Muscled discrete bottom

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Metrics details. Considering a large class of muscle contraction models ing for actin—myosin interaction, we present a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances. Moreover, we propose a complete discretization strategy for which we are also able to obtain discrete versions of the thermodynamic balances and other properties.

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Our major objective is to show how the thermodynamics of such models can be tracked after discretization, including when they are coupled to a macroscopic muscle formulation in the realm of continuum mechanics. Our approach allows to carefully identify the sources of energy and entropy in the system, and to follow them up to the numerical applications. The modeling of the active mechanical behavior of muscles has been the object of intense research since the seminal work of Huxley [ 12 ] modeling the attachment-detachment process in the actin—myosin interaction responsible for sarcomere contraction.

Then, numerous extensions—mostly based on refinements of the chemical process introduced by Huxley—of the model have been proposed in order to take into different time scales Muscled discrete bottom the actin—myosin interaction. In particular several models have been developed to for the power stroke phenomenon [ 451320 ]. In parallel, the question of the thermodynamic balances associated with the chemical machinery was intensively studied, notably with the fundamental contributions of Hill [ 1011 ].

Note that these models are specific cases of molecular motors models without the natural diffusion introduced by the Fokker—Plank equation [ 231418 ]. Moreover, we present how these microscopic models can be incorporated into a macroscopic model of muscle fibers in the spirit of [ 2 ] with the aim of following these thermodynamic balances at the macroscopic level for the continuous-time dynamics but also after adequate time discretization.

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This last part is general with respect to the chemical microscopic model of interest and could also be extended to similar types of models [ 319 ], or those mixing mechanical and chemical modeling elements, for instance [ 11722 ]. The outline of the paper is as follows. The first section presents the modeling ingredients of the microscopic models of actin—myosin interaction and we derive in a second section the fundamental properties of these models with the associated thermodynamic balances, up to the coupling with the macroscopic mechanical formulation. The third section then describes the discretization scheme and justifies its thermodynamic compatibility.

Finally, the last section illustrates our with numerical investigations. Muscles are multi-scale structures in which motion is initiated at the cellular level by the relative sliding between two types of filaments: actin filaments and myosin filaments. At the Muscled discrete bottom of the myosin filament, myosin he can bind to the actin filament.

The actin filament has a periodic structure with regularly spaced attachment sites. The interaction between myosin he and actin sites occurs in a cyclic manner [ 16 ], see Fig. The cycle includes attachment and detachment of the myosin head to and from an actin site and a conformation change of the attached myosin head called the power stroke. The detachment stage requires an energy input obtained from ATP molecules buffered inside the cell.

Lymn—Taylor cycle representation. Each stage of the cycle can be seen as a change of chemical state. Different levels of description of the actin—myosin interaction can be considered benefiting from the fact that the power stroke occurs much faster than the attachment Muscled discrete bottom detachment processes. In his seminal work [ 12 ], Huxley describes the myosin head with two chemical states representing the attached and detached configurations.

Each myosin can interact with its closest actin site only. Top: definition of the transitions between the attached state A and the detached state D. Bottom left: model parametrization, representation in the detached state. Bottom right: model parametrization, representation in the attached state. Transition rates between the states satisfy the detailed balance, i. A schematic of the model is presented in Fig. In a nutshell, the parameter functions must satisfy.

This implies energetically that. Actin sites and myosin he are located at discrete locations separated by regular intervals along their respective filaments. The spatial periodicities are, however, different on each filament. These extensions can use more than two states to describe the myosin head and allow interactions with an arbitrary of attachment sites. A complete directed graph is a set of vertices connected by edges, in which: edges have a direction; for each edge of the graph, the edge connecting the same vertices in the inverse direction also belongs to the graph; no vertex is connected to itself.

Here, we use superscripts in the edges definition to denote that there are two reactions between the same vertices. We extend the definition of the ratio of he in detached states by periodicity, i. The system dynamics is governed by. A specific representative—denoted PL95—of this family of models has been derived in [ 20 ] with the aim of ing for the energetics of muscle contraction. It describes the myosin head with five states arranged in two cycles of chemical reactions, see Fig. A first long cycle cycle a is meant to represent a complete power stroke, while a short cycle cycle b allows the myosin head to cycle at small or zero sliding velocity with incomplete power stroke.

The vertex indices are given in red. The transition indices are given in blue. Moreover, it is assumed in this model that the myosin can attach to an arbitrary of actin sites, hence it is also multi-site. Therefore, we can rewrite the system 2 in the form of a single equation. However, the dynamics 2 is a first-order transport equation associated with only one boundary condition.

Therefore, we can either consider one single Dirichlet boundary condition at one end of the interval—i. As the first option yields a periodic solution, it is clear that the Muscled discrete bottom options are equivalent. In fact, closed-form expressions can be obtained for the solution.

We want to check that the solution has values consistent with ratio quantities. Again, we rely on the solution obtained by the method of characteristic lines 9. We consider a system made of a population of myosin he and define the average energy per myosin head, namely. Then, computing the time derivative, we obtain. The energy balance can be interpreted as follows: the energy brought by ATP is for one part converted into work, the other part being dissipated as heat production.

The system remains at a constant temperature, the outside environment playing the role of a thermostat. We introduce the Helmholtz free energy. Using again integrations by part for the transport terms and the boundary properties of the solution, we obtain. Combining Eqs. The model will be compatible with the second principle if this entropy production is always positive.

Using the relation 1 deduced from the detailed balance, we recall that. As a consequence, we have two cases. Proceeding in the same way for the second reaction, we finally have. We can summarize this property using 15 by the free energy balance. We want to establish the thermodynamic balances associated with this model. In particular, conservation of matter here re. The first principle then naturally re. We have, as in 23the free energy balance. The thermodynamic properties of these classes of models are very useful when coupling them with a macroscopic model, typically to represent a muscle fiber, as it will ensure a global consistent thermodynamic balance between macroscopic and microscopic contributions.

Let us consider, indeed, a macroscopic model of muscle fiber modeled in the realm of non-linear continuum mechanics, as large deformations Muscled discrete bottom occur in muscle fibers. The fiber as shown in Fig. Following [ 2 ], which extends the classical Hill-Maxwell scheme [ 9 ] to nonlinear behavior, we gather all the constitutive ingredients by defining an adequate rheological scheme—presented in Fig. The lower branch represents a 3D passive matrix, associated with the cellular envelope and the extracellular matrix. Each branch contains elastic and viscous constituents, respectively visualized by springs and dashpots, with specific constitutive equations given below.

Fiber rheology combining a 1D active element upper branch and a 3D visco-hyperelastic element lower branch. However, we will depart from [ 2 ] for the series branch.

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In fact, the natural view of muscle fibers made of a succession of active and passive segments points to a one-dimensional homogenization type of rheological interpretation. Then, the length change of the half-sarcomere can be decomposed into. Therefore, we can summarize as. Moreover, 28 must be complemented by relationships between 3D and 1D quantities.

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Considering the component of the Green-Lagrange strain tensor in the fiber direction, we directly have. Hence, the associated contribution in the second Piola-Kirchhoff stress tensor re. This velocity is independent of the microscopic variable swhich justifies our above study.

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We thus get. Then, we decompose. Then, using the rheological rules we find. We now present the proposed discretization scheme for the muscle contraction models. Classical schemes are sufficient for our purposes, and the main originality of this work is to show their compatibility with discrete versions of the thermodynamical principles. Nevertheless, for the sake of completeness, some basic properties of the schemes are quickly re-established before focusing on thermodynamics.

The discretization scheme then re. Numerically, we use a finite value defined as given in 36cand we prove in the following section that this choice does not affect the convergence of the scheme. For the numerical scheme 35we also need to prescribe adequate boundary conditions. This property is only satisfied approximately, or asymptotically when the spatial discretization length goes to zero.

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We first present the basic—but essential—properties of the proposed scheme. This is done using classical strategies for the analysis of transport equations schemes see for instance [ 21 ]. Therefore, the proposed numerical scheme preserves the adequate positivity and boundedness. Note that this solution satisfies a Dirichlet boundary condition on one side of the simulation interval i. We define, as usual, the convergence error by.

The numerical scheme is thus consistent at the first order with the continuous Eq. Note that a well-known result by Godunov states that we cannot have more than first-order convergence in time with a discrete Muscled discrete bottom that satisfies the positivity and boundedness property [ 6 ]. The stability analysis coupled to the consistency analysis gives directly the convergence error.

Indeed, we find. Our objective is more ambitious than numerical convergence, as we want in fine to establish thermodynamic balances at the discrete level. In this respect, let us first consider the energy balance. We recall that the average energy of a myosin head is given at the continuous level by With the notation.

Let us now establish a discrete entropy balance. In this respect, we introduce the discrete entropy. We then rewrite the calculation in a manner that closely follows the calculation in the continuous case. We have. Developing the expression of the chemical potentials, performing Abel transformations and using the periodic boundary conditions, we obtain.

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We finally find. Comparing 41 with the formal expression of the second principle 18we define the discrete entropy creation by. The developments are, however, not straightforward because the multi-site assumption implies an infinite of attachment and detachment fluxes, which has to be properly integrated into the discrete thermodynamical balances. The last term vanishes with the periodic boundary conditions. Performing an Abel transformation on the penultimate term and defining the discrete force as. We can derive a full discretized version of the macroscopic model presented in Fig.

Here, we will rely—as in [ 2 ]—on mid-point rules for the discretization of the PVW, with additional corrections in order to guarantee the energy balance. First, as recommended in [ 27815 ], we will consider the following non-standard—albeit classical—mid-point quantities. If the 1D elastic element is chosen nonlinear hyperelastic, the corresponding term in 50c has to be treated as proposed for the 3D elastic element in The balance associated with the hyperelastic contribution is handled by Muscled discrete bottom choice in 49and the viscous part directly gives a negative contribution, so that we have.

These illustrations serve several purposes. We first want to demonstrate that the thermodynamics identities established at the discrete level are satisfied in the numerical simulations. Then, we want to show that the ability to compute the thermodynamical balances numerically allows to gain additional insight into the physiology of muscle contraction. We choose the energy levels and transition rates as follows.

Muscled discrete bottom

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