# Thermodynamic properties of muscle contraction models and associated discrete-time principles

- 114 Downloads

## Abstract

Considering a large class of muscle contraction models accounting 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. 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.

## Keywords

Muscle contraction Sliding filaments Thermodynamically consistent time-discretization Clausius–Duhem inequality## Abbreviations

- ATP
Adenosine triphosphate

- PL95
Piazzesi–Lombardi’95

- PVW
Principle of virtual work

## Mathematics Subject Classification

74F25 74H15 65M12 35Q79 92C45## Introduction

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 previous model have been proposed in order to take into account different time scales of the actin–myosin interaction. In particular several models have been developed to account for the power stroke phenomenon [4, 5, 13, 20]. In parallel, the question of the thermodynamic balances associated with the chemical machinery was intensively studied, notably with the fundamental contributions of Hill [10, 11]. Note that these models are specific cases of molecular motors models without the natural diffusion introduced by the Fokker–Plank equation [2, 3, 14, 18]. In this paper, our objective is to develop a formalism allowing to derive these thermodynamic balances for Huxley’s model and its extensions with an additional tracking of these balances at the discrete level after time-discretizing the model dynamics. 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. This last part is general with respect to the chemical microscopic model of interest and could also be extended to similar types of models [3, 19], or those mixing mechanical and chemical modeling elements, for instance [1, 17, 22].

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 results with numerical investigations.

## Modeling of muscle contraction

### Physiology of muscle contraction

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 and detachment processes.

### Huxley’57 model

Considering, in a population of myosin heads, the subset of heads with rest position located at distance \( s \) from their nearest attachment site, we define by \( { a(t,s) } \) the ratio of actually attached heads at time \( t \). Equivalently, the ratio of detached heads is denoted by \( d(s,t) = 1 - a(s,t)\), due to the assumption that both types of filaments are rigid. The sliding velocity \( v_c \) between the filaments is a macroscopic variable, hence independent of *s* and often quasi-static with respect to the microscopic time scales. We refer to “Coupling with a macroscopic model of muscle fiber” section for an illustrating example of coupling between a macroscopic formulation and such a microscopic model.

The detached state is associated with a constant energy level \( w_0 \) and the attached state with an energy \( w_1 \) that depends on the distance \( s \)—the myosin head bound to actin is modeled as an elastic spring. This is where mechanics enters the model, and we point out that we extend here the original Huxley’57 model by allowing the spring to have a non-linear behavior. The myosin head is brought back to the initial detached energy level by the ATP energy input \( \mu _T \).

### Extension of Huxley’57 model

To obtain a behavior closer to physiology, and in particular to capture the power stroke, various extensions of Huxley’57 model have been proposed [4, 5, 13, 20]. These extensions can use more than two states to describe the myosin head and allow interactions with an arbitrary number of attachment sites. In this section, our objective is to present these models in a general form, albeit close to the initial 2-state Huxley’s model, in particular concerning their general mathematical and mechanical properties.

#### Multi-state models

#### Multi-site models

#### Example of a multi-state, multi-site model:Piazzesi–Lombardi’95

An energy \( \mu _T \) is brought to the myosin head by ATP in the transitions \( 2 \rightarrow 5 \) and \( 3 \rightarrow 4 \).

We denote by \( w_q \) the energy associated with the state of vertex \( q \).

## Model properties based on thermodynamics principles

### From conservation of matter to boundary conditions and monotonicity properties

*s*,

*t*), \( a(s,t) + d(s,t) = 1\) as soon as we choose our initial condition \(\forall s \in [s^-, s^+],\,a^0(s) \in [0,1]\) and \( d^0(s) = 1-a^0(s)\), since

#### Boundary values

As explained in our model presentation, we expect the myosin head to only interact with the nearest actin site, which imposes that the probability of being attached must vanish on the boundaries of the interval \( [s^-, s^+] \). 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.e., in \( s^- \) if \( v_c > 0 \) and \( s^+ \) if \( v_c < 0 \)—and then rely on the conditions (3) to obtain the proper value of the solution at the other end—as a property—or alternatively consider periodic boundary conditions. As the first option yields a periodic solution, it is clear that the two options are equivalent. However, they differ at the discrete level, in which case we will have to make a choice, see “A numerical scheme for Huxley’57 model” section.

#### Positivity and boundedness properties

We want to check that the solution has values consistent with ratio quantities. More specifically, we want that, with an initial condition \( a^0(s) \in [0,1] \), the property \( a(s,t) \in [0,1] \) holds. Again, we rely on the solution obtained by the method of characteristic lines (9). As the transition rates and the initial condition are positive, we find that \( a(s,t) \ge 0 \). Then, noting that \( 1 - a(s,t) \) is governed by an equation of the same form as (7) with the initial condition \( 1 - a^0(s) \ge 0 \), we similarly deduce that \( a(s,t) \le 1 \).

### First principle

*s*tends to \(s^{-}\) and \(s^{+}\), with the physical interpretation that no finite energy (\(w_1 a\)) is stored and no detachment flux (\(k_{2} a\) and \(k_{-1} a\)) occurs at the ends of the interval. We will see in “Numerical results and discussion” section that this assumption is easily satisfied in practice when (3) holds. Then, computing the time derivative, we obtain

### Second principle

### Extension to multi-state, multi-site models

Let us now consider the Piazzesi–Lombardi’95 model. We want to establish the thermodynamic balances associated with this model.

#### First principle

#### Second principle

### Coupling with a macroscopic model of muscle fiber

*s*, which justifies our above study. Note, however, the dependency of \(a(\underline{\mathrm {x}},s,t)\) on \(\underline{\mathrm {x}}\), which means that the microscopic model must be solved everywhere in the domain, i.e. at all numerical quadrature points in numerical simulations.

## Discretization and thermodynamic principles at discrete level

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.

### A numerical scheme for Huxley’57 model

Note that the exact aggregated detachment rate goes to infinity on the boundary of the interval \( [s^-, s^+] \). Numerically, we use a finite value defined as given in (36c), and we prove in the following section that this choice does not affect the convergence of the scheme.

For the numerical scheme (35), we also need to prescribe adequate boundary conditions. As the analytical solution of (7) vanishes on the boundaries of the interval \( [s^-, s^+] \), we here again can choose: either a Dirichlet condition on one side, and check the consistency on the other side, or choose periodic boundary conditions and again ensure the consistency on the boundary of the interval \( [s^-, s^+] \). From a numerical point of view, it is in fact more convenient for energy estimates to choose periodic boundary conditions for \( a \), i.e. \( a_0^n = a_\ell ^n \). Note that, with this choice, we do not strictly have \( a_0^n = a_\ell ^n = 0 \). This property is only satisfied approximately, or asymptotically when the spatial discretization length goes to zero.

### Some fundamentals properties

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]).

#### Uniform positivity and boundedness

*i.e.*for \( \varvec{a}\in \mathbb {R}^\ell , \mathbb {D} \varvec{a}\ge 0 \Rightarrow \varvec{a}\ge 0 \) (where we use the convention that a vector is positive if all its coefficients \( a_i \) are positive). Let us then take \( \varvec{a}\in \mathbb {R}^\ell \) such that \( \mathbb {D} \varvec{a}\ge 0 \). We have \( \forall i \in [\![1 , \ell ]\!] \)

#### Consistency

#### \(L^{2}\) *-Stability*

#### Convergence

### First principle

*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 (11). Similarly to (36), we assign a finite value to the energy of the attached state on the boundary of the interval \( [s^-, s^+] \). With the notation

### Second principle

### Extension to multi-state, multi-site models

#### Mass conservation

#### First principle

#### Second principle

### Discretization of the macroscopic model coupling

## Numerical results and discussion

In this section, our goal is to illustrate the analysis of the discrete system presented in the previous section for the Huxley’57 model and the Piazzesi–Lombardi’95 model, which we chose as a representative of the multi-site, multi-state models. 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. Additionally, for the Piazzesi–Lombardi’95 model, we compare our simulation results with that obtained in the original paper as a further validation of our approach.

### Huxley’57 model

Model parameters used in the simulations with the Huxley’57 model

Model parameters | |
---|---|

\(\mu \hbox {T}\) | \(100\hbox { zJ}\) |

\(s^{+}\) | \(20\hbox { nm}\) |

\(s^{-}\) | \(-20\hbox { nm}\) |

\(s^{*}\) | 0 |

\(\tilde{s}\) | \(9\hbox { nm}\) |

\({\bar{s}}\) | \(5\hbox { nm}\) |

\(\kappa _{w}\) | \(1.1\hbox { pN nm}^{-1}\) |

\(k_{\mathrm{max}}\) | \(41.3 \times 10^{-3}\hbox { ms}^{-1}\) |

\(k_{\mathrm{min}}\) | \(10 \times 10^{-3}\hbox { ms}^{-1}\) |

\(k_{\mathrm{mid}}\) | \(30 \,k_{\mathrm{max}}\) |

\(\lambda _{1}\) | \(6.21 \times 10^{-5}\hbox { nm}^{-8}\) |

\(\lambda _{2}\) | \(3\, \lambda _{1}\) |

\(\alpha _{w}\) | \(\frac{1}{2} \kappa _{w}\,({\tilde{s}} - {\bar{s}})^{2}\) |

\(\alpha _{{k}_{1}}\) | \(k_{\mathrm{mid}}/ \lambda _{1}\) |

\(\alpha _{{k}_{2}}\) | \(k_{\mathrm{mid}}\) |

The asymptotic properties of the chosen transition rates and of the associated solutions can be easily obtained using the analytical solution (9) and the theorem of dominated convergence.

We consider two simulation cases. First, we simulate the tension rise in isometric conditions (\(v_c = 0\)). Then, we compute the muscle response in contraction at constant shortening velocity (\(v_c < 0 \)) starting from the isometric steady-state solution.

#### Validation of the thermodynamical identities at discrete level

#### Tension rise

In our first illustration of the results obtained for the Huxley’57 model, we simulate the tension rise in isometric conditions (\(v_c = 0\)). We initialize all heads in the detached state and let the myosin heads evolve. Along the usual tension evolution, our scheme allows us to compute the thermodynamic fluxes associated with muscle contraction—see Fig. 9. In the steady-state regime, the energy input remains positive and heat is dissipated. The force is sustained through the continuous cycling of the myosin heads in interaction with the actin filament. This process is fueled by the energy brought by ATP. We see here the active nature of muscle contraction. Force is produced when the muscle is supplied with energy. Naturally, as the velocity is zero no work is produced.

#### Constant velocity contraction

We now show a second illustrative example with a contraction at constant shortening velocity (\(v_c < 0 \)) starting from the isometric steady-state solution. The simulation results are presented in Fig. 10. After a transient phase, the system reaches a permanent regime in which the classical force-velocity curve is measured [9] (note that in the original experimental protocol force and not length is controlled). In this regime, we observe the energy mechano-transduction performed by the molecular motors: the energy input brought by ATP is for one part converted into work produced by the system (\( \mathcal {W} < 0 \)), and for the other part dissipated by entropy production.

### Piazzesi–Lombardi’95 model

The Piazzesi–Lombardi’95 model reproduces the physiology of muscle contraction more precisely. In particular, it is able to capture the power stroke fast dynamics observed in length step experiments.

We simulate such an experiment starting from the isometric steady state with a length step of \(8\hbox { nm}\). As in the experimental conditions, the length step is made by a ramp of duration \(100 \upmu \hbox {s}\). Note that, here, the compliance of the myosin and actin filaments is neglected as in the original paper. We choose the energy levels as defined in [20]. We use modified transition rates to ensure that detachment rates diverge at infinity. The energy brought by ATP is set to \(50\hbox { zJ}\) following the model assumption that an ATP molecule can be used for the detachment of several myosin heads.

## Concluding remarks

Considering a large class of muscle contraction models based on actin-myosin interaction—i.e. the Huxley’57 model and various extensions thereof, including the Piazzesi–Lombardi’95 model—we have presented a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances. Moreover, we have proposed a complete discretization strategy for which we were also able to obtain discrete versions of the thermodynamic balances and other properties. In addition, we have also shown how these models can be coupled with a macroscopic continuum mechanics formulation in such a way that these balances carry over to the macroscopic level, including for the discrete versions of the models. As muscle energetics are of major relevance in physiology, this is an important achievement, both from a fundamental and numerical point of view. This paves the way, indeed, for detailed numerical studies of energy exchanges in various applications, such as with a complete realistic heart model.

## Notes

### Author's contributions

FK was the main investigator in this work; he also performed the numerical implementation and simulations, and initiated the writing. DC was in a position of secondary supervision, and is responsible for some detailed technical aspects and writing. PM was in a position of primary supervision, and proposed initial ideas on topic and discretization strategy; he’s also responsible for some detailed technical aspects and writing. In addition, DC and PM devised the proposed integration of muscle models in continuum mechanics and the associated discretization ingredients. All authors read and approved the final manuscript.

### Acknowledgements

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

Not applicable.

### Funding

Not applicable.

## References

- 1.Caruel M, Moireau P, Chapelle D. Stochastic modeling of chemical–mechanical coupling in striated muscles. Biomech Model Mechanobiol. 2018;1:1–25.Google Scholar
- 2.Chapelle D, Le Tallec P, Moireau P, Sorine M. Energy-preserving muscle tissue model: formulation and compatible discretizations. J Multiscale Comput Eng. 2012;10:189–211.CrossRefGoogle Scholar
- 3.Chipot M, Hastings S, Kinderlehrer D. Transport in a molecular motor system. ESAIM: M2AN. 2004;38(6):1011–34.MathSciNetCrossRefGoogle Scholar
- 4.Eisenberg E, Hill TL. A cross-bridge model of muscle contraction. Progr Biophys Mol Biol. 1978;33(1):55–82.Google Scholar
- 5.Eisenberg E, Hill TL, Chen Y. Cross-bridge model of muscle contraction. Quantitative analysis. Biophys J. 1980;29(2):195–227.CrossRefGoogle Scholar
- 6.Godunov SK. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Matematicheskii Sbornik. 1959;89(3):271–306.MathSciNetzbMATHGoogle Scholar
- 7.Gonzales O. Exact energy and momentum conserving algorithm for general models in nonlinear elasticity. Comput Meth Appl Mech Eng. 2000;190(13–14):1763–83.MathSciNetCrossRefGoogle Scholar
- 8.Hauret P, Le Tallec P. Energy controlling time integration methods for nonlinear elastodynamics and low velocity impact. Comput Meth Appl Mech Eng. 2006;195:4890–916.MathSciNetCrossRefGoogle Scholar
- 9.Hill AV. The heat of shortening and the dynamic constants of muscle. In: Proceedings of the Royal Society of London, 1938.Google Scholar
- 10.Hill TL. Free energy transduction in biology. Cambridge: Academic Press; 1977.Google Scholar
- 11.Hill TL. Free energy transduction and biochemical cycle kinetics. New York: Dover; 2004.Google Scholar
- 12.Huxley AF. Muscle structure and theories of contraction. Progr Biophys Chem. 1957;7:255–318.CrossRefGoogle Scholar
- 13.Huxley AF, Simmons RM. Proposed mechanism of force generation in striated muscle. Nature. 1971;233:533.CrossRefGoogle Scholar
- 14.Julicher F, Ajdari A, Prost J. Modeling molecular motors. Rev Mod Phys. 1997;69(4):1269–81.CrossRefGoogle Scholar
- 15.Le Tallec P, Hauret P. Energy conservation in fluid structure interactions. In: Kuznetsov Y, Neittanmaki P, Pironneau O, eds. Numerical methods for scientific computing—variational problems and applications, 2003.Google Scholar
- 16.Lymn RW, Taylor EW. Mechanism of adenosine triphosphate hydrolysis by actomyosin. Biochemistry. 1971;10(25):4617–24.CrossRefGoogle Scholar
- 17.Marcucci L, Washio T, Yanagida T. Including thermal fluctuations in actomyosin stable states increases the predicted force per motor and macroscopic efficiency in muscle modelling. PLoS Comput Biol. 2016;12(9):e1005083.CrossRefGoogle Scholar
- 18.Mirrahimi S, Souganidis PE. A homogenization approach for the motion of motor proteins. Nonlinear Differ Equ Appl. 2012;20(1):129–47.MathSciNetCrossRefGoogle Scholar
- 19.Peskin CS. Mathematical aspects of heart physiology. Courant Institute of Mathematical Sciences—NYU, 1975.Google Scholar
- 20.Piazzesi G, Lombardi V. A cross-bridge model that is able to explain mechanical and energetic properties of shortening muscle. Biophys J. 1995;68:1966–79.CrossRefGoogle Scholar
- 21.Richtmyer RD, Morton KW. Difference methods for initial-value problems. Interscience tracts in pure and applied mathematics. Geneva: Interscience Publishers; 1967.Google Scholar
- 22.Sheshka R, Truskinovsky L. Power-stroke-driven actomyosin contractility. Phys Rev. 2014;89:1.Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.