lung and abdomen. As such, the force developed by diaphragmatic contraction is the sum of the forces developed in each part, whereas the volume displaced is the same for each part and equal to the total volume displaced. The costal part of the diaphragm is in series with the intercostal and accessory (IA) muscles between the lung and RC, whereas the crural part is in parallel. The volume displaced by simultaneous contraction of the crural part and IA is the sum of volumes displaced by each part. The action of pleural and abdominal pressure [acting through the area of apposition (Aap) of the diaphragm to RC] can be modelled as a summing junction between IA and RC. With hyperinflation the costal part acts more and more in parallel with both IA and the crural part, whereas Aap diminishes, so that the ability to develop large forces decreases independently of the muscles' force-length relationships. The model also predicts that the factors determining the length of the costal and crural parts are different. Finally, the parallel and serial arrangement of the inspiratory musculature allows for increases in maximum power, maximum force, and maximum velocity by appropriate recruitment of the various muscle groups.     

Mammalian cardiac muscle thick filaments: their periodicity and interactions with actin

Cardiac muscle has been extensively studied, but little information is available on the detailed macromolecular structure of its thick filament. To elucidate the structure of these filaments I have developed a procedure to isolate the cardiac thick filaments for study by electron microscopy and computer image analysis. This procedure uses chemical skinning with Triton X-100 to avoid contraction of the muscle that occurs using the procedures previously developed for isolation of skeletal muscle thick filaments. The negatively stained isolated filaments appear highly periodic, with a helical repeat every third cross-bridge level (43 nm). Computed Fourier transforms of the filaments show a strong set of layer lines corresponding to a 43-nm near-helical repeat out to the 6th layer line. Additional meridional reflections extend to at least the 12th layer line in averaged transforms of the filaments. The highly periodic structure of the filaments clearly suggests that the weakness of the layer lines in x-ray diffraction patterns of heart muscle is not due to an inherently more disordered cross-bridge arrangement. In addition, the isolated thick filaments are unusual in their strong tendency to remain bound to actin by anti-rigor oriented cross-bridges (state II or state III cross-bridges) under relaxing conditions.     

Formation and destabilization of actin filaments with tetramethylrhodamine-modified actin

Actin labeling at Cys(374) with tethramethylrhodamine derivatives (TMR-actin) has been widely used for direct observation of the in vitro filaments growth, branching, and treadmilling, as well as for the in vivo visualization of actin cytoskeleton. The advantage of TMR-actin is that it does not lock actin in filaments (as rhodamine-phalloidin does), possibly allowing for its use in investigating the dynamic assembly behavior of actin polymers. Although it is established that TMR-actin alone is polymerization incompetent, the impact of its copolymerization with unlabeled actin on filament structure and dynamics has not been tested yet. In this study, we show that TMR-actin perturbs the filaments structure when copolymerized with unlabeled actin; the resulting filaments are more fragile and shorter than the control filaments. Due to the increased severing of copolymer filaments, TMR-actin accelerates the polymerization of unlabeled actin in solution also at mole ratios lower than those used in most fluorescence microscopy experiments. The destabilizing and severing effect of TMR-actin is countered by filament stabilizing factors, phalloidin, S1, and tropomyosin. These results point to an analogy between the effects of TMR-actin and severing proteins on F-actin, and imply that TMR-actin may be inappropriate for investigations of actin filaments dynamics.     

Variable patterns of actin filaments in mammalian skeletal muscle

Summary The ultrastructural organization of myofilaments in skeletal muscle was studied in four mammalian species (mouse, rat, hamster, goat). In all these species, myofibrils showing irregularly distributed arrays of a variable number of actin filaments (from 6 to 11) were observed. The proportion of such myofibrils and the predominant patterns of actin filaments varied from one species to another. These results are in agreement with those previously reported for human skeletal muscle.     

Direction and speed of actin filaments moving along thick filaments isolated from molluscan smooth muscle

The active movement of fluorescence-labeled actin filaments along thick filaments isolated from molluscan smooth muscle was observed. Along a single thick filament, actin filaments moved toward the center of the thick filament at the speed of 1.19 +/- 0.38 microns s-1 (mean +/- SD, n = 42) and detached themselves from it upon reaching the central zone. Movement of actin also occurred in the opposite direction, i.e., away from the center, albeit at a much lower velocity (0.09 +/- 0.07 microns s-1, n = 17). Thus, the thick filament shows functional bipolarity in terms of velocity but does not determine the direction of the movement.     

Extra actin filaments at the periphery of skeletal muscle myofibrils.

Myofibrils isolated from a variety of vertebrate muscle fibers have a set of peripheral filaments associated with the periphery of the Z line free to move away from the surface of the myofibril. Decoration with myosin subfragment 1 shows that these are actin filaments.     

A simple model for the cooperative stabilisation of actin filaments by phalloidin and jasplakinolide

The stabilisation of magnesium actin filaments by phalloidin and jasplakinolide was studied using the method of differential scanning calorimetry. The results showed that actin could adapt three conformations in the presence of drugs. One conformation was adapted in direct interaction with the drug, while another conformation was identical to that observed in the absence of drugs. A third conformation was induced through allosteric inter-protomer interactions. The effect of both drugs propagated cooperatively along the actin filaments. The number of the cooperative units determined by using a quantitative model was larger for jasplakinolide (15 actin protomers) than for phalloidin (7 protomers).     

Cofilin increases the bending flexibility of actin filaments: implications for severing and cell mechanics

We determined the flexural (bending) rigidities of actin and cofilactin filaments from a cosine correlation function analysis of their thermally driven, two-dimensional fluctuations in shape. The persistence length of actin filaments is 9.8 μm, corresponding to a flexural rigidity of 0.040 pN μm². Cofilin binding lowers the persistence length ∼5-fold to a value of 2.2 μm and the filament flexural rigidity to 0.0091 pN μm². That cofilin-decorated filaments are more flexible than native filaments despite an increased mass indicates that cofilin binding weakens and redistributes stabilizing subunit interactions of filaments. We favor a mechanism in which the increased flexibility of cofilin-decorated filaments results from the linked dissociation of filament-stabilizing ions and reorganization of actin subdomain 2 and as a consequence promotes severing due to a mechanical asymmetry. Knowledge of the effects of cofilin on actin filament bending mechanics, together with our previous analysis of torsional stiffness, provide a quantitative measure of the mechanical changes in actin filaments associated with cofilin binding, and suggest that the overall mechanical and force-producing properties of cells can be modulated by cofilin activity.     

The pause in firing of a primary muscle spindle at the transition between dynamic and static stretch: A mathematical model

A mathematical model to offer an explanation of the pause in firing of a primary muscle spindle at the transition between dynamic and static stretch is suggested. It is proposed that the firing frequency is proportional to a generator current at the site of the transduction (presumably the five terminal endings) and that the receptor potential is a version of this current filtered by the passive cable properties of the nerve between site of transduction and site of measurement.A mathematical expression is derived for the receptor potential from data in the literature. From this and an equivalent circuit of the nerve, an expression for the generator current is derived.This work was partially supported by National Institutes of Health Training Grant, Presbyterian-St. Luke's Hospital grant-in-aid and National Science Foundation. It was done in partial fulfillment of the requirements for the degree of Doctor of Philosophy (HBN) at the University of Illinois at the Medical Center.     

Interaction of tropomyosin-troponin with actin filaments

The assembly of actin filaments with tropomyosin-troponin was investigated by means of light scattering. Binding curves of tropomyosin-troponin [consisting of all three subunits (holotroponin)] and of tropomyosin-troponin-T-I to actin filaments were analyzed by separating the affinity of tropomyosin-troponin for actin filaments and the affinity for the end-to-end contact of tropomyosin molecules. Under the experimental conditions (42.4 degrees C, 300 mM KCl), tropomyosin-holotroponin in the absence of calcium and tropomyosin-troponin-T-I had similar affinities for actin filaments whereas tropomyosin-holotroponin in the presence of calcium was found to bind more weakly. Tropomyosin-holotroponin and tropomyosin-troponin-T-I bound about 200-300-fold more strongly to binding sites with adjacent tropomyosin-troponin units than to isolated sites on actin filaments. The equilibrium constant for isolated association with actin filaments was more than 2-fold higher for tropomyosin-holotroponin in the absence of calcium (15 400 M-1) and tropomyosin-troponin-T-I (17 500 M-1) than for tropomyosin-holotroponin in the presence of calcium (6600 M-1). Binding curves of mixtures of tropomyosin-holotroponin in the presence of calcium and of tropomyosin-troponin-T-I were measured and analyzed on the basis of a model of cooperative binding of two types of large ligands to a one-dimensional homogeneous lattice. The results provided information on the strength of the end-to-end contacts of tropomyosin-troponin units in different positions on an actin filament. It was found that a tropomyosin-troponin unit binds adjacently to another unit in a different position on an actin filament about 2-fold more weakly than adjacent to a unit in the same position. With the aid of these results, it was possible to obtain information of the equilibrium distribution of tropomyosin-troponin in the two positions on actin filaments. Generation of a sequence of tropomyosin-troponin units in a different position on actin filaments was found to be 4-fold less favored than elongation of an existing sequence (cooperativity parameter sigma = 1/4). Shifting of tropomyosin-troponin on actin filaments appears to be accompanied by small free-energy changes in the various interactions of the components of actin-tropomyosin-troponin filaments and not to be an all-or-none reaction