Biology subjects

Louvet, S.

Publications and source records attributed to Louvet, S..

6 recordsLinked to original sources

bioRxiv2019

Mechanical model of muscle contraction. 3. The orientation of the levers belonging to the myosin heads in working stroke follows the same uniform law in all half-sarcomeres of an isometrically stimulated fiber

A myosin II head is modelled during the working stroke (WS) by three rigid segments articulated between them: the motor domain (S1a), the lever (S1b) and the rod (S2). Hypothesis 4 introduced in accompanying Paper 2 states that the lever of a WS head moves in a fixed plane where the position of S1b is characterized by the angle {theta}. This assumption allows the geometrization of a cross-bridge, i.e. the poly-articulated chain consisting of five rigid segments: the actin filament (Afil), S1a, S1b, S2, and the myosin filament (Mfil). The equations established in Paper 2 are operative to calculate the number of heads potentially in WS for a Mfil surrounded by six Afil. In addition, the value of the angles {theta} of the levers belonging to these WS heads is accessible. This census leads to an integer number (Np) of angular positions ({theta}i) distributed discretely between {theta}up and {theta}down, the two values that delimit {theta} during the WS. The number of Mfil per half-sarcomere (hs) is estimated between 400 and 2000 depending on the typology, figures that induce Gaussian variability for each of the Np values {theta}i calculated for a single Mfil. By summing the Gaussian Np densities and after normalization, we obtain a probability density (dG) of the continuous variable {theta} between {theta}up and {theta}down. The function dG is calculated for a random length of a hs between 1 and 1.1 m where the binding rate of the myosin heads is maximum. From this reference length, the hs is shortened 11 times with a step of 1 nm, i.e. a total of 11 nm. For each shortening, a count of the new {theta}i positions is performed, which leads to a new probability density dG. The classic statistical law that approximates these 12 distributions of {theta} is the Uniform law between {theta}up and {theta}down. Other conditions and values given to the data of the algorithmic procedure lead to a similar result, hence the formulation of hypothesis 5: the distribution of the angle {theta} follows an identical uniform law in all the hs of a muscle fiber stimulated in isometric conditions.

biophysics↗

bioRxiv2019

Mechanical model of muscle contraction. 1. Force-velocity relationship

The two parameters that determine the functionality of a skeletal muscle fiber are the tension (T) exerted at its two endpoints and the shortening speed (V), two mechanical characteristics. We established a relationship between T and V by developing a theoretical model of muscle contraction based on the swinging lever arm hypothesis. At the nanoscale, force and movement are generated by the myosin II heads during the working stroke (WS). The change in conformation of a myosin head during the WS is characterized by the rotation of the lever correlated to the linear displacement of the motor domain. The position of the lever is marked by the angle {theta}. The maximum variation of {theta} between the two limits {theta}up and {theta}down relating to the two positions up and down is usually given equal to 70{degrees}. When the angle {theta} is between {theta}up and {theta}down, the WS is triggered in three modes, fast, slow or very slow. During the isometric tetanus plateau, {theta} is uniformly distributed between the two angles {theta}up and {theta}T separated by a usual difference of 50{degrees}. Consequently during isometric tetanus plateau there is a 20{degrees} interval between {theta}T and {theta}down where no head is found in WS. We link this absence to the slow detachment of the heads whose orientation of the levers is between {theta}T and {theta}down during the rise to the isometric tetanus plateau. The equation between T and V refers to these four occurrences: fast, slow or very slow initiations of the WS between {theta}up and {theta}down, then slow detachment between {theta}T and {theta}down. The equation is constructed from the geometric data of the myosin head and the time constants of the cross-bridge cycle reactions associated with these four events. The biphasic aspect of the curve is explained by the slow detachment that occurs only at very slow speeds. An additional term, derived from the viscosity present as soon as the velocity increases completes the equation. An adequate fit between the model and examples from the physiological literature is found (r2 > 99%).

biophysics↗

bioRxiv2019

Mechanical model of muscle contraction 2. Kinematic and dynamic aspects of a myosin II head during the working stroke

The condition of a myosin II head during which force and movement are generated is commonly referred to as Working Stroke (WS). During the WS, the myosin head is mechanically modelled by 3 two by two articulated segments, the motor domain (S1a) strongly fixed to an actin molecule, the lever (S1b) on which a motor moment is exerted, and the rod (S2) pulling the myosin filament (Mfil). When the half-sarcomere (hs) is shortened or lengthened by a few nanometers, it is assumed that the lever of a myosin head in WS state moves in a fixed plane including the longitudinal axis of the actin filament (Afil). As a result, the 5 rigid segments, i.e. Afil, S1a, S1b, S2 and Mfil, follow deterministic and configurable trajectories. The orientation of S1b in the fixed plane is characterized by the angle {theta}. After deriving the geometric equations singularizing the WS state, we obtain an analytical relationship between the hs shortening velocity (u) and the angular velocity of the lever [Formula]. The principles of classical mechanics applied to the 3 solids, S1a, S1b and S2, lead to a relationship between the motor moment exerted on the lever (MB) and the tangential force dragging the actin filament (TA). We distinguish {theta}up and {theta}down, the two boundaries framing the angle {theta} during the WS, relating to up and down conformations. With the usual data assigned to the cross-bridge elements, a linearization procedure of the relationships between u and [Formula], on the one hand, and between MB and TA, on the other hand, is performed. This algorithmic optimization leads to theoretical values of {theta}up and {theta}down equal to +28{degrees} (-28{degrees}) and -42{degrees} (+42{degrees}) respectively with a variability of {+/-}5{degrees} in a hs on the right (left), data in accordance with the commonly accepted experimental values for vertebrate muscle fibers.

biophysics↗

bioRxiv2019

Mechanical model of muscle contraction. 4. Theoretical calculations of the tension during the isometric tetanus plateau and the tension exerted at the end of phase 1 of a length step

Hypothesis 4 presented in accompanying Paper 2 states that the lever of a myosin II head in working stroke (WS) moves in a fixed plane, the orientation of the lever being defined by the angle {theta}. From this conjecture can be deduced the hypothesis 5 developed in accompanying Paper 3: the distribution of {theta} is identical and uniform in each half-sarcomere (hs) of a muscle fiber stimulated under isometric conditions. We propose a sixth hypothesis that establishes a linear relationship between the {theta} angle and the motor moment ([M]) exerted on the lever. These three hypotheses lead to calculations of the tension during isometric tetanus plateau (T0) and the tension applied at the end of phase 1 of a length step when the only internal actions are the forces of elastic origin produced by the myosin heads in WS (T1Elas). However, the T1Elas values are higher than those observed experimentally. The model introduces the presence of viscosity as the seventh hypothesis. The internal actions resulting from the coupling of the elasticity of the WS heads and the viscosity make it possible to explain all the observed phenomena that contribute to the phase 1 of a length step. An adequate adjustment between the theoretical tension from the model (T1) and the tension representative of the end of phase 1 exposed in examples from the physiological literature is proven (r2 > 98%). Other parameters such as stiffness (e), compliance (C) and strain (Y) are deduced; their investigation enables the construction of an analytical "nanoscope" by means of which the uniform density of {theta} is explored. The equations for T0, T1, e, C and Y explain and predict the influence of factors such as the duration of phase 1, the initial length of the sarcomere, the concentration of calcium, the presence of an inhibitor, the tension rise to the isometric tetanus plateau, relaxation after tetanization or shortening at constant speed. The results obtained during a slack-test are indicated by the model, the slack of the fiber being interpreted as an event of purely viscous origin.

biophysics↗

bioRxiv2019

Mechanical model of muscle contraction. 5. Tension rise after phase 1 of a length step

The theoretical approaches developed in accompanying Papers 1 to 3 lead to calculations of the isometric tetanus tension (T0) and the minimum tension (T1) observed at the end of phase 1 of a length step where the fiber is shortened (see Paper 4). During the next three phases, the time rise of the tension (T), from T1 to T0, is determined for any step (see Supplement S5.K). The tension T is expressed as a master equation which is the sum of five terms: (a) T1, (b) a positive or zero contribution resulting from the relaxation induced by the disappearance of the viscosity forces present during phase 1, (c) a positive contribution of elastic origin resulting from the new myosin II heads initiating a working stroke (WS) in the blank areas, (d) a negative contribution caused by the fast detachment of the heads still strongly attached and whose orientation of the levers is beyond the up position, (e) a negative contribution caused by the slow detachment of WS heads whose orientation of the levers is close to the up position. An agreement between the model equation and the experimental results referenced in the physiological literature is proven (r2>97.5%). The kinetics of each of the theoretical curves make it possible to distinguish phases 2, 3 and 4 characteristics of the tension rise to T0. The criteria defined to describe the tension at the end of phase 2 (T2) are applied to the master equation. There is an adequate adjustment between the theoretical and experimental T2 values for shortenings less than 8 nm in modulus (r2 > 97%).

biophysics↗

bioRxiv2019

Mechanical model of muscle contraction. 6. Calculations of the tension exerted by a skeletal fiber during a shortening staircase

Accompanying Paper 1 tests a theoretical relationship between force and shortening velocity of a muscle fiber without justifying its validity. Paper 2 determines the kinematics and dynamics of a myosin II head during the working stroke (WS). Paper 3 imposes the Uniform law as a density representative of the orientation of the levers belonging to the WS heads. By support of these works, Papers 4 and 5 put into equation the evolution of the tension during the four phases of a length step. The present paper closes all six articles by imposing two tasks on itself. The first purpose is to apply the theoretical elements developed for a length step to a succession of identical length steps, otherwise known as shortening staircase. With the values of the geometric and temporal parameters assigned to a myosin head in Papers 1 to 5, a correct adjustment is established between the theoretical tension deduced from our model and the experimental tension published in 1997 by a team of Italian researchers relating to nine shortening staircases performed on the same fiber. In particular, we obtain the equation of the tension reached at the time end of the step (T*) which remains constant step by step as soon as the shortening of a half-sarcomere exceeds 17 nm. The second objective is to find and explain the equation of the Force-Velocity curve introduced ex abrupto into Paper 1: by decreasing the size and duration of the steps, the staircase tends towards a constant slope line corresponding to a continuous speed shortening. By applying the methods of infinitesimal calculus to the different formulations leading to T*, we deduce the Force-Velocity relationship (see Supplement S6.L). And the circle is complete.

biophysics↗