4  Microscale: the contractile unit

4.1 Background: muscle contractile units and models

4.1.1 The muscle contractile unit

Figure 4.1: Schematic unidimensional representation of a contractile unit. Adapted from Chaintron, Caruel, and Kimmig, “Modeling actin-myosin interaction: Beyond the huxley–hill framework,” 2023, CC BY 4.0

The representative structure of the contractile apparatus above the single molecular motor is the contractile unit, a bundle of ~300 myosin motors protruding from the same myosin half-thick filament and interacting with six surrounding actin thin filaments. In the cross-section, each thick filament is at the center of a hexagon formed by the surrounding thin filaments, see Figure 1.2. Even though the structure of the contractile is three-dimensional, it is usually represented as a 1D chain of motors interacting with a single actin filament like in Figure 4.1. Within the sarcomeres, the contractile units are linked by cytoskeletal proteins whose role will be presented in Chapter 5.

Since the molecular motors are connected to a set of common backbones, they necessarily interact. Therefore, a change of state of one element can change the position of the backbone, which can affect the state other elements at short range or long range. Studying the mechanisms of these interactions and their effects on the mechanical ouput of the contractile unit is the topic of this Chapter. One important question raised is to what extent finite size effect impact the response of the contractile unit. This question is intimately linked to the potential limitation of the usual mean-field approach mentioned in Section 3.4.4 and Section 3.7.1. swift force decay giving The main problem with this intermediate scale is the lack of specific experimental data since classical experimental setups usually involve more macroscopic samples or single molecules. To study the specifics of the contractile unit behavior, one has to resolve to artificial reconstruction.

Recently the team lead by Dr. P. Bianco from the PhysioLab (University of Florence, Italy) has succeeded in reconstructing a minimal functional contractile unit (called the nanomachine) out of purified actin and myosin proteins, able to reproduce the performance of the functional unit of the muscle.¹ A dual laser optical trap holds a bead to which an actin filament is glued. This actin filament is brought to the vicinity of the side of a microneedle covered with myosin motors.

¹ Pertici et al., “A myosin II nanomachine mimicking the striated muscle,” 2018; Pertici et al., “A Myosin II-Based Nanomachine Devised for the Study of Ca2+-Dependent Mechanisms of Muscle Regulation,” 2020.

² Plaçais et al., “Spontaneous Oscillations of a Minimal Actomyosin System under Elastic Loading,” 2009; Walcott, Warshaw, and Debold, “Mechanical Coupling between Myosin Molecules Causes Differences between Ensemble and Single-Molecule Measurements,” 2012; Kalganov et al., “Forces measured with micro-fabricated cantilevers during actomyosin interactions produced by filaments containing different myosin isoforms and loop 1 structures,” 2013; Kaya et al., “Coordinated force generation of skeletal myosins in myofilaments through motor coupling,” 2017.

Previous works on similar systems, reported interesting dynamical regimes with various spontaneous oscillations motifs.² It should be mentioned, however, that these observations result from a non-physiological loading setup since the alignment between the actin filament and the array of motors cannot be maintained upon the detachment or motors. In such condition, the detachment of the first motor, generates a cascade of detachments resulting in swift force decay giving and an overall sawtooth oscillatory pattern. This behavior is not observed in the nanomachine developed by the PhysioLab, since the motors are kept close to the actin filament. With this setup, the systems can generate an isometric tension, i.e. a steady force, with a minimum of 8 to 10 motors.

Such artificial contractile unit represents a unique tool for understanding the most basic principles of muscle physiology, with full control on the nature of the proteins involved, their number, and their chemical environment etc. It could be used for instance to test the effect of drugs or mutations on the basic mechanical output of this minimal system, which represents a major asset for the development of new drugs.

4.1.2 Contractile unit modeling

Collective dynamics of molecular motors within a contractile unit has been studied since the late 1990s notoriously by J. Prost and J.F. Joanny’s group at Institut Curie (Paris).³ They have shown in particular that the mechanical feedback induced by the presence of a shared mechanical load or a shared elastic backbone can lead to rich dynamical behaviours involving spontaneous oscillations.

³ Jülicher and Prost, “Cooperative molecular motors,” 1995; Jülicher and Prost, “Spontaneous oscillations of collective molecular motors,” 1997; Plaçais et al., “Spontaneous Oscillations of a Minimal Actomyosin System under Elastic Loading,” 2009; Guérin et al., “Coordination and collective properties of molecular motors: theory,” 2010; Guérin, Prost, and Joanny, “Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models,” 2011.

⁴ Guérin, Prost, and Joanny, “Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models,” 2011.

⁵ In this respect, this formulation slightly differs from the classical Huxley-Hill model, because the latter considers four rates see Jülicher, Ajdari, and Prost, “Modeling molecular motors,” 1997 and Section 3.2.1.

A typical model of this type is a direct analog to the Huxley-Hill model (see Section 3.2.1), which can be formulated as follows.⁴. Considering the mean-field asumption mentioned in Section 3.4.4, let $\rho (x,t)$ be the density of attached molecular motors such that $\rho (x,t)\mathrm{d}x$ is the number of motors in the attached state at time $t$ whose positions are inside any interval of the type $[x+n\ell ,x+n\ell +dx]$, where $\ell$ is the distance between consecutive binding sites and $n$ is an integer. The myosin filament is considered rigid, and its position is denoted by $X$. The attachment and detachment rates are denoted by $f$ and $g$, respectively, like in the two state Huxley Hill model, see Section 3.2.1. Notice that to maintain the system out of equilibrium these two rates should not satisfy the detailed balance.⁵

Writing the conservation of the myosin motors leads to the following partial differential equation $${\partial }_t\rho -\dot{X}{\partial }_x\rho =-g(x)\rho +f(x)(1/\ell -\rho ).$$ The contractile unit is submitted to an external force $F_e$ and experience a viscous drag $-\xi \dot{X}$. These external forces balance the force produced by the attached molecular motors $F_m$: $$\xi \dot{X}=F_e+F_m,$$ with $F_m=N{\int }_0^{\ell }\rho (x,t){\partial }_{x}u(x)\mathrm{d}x$, where $u$ denotes the interaction potential with actin and $N$ is a scaling factor accounting for the volume density of motors. When the external force $F_e$ is imposed, the force balance creates feedback on the population dynamics. The detailed analysis of the consequences of this feedback in various regimes can be found in the work of Guérin et al.⁶. A particularly interesting point being the broad variety of observable oscillatory responses depending on parameter values.⁷

⁶ “Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models,” 2011.

⁷ Variants of the above model were also studied in Guérin, Prost, and Joanny, “Dynamic Instabilities in Assemblies of Molecular Motors with Finite Stiffness,” 2010 and Plaçais et al., “Spontaneous Oscillations of a Minimal Actomyosin System under Elastic Loading,” 2009.

4.2 Challenges

To put our work in perspective, we raise here a list of challenges relevant to the scale of the contractile unit.

1. As explained in Section 3.1.3, the use of synthetic contractile systems, like in the single molecule experimental setup, allows studying the most basics acto-myosin interaction mechanism. The nanomachine developed by Pertici et al.⁸, brings this experimental approach to a new level by allowing to reconstruct a minimal functioning contractile unit that was shown to behave as a scaled down muscle. The first challenge is to formulate a model of this class of experiments, where a finite number of interacting molecular motors are involved.
2. Simple mechanical feedback mediated by a connection to a common rigid backbone has already shown to trigger a broad range of dynamical regimes. However, the elastic interactions might be more complex, mixing short and long-range components. For instance, the actin and myosin filaments are in fact cross-linked not only by active molecular motors but also by the supposedly passive Myosin Binding Protein C (MyBP-C) and titin.⁹
3. The topic has not been mentioned yet but regulation mechanisms may be influenced by the mechanical interactions between the myosin motors within contractile units. These Regulation mechanisms are crucial in heart tissue undergoing rapid activation-deactivation cycles. During a heartbeat, full activation is never reached, keeping the percentage of attached motors below 15%. Although the cardiac tissue contains many motors (supporting mean-field approaches), individual contractile units may then involve at most 44 locally coupled attached motors, which questions the usual mean-field descriptions as associated models may fail to capture finite-size effects. For example, when mean-field models have multiple stable equilibria, finite-size systems often exhibit abrupt transitions between the equilibria over timescales that are exponential in (N ), which mean-field models do not reproduce.

⁸ Pertici et al., “A myosin II nanomachine mimicking the striated muscle,” 2018.

⁹ Tamborrini et al., “Structure of the native myosin filament in the relaxed cardiac sarcomere,” 2023 and Chapter 5

4.3 Contributions

Associated publications

The majority of our work on contractile unit results from Caruel, Allain, and Truskinovsky, “Muscle as a Metamaterial Operating Near a Critical Point,” 2013, Caruel, Allain, and Truskinovsky, “Mechanics of collective unfolding,” 2015; Caruel and Truskinovsky, “Statistical mechanics of the Huxley-Simmons model,” 2016 and Caruel and Truskinovsky, “Bi-stability resistant to fluctuations,” 2017. The results mentionned here have already been the object of a review in Sec. 2 of Caruel and Truskinovsky, “Physics of muscle contraction,” 2018. In Section 4.3.7 we will present an application of the results obtained in the context of muscle contraction to the understanding of the molecular processes driving neurotransmission.

4.3.1 Fast transient response of muscle fibers

Our contribution focuses on the fast timescale response of a muscle fiber submitted to swift load changes.¹⁰ The experiment consists in stimulating a muscle fiber up to isometric tetanus, where the tension reaches a steady state value denoted by $T_0$, see Figure 4.2 (a). At $t=0$, a load step is applied, either in length (soft device) or in tension (hard device). The first milliseconds of the response are illustrated in Figure 4.2 Simulaneously to a length step [see (a)], the tension drops to a level $T_1$ (phase 1) before rising up to a level $T_2$ within a few milliseconds (phase 2). In response to a force step, the fiber shortens simultaneously by an amount $\delta z_1$ (phase 1) and then more slowly by an amount $\delta z_2$.

¹⁰ Podolsky, Nolan, and Zaveler, “Cross-bridge properties derived from muscle isotonic velocity transients.” 1969; Huxley and Simmons, “Proposed mechanism of force generation in striated muscle,” 1971.

Figure 4.2: Fast timescale response of an activated muscle fiber, preveviously held in isometric conditions, consubmitted to swift load changes in hard (a, imposed shortening) and soft (b, imposed force drop) devices. The length changed are indicated in nanometer per half-sarcomere (nm/hs). Adapted from Caruel and Truskinovsky, “Physics of muscle contraction,” 2018, data from Piazzesi et al., “Mechanism of force generation by myosin heads in skeletal muscle,” 2002; Piazzesi, Lucii, and Lombardi, “The size and the speed of the working stroke of muscle myosin and its dependence on the force,” 2002.

Importantly, in these fast timescale regimes the attachment-detachment events of individual myosin motors can be ignored, which means that the number of cross-bridges can be considered constant. This strong assumption is supported by several experimental studies showing that the first detachment events following the application of a quick change in load occur after a few milliseconds.¹¹ In the interval, the number of cross-bridges is constant which allows detecting the conformational change in the attached motors.

¹¹ Reconditi et al., “The myosin motor in muscle generates a smaller and slower working stroke at higher load,” 2004.

In the following, we summarize our contribution to the understanding of the mechanics of this quick response, with a particular emphasis on the role played by the long-range elastic interactions on the syncronization of the power-strokes.

Figure 4.3: Mechanical model of a contractile unit with $N$ cross-bridges assembled in parallel and connected in series to a common elastic backbone with stiffness ${\lambda }_b$. $z$: total elongation of the contractile unit; $y$: elongation of the cluster of cross-bridges; $\mathrm{\Sigma }$: tension generated by the system. Figure adapted from Caruel and Truskinovsky,Caruel and Truskinovsky, “Physics of muscle contraction,” 2018.

4.3.2 Mechanical model of a contractile unit

The mechanical model proposed for a contractile unit is represented in Figure 4.3. The main assumption of this model is to consider a parallel arrangement of cross-bridges. This simplified representation, where the contribution of the myosin and actin filaments to the elastic response is lumped into a linear series spring, is supported by the work of Ford, Huxley and Simmons, and Linari et al.¹²

¹² Ford, Huxley, and Simmons, “The relation between stiffness and filament overlap in stimulated frog muscle fibres.” 1981; Linari et al., “The Stiffness of Skeletal Muscle in Isometric Contraction and Rigor: The Fraction of Myosin Heads Bound to Actin,” 1998. This mechanical representation is valid at high load where the number of bound myosin motors is high. At low load this model has to be modified to incorporate the contribution of Titin, see Pertici, Caremani, and Reconditi, “A mechanical model of the half-sarcomere which includes the contribution of titin,” 2019; Powers et al., “Contracting striated muscle has a dynamic I-band spring with an undamped stiffness 100 times larger than the passive stiffness,” 2020; Squarci et al., “Titin activates myosin filaments in skeletal muscle by switching from an extensible spring to a mechanical rectifier,” 2023.

¹³ Huxley and Simmons, “Proposed mechanism of force generation in striated muscle,” 1971; Marcucci and Truskinovsky, “Mechanics of the power stroke in myosin II,” 2010.

In the contractile unit model, each cross-bridge is a represented as a bistable snap-spring with the two configurations of the spring representing the pre- and post-power stroke conformation, respectively. As mentioned in Section 3.4.2, the dynamics of the conformational change can be described either using a jump process between two discrete state as formulated by Huxley and Simmons, or using a continuous Langevin dynamics in a double well potential as in the work of Marcucci and Truskinovsky.¹³ Both approaches were analysed. The advantage of the former is that it is fully analytic.

4.3.3 Purely mechanical response of a contractile unit

Associated reference

The detailed analysis of the purely mechanical properties of the model is available in Caruel, Allain, and Truskinovsky, “Mechanics of collective unfolding,” 2015.

The mechanical behavior of the contractile model can be studied without considering the effect of temperature, shedding light on the mechanical origin of the interaction between the cross-bridges. In this context, under a given load, the system will equilibrate in local energy minima representing particular microstates. In the model represented in Figure 4.3, the cross-bridge can be interchanged without changing the energy of the microstate. Hence, the energy of a microstate is fully characterized by the fraction $p$ of cross-bridges in the post-power stroke conformation.

Figure 4.4: Equilibrium response of the contractile unit model with a discrete power-stroke representation at zero temperature and without an elastic backbone (${\lambda }_b\to \mathrm{\infty }$). [(a) and (b)] Tension-elongation relations corresponding to the metastable states (gray) and along the global minimum path (thick lines), in hard (a) and soft (b) devices. (c–e) [respectively (f–h)] Energy levels of the metastable states corresponding to $p=0,0.1,\dots ,1,$ at different elongations $y$ (respectively tensions $\sigma$). Corresponding transitions (E $\to$ B, P $\to$ Q, …) are shown in (a) and (b). Reprinted from Caruel and Truskinovsky, “Physics of muscle contraction,” 2018.

To illustrate the behavior of the system in the absence of temperature, we first consider a system with a rigid backbone (consider ${\lambda }_b\to +\mathrm{\infty }$ in the model of Figure 4.3). The energy and the tension of all metastable states (each characterized by a value of $p$) can be computed as function of the loading in hard and soft devices, see Figure 4.4 . In both cases, the configuration that minimizes the energy is homogenous ($p=0$ or $p=1$), see [(a) and (b)]. The global equilibrium response is therefore characterized by a sharp transition at $y=y_0$ in hard device and $\sigma ={\sigma }_0$ in soft device.

The difference between the two responses can be investigated by representing the energy of the metastable states as function of $p$ at these transition points, see [(d)-(g)]. In the hard device case at $y=y_0$ (d), all configurations have the same energy. However, in the soft device case at $\sigma ={\sigma }_0$ (g), mixed states ( $0<p<1$ ) have energies higher than the homogenous states. The origin of this difference is the long range elastic interaction: In the soft device case, isolated pre- and post- power stroke cross-bridges would have different lengths [corresponding to points B and E in (b)], but within the cluster, they are constrained to an intermediate elongation which necessarily compresses (respectively elongates) pre-power stroke (respectively post-power stroke) cross-bridges. Hence, the increased energy of the mixed configurations. This simple mechanical feedback is of long-range nature: all cross-bridges are affected equally by the conformational change of a single one.¹⁴ This generic mechanical cross-talk may be present in many other biological systems, ranging from epigenetics to hair-cells.¹⁵ We now present how temperature interferes with these purely mechanical long-range interactions.

¹⁴ More results are available in Caruel, Allain, and Truskinovsky, “Mechanics of collective unfolding,” 2015 and a similar study is carried out when an elastic backbone is present (mixed device) in Caruel and Truskinovsky, “Physics of muscle contraction,” 2018.

¹⁵ Caruel and Truskinovsky, “Physics of muscle contraction,” 2018.

4.3.4 Equilibrium statistical mechanics in length clamp

The first model of the fast transient response of muscle fibers was proposed by Huxley and Simmons to interpret the experimental results obtained for imposed length steps.¹⁶ The model has been reformulated by Caruel and Truskinovsky¹⁷ and its statistical mechanical properties were derived by drawing on a formal analogy with a paramagnetic Ising model. The power stroke being modeled as a two state jump process, each cross-bridge is analog to an “elastic” spin pertaining to a parallel bundle confined between rigid backbones (consider the limit ${\lambda }_b\to \mathrm{\infty }$ in Figure 4.3). In such system, the spins are uncoupled, which implies that the properties of the bundle of cross-bridge are obtained by simply rescaling the properties of a single unit.

¹⁶ Huxley and Simmons, “Proposed mechanism of force generation in striated muscle,” 1971.

¹⁷ “Statistical mechanics of the Huxley-Simmons model,” 2016.

One of the most important characteristics of the thermal equilibrium of this model is illustrated in Figure 4.5. The response depends on the non-dimensional ratio between elastic and thermal energies $\beta =\kappa a^2/(k_{\text{B}}T)$, where $\kappa$ is the stiffness of the myosin head, $a$ is the power-stroke characteristic length, $k_{\text{B}}$ is the Boltzmann constant and $T$ is the absolute temperature. A salient feature of the system is the presence of a pseudo-critical temperature $\beta ={\beta }_c=4$, beyond which the energy becomes a non-convex function of the applied elongation $y$ and the tension-elongation curve shows a region of negative stiffness.

4.3.5 Equilibrium statistical mechanics in soft and mixed devices

The analysis of the Huxley and Simmons’ model in a length clamp setup (hard device) shows that the system does not experience any phase transition. However, the pseudo-crtical temperature $\beta =4$ signals the presence of a genuine second order phase transition if one considers the system in a force clamp setup (soft device). This can be anticipated by observing on Figure 4.5 that, for $\beta >4$ there exist a load interval $[{\sigma }_{-},{\sigma }_{+}]$ within which the same load $\sigma$ can correspond to three different elongations. In the soft device condition, the Huxley and Simmons’ model is in fact a direct analog to the Curie-Wiess model.

Figure 4.5: Thermal equilibrium response of the Huxley and Simmons model (“Proposed mechanism of force generation in striated muscle,” 1971) in a length clamp setup, where the elongation $y-y_0$ is imposed. (a) free energy (b) normalized tension. The different curves correspond to different values of the non-dimensional parameter $\beta =\kappa a^2/(k_{\text{B}}T)$ representing the ratio between the characteristic elastic energy and the characteristic thermal energy in the system. Above the “critical” temperature $\beta =4$ (dashed and dot-dashed lines), the system shows a region of negative stiffness. Adapted with permission from Caruel and Truskinovsky, “Statistical mechanics of the Huxley-Simmons model,” 2016, ©️2016 by the American Physical Society.

The phase transition comes directly from the competition between the long range mechanical interactions which tend to increase the free energy of mixed states (see Section 4.3.3) and the entropy which tends to decrease the free energy of these states. Hence, the mechanical feedback dominates at low temperature but diminishes when the thermal energy allows the cross-brigdes to easily flip conformation thanks to fluctuations.

Figure 4.6: Phase transition of the Huxley and Simmons model (“Proposed mechanism of force generation in striated muscle,” 1971) loaded in a soft device. The fraction of cross-bridge in the post-power-stroke conformation is denoted by $p$. (a) pitchfork bifurcation showing the location of the equilibrium free energy minima (A and C) and local maximum (B). The inserts show the free energy for $\beta =2$ (left) and $\beta =5$ (right). (b) and (c) show $⟨p⟩$ and the normalized tension $\sigma /{\sigma }_0$ in the post-bifurcation regime where for a given load (here $\sigma ={\sigma }_0$) two stable configurations (A and C) exist. [(d)-(f)] shows a simulation of the system for $\sigma ={\sigma }_0$ before (d), at (e) and after (f) the bifurcation. Adapted from Caruel and Truskinovsky, “Physics of muscle contraction,” 2018.

A detailed analysis of this phase transition and its consequences on the thermal equilibrium properties and dynamical response to load steps has been published in Caruel and Truskinovsky, “Bi-stability resistant to fluctuations,” 2017. The main results in the thermodynamic limit and when the load $\sigma ={\sigma }_0$ are summarized in Figure 4.6. The free energy is represented as a function of the fraction $p$ of cross-bridges in the post-power-stroke configuration. Before the bifurcation ($\beta <4$), thermal agitation prevails, resulting into a single stable configuration around $p=1/2$ (see B). After the bifurcation, the cross-bridge segregate into metastable relatively homogenous states (see A and C). In this regime, the conformational changes are synchronized, but as the barrier between the states A and B (see insert in (a)) increases with $N$ synchronization is exponentially slower for large clusters.

The above analysis can be extended to the situation where the stiffness of the backbone is finite, see Figure 4.3. In this case, the bundle of cross-bridges is loaded in a mixed device.¹⁸ The new parameter ${\lambda }_b={\kappa }_f/(N\kappa )$ represents the stiffness of the filament ${\kappa }_f$ relative to the combined stiffness of the $N$ cross-bridges forming the bundle. The hard and soft devices limits response can be viewed as the limit ${\lambda }_b\to +\mathrm{\infty }$ and ${\lambda }_b\to 0$ ($z\to \mathrm{\infty }$), respectively. For the Huxley and Simmons’ “spin” model, the critical temperature now reads ${\beta }_c=4(1+{\lambda }_b)$ and therefore depends on $N$. This result shows the importance of considering finite size contractile units.

¹⁸ The details of this analysis can be found in Caruel and Truskinovsky, “Physics of muscle contraction,” 2018

Figure 4.7: Summary of the thermal equilibrium response of the Huxley and Simmons (“Proposed mechanism of force generation in striated muscle,” 1971) model in hard, soft and mixed device. The phase diagram (a) show the phases boundaries as function of the nondimensional parameters $\beta$ (temperature), and ${\lambda }_b$ (backbone elasticity). The pure hard and soft device behaviors corresponds to the lines at ${\lambda }_b\to \mathrm{\infty }$ and ${\lambda }_b=0$, respectively. The typical responses in each phase (labelled A-F) are shown on the rignt panels. The top row of these panels illustrates the free energies in thermal equilibrium (thick lines) and the metastable and unstable branches (dotted lines). The bottom row shows the corresponding tension-elongation relation. Adapted from Caruel and Truskinovsky, “Physics of muscle contraction,” 2018.

The consequence of this phase transition is summarized in Figure 4.7. The phase diagram (a) shows three phases in the $(\beta ,{\lambda }_b)$ space, labelled I, II and III. We recall that the pure hard and soft device loadings correspond to the limits ${\lambda }_b\to \mathrm{\infty }$ and ${\lambda }_b=0$, respectively. Phase I is the region where thermal effects dominates. In a hard device [(a) and (b)], the systems does not show negative stiffness and no metastability. The first phase transition is at $\beta =4$ and concerns the soft device response. In the post-bifurcation regime (phase II), the hard device response is charaterized by negative stiffness and the soft device response shows metastability (dotted lines in D) with an equilibrium response that includes a jump (thick lines in D), see Figure 4.6. In the third phase (III) that appears for $\beta >4(1+{\lambda }_b)$ in a mixed device, the system also shows metastability [dotted lines in (F)] and a jump in the equilibrium response [thick lines in (F)].

Overall this study reveals that the response of a system of parallel cross-bridges depends on the type of loading. This finding may affect the behavior also at larger scales. This point is discussed further in Chapter 5.

To match the experimental results obtained from the fast load changes, it is necessary to consider a regularized version of the Huxley and Simmons’ model as explained in Section 3.4.2. Qualitatively, the behavior of the regularized model is similar to the Huxley and Simmons’ model, though it partially looses its analytical transparency.¹⁹ The mixed device model of the system calibrated on the experimental data showed that the contractile unit might operate close to the critical line (between phase II and III using the notations of Figure 4.7).²⁰

¹⁹ We refer to Section 2.2 of Caruel and Truskinovsky, “Physics of muscle contraction,” 2018 for the details.

²⁰ This finding was published in Caruel, Allain, and Truskinovsky, “Muscle as a Metamaterial Operating Near a Critical Point,” 2013.

²¹ Borja da Rocha and Truskinovsky, “Functionality of Disorder in Muscle Mechanics,” 2019.

A follow-up of this work was proposed by Borja da Rocha and Truskinovsky, where quenched desordered was added to the model adding third dimension to the phase diagram. Again, the author showed that the system was poised to criticality.²¹

4.3.6 Effective dynamical model of a contractile unit

The above analysis has shown that the presence of long-range interactions at the level of individual contractile units may create metastable states associated with power stroke synchronization. The timescale for the synchronized depends exponentially on the number of units in the cluster. These results show a limitation of the classical mean-field approach, which cannot capture finite size effect. A legitimate question at this stage is wether it is possible to derive another effective model of a contractile unit that would be faster to simulate than the fully detailed model but still capturing finite size effects. A preliminary attempt was made in Caruel, “Mechanics of Fast Force Recovery in striated muscles,” 2011 (Chap. 6) by trying to project the high dimensional dynamics of the contractile unit model introduced in Section 4.3.2 on a low dimension manifold.

For the classical spin model a natural collective variable could be the fraction $p$ of cross-bridges in the post-power-stroke conformation. An effective one-dimensional energy landscape was derived in Caruel, Allain, and Truskinovsky, “Mechanics of collective unfolding,” 2015, for the soft device (imposed load) case, and a “coarse grained” dynamical model was proposed and validated numerically in pre- and post-bifurcation regimes in Caruel and Truskinovsky, “Bi-stability resistant to fluctuations,” 2017.

Results for the regularized model are still preliminary, since the choice of the collective variable is less obvious. Two strategies were tested in Caruel, “Mechanics of Fast Force Recovery in striated muscles,” 2011 (Chap. 6). The first one uses the cluster elongation $y$ (see Figure 4.3) as a collective variable and assumes that the conformations of the cross-bridges relaxes quickly to equilibrium (adiabatic elimination). The second strategy is based on a weak coupling approximation. It this approximation, it is assumed that the cross-bridges are independent conditionally on the position of the backbone.
The latter approximation seems more promising but a more rigorous mathematical treatment is required.

4.3.7 Application of the contractile unit model to neurotransmission

Associated publications

This work was published in Manca et al., “SNARE machinery is optimized for ultrafast fusion,” 2019 and a follow-up study was recently issued in Caruel and Pincet, “Dual-Ring SNAREpin Machinery Tuning for Fast Synaptic Vesicle Fusion,” 2024.

A collaboration was initiated with the groups of J.E. Rothman (Yale University) and F. Pincet in 2015 on the mechanical modeling of the molecular processes driving the synaptic neurotranmitters release.

Information is transmitted from one neuron to the next in chemical form through the synaptic cleft that separates the membranes of the neurons. The neurotransmitters are “stored” inside the emiting neuron within vesicles that are docked near the membrane. The incomming eletrical signal triggers the massive release of $\mathrm{Ca}{\phantom{A}}^{2+}$ ions in the cytoplasm of the emiting neuron, which results in the fusion of the vesicles membranes with the neuron membrane, and the release of their content in the cleft. The key step of the process is the fusion. This molecular topological transformation requires a large amount of energy (about 25 k_BT): a spontaneous fusion would take more than 1 s, while it happens in less than 1ms in the brain.

The physiological speed is achieved thanks to a bundle of bistable proteins called SNAREpins,²² that bridge the vesicle and the neuron membrane prior to fusion, see Figure 4.8 (a). Their collective conformational change ressembles a zippering that pulls the two membranes toward each other, see Figure 4.8 (a).

²² soluble N-ethylmaleimide–sensitive factor attachment protein receptors. The role played by these proteins in membrane fusion in living systems has been discovered by J.E. Rothman, who won the Nobel-Prize in 2013.

Figure 4.8: Mechanical modeling synaptic vesicle fusion mediated by SNAREpins. (a) Two SNAREpins bridging the vesicle and the neuron membranes before fusion. One SNAREpins is unzippered (n) and the other is zippered (c). (b) mechanical model of 4 SNAREpins between rigid membranes. (c) free energy landscape of the system. Black, dotted: fusion barrier without SNAREs; dashed line, energy landscape of a single SNAREpin; solid red and dot dashed blue: total free energy with one and four SNAREpins, respectively. (d) Influence of the number of SNAREpins on the simulated passage times over the zippering ($n\to c$, squares) and fusion (circle) barriers, and on th fusion time (red triangles). Adapted from Manca et al., “SNARE machinery is optimized for ultrafast fusion,” 2019 ©️2019 CC BY-NC-ND 4.0.

The system evolves under no external forces other than the viscous drag on the vesicle and the thermal noise. Using a slightly modified version of the Huxley and Simmons’ contractile unit model, Manca et al. predicted that the time to reach fusion was a non-monotone function of the number of SNAREpins $N$ showing an optimum around $N=4$, see Figure 4.8 (d). The fusion model is the combination of two processes:

1. the collective zippering ($n\to c$), whose timescale augments with $N$, due to an increasing energy barrier [see Figure 4.8 (c) and (d)],
2. overcoming the fusion energy barrier, whose timescale decreases with $N$ since more SNAREpins can exert more force.

The counterintuitive finding that more SNAREpins actually slow down the process is directly linked to the result of Section 4.3.5: Under a fixed load (here 0 force), the long-range interaction generates internal frustration that increases the energy of mixed microstate. The result of this feedback is an energy barrier between the two homogenous states (fully zippered or fully unzippered) that increases linearly with $N$, which makes the zippering process exponentially slower for larger clusters, see Figure 4.8 (c). The prediction of the model is in accordance with the actual molecular structure of the fusion machinery as explained in the recent review by Rothman et al. “Turbocharging synaptic transmission,” 2023.

Recent work by Bera et al., suggests that the SNAREpins forms a double ring structure.²³ In a follow-up study²⁴ that considers a dual ring configuration, we show that the systems still shows an optimal configuration but that the relative positioning of the two rings on the vesible may modify the fusion time by few orders of magniture. Hence, this dual ring machinery has to be finely tuned as well.

²³ Bera et al., “Synaptophysin chaperones the assembly of 12 SNAREpins under each ready-release vesicle,” 2023.

²⁴ Caruel and Pincet, “Dual-Ring SNAREpin Machinery Tuning for Fast Synaptic Vesicle Fusion,” 2024.

4.4 Perspectives

A natural follow-up of our work on the fast transient response of muscle fiber is to use our contractile unit model in combination with a molecular motor model that includes the attachment-detachment process. The simplest approach is to consider a two state, attached-detached, model and implement for instance the reformulation of the Huxley’57 model introduced by Chaintron, Caruel and Kimmig.²⁵ The framework would then be quite similar to the one used by Guérin et al,²⁶, but with a finite-size model that was calibrated and validated on physiological data. We will then be able to observe wether the different dynamical regimes prediced could be observed in realistic muscle contractile units. Another option are motor models proposed by Caruel, Moireau and Chapelle²⁷ by Chaintron et al.²⁸ or the fully mechanical model presented in Sheshka and Truskinovsky²⁹

²⁵ “Modeling actin-myosin interaction: Beyond the huxley–hill framework,” 2023.

²⁶ “Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models,” 2011.

²⁷ “Stochastic modeling of chemical–mechanical coupling in striated muscles,” 2019.

²⁸ “A jump-diffusion stochastic formalism for muscle contraction models at multiple timescales,” 2023.

²⁹ “Power-stroke-driven actomyosin contractility.” 2014; see also Caruel and Truskinovsky, “Physics of muscle contraction,” 2018, Sec. 4.2

³⁰ Dawson and Gärtner, “Large deviations and tunnelling for particle systems with mean field interaction,” 1986.

A challenging research objective using these models will be to characterize finite-size effects, reconsider the mean-field hypothesis and suggest an enhanced reduced model of the contractile unit that is able to capture its metastable behavior. Precisely describing this behavior, even in simple models like the continuous Curie-Weiss model, remains an open question in probability and statistical physics.³⁰ Therefore, this perpective will necessitate advanced mathematical developments.

BERA, Manindra, RADHAKRISHNAN, Abhijith, COLEMAN, Jeff, K. SUNDARAM, R. Venkat, RAMAKRISHNAN, Sathish, PINCET, Frederic and ROTHMAN, James E., 2023. Synaptophysin chaperones the assembly of 12 SNAREpins under each ready-release vesicle. Proceedings of the National Academy of Sciences of the United States of America. 2023. Vol. 120, no. 45, p. e2311484120. DOI 10.1073/pnas.2311484120.

BORJA DA ROCHA, Hudson and TRUSKINOVSKY, Lev, 2019. Functionality of Disorder in Muscle Mechanics. Physical Review Letters. 2019. Vol. 122, no. 8, p. 088103. DOI 10/gqj7sz.

CARUEL, Matthieu, 2011. Mechanics of Fast Force Recovery in striated muscles. Ecole Polytechnique.

CARUEL, Matthieu, ALLAIN, Jean-Marc and TRUSKINOVSKY, Lev, 2013. Muscle as a Metamaterial Operating Near a Critical Point. Physical Review Letters. 2013. Vol. 110, no. 24, p. 248103. DOI 10/gmtzn5.

CARUEL, Matthieu, ALLAIN, Jean-Marc and TRUSKINOVSKY, Lev, 2015. Mechanics of collective unfolding. Journal of the Mechanics and Physics of Solids. 2015. Vol. 76, p. 237–259. DOI 10/f639qf.

CARUEL, Matthieu, MOIREAU, Philippe and CHAPELLE, Dominique, 2019. Stochastic modeling of chemical–mechanical coupling in striated muscles. Biomechanics and Modeling in Mechanobiology. 2019. Vol. 18, no. 3, p. 563–587. DOI 10.1007/s10237-018-1102-z.

CARUEL, Matthieu and PINCET, Frédéric, 2024. Dual-Ring SNAREpin Machinery Tuning for Fast Synaptic Vesicle Fusion. Biomolecules. 2024. Vol. 14, no. 5, p. 600. DOI 10.3390/biom14050600.

CARUEL, Matthieu and TRUSKINOVSKY, Lev, 2016. Statistical mechanics of the Huxley-Simmons model. Physical Review E. 2016. Vol. 93, no. 6, p. 062407. DOI 10/gkpp6d.

CARUEL, Matthieu and TRUSKINOVSKY, Lev, 2017. Bi-stability resistant to fluctuations. Journal of the Mechanics and Physics of Solids. 2017. Vol. 109, p. 117–141. DOI 10/gcgxxs.

CARUEL, Matthieu and TRUSKINOVSKY, Lev, 2018. Physics of muscle contraction. Reports on Progress in Physics. 2018. Vol. 81, no. 3, p. 036602. DOI 10/gf8wq6.

CHAINTRON, Louis-Pierre, CARUEL, Matthieu and KIMMIG, François, 2023. Modeling actin-myosin interaction: Beyond the huxley–hill framework. MathematicS In Action. 2023. Vol. 12, no. 1, p. 191–226. DOI 10.5802/msia.38.

CHAINTRON, Louis-Pierre, KIMMIG, François, CARUEL, Matthieu and MOIREAU, Philippe, 2023. A jump-diffusion stochastic formalism for muscle contraction models at multiple timescales. Journal of Applied Physics. 2023. Vol. 134, no. 19, p. 194901. DOI 10.1063/5.0158191.

DAWSON, Donald A and GÄRTNER, J, 1986. Large deviations and tunnelling for particle systems with mean field interaction. CR Math. Rep. Acad. Sci. Canada. Online. 1986. Vol. 8, no. 6, p. 387–392. Available from: https://mathreports.ca/article/large-deviations-and-tunnelling-for-particle-systems-with-mean-field-interaction/

FORD, L E, HUXLEY, A F and SIMMONS, R M, 1981. The relation between stiffness and filament overlap in stimulated frog muscle fibres. The Journal of Physiology. Online. 1981. Vol. 311, no. 1, p. 219–249. DOI 10/gmjmmd.

GUÉRIN, Thomas, PROST, Jacques and JOANNY, Jean-François, 2011. Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models. Eur. Phys. J. E. 2011. Vol. 34, no. 6, p. 60. DOI 10/dwphkc.

GUÉRIN, Thomas, PROST, Jacques, MARTIN, Pascal and JOANNY, Jean-François, 2010. Coordination and collective properties of molecular motors: theory. Current Opinion in Cell Biology. 2010. Vol. 22, no. 1, p. 14–20. DOI 10/c4hrnm.

GUÉRIN, Thomas, PROST, J. and JOANNY, J.-F., 2010. Dynamic Instabilities in Assemblies of Molecular Motors with Finite Stiffness. Physical Review Letters. Online. 2010. Vol. 104, no. 24, p. 248102. DOI 10/cmftsw.

HUXLEY, Andrew F and SIMMONS, Robert M, 1971. Proposed mechanism of force generation in striated muscle. Nature. 1971. Vol. 233, no. 5321, p. 533–538. DOI 10.1038/233533a0.

JÜLICHER, F and PROST, J, 1997. Spontaneous oscillations of collective molecular motors. Physical Review Letters. 1997. Vol. 78, no. 23, p. 4510–4513. DOI 10.1103/physrevlett.78.4510.

JÜLICHER, Franck and PROST, Jacques, 1995. Cooperative molecular motors. Physical Review Letters. 1995. Vol. 75, no. 13, p. 2618–2621.

JÜLICHER, Frank, AJDARI, Armand and PROST, Jacques, 1997. Modeling molecular motors. Reviews of Modern Physics. 1997. Vol. 69, no. 4, p. 1269–1282. DOI 10/dvk9kt.

KALGANOV, Albert, SHALABI, Nabil, ZITOUNI, Nedjma, KACHMAR, Linda Hussein, LAUZON, Anne-Marie and RASSIER, Dilson E., 2013. Forces measured with micro-fabricated cantilevers during actomyosin interactions produced by filaments containing different myosin isoforms and loop 1 structures. Biochimica et Biophysica Acta (BBA) - General Subjects. 2013. Vol. 1830, no. 3, p. 2710–2719. DOI 10.1016/j.bbagen.2012.11.022.

KAYA, Motoshi, TANI, Yoshiaki, WASHIO, Takumi, HISADA, Toshiaki and HIGUCHI, Hideo, 2017. Coordinated force generation of skeletal myosins in myofilaments through motor coupling. Nat Commun. 2017. Vol. 8, no. 1, p. 16036. DOI 10/gbkxhq.

LINARI, Marco, DOBBIE, Ian, RECONDITI, Massimo, KOUBASSOVA, Natalia, IRVING, Malcolm, PIAZZESI, Gabriella and LOMBARDI, Vincenzo, 1998. The Stiffness of Skeletal Muscle in Isometric Contraction and Rigor: The Fraction of Myosin Heads Bound to Actin. Biophysical Journal. Online. 1998. Vol. 74, no. 5, p. 2459–2473. DOI 10/bkzvq6.

MANCA, Fabio, PINCET, Frederic, TRUSKINOVSKY, Lev, ROTHMAN, James E., FORET, Lionel and CARUEL, Matthieu, 2019. SNARE machinery is optimized for ultrafast fusion. Proceedings of the National Academy of Sciences. 2019. Vol. 116, no. 7, p. 2435–2442. DOI 10/gmn5g3.

MARCUCCI, L. and TRUSKINOVSKY, L., 2010. Mechanics of the power stroke in myosin II. Physical Review E. 2010. Vol. 81, no. 5, p. 051915. DOI 10/c9xrzx.

PERTICI, Irene, BIANCHI, Giulio, BONGINI, Lorenzo, LOMBARDI, Vincenzo and BIANCO, Pasquale, 2020. A Myosin II-Based Nanomachine Devised for the Study of Ca2+-Dependent Mechanisms of Muscle Regulation. Int J Mol Sci. 2020. Vol. 21, no. 19, p. 7372. DOI 10.3390/ijms21197372.

PERTICI, Irene, BONGINI, Lorenzo, MELLI, Luca, BIANCHI, Giulio, SALVI, Luca, FALORSI, Giulia, SQUARCI, Caterina, BOZÓ, Tamás, COJOC, Dan, KELLERMAYER, Miklós S. Z., LOMBARDI, Vincenzo and BIANCO, Pasquale, 2018. A myosin II nanomachine mimicking the striated muscle. Nat Commun. 2018. Vol. 9, no. 1, p. 3532. DOI 10/gd69s3.

PERTICI, Irene, CAREMANI, Marco and RECONDITI, Massimo, 2019. A mechanical model of the half-sarcomere which includes the contribution of titin. Journal of Muscle Research and Cell Motility. Online. 2019. Vol. 40, no. 1, p. 29–41. DOI 10/gmtzrp.

PIAZZESI, Gabriella, LUCII, Leonardo and LOMBARDI, Vincenzo, 2002. The size and the speed of the working stroke of muscle myosin and its dependence on the force. The Journal of Physiology. 2002. Vol. 545, no. 1, p. 145–151. DOI 10/dtj3gc.

PIAZZESI, Gabriella, RECONDITI, Massimo, LINARI, Marco, LUCII, Leonardo, SUN, Yin-Biao, NARAYANAN, Theyencheri, BOESECKE, Peter, LOMBARDI, Vincenzo and IRVING, Malcolm, 2002. Mechanism of force generation by myosin heads in skeletal muscle. Nature. 2002. Vol. 415, no. 6872, p. 659–662. DOI 10/frsg7p.

PLAÇAIS, P.-Y., BALLAND, M., GUÉRIN, T., JOANNY, J.-F. and MARTIN, P., 2009. Spontaneous Oscillations of a Minimal Actomyosin System under Elastic Loading. Physical Review Letters. 2009. Vol. 103, no. 15, p. 158102. DOI 10/cdjrzv.

PODOLSKY, R. J., NOLAN, A C and ZAVELER, S A, 1969. Cross-bridge properties derived from muscle isotonic velocity transients. Proceedings of the National Academy of Sciences of the United States of America. 1969. Vol. 64, no. 2, p. 504–511.

POWERS, Joseph D., BIANCO, Pasquale, PERTICI, Irene, RECONDITI, Massimo, LOMBARDI, Vincenzo and PIAZZESI, Gabriella, 2020. Contracting striated muscle has a dynamic I-band spring with an undamped stiffness 100 times larger than the passive stiffness. The Journal of Physiology. 2020. Vol. 598, no. 2, p. 331–345. DOI 10/gm3sgx.

RECONDITI, Massimo, LINARI, Marco, LUCII, Leonardo, STEWART, Alex, SUN, Yin-Biao, BOESECKE, Peter, NARAYANAN, Theyencheri, FISCHETTI, Robert F., IRVING, Tom, PIAZZESI, Gabriella, IRVING, Malcolm and LOMBARDI, Vincenzo, 2004. The myosin motor in muscle generates a smaller and slower working stroke at higher load. Nature. Online. 2004. Vol. 428, no. 6982, p. 578–581. DOI 10/dfh49n.

ROTHMAN, James E., GRUSHIN, Kirill, BERA, Manindra and PINCET, Frederic, 2023. Turbocharging synaptic transmission. FEBS Letters. Online. 2023. Vol. 597, no. 18, p. 2233–2249. DOI 10.1002/1873-3468.14718.

SHESHKA, Raman and TRUSKINOVSKY, Lev, 2014. Power-stroke-driven actomyosin contractility. Phys Rev E. 2014. Vol. 89, no. 1, p. 012708–12. DOI 10.1103/physreve.89.012708.

SQUARCI, Caterina, BIANCO, Pasquale, RECONDITI, Massimo, PERTICI, Irene, CAREMANI, Marco, NARAYANAN, Theyencheri, HORVÁTH, Ádám I., MÁLNÁSI-CSIZMADIA, András, LINARI, Marco, LOMBARDI, Vincenzo and PIAZZESI, Gabriella, 2023. Titin activates myosin filaments in skeletal muscle by switching from an extensible spring to a mechanical rectifier. Proceedings of the National Academy of Sciences. Online. 2023. Vol. 120, no. 9, p. e2219346120. DOI 10.1073/pnas.2219346120.

TAMBORRINI, Davide, WANG, Zhexin, WAGNER, Thorsten, TACKE, Sebastian, STABRIN, Markus, GRANGE, Michael, KHO, Ay Lin, REES, Martin, BENNETT, Pauline, GAUTEL, Mathias and RAUNSER, Stefan, 2023. Structure of the native myosin filament in the relaxed cardiac sarcomere. Nature. Online. 2023. P. 1–9. DOI 10.1038/s41586-023-06690-5.

WALCOTT, Sam, WARSHAW, David M. and DEBOLD, Edward P., 2012. Mechanical Coupling between Myosin Molecules Causes Differences between Ensemble and Single-Molecule Measurements. Biophysical Journal. 2012. Vol. 103, no. 3, p. 501–510. DOI 10/f37d3b.