Living cells move thanks to assemblies of actin filaments and myosin motors that range from very organized striated muscle tissue to disordered intracellular bundles. The mechanisms powering these disordered structures are debated, and all models studied so far predict that they are contractile. We reexamine this prediction through a theoretical treatment of the interplay of three well-characterized internal dynamical processes in actomyosin bundles: actin treadmilling, the attachement-detachment dynamics of myosin and that of crosslinking proteins. We show that these processes enable an extensive control of the bundle’s active mechanics, including reversals of the filaments’ apparent velocities and the possibility of generating extension instead of contraction. These effects offer a new perspective on well-studied in vivo systems, as well as a robust criterion to experimentally elucidate the underpinnings of actomyosin activity.
Many cellular functions, from motility to cell division, are driven by myosin motors exerting forces on actin filaments held together by crosslinking proteins. This wide variety of processes is powered by an equally wide range of actomyosin structures, many of which do not display any apparent spatial organization of their components Verkhovsky et al. (1995); Cramer et al. (1997); Medalia et al. (2002); Kamasaki et al. (2007). While these structures are overwhelmingly observed to contract Murrell et al. (2015), the mechanisms underlying this contraction are unclear, as individual myosin motors can in principle elicit extension just as easily as contraction [Fig. 1(a-b)] Hatano (1994); Sekimoto and Nakazawa (1998); Lenz et al. (2012a); Pinto et al. (2013).
Recent investigations into this breaking of symmetry between contraction and extension have focused on two classes of models. The first of these classes is based on the idea that mechanical nonlinearities, e.g., the buckling of individual filaments under compression could suppress the propagation of extensile forces and thus favor contraction Dasanayake et al. (2011); Lenz et al. (2012b); Ronceray et al. (2016). By contrast, in the second type of models the spatial self-organization of the bundle’s motors and crosslinks along undeformable, rod-like actin filaments leads to contraction Kruse and Sekimoto (2002); Zumdieck et al. (2007); Zemel and Mogilner (2009); Oelz et al. (2015). So far, opportunities to discriminate between these two models experimentally remain very limited for lack of a clear theoretical prediction setting one apart from the other.
Here we provide such a prediction, namely that the self-organization mechanisms imply that actomyosin bundles robustly extend if taken to certain parameter regimes. This stark qualitative change from contraction to extension is easily detectable experimentally, and is not expected in mechanical nonlinearities models. Our prediction crucially rests on a simultaneous treatment of the filament, motor and crosslink dynamics; previous studies only involved partial treatments. The coupled dynamics of these elements induces a spatial organization of motors and crosslinks along the filaments, and our predicted switch between contraction and extension is driven by a localization of the motors and crosslinks to the filament ends, as illustrated in Fig. 1(c). We characterize the experimental regimes where either behavior is expected, and find that extension arises when the motor run-length and unbinding rate are relatively large compared to the filament length and the crosslink unbinding rate, respectively. Our study moreover identifies simple, widely applicable ideas to understand self-organization in active filament-motor systems.
We consider a bundle of polar filaments of length aligned in the -direction and subjected to periodic boundary conditions. The filaments are rigid, ruling out contraction arising from mechanical nonlinearities Lenz et al. (2012a). A filament may point in the direction of positive or negative , and maintains this polarity throughout the dynamics. At steady-state, filaments constantly grow from their barbed ends and shrink from their pointed ends at a fixed velocity , a phenomenon known as “treadmilling” throughout which their length remains constant Alberts et al. (2015) [Fig. 2(a)]. Motors and crosslinks constantly bind and unbind from filaments, and we denote by () and () the average motor (crosslink) unbinding time and equilibrium density [Fig. 2(b)].
Once bound to a filament, motors slide towards its barbed end with a velocity . The value of is set by a competition between the propulsive forces of the motors and the restoring forces of the crosslinks, and is to be determined self-consistently at a later stage of the calculation. In a mean-field description (valid for filaments interacting with many neighbors through many motors and crosslinks), this results in the pattern of motion illustrated in Fig. 2(c).
Focusing on a single right-pointing filament, the combined effect of motor motion and actin treadmilling implies that motors move with a velocity relative to the growing barbed end. Denoting by the distance between the motor and the barbed end [Fig. 2(c)], this implies that the number of bound motors per unit filament length satisfies the reaction-convection equation:
where is the motor current in the reference frame of the barbed end, and represents the attachment rate of unbound motors from the surrounding solution. Newly polymerized actin in does not yet have any motors bound to it, implying if ; likewise if . Motors bound to two filaments of opposing polarities exert forces on each filament, and we denote by the longitudinal force per unit length exerted by the motors on a right-pointing filament. For independent motors operating close to their stall force (i.e., motors whose velocity is essentially controlled by the external crosslink restoring forces), is proportional to the local motor density through . Note that motors do not induce internal forces in pairs of filaments with identical polarities, which we thus need not consider here.
The density of crosslinks of age bound in at time satisfies the conservation equation
with . Since the crosslink attachment points do not slide on the actin, their advection relative to the barbed end is entirely due to treadmilling and the crosslink current reads . The term in Eq. (2) can be viewed as an advection term along the coordinate , which account for the fact that the age of a bound crosslink increases linearly with time . While attached crosslinks are thus advected towards increasing , newly attached crosslinks all have age by definition, which we enforce through the delta function in the source term . As motor forces tend to slide filaments of opposing polarities respective to one another, they are opposed by the restoring forces of the crosslinks, which tend to keep filaments stationary with respect to one another. To describe this competition, we assimilate crosslinks to Hookean springs with elastic constant . The average extension of a crosslink bound to two antiparallel filaments is equal to zero at the time of its binding (denoted as ), but increases as as the filaments slide respective to one another [Fig. 2(d)]. As each crosslink exerts a Hookean force on the filament, the crosslink force per unit filament length is obtained by summing this force over all filament ages, yielding .
Equations (3) describe a depletion of motors and crosslinks close to the filament ends, with associated depletion lengths and , as illustrated in Fig. 3. The crosslink depletion results from the finite time required to decorate newly polymerized actin with crosslinks, while the motor depletion arises from the time required to dress a newly created filament overlap with motors. This delay may result from the motor binding time as discussed above, or from a rearrangement time required for an already-present motor to properly engage the filament. Provided the filament length is much larger than these depletion lengths, the motor force and crosslink friction asymptotically go to the constant values and far from the filament ends as the motor and crosslink densities go to their equilibrium values. We denote by the speed at which these asymptotic forces balance each other, which characterizes the hypothetical motion of infinite-length filaments.
By contrast, shorter filaments undergo both a smaller overall driving force and a smaller friction. Depletion thus affects the velocity , while itself affects motor depletion as described by Eq. (3a). Rescaling all lengths by and times by , we henceforth denote dimensionless variants of previously introduced variables with a tilde and determine by demanding that the total force exerted on a single filament vanishes. Defining , we insert Eqs. (3) into this condition and obtain a transcendental equation for :
where and are two constants and [see Fig. 4(a)]. As and , Eq. (4) gives rise to three regimes illustrated in Fig. 4(b-c): one where translocation by the motors is faster than treadmilling (), one where treadmilling is faster than translocation () and one where one solution coexists with two solutions. We determine the stability of these solutions by perturbing by a small quantity and assessing whether the overall force exerted on the filament tends to amplify or suppress this perturbation. We find that all unique solutions are stable (i.e., ). In the three-solutions regime, the smaller of the two solutions is unstable. The bundle thus chooses one of the other two, resulting in two coexisting stable solutions of opposing signs as illustrated in Fig. 4(d). As for any first-order (discontinuous) transition, bundles in this parameter regime will select either value of depending on their initial condition, and any switching from one to the other involves hysteresis.
We now turn to the contractile/extensile character of a bundle comprised of filaments per unit length. A filament in this bundle is subjected to a total force per unit length at location , where denotes the number of interacting neighbors of a filament. As the filament tension vanishes at the filament ends , its tension in thus reads . The contractile or extensile character of our bundle is revealed by its integrated tension across any plane. In thick bundles, this plane is intersected by a large number of filaments (namely ) each intersecting the plane at a random coordinate that is uniformly distributed between and . As a result, the bundle tension is given by the average . Defining , the respective contributions of the motors and crosslinks to the dimensionless bundle tension are
where the function is illustrated in Fig. 4(a). As shown in Fig. 5(a), these expressions can result in either sign for depending on the values of and . As the periodic boundary conditions used here confine the bundle to a fixed length, a bundle with a propensity to extend develops a negative tension (i.e., is compressed), while denotes a contractile (tense) bundle. These two behaviors respectively correspond to the situations illustrated in Fig. 3(b-c) and Fig. 3(d). We illustrate the regimes in Fig. 5(b) as a function of the original dimensionless parameters , and . As some parameter values yield coexisting metastable values of , so can they allow for both contractile and extensile steady states. However, despite this ambiguity at intermediate parameter values, Fig. 5(b) shows that the self-organization mechanism investigated here results in unambiguous extension for broad ranges of parameters.
This transition from contractile to extensile behavior upon an increase in the crosslink detachment time can be rationalized by an enlarged crosslink depletion zone in the vicinity of the filament barbed ends (Fig. 3). This implies a localization of crosslinks towards the filament pointed ends, resulting in an extensile “anti-sarcomere” organization [Fig. 1(c), right], in contrast with the contractile “sarcomere” structures [Fig. 1(c), left] found in our highly organized striated muscle. The emphasis of this mechanism on barbed end depletion suggests that it will not be significantly affected if pointed end assembly proceeds through severing Theriot (1997) rather than depolymerization. We also predict another transition, whereby a further increase of in the extensile phase causes the variable to change sign through a first-order (for small ) or a second-order (for large ) transition [Fig. 5(b)]. Indeed, the enhanced crosslink depletion associated with a large tends to reduce the friction between filaments, resulting in faster motor motion and thus in a situation where motor sliding outpaces treadmilling (). Both transitions could be directly observed by manipulating the actin dynamics or motor composition in current in vitro assays Thoresen et al. (2013); Reymann et al. (2012); Murrell and Gardel (2012) and possibly in cells Pinto et al. (2012). Such changes could also be at work in smooth muscle, where the number of myosins in individual thick filaments is regulated dynamically Seow (2005). The experimental relevance of these transitions is illustrated by a dashed line in Fig. 5(b), which shows that both transitions can be probed by varying between and while holding , Erdmann et al. (2013), Howard (2001), and Miyata et al. (1996) fixed. In addition, the magnitude of the forces and velocities predicted here are on par with those found in vivo, e.g., in the cytokinetic ring of fission yeast. Indeed, setting , , Wu and Pollard (2005), Rief et al. (1999), as in a hexagonal packing and , we find a contractile force comparable with the force required for fission extrapolated from the required cleavage force in echinoderm eggs Rappaport (1967) to a yeast ring with radius Zumdieck et al. (2007). We also find a characteristic velocity similar to that of ring contraction ().
Overall, our prediction that self-organized force generation entails a robust extensile regime provides a stringent test to validate or invalidate this model in specific experiments. For instance, the bundles of Refs. Thoresen et al. (2011, 2013) contract despite the fact that , which contradicts the self-organization prediction and thus validates the fact that they are dominated by mechanical nonlinearities. Conversely, stiff microtubules systems where buckling nonlinearities are strongly suppressed extend in vitro when the filament polymerization/depolymerization dynamics is blocked Sanchez et al. (2012); Keber et al. (2014) and contract in more complex in vivo situations Foster et al. (2015), consistent with the self-organization model. In addition, both extension and contraction have also been reported in stiff, non-buckling actin bundles Stam et al. (2017). Beyond pure contraction or extension, transitions between these two possibly coexisting (as in the multiple-solution regime of Fig. 4) metastable states could help understand several in vivo behaviors involving alternating contractions and expansions of the actomyosin cortex. This includes cell area oscillations observed during Drosophila, C. elegans, and Xenopus development Roh-Johnson et al. (2012); Kim and Davidson (2011) or propagating actomyosin contractility waves Allard and Mogilner (2013). It would also be interesting to see how the mechanisms described here apply to the more complex geometry of two- or three-dimensional actomyosin assemblies Lenz (2014), and to connect our self-organization mechanisms to the onset of positional ordering in muscle-like bundles Friedrich et al. (2012); Hu et al. (2017). Finally, the fundamental principles for the dynamical depletion of motors and crosslinks described here could serve as guiding principles in our nascent understanding of self-organized contractility in the cytoskeleton.
Acknowledgements.I thank Alex Mogilner for sharing Ref. Oelz et al. (2015) before publication, Pierre Ronceray for enlightening discussions and Samuel Cazayus-Claverie, Michael Murrell, Guglielmo Saggiorato and Danny Seara for comments on the manuscript. This work was supported by Marie Curie Integration Grant PCIG12-GA-2012-334053, “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM), ANR grant ANR-15-CE13-0004-03 and ERC Starting Grant 677532. My group belongs to the CNRS consortium CellTiss.
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