4 Microscale: the contractile unit
4.1 Background: muscle contractile units and models
4.1.1 The muscle contractile unit
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 forming a hexagon if observed in the transverse direction, see Figure 1.2 and Figure 5.2. Even though the strcuture 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 structural 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, and can therefore 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.
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 (Pertici et al. 2018; Pertici et al. 2020). A dual laser optical trap holds a bead to which an actin filament is glued. This actin filament is brought in the viciinity of the side of a microneedle covered with myosin motors.
Previous works on similar systems, reported interesting dynamical regimes with various spontaneous oscillations motifs, see in particular the work of Plaçais et al. (2009); Walcott, Warshaw, and Debold (2012), Kalganov et al. (2013), and Kaya et al. (2017). 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 motors, generate a cascade of detachment resulting in swift force decay giving and 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.
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) (Jülicher and Prost 1995, 1997; Plaçais et al. 2009; Guérin et al. 2010; Guérin, Prost, and Joanny 2011). It has been 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 for instance spontaneous oscillations.
A typical model of this type can be formulated as follows (Guérin, Prost, and Joanny 2011). 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.1
1 In this respect, this formulation slightly differs from the classical Huxley-Hill model, because the latter considers four rates see (Jülicher, Ajdari, and Prost 1997) and Section 3.2.1.
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 a feedback on the population dynamics. The detailed analysis of the consequence of this feedback in various regimes that depend on the paramters values can be found in (Guérin, Prost, and Joanny 2011), a particularly interesting point being the broad variety of oscillatory regimes that can potentially be observed. Variants of the above model were also studied in (Guérin, Prost, and Joanny 2010; Plaçais et al. 2009).
Our research also aims at modeling the mechanical interaction between motors induced by the presence of long-range elastic connectors, the myofilaments, within the contractile structure. The difference with the above work lays into the efforts made toward a quantitative physiologically relevant description of the force generation process to see in particular if the behaviors predicted by earlier physical models can be observed in physiological regimes.
4.2 Challenges
To put our work in perspective, we raise here a list of challenges relevant to the scale of the contractile unit.
- As explained in Section 3.1.3, the use of synthetic contractile systems, like in the single molecule experimental setup, allows to study the most basics acto-myosin interaction mechanism. The nanomachine developed at the PhysioLab, 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 by Pertici et al. (2018). The first challenge is to formulate a model of this class of experiments, where a finite number of interacting molecular motors are involved.
- 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 for instance short and long-range components. For instance, the actin and myosin filament 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, see (Tamborrini et al. 2023) and Chapter 5. The second challenge is thus to provide a model of this elastic network.
- Just like a low dimensional model of a single molecule is necessary to simulate the effective behavior of the myosin protein over physiological timescale, it is likely that a model reduction will be necessary to replace the simulation of hundreds of motors forming a contractile unit. The formulation of such model in the presence of internal interactions in a rigorous way may reveal a difficult mathematical challenge.
- The topic has not been mentioned yet but regulation mechanisms may be influenced by the mechanical interactions between the myosin motors within a contractile unit. The question of regulation will be addressed in more details in Chapter 5.
4.3 Contributions
The majority of our work on contractile unit results from our collaboration with L. Truskinovsky and J.M. Allain (Caruel, Allain, and Truskinovsky 2013, 2015; Caruel and Truskinovsky 2016, 2017). The results mentionned here have already been the object of a review, see (Caruel and Truskinovsky 2018, sec. 2). 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 the neurotransmission.
4.3.1 Fast transient response of muscle fibers
Our contributions focus on the fast timescale response of a muscle fiber submitted to swift load changes (Podolsky, Nolan, and Zaveler 1969; Huxley and Simmons 1971). Briefly, 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). A \(t=0\), a load step is applied, either in tension (soft device) or in length (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 \(\delta z_{2}\).
Importantly, in these fast timescale regimes the attachment-detachment events of individual myosin motors can be ignored2, which means that the number of cross-bridges can be consiered constant.
2 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 millisecond Reconditi et al. (2004). In the interval, the number of cross-bridges is constant which allows measuring detect the conformational change in the attached motors.
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.
4.3.2 Mechanical model of a contractile unit
3 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 2019; Powers et al. 2020; Squarci et al. 2023)
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 myosfilament to the elastic response is lumped into a linear series element, is supported by the work of Ford, Huxley, and Simmons (1981) and Linari et al. (1998) 3.
In the contractile unit model, each cross-bridge is a represented as a bistable snap-spring. The two configuration of the spring represents the pre- and post-power stroke conformation. 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 (1971), or using a continuous Langevin dynamics in a double well potential as in (Marcucci and Truskinovsky 2010). 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
The detailed analysis of the purely mechanical properties of the model is available in (Caruel, Allain, and Truskinovsky 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 each the post-power stroke conformation.
To illustrate the behavior of the system in the absence of temperature, Caruel, Allain, and Truskinovsky (2015) considered a system with a rigid backbone (consider \(\lambda_{b}\to +\infty\) 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 hard and soft device, 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 response can be investigated by representing the energy of the metastable states as function of \(p\) at these transitions, see [(d)-(g)]. In the hard device case at \(y=y_0\) (d), all configuration have the same energy. However, in the soft device case at \(\sigma=\sigma_0\) (g), mixed states (\(0<p<1\)) have a higher energy 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)]. 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.
More results are available in (Caruel, Allain, and Truskinovsky 2015) and a similar study is carried out when an elastic backbone is present (mixed device) in (Caruel and Truskinovsky 2018). In the discussion of (Caruel and Truskinovsky 2018), we show that this generic mechanical cross-talk may be present in many other biological systems, ranging from epigenetics to hair-cells. We now present how temperature interferes with these purely mechanical long-range interactions.
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 (1971) to interpret the experimental results obtained for imposed length steps. The model was reformulated by Caruel and Truskinovsky (2016) 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 \(\lambda_{b}\to\infty\) 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.
One of the most important characteristics of the thermal equilibrium of this model is illustrated in Figure 4.5. The response depends on the 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 (1971) 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.
This 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 tend to decreses the free energy of these states. Hence, the mechanical feedback dominates at low temperature but becomes negigible when the thermal energy allows the cross-brigde to easily flip conformation thanks to high fluctuations.
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 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 \(N\). Therefore, synchronization tends to be 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. The details of this analysis can be found in (Caruel and Truskinovsky 2018). 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 \(\lambda_{b}\to + \infty\) and \(\lambda_{b}\to 0\) (\(z\to\infty\)), respectively. For the Huxley and Simmons (1971) “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 also finite size contractile units.
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 loading correspond to the limits \(\lambda_{b}\to\infty\) 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 (1971) model as explained in Section 3.4.2. Qualitatively, the behavior of the regularized model is similar to the Huxley and Simmons (1971) model, though it partially looses its analytical transparency. We refer to (Caruel and Truskinovsky 2018, sec. 2.2) for the details. 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). This finding was published in (Caruel, Allain, and Truskinovsky 2013).
A follow-up of this work was proposed by Borja da Rocha and Truskinovsky (2019), where quenched desorded was added to the model adding third dimension to the phase diagram. Again, the author showed that the system was poised to criticality.
4.3.6 Effective dynamical model of a contractile unit
A preliminary attempt was made in (Caruel 2011, chap. 6) to derive an effective model of a contractile unit, the goal being to project the high dimensional dynamics of the contractile unit on a low dimension manifold. So far only the fast transition behavior at constant \(N\) was considered.
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 (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 (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 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
This work was published in (Manca et al. 2019) and a follow-up study was recently issued by Caruel and Pincet (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. When the eletrical signal arrives, it triggers the massive release of \(\ce{Ca^2+}\) ions, which results in the fusion of the vesicles membranes with the neuron membrane, and releases 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 kBT): a spontaneous fusion would take more than 1s, while it happens in less than 1ms in the brain.
The physiological speed is achieved thanks to a bundle of bistable proteins called SNAREpins4, that are bridging the vesicle and the neuron membrane prior to fusion, see Figure 4.8. Their collective conformational change ressembles a zippering that pulls the two membranes toward each other, see Figure 4.8 (a).
4 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.
The system evolves under no external forces other than the viscous drag on the vesicle and the thermal noise. We prediced in (Manca et al. 2019) that the time to reach fusion was a non-monotone function of the number of SNAREpins \(N\) showing an optimum arount \(N=4\), see Figure 4.8 (d). The fusion model is the combination of two processes:
- the collective zippering (\(n\to c\)), whose timescale augments with \(N\), due to an increasing energy barrier [see Figure 4.8 (c) and (d)],
- 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 with \(N\), which makes the zippering process 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 by the recent review by Rothman et al. (2023).
The follow-up study (Caruel and Pincet 2024)is based on the recent work by Bera et al. (2023), suggesting that the SNAREpins forms a double ring structure. In our first study, only one ring was considered. We show in our new publication 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.
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 consider for instance the reformulation of the Huxley (1957) model proposed by Chaintron, Caruel, and Kimmig (2023). The framework would then be quite similar to the one used by Guérin, Prost, and Joanny (2011), but with a 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.
The next step is to use a model that involves both the power-stroke and the attachment-detachment process. We here have the choice between the hybrid molecular motor model proposed in (Caruel, Moireau, and Chapelle 2019; Chaintron et al. 2023) or the fully mechanical model proposed by Sheshka and Truskinovsky (2014),see also (Caruel and Truskinovsky 2018, sec. 4.2).
The development of these contractile unit models is highly motivated by our collaboration with the PhysioLab and the oportunity to use the nanomachine platform presented in (Pertici et al. 2018) to improve our understanding of the basic functioning of finite size groups of molecular motors. Among the remaining tasks is the modeling of the dispersion of the motor orientation that is inherent to the nanomachine since the molecular motors are randomly positioned on the glass surface facing the actin filament. This dispersion could have an effect similar to the quenched disorder discussed by Borja da Rocha and Truskinovsky (2019).