Design principles for selective self-assembly of active networks
Living cells dynamically modulate the local morphologies of their actin cytoskeletons to perform biological functions, including force transduction, intracellular transport, and cell division. A major challenge is to understand how diverse structures of the actin cytoskeleton are assembled from a limited set of molecular building blocks. Here we study the spontaneous self-assembly of a minimal model of cytoskeletal materials, consisting of semiflexible actin filaments, crosslinkers, and molecular motors. Using coarse-grained simulations, we demonstrate that by changing concentrations and kinetics of crosslinkers and motors we can generate three distinct structural phases of actomyosin assemblies: bundled, polarity-sorted, and contracted. We introduce new metrics to distinguish these structural phases and demonstrate their functional roles. We find that the binding kinetics of motors and crosslinkers can be tuned to optimize contractile force generation, motor transport, and mechanical response. By quantitatively characterizing the relationships between modes of cytoskeletal self-assembly, the resulting structures, and their functional consequences, our work suggests new principles for the design of active materials.
August 2, 2019
Mechanical functions of living cells are determined by dynamic restructuring of the actin cytoskeleton, a highly conserved cellular machinery composed of filamentous actin (F-actin), myosin molecular motors, and crosslinking proteins Murrell et al. (2015). An enormous variety of F-actin binding proteins exist, with a wide range of physico-chemical properties Michelot and Drubin (2011) that can combine with F-actin to assemble function-specific, cellular structures. Spatiotemporal control over these structures is essential for coordinated force generation during cell migration Pollard and Borisy (2003); Lomakin et al. (2015), cell adhesion Parsons et al. (2010), cytokinesis Sedzinski et al. (2011), and intracellular transport Munro et al. (2004); Tabei et al. (2013). A quantitative understanding of how diverse cytoskeletal structures are assembled from a limited set of molecular building blocks presents an outstanding challenge at the interface of soft matter physics and cell biology.
Given the many interconnected molecular processes within cells, it is experimentally difficult to controllably study how variations in molecular-scale properties affect emergent actin network structures and function. To this end, in vitro biochemical studies have enabled major inroads to be made into uncovering how biochemical compositions and relative amounts of actin filaments and molecular motors determine network architectures and mechanical properties Bendix et al. (2008); Kohler et al. (2011); Alvarado et al. (2013); Murrell and Gardel (2012, 2014); Ennomani et al. (2016); Linsmeier et al. (2016); Chugh et al. (2017); Stam et al. (2017). Additionally, it has been shown how the mechanical response of actin networks can be tuned by varying the types and amounts of crosslinkers involved Gardel et al. (2004); Kasza et al. (2009); Freedman et al. (2017); Schmoller et al. (2009); Weirich et al. (2017); Lieleg et al. (2009).
Simulations complement experiments and theories Kruse et al. (2005); Lenz et al. (2012); Prost et al. (2015) by allowing both precise control of the physical properties of constituents and examination of microscopic mechanisms. These features have enabled simulations to provide insights into contractility Dasanayake et al. (2011); Wang and Wolynes (2012); Ennomani et al. (2016); Belmonte et al. (2017); Stam et al. (2017), viscoelastic properties Kim et al. (2009); Head et al. (2011), transport Scholz et al. (2016), and structural rearrangements Cyron et al. (2013); Bidone et al. (2017). While the structural phase diagram of filaments with passive crosslinkers has been mapped and shown to exhibit homogeneous isotropic gel, bundled, clustered, and lamellar phases Cyron et al. (2013), to the best of our knowledge, a corresponding study of active materials has not been performed previously. It remains to determine the network structures accessible to mixtures of filaments, crosslinkers, and motors, the conditions under which they are formed, and their relations to functions.
In this Letter, we map the non-equilibrium structural phases of cytoskeletal networks consisting of F-actin, crosslinkers, and myosin motors. We observe homogeneous, bundled, contracted, and polarity-sorted actin networks and characterize the parameters that control their assembly. In doing so, we introduce order parameters that can classify the extent to which networks contract into dense aggregates, sort F-actin by polarity, or bundle filaments into force propagating networks. Using these metrics, we demonstrate how actin networks can be selectively optimized for specific mechanical functionalities, by systematic variations in network composition, filament length, and kinetic properties of actin binding proteins.
Coarse-grained model. To study the spontaneous self-assembly of cytoskeletal structures at experimentally relevant length and time scales (microns and minutes), we use AFINES, a simulation framework we recently developed Freedman et al. (2017). In brief, actin filaments are modeled as polar worm-like chains (represented by beads connected by springs) with defined barbed and pointed ends (Figure 1); crosslinkers are modeled as linear springs with ends (heads) that can stochastically bind and unbind from F-actin via a kinetic Monte Carlo procedure that preserves detailed balance; molecular motors are modeled as active crosslinkers such that once bound, they walk toward the barbed end of F-actin at a load-dependent speed. We use Brownian dynamics to evolve the positions of constituents in 2D. Restriction to 2D is consistent with the fact that in vitro reconstitutions of actomyosin networks are nearly flat Schmoller et al. (2009); Murrell and Gardel (2012); Stam et al. (2017). To enable rearrangement in 2D, we neglect excluded volume, which is reasonable since the F-actin density in our simulations is well below the isotropic to nematic transition, so the network connectivity dominates the dynamics. The model is described in detail in Appendix S1, and Table S1 lists all simulation parameters.
Self-assembly and characterization of actin network structures. We observe three distinct network architectures formed from initially disordered mixtures of F-actin, motors, and crosslinkers: bundled, polarity-sorted, and contracted. Examples of each structure are shown for simulations of 500 10 m long filaments in the highlighted corners of Figure 1. When F-actin is mixed with crosslinkers, a bundled network forms as crosslinkers bind proximal F-actin into thick bundles, which in turn are joined at intersections to yield a well-connected mesh. When F-actin is mixed with motors, barbed ends aggregate to form a polarity-sorted network. Combining F-actin with both motors and crosslinkers results in macroscopic contraction of the actin filaments into dense and disconnected aggregates.
To systematically explore how varying constituent network properties affects structure formation, we introduce physical order parameters to quantify each of the observed structural phases. We compute the spatial extent of F-actin aggregation using the radial distribution function, , where is the probability that two actin beads are separated by a distance in the range , (here, ) and is the number density of actin beads. For a homogeneous network, at all distances (Figure 2A; the small peaks at integer arise from the spacing of beads within actin filaments). In contrast, for contracted networks, for , indicating F-actin exceeds the bulk density.
While Figure 2A shows that actin filaments are nearly uniformly distributed in a polarity-sorted network, Figure 1 indicates that their barbed ends (blue) are concentrated. To quantify their aggregation specifically, we compute the ratio , where is the distance between filament barbed ends. Figure 2B shows that in polarity-sorted networks F-actin barbed ends are aggregated (and have a secondary peak at , the rest length of motors). In contracted networks, barbed ends also aggregate to a higher degree than bundled networks, indicating a degree of polarity sorting.
Figure 2A indicates that bundled networks aggregate at smaller length scales, , corresponding to the crosslinker rest length . To quantify the degree of bundling, and to distinguish it from contractility, we measure the distribution of network pore sizes by a procedure that is similar in spirit but simpler than that in Ref. 35. Namely, we grid the simulation box into bins and compute how many filaments pass through each bin. For each empty bin, we determine the lengths of the contiguous stretches of empty bins that contain it in the vertical and horizontal directions (Figure S2). We average these lengths over all empty bins to obtain an average mesh size for each network structure.
In Figure 2C, the distributions of mesh sizes for polarity-sorted and homogeneous networks are similar, indicating that the former does not coarsen significantly. The bundled and contracted networks exhibit larger pore sizes; indeed, contracted networks exhibit pore sizes spanning the simulation region, indicating that the network has ripped apart. We can distinguish these cases by normalizing the mesh size by (), which quantifies the extent of aggregation. The inset shows that, while the contracted networks initially bundle, at long times this effect is small compared to aggregation. In contrast, bundled networks have a continuously increasing normalized mesh size (Figure 2C, inset); we take this measure as a metric for the degree of bundling.
To examine the relationship between actin network structure and contractility, we use the metric , where is the actin’s velocity field, and indicates spatial averaging (Appendix S3) Freedman et al. (2017). As shown in Figure 2D, becomes significantly more negative for aggregating networks than for bundling or polarity-sorted networks. Comparison with Figure 2A shows that extensive contractility is associated with large .
Using these order parameters, we map the structural phase space of actomyosin networks as functions of motor and crosslinker densities and their binding affinities (Figure 3). Consistent with Figure 1, networks are contracted when motor and crosslinker densities are high (Figure 3A), polarity-sorted when only motor density is high (Figure 3B), and bundled when only crosslinker density is high (Figure 3C). Interestingly, while high motor densities inhibit bundling, a small population of motors () enhances filament bundling.
Non-monotonic trends in binding kinetics and filament length. Finally, we modulate molecular-level interaction parameters between F-actin and its binding partners (crosslinkers, motors) to dissect their relative roles in building different structures. These parameters are hard to change independently in experiment. At fixed motor and crosslinker densities, we find that cytoskeletal structures can be tuned by varying the dissociation constants, (Figure 3D-F). The trends are non-monotonic, in contrast to those in the densities (Figure 3A-C). In particular, contraction is highest for intermediate values of (Figure 3D), and bundling is highest for low values of with low or high (Figure 3F). Notably, this non-monotonic trend only arises for a fixed simulation time, . For a fixed value of , changing modulates structure formation in a monotonic manner, akin to changing (Figure S3). The non-monotonic trends are important, however, because many structures in cells must form transiently due to competing kinetic processes, such as actin filament turnover Munro et al. (2004); Fritzsche et al. (2016).
As the length () of F-actin varies considerably within cells Mohapatra et al. (2016), we tested how varying and in tandem affects structure formation. In Figure 4A, we show that when mixed with both motors and crosslinkers, longer actin filaments yield larger contracted aggregates. This is due to an increase in network connectivity for longer filaments; filaments that are too short do not undergo significant aggregation. As in Figure 3B, the dependence of aggregation on binding affinity is non-monotonic whenever there is significant aggregation (i.e., ). In Figure 4B, we show that for networks with only motors, at low , increasing filament length promotes polarity sorting. By contrast, high , completely suppresses polarity sorting due to the short lifetime of motor attachment. As evident from the representative network structures in Figure S6, making the filaments short also suppresses polarity sorting. Both the large number of filaments and their rapid diffusion favor mixing over the consistent motor-filament interactions needed for sorting.
For networks with only crosslinkers, the mesh size (Figure 4C) is non-monotonic with respect to both filament length and crosslinker affinity. From example structures (Figure S7), we see that low crosslinker affinity and short filament lengths cause mixing for the same reasons as immediately above, such that stable crosslinked networks do not form. Conversely, assemblies with high crosslinker affinity or long filament lengths form crosslinked networks, but they rearrange slowly, so further coarsening is impeded, and the mesh size remains small. As these non-monotonic trends only occur for m, they are more likely to impact structures with longer actin filaments found in budding yeast Chesarone-Cataldo et al. (2011), stereocilia Lin et al. (2005), filopodia, or in vitro reconstituted networks Weirich et al. (2017); Stam et al. (2017). Structures with shorter filaments, as found in lamellipodia or the actin cortex ( m) Salbreux et al. (2012); Chugh et al. (2017), are less likely to have a finite binding affinity that maximizes contractility.
Structure of networks tunes transport and force propagation. While the structures of contracted, polarity-sorted, and bundled networks clearly differ, their consequences for biophysical functions are not immediately apparent. To determine how these structures influence motor transport Brawley and Rock (2009); Kohler et al. (2011), we fix the actin structures and follow the dynamics of the motors. This facilitates obtaining well-converged statistics for the dynamics and interpretation in terms of individual trajectories. While the apparent scaling of the mean-squared displacement is consistent with simple diffusion (Figure S8A), as observed experimentally Burov et al. (2013), sample trajectories (Figure S8C) indicate that motors in contracted and polarity-sorted structures spend significant amounts of time trapped in aggregates of barbed ends. We quantify this caging effect using a previously defined metric that can distinguish different kinds of motion Burov et al. (2013). Namely, we compute the velocity of a motor with center of mass position at time as and evaluate the distribution of angles, , between consecutive velocity vectors and , at different values of .
We find that for all structures, there is at least one time scale in which the distribution has a broad peak at (Figure 5A), indicating that motors are reversing direction, consistent with confinement. For polarity-sorted networks barbed ends are most tightly aggregated, and thus motors exhibit caging at all time scales measured. Contracted networks are partially polarity-sorted, so filaments can direct motors both in and out of aggregates, making the caging more spatially extended. Because it takes longer to explore the extended length scale, the caging manifests only at s. Bundled networks show caging effects most prominently at intermediate time scales. In this case, the caging corresponds to motors cycling between oppositely oriented filaments, which can give rise to apparently glassy dynamics Scholz et al. (2016).
Next, we evaluated how structural rearrangements in actin networks affect their ability to propagate mechanical forces over long length scales. To this end, we subjected the final network configuration to a shear strain of magnitude (Appendix S8) and measured the resulting strain energy. In Figure 5B, we show the strain dependence of the total strain energy density, , where , , and are the potential energies of the F-actin, motors, and crosslinkers, respectively, (Appendix S1) and is the simulation box volume, assuming a thickness of . Bundled networks exhibit a quadratic dependence on strain, indicating a solid-like material response. By contrast, contracted and polarity-sorted networks are fragile, with linear dependences of energy density on strain.
Discussion. Our coarse-grained simulations show how modulating the abundance and properties of cytoskeletal constituents can tune emergent network structures. For finite times, we find that motor and crosslinker binding affinities, as well as filament lengths, have optimal values for maximizing contractility, bundling, and polarity sorting. While there are still many unanswered questions on the self-organized mechanical behavior of cytoskeletal materials, our work takes an important step forward by linking kinetics to the selection of network structures with specific mechanical functions. We expect that the order parameters that we introduced to characterize network structure can also be applied to interpretation of experimental imaging data. Further simulations using our modeling framework can shed light on the structure-function relations in other active polymer assemblies, including networks of microtubules, kinesin, and dynein Sanchez et al. (2012); Foster et al. (2015). From the perspective of materials design, our work demonstrates how a limited set of molecular building blocks can self-assemble diverse active materials. This presents the prospect that theory and simulation can be used to guide the design of active biomimetic materials with desired collective mechanical properties.
Acknowledgements.We thank M. Gardel, J. Weare, S. Stam, and K. Weirich for helpful conversations. This research was supported in part by the University of Chicago Materials Research Science and Engineering Center (NSF Grant No. 1420709). S.L.F. was supported by the DoD through the NDSEG Program. G.M.H. was supported by an NIH Ruth L. Kirschstein NRSA award (1F32GM113415-01). S.B. was supported by a UCL Strategic Fellowship. Simulations resources were provided by the Research Computing Center at the University of Chicago.
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Supplemental Materials for
Design principles for selective self-assembly of
Simon L. Freedman, Glen M. Hocky, Shiladitya Banerjee, Aaron R. Dinner
Appendix S1 AFINES simulation
In AFINES, actin filaments, myosin motors, and passive crosslinker proteins are modeled as coarse grained entities. Actin filaments are treated as worm-like chains of beads connected by harmonic springs (links) and angular harmonic springs. Thus, the internal forces on an actin filament can be obtained from the gradient of the potential energy :
where is the position of the bead on a filament, is the angle between the and links, is the stretching force constant, is the bending modulus, and is the equilibrium length of a link. The persistence length of the filament is then , where is Boltzmann’s constant and is the temperature.
We model crosslinkers as Hookean springs, with two ends (heads) that can stochastically bind to and unbind from filaments. Thus, the potential energy of a crosslinker is
where is the crosslinker stiffness, is its rest length, is the position of head , is if head is bound and otherwise, and is the rate constant for binding (unbinding). When a crosslinker is bound, it moves with the filament to which it is bound. When both crosslinker heads are bound, its tensile force, is propagated onto the filament beads neighboring each bound head at position via the lever rule,
where is the force on the filament bead at position .
Binding and unbinding are governed by a Monte Carlo procedure constructed to satisfy detailed balance in the absence of motors. At each timestep of duration , an unbound crosslinker head becomes bound to the nearby filament with probability , where is defined as follows. The closest point on the filament is identified, and the change in energy associated with moving the head to it, is computed; . When a head becomes bound, its displacement, in the frame of reference of the filament link to which it attached, is stored as . Later, the head can become unbound with probability , where and is the energy that would result from applying a displacement of to the head.
We model a motor similarly to a crosslinker, in that it is a Hookean spring, can bind to and unbind from filaments, and propagates force onto them. Thus, their potential energy is identical to Equation S2 with the subscript replacing the subscript . Additionally, a bound motor head moves towards the barbed end of the actin filament to which it is bound at a load dependent velocity
where is the unloaded motor speed, is the tensile force on the motor, and is the tangent to the filament at the point where the motor is bound; points toward the pointed end of the filament.
We simulate the system using Brownian dynamics such that the position of an actin bead, motor head, or crosslinker head at time is generated by the equation
where is the gradient of the potential of the particle, is a vector of random numbers drawn from the standard normal distribution, and we use the Stokes relation in the damping term, where is the size of the particle, and is the dynamic viscosity of its environment leimkuhler2013S (). We simulate the system in and use periodic boundary conditions to limit boundary effects. A complete list of model parameters used for Figures 4, 3, 2 and 1 is provided in Table S1 and additional methods used for Figure 5 are described below (Appendices S8 and S7).
|Symbol||Description (units) [ref]||Value|
|link density ()|
|number of links per filament ()|
|link rest length () odijk1983S ()|
|stretching force constant (pN)|
|bending modulus (pN) ott1993S ()|
|rest length () niederman1975S ()|
|maximum attachment rate (s)|
|maximum detachment rate (s)|
|maximum end detachment rate (s)|
|unloaded speed (s) kron1986S ()|
|stall force of myosin (pN) veigel2003S ()|
|rest length (filamin) () ferrer2008S ()|
|maximum attachment rate ()|
|maximum detachment rate ()|
|dynamics timestep (s)|
|maximum simulated time (s)|
|,||length and width of assay ()|
|grid density ()|
|dynamic viscosity (Pas)|
Appendix S2 Varying motor and crosslinker densities
Appendix S3 Actin velocity field calculation
Contraction of actin networks is typically calculated by identifying sinks in the divergence of the actin velocity field. To construct this field, we first calculate the velocity of actin beads where is the position of an actin bead at time , and is the lag time. To reduce noise, we calculate , the average velocity in every bin of size . We then interpolate the velocity field using Gaussian Radial Basis Functions (RBF), such that the velocity at any position is
where is the number of bins with at least actin beads, and are the weights of the basis functions, determined by solving the equation (using the scipy.interpolate.Rbf package scipyS ()). For Figure 2B, we used a lag time of s, a threshold of actin beads in a local box of size and a Gaussian width of , as we have found these interpolation values robustly capture the motion of the actin freedman2017S ().
Appendix S4 Mesh Size
We show an example of the results of the mesh size calculation described in the main text on a bundled network at s.
Appendix S5 Effects of relaxation times on network structures
To understand how the structure-parameter relations that we elucidated depended on the observation time, we measured the magnitude of contraction at a fixed value of . We find that fixing yields a monotonic decrease in contraction (Figure S3A), indicating that for long times, total contraction (as opposed to the rate of contraction) is maximized by increasing the affinity of motors and crosslinkers with F-actin. In contrast to contraction, for networks without crosslinkers, polarity sorting decreases monotonically with decreasing motor affinity (Figure S3B). However, for networks without motors, at fixed times bundling is maximized for intermediate , while at long times it is maximized at low (Figure S3C). Our results indicate that to attain specific structures within a fixed time scale, there are optimal choices for motor and crosslinker affinities, while at long times affects structure formation similarly to .
Appendix S6 Structure Examples
Appendix S7 Effects of structure on motor transport
To quantify the ability of networks to direct motor transport, we added motors at random locations to all three structural phases (i.e., their structures at s) and simulated for an additional s. In Figure S8A-B, we plot the mean squared displacement as a function of lag time, , and show that for all phases, this quantity increases linearly, indicating that the motor trajectories are, on average, diffusive. We performed this measurement with the motors described above (Appendix S1), as well as ones that detach faster from the barbed ends of filaments; i.e., at the barbed end they had a detachment rate of freedman2017S (). This second set of measurements was done to ascertain that the observed motor caging was a consequence of network structure, and not a consequence of dwelling at the barbed ends of the filaments, given that we assume a uniform detachment rate. Both types of motors display caging effects in their trajectories (Figure S8C), which we have quantified using the distribution of angles between consecutive velocity measurements at varying lag times (Figure 5A and Figure S9). Motors with exhibit an additional spike of their distribution at due to barbed ends acting as local attractors, between which motors move back and forth.
Appendix S8 Simulating shear
To measure the stiffness of self-assembled actin networks, we simulate a controlled strain experiment and shear the final network configuration by a total strain of in a fixed amount of time s. This is accomplished by supplementing the Brownian Dynamics described in Appendix S1 by explicitly shifting the actin bead position such that where is the simulation cell height, and is the amount of time for a small shear evans1984S (). Additionally, the boundary conditions follow the Lees-Edwards convention during the shear lees1972S (). As described in freedman2017S (), we do not perform this shift at every time step; rather where is the simulation time step and is a suitable amount of time for the simulation to relax from the large external force imposed by the shear. In Figure 2F, we used s and s.
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