1  Context and Positioning

In this chapter, we give the essential background for understanding the principles of muscle contraction. Some elements of this presentation will be refined in appropriate chapters.

1.1 Muscle contraction in brief

1.1.1 The contractile structure

Figure 1.1: Hierarchical structure of the muscle tissue, from Britannica (2015)

The human body contains three types of muscles: the skeletal muscles, responsible for motion, the cardiac muscle responsible for the blood circulation, and the smooths muscles found essentially in the digestive apparatus. This manuscript focuses on the group of striated muscles which contains the skeletal and caridac muscles.

The striated muscle tissue consists in a hierarchy of bundled fibers, with the highest level corresponding to the tissue itself, see Figure 1.1. The skeletal muscles contain fascicles¹ (diameter \(\sim\) 1 mm ), each one grouping about 10–100 myocytes (diameter 15 μm ). The myocytes, also called muscle fibers, are the fibrilar cells responsible for the contraction. Their cytoplasm contains mostly a parallel arrangement of 2 μm diameter myofibrils, in addition to the classical cell organelles (nucleus, mitochondria etc.).

¹ the fascicles structures are specific to skeletal muscles.

Figure 1.2: Side view of the contractile apparatus inside a myofibril. The cross-section shows the hexagonal lattice formed by the myosin (thick) and actin (thin) filaments. Adapted from (Caruel and Truskinovsky 2018).

The core of the contractile machinery is embedded in the myofibrils, see Figure 1.2. It consists of a highly regular longitudinal succession of \(\sim\) 2 μm-long sarcomeres, which are connected by the Z-lines. A sarcomere itself is constituted of two antagonist parallel arrays of \(\sim\) 1500 contractile units facing each other on both sides of the M-lines.

The cross-section of a sarcomere reveals a crystalline hexagonal lattice of myofilaments of actin (thin) and myosin (thick) running parallel to the direction of the fiber. The unit cell of this lattice thus contains one thick filament and two thin filaments. When the muscle contracts, the myosin proteins that constitute the thick filaments induce a relative sliding of the surrouding actin filaments that results in the shortening of the sarcomeres. This active shortening is due to the metabolic activity of the myosin proteins which act as molecular motors.

The contractile structure also involves other structural proteins (see M-lines, Z-disks and titin in Figure 1.2) that will be presented in more details in Section 5.1.1. We will now focus on the molecular mechanism of force generation by the molecular motors.

1.1.2 Molecular mechanism of contraction

Figure 1.3: Lymn–Taylor cycle. A myosin protein is represented by a spring with a rotating head (the second head is not represented). The myosin head can attach to specific site located on a surrounding actin filament (step \(1\to 2\)). While the head is attached (steps \(2\to 3\)) it undergoes a conformational change, the power-stroke (or working stroke) that produces force. After the head detachment (step \(3\to 4\)), this conformational change is reversed (steps 4 \(\to\) 1). Adapted from (Caruel and Truskinovsky 2018).

The thick filament is a bundle of 600 myosin proteins connected by their tails. They form into two antagonists contractile units of 300 individuals each, see Figure 1.2. A single myosin protein has a pair of heads that points radially, from the myosin filament towards the surrounding thin filaments. It is the cyclic interaction between myosin heads and actin that produces the active force necessary to shorten the sarcomeres. The actin-myosin interaction cycle is referred to as the Lymn and Taylor (1971) cycle², see Figure 1.3.

² Here we present the classical Lymn-Taylor cycle, which contains four steps. Since their fundamental 1971 publication, numerous tudies on the molecular mechanism of force generation have contributed to the refinement of this cycle. See for instance the review by Houdusse and Sweeney (2016).

³ see Chapter 4 for more details about the activation process

When the muscle contraction is activated³ the myosin heads can attach to specific actin binding sites regularly positioned on the actin filaments (step \(1\to 2\)). While attached, the actin-myosin complex is called a cross-bridge.

A fundamental characteristic of the myosin protein is its ability to undergo a conformational change, the working- (or power-) stroke (step \(2\to 3\)) that results, thanks to a lever-arm, into a 10 nm relative displacement of the two bridged filaments in absence of impairing force, see (Huxley and Simmons 1971; Rayment, Holden, et al. 1993; Rayment, Rypniewski, et al. 1993).

After the working stroke has been executed, the myosin detaches from actin (step \(3\to 4\)). In the detached state, the working stroke conformational change is reversed, waiting to be triggered again upon the next attachment (step \(4\to 1\)). This cocking mechanism requires the hydrolysis of one molecule of Adenosine TriPhosphate (ATP). The hydrolysis products, Adenosine DiPhosphate (ADP) and inorganic phosphate (P_i) are released from wihin the molecule alongside the power-stroke in the attached state⁴. For more details about the molecular mechanisms of the enzymatic activity of myosin we refer to Houdusse and Sweeney (2016).

⁴ The exact sequence of events leading to the departure of P_i from the myosin active site and their relationship with the power-stroke not fully resolved. See (Caremani et al. 2013).

The collective actin-myosin interactions, which generate antagonistically oriented working strokes inside each sarcomere, is ultimately responsible for the macroscopic contraction of the muscle fibers.

1.2 Why study muscle contraction? The example of cardiomyopathies

Pathological alterations of the complex contractile apparatus are involved in the development of genetic cardiomyopathies, which, in the most severe cases, lead to Acute Heart Failure (AHF) and sudden death.

The causes of AHF are diverse, which makes it particularly difficult to treat (Rossignol et al. 2019; George et al. 2014). The most prevalent genetic cardiomyopathy degenerating into AHF is the Hypertrophic Cardiomyopathy (HCM), affecting 1 individual per 500 (Maron and Maron 2013). This pathology is often associated with mutations of the myosin protein that result in an increase of the tissue contractility (Morita et al. 2008). Another severe genetic cardiomyopathy leading to AHF is the Dilated Cardiomyopathy (DCM) which accompanies the deterioration of the mechanical properties of the M and Z lines and titin, resulting in the loss of register of the contractile units inside the fibrils (see Figure 1.2), and abnormal inflation of the tissue over time (Hinson et al. 2015; Lange et al. 2019; Gerull et al. 2002; Granzier et al. 2005).

The mechanisms underlying HCM and DCM are not well understood. The connection between protein alterations at various biological levels and the resulting observable effects on heart function has not been clearly established.

A promising therapeutic approach in AHF, HCM, DCM and other cardiomyopathies is the use of small effector molecules, such as Omecamtiv Mecarbil and Mavacamten, that specifically modulate myosin activity (reviewed in Day, Tardiff, and Ostap 2022; RabieeRad et al. 2023). For instance, the clinical trials have shown that cardiac function of patients with severe cardiomyopathies can be improved within minutes of inoculation with Omecamtiv Mecarbil. It is worth noticing that Mavacamten has recently been approved by the U.S. Food and Drug Administration (Nag et al. 2023).

Despite the proven clinical efficacy of these molecules, there remains an unclear understanding of their impact on the contraction process, potentially complicating the adjustment of prescriptions. For this reason, and because of their promising impact on deseases, these drugs are under intense experimental scrutiny at the structural, cellular and organ (or body) scales (Houdusse et al. 2024).

1.3 Some motivations

Between the nano- and the macro-scale, a full understanding of how the drugs or pathologic mutation modify the contraction process is still lacking, even at the most basic mechanical level. Practitioners could leverage this understanding in their clinical decisions. For this reason, mutations leading to cardiomyopathies and the associated treatments are considered to be priority research subjects (Velden et al. 2015).

Existing experimental studies provide valuable information on the effect of mutation and drugs at specific scales of the tissues. The rationale of research is to leverage this knowledge to design a unified modeling framework to emulate the genotype to phenotype relationship in healthy and diseased muscle tissue. Furthermore, the knowledge of the mechanical and physiological pathways explaining this relationship, would open the possibility to engineer small protein effectors based on the simulation of their desired macroscopic effect.

Another potential outcome of the research efforts in understanding the physical principles governing muscle contraction is to design artificial actuators leveraging similar principles. As an example, we may consider elactromechanical actuators being used in machines or vehicles. In some applications, in partiular aircraft engineering, critical functions can be impaired by the malfunction of a single actuator. In such situations, critical motions are often ensured by redundant actuators for savety reasons. However, in some cases, redundancy is not permitted, either because of steric hindering (e.g. for helicopters) or because of weight cost (e.g. for spatial application).

Muscle resilience originates from its multiscale structure and multileveled neuronal command. The structure and command strategies may be advantageously mimicked to create a new kind of resilient actuator. Moreover, recent advancements in nanomachines constructed from purified muscle proteins (Pertici et al. 2018; Saper and Hess 2020) suggest the potential design of actuators assembled from genuine proteins.

In summary, the goal of our research is to provide models of the contraction at different scales that can be used

• to understand the specific impact of pathological mutations and drugs on the mechanisms of contraction,
• to design artificial devices that mimicks the outstanding properties of muscle cells and tissue.

1.4 Four scales

We view the modeling framework as the integration of four projects, each being specific to one of the following scales:

• the nanoscale,
• the microscale,
• the mesoscale,
• the macroscale.

In the following paragraphs we introduce briefly the context of each scale. More details are available in the dedicated chapters.

1.4.1 Nanoscale: the molecular motor

The typical size of a protein is just a few nanometers. At this scale, the objective is to understand and model:

• The mechanics of allosteric enzymatic activity in myosin, specifically its ability to catalyze ATP hydrolysis within an active site of the molecule, and use this reaction to trigger a significant conformational change at a distant site within the molecule.
• The physical principles underlying the interaction between a single myosin protein and an actin filament.

Our goal at the nanoscale is to provide a physiologically relevant dynamical model of the actin-myosin interaction, based on a minimal set of descriptive variables.

1.4.2 Microscale: the contractile unit

A contractile unit is a bundle of molecular motors sharing the same myosin filament backbone and interacting with the surrounding actin filaments. At this scale, the objective is to understand how the molecular motors cooperate. The cooperative mechanisms emerge from the elastic coupling that is mediated by the filament themselves (Caruel and Truskinovsky 2018). Such coupling plays a fundamental role not only in the basic force production but also in the activation and regulation of this process (Brunello and Fusi 2024).

Our goal is to understand how the interconnexions between the molecular motors at the scale of a prototypical contractile unit affect their collective functionning, in particular the force production and its regulation.

1.4.3 Mesoscale: the sarcomere and the myofibril

Stacks of contracile units are arranged parallel to form sarcomeres which are, in turn, assembled in series within muscle fibrils and fibers, see Figure 1.2. It is the synchronized contraction of all activated contractile units that enables the macroscopic shortening of the muscle fibers.

The mechanical coupling between the contractile units is facilitated by scaffolding structures (such as Z-lines, M-lines, and Titin), which maintain the entire hierarchy in alignment and preserve its integrity. These structures also play a fundamental role in the regulation of the contraction (Brunello and Fusi 2024). Finally, it is now clear that mutations affecting the mechanical properties of the M-lines, Z-disks and titin structural proteins that constitute this elastic network, are involved in the development of cardiomyopathies, potentially via an alteration of their ability to maintain the sarcomeres in register (Hinson et al. 2015; Lange et al. 2019; Gerull et al. 2002; Granzier et al. 2005; Herwig et al. 2020; Wadmore, Azad, and Gehmlich 2021).

Finally, the cardiac tissue being a composite material with active stress fibers embedded in the extracellular matrix, the mesoscale structure is also characterized by the orientation of the fibers within the tissue. Microscope observations show that the fiber orientation distribution in healthy and pathological tissues may greatly differ (Hoshino et al. 1983).

Our goals at the mesoscale are to (i) provide a physiologically valid model of the network of sarcomeric structural proteins and study how its properties affect the collective functioning of contractile units, and (2) to formulate a homogenized macroscopic behavior law of the tissue that takes into account its composite microstructure.

1.4.4 Macroscale: muscle fiber and muscle tissue

The macroscale is the scale of the contracting tissue itself. Macroscopic data include the typical continnum mechanics fields of displacement, deformation, stresses etc. In the case of the heart they also include the internal pressure, the ventricules or atria volumes and the blood flow rates in and out the cavities.

The challenge is to provide a model that can reproduce these macroscopic observables using ingredients reminiscent of the actual physiological process. Since these processses are complex, the models may involve many parameters that end up to be difficult to calibrate. Another difficulty lies in the numerical cost of these models which can impair their use in clinical contexts.

1.5 Positionning

In this manuscript, we will position our work mainly with respect to research on the heart contraction in health and disease. However, the models that will be presented can be used with other types of muscle as well.

1.5.1 Experimental research

Muscle contraction mechanisms can be studied experimentally at all scales.

Nanoscale, molecular motors. 3D Molecular structures of the various myosin conformations involved in the contraction cycle can be resolved from X-Ray crystallography and/or Cryogenic Electron Microscopy, both in control and in the presence of a mutation or a small effector, giving insight into how they could influence the molecular mechanism of force generation (Robert-Paganin et al. 2020). These crystallographic structures can be viewed as high resolution snapshots of the protein in different conformations (Houdusse and Sweeney 2016). It is then possible to use molecular dynamics simulation to get insight into the force generation mechanism over short timescales.

The properties of isolated molecular motors can also be probed using mechanical tests in optical tweezers (Arbore et al. 2019; Woody et al. 2018; Woody et al. 2019; Yanagida et al. 2000)

Finally, taking advantage of the crystalline nature of the muscle myofibrils, X-ray diffraction can be used in-situ to probe nanoscale structural change in real time, while performing a mechanical test on a macroscopic sample (Piazzesi et al. 2007). In the latter type of experiment, the average behavior of a motor can be monitored with nanometer resolution while it interacts with other motors in near to physiological conditions.

Micro-scale, contractile units

Interactions between molecular motors within contractile units can be observed in artificial preparations (Marston 2022).

In in-vitro motility assays for instance, large groups of myosin motors are fixed on a plane surface and moving actin filament put on to of them (Warshaw 1996; Holzbaur and Goldman 2010). Contractile units involving a controled number of motors can also be manipulated in vitro. Recently the team lead by Dr. P. Bianco from the PhysioLab (University of Florence, Italy) has succeeded in reconstructing a minimal functional contractile unit out of purified actin and myosin proteins, able to reproduce the performance of the functional unit of the muscle (Pertici et al. 2018; Pertici et al. 2020; Buonfiglio et al. 2024), see also (Kaya et al. 2017; Cheng, Leite, and Rassier 2020).

These experimental setups provide ideal platforms for testing the basic effect of new drugs and for designing artificial biomimetic devices.

Meso-scale: Inter-sarcomere dynamics and regulation.

Over the past two decades, experimental studies at the scale of single fibrils or fibers have revealed the existence of non-uniformities (non-affine behavior) of the sarcomere lengths during contraction. The direct observation can come from single fibrils in an optical tweezer setup or from single cell preparation using fast confocal microscopy. Recent reviews of these studies reports on the important role played by the elastic network that connects the sarcomeres together in tailoring these non-uniformities (Leite and Rassier 2020; Linke 2023; Herzog and Schappacher-Tilp 2023).

It is also hypothesized that these sarcomeric structural proteins participate in fundamental regulation pathways involving mechanical feedback-loops: the force transmitted via the elastic scaffold may enhance contractile units’ activation (Ait-Mou et al. 2016; Linke and Krüger 2010; Caremani et al. 2022; Brunello and Fusi 2024).

The molecular structure of main proteins of the network can be infered from X-ray cristallography directly during the contraction. This approach allowed understanding of how the elastic network react to activation signals (Squarci et al. 2023).

Finally, the distribution of fiber orientation in the tissue can be quantified post-mortem using standard microscopy (Hoshino et al. 1983) or using diffusion-weighted MR imaging (DW-MRI) (Damon et al. 2017).

Macro-scale: experiments on intact trabeculae allow to test the effect of drugs and mutations on the tissue contraction.

The majority of experimental data are obtained from mechanical tests performed on multicellular preparations. For skeletal muscles, in situ experiments are performed either on single fibers or directly on intact muscle preparations. For the heart muscle, a large body of experimental data is obtained from mechanical tests performed on multicellular preparations—typically the pillar-shaped cardiac trabecula⁵—isolated from mammals ventricles [de Tombe and ter Keurs (1990);Ait Mou et al. (2018);Caremani et al. (2016);Pinzauti et al. (2018);].

⁵ the trabeculae muscle (or trabeculae carneae) are muscular columns that projects from the inner surface of the right and left ventricles. They may form simple ridges, or be fixed at both extremities. Papillary muscles are examples of trabeculae that holds the tendinous chords holding the cusps of the valves. Trabeculae muscles are used in experiments because they are essentially one dimensional objects with aligned fibers.

The mechanisms of contraction can then be tested in a wide range of conditions by varying the content of the bathing solution, the temperature and the mechanical loading. As mentioned above, this type of experiment, can be coupled with nanometer resolution X-ray diffraction measurements (Piazzesi et al. 2007).

At the scale of the organ, macroscopic deformation maps can be recovered from tagged cine MRI (Chabiniok et al. 2016). Catheter pressure probes can also be inserted in the heart to record pressure-volume loops (Bastos et al. 2020; Protti et al. 2024; Foëx and Leone 1994).

1.5.2 Mechanical models and heart simulations

The interest of having a physiologically relevant model at organ-scale is obvious: it allows to simulate the contraction and have access to various indicators that can be compared to clinical data on the one hand but also estimates of internal parameters that are not directly accessible by macroscale measurements, such as internal stresses.

To link these internal parameters to the macroscopic observables, comprehensive models of the heart covering all physiological aspects of its functioning are already available. Large international consortium are already using this kind of models to study cardiomyopathies and their treatments, see for instance the work of the SilicoFCM consortium or the MATHCARD project.

In the last decade, computer methods have been developed to simulate realistic muscle behavior, especially for the heart (Chabiniok et al. 2016; Stojanovic et al. 2019; Regazzoni, Dedè, and Quarteroni 2020; Sugiura et al. 2012). Essential underlying model elements are the force generation by molecular acto-myosin motors, the activation processes regulating contraction, the perfusion mechanisms (for the heart), and the passive viscoelastic properties of the tissue.

All the existing approaches share similar background based on three essential assumptions.

1. The molecular motors’ behavior can be represented as a jump process between a set of states characterizing the conformation of the myosin motors (power-stroke), its attachment to actin (attached or detached) and its biochemical state (ATP hydrolysis stage within the active site). The transition rates between these states are dependent on conformational and positional mechanical degrees of freedom. Hence, the designation chemical-mechanical models.
2. The bulk density of motors within the tissue is sufficiently large for a mean-field description to be valid. The motors population dynamics is then governed by a (potentially large) system of Partial Differential Equations on the probability distributions characterizing each “chemical state” of the motors.Mathematical simplifications (surrogate models) and associated numerical schemes have been developed to make this system of governing equations suitable for organ-scale finite elements simulations (Kimmig and Caruel 2020; Milićević et al. 2022; Regazzoni et al. 2022)
3. The equations describing the population of motors are directly coupled to the equilibrium laws of continuum mechanics, using rheological lumped elements that represent the contribution of sarcomeric structural proteins to the passive viscoelastic properties of the tissue. It is usually assumed that active fibers are homogenous and that a unique fiber direction can be defined at each material point.

In view of these modeling hypotheses, we can point several limitations of the existing frameworks.

1. The chemical-mechanical model of the molecular motor functioning is in fact an asymptotic represenation of a high-dimensional stochastic system describing the motion of interconnected atoms or amino-acids. There is no systematic derivation such asymptotic model reduction, which is in fact valid only if the so-called states represent deep enough energy wells. There is also no formal link between the chemical-mechanical model and the underlying structural mechanisms and conformational changes.
2. The mean-field assumption made to describe a large population of molecular motors neglects the mechanical interactions that can exist at the level of single contractile units, i.e. where a finite size system of molecular motors are coupled via the myofilaments. It also corresponds to an asymptotic limit that would need to be mathematically justified.
3. Using the assumption of a direct coupling between the molecular scale processes and the macroscopic mechanical balance laws, current models consider that a muscle fiber can be viewed as a homogenous 1D continuum, without detailed modeling of the sarcomeric proteins at the mesoscale and without questioning the validity of having a single fiber orientation per material point. However, as mentioned in the previous sections, experiments have shown that the structural proteins play a role in maintaining contractile units in register, notably during activation (Squarci et al. 2023; Moo and Herzog 2018). Alteration of their mechanical properties may lead to non-affine deformations which could contribute to the development of cardiomyopathies (Lange et al. 2019). Direct observation have also shown that in pathological situation, the definition of a clear fiber orientation was not possible (Hoshino et al. 1983).

For these reasons, there is no truly multiscale framework linking the molecular structural alterations induced by cardiomyopathies and their associated pharmacology, with their consequences on the heart.

In the folowing chapters, we will review the different scales of modeling and present an overview of the current state of research, of our contribution to this research, and sketch further developments.

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