A design framework for actively crosslinked filament networks
AA design framework for actively crosslinked filament networks
Sebastian F¨urthauer, Daniel J. Needleman, and Michael J. Shelley
1, 3 Center for Computational Biology, Flatiron Institute, New York, NY 10010, USA Paulson School of Engineering & Applied Science and Department of Molecular & Cellular Biology,Harvard University, Cambridge, MA 02138, USA Courant Institute, New York University, New York, NY 10012, USA
Living matter moves, deforms, and organizes itself. In cells this is made possible by networksof polymer filaments and crosslinking molecules that connect filaments to each other and that actas motors to do mechanical work on the network. For the case of highly cross-linked filamentnetworks, we discuss how the material properties of assemblies emerge from the forces exerted bymicroscopic agents. First, we introduce a phenomenological model that characterizes the forcesthat crosslink populations exert between filaments. Second, we derive a theory that predicts thematerial properties of highly crosslinked filament networks, given the crosslinks present. Third, wediscuss which properties of crosslinks set the material properties and behavior of highly crosslinkedcytoskeletal networks. The work presented here, will enable the better understanding of cytoskeletalmechanics and its molecular underpinnings. This theory is also a first step towards a theory of howmolecular perturbations impact cytoskeletal organization, and provides a framework for designingcytoskeletal networks with desirable properties in the lab.
I. INTRODUCTION
Materials made from constituents that use energy to move are called active. These inherently out of equilibriumsystems have attractive physical properties: active materials can spontaneously form patterns [1], collectively move[2–4], self-organize into structures [5, 6], and do work [7]. Biology, through evolution, has found ways to exploit thispotential. The cytoskeleton, an active material made from biopolymer filaments and molecular scale motors, drivescellular functions with remarkable spatial and temporal coordination [8, 9]. The ability of cells to move, divide, anddeform relies on this robust, dynamic and adaptive material. To understand the molecular underpinnings of cellularmechanics and design similarly useful active matter systems in the lab, a theory that predicts their behavior fromthe interactions between their constituents is needed. The aim of this paper, is to address this challenge for highlycrosslinked systems made from rigid rod-like filaments and molecular scale motors.The large-scale physics of active materials can be described by phenomenological theories, which are derived fromsymmetry considerations and conservation laws, without making assumptions on the detailed molecular scale inter-actions that give rise to the materials properties [10–12]. This has allowed exploring the exotic properties of activematerials, and the quantitative description of subcellular structures, such as the spindle [6, 13] (the structure that seg-regates chromosomes during cell division) and the cell cortex[14–17] (the structure that provides eukaryotic cells withthe ability to control their shape), even though the microscale processes at work often remain opaque. In contrast,understanding how molecular perturbations affect cellular scale structures requires theories that explain how materialproperties depend on the underlying molecular behaviors. Designing active materials with desirable properties in thelab will also require the ability to predict how emergent properties of materials result from their constituents [18].Until now, the attempts to bridge this gap have relied heavily on computational methods [19–21], or were restrictedto sparsely crosslinked systems [22–26], one dimensional systems [27, 28], or systems with permanent crosslinks [29].Our interest here are cytoskeletal networks, which are in general highly crosslinked by tens to hundreds of transientcrosslinks linking each filament into the network. In this regime, the forces generated by different crosslinks in thenetwork balance against each other, and not against friction with the surrounding medium, as they would in a sparselycrosslinked regime [30].We derive how the large scale properties of an actively crosslinked network of cytoskeletal filaments depend onthe micro-scale interactions between its components. This theory generalizes our earlier work on one specific typeof motor-filament mixture, XCTK2 and microtubules [30, 31], by introducing a generic phenomenological model todescribe the forces that crosslink populations exert between filaments.The structure of this paper is as follows. In section II, we discuss the force and torque balance for systems ofinteracting particles, and specialize to the case of interacting rod-like filaments. This will allow us to introducekey concepts of the continuum description, such as the network stress tensor. Next, in section III, we presenta phenomenological model for crosslink interactions between filaments, that can describe the properties of manydifferent types of crosslinks in terms of just a few parameters, which we call crosslink moments. In section IV, wederive the continuum theory for highly crosslinked active networks and obtain the equations of motion for thesesystems. Finally, in section V we give an overview of the main predictions of our theory and discuss the consequences a r X i v : . [ c ond - m a t . s o f t ] S e p of specific micro-scale properties for the mechanical properties of the consequent active material. We summarize andcontextualize our findings in the discussion section VI. II. FORCE AND TORQUE BALANCE IN SYSTEMS OF INTERACTING ROD-LIKE PARTICLES
We start by discussing the generic framework of our description. In this section we give equations for particle,momentum and angular momentum conservation and introduce the stress tensor, for generic systems of particles withshort ranged interactions. We then specialize to the case of interacting rod-like filaments, which form the networksthat we study here.
A. Particle Number Continuity
Consider a material that consists of a large number N of particles, that are characterized by their center of masspositions x i and their orientations p i , where | p i | = 1 is an unit vector and i is the particle index. We define theparticle number density ψ ( x , p ) = (cid:88) i δ ( x − x i ) δ ( p − p i ) . (1)Here and in the following δ ( x − x i ) has dimensions of inverse volume, while δ ( p − p i ) is dimensionless. Ultimately,our goal is to predict how ψ changes over time. This is given by the Smoluchowski equation ∂ t ψ ( x , p ) = −∇ · ( ˙ x ψ ) − ∂ p · ( ˙ p ψ ) , (2)where ˙ x ψ = (cid:88) i ˙ x i δ ( x − x i ) δ ( p − p i ) (3)and ˙ p ψ = (cid:88) i ˙ p i δ ( x − x i ) δ ( p − p i ) (4)define ˙ x and ˙ p , the fluxes of particle position and orientation. The aim of this paper is to derive ˙ x and ˙ p , from theforces and torques that act on and between particles. B. Force Balance
Each particle in the active network obeys Newton’s laws of motion. That is˙ g i = (cid:88) j F ij + F (drag) i , (5)where g i is the particle momentum, and F ij is the force that particle j exerts on particle i . Moreover, F (drag) i is thedrag force between the particle i and the fluid in which it is immersed. Momentum conservation implies F ij = − F ji .We are interested in systems where the direct particle-particle interactions are short ranged. This means that F ij (cid:54) = 0only if | x i − x j | < d , where d is an interaction length that is small (relative to system size).The momentum density is defined by g = (cid:88) i δ ( x − x i ) g i (6)which, using Eq. (5), obeys ∂ t g + ∇ · (cid:88) i δ ( x − x i ) v i g i = (cid:88) i,j δ ( x − x i ) F ij + (cid:88) i δ ( x − x i ) F (drag) i , (7)where v i = ˙ x i is the velocity of the i -th particle. The terms on the left hand side of Eq. (7) are inertial, and in theoverdamped limit, relevant to the systems studied here, they are vanishingly small. Interactions between particlesare described by the first term on the right hand side of Eq. (7) and generate a momentum density flux Σ (the stresstensor) through the material. To wit, using that d is small, so that particle-particle interactions are short-ranged,gives (cid:88) i,j δ ( x − x i ) F ij = 12 (cid:88) i,j ( δ ( x − x i ) − δ ( x − x j )) F ij = −∇ · (cid:88) i,j δ ( x − x i ) x i − x j F ij + O ( d )= ∇ · Σ . (8)where Σ = − (cid:88) i,j δ ( x − x i ) x i − x j F ij + O ( d ) . (9)Note that Eq. (9) does not necessarily produce a symmetric stress tensor. Force couples for which F ij and x i − x j are not parallel generate antisymmetric stress contributions, since these couples are not torque free. We discuss howto reconcile this with angular momentum conservation in Appendix C. The drag force density is f = (cid:88) i δ ( x − x i ) F (drag) i , (10)and after dropping inertial terms, the force balance reads ∇ · Σ + f = , (11)and the total force on particle i obeys (cid:88) j F ij + F (drag) i = . (12)This completes the discussion of the force balance of the system. We next discuss angular momentum conservation. C. Torque Balance
The total angular momentum of particle i , (cid:96) (tot) i = (cid:96) i + x i × g i , (13)is conserved, where (cid:96) i is its spin angular momentum and its x i × g i its orbital angular momentum. Newton’s lawsimply that ˙ (cid:96) i = (cid:88) j T ij + T (drag) i , (14)where T ij is the torque exerted by particle j on particle i , in the frame of reference moving with particle i ,nd T (drag) i is the torque from interaction with the medium, in the same frame of reference. Importantly, since the total angularmomentum is a conserved quantity, the total torque transmitted between particles T ij + x i × F ij = − T ji − x j × F ji is odd upon exchange of the particle indices i and j . Taking a time derivative of Eq. (13) and using Eq. (5) leads tothe torque balance equation for particle i (cid:88) j ( T ij + x i × F ij ) + T (drag) i + x i × F (drag) i = , (15)and thus (cid:88) j T ij + T (drag) i = , (16)where we ignored the inertial term v i × g i and used Eq. (12). The angular momentum fluxes associated with spin,orbital and total angular momentum are discussed in Appendix C for completeness. D. The special case of rod-like filaments
FIG. 1: a/ Interaction between two cytoskeletal filament i and j via a molecular motor. Filaments are characterized by theirpositions x i , x j , their orientations p i , p j , and connect by a motor between arc-length position s i , s j . A motor consist of twoheads that can be different (circle, pentagon) and are connected by a linker (black zig-zag) of lengt R b/ The total force onfilament i is given by the sum of the forces exerted by all a (circle) and b (pentagon) heads, which connect the filament into thenetwork. The shaded area shows all geometrically accessible positions that can be crosslinked to the central (black) filament. We now specialize to rod-like particles, such as the microtubules and actin filaments that make up the cytoskeleton.In particular, we calculate the objects F ij , T ij , and Σ from prescribed interaction forces and torques along rod-likeparticles.
1. Forces
Again, filament i is described by it center of mass x i and orientation vector p i . All filaments are taken as havingthe same length L , and position along filament i is given by x i + s i p i , where s i ∈ [ − L/ , L/
2] is the signed arclength.We consider the vectorial momentum flux from arclength position s i on filament i to arclength position s j on filament j f ij = f ij ( s i , s j ) . (17)where f ij = − f ji and having dimensions of force over area, i.e. a stress. Here we focus on forces generated bycrosslinks; see Fig. 1 (a). The total force between two particles is F ij = (cid:98) δ ( x − x j − s j p j ) f ij (cid:101) ij Ω( x i + s i p i ) , (18)where the brackets (cid:98)· · · (cid:101) ij Ω( x i ) denote the operation (cid:98) φ (cid:101) ij Ω( x i ) = L (cid:90) − L ds i L (cid:90) − L ds j (cid:90) Ω( x i ) d x φ, (19)where φ is a dummy argument and Ω is a sphere whose radius is the size of a cross-linker (i.e., d , the interactiondistance). With the definition Eq. (19), the operation (cid:98)· · · (cid:101) ij Ω( x i + s i p i ) integrates its argument over all geometricallypossible crosslink interactions, between filaments i and j ; see Fig. 1 (b). By Taylor expanding and keeping terms upto second order in the filament arc length ( s i , s j ), we find F (tot) i = (cid:88) j s i p i − s j p j ) · ∇ + ( s i p i p i + s j p j p j ) : ∇∇ δ ( x − x j ) f ij ij Ω( x i ) + F (drag) i (20)and the network stress Σ = − (cid:88) i,j (cid:22) δ ( x − x i ) δ ( x (cid:48) − x j )( x i − x j + s i p i − s j p j ) f ij (cid:25) ij Ω( x i ) (21)where we used that f ij = − f ji .
2. Torques
Similarly, the angular momentum flux that crosslinkers exert between filaments can be written as t ij = ¯ t ij ( s i , s j ) + s i p i × f ij , (22)which dimensionally is a torque per unit area. Thus T ij = (cid:98) δ ( x − x j − s j p j ) t ij (cid:101) Ω( x i + s i p i ) (23)which leads to T (tot) i = T (drag) i + (cid:88) j δ ( x − x j ) (¯ t ij + s i p i × f ij )+( s i p i − s j p j ) · ∇ δ ( x − x j ) (¯ t ij + s i p i × f ij )+ ( s i p i p i + s j p j p j ) : ∇∇ δ ( x − x j )¯ t ij ij Ω( x i ) (24)In the following we will consider crosslinks for which ¯ t ij = 0, for simplicity. III. FILAMENT-FILAMENT INTERACTIONS BY CROSSLINKS AND COLLISIONS
We next discuss how filaments in highly crosslinked networks exchange linear and angular momentum. Two typesof interactions are important here: interactions mediated by crosslinking molecules, which can be simple static linkersor active molecular motors, and steric interactions. We start by discussing the former.
A. Crosslinking interactions
To describe crosslinking interactions, we propose a phenomenological model for the stress f ij that crosslinkers exertbetween the attachment positions s i and s j on filaments i and j . f ij = K ( s i , s j , t ) ( x i + s i p i − x j − s j p j )+ γ ( s i , s j , t ) ( v i + s i ˙ p i − v j − s j ˙ p j )+ [ σ ( s i , s j , t ) p i − σ ( s j , s i , t ) p j ] , (25)The first term in this model, with coefficient K , is proportional to the displacement between between the attachmentpoints, x i + s i p i − x j − s j p j , and captures the effects of crosslink elasticity and motor slow-down under force. Thesecond term, with coefficient γ , is proportional to v i + s i ˙ p i − v j − s j ˙ p j , and captures friction-like effects arisingfrom velocity differences between the attachment points. The last terms are motor forces that act along filamentorientations p i and p j , with their coefficients σ having dimensions of stress. Additional forces proportional to therelative rotation rate between filaments, ˙ p i − ˙ p j , are allowed by symmetry, but are neglected here for simplicity.In general, the coefficients K , γ , and σ are tensors that depend on time, the relative orientations between microtubule i and j and the attachment positions s i , s j on both filaments. In this work, we take them to be scalar and independentof the relative orientation, for simplicity. Generalizing the calculations that follow to include the dependences of K, γ and σ on p i and p j is straightforward but laborious and will be discussed in a subsequent publication. We emphasizethat Eq. (25) is a statement about the expected average effect of crosslinks in a dense local environment and is not adescription of individual crosslinking events.Inserting Eq.(25) into Eqs. (20, 21, 24) we find that the stresses and forces collectively generated by crosslinksdepend on s ij -moments of the form X nm ( x ) = (cid:98) X ( s i , s j ) s ni s mj (cid:101) ij Ω( x ) , (26)where X = K , γ , or σ . We refer to these as crosslink moments. In principle, given Eqs.(20, 21, 25) only the moments X , X , X , X , X , X , X , and X , contribute to the stresses and forces in the filament network. We furthernote that X , X and X are O ( L ), and can thus be neglected without breaking asymptotic consistency. Moreover, X and X can be expressed in terms of lower order moments since X = X + O ( L ) = ( L / X + O ( L ).Finally, by construction K ( s i , s j ) = K ( s j , s i ) and γ ( s i , s j ) = γ ( s j , s i ), and thus γ = γ ≡ γ and K = K ≡ K . To further simplify our notation, we introduce X = X . Explicit expressions for the seven crosslinking momentsthat contribute to the continuum theory are given in the Appendix B. In summary, in the long wave length limit allforces and stresses in the network can be expressed in terms of just a few moments, K , K , γ , γ , σ , σ , σ . Howdifferent crosslinker behavior set these moments will be discussed in SectionV. B. Sterically mediated interactions
In addition to crosslinker mediated forces and torques, steric interactions between filaments generate momentum andangular momentum transfer in the system. We model steric interactions by a free energy E = (cid:82) V e ( p i , · · · , x i , · · · ) d x which depends on all particle positions and orientations. The steric force is¯ F i = − δEδ x i , (27)and the torque acting on it is ¯ T i = − δEδ p i . (28)This approach is commonly used throughout soft matter physics [32, 33]. Common choices for the free energy density e are the ones proposed by Maier and Saupe [34], or Landau and DeGennes [35]. IV. CONTINUUM THEORY FOR HIGHLY CROSSLINKED ACTIVE NETWORKS
In the previous sections we derived a generic expression for the stresses and forces acting in a network of filamentsinteracting through local forces and torques, and proposed a phenomenological model for crosslink-driven interactionsbetween filaments. We now combine these two and obtain expressions for the stresses, force, and torques actingin a highly crosslinked filament network, and from there derive equations of motion for the material. We start byintroducing the coarse-grain fields in terms of which our theory is phrased.
A. Continuous Fields
The coarse grained fields of relevance are the number density, ρ = (cid:88) i δ ( x − x i ) , (29)the velocity v = (cid:104) v i (cid:105) , the polarity P = (cid:104) p i (cid:105) , the nematic-order tensor Q = (cid:104) p i p i (cid:105) , and the third and fourth ordertensors T = (cid:104) p i p i p i (cid:105) , and S = (cid:104) p i p i p i p i (cid:105) . Here the brackets (cid:104)·(cid:105) signify the averaging operation ρ (cid:104) φ i (cid:105) = (cid:88) i δ ( x − x i ) φ i , (30)where φ i is a dummy variable. Furthermore, we define the tensors j = (cid:104) p i ( v i − v ) (cid:105) , J = (cid:104) p i p i ( v i − v ) (cid:105) , H = (cid:104) p i ˙ p i (cid:105) ,and the rotation rate ω = (cid:104) ˙ p i (cid:105) . B. Stresses
The presence of crosslinkers generates stresses in the material which, through Eq. (21), depends on the crosslinkingforce density Eq. (25). Following the nomenclature from Eq. (25), we write the material stress as Σ = Σ ( K ) + Σ ( γ ) + Σ ( V ) + ¯Σ , (31)where Σ ( K ) = − ρ K (cid:18) α I + L Q (cid:19) , (32)is the stress due to the crosslink elasticity, Σ ( γ ) = − ρ (cid:18) η ∇ v + γ j + γ L H (cid:19) , (33)is the viscous like stress generated by crosslinkers, and Σ ( V ) = − ρ ( ασ ∇ P + σ Q − σ PP ) (34)is the stress generated by motor stepping. Here, we defined the network viscosity η = αγ and α = R .Finally, the steric (or Ericksen) stress obeys the Gibbs Duhem Relation ∇ · ¯ Σ = ρ ∇ µ + ( ∇E ) : Q . (35)where µ = − δeδρ is the chemical potential, and E = − δeδ Q is the steric distortion field. An explicit definition of ¯ Σ andthe derivation of the Gibbs Duhem relation are given in Appendix (D). Note that for simplicity, we chose that thesteric free energy density e depends only on nematic order and not on polarity. C. Forces
We now calculate the forces acting on filament i . The total force F i on filament i is given by F i = F ( K ) i + F ( γ ) i + F ( V ) i + ¯ F i + F (drag) i , (36)where F ( K ) i = ( ∇ ρ ) · L K ( p i p i − Q ) − ρ ∇ · Σ ( K ) , (37)is the elasticity driven force F ( γ ) i = γ ρ ( v i − v ) + γ ρ ( ˙ p i − ω )+ γ ( ∇ ρ ) · [ p i ( v i − v ) − j − P ( v i − v )]+ L γ ( ∇ ρ ) · [ p i ˙ p i − H ]+ L γ ( ∇∇ ρ ) : [ p i p i ( v i − v ) − J + Q ( v i − v )] − ρ ∇ · Σ ( γ ) . (38)is the viscous like force, and F ( V ) i = ρσ ( p i − P )+ ( ∇ ρ ) · [ σ ( p i p i − Q ) − σ ( p i P + Pp i − PP )]+ L σ ( ∇∇ ρ ) : [ p i p i p i + Q p i − p i p i P − T ] − ρ ∇ · Σ ( V ) . (39)is the motor force. Finally, ¯ F i = − ∇E ρ : ( p i p i − Q ) − ρ ∇ · ¯ Σ , (40)is the steric force on filament i , where we again chose e to only depend on nematic order and not on polarity. D. Crosslinker induced Torque
We next calculate the torques acting on filament i . The total torque acting on filament i is T i = T ( γ ) i + T ( V ) i + ¯ T i + T (drag) i (41)Note, that crosslinker elasticity does not contribute. Here T ( γ ) i = γ ρ p i × ( v i − v ) + L γ ρ p i × ˙ p i + L γ p i × ( p i · ∇ ρ ) ( v i − v ) − L γ ρ p i × ( p i · ∇ v ) (42)and T ( V ) i = − ρ p i × (cid:18) σ P + L σ p i · ∇ P (cid:19) − L σ p i × ( p i · ∇ ρ ) P (43)are the viscous and motor torques, respectively. Steric interactions contribute to the torque¯ T i = p i × E ρ · p i . (44) E. Equations of Motion
To find equations of motion for the highly crosslinked network, we use Eqs. (36, 37, 38, 39), and obtain v i − v = − σ γ ( p i − P ) − ργ (cid:16) F (drag) i − f /ρ (cid:17) + O (cid:0) L (cid:1) , (45)which will be a useful low-order approximation to v i − v . Note too that we have dropped steric forces, since ∇E /ρ scales with the inverse of the system size, which is much larger than L . Using Eq. (45) in Eq. (41) we find the equationof motion for filament rotations, ˙ p i = ( I − p i p i ) · p i · U + γ L ρ p i · E + γ L A ( P ) P , (46)where we neglect drag mediated terms, which are subdominant at high density, for simplicity. A detailed calculation,and expressions which includes drag terms, is given in Appendix A. Here, U = ∇ v + σ γ ∇ P , (47)is the active strain rate tensor, which consists of the consists of the strain rate and vorticity ∇ v and an active polarcontribution ∇ P . Moreover A ( P ) = σ − σ γ γ . (48)is the polar activity coefficient. The filament velocities are given by v i − v = − σ γ ( p i − P ) − γ γ (( p i − P ) · U − ( p i p i p i − T ) : U ) − γ L ρ γ (( p i − P ) · E − ( p i p i p i − T ) : E )+ 12 γ L γ A ( P ) ( p i p i − Q ) · P , (49)where we used Eqs. (45, 46) in Eq. (36). In Eq. (49), we ignored terms proportional to density gradients, for simplicity.The full expression is given in Appendix A. After some further algebra (see Appendix A), we arrive at an expressionfor the material stress in terms of the current distribution of filaments, Σ = − ρ (cid:16) χ : U + αK I + A ( Q ) Q − A ( P ) T · P (cid:17) + Σ (S) , (50)where χ αβγµ = ηδ αγ δ βµ + L γ ( Q αγ δ βµ − S αβγµ ) , (51)is the anisotropic viscosity tensor, A ( Q ) = σ − σ γ γ + L K (52)is the nematic activity coefficient, and Σ (S) αβ = ¯ Σ αβ − ( Q αγ δ βµ − S αβγµ ) E γµ . (53)is the steric stress tensor. Together Eqs. (2, 46, 49, 50) define a full kinetic theory for the highly crosslinked activenetwork. V. DESIGNING MATERIALS BY CHOOSING CROSSLINKS
Eqs. (2, 46, 50, 49) define a full kinetic theory for highly crosslinked active networks. This theory has the sameactive stresses known from symmetry based theories for active materials[7, 11, 36] and thus can give rise to the samerich phenomenology. Since our framework derives these stresses from microscale properties of the constituents of thematerial it enables us to make predictions on how the microscopic properties of the network constituents affect itslarge scale behavior. We first discuss how motor properties set crosslink moments in Eq. (25). We then study howthese crosslink properties impact the large scale properties of the material.0
A. Tuning Crosslink-Moments
FIG. 2: (a, b) Populations of crosslink heads are characterized by the density with which they bind a filament along its arclength s and the speed at which they move when force free. Two different head types, one with non-uniform speed but uniform density(a) another with uniform speed and non-uniform density (b) are shown. In (c) we list some possible crosslink heads. Red andBlue lines illustrate the change of crosslink speed and density with s , respectively. In (d) we illustrate example crosslinks whichconsist of two heads and a linker. The coefficients in Eq. (25) arise from a distribution of active and passive crosslinks that act between filaments.Consider an ensemble of crosslinking molecules, each consisting of two heads a and b , joined by a spring-like linker;see Fig. (2). For any small volume in an active network, we can count the number densities ξ a ( s ), and ξ b ( s ) of a and b heads of doubly-bound crosslinks that are attached to a filament at arc-length position s . In an idealized experiment ξ a ( s ) and ξ b ( s ) could be determined by recording the positions of motor heads on filaments. The number-density ξ ab ( s i , s j ) of a heads at position s i on microtubule i connected to b heads at position s j on microtubule j is thengiven by ξ ab ( s i , s j ) = ξ a ( s i ) ξ b ( s j ) N ( i ) b ( s i ) (54)where N ( i ) b ( s i ) counts the b heads that an a -head attached at position s i on filament i could be connected to giventhe crosslink size. It obeys N ( i ) b ( s i ) = (cid:88) k (cid:54) = i L/ (cid:90) − L/ ds k (cid:90) Ω( x i + s i p i ) d x ξ b ( s k ) δ ( x k + s k p k − x ) . (55)Analogous definitions for ξ ba ( s i , s j ) and N ( i ) a ( s i ) are implied. It follows naturally that ξ ( s i , s j ) = ξ ab ( s i , s j )+ ξ ba ( s i , s j )is the total number density of crosslinks acting between filaments i and j at the arclength positions s i , s j .Now let V a ( s ) , V b ( s ) be the load-free velocities of motor-heads a, b moving along filaments. Here, V a ( s ) , V b ( s ) arefunctions of the arc-length position s . Like ξ a and ξ b , they are in principle measurable. With these definitions, the1force per unit surface that attached motors exert is f ij = − Γ ξ ( s i , s j ) ( v i + s i ˙ p i − v j + s j ˙ p j ) − κξ ( s i , s j ) ( x i + s i p i − x j + s j p j ) − Γ ([ ξ ab ( s i , s j ) V a ( s i ) + ξ ba ( s i , s j ) V b ( s i )] p i )+Γ ([ ξ ab ( s j , s i ) V a ( s j ) + ξ ba ( s j , s i ) V b ( s j )] p j ) , (56)where Γ is an effective linear friction coefficient between the two attachment points and κ is an effective springconstant. They depend on the microscopic properties of motors, filaments, and the concentrations of both and theirregulators. In general, Γ and κ are second rank tensors, which depend on the relative orientations of filaments. Herewe take them to be scalar, for simplicity and consistency with earlier assumptions. By comparing to Eq. (25) weidentify γ ( s i , s j ) = − Γ ξ ( s i , s j ) , (57) K ( s i , s j ) = − κξ ( s i , s j ) , (58)and σ ( s i , s j ) = − Γ ξ ab ( s i , s j ) V a ( s i ) + Γ ξ ba ( s i , s j ) V b ( s i ) . (59)Using Eqs. (57, 58, 59), we now discuss some important classes of crosslinking molecules. We consider crosslinkswhose heads can be motile or non-motile, the binding and walking properties can act uniformly or non-uniformlyalong filaments, and the two heads of the crosslink can be the same (symmetric crosslink) or different (non-symmetriccrosslink). Figure 2 maps how varying crosslink types can be constructed, while Table I lists the moments to whichdifferent classes of crosslinks contribute. Non-motile crosslinks are crosslinks that do not actively move, i.e. V a = V b = 0. Examples of non-motile crosslinksin cytoskeletal systems are the actin bundlers such as fascin, or microtubule crosslinks such as Ase1p [8]. Whilethese types of crosslinks are not necessarily passive, since the way they binding or unbind can break detailed balance,that their attached heads do not walk along filaments implies that σ = σ = σ = 0. Non-motile crosslinkschange the material properties of the material by contributing to the crosslink moments γ , γ and K , K . Somenon-motile crosslinks bind non-specifically along filaments they interact with, giving uniform distributions. For these γ = K = 0. Others preferentially associate to filament ends, and thus bind non-uniformly. For these γ and K are positive. Note that the two heads of a non-motile crosslink can be identical (symmetric) or not (non-symmetric).Given the symmetric structure of Eqs (57, 58) mechanically a non-symmetric non-motile crosslink behaves the sameas a symmetric non-motile crosslink. and Symmetric Motor crosslinks are motor molecules whose two heads haveidentical properties, i.e. V a = V b = V and ξ a = ξ b = ξ . Examples are the microtubule motor molecule Eg-5 kinesin,and the Kinesin-2 motor construct popularized by many in-vitro experiments[37]. Symmetric motors contribute tothe large-scale properties of the material by generating motor forces. In particular they contribute to the crosslinkmoments σ , σ , and σ . From Eq. (59) it is easy to see that σ = V γ + V γ /L , where we defined the moments ofthe motor velocity V ( s i , s j ) using Eq. (26). Some symmetric motor proteins preferentially associate to filament ends,and display end-clustering behavior, where their walking speed depends on the position at which they are attachedto filaments. Motors that do either of these also generate a contribution to σ and σ . Since both motor heads areidentical we have σ = σ ≡ σ and from Eq. (59) we find that σ = γ V + V γ . Non-Symmetric motor crosslinks are motor molecules whose two heads have differing properties. An example isthe microtubule-associated motor dynein, that consists of a non-motile end that clusters near microtubule minus-endsand a walking head that binds to nearby microtubules whenever they are within reach [20, 38]. A consequence ofmotors being non-symmetric is that σ (cid:54) = σ . Since non-symmetric motors can break the symmetry between the twoheads in a variety of ways we spell out the consequences for a few cases. Let us first consider a crosslinker with onehead a that acts as a passive crosslink ( V a = 0) and a second head b that acts as a motor, moving with the steppingspeed V b = V . For such a crosslink σ = γ V /
2. If both heads are distributed uniformly along filaments and their V is position independent then σ = σ = 0. If the walking b -head is distributed nonuniformly ( ξ b = ξ b ( s ) , ξ a =constant) then σ = γ V and σ = 0. Conversely, if the static a -head has a patterned distribution ( ξ a = ξ a ( s ) , ξ b =constant) then σ = γ V , σ = 0. Finally, we note that if both heads are distributed uniformly along the filament( ξ a = ξ b =constant), but the walking b -head of the motor changes its speed as function of position then σ = V γ / σ = 0.Note that stresses and forces are additive. Thus it may be possible to design specific crosslink moments by designingmixtures of different crosslinkers. For instance mixing a non-motile crosslink that has specific binding to a filamentsolution might allow to change just γ and γ in a targeted way. We will elaborate on some of these possibilities inwhat follows.2 γ , K γ , K σ σ σ symmetricuniformnon-motile yes no no no nonon-symmetricuniformnon-motile yes yes no no nosymmetricuniformmotor yes no yes no nosymmetricnon-uniformmotor yes yes yes yes σ = σ yes σ = σ non-symmetricnon-uniformmotor yes yes yes yes yesTABLE I: Table summarizing which crosslink moments different crosslink types generate. B. Tuning viscosity
We now discuss how microscopic processes shape the overall magnitude of the viscosity tensor χ . From Eq. (51)and remembering that η = 3 R / γ , it is apparent that the overall viscosity of the material is proportional to thenumber of crosslinking interactions and their resistance to the relative motion of filaments, quantified by the frictioncoefficient ρ γ . Furthermore, γ itself scales as the squared filament length L , and the cubed crosslink size R (seethe definition in Appendix B), which, with ρ , sets the overall scale of the viscosity as ρ L R .We next show how micro-scale properties of network constituents shape the anisotropy of χ ; see Eq. (51). Tocharacterize this we define the anisotropy ratio a as a = L γ η = 518 L R , (60)which is the ratio of the magnitudes of the isotropic part of χ αβγµ , that is ηδ αγ δ βµ , and its anisotropic part γ L / Q αγ δ βµ − S αβγµ ). Most apparently the anisotropy ratio will be large if the typical filament length L islarge compared to the motor interaction range R . This is typically the case in microtubule based systems, as mi-crotubules are often microns long and interact via motor groups that are a few tens of nano-meters in scale [8].Conversely, in actomyosin systems filaments are often shorter (hundreds of nano-meters) and motors-clusters calledmini-filaments, can have sizes similar to the filament lengths [8]. The anisotropy of the viscous stress is not exclusiveto active systems and has been described before in the context of similar passive systems, such as liquid crystals andliquid crystal polymers [33–35].3 Mixture Active Strain, σ /γ Active Pressure, Π (A)
Axial stress, ¯ S σ γ = V no σ γ = V no σ γ = V a + V b P |01/21 ( A ) , S ( + | P | ) / + ( X )0 / ( M )0 / V slide ( M )0( M )0 + ( X )0 no+or+ ( X )0 / ( M )0 / V slide ( M )0( M )0 + ( X )0 P |01/21 ( A ) , S | P | TABLE II: Active pressure and strain generated by different crosslink types and mixtures. In the plots pertaining to the activestrain rate γ = γ ( M )0 + γ ( X )0 where γ ( M )0 denotes the is the part of γ induced by mobile and γ ( X )0 denotes contribution fromnon-mobile crosslinkers. The filament sliding velocity expected in a stress free system is V slide = σ /γ and is given unitsof the force free speed of immobile crosslinks and describe the expected speed of filament sliding in the material. Moreover,¯ S = | Π (A) /q | is the magnitude of the motor-stepping induced axial stress, i.e of the axial stress in the limit K → C. Tuning the active self-strain
The viscous stress in highly crosslinked networks is given by χ : U , where U = ∇ v + ( σ /γ ) ∇ P takes the role ofthe strain-rate in passive materials, but with an active contribution ( σ /γ ) ∇ P . Thus, internally driven materialscan exhibit active self-straining.In particular a material in which each filament moves with the velocity v i = − σ /γ p i + C , where C is a constantvector that is sets the net speed of the material in the frame of reference, has U = 0, and thus zero viscous stress.In such a material filaments can slide past each other at a speed σ /γ without stressing the material. Notably, thesliding speed is independent of the local polarity and nematic order of the material [30].The crosslink moments that contribute to the active straining behavior are σ and γ . In active filament networkswith a single type of crosslink σ /γ (cid:39) V , regardless of crosslink concentration. Thus for single-crosslinker systems,the magnitude of self-straining is independent of the motor concentration [30].4Self-straining can be tuned in mixtures of crosslinks. For instance the addition of a non-motile crosslinker canincrease γ , while leaving σ unchanged. In this way self-straining can be relatively suppressed. In table II we plot theexpected active strain-rate for materials actuated by mixtures of immotile and motor crosslinks. In such a material γ = γ ( M )0 + γ ( X )0 where γ ( M )0 denotes the part of γ induced by motile crosslinkers and γ ( X )0 denotes that fromnon-motile crosslinkers. The resulting velocity V slide with which a filament slides through the material will scale as V slide (cid:39) γ ( M )0 / ( γ ( M )0 + γ ( X )0 ); see Table II. D. Tuning the Active Pressure
Many active networks spontaneously contract [38] or expand [37]. We now study the motor properties that enablethese behaviors.An active material with stress free boundary conditions, can spontaneously contract if its self-pressure,Π = Tr (cid:0) Σ + ρ χ : U (cid:1) . (61)is negative. Conversely the material can spontaneously extend if Π is positive. We can also writeΠ = Π (A) + Π (S) (62)where Π (S) = Tr (cid:0) Σ (S) (cid:1) is the sterically mediated pressure, and Π (A) is the activity driven pressure (or active pressure)given by Π (A) = − ρ (cid:16) αK + A ( Q ) − A ( P ) | P | (cid:17) . (63)see Eq. (50). Here and in the following we approximated Tr( T · P ) (cid:39) | P | for simplicity. We ask which properties ofcrosslinks set the active pressure and how its sign can be chosen.We first discuss how interaction elasticity impacts the active pressure Π ( A ) in the absence of motile crosslinks, i.e.when σ = σ = σ = 0. In this case, Eq. (63) simplifies to Π (A) = − ρ ( α + L / K , where we used Eq. (52).Thus, even in the absence of motile crosslinks, active pressure can be generated. This can be tuned by changingthe effective spring constant K . We note that Π (A) + Π (S) = 0 when crosslink binding-unbinding obeys detailedbalance and the system is in equilibrium. The moment K can have either sign when detailed balance is broken.Microscopically this effect could be achieved, for instance, by a crosslinker in which active processes change the restlength of a spring-like linker between the two heads once they bind to filaments.We next discuss the contributions of motor motility to the active pressure. To start, we study a simplified apolar(i.e. P = 0) system where K = 0. In such a system the active pressure is given byΠ (A) = − ρ (cid:18) σ − σ γ γ (cid:19) (64)We ask how motor properties set the value and sign of this parameter combination.We first point out that generating active pressure by motor stepping requires that either σ or γ are non-zero.This means that generating active pressure requires breaking the uniformity of binding or walking properties alongthe filament. A crosslink which has two heads that act uniformly can thus not generate active pressure on its own.However, when operating in conjunction with a passive crosslink that preferentially binds either end of the filament,the same motor can generate an active pressure. This pressure will be contractile if the non-motile crosslinks couplethe end that the motor walks towards ( γ and σ have the same sign) and extensile if they couple the other ( γ and σ have opposite signs). In summary, a motor crosslink that acts the same everywhere along the filaments it couplesdoes not generate active pressure on its own. However, it can do so when mixed with a passive crosslink that actsnon-uniformly.We next ask if a system with just one type of non-uniformly acting crosslink can generate active pressure. To start,consider symmetric motor crosslinks , i.e a motor consisting of two heads with identical (but non-uniform) properties.We then have σ = σ = γ V + γ V and σ = V γ + V γ /L . Using this in Eq. (64) and dropping the termproportional to γ (higher order in this case) we find that such symmetric motor crosslinks generate no contributionto the active pressure when operating alone. However when operating in concert with a non-motile crosslink, evenone that binds filaments uniformly, they can generate and active pressure. The sign of the active pressure is set bythe particular asymmetry of motor binding and motion. The system is contractile if motors cluster or speed up nearthe end towards which they walk, and extensile if they cluster or accelerate near the end that they walk from. Ourprediction that many motor molecules can only generate active pressure in the presence of an additional crosslink,5might explain observation on acto-myosin gels, which have been shown to contract only when combined a passivecrosslink operate in concert with the motor myosin [39].We next ask if non-symmetric motor crosslinks can generate active pressure. Consider a crosslink with one immobileand one walking head. For such a crosslink σ = γ V /
2. If the immobile head preferentially binds near one filamentend, while the walking head attaches everywhere uniformly, then σ = γ V and σ = 0. For such a motor we predictan active pressure proportional to V /
2. The active pressure will be contractile if the static ends bind near the endthat the motor head walks to and extensile if the situation is reversed. The motor dynein has been suggested to consistof an immobile head that attaches near microtubule minus ends and a walking head that grabs other microtubulesand walks towards their minus ends. Our theory suggests that this should lead to contractions, which is consistentwith experimental findings [39].After having discussed the effects of motor stepping on the active pressure in systems with P = 0, we ask howthe situation changes in polar systems. In polar system an additional contribution, − ( σ − γ γ σ ) | P | , exists. Forsymmetric motors, where σ = σ this implies that the active pressure generated by a network of symmetric motorsand passive crosslinks is strongest in apolar regions of the system and subsides in polar regions, since the polarand apolar contributions to the active stress appear in Eq. (50) with opposite signs. We plot the magnitude of theactive pressure Π (A) (cid:39) − | P | as a function of | P | in Table II. This is reminiscent of the behavior predicted in theframeworks of a sparsely crosslinked system in [21]. In contrast the effects of non-symmetric motors can be enhancedin polar regions. Consider again, the example of a motor with one static head that preferentially binds near one ofthe filament ends and a mobile head that acts uniformly. For this motor σ = γ V and σ = 0 and σ = γ V / (A) (cid:39) (1 + | P | ) /
2, see the table II for a plot of the active pressure Π (A) as a function of | P | . This is reminiscent ofthe motor dynein in spindles, which is though to generate the most prominent contractions near the spindle poles,which are polar [40].Finally, we ask how filament length affects the active pressure. Looking at the definitions of the nematic and polaractivity Eqs. (52, 48) and remembering the definition and scaling of the coefficient in there (see App. (B)), we noticethat the active pressure scales as L . Since the viscosity scaled with L , this predicts that systems with shorterfilaments contract slower than systems with longer filaments. This effect has observed for dynein based contractionsin-vitro [20]. E. Tuning axial stresses, buckling and aster formation
Motors in active filament networks generate anisotropic (axial) contributions to the stress, which can lead to largescale instabilities in materials with nematic order [3, 26, 36, 41]. At larger active stresses, nematics are unstable tosplay deformations in systems that are contractile along the nematic axis, and to bend deformations in systems thatare extensile along the nematic axis [7, 36]. In both cases, the instabilities set in when the square root of ratio of theelastic (bend or splay) modulus that opposes the deformation to the active stress - also called the Fr´eedericksz length- becomes comparable to the systems size. We now discuss which motor properties control the emergence of theseinstabilities, and how a system can be tuned exhibit bend or splay deformations. For this we ask how axial stresses,which are governed by the activity parameters A ( Q ) and A ( P ) , are set in our system.The magnitude S of the axial stress along the nematic axis is given by S = − ρ q (cid:16) A ( Q ) − A ( P ) | P | (cid:17) , (65)where we defined the nematic order parameter q , as the largest eigenvalue of Q −
Tr( Q ) I /
3; see Eq. (50). The axialstress is contractile along the nematic axis if S is positive and extensile if S is negative. Comparing Eqs. (65, 63) wefind that S = q (Π (A) + ρ αK ) and in the limit where K →
0, where motor elasticity is negligible, S = q Π (A) . Wediscussed how Π (A) is set for different types of crosslinks in the previous section; see Table II.The prototypical active nematic [37] which consists of apolar bundles of microtubules actuated by the kinesin motorsand is axial extensile. In our theory, an axial extensile stress (i.e. S <
0) in an apolar system ( P = 0) implies that A ( Q ) = σ − σ γ γ + L K >
0. This can be achieved either by crosslinks that act uniformly (i.e. σ − σ γ γ = 0) andgenerate a spring like response that induces K > σ − σ γ γ >
0. The latter implies either a non-symmetric motor crosslinks, or the presence of morethan one kind of crosslinks, as was discussed more extensively earlier in the context of active pressure. At high enoughactive stress we expect systems with negative S to become unstable towards buckling. This has been observed in[42, 43].Conversely axial contractile behavior can be achieved if either K < σ − σ γ γ <
0. At high enough activestress, such systems can become unstable towards an aster forming transition, as seen in [38].6Note that S (cid:39) Π (A) + ρ αK , implies that S and Π (A) need not be the same if K (cid:54) = 0. In particular when Π (A) and K have opposite signs systems can exist, which are axially extensile while being bulk contractile and vice versa.We finally note that the magnitude of axial stresses changes if the system transitions from apolar to polar, if theorigin of the axial stresses is motor stepping but not if the origin of the axial stresses is the effective spring likebehavior of motor, since A ( Q ) , but not A ( P ) , depends on K , see Eqs. (48, 52). In systems in which the active stress isgenerated by the stepping of symmetric motor-crosslink, | S | is highest nematic apolar phase ( | P | = 0), while systemsmade from non-symmetric crosslinks generate the most stress when polar ( | P | = 1); see Table II. This opens thepossibility that a system can overcome the threshold towards an instability when its other dynamics drives it fromnematic apolar to polar arrangements or vice versa. We suggest that the buckling instabilities discussed in [42, 43]should be interpreted in this light. VI. DISCUSSION
In this paper, we asked how the properties of motorized crosslinkers that act between the filaments of a highlycrosslinked polymer network set the large scale properties of the material.For this, we first develop a method for quantitatively stating what the properties of motorized crosslinks are. Weintroduce a generic phenomenological model for the forces that crosslink populations exert between the filamentswhich they connect; see Eq. (25). This model describes forces that are (i) proportional to the distance ( K ), and (ii)the relative rate of displacement ( γ ). Finally (iii) it describes the active motor forces ( σ ) that crosslinks can exert.Importantly, forces from crosslinkers ( K, γ, σ ) can depend on the position on the two filaments which they couple.This allows the description of a wide range of motor properties, such as end-binding affinity, end-dwelling, and eventhe description of non-symmetric crosslinks that consist of motors with two heads of different properties.We next derived the stresses and forces generated on large time and length scale, given our phenomenologicalcrosslink model. We find that the emergent material stresses depend only on a small set of moments; see Eq. (26)of the crosslink properties. These moments are effectively descriptions of the expectation value of the force exertedbetween two filaments given their positions and relative orientations. The resulting stresses, forces, and filamentreorientation rates (Eqs. (50, 49, 46)) recover the symmetries and structure predicted by phenomenological theoriesfor active materials, but beyond that provide a way of identifying how specific micro-scale processes set specificproperties of the material.We discussed how four key aspects of the dynamics of highly crosslinked filament networks can be tuned by themicro-scale properties of motors and filaments. In particular we discussed how (i) the highly anisotropic viscosity ofthe material is set; (ii) how active self-straining is regulated; (iii) how contractile or extensile active pressure can begenerated; (iv) which motor properties regulate the axial active nematic and bipolar stresses, which can lead to largescale instabilities.Our theory makes specific predictions for the effects of distinct classes of crosslinkers on cytoskeletal networks.Intriguingly these predictions suggest explanations for phenomena experimentally seen, but currently poorly under-stood.Experiments have shown that mixtures of actin filaments and myosin molecular motors can spontaneously contract,but only in the presence of an additional passive crosslinker [39]. Our theory allows us to speculate on explanationsfor this observation. In the crosslink classification that we introduced, myosin, which form large mini-filaments, is asymmetric motor crosslink; see Fig. (2). We find that symmetric motor crosslinks, which have two heads that act thesame can generate contractions only in the presence of an additional crosslinker that helps break the balance between γ /γ σ and σ in the active pressure; see Eq. (64) and Table II. Further work will be needed to explore whetherthis connection can be made quantitative.A second observation that was poorly understood prior to this work is the sliding motion of microtubules in meioticXenopus spindles, which are the structures which segregate chromosomes during the meiotic cell division. Thesespindles consist of inter-penetrating arrays of anti-parallel microtubules, which are nematic near the chromosomes,and highly polar near the spindle poles. In most of the spindle the two anti-parallel populations of microtubules slidepast each other, at near constant speed driven by the molecular motor Eg-5 Kinesin, regardless of the local networkpolarity. Our earlier work [30] showed that active self straining explains this polarity independent motion. The theorythat we develop here provides the tools to explore the behavior of different motors and motor mixtures which willallow us to investigate the mechanism by which different motors in the spindle shape its morphology. This will helpto explain complex behaviors of spindles such as the barreling instability [13] that gives spindles their characteristicshape or the observation that spindles can fuse [44].Our theory provides specific predictions on how changing motor properties can change the properties of the materialwhich they constitute, it can enable the design of new active materials. We predict the expected large scale properties ofa material, in which an experimentalist had introduced engineered crosslinks with controlled properties. With current7technology, an experimentalist could engineer a motor that preferentially attaches one of its heads to a specifiedlocation on a filament, while its walking head reaches out into the network. Or, as has already been demonstratedin studies by the Surrey Lab [45] the difference in the rates of filament growth and motor walking speeds, could beexploited to generate different dynamic motor distributions on filaments. This design space will provide ample roomto experimentally test our predictions, and use them to engineer systems with desirable properties. Finally recentadvances in optical control of motor systems [46] could be used to provide spatial control.The theory presented here does however make some simplifications. Importantly, we neglected that the distributionof bound crosslinks on filaments themselves in general depends on the configuration of the network. This means thatthe crosslink moments can themselves be functions of the local network order parameters. Effects like this have beenargued to be important for instance when explaining the transition from contractile to extensile stresses in orderingmicrotubule networks [47] and the physics of active bundles [28]. Such effects can be recovered when making theinteractions K, γ, σ in the phenomenological crosslink force model Eq. (25) functions of p i , and p j . This will be thetopic of a subsequent publication.In summary, in this paper we derived a continuum theory for systems made from cytoskeletal filaments and motorsin the highly crosslinked regime. Our theory makes testable predictions on the behavior of the emerging system,provides a unifying framework in which dense cytoskeletal systems can be understood from the ground up, andprovides the design paradigms, which will enable the creation of active matter systems with desirable properties inthe lab. Acknowledgements
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Elife , 9:e51751, 2020. Appendix A: Detailed derivation of the equations of motion
In the following we derive the equations of motion for the highly crosslinked active network. We start by usingEq. (45, 24) and obtain ˙ p i = ( I − p i p i ) · (cid:26) p i · ( U + 12 γ L E ρ ) + 12 γ L A ( P ) P (cid:27) − ρ ¯ T (drag) i , (A1)The torque due to drag with the medium is¯ T ( drag ) i = T ( drag ) i + ( I − p i p i ) · (cid:18) γ L + p i · ∇ ρρ (cid:19) (cid:18) F (drag) i − ρ f (cid:19) . (A2)This implies ω = P · ( U + 12 γ L E ρ ) − T : ( U + 12 γ L E ρ )+ 12 L A ( P ) ( P − Q · P ) − ρ ω (drag) , (A3)where ω (drag) = (cid:68) ¯ T ( drag ) i (cid:69) (A4)and H = Q · ( U + 12 γ L E ρ ) − S : ( U + 12 γ L E ρ )+ 12 γ L A ( P ) ( PP − T · P ) − ρ H (drag) , (A5)where H (drag) = (cid:68) p i ¯ T ( drag ) i (cid:69) . (A6)Furthermore we note that j = σ γ ( PP − Q ) + 1 γ ρ j (drag) + O (cid:0) L (cid:1) , (A7)and J = σ γ ( Q P − T ) + 1 γ ρ J (drag) + O (cid:0) L (cid:1) , (A8)where j (drag) = − γ (cid:28) p i (cid:18) F (drag) i − ρ f (cid:19)(cid:29) (A9)and J (drag) = − γ (cid:28) p i p i (cid:18) F (drag) i − ρ f (cid:19)(cid:29) . (A10)0Putting all of this together, we arrive at an expression for the networks stress in terms of the current distribution offilaments, Σ = − ρ ( χ : U + αK I ) − ρ (cid:16) A ( Q ) Q − A ( P ) T · P (cid:17) − ¯ χ : E + ¯ Σ + ρ Σ (drag) . (A11)where Σ (drag) = − γ L H (drag) − γ j (drag) (A12)and at a similar equation for the motion of filament i v i − v = − σ γ ( p i − P ) − γ γ (cid:32) ( p i − P ) · ( U + γ L E ρ ) − ( p i p i p i − T ) : ( U + γ L E ρ ) (cid:33) + 1 γ γ L γ A ( P ) ( p i p i − Q ) · P − ∇ ργ ρ · (cid:34) A Q ( p i p i − Q ) − A ( P ) ( p i P + Pp i − PP ) (cid:35) − (cid:18) p i p i − Q ) : ∇ E ρ (cid:19) − ρ v (drag) (A13)where 1 ρ v (drag) = 1 ργ F (drag) i + O (cid:0) /ρ (cid:1) . (A14) Appendix B: Crosslink Moments
The crosslink moment which enter the hydrodynamic descriptions are defined from moments of crosslinker mediatedfilament-filament forces. Specifically, K = (cid:98) K ( s i , s j ) (cid:101) ij Ω( x i ) , (B1) K = (cid:98) s i K ( s i , s j ) (cid:101) ij Ω( x i ) , (B2) γ = (cid:98) γ ( s i , s j ) (cid:101) ij Ω( x i ) , (B3) γ = (cid:98) s i γ ( s i , s j ) (cid:101) ij Ω( x i ) , (B4) σ = (cid:98) σ ( s i , s j ) (cid:101) ij Ω( x i ) , (B5) σ = (cid:98) s i σ ( s i , s j ) (cid:101) ij Ω( x i ) , (B6)and σ = (cid:98) s j σ ( s i , s j ) (cid:101) ij Ω( x i ) . (B7) Appendix C: Angular Momentum Fluxes and antisymmetric stresses
The spin and orbital angular momenta obey the continuity equations˙ (cid:96) i = (cid:88) j T ij + T (drag) i (C1)1and x i × ˙ g i = (cid:88) j x i × F ij + x i × F (drag) i , (C2)where we used Eq. (5) and that ˙ x i is parallel to g . We and introduce the densities of spin and orbital angularmomentum which are (cid:96) = (cid:88) i δ ( x − x i ) (cid:96) i , (C3)and (cid:96) (orb) = (cid:88) i δ ( x − x i ) x i × g i , (C4)respectively. They obey continuity equations ∂ t (cid:96) + ∇ · (cid:88) i ( δ ( x − x i ) v i (cid:96) i ) = (cid:88) i,j δ ( x − x i ) T ij + τ, (C5)where τ = (cid:88) i δ ( x − x i ) T (drag) i (C6)and ∂ t (cid:96) (orb) + ∇ · (cid:88) i δ ( x − x i ) v i x i × g i = (cid:88) i,j δ ( x − x i ) x i × F ij + x × f . (C7)The first term on the right hand side of Eq. (C7) describes the orbital angular momentum transfer by crosslinkinteractions. It can be rewritten as the sum of an orbital angular momentum flux M (orb) and a source term relatedto the antisymmetric part of the stress tensor Σ , (cid:88) i,j δ ( x − x i ) x i × F ij = (cid:88) i,j δ ( x − x i ) x i + x j × F ij + (cid:88) i,j δ ( x − x i ) x i − x j × F ij = ∇ · M (orb) + 2 σ a + O ( d ij ) , (C8)where the orbital angular momentum flux is M (orb) = − (cid:88) i,j δ ( x − x i ) x i − x j (cid:18) x i + x j × F ij (cid:19) (C9)and σ a = (cid:88) i,j δ ( x − x i ) x i − x j × F ij , (C10)which is the pseudo-vector notation for the antisymmetric part of the stress Σ such that in index notation, σ aα = 12 (cid:15) αβγ Σ βγ , (C11)2where used the Levi-Civita symbol ε αβγ and summation over repeated greek indices is implied.Similarly, the first term on the right hand side of Eq. (C5) describes the spin angular momentum transfer bycrosslink interactions. It can be rewritten as the sum of an orbital angular momentum flux M and a source termrelated to the antisymmetric part of the stress tensor Σ , (cid:88) i,j δ ( x − x i ) T ij = (cid:88) i,j δ ( x − x i ) (cid:18) T ij + x i − x j × F ij (cid:19) − (cid:88) i,j δ ( x − x i ) x i − x j × F ij = ∇ · M − σ a + O ( d ij ) , (C12)where the spin angular momentum flux M = − (cid:88) i,j δ ( x − x i ) x i − x j (cid:18) T ij + x i − x j × F ij (cid:19) (C13)After defining the total and spin angular momentum fluxes as M (tot) = M + M (orb) = − (cid:88) i,j δ ( x − x i ) x i − x j T ij + x i × F ij ) , (C14)we finally write down the statements of angular momentum conservation ∇ · M (tot) + x × f + τ = 0 , (C15)spin angular momentum continuity ∇ · M − σ a + τ = 0 , (C16)and orbital angular momentum continuity ∇ · M (orb) + 2 σ a + x × f = 0 , (C17)where we dropped inertial terms. We note that the antisymmetric stress Σ a acts to transfer spin to orbital angularmomentum. Importantly, the total angular momentum is conserved as evident from the form of Eq. (C15). Appendix D: The Ericksen Stress
In this appendix we derive the effects of steric interactions on the system. As stated in the main text, stericinteractions are best described in terms of a potential e ( x i , p i ), which depends on all particle positions and orientations.The steric free energy of the system is E = (cid:82) V ed x where V is the volume of the system. For the treatment to followwe shall assume the steric interactions do not depend on the polar, but only on the nematic order of the system. Thena generic variation of the systems free energy can be written as δE = (cid:90) ∂ V (cid:18) eu γ + ∂e∂ ( ∂ γ Q αβ ) δQ αβ (cid:19) dS γ − (cid:90) ∂ V ( µδρ + E αβ δQ αβ ) (D1)where we defined the chemical potential µ = − ∂e∂ρ (D2)3and the distortion field E αβ = − ∂e∂Q αβ + ∂ γ ∂e∂ ( ∂ γ Q αβ ) , (D3)and introduced the infinitesimal deformation field u . Now, any physically well defined free energy density needs toobey translation invariance. Thus δE = 0 for any pure translation, which is the transformation where δρ = − u γ ∂ γ ρ , δQ αβ = − u γ ∂ γ Q αβ , u γ is a constant. Thus ∂ β (cid:18) ( e + Q µν E µν + µρ ) δ αβ − ∂e∂ ( ∂ β Q γµ ) ∂ α Q γµ (cid:19) = ρ∂ α µ + Q µν ∂ α E µν , (D4)which is the Gibbs-Duhem relation used in the main text, where¯Σ αβ = ( e + Q µν E µν + µρ ) δ αβ − ∂e∂ ( ∂ β Q γµ ) ∂ α Q γµ ..