Velocity of contraction at different external loads.

<p>Applied tension is reduced or increased with respect to isometric tension at a given time. Simulated Tension vs. Time traces are shown in the insert. Tensions are normalized to the isometric tension, . For different tensions, velocities are calculated from the linear portion of the shortening traces and normalized to the maximum velocity. The simulated velocities vs. tensions (blue triangles) are compared with experimental data (black dots) from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Piazzesi5" target="_blank">[35]</a>. In the concentric region (), the simulations fit the experimental data well even if the velocity is underestimated in the central part (see text). Experimental data for the eccentric region () are not shown, but a qualitative correspondence can be observed with a plateau region at high tensions. Mean valuesstandard deviation over 11 trials are shown. Where error bars are not visible, errors are smaller than the symbol width.</p

From Single Molecule Fluctuations to Muscle Contraction: A Brownian Model of A.F. Huxley's Hypotheses

<div><p>Muscular force generation in response to external stimuli is the result of thermally fluctuating, cyclical interactions between myosin and actin, which together form the actomyosin complex. Normally, these fluctuations are modelled using transition rate functions that are based on muscle fiber behaviour, in a phenomenological fashion. However, such a basis reduces the predictive power of these models. As an alternative, we propose a model which uses direct single molecule observations of actomyosin fluctuations reported in the literature. We precisely estimate the actomyosin potential bias and use diffusion theory to obtain a Brownian ratchet model that reproduces the complete cross-bridge cycle. The model is validated by simulating several macroscopic experimental conditions, while its interpretation is compatible with two different force-generating scenarios.</p> </div

Predicted efficiency at different velocities

<p>Simulation of the efficiency of contraction, as predicted by the model compared to experimental data (circles from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Barclay1" target="_blank">[38]</a>, green line). Efficiency is computed as tension times velocity divided by the chemical energy consumed <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Parmeggiani1" target="_blank">[37]</a> (triangles) and following the method described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Sekimoto1" target="_blank">[39]</a> (squares). Mean valuesstandard deviation over 11 trials are shown. Where error bars are not visible, errors are smaller than the symbol width.</p

Bias of actomyosin energy.

<p>The diffusion of a particle linked to an external micro-needle in a periodic potential tilted by a constant bias, , is simulated, and the ratio of forward, , and backward, , jumps for different is reproduced and compared to the experimental data. Red squares, ; green triangles, ; and blue rumble, . Black dots, experimental data from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Kitamura2" target="_blank">[15]</a>. Continuous lines are the corresponding exponential fits. Simulations show that a bias higher than corresponds to the observed ratio. A value of is chosen as most probable due to the boundary conditions (see text). Insert: Experimental oscillating sub-steps as expected from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Huxley1" target="_blank">[1]</a> during the attached state (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042.s002" target="_blank">fig. S2</a>). Figure adapted from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Kitamura2" target="_blank">[15]</a>.</p

Microscopic behaviour of the model

<p>Upper: Simulated mean strain per motor during the steady state phase of isotonic contraction at different external tensions compared with experimental data from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Piazzesi3" target="_blank">[8]</a>. Mean strain is well fitted at high , but overestimated at low , where the low number of attached motors may have compromised the experimental analysis, as observed in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Piazzesi3" target="_blank">[8]</a>. Lower: Relative number of attached motors at different loads during the steady state phase of isotonic contraction. Simulation, triangles; experimental data, circles. Mean valuesstandard deviation over 11 trials are shown. Where error bars are not visible, errors are smaller than the symbol width.</p

Cross-bridge cycle (left) and half-sarcomere model (right).

<p>Left side: Four states, two detached (upper) and two attached (lower), are considered. In the force generating step, the actomyosin complex oscillates between stable states either by rotating the lever arm domain (scenario 1) or by sliding the myosin head along the actin filament (scenario 2). Right side: Mechanical representation of the half-sarcomere as many parallel myosin heads. In the detached state, thermal fluctuations by a myosin head are constrained only by an asymmetric elastic element <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Kaya1" target="_blank">[29]</a>; the absence of other interactions is represented by the flat energy landscape. In the attached state, the head also experiences an actomyosin complex energy landscape, , which has a periodicity of and four minima or stable states (see text). The jump process between attached and detached states is driven by the rate functions and () to generate a flashing Brownian ratchet.</p

Fast tension recovery.

<p>Simulations and experimental data for a small and fast length step, , applied to a muscle fiber in isometric contraction. The simulated tensions after the imposed step, (red triangles), and after actomyosin re-equilibration, (blue triangles), are shown and compared to the experimental results (circles; data from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Piazzesi4" target="_blank">[33]</a>). All tensions are normalized with respect to isometric tension . Simulated Tension vs. Time traces are shown in the insert at different , is estimated by the tangent method <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040042#pone.0040042-Ford1" target="_blank">[41]</a>. Mean valuesstandard deviation over 11 trials are shown. Where error bars are not visible, errors are smaller than the symbol width.</p

Calculation of small step frequency from the distant binding state.

<p>The distribution of small steps, <i>s</i>, from the distant binding state was constant (0.14) against ATP concentration. Errors for the frequency were calculated using errors for each apparent stepping rate, the law of error propagation and the freeware Maxima.</p

Spontaneous Detachment of the Leading Head Contributes to Myosin VI Backward Steps

<div><p>Myosin VI is an ATP driven molecular motor that normally takes forward and processive steps on actin filaments, but also on occasion stochastic backward steps. While a number of models have attempted to explain the backwards steps, none offer an acceptable mechanism for their existence. We therefore performed single molecule imaging of myosin VI and calculated the stepping rates of forward and backward steps at the single molecule level. The forward stepping rate was proportional to the ATP concentration, whereas the backward stepping rate was independent. Using these data, we proposed that spontaneous detachment of the leading head is uncoupled from ATP binding and is responsible for the backward steps of myosin VI.</p> </div

Including Thermal Fluctuations in Actomyosin Stable States Increases the Predicted Force per Motor and Macroscopic Efficiency in Muscle Modelling

<div><p>Muscle contractions are generated by cyclical interactions of myosin heads with actin filaments to form the actomyosin complex. To simulate actomyosin complex stable states, mathematical models usually define an energy landscape with a corresponding number of wells. The jumps between these wells are defined through rate constants. Almost all previous models assign these wells an infinite sharpness by imposing a relatively simple expression for the detailed balance, i.e., the ratio of the rate constants depends exponentially on the sole myosin elastic energy. Physically, this assumption corresponds to neglecting thermal fluctuations in the actomyosin complex stable states. By comparing three mathematical models, we examine the extent to which this hypothesis affects muscle model predictions at the single cross-bridge, single fiber, and organ levels in a <i>ceteris paribus</i> analysis. We show that including fluctuations in stable states allows the lever arm of the myosin to easily and dynamically explore all possible minima in the energy landscape, generating several backward and forward jumps between states during the lifetime of the actomyosin complex, whereas the infinitely sharp minima case is characterized by fewer jumps between states. Moreover, the analysis predicts that thermal fluctuations enable a more efficient contraction mechanism, in which a higher force is sustained by fewer attached cross-bridges.</p></div