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

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Physiology of muscle contraction

In this chapter, we present a literature review of the experimental works targeting the actin-myosin interaction and its regulations with the aim of compiling the relevant infor-mation for the development of the models. We first give a general qualitative review of the heart function and the associated underlying mechanisms following a top-down ap-proach. In a second part, we focus on the quantitative properties of the cardiac muscle. We first analyze the experimental results aiming at characterizing the interaction between myosin heads and actine sites and then focus on the characterization of the regulation mechanisms.
In this chapter, we give a synthetic review of the structure of muscles in animals and the physiology of muscle contraction with some personal interpretation.
Our presentation will be organized in two main parts. We will start by a qualitative description of the muscle excitation-contraction mechanism, mainly focused on the phys-iology of cardiac muscles. Skeletal muscles, which share essential features with cardiac muscles, will be briefly introduced as a variation of the cardiac muscle and sometimes used as a benchmark for cardiac muscles, although the characterization of skeletal muscles has historically been performed first. Then, we will present the state of the art on the quantitatively properties of cardiac muscles.
A particular emphasis will be put on the levels of physiology that will be used to support the development of our models (actin-myosin interaction, thin filament activation, microscopic mechanisms underlying the Frank-Starling macroscopic regulation).
Our presentation of general physiology mainly relies on [Silverthorn et al., 2009]. For more details on the sarcomere structure we refer to [Craig and Padrón, 2004] and to [Craig and Woodhead, 2006] for a detailed description of the myofilaments. A presentation of the excitation mechanisms can by found in [ Bers, 2002]. Our review of the quantitative properties of the excitation will be two-fold. First, to present the actin-myosin interac-tion, we use the extensive work of the group of Vincenzo Lombardi at the University of Florence. Second, the description of the thick and thin filaments activation and the as-sociated regulation is mainly supported by the works of Henk ter Keurs, Peter de Tombe and their co-authors (see e.g. [de Tombe et al., 2010; de Tombe and ter Keurs, 2016]). The physiological latter mechanisms are at the origin of one of the main regulation at organ level called the Frank-Starling mechanism, which makes the left ventricular pressure at the end of the contraction vary as a function of the ventricular volume. For a general presentation on this topic, we refer to the reviews by Allen and Kentish [1985], ter Keurs [1996] de Tombe et al. [2010] and Sequeira and Velden [2017].

Description of muscle contraction

To qualitatively present the various mechanisms involved in muscle contraction, we follow a top-down approach from the macroscopic description of the tissue to the nanometric events occurring at the level of individual proteins.

Anatomy of muscles

Muscles are one of the four types of soft tissues in animals along with connective tissues, nervous tissues and epithelial tissues. Their function is to transform a signal from the nervous system into a mechanical force. They are made of the association of proteins taking care of the structure integrity, the generation of force and the regulation of the generated force. There are two categories of muscles in animals: striated muscles and smooth muscles.

Different types of muscles in human body

Striated muscles, which group together cardiac and skeletal muscles, contract in response to electrical pulse signals resulting from conscious or unconscious control. Their normal functioning is a succession of contraction and relaxation phases paced by the electrical stimulations. Smooth muscles are found in the walls of organs such as the stomach or blood vessels. As opposed to striated muscles, they contract under unconscious control and generate near permanent contractions. Only striated muscles will be studied in this manuscript and we will restrict the presentation to this type of muscle, and in particular to cardiac muscles, in what follows.

Multiscale structure of striated muscle in space

Cardiac muscle are a multi-scale structure in space. They are made of an arrangement of cardiac cells, called cardiomyocytes, which form an array of branching fibers. The cells

Anatomy of a cardiomyocyte

Cardiomyocytes contain all the materials needed to perform the activation-contraction coupling (also called the excitation-contraction coupling). It is the transformation of a local electrical signal into a mechanical force.
The cell membrane contains ion channels and ion exchangers that allow the transfer of ions inwards and outwards. These elements react as a function of the membrane electrical potential and are affected by the ions concentrations on both sides of the membrane. The ions exchangers of interest for the muscle contraction are the calcium sodium exchangers (denoted NCX). They exchange one Ca2+ ion against three Na+ ions. The NCX has a threshold potential that depends on the calcium and sodium concentrations. When the membrane potential is higher than the NCX potential, calcium is brought into the cell, and vice versa.
The sarcomeres in series span across the cell and are linked on the boundaries to the intercalated disks in order to transmit the force to the neighboring cells.
The cell is also equipped with a reservoir of calcium ion called the sarcoplasmic retic-ulum (sarcoplasmic reticulum). It captures calcium ions from the cell cytosol with ATP-driven Ca2+ -pumps and releases them quickly and in a large amount when activated. The activation is triggered by inwards calcium fluxes detected by receptors denoted RyRs. The RyRs also act as release channels for calcium ions. The released calcium ions are captured by the sarcomere, where they trigger the contraction. Under the action of the calcium Ca2+-pumps, calcium is liberated by the sarcomere and uptaken by the sarcoplasmic retic-ulum. The transfer of ions through the membranes is an active process in the sense that it consumes energy brought by ATP. It uses between 30% and 40% of the energy consumed by the cell [Barclay, 2015].
The structure of a cardiomyocyte is schematically displayed in Figure 1.2.
We now define the calcium concentrations that will be used through the entire manuscript. The external calcium concentration is denoted by [Ca2+]ext. In the cell, we distinguish between the membrane ion concentration, which is related to the ions linked to the mem-brane and the intracellular calcium concentration depending on the amount of ion in the cytosol. The intracellular calcium concentration is denoted by [Ca2+]i.

Anatomy of a sarcomere and the myofilaments

The elementary unit of the contractile apparatus is the sarcomere. The sarcomere is a highly organized structure (see Figure 1.3). It has a longitudinal symmetry and can thus be seen as two half-sarcomeres contracting in opposite directions. A sarcomere is delimited by protein components appearing as thick black lines on microscope images called Z-disks. Our description of the sarcomere structure relies on Craig and Padrón [2004], completed by [Kobayashi et al., 2008] for the description of the thin filament.
The sarcomere is mainly composed of two family of parallel filaments overlapping themselves in a crystalline-like arrangement (see Figure 1.3). The first family is made of the actin filaments, which are anchored on the Z-disks. The second family is made of myosin filaments spanning in both directions from the sarcomere median plan on which they are linked. In the middle of the sarcomere, a protein structure connects the thick filaments. It is the M-line that appears in the microscope image.
The actin filaments form the so-called thin filaments, which are polymers made of actin monomers forming a double helix (see illustration in Figure 1.4.). The spatial periodicity of the double helix is 37–40 nm, while the actin monomer is 5:5 nm long. Each actin monomer has a myosin binding site. At rest, these binding sites are covered by a tropomyosin molecule (Tm), which prevents myosin binding. Tm is a 40 nm long polymer that is wrapped around the actin filaments. The thin filament is thus covered by a set of Tm that overlap in their head and tail regions. The part of the thin filament that is covered by one Tm molecule is called a Regulatory Unit (RU). A regulatory unit thus includes seven actin monomers and has a length of 38:5 nm. In each RU, a troponin complex (Tn) is bound to Tm and block its position in front of the actin sites. The troponin complex is made of three proteins: the troponin-T (TnT) that actually binds to Tm, the troponin-C (TnC) that binds to calcium and a troponin-I (TnI). At rest, a domain of TnI binds to actin, which strongly fixes the Tm with respect to actin. In the presence of calcium, the affinity of TnI for another part of the troponin complex increases, TnI detaches from the actin filament. The tropomyosin is liberated and this allows the access to some actin sites of the regulatory unit for the myosin heads. Note that due to the varying orientation of the actin site along the actin filament helix, it is likely that the myosin head can not bind to all actin sites but only to those that have the appropriate orientation. The question of how many actin
The myosin filaments form the the so-called thick filaments . They contain 294 myosin molecules [Reconditi et al., 2017], each of which is made of a head, which is exposed towards the outside of the filaments, and a tail, which is anchored in the backbone of the filament. The heads are organized in a regular array forming helices. The periodicity of the helix is 43 nm, meaning that the myosin heads are separated by this distance along the longitudinal direction. Along the helix direction, the myosin heads are separated by 14:3 nm [Craig and Padrón, 2004] (see Figure 1.5). In the central region of the thick filament, there is no myosin head; it is called the bare zone. The density of thick filament in a cross section is 424 filaments/m2 [Pinzauti et al., 2018]. The thin and thick filaments are regrouped under the term myofilaments.
In addition to these two types of filaments, another element, called titin, plays an important role in muscle contraction. It is a giant protein spanning from the Z-disks, wrapping around the thin filament and anchored in the M-line. Titin allows the sarcomere to recover its original length in the relaxation phase. Titin also contributes to the passive stiffness of the tissue [ Fukuda et al., 2008; Mateja et al., 2013; Methawasin et al. , 2014]. In addition, it may play a role in many regulatory mechanisms such as the on-off transition (see Section and the modulation of the thin filament sensitivity to calcium (see Section
The rest length of the sarcomere is called the slack length. The slack length of cardiac sarcomeres, is commonly reported to be 1:85 µm [ter Keurs et al., 1980] for rat and 1:70 µm for humans [van der Velden et al., 2000]. We can note that a negligible stiffness is measured around the slack length which explains the difficulty to define the stress-free state of cardiac fibers [Caruel et al., 2013].

Anatomy of a myosin head

As already mentioned above, the myosin has two main components: its tail and its head. The head is named the S1 domain and the tail can itself be split into two parts: the light meromyosin (LMN), which is anchored in the myosin filament, and the S2 domain that sticks out of the filament core (see Figure 1.6). Electron-microscopy and X-ray crystallographic analyses reveal the complex 3D structure of the myosin heads [Rayment et al., 1993; Irving et al., 2000]. The myosin head has two stable positions corresponding to two conformations of the protein. The angle between the head and the tail changes between these two conformations by 70°. It corresponds to a longitudinal displacement (projected along the filament direction) of 11 nm [Irving et al., 2000] (see Figure 1.6). Note that this value is sometimes also reported to be 8 nm [Kaya and Higuchi, 2010].
When bound together the actin site and the myosin head form a cross-bridge. Ex-periments on individual molecules show that the cross-bridge has elastic properties. The compliance is concentrated in the head or at the connection point between the actin site and the myosin head, the tail part of the myosin being much stiffer [Kaya and Higuchi, 2010].
When a myosin head is attached, the change between the two stable conformations generates a force and the transition is then called the power stroke. The initial and final conformation are naturally referred to as pre-power stroke and post-power stroke, respectively.


Origin of contraction – the actin-myosin interaction

The actin-myosin interaction occurs in a cyclic manner. A commonly accepted description of this cycle has been proposed by Lymn and Taylor [1971]. It is illustrated in Figure 1.7. The Lymn-Taylor cycle is composed of four stages:
• in the first stage, the myosin head in the pre-power stroke conformation attaches to an actin site.
• in the second stage, the myosin undergoes a conformation change, which generate a force. This is the power stroke.
• in the third stage, the myosin head, which is now in the post-power stroke conforma-tion, detaches. The use of chemical energy brought by ATP molecules is necessary for the detachment. In each cycle, this consumes an energy of 100 zJ [Barclay, 2015].
• in the fourth stage, the detached myosin head recovers its power stroke capability by transitioning back to the pre-power stroke conformation. The myosin head can then enter in a new cycle.
Note that experiments in solution indicate that the power stroke is a fast step – its duration is of the order of 1 ms – compared to the whole cycle duration, which is of the order of 100 ms [Linari et al. , 2010]. Conclusions drawn for experiments in solution must be transposed with care to intact muscle, where geometric and mechanical constraints are drastically different. Nevertheless, this still shows a separation of time scales between a fast power stroke stage compared to the rest of the cycle comprising the attachment-detachment process. A confirmation of this property is given by mechanical experiments on individual cardiac muscle cells, which display the same separation of time scales (see Section The Lymn-Taylor cycle gives a correct picture of the interaction between myosin heads and actin sites but there is still a lot of unknowns in this molecular inter-action. Intense research activities are dedicated to the description of all the structural changes occurring during this cycle [Houdusse and Sweeney, 2016].

Thick filament activation

In the above presentation of the interaction between a myosin head and an actin site, we suppose that the myosin head is always available for attachment and that the actin site is always accessible for the myosin head. However, in physiological conditions, this may not be the case.
It has recently been observed in skeletal muscles with X-ray crystallographic measure-ments, that the myosin head can be in a position where the S2 domain does not point out of the thick filament backbone anymore, but is instead aligned along the filament back-bone [Reconditi et al., 2011] and the myosin heads is folded back towards the M-line. This state is called off-state, as opposed to on-states when the S2 domain points towards the thin filament. Additional investigation with skeletal muscles, showed that the on-off tran-sition is triggered by a mechanosensing mechanism [Linari et al., 2015]. More recently, the off-state was also observed in cardiac muscles but only at rest and in isometric conditions [Reconditi et al., 2017].
Moreover, when stretching the sarcomere, the level of activation of the thick filament varies (see Section

Thin filament activation

As for the myosin filament, the level of activation of the thin filament is controlled and regulated. The cardiac cycle is an alternation of contractions and relaxations, both of which are solely triggered, in physiological conditions, by the presence or the absence of Ca2+ ions in the cytosol, which serve to activate the thin filament. Pacemaker cells in the atria create a depolarization wave that propagates in the heart in a controlled manner. When the depolarization wave reaches a cardiomyocyte, it induces the release of calcium in the cytosol of the contractile cells. The calcium ions then interact with the thin filament to activate the actin sites and allow the binding of myosin heads. This activation signal transduction is regulated extrinsically by the neuroendocrine system (see Section 1.1.5) and intrinsically by the extension at the scale of the sarcomere (see Section
The variations of the level of thin filament activation act as a command to switch the muscle cell between the contraction and relaxation phases.
In this section we present the different stages of the signal transduction between the pacemaker signal to the activation of the actin site keeping our top-down approach. We then introduce the regulation mechanisms. Note that an illustration of the events of the activation process occurring at the cell level is presented in Figure 1.2.

Macroscopic activation signal propagation

We will first consider the basal behavior, that is in the absence of the extrinsic regulation (see [Bers, 2002] for more details). At the level of the sinoatrial node (in the right atrium) special cardiac muscles cells, called pacemaker cells, generate an electrical signal. The proteins across the cell membrane allow a inflow of calcium which results in increasing the amount of positive charges inside the cell. As a result, the originally negative membrane potential is increasing. When the potential reaches a threshold, the membrane depolarizes. In response to depolarization, a sequence of events involving the movement of ions into and out of the cell occurs, thus changing the membrane potential. This behavior is spontaneous (is not necessarily triggered by a signal from the nervous system) and occurs at a constant frequency (60–80 bpm). The sinoatrial mode is a natural pacemaker.
As part of the sequence of events resulting from the depolarization, ion currents appear toward the neighboring cells thus changing their own membrane potential. At some point the potential reaches a threshold and the neighboring cells depolarizes. The depolarization of the pacemaker cells thus triggers the depolarization of the surrounding cells in the atrium. The depolarization wave propagates and reaches the atrioventricular node which acts like a synchronizer.
The depolarization wave propagates then through the bundle of His and the Purkinje fibers to the apex of the heart and triggers the depolarization of the contractile cells. From the apex, the depolarization spreads to the ventricular contracting cells and travels to the top of the ventricle (towards the valves).

Interactions at the cell membrane – the action potential

We will now describe in more detail the sequence of events occurring in the contractile cells leading to the action potential.
At rest, the contractile cell membrane is polarized. It contains more negative charges inside the cell than the outside environment which creates a negative potential. ATP-fueled pumps maintain this potential by bringing sodium and calcium ions out and potassium ions in.
The activation is initiated from the Purkinje fibers or a neighboring cell. In response to the stimulation, the sodium ion channels open causing a large flux of sodium ions towards the inside of the cell. The potential increases; the cell is depolarizing (phase 0). Following the depolarization (phase 1), the potassium channels open and a small amount of potassium ions goes out of the cell, and causes a slight decreases of the potential. It is the early repolarization (phase 2). In a subsequent step, the fluxes of calcium ions into the cell and potassium ions out of the cell balance, causing a plateau phase in the evolution of the membrane potential (phase 3). In the last phase, calcium channels close and potassium ions keep leaving the cell. The potential decreases to its original value; this is the repolarization of the cell (phase 4). The shape of the action potential in a contractile cell is presented in Figure 1.8(a). Note that the action potential in pacemakers cell has a different shape. Note also that the superposition of the action potentials gives rise to the macroscopic ECG signal.
The sodium channels that intervene in the early phase of the action potential need time after being activated to reach their initial configuration and be active again [Silverthorn et al., 2009]. The time during which the sodium channel cannot be activated is called the refractory period. In this period an electrical stimulation will have no effect on the sodium channels and thus will not trigger an action potential. In cardiac muscles the refractory period ends as the force relaxation is well advanced. Therefore, a new contraction cannot be initiated until the current contraction is finished. As a result, tetanus is prevented under physiological conditions. Preventing tetanus in cardiac muscle is physiologically relevant because the cardiac cycle consists in alternating periods of contraction (ejection phase) and relaxation (filling phase). A tetanised state would thus stop the cardiac cycle. However, note that under specific non-physiological conditions, a tetanised state is still possible in cardiac muscle (see Section

Table of contents :

Introduction (English)
Introduction (Français)
1 Physiology of muscle contraction 
1.1 Description of muscle contraction
1.1.1 Anatomy of muscles
1.1.2 Origin of contraction – the actin-myosin interaction
1.1.3 Thick filament activation
1.1.4 Thin filament activation
1.1.5 Neuroendocrine regulation
1.1.6 Additional information
1.1.7 Skeletal muscles
1.2 Passive properties of muscle cells
1.3 Characterization of fast muscles activation-contraction coupling
1.3.1 Muscle twitch contraction
1.3.2 Actin-myosin interaction
1.3.3 Activation of the thin filament
1.4 Characterization of the muscle active contraction regulation
1.4.1 Effect of the regulation at the scale of the sarcomere
1.4.2 Origin of the observed regulation mechanisms
1.4.3 Regulation by the neuroendocrine system
1.4.4 Link with the Frank-Starling mechanism
1.5 Limitations of our presentation
1.5.1 Influence of temperature
1.5.2 Variability between species
1.5.3 Isolation from the neuroendocrine regulation
1.5.4 Conclusion of the limitations
1.6 Appendix
1.6.1 Hill’s cooperativity models
2 Thermodynamic properties of muscle contraction models and associated discrete-time principles 
2.1 Introduction
2.2 Modeling of muscle contraction
2.2.1 Physiology of muscle contraction
2.2.2 Huxley’57 model
2.2.3 Extension of Huxley’57 model
2.3 Model properties based on thermodynamics principles
2.3.1 From conservation of matter to boundary conditions and monotonicity properties
2.3.2 First principle
2.3.3 Second principle
2.3.4 Extension to multi-state, multi-site models
2.3.5 Coupling with a macroscopic model of muscle fiber
2.4 Discretization and thermodynamic principles at discrete level
2.4.1 A numerical scheme for Huxley’57 model
2.4.2 Some fundamentals properties
2.4.3 First principle
2.4.4 Second principle
2.4.5 Extension to multi-state, multi-site models
2.4.6 Discretization of the macroscopic model coupling
2.5 Numerical results and discussion
2.5.1 Huxley’57 model
2.5.2 Piazzesi-Lombardi’95 model
2.6 Concluding remarks
2.7 Appendix
2.7.1 Numerical scheme for negative sliding velocities
3 Hierarchical modeling of force generation in cardiac muscle 
3.1 Introduction
3.2 Hierarchy of models
3.2.1 Population models
3.2.2 Macroscopic models
3.3 Results
3.3.1 Population models
3.3.2 Macro models
3.4 Discussion
3.4.1 Limitations of the models
3.4.2 Limitations of our calibration
3.5 Conclusion
3.6 Appendices
3.6.1 Reference calibration
3.6.2 Asymptotic calculation of the T1-curve
3.6.3 Asymptotic calculation of the T2-curve
3.6.4 Computation of the PSE tension
3.6.5 Extrapolation of the rate of phase II in length control conditions for cardiac cells
4 Activation-contraction coupling in a multiscale heart model capturing the Frank-Starling effect
4.1 Introduction
4.2 Physiological review
4.2.1 Frank-Starling effect
4.2.2 Evidence of a regulation mechanism intrinsic to the thick filament
4.2.3 Dynamics of the regulation
4.3 Model presentation
4.3.1 Contraction model – Huxley’57 model family
4.3.2 Incorporation of the variations in myosin availability level in the model equations
4.3.3 Comparison with previous formulations
4.4 Thermodynamics
4.4.1 Myosin heads conservation
4.4.2 First principle
4.4.3 Second principle
4.4.4 Coupling with a macroscopic model
4.5 Range of validity and limitations
4.5.1 Impact of the homogenized description in the pool model
4.5.2 Comparison with individual description of the myosin heads
4.5.3 Comparison with previously proposed formulation
4.6 Discretization
4.6.1 Microscopic numerical scheme
4.6.2 Numerical illustration
4.6.3 Link with discrete macro model: a multi-time step strategy
4.7 Physiological simulation of a heart beat
4.7.1 Modification to account for the thin filament activation
4.7.2 Model calibration
4.7.3 Numerical results
4.8 Conclusion
4.9 Appendix
4.9.1 Some properties of the Chapelle’12 Frank-Starling model
4.9.2 Proof of the equivalence between the random exchange model and the homogenized pool model
4.9.3 Proof of the discrete thermodynamics identities
4.9.4 Numerical scheme for negative sliding velocities
4.9.5 Validation of the discrete thermodynamics balance illustration
4.9.6 Moment equation
5 Varying thin filament activation in the framework of the Huxley’57 model
5.1 Abstract
5.2 Introduction
5.3 Physiological review
5.3.1 Activation of the thin filament
5.3.2 Regulation mechanisms
5.4 Model presentation
5.4.1 Actine-myosin interaction and thick filament activation
5.4.2 Thin filament activation
5.4.3 Model steady-state
5.5 Model calibration and validation
5.6 Conclusion
5.7 Appendix
Conclusions and perspectives


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