Hydrodynamic Theory of Active Matter Active Polar Gels J F Joanny, F Jülicher, K Kruse Et Al

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Rep. Prog. Phys. Rep. Prog. Phys. 81 (2018) 076601 (27pp) https://doi.org/10.1088/1361-6633/aab6bb

81 Review 2018 Hydrodynamic theory of active matter

© 2018 IOP Publishing Ltd Frank Jülicher1 , Stephan W Grill2 and Guillaume Salbreux1,3 RPPHAG 1 Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzerstr. 38, 01187 Dresden, Germany 2 BIOTEC, Technical University Dresden, Tatzberg 47/49, D-01307 Dresden, Germany 076601 3 The Francis Crick Institute, 1 Midland Road, London NW1 1AT, United Kingdom

E-mail: [email protected] F Jülicher et al Received 10 December 2017, revised 6 March 2018 Accepted for publication 15 March 2018 Hydrodynamic theory of active matter Published 5 June 2018

Corresponding Editor Professor Erwin Frey Printed in the UK Abstract ROP We review the general hydrodynamic theory of active soft materials that is motivated in particular by biological matter. We present basic concepts of irreversible thermodynamics of spatially extended multicomponent active systems. Starting from the rate of entropy 10.1088/1361-6633/aab6bb production, we identify conjugate thermodynamic fluxes and forces and present generic constitutive equations of polar active fluids and active gels. We also discuss angular 1361-6633 momentum conservation which plays a role in the the physics of active chiral gels. The irreversible thermodynamics of active gels provides a general framework to discuss the physics that underlies a wide variety of biological processes in cells and in multicellular 7 tissues.

Keywords: **, active matter, hydrodynamics, nonequilibrium processes, active gels, molecular motors, irreversible thermodynamics (Some figures may appear in colour only in the online journal)

1. Introduction filaments driven by the chemical free energy provided by ATP hydrolysis. These movements proceed even in the presence of Cells and tissues provide extraordinarily complex examples an external load and the motors can perform mechanical work for active condensed matter far from thermodynamic equi- [3]. Dynamic processes on the cell scale arise as collective librium. Cells are inherently dynamic and exhibit force behaviors of many molecular components, including motor generation by active molecular processes and the genera- proteins, in the cell cytoskeleton. tion of spontaneous movements [1]. Active processes on the The cytoskeleton governs the material properties of cells molecular scale are driven by chemical free energy provided and their response to external forces [4]. Filaments of the by metabolic processes [2]. Examples are cell motility such cytoskeleton turn over by polymerization and depolymeri- as the gliding of cells on a solid substrate, and cell division. zation processes [5]. They can be cross-linked by a number An important phenomenon is the ability of cells to gener- of associated proteins. A first class of cytoskeletal filaments ate forces and to control their shape. The prototype example are microtubules, which are stiff filaments that can become for active molecular processes are motor molecules which tens of micrometers long. They play a key role to organize the interact with filaments of the cytoskeleton [3]. Motors bind mitotic spindle and they form the basis of motile cilia. Kinesin to specific filaments. These have two different ends and thus and dynein motor proteins exert forces and movements along display a polar asymmetry, providing a local direction for the microtubules. A second important class of cytoskeletal fila- motion of motors. Motor molecules catalyze the hydrolysis of ments that can form active materials are actin filaments. Actin Adenosinetriphospate (ATP) to Adenosinediphospate (ADP) filaments are semiflexible, less stiff and typically shorter than and inorganic phosphate. ATP serves as a chemical fuel, and microtubules. They often form gel-like materials in the cell, motor molecules generate spontaneous movements along assembling into contractile bundles such as stress fibers. In

1361-6633/18/076601+27$33.00 1 © 2018 IOP Publishing Ltd Printed in the UK Rep. Prog. Phys. 81 (2018) 076601 Review the presence of myosin motor proteins and ATP, mechanical components in a given system. In section 3, we define ther- stresses and movements are generated in actin filaments sys- modynamic quantities and discuss the balance of entropy tems. The actin cytoskeleton also plays a key role to deter- and free energy. To build the hydrodynamic theory of active mine the mechanical properties of cells and the cell shape. matter, we identify pairs of conjugate thermodynamic fluxes It forms a thin layer of an actin gel below the outer plasma and forces that determine the rate of entropy production. membrane of cells, the cell cortex, which generates contractile Section 4 presents the constitutive relations for active matter tension that governs cell surface tension. The actin cytoskel- using linear response theory and symmetry arguments. The eton thus forms a complex material based on a network of cases of polar active gels, active nematics and active chiral filaments, motors and other molecular components. This net- gels are discussed. Examples for the application of contin- work is dynamically organized in space and time and governs uum theories to cellular processes and tissue dynamics are cellular material properties and cell mechanics. In many situ- discussed in section 5. The paper concludes with a discussion ations, this material is anisotropic. When filaments align on in section 6. average, the network can be locally characterized by polar or nematic order [6], see figure 1. The associated order param 2. Conservation laws eter can couple to force generation and movement [7–12]. Cytoskeletal networks thus are important examples for active We construct the general theory of active matter starting materials with complex dynamics and unconventional mat from fundamental conservation laws. We consider a spatially erial properties. This class of systems has been termed active extended macroscopic system of soft condensed matter in gels, see figure 2. These are gel-like materials that are subject the presence of chemical reactions. We use a coarse-grained to active processes that give rise to active stresses and sponta- description that is based on a large number of local volume neous flows [10, 11, 13–18]. Similar concepts describe mat elements in which densities of molecular components, of the erial properties of multicellular systems and tissues, and it has momentum and of the energy are defined. We consider a con- been shown that tissues can also exhibit properties of active tinuum limit that is valid on large length scales as compared matter, both as active solids or as active liquids [19–31]. to the size of volume elements. The microscopic equations of An important conceptual approach to active biological motion of the molecular components satisfy several conserva- matter is to identify general physical principles and robust tion laws. These conservation laws are generally valid both in properties of active matter. This can be achieved using a and out of thermodynamic equilibrium and at all scales. On a hydrodynamic approach that captures the slow dynamics of coarse grained level they can be expressed as continuity equa- modes that correspond to conservation laws or broken symme- tions for conserved densities. tries [17, 32, 33]. Here we review the general hydrodynamic Mass conservation. First, we consider mass conservation theory of active gels and fluids based on irreversible thermo- which can be expressed as dynamics. We discuss the basic thermodynamic concepts and derive expressions for entropy production in such systems. ∂tρ + ∂α(ρvα)=0. (1) Using linear constitutive relations between generalized thermodynamic fluxes and forces we obtain a generic theory of Here ρ is the mass density or mass per volume. We use a sum- active fluids and gels with polar or nematic asymmetries. This mation convention where summation over repeated vector hydrodynamic theory generalizes the hydrodynamics and indices is implied. Equation (1) defines the local center-of- statistical mechanics of liquid crystals [6, 34–39] to systems mass velocity vα. The mass flux is given by the momentum maintained away from thermodynamic equilibrium by chemi- density gα = ρvα. cal processes driven by fuels provided in external reservoirs. Energy conservation. Energy conservation implies that the Coupling soft materials to chemical reactions leads to mech- energy density e in local volume elements obeys ano-chemical couplings and the transduction of chemical free e energy to active mechanical stresses and spontaneous material ∂te + ∂αJα = 0, (2) flows. In addition to active polar and nematic systems, we also e e discuss active processes in chirally asymmetric systems. The where Jα = evα + jα is the energy flux. Here we have intro- e hydrodynamic approach shows that chiral active processes duced the relative energy flux jα in a reference frame that such as motors interacting with chirally asymmetric filaments moves with the local volume elements. can give rise to pronounced chirality at larger scales and active Momentum conservation. Momentum conservation can be chiral hydrodynamic processes. In order to discuss active chi- expressed as a continuity equation for the momentum balance ral processes, the conservation of angular momentum plays gα which reads an important role. This is different from hydrodynamics of tot ∂tgα ∂βσαβ = 0. (3) passive fluids, where intrinsic rotations do not play significant − roles on long time scales. Here, the momentum flux is the negative total stress σtot . − αβ The paper is organized as follows. After the introduction, We write σtot = σ g v , where σ denotes the stress in αβ αβ − α β αβ we present in section 2 the key conservation laws of mass, a reference frame moving with local volume elements. energy, momentum and angular momentum. Chemical reac- Angular momentum conservation. Similarly, angular tions specify the number of conservation laws for molecular momentum conservation can be expressed as

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Figure 1. (a) Coarse graining of an active gel. The gel is a dynamic meshwork of filaments that interact with molecular motors. Filaments have an average direction that can be characterized by a polarity vector p. Motors consume the fuel ATP which is hydrolyzed to ADP and inorganic phosphate. The chemical free energy provided by this reaction drives force generation of motors. Averaging filament polarity (red), and center of mass movements (blue) in small volume elements provides the basis for a coarse grained hydrodynamic description on large scales. (b) The hydrodynamic theory of active gels is based on broken symmetries that are characterized by order parameters. A filament system can be isotropic or it can for example exhibit polar or nematic anisotropies. Since individual filaments are chiral, chiral asymmetries in active gels can also play a role.

tot tot ∂tl ∂γ M = 0. (4) αβ − αβγ tot tot Here, the antisymmetric tensor lαβ = αβγ lγ describes the tot angular momentum density and lγ is the angular momentum density pseudo vector. We denote the total angular momentum tot flux by Mαβγ, which is antisymmetric in the first two indices. The angular momentum density can be decomposed in an intrinsic or spin angular momentum density lαβ and an orbital contribution which corresponds to the angular momentum of the center-of-mass motion: ltot = l +(x g x g ), αβ αβ α β − β α where the position vector is denoted xα. We also decompose the angular momentum flux into an intrinsic or spin angular π momentum flux Mαβγ and the flux of orbital angular momentum as Mtot = Mπ + x σtot x σtot . αβγ αβγ α βγ − β αγ Using these definitions, angular momentum conservation Figure 2. (a) The cell cytoskeleton is a filament network in which can be rewritten in the form [33, 39] molecular motors can act as active cross-linkers. When a filament pair (red arrows) interacts with a small aggregate of molecular motors (green), the motors introduce a force and a counter force tot tot a ∂t(xαgβ xβgα) ∂γ (xασβγ xβσαγ )=2σαβ (5) in the network in the presence of the fuel ATP. The material can be − − − maintained away from thermodynamic equilibrium if the fuel and π a reaction products are kept at constant concentration by external ∂tlαβ ∂γ M = 2σ (6) − αβγ − αβ reservoirs. (b) The force exerted by motor on a pair of filaments where introduces a force dipole in the material. On larger scales, the average density of force dipoles corresponds to the active stress. 1 σa = (σtot σtot ) (7) αβ αβ βα i 2 − ∂tni + ∂αJα = ri (8) i i is the antisymmetric part of the stress. Note that equations (5) where Jα = nivα + jα are particle currents and we have i and (6) follow directly from equations (3) and (4). defined the relative fluxes jα. In the absence of chemical reac- Particle number balance—chemical reactions. In addition tions, these densities are all conserved. Chemical reactions to mass balance, the particle number densities of molecular contribute to sources ri in equation (8). We denote chemical species can obey conservation laws. We write balance equa- reactions by the index I = 1, ..., M, where M is the num- tions for the particle number densities ni of molecular species ber of reactions. These chemical reactions can in general be i with i = 0, ..., N: expressed in the form

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N volume elements equilibrate at times that are short compared I ai Ai 0. (9) to the slow dynamics of hydrodynamic modes at large scales. i=0 We discuss the equilibrium thermodynamics of local volume Here, Ai denotes the chemical symbol of molecule species i. elements of volume V at temperature T containing Ni = niV I particles of type i with free energy density F = fV and entropy The numbers ai are the stoechiometric coefficients for reac- N I S = sV , where f and s denote the densities of free energy and tion I which obey i=0 ai mi = 0, where mi are the masses I entropy, respectively. We have s = ∂f /∂T and the chemi- of molecules of species i. For ai > 0, molecular species i is a − I cal potentials are given by µi = ∂f /∂ni in each local volume reaction substrate, for ai < 0, species i is a reaction product. I element. The system as a whole is in general not at a global For ai = 0 the molecular species i is not taking part in reac- thermodynamic equilibrium. However its total free energy tion I. The rates ri at which molecular species i is generated by chemical reations can thus be written as a sum over contrib F = d3xf and entropy S = d3xs are always well defined utions from different reactions as the sum of the contributions of locally equilibrated volume elements. M For such a non equilibrium system, the total entropy S in r = aIrI, (10) i − i general increases over time according to the second law of I=1 I thermodynamics. Accordingly, we can express changes of where we have introduced the chemical fluxesr , which are entropy in the system with a balance equation for the entropy the rates at which chemical reactions I proceed. i density: The currents jα describe relative movements of differ- s ent molecular species. The mass density can be written as ∂ts + ∂αJα = θ (14) N ρ = i=0 mini. Mass conservation (1) then implies where θ>0 denotes the entropy production rate per unit vol- s s N N ume due to irreversible processes and Jα = svα + jα is the i mijα = 0 and miri = 0. (11) entropy flux. Here we have introduced the relative entropy flux js . We can also write a balance equation for the free i=0 i=0 α We can therefore express the current and the source rate of energy density f one molecular species (we choose the solvent i = 0) by the f ∂tf + ∂αJ = θf (15) remaining currents and rates: α where we have introduced the free energy flux J f and the N α 0 1 i source of free energy θf . The free energy density obeys the jα = mijα (12) −m0 local thermodynamic relation f = e − Ts between densi- i=1 ties of energy, entropy and free energy. We express the free f N energy flux also as a convected part and the flux jα relative 1 f f r0 = miri. (13) to the center of mass, Jα = f vα + jα. The total energy flux −m0 i=1 is then the sum of free energy transport and heat transport 0 e f Q. Here, we define the heat flux as The reaction rate r0 and flux j are thus not independent Jα =(f + Ts)vα + jα + jα α Q s variables and can be eliminated. In addition to overall mass jα = Tjα, which is the entropy flux with respect to the center conservation, for N + 1 molecular species and M linearly of mass motion (note that other definitions of the heat flux are independent chemical reactions there are thus N − M conser- also sometimes used [33]). Thus, the relative free energy flux f e Q vation laws for molecule number densities. jα = jα jα is the part of the relative energy flux that is not in the form− of heat. The projected free energy flux at the system boundaries j f n can be interpreted as work performed − α α per unit area by the environment on the system. Here nα is a 3. Irreversible thermodynamics of active matter unit vector normal to the boundary pointing outwards. In isothermal systems which are maintained everywhere at temper 3.1. Balance of entropy and free energy ature T by a thermostat, the local reduction rate of free energy In order to understand the dynamics of a complex system of is directly related to the rate of entropy production: Tθ = θf , − condensed matter, one has in principle to solve microscopic which follows using equation (2), see equation (A.22). In the equations of motion. At thermodynamic equilibrium, the following, we will limit our discussion for simplicity to this behaviours of a many-particle system can be understood to a isothermal case. The irreversible thermodynamics of a multi- large extent from the knowledge of thermodynamic potentials component fluid, taking into account temperature variations, which are functions of only a few thermodynamic variables. is reviewed in appendix A. Here we employ the framework of irreversible thermodynamics in order to discuss the dynamics of a complex system in the 3.2. Equilibrium thermodynamics of a polar fluid vicinity of a thermodynamic equilibrium [33]. More precisely, we describe the case where a system is locally at equilibrium We consider a polar fluid that is characterised by a local vec- while being maintained away from equilibrium globally. Such torial anisotropy described by the vector pα. In the context of local thermodynamic equilibrium can be considered if small the cell cytoskeleton, the vector

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n pα = pα (16) The generalized Gibbs–Duhem relation (22) shows that the equilibrium stress obeys a force balance condition ∂ σe = 0 could for example correspond to the average in a local volume β αβ n at thermodynamic equilibrium when ∂αµi and hβ vanish. element of vectors pα that define the directions of individual However, in an out of equilibrium situation, the equilibrium filaments n in a filament network, see figure 1. Considering a stresses do not obey a force balance and out of equilibrium local equilibrium, we denote the free energy per volume in the contributions to the stress must exist. reference frame moving with the center of mass velocity vα by f0( pα, ∂βpα, ni, T), which in general depends on the local polarity anisotropy pα. For inhomogeneous systems, we also 3.4. Rotation invariance and equilibrium angular have to consider the dependence of the free energy on gradi- momentum flux ents of the polarity field ∂αpβ. The total free energy then reads Rotation invariance of the free energy provides an additional g2 condition that is related to angular momentum conservation. F = d3x α + f ( p , ∂ p , n ) , (17) 2ρ 0 α α β i The change of free energy under an infinitesimal rotation reads and we define the total free energy density 3 e ∂f0 f ( pα, ∂αpβ, ni, gα)= δF = d x (σβα hβpα)βγαθγ + dSδ pαβγαθγ . g2 /(2ρ)+f ( p , ∂ p , n ) which also contains the kinetic − ∂(∂δpβ) α 0 α α β i (23) energy of center-of-mass motion. The conjugate field to the Equation (23) describes the vanishing net torque and we iden- polarity pα is hα = δF0/δpα where the functional derivative can be expressed− as tify the equilibrium intrinsic angular momentum flux ∂f ∂f e ∂f0 ∂f0 0 0 Mαβγ = pα pβ. (24) hα = + ∂β (18) ∂(∂γ pβ) − ∂(∂γ pα) −∂pα ∂(∂βpα)

3 From the condition that δF in equation (23) must vanish for and F0 = d xf0. Furthermore, we introduce the total chemi- tot any small rotations θβ, we obtain cal potential µi = ∂F/∂ni with e e,a ∂γ Mαβγ = 2σ +(pαhβ pβhα). (25) 1 αβ − µtot = m v2 + µ (19) i −2 i α i Note that the Ericksen stress can have an antisymmetric part e,a e e where µi = ∂f0/∂ni is the conventional chemical potential. σαβ =(σαβ σβα)/2. At equilibrium with hα = 0 and − π e In an anisotropic system, the stress at equilibrium is not an ∂tlαβ = 0, we have Mαβγ = Mαβγ and the torque balance e e,a isotropic pressure but an anisotropic stress. The equilibrium equation (6) reads ∂γ Mαβγ = 2σαβ. stress tensor can be obtained from considering translation invariance. 3.5. Entropy production rate of a polar fluid Rate of change of the free energy. For an isothermal system, 3.3. Translation invariance of the free energy: the equilibrium stress tensor the rate of entropy production obeys TdS/dt = dF/dt is related to the rate of change of the free energy. We− derive in Using the invariance of the free energy with respect to several steps a general expression for the rate of change of the translations in space results in a generalized Gibbs–Duhem free energy for the case of a polar isothermal system. Within a equation and an expression for the equilibrium stress. The free volume V of fixed shape we have energy change due to an infinitesimal translation by a vector dF 3 tot ∂f0 δx is given by (see appendix B) = d x vα∂ gα + µ ∂ n hα∂ pα + dSα ∂ pβ α dt t i t i − t ∂(∂ p ) t V α β 3 tot 3 tot tot i tot δF = d x (∂βvα)gα +(∂βµ )ni + hα∂βpα δxβ = d x( (∂ v )σ + µ r +(j + n v )∂ µ h ∂ p ) i − β α αβ i i α i α α i − α t α V tot ∂f0 tot tot i ∂f0 + dSα ( f vγ gγ µi ni)δαβ ∂βpγ δxβ. + dSα vβσβα µi Jα + ∂tpβ , (26) − − − ∂(∂αpγ ) − ∂(∂αpβ) (20) Here and in the following we imply summation over repeated where we have used the conservation laws for mass and indices i from i = 0 to N. The area element dSα is a vector momentum. Using the Gibbs–Duhem relation (22), we can pointing in the direction normal to the surface in outward rewrite this as direction. From the surface contribution we identify the equilibrium stress, also called Ericksen stress dF dpα = d3x( ∂ v (σ σe )+µ r + ji ∂ µ h ) dt − β α αβ − αβ i i α α i − α dt ∂f0 σe =(f µ n )δ ∂ p . αβ 0 i i αβ α γ (21) i ∂f0 dpβ − − ∂(∂βpγ ) + dS v σ v f µ j + (27) α β βα − α − i α ∂(∂ p ) dt α β Translational invariance δF = 0 implies the Gibbs–Duhem relation e where d/dt = ∂t + vγ ∂γ is a convected time ∂βσαβ =(∂αµi)ni + hβ∂αpβ. (22) tot − derivative and σαβ = σαβ + vαgβ. We can

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decompose the velocity gradient ∂ v = v ω in a The first term describes the mechanical work of surface forces. β α αβ − αβ symmetric part vαβ =(1/2)(∂αvβ + ∂βvα) and the vorticity The second term accounts for the work of surface torques. The ωαβ =(1/2)(∂αvβ ∂βvα) and write remaining terms describe chemical work on the surface and − the work associated with changes of the polarity field. dF = d3x( v (σs σe,s )+ω (σa σe,a + h p ) Rate of entropy production. The form of equation (31) dt − αβ αβ − αβ αβ αβ − αβ α β ensures that all bulk contributions to free energy changes i Dpα involve conjugate pairs of thermodynamic fluxes and + µiri + jα∂αµi hα ) − Dt of generalized forces that all vanish for thermodynamic i ∂f0 dpβ equilibrium states. Equilibrium states correspond either + dSα vβσβα vαf µijα + − − ∂(∂αpβ) dt to systems with zero or constant velocity vα or to situa- (28) tions with constant vorticity ωαβ corresponding to a spin- s where σαβ =(σαβ + σβα)/2 the symmetric part of the stress ning equilibrium system. For nonequilibrium situations the and we have introduced the convected corotational time entropy production reads

derivative M d,s 1 d I I Dpα Tθ = vαβσαβ + (∂γ ωαβ)Mαβγ + r ∆µ = ∂tpα + vγ ∂γ pα + ωαβpβ (29) 2 Dt I=1 N of the vector pα, which is the time derivative evaluated in a co- i Dpα j ∂αµ¯ + hα . (35) rotating and co-moving reference frame. We still have to use − α i Dt i=1 angular momentum conservation to obtain the final expres- Here we have introduced the the relative chemical potentials sion that allows us to identify the local entropy production rate. In the free energy (17) we do not take into account the mi µ¯i = µi µ0, (36) kinetic energy associated with intrinsic angular momentum. − m0 This corresponds to the case where the density of intrinsic and the chemical free energy change associated with reaction I angular momentum can be neglected, l = 0. In this simple N N αβ is ∆µI = aIµ . From mass conservation m aI = 0 case angular momentum conservation implies i=0 i i i=0 i i it follows that π e a e,a N ∂γ Mαβγ ∂γ Mαβγ = 2(σαβ σ ) ( pαhβ pβhα), − − αβ − − I I ∆µ = a µ¯i . (37) (30) i i=1 where we have used equation (24). Expression (30) permits us From equation (13) we also have N ji µ = N ji µ¯ and to eliminate the antisymmetric stress from equation (28). The i=0 α i i=1 α i thus only N diffusion fluxes dissipate. The contributions of rate of change of the free energy finally reads the chemical reactions I = 1 ...M to entropy production are dF 3 d,s 1 d i Dpα written explicitely. Each chemical reaction I is driven by the = d x( vαβσ (∂γ ωαβ)M + µiri + j ∂αµi hα ) dt − αβ − 2 αβγ α − Dt I chemical free energy changes ∆µ associated with the corre 1 π i ∂f0 Dpβ M I I N + dSα vβσβα + Mβγαωβγ vαf µijα + . sponding reaction. Note that I=1 r ∆µ = i=1 riµ¯i. 2 − − ∂(∂αpβ) Dt − (31) 3.6. Irreversible thermodynamics of a nematic fluid Here we have introduced the symmetric deviatoric stress If the local order of filaments is not polar but nematic, the σd,s = σs σe,s (32) αβ αβ − αβ relevant order parameter becomes a traceless symmetric and the deviatoric part of the angular momentum flux tensor

d π e n n 1 n n Mαβγ = Mαβγ Mαβγ . (33) Q = p p p p δ , (38) − αβ α β−3 γ γ αβ The rate of change of the free energy given by equation (31) where the average is over the directions pn of individual fil- has a bulk and a surface contribution. The bulk term vanishes α aments in a local volume element. The free energy density at thermodynamic equilibrium and can be identified with the 2 g /(2ρ)+f0(ni, Qαβ, ∂γ Qαβ) now depends on the nematic free energy source 3 . In isothermal systems it is directly α d xθf δF0 order parameter and its gradients. The field Hαβ = δQ , related to entropy production Tθ = θf . The surface contrib − αβ ution describes the mechanical and chemical− work performed conjugate to the nematic order, is given by f on the system at the surface. It is of the form dSαJα with ∂f ∂f − 0 0 J f = f + j f . From equation (31), we thus identify the rela- Hαβ = + ∂γ . (39) α vα α −∂Qαβ ∂(∂γ Qαβ) tive free energy flux Here, the tensor Hαβ is traceless because only traceless variations of Qαβ are permitted when determining the functional f 1 π i ∂f0 Dpβ jα = vβσβα ωβγMβγα + µijα . (34) derivative. The equilibrium stress then reads − − 2 − ∂(∂αpβ) Dt

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∂f e el e 0 ∂αP = ni∂αµi σβγ∂αu˜βγ. (46) σ =(f0 µini)δαβ ∂αQγν (40) αβ − − ∂(∂ Q ) − β γν The rate of change of the free energy change is and the Gibbs–Duhem relation is given by dF 3 e i el Du˜αβ e = d x( ∂βvα(σαβ + P δαβ)+µiri + jα∂αµi + σαβ ) ∂βσ =(∂αµi)ni + Hγν∂αQγν. (41) dt − Dt − αβ i Invariance under infinitesimal rotations implies the relations + dSα vβσβα vαf µij , (47) − − α (B.6) and (B.7). The rate of change of free energy then reads where the convected corotational time derivative of u˜αβ is dF 1 = d3x( v σd,s (∂ ω )Md + µ r defined in equation (43). dt − αβ αβ − 2 γ αβ αβγ i i i DQαβ + jα∂αµi Hαβ ) 3.8. Irreversible thermodynamics of systems with intrinsic − Dt rotations 1 + dS v σ + Mπ ω v f α β βα 2 βγα βγ − α So far, we have simplified angular momentum conservation and have ignored the density lαβ = αβγ lγ of intrinsic angular i ∂f0 DQβγ µijα + (42) momentum in the free energy. In general, the free energy den- − ∂(∂αQβγ) Dt sity in the center of mass reference frame f0( pα, ∂βpα, ni, lαβ) depends on the density of intrinsic angular momentum lαβ where the convected co-rotational time derivative of a tensor [41]. This dependence accounts for the kinetic energy contrib is defined as [40] utions that are associated with intrinsic rotations. The intrinsic

DQαβ rotation rate is defined as = ∂ Q + v ∂ Q + ω Q + ω Q . (43) Dt t αβ γ γ αβ αγ γβ βγ αγ ∂f0 Ωαβ = 2 , (48) Note that there exist several definitions of corotational time ∂lαβ derivatives of a tensor. Equation (43) is often called upper convected time derivative. It represents a material time deriva- with Ωαβ = αβγ Ωγ and Ωαβlαβ = 2Ωαlα. The derivation of tive taken in a reference frame that is convected and co-rotated the rate of entropy production taking into account the kinetic with the center of mass motion. energy of intrinsic rotations is given in appendix B. The entropy production rate reads 1 3.7. Irreversible thermodynamics of a visco-elastic gel Tθ = v σd,s + (Ω ω )σd,a + (∂ ω )Md αβ αβ αβ − αβ αβ 2 γ αβ αβγ So far we have discussed the thermodynamics of fluids with M N Dpα polar or nematic order. Cytoskeletal networks are gel-like + rI∆µI ji ∂ µ¯ + h . (49) − α α i α Dt materials which in general have visco-elastic properties. I=1 i=1 Such visco-elastic material properties can also be captured by the generic thermodynamic approach. We thus discuss Here, the deviatoric antisymmetric stress is defined as the physics of viscoelastic systems for simplicity for the 1 1 σd,a = σa σe,a ( p h p h ) ∂ Md . (50) case of a viscoelastic gel or filament network that is isotro- αβ αβ − αβ − 2 α β − β α − 2 γ αβγ pic at equilibrium. Such materials can exhibit transient elas- The total stress is then given by el tic shear stress σαβ and elastic shear strain u˜αβ, which is a 1 1 traceless symmetric tensor describing local network anisot- σtot = σd,s + σd,a ρv v + σe + ( p h p h )+ ∂ Md . αβ αβ αβ − α β αβ 2 α β − β α 2 γ αβγ ropies. The elastic stress relaxes via network rearrangements (51) within a finite relaxation time. To describe the thermodynam- The total stress has an antisymmetric part ics of such systems, we introduce a thermodynamic ensemble σa = σe,a + 1 ( p h p h )+ 1 ∂ Md + σd,a. Often with prescribed strain and corresponding free energy density αβ αβ 2 α β − β α 2 γ αβγ αβ 2 only the reactive or equilibrium part of this antisymmetric f = gα/(2ρ)+f0(ni, u˜αβ) which is a function of densities ni e,a 1 and of the prescribed shear strain . The conjugate thermo- stress σ + ( pαhβ pβhα) is discussed. However in gen- u˜αβ αβ 2 − dynamic field eral dissipative contributions to the antisymmetric stress can exist [41, 42]. el ∂f0 σαβ = (44) ∂u˜αβ 3.9. Conjugate pairs of thermodynamic fluxes is the traceless elastic stress associated with u˜αβ. The equilib- and generalized forces rium stress σe = Peδ is isotropic and given by αβ − αβ e By combining the different cases discussed above, we can σ =(f0 niµi)δαβ (45) αβ − now extend equation (35) and write a general form of the where Pe is the equilibrium hydrostatic pressure. It obeys the entropy production rate. We decompose the symmetric s Gibbs–Duhem relation part of the stress tensor σαβ =(σαβ + σβα)/2 in the form

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s s s Table 1. List of the pairs of conjugate thermodynamic fluxes J σαβ =˜σαβ + σδαβ, where σ˜αβ is the traceless symmetric n and forces F . The signatures of the thermodynamic forces F shear stress and σ =(1/d)σs denotes the isotropic stress with n n γγ under time reversal are given. The time reversal signature of the d the number of space dimensions. Similarly, the deviatoric n corresponding thermodynamic flux Jn is n. Scalars, vectors and part of the symmetric stress that enters the total stress in equa- tensorial objects differ in their transformation− behaviours under d,s d d coordinate rotations. tion (51) is decomposed as σαβ =˜σαβ + σ δαβ in a traceless d d d e Time-reversal part σ˜αβ and the isotropic part σ , with σ = σ + P , where Flux J force F signature of Fn Rotation symmetry Pe = (1/d)σe,s denotes the equilibrium pressure. Finally, n ↔ n n − γγ we decompose the symmetric part of the velocity gradient d,s −1 Traceless σ˜αβ v˜αβ 1 ˜ ↔ symmetric tensor vαβ = v˜αβ +(∂γ vγ ) d δαβ in a traceless rate of pure shear vαβ σd ∂ v −1 Scalar and local expansion rate, which is the divergence ∂γ vγ of the ↔ γ γ Dpα Vector flow velocity. The total entropy production rate then reads hα +1 Dt ↔ DQαβ +1 Traceless d,s d,a Hαβ Tθ = v σd + v˜ σ˜ + (Ω ω )σ Dt ↔ symmetric tensor γγ αβ αβ αβ − αβ αβ i Vector M N jα ∂αµ¯i +1 1 d I I i I ↔−I + (∂γ ωαβ)M + r ∆µ j ∂αµ¯i r ∆µ +1 Scalar 2 αβγ − α ↔ I=1 i=1 Md 1 ∂ ω −1 Rank three tensor αβγ ↔ 2 γ αβ Dpα DQαβ el Du˜αβ d,a −1 Antisymmetric + h + H σ , (52) σαβ (Ωαβ ωαβ) α Dt αβ Dt − αβ Dt ↔ − tensor

D˜uαβ σel +1 Traceless symmetric where we consider the isothermal case for simplicity. The Dt ↔− αβ tensor rate of entropy production (52) is a sum of independent

contributions which are products of a generalized thermo- which ensures that all Jn vanish when the Fn are zero. This dynamic force Fn and a conjugate thermodynamic flux Jn of linear relation between thermodynamic fluxes and generalized the form forces defines the constitutive relations that describe the properties of a given material or system. Here we have introduced Tθ = JnFn. (53) d r the matrix of Onsager coefficients Onm = Onm + Onm which n d can be decomposed in a dissipative part Onm and a reactive Here, the index n specifies a thermodynamic variable or a r d part Onm. Only the dissipative part Onm with the Onsager sym- tensor or vector component of a given variable. From equa- d d metry property Onm = Omn contributes to entropy production tion (52) we can identify the set of conjugate pairs of thermo- and dissipation. The reactive part is antisymmetric and obeys dynamic fluxes and generalized forces given in table 1. r r Onm = Omn. Because of this antisymmetry the reactive coef- The generalized forces and fluxes can be characterized by ficients− do not contribute to entropy production and describe their signature ε under time reversal. We denote by n the time reversible phenomena. Time reversal symmetry implies that reversal signatures of the generalized thermodynamic forces d the dissipative coefficients Onm are nonzero if the generalized Fn, which are given in table 1. For example, under the trans- force Fn and the flux Jm have opposite time reversal signature formation t t, we have v˜αβ v˜αβ which corresponds →− →− (and thus Fn and Fm have the same signature): n m = 1 and to = 1 while hα hα and h = 1. Because the entropy v − → vanish if Fn and the flux Jm have the same signature. Similarly, production θ has signature θ = 1, the time reversal sig- r − the reactive coefficients Onm are nonzero only in the case nature of the conjugate thermodynamic fluxes Jn is given by where Fn and Jm have the same time reversal signature (and n, opposite to those of Fn. The time reversal signatures of − thus Fn and Fm have opposite signature) or n m = 1. thermodynamic forces are listed in table 1. − d In addition to time reversal symmetry, the coefficients Onm r and Onm have to reflect the spatial symmetries of a given sys- 4. Constitutive relations and hydrodynamic tem. This is called the Curie principle. The Curie principle equations provides a guide to determine the form of the matrix Onm based on the symmetries of a system [47]. In particular, the 4.1. Onsager relations Curie principle demands that the transformation properties of

At thermodynamic equilibrium all thermodynamic fluxesJ n scalars, vectors and tensors under coordinate changes have and generalized forces Fn vanish and no entropy is produced, to be respected. The thermodynamic fluxes Jn can be scalars, Tθ = 0. Irreversible thermodynamics provides a systematic vectors or tensors, see table 1. If a particular constitutive equa- expansion of the fluxes Jn in terms of the generalized forces tion describes a scalar flux, this flux can involve vector quanti- Fn in the vicinity of equilibrium. These relations define the ties only via scalar products. Similarly, a thermodynamic flux dynamic properties of the system and represent constitutive which is a vector can be constructed from other vectors or relations characterizing the material properties. To linear from the contraction of a second rank tensor with a vector. order, we have [43–46] The coefficients Onm can themselves be vectors or tensors if the system exhibits vectorial or tensorial asymmetries such as polar or nematic order parameters. We will now discuss J = O F , n nm m (54) the constitutive relations of different classes of active systems m 8 Rep. Prog. Phys. 81 (2018) 076601 Review

using the Curie principle to express the Onsager relations polarity changes. The coefficientχ couples elastic stresses given by equation (54). to polarity while the coefficient λI can serve as a Lagrange 2 multiplier to impose the constraint pα = 1. We included the coefficientsλ I, λI and λI of second order terms in that can 4.2. Active gels 1 2 3 pα exist in polar systems. While a term proportional to pγ ∂αpγ I Active polar gels. Using this formalism, we can now con- described by the coefficient λ3 also exists in passive polar sys- I I struct the theory of active polar gels. We consider the sim- tems, the other two coefficients λ1 and λ2 are specific to active ple case of an incompressible medium with divergence free systems and generate interesting physics [12, 17, 50]. The i flow, ∂γ vγ = 0. In this case, the isotropic stress σ = P or coefficients κ describe the coupling between diffusion fluxes the hydrostatic pressure P becomes a Lagrange multiplier− and polarity. The matrix Λij describes diffusion coefficients to impose the incompressibility constraint. We are left with and cross-diffusion and we take into account the coupling of d,s elastic stresses to diffusion fluxes via the coefficients and the pairs of conjugate variables σ˜ v˜αβ, Dpα/Dt hα, χi αβ ↔ ↔ iI ji ∂ µ¯ , rI ∆µI and Du˜ /Dt σel . Here we χ1i and couplings to polarity gradients via the coefficients 1 α ↔− α i ↔ αβ ↔− αβ iI iI neglect for simplicity dissipative contributions to the antisym- and 2 . The coefficients λ describe molecular fluxes gener- metric stress which are irrelevant in the hydrodynamic limit, ated by chemical reactions such as the effects of molecular see also section 4.4 below and appendix C.3 [39, 41]. motors. The coefficient Γ describes the relaxation of elastic I Writing terms obeying scalar, vectorial or tensorial sym- strain and ψ captures the generation of active elastic stresses ¯ IJ metries only, the Onsager relations (54) of an active polar gel by chemical reactions. Finally, the coefficients Λ describe read [11, 16, 18, 48, 49] the chemical reaction rates. Note that in the constitutive equations (55)–(59), the component indices run from i = 1, ...M d,s el ν1 2 I I and do not include the solvent i = 0. The solvent flux j0 fol- σ˜αβ = σαβ + 2ηv˜αβ + ( pαhβ + pβhα pγ hγ δαβ)+ζ ∆µ q˜αβ α 2 − 3 lows from other fluxes through the mass conservation equa- νi 2 ( pβ∂αµ¯i + pα∂βµ¯i pγ ∂γ µ¯iδαβ) (55) tion (equation (12)). − 2 − 3 Note that the elastic stress σel enters in equation (55) − αβ with Onsager coefficient −1 and v˜αβ in equation (59) with Dpα 1 el I I = hα ν1pβv˜αβ χpβσαβ + λ pα∆µ coefficient 1, consistent with a reactive coupling. The magni- Dt γ1 − − I I I I I I i i tude of the dimensionless coefficients can be chosen without + λ1∆µ pα∂γ pγ + λ2∆µ pγ ∂γ pα + λ3∆µ pγ ∂αpγ +κ ∂αµ¯ loss of generality to be 1. The total stress is given by (56) ± i ij i el i iI I i el tot e 1 d,s j = Λ ∂αµ¯j χ pβσ ν pβv˜αβ + λ pα∆µ + χ ∂βσ σ = ρv v +˜σ Pδ + ( p h p h )+˜σ , α − − αβ − 1 αβ αβ α β αβ αβ α β β α αβ iI I iI I i − − 2 − + 1 ∆µ pα∂γ pγ + 2 ∆µ pγ ∂γ pα κ hα (57) (61) − e where σ˜αβ denotes the traceless part of the equilibrium stress σe defined in equation (21). rI = Λ¯ IJ∆µJ ζIq˜ v˜ ψIq˜ σel +λIp h αβ − αβ αβ − αβ αβ α α Hydrodynamic limit. In the hydrodynamic limit, the system I I + λ1hαpα∂γ pγ + λ2hαpγ ∂γ pα is governed by the slow dynamics of hydrodynamic modes + λI h p ∂ p λiIp ∂ µ¯ iI(∂ µ¯ ) p ∂ p which follow from conservation laws. The relaxation time 3 α γ α γ − α α i − 1 α i α γ γ iI of these modes diverges in the limit of long wave-lengths. 2 (∂αµ¯i)pγ ∂γ pα (58) − Because the strain variable relaxes on a finite time scale, in the hydrodynamic limit it has relaxed to a steady state value Du˜αβ χ 2 el el with Du˜αβ/Dt = 0. The elastic stress σαβ can then be deter- = v˜αβ Γσαβ + ( pαhβ + pβhα pγ hγ δαβ) Dt − 2 − 3 mined from equation (59). The gel then behaves as a viscous I I + ψ ∆µ q˜αβ fluid. On long length scales all terms proportional to spatial χi 2 gradients can be neglected. After eliminating the elastic stress ( pα∂βµ¯i + pβ∂αµ¯i pγ ∂γ µ¯iδαβ) el − 2 − 3 σαβ in equation (55), we arrive at the hydrodynamic theory of i 1 an active polar fluid [11, 16, 18, 48] χ1(∂α∂βµ¯i ∂γ ∂γ µ¯iδαβ) (59) − − 3 νeff 2 σ˜d,s = 2ηeffv˜ + 1 ( p h + p h p h δ )+ζI ∆µIq˜ αβ αβ 2 α β β α − 3 γ γ αβ eff αβ where we have have defined the traceless symmetric tensor (62)

1 2 2 q˜αβ = pαpβ pγ pγ δαβ. (60) Dpα 1 χ pβpβ χ eff − 3 = hα pαhβpβ ν1 pβv˜αβ Dt γ − 2Γ −6Γ − 1 We have introduced dissipative and reactive Onsager coef- I I I I + λ pα∆µ + λ1∆µ pα∂γ pγ ficients that obey symmetric relations: The viscosityη , the ψIχ polarity-flow-coupling coefficient ν , the coefficients ζI + λI ∆µIp ∂ p + λI ∆µIp ∂ p ∆µIp q˜ 1 2 γ γ α 3 γ α γ − Γ β αβ describing the generation of active stress by reaction I, a (63) coefficient νi coupling the chemical potential gradients to ji = Λij∂ µ¯ j + λiIp ∆µI (64) the stress as well as γ1 describing dissipation associated to α − α α

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I J I ¯ IJ ψ ψ J I r =(Λ q˜αβq˜αβ)∆µ ζeffq˜αβv˜αβ act 1 n n n − Γ − σαβ = f apαpβ (70) I V χψ I I n q˜αβpαhβ + λ pαhα + λ hαpα∂γ pγ − Γ 1 where the sum is over motor-induced force dipoles in the vol- I I iI (65) + λ2hαpγ ∂γ pα + λ3hαpγ ∂αpγ λ pα∂αµ¯i. ume V. The anisotropic part of this active stress is given in − equation (66) to linear order in the chemical force ∆µI . In Here we have introduced the effective gel viscosity the context for motor protein generated active stress the ATP ηeff = η + 1/(2Γ) as well as the effective Onsager coeffi- hydrolysis reaction drives the generation of the active stress eff I I I I cient ν1 = ν1 + χ/Γ and the coefficient ζeff = ζ + ψ /Γ and ∆µ corresponds to the chemical potential difference of describing active stresses. For simplicity, we have consid- the fuel ATP and its reaction products. i i i i iI ered the case χ = 0, χ1 = 0, ν = 0, κ = 0, and k = 0 Visco-elastic gel. On shorter time scales, the active gel with k = 1, 2. el exhibits elastic behaviors. Writing σαβ = Ku˜αβ, where K is Active stress. A key feature of the constitutive equations of an elastic modulus, The equation (59) for the elastic stress active polar gels given by equations (55) and (62) is the trace- becomes [49] less and symmetric contribution to the stress D 1 χ 2 act I I (1 + τ )σel = v˜ + ( p h + p h p h δ ) σ˜αβ = ζ ∆µ q˜αβ (66) Dt αβ Γ αβ 2Γ α β β α − 3 γ γ αβ due to active chemical processes. This active stress plays a ψI∆µI + q˜ (71) fundamental role for the dynamics of active gels and fluids Γ αβ and implies that in the absence of external stresses the gel can undergo spontaneous shear. Furthermore the system can where τ = 1/(KΓ) is a Maxwell relaxation time. Equation (71) perform mechanical work against external stresses which thus is the generalization of a Maxwell model of viscoelastic are powered by chemical energy related to chemical poten- gels to the case of polar and active gels. Permeation of this tial differences ∆µI . Because of Onsager symmetries of the Maxwell gel by solvent is captured in this theory when we i coupling coefficients, the coefficients ζI also enter the chemi- include the coefficient χ1 from equation (57) in equation (64) cal reaction rates (58) and (65), which effectively become for i = g, where g denotes the gel material. We then have mechanosensitive. g jg = Λgj∂ µ¯ + λgIp ∆µI + χ ∂ σel . (72) In the biological context, the molecular processes that gen- α − α j α 1 β αβ g g g g erate active stresses are often mediated by molecular motors. Using Jα = jα + ngvα and Jα = ngvα, this can be written as These motors are fueled by the hydrolysis reaction of ATP to a permeation equation ADP and inorganic phosphate. This hydrolysis reaction there- λg(vg v )= Λ¯ gj∂ µ¯ + λ¯gIp ∆µI + ∂ σel , (73) fore is the dominant chemical reaction I that couples via the α − α − α j α β αβ I act el coefficient ζ to active stress. The active stress σαβ emerges which describes the effects of the elastic force density ∂βσαβ from a large number of force generating events that occur pushing the fluid through the gel [51]. Here, the permeation in the anisotropic material. A small assembly of molecular g g ¯ gj gj g coefficient is λ = ng/χ1. Furthermore, Λ =Λ /χ1 and motors that locally interacts for example with two aligned ¯gI gI g ¯gI I λ = λ /χ1. The contribution λ pα∆µ describes the force filaments in the presence of ATP generates a pair of equal and density due to the action of chemical reactions or molecular opposite forces f npn in the material, see figures 2(a) and (b). motors. ± α Here pn are unit vectors, characterizing the opposite orien- In the long-time limit, the elastic stress is governed by gel ± α tations of the two aligned filaments and n is an index over viscosity and active stresses and is of the form [49] such pairs of aligned filaments that are cross-linked by active 2 motors. This force and counter force correspond to a force el g g g σαβ ηg(∂αvβ + ∂βvα ∂γ vγ δαβ) dipole with force density − 3 ψI∆µI χ 2 n n a n a n + q˜αβ + ( pαhβ + pβhα pγ hγ δαβ). φα = f p δ(x x0 + p ) δ(x x0 p ) (67) Γ 2Γ − 3 α − 2 − − − 2 (74) at position x0 in the material. Here a denotes the distance at Gel viscosity and gel permeation coefficient introduce a g 1/2 which force and counter-force act. On large length scales or characteristic length =(ηg/λ ) . On length scales large in a continuum limit, this can be expanded in the microscopic el compared to , the contribution ∂βσαβ can be neglected in length a. To first order it corresponds to the point dipole equation (72) and we recover in the hydrodynamic limit the n n n n φα af pαpβ∂βδ(x x0)=∂βσαβ (68) form of equation (64). On short length scales, however, the − g g friction due to permeation λ (vα vα) can be neglected. with the corresponding stress [15] In this limit, the permeating fluid −can be ignored and equa- g n n n tion (73) for λ = 0 describes the force balance of the gel σ = fap p . (69) αβ α β alone. In this limit a simplified one-component model of only In a continuum description, the density of these force dipoles the gel [11, 16, 52] can be used to capture the main features in a local volume elements V is the active stress of the active gel.

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4.3. Active nematics d,a η¯ σαβ = 2η (Ωαβ ωαβ)+ pγ ∂γ ωαβ Nematic liquid crystals are characterized by the tensor order − 2 ν2 parameter Q , which is traceless and characterizes local + ( pαhβ pβhα)+ν3αβγ hγ αβ 2 − anisotropies. In the presence of active processes and on I I (81) + ζa∆µ pγ αβγ time scales when elastic stresses have relaxed, we consider d,s the conjugate variables σ˜αβ v˜αβ, DQαβ/Dt Hαβ, i I I ↔ ↔ Dpα 1 jα ∂αµ¯i and r ∆µ . We again describe an incom- = hα ν1pβv˜αβ + ν2pβ(Ωαβ ωαβ) ν3αβγ (Ωβγ ωβγ) ↔− ↔ Dt γ1 − − − − pressible system and the pressure P serves as a Lagrange ν¯ ν¯ ν¯ + 2 p p ∂ ω 3 p ∂ ω 4 p ∂ ω multiplier to impose incompressibility. An active nematic 2 γ β γ αβ − 2 γβα δ δ γβ − 2 γβδ α δ γβ I I I I I I fluid is then characterized by the following constitutive rela- + λ pα∆µ + λ1∆µ pα∂γ pγ + λ2∆µ pγ ∂γ pα (82) tions [53, 54]

d d,s I I Mαβγ = κ∂γ ωαβ +¯ηpγ (Ωαβ ωαβ) σ˜αβ = 2ηv˜αβ + ν1Hαβ + ζ ∆µ Qαβ (75) − ν¯ + 2 p ( p h p h )+¯ν h p +¯ν p h 2 γ α β − β α 3 αβδ δ γ 4 αβγ δ δ DQ 1 I I I I αβ I I + ζ1∆µ αβγ + ζ2∆µ αβν pν pγ = Hαβ ν1v˜αβ + λ Qαβ∆µ (76) Dt γ − I I I I 1 + ζ3∆µ (αγδpδpβ βγδpδpα)+ζ4∆µ (δαγ pβ δβγpα) − − (83) i ij iI I jα = Λ ∂αµ¯j + ∆µ ∂βQαβ (77) I IJ J I I I − r =Λ¯ ∆µ ζ q˜αβv˜αβ + λ pαhα + λ hαpα∂γ pγ − 1 I I I IJ J I I iI + λ2hαpγ ∂γ pα + ζa(ωαβ Ωαβ) pγ αβγ r = Λ¯ ∆µ ζ Qαβv˜αβ + λ QαβHαβ (∂αµ¯i)∂βQαβ, − − − (78) ∂γ ωαβ I I I ζ1αβγ + ζ2αβν pν pγ + ζ3(αγδpδpβ βγδpδpα) where we have introduced phenomenological coupling coef- − 2 − I ficients ν , γ , λI, iI. They are related to coefficients with the +ζ4(δαγ pβ δβγpα) . (84) 1 1 − same name in the polar case given in equations (55)–(59), but correspond here to a nematic system and do not have the same Here, we have introduced new Onsager coefficients that values. The total stress is given by describe antisymmetric stresses, angular momentum fluxes and active chiral terms. The dissipative coefficients η , η¯, κ σtot = ρv v +˜σe Pδ + Q H Q H +˜σd,s , and the reactive coefficients ν and ν¯ also exist in nonchiral αβ α β αβ − αβ αγ βγ − βγ αγ αβ 2 2 passive systems and are irrelevant in the hydrodynamic limit. (79) e with the equilibrium stress σαβ defined in (40). The chiral terms described by the coefficients ν3, ν¯3 and ν¯4 are passive and also irrelevant in the hydrodynamic limit. Active I I I I chiral terms are described by the coefficients ζa, ζ1, ζ2 and ζ3. 4.4. Active chiral systems I The active term with coeffient ζ4 also exists in nonchiral active The molecular building blocks of biological systems are chi- systems [42]. a,d ral. In particular, the filaments of the cytoskeleton form helical What are the roles of the new variables Ωαβ, σαβ and structures along which molecular motors move. As a conse- d a,d Mαβγ? The total antisymmetric stress depends on σαβ and quence active processes that generate forces and movements d Mαβγ through equation (50). Therefore both the dissipative at molecular scales are chirally asymmetric. This implies that antisymmetric stress and the angular momentum flux enter the in the constitutive material equations terms are allowed that momentum conservation equation (3), which generalizes the are chirally asymmetric. Such terms can involve the totally Stokes equation and determines the velocity field. These new antisymmetric pseudo tensor αβγ which can couple antisym- variables thus capture the effects of active chiral processes metric tensors with vectors. on the flow field. Using the angular momentum conservation Active chiral processes generate torques. More precisely, equation (6) and the definition of the dissipative antisym- they introduce torque dipoles in the active gel which con- metric stress (50), one obtains an equation for the angular tribute to fluxes of angular momentum. To capture the phys- momentum density (appendix C.2) ics of such active torque dipoles, we include the variables describing intrinsic rotations and angular momentum fluxes a,d ∂tlαβ + ∂γ (vγ lαβ) (Ωαlβ Ωβlα)= 2σ . (85) in the discussion of active chiral systems [42, 55]. We thus − − − αβ consider an active chiral polar fluid and focus on the pairs of Therefore, the dissipative antisymmetric stress σa,d provides d,s I I conjugate variables σ˜αβ v˜αβ, Dpα/Dt hα, r ∆µ , a source for the angular momentum density and influences d ↔d,a ↔ ↔ M ∂γ ωαβ/2 and σ (Ωαβ ωαβ). the intrinsic rotation rate Ω . In passive fluids or liquid αβγ ↔ αβ ↔ − αβ The constitutive relations then read crystals, the intrinsic rotation rate approaches the vorticity field, Ωαβ ωαβ in the hydrodynamic limit and Ωαβ ωαβ d,s ν1 2 I I − σ˜ = 2ηv˜αβ + ( pαhβ + pβhα pγ hγ δαβ)+ζ ∆µ q˜αβ becomes irrelevant. As we show now, the intrinsic rotation αβ 2 − 3 (80) mismatch Ω ω can persist in active chiral systems. αβ − αβ

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We consider the limit where the angular momentum den- and using a redefined angular momentum density lαβ and sity lαβ is small and can be neglected in equation (85) as angular momentum flux Mαβγ with lαβ = 0 and Mαβγ = 0, compared to terms proportional to the rotational viscosity η the corresponding redefined stress tensor σ becomes sym- a,d αβ that contribute to σαβ. In that case equation (85) reduces to metric [39], see appendix C.3. With these redefinitions one a,d σαβ = 0. From equation (81) we then obtain an equation for obtains a hydrodynamic theory in which the stress tensor the intrinsic rotation rate mismatch: is symmetric. In the case of passive systems with ∆µI = 0, this theory with symmetric stress tensor σαβ contains the η¯ ν2 Ωαβ ωαβ = pγ ∂γ ωαβ ( pαhβ pβhα) same physics on large scales as the theory presented in the − − 4η − 4η − previous sections that is based on asymmetric stress ten- I ν3 ζa I sors. However, differences can arise on smaller scales. αβγ hγ ∆µ pγ αβγ . (86) − 2η − 2η Importantly, for an active system with active chiral terms differences between both approaches can arise even in the The intrinsic rotation mismatch corresponds to differences hydrodynamic limit. This is because a theory based on a between the intrinsic rotation rate and the local rotation vor- symmetrized stress tensor misses the physics of internal ticity. Here, terms on the right-hand side correspond to differ- rotations and it does not contain the internal rotation mis- ent processes generating a rotation mismatch when the fluid match Ωαβ ωαβ. However this term can be critical: while is out of equilibrium. The last term corresponds to an active for passive systems− this internal rotation mismatch vanishes term describing chemical reactions driving such rotation devi- in the hydrodynamic limit, this is not so in the presence of ations. Note that equation (86) shows that for active systems active chiral processes where it can persist at long times as I with ζa nonzero, the rotation mismatch Ωαβ ωαβ can persist described by equation (86). in the hydrodynamic limit and for long times.− This is a novel and original feature of active chiral systems. In situations where the intrinsic rotation rate mismatch is 4.5. Fluctuations and noise not experimentally accessible or not of interest, equation (86) We have presented a systematic procedure to construct consti- can be used to eliminate the intrinsic rotation rate Ωαβ from tutive equations for polar, nematic and visco-elastic active gels the constitutive equations. The equations for the rate of polar- based on the principles of irreversible thermodynamics and ity change and for the dissipative part of the angular momen- Onsager relations governing conjugate thermodynamic fluxes tum flux then take the form and forces. In this framework, one can systematically add the 2 2 2 Dpα 1 ν pγ pγ + 4ν ν effects of fluctuations, taking into account the fluctuation dis- = ( + 2 3 )δ 2 p p h ν p v˜ Dt γ 4η αβ − 4η α β β − 1 β αβ sipation theorem that applies at thermodynamic equilibrium. 1 eff eff Realistic biological systems operate far from thermodynamic ν¯2 ν¯3 ν¯4 + pγ pβ∂γ ωαβ γβαpδ∂δωγβ γβδpα∂δωγβ equilibrium and the fluctuation dissipation theorem is broken. 2 − 2 − 2 I,eff I I I I I However introducing noise according to rules from irrevers- + λ pα∆µ + λ1∆µ pα∂γ pγ + λ2∆µ pγ ∂γ pα (87) ible thermodynamics is a useful starting point [56]. Far from equilibrium specific arguments are needed to characterize the η¯2 ν¯eff properties of the nonequilibrium noise [57]. Md = κδ p p ∂ ω + 2 p ( p h p h ) αβγ γδ − 4η γ δ δ αβ 2 γ α β − β α The linear response relations (54) can be extended to eff include noise terms which then leads to noisy hydrodynamic +¯ν pγ hδαβδ 3 equations. The extended linear response reads [32] I I I,eff I +¯ν4αβγ pδhδ + ζ1∆µ αβγ + ζ2 ∆µ αβν pν pγ I I Jn = OnmFm + ηn(t). + ζ3∆µ (αγδpδpβ βγδpδpα) (89) − m I I + ζ4∆µ (δαγ pβ δβγpα), − (88) The noises ηn(t) are Gaussian stochastic variables with average η (t) = 0 and variance where we have introduced the effective coefficients n eff eff I,eff I I d ν¯2 =¯ν2 ην¯ 2/(2η ), ν¯3 =¯ν3 ην¯ 3/(2η ), λ = λ + ν3ζa/η ηn(t)ηm(t ) = 2kBTOnmδ(t t ) (90) I,eff − I I − − and ζ2 = ζ2 ηζ¯ a/(2η ). Eliminating the rotation mismatch d − where O =(Omn + Onm)/2 is the symmetric part of Omn thus leads to a renormalization of phenomenological coeffi- mn and the fluctuation dissipation theorem has been used to set cients. In addition, new anisotropies are introduced in the term the noise amplitude. Higher order correlations follow from the proportional to in equation (87) and proportional to hβ ∂δωαβ Gaussian statistics of these variables. in equation (88). These terms are allowed by symmetry in We illustrate this principle by adding appropriate noise equations (82) and (83) but were neglected for simplicity. The terms to the constitutive equations of a polar active gel (55)– corresponding antisymmetric part of the total stress is given (59) which read [56] a e,a 1 1 d by σαβ = σαβ + 2 ( pαhβ pβhα)+ 2 ∂γ Mαβγ. − d,s el ν1 2 I I It is also possible to derive the constitutive equations of σ˜ = σ + 2ηv˜αβ + ( pαhβ + pβhα pγ hγ δαβ)+ζ ∆µ q˜αβ αβ αβ 2 − 3 an active chiral fluid using symmetric stresses only. In fact, νi 2 (σ) (91) ( pβ∂αµ¯i + pα∂βµ¯i pγ ∂γ µ¯iδαβ)+ξ using a specific redefinition g of the momentum density − 2 − 3 αβ

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( ) Dpα 1 ( p) k j k el I I ξα (t, x)ξ (t , x ) = 2kBTκ δαβδ(t t )δ(x x ) = hα ν1pβv˜αβ χpβσαβ + λ pα∆µ β − − − Dt γ1 − − (103) I I I I + λ ∆µ pα∂γ pγ + λ ∆µ pγ ∂γ pα 1 2 i( j) (u) i I I i i ( p) ξα (t, x)ξβγ (t , x ) = kBTχ [δαβpγ + δαγ pβ + λ ∆µ pγ ∂αpγ +κ ∂αµ¯ + ξ 3 α (92) 2 δβγpα]δ(t t )δ(x x ) i ij j i el i − 3 − − jα = Λ ∂αµ¯ χ pβσαβ ν pβv˜αβ − − − i 2 iI I i el kBTχ [δαβ∂γ + δαγ ∂β δβγ∂α]δ(t t )δ(x x ) + λ pα∆µ + χ1∂βσαβ − 1 − 3 − − (104) + iI∆µIp ∂ p + iI∆µIp ∂ p κih + ξi( j) 1 α γ γ 2 γ γ α− α α (93) k( j) I(r) kI kI ξα (t, x)ξ (t , x ) = 2kBT λ pα + 1 pα(∂γ pγ ) I ¯ IJ J I I el I r = Λ ∆µ ζ q˜αβv˜αβ ψ q˜αβσαβ+λ pαhα kI − − +2 p γ (∂γ pα) + λI h p ∂ p + λI h p ∂ p 1 α α γ γ 2 α γ γ α δ(t t )δ(x x ) (105) × − − + λI h p ∂ p λiIp ∂ µ¯ iI(∂ µ¯i) p ∂ p 3 α γ α γ − α α i − 1 α α γ γ iI i I(r) (u) I(r) I (∂αµ¯ ) pγ ∂γ pα + ξ ξ (t, x)ξ (t , x ) = 2k Tψ q˜αβδ(t t )δ(x x ). (106) − 2 (94) αβ B − − The noisy constitutive relations characterized by the noise Du˜αβ el χ 2 = v˜αβ Γσ + ( pαhβ + pβhα pγ hγ δαβ) correlations equations (101) (106) together with the conser- Dt − αβ 2 − 3 – vation laws provides stochastic dynamic equations for active + ψI∆µIq˜ αβ gels in the vicinity of thermal equilibrium. Note that some i χ i i 2 i noise correlations depend on the polarity field pα(x, t) and ( pα∂βµ¯ + pβ∂αµ¯ pγ ∂γ µ¯ δαβ) − 2 − 3 the noise is therefore state dependent or multiplicative. In i i 1 i (u) this case, the specific conventions that are used when defining + χ1(∂α∂βµ¯ ∂γ ∂γ µ¯ δαβ)+ξαβ . (95) − 3 time integrals of these stochastic variables become relevant. (σ) Using Stratonovich or Ito conventions requires the addition of The Gaussian noises are the stress fluctuation ξ , the polar- αβ compensating drift terms in the stochastic differential equa- ( p) i( j) ity fluctuations ξα , the current fluctuations ξα , the reaction tions in order to satisfy the fluctuation dissipation theorem. I(r) (u) rate fluctuations ξ and the strain fluctuations ξαβ. They all When using the isothermal convention, such compensating vanish on average and have the correlations terms do not occur in the Langevin equations [58–60].

(σ) (σ) 2 ξ (t, x)ξ (t , x ) = 2k Tη δαγ δβδ + δαδδβγ δαβδγδ 5. Nonequilibrium processes in cells and tissues αβ γδ B − 3 δ(t t )δ(x x ) (96) 5.1. Active gels in cell biology − − The hydrodynamic theory presented in the previous sec- ( p) ( p) 2kBT ξα (t, x)ξβ (t , x ) = δαβδ(t t )δ(x x ) (97) tions provides a general framework to capture key aspects γ1 − − of material properties and the dynamics of biological matter. k( j) l( j) kl On mesoscopic scales, cellular materials and in particular the ξ (t, x)ξ (t , x ) = 2k TΛ δαβδ(t t )δ(x x ) α β B cell cytoskeleton can be considered as active gels. Such sys- − − (98) tems can exhibit unconventional material properties of active I(r) J(r) IJ ξ (t, x)ξ (t , x ) = 2k TΛ¯ δ(t t )δ(x x ) (99) B − − matter. In situations where the filaments are short compared to mesoscopic length scales of cellular structures, a hydro- (u) (u) ξ (t, x)ξ (t , x ) = 2kBTΓ[δαγ δβδ + δαδδβγ dynamic or continuum approach can capture the key features αβ γδ 2 of intracellular flows and force generation. Actin filaments δαβδγδ δ(t t )δ(x x ). −3 − − together with myosin molecular motors and many associated (100) proteins form very dynamic gels with contractile properties. There are further dissipative couplings which imply the Early continuum approaches to cellular gels that are perme- cross-correlations ated by a solvent were based on two phase descriptions [13]. In many situations, filaments are organized in anisotropic ( p) (u) ξα (t, x)ξ (t , x ) = kBTχ[δαβpγ arrangements that can be polar or nematic. Here, the full prop- βγ 2 erties of active polar or active nematic gels become relevant. + δαγ pβ δβγpα]δ(t t )δ(x x ) − 3 − − Cell locomotion and cellular shape changes are key exam- (101) ples for processes that are governed by the mechanics and ( p) I(r) I I hydrodynamics of active gels. Cells crawling on a solid sub- ξ (t, x)ξ (t , x ) = 2k T λ pα + λ pα(∂γ pγ ) α B 1 strates such as fish keratocytes have been studied extensively +λI p (∂ p )+λI p (∂ p ) 2 γ γ α 3 γ α γ [2, 52, 61–63]. The emerging motion of such cells is a com- δ(t t )δ(x x ) (102) plex biophysical process involving the interplay of adhesion, × − −

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Figure 3. The cell cortex can be represented as a thin layer of an active gel of thickness h below the cell membrane. (a) In the simplest case, the average filament polarity vector pα is normal to the surface. By integrating over the thin dimension z one can obtain an effective act two-dimensional hydrodynamic theory of the system in the x − y plane with an isotropic active tension T δij in the plane. (b) In the general case, a normal component p and a tangential component p of average filament orientation with respect to the surface exist. In this act ⊥ case, the active tension Tij is anisotropic. active contractile stresses in an actomyosin gel that undergoes gel in which the dynamics of filaments maintains a preferred polymerization and depolymerization. Importantly, all these film height h, vz becomes proportional to the net flow veloc- processes are organized in space and time. Continuum theo- ity of material leaving the gel and entering the cytosol after ries of active gels allow to capture the essential force balances depolymerisation. involved in lamellipodium protrusion and can explain the ori- In order to discuss the dynamic equations of the active film gin of retrograde flows as a consequence of contractile active we write the constitutive equations of an active gel in the long stresses in the polarized actomyosin gel [52]. time limit (62) in d = 2. For simplicity, we consider the case The actomyosin cell cortex is an important intracellular where the system is isotropic in the plane of the film and we structure where continuum approaches to active gels have therefore do not introduce a polarity field pα and the conjugate been useful for describing cytoskeletal deformations and field hα in the plane, corresponding to figure 3(a). Because flows [53, 64–70]. The cell cortex is a thin layer of actomyo- the active gel turns over and material can escape in the third sin gel associated with the cell membrane [4]. The height of dimension the active gel is compressible in two dimensions. the layer is small compared to its lateral extents, and an effec- The film tension tensor can be defined as an integral of the tive two-dimensional theory of the dynamic cell cortex can be stress over the film thickness

obtained [53, 64, 71] with equations for the densities of actin h filaments and actin monomers that are contained in the bal- Tij = dzσij. (108) ance equations equation (8). 0 To illustrate the basic ideas, we consider a thin layer of an The stress tensor in two dimensions Tij, where i, j = x, y are active gel on a solid support. The active film equations can indices in two dimensions has units of energy per area. The be found most easily by writing the active gel theory in two constitutive equation for the active film read dimensions. Alternatively, one can derive these equations tak- 1 ¯ 2d ing the thin film limit of a three dimensional gel layer of thick- Tij = 2η(vij vkkδij)+ηbvkkδij +(ζ∆µ P )δij. (109) ness h v /k, see figure 3. This thickness is set by a balance − 2 − ∼ p of filament polymerisation with velocity vp and depolymerisa- Here, we have introduced the two-dimensional shear and 2d tion which eliminates filaments at a rate k. The planar geom- bulk viscosities η¯ and ηb, respectively, P is the two dimen- etry below the membrane implies that an axis of anisotropy sional pressure and the two-dimensional active tension is is provided by the normal to the surface. In the simplest case, Tact = ζ¯∆µ. Material balance of the film can be expressed as the filaments are on average oriented normal to the surface, as equation for its two dimensional actin density ρ see figure 3(a). In general, one can distinguish a normal and a dρ tangential component of the average polarity, see figure 3(b). = ρ∂kvk k(ρ ρ0) (110) While in three dimensions we usually consider the active dt − − − material to be incompressible, no incompressibility condition where d/dt denotes a convected time derivative. The film exists in the effective film equations in two dimensions. This turns over at a rate k by polymerization and depolymeriza- is because material can be exchanged between the two dimen- tion and has a steady state area density ρ0. The film pressure 2d sional film and its surroundings. Integrating the incompress- is P = χ(ρ ρ0)/ρ0, where χ is a compressibility in two − ibility condition ∂αvα = 0 along the z-axis perpendicular to dimensions. On times long compared to the turnover rate k, the film, we have the density is stationary, dρ/dt 0 and we have vz h(∂xv¯x + ∂yv¯y) . (107) 2d χ − P ∂kvk. (111) −k Thus, the resulting average two-dimensional flow field v¯i with i = x, y is not divergence free. The divergence ∂iv¯i in two Therefore, the 2d pressure effectively renormalizes the bulk dimensions defines a velocity vz that for a gel with constant viscosity in equation (109) and we have density corresponds to the rate of height changes. For a thin

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1 processes therefore modify and control active and passive T = 2η(v v δ )+¯ηv δ + Tactδ (112) ij ij − 2 kk ij kk ij ij material properties in the cell cortex. This in turn implies that couplings and feedbacks exist between active mechanical with the effective two dimensional bulk viscosity processes in the material and chemical processes. These cou- η¯ = η + χ/k. The force balance can then be written as b plings can arise through mechano-sensing, but also through ∂jTij = γvi (113) advective transport where the regulators that control molecular processes in the gel are transported by gel flows [75 77]. where γ is a coefficient describing friction forces with respect – From the point of view of physics, morphogenesis in biologi- to a substrate. cal systems is fundamentally a mechano-chemical process. A simple case is an effectively one-dimensional geom- On the scale of a cell, such mechano-chemical processes cre- etry, where gradients are all in x − direction and vanish in the ate patterns and cell flows that break cellular symmetries and y − direction. In this case, T =(η +¯η)∂ v + Tact. Using xx x x can lead for example to the emergence of cell polarity and cell ∂ T = γv , we finally obtain the hydrodynamic equation for x xx x chirality [64, 78]. On a multicellular level, structures such as the flow velocity in x − direction tissues, organs and eventually organisms form via the collec- act 2 ∂xT = γvx (η +¯η)∂x vx. (114) tive organization of large numbers of cells in space and time. − The dynamics of tissues also involves a coupling of active It is interesting to consider the simple case where the active mechanical events with regulatory pathways and chemical stress is piecewise constant with Tact = T for x < 0 and A signals. Thus, tissue morphogenesis and the development Tact = T for x > 0. In this case, ∂ Tact =(T T )δ(x) and P x P A of organisms from a fertilized egg are examples of specific the flow field is given by − mechano-chemical patterning processes that serve to build tis-

/2 x / sue-scale structures. These processes are guided by upstream v =(T T ) e−| | . (115) x P − A η +¯η genetic programs that define the dynamic rules of the cellular behaviors, ensuring that appropriate structures are formed at Thus, gradients of active stresses induce flow patterns in the the right place and time. active film governed by the hydrodynamic length Mechano-chemical pattern and structure formation pro- 1/2 vides a paradigm for biological morphogenesis. This mech η +¯η = . (116) anism of pattern formation is qualitatively different from γ classical concepts, such as pattern formation via reaction– The magnitude of these flows increases linearly with this diffusion processes and Turing patterns. These are essen- length scale. tially chemical in nature and do not take into account force Actomyosin flows that serve to establish cell polarity in balances and actively generated material flows and deforma- the Caenorhabditis elegans embryo can be described within tions [79–81]. Hence, mechano-chemical self-organization this framework, by considering the spatial distribution of represents a novel class of pattern formation processes in active stress generated by myosin molecular motors in the Physics. General and simple features of mechano-chemical thin cortical layer together with viscous stresses as well as pattern formation can be best studied in minimal systems and external friction forces acting on the cortex when it moves simplified models. An example consists of a thin active gel relative to the outer egg shell [64]. Flows within an actin ring or fluid in which a regulator of active stress diffuses and is powering zebrafish epiboly can be understood with a similar convected by flows. The behavior of this system is character- approach [72]. Myosin induced cortical flows also provide a ized by a Peclet number that measures the ratio of diffusive physical mechanism for stable bleb amoeboid motion [73] and convected transport. Steady states exist in which con- and for the adhesion independent migration of cells in narrow centration and stress profiles are homogeneous and no flows tubes [74]. Furthermore cell division is driven by actin fila- occur. Homogeneous quiescent states can become unstable ments assembling into a contractile ring [53, 70, 71]. Actin with respect to wave-like perturbations of the regulator con- filaments are aligned along the cell equator, and this align- centration field. Increased concentrations of the regulator ment can be generated by actomyosin flows converging upon lead to increased active stress, and beyond a critical coupling the equatorial region and compressing the network which in strength between regulator concentration and active stress, turn will generate alignment. A continuum theory of an active flows can arise spontaneously. These flows transport stress nematic gel can capture this flow-alignment coupling and the regulators towards regions where stress is already increased. corresponding dynamics as measured in the one cell stage This gives rise to an instability. Beyond this instability, the Caenorhabditis elegans embryo [70]. system reaches a steady state exhibiting a spatial concentra- Active stresses in the actin cytoskeleton of cells are con- tion profile and a flow patterns [76]. Introducing two dif- trolled by regulatory pathways. These pathways are based on fusing species which activate and inhibit active stress can sets of regulatory molecules that typically diffuse in the cell. lead to oscillatory patterns. Such dynamic instabilities might Specific regulators control molecular processes in the acto- also be at the heart of transient aggregations of myosin into myosin gel. They can for example regulate myosin density so-called foci that are often observed in highly contractile and thereby control overall myosin activity and the amount actomyosin cortices during the development of organisms of myosin-based force generation. In general, regulatory [82–84].

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5.2. Continuum theories of tissues d 1 + τ σ =¯ηvc P , (120) dt γγ − h The hydrodynamic equations governing the dynamics of active gels discussed in this review are generic and depend where the coefficient η¯ describing the pressure dependence only on the symmetries and conservation laws that character- of k k plays the role of the bulk viscosity. The relaxation d − a ize a given system. Beside the cell cytoskeleton, which is a time for isotropic stresses is τ =¯η/χ. biological example of an active gel, tissues are also soft active When considering the anisotropic stress, cell division and materials. Furthermore, tissues can exhibit vectorial, nematic apoptosis events generally introduce force dipoles that can or chiral asymmetries similar to the asymmetries encountered relax existing elastic stresses in the cytoskeleton. As a result, the equations for active gels can also be applied to tissue dynamics. However there are a D c σ˜αβ = 2Evαβ n(ddkd + daka)q˜αβ. (121) few specific considerations. Dt − Tissues are assemblies of many cells which adhere to each Here, dd and da denote the magnitudes of force dipoles asso- other and which are often also associated to an extracellular ciated with individual division and apoptosis events, see matrix. The extracellular matrix is an extracellular soft poly- equation (69). The traceless tensor q˜αβ describes the average meric material, the components of which are secreted by cells orientation of these force dipoles. Taking into account that [85, 86]. Cells exhibit anisotropies such as cell polarity or they existing stress anisotropies bias the orientation of division may take elongated shapes, which correspond to a nematic events, we use q˜ q˜0 +˜σ /σ , where q˜0 captures αβ αβ αβ 0 αβ anisotropy. In a continuum approach, tissue anisotropies can stress independent anisotropies of division and apoptosis thus be captured by nematic or polar order parameter fields events. We then arrive at a viscoelastic constitutive equa- [25, 87–89]. In addition, tissues generate active stresses and tion for the anisotropic stress flows. Cells may exert contractile stresses which often are mediated by the cellular acto-myosin cytoskeleton. For exam- D 1 + τ σ˜ = 2ηv˜c σ q˜0 , (122) ple, stress fibers in cells can contract and induce a force dipole a Dt αβ αβ − 0 αβ and thus an active stress in the tissue [90]. Growth processes which takes the form of an active visco-elastic material with and in particular cell growth is typically associated with iso- 1 relaxation time τ − = n(d k + d k )/σ of anisotropic tropic active stresses [22]. Cell division is an anisotropic pro- a d d a a 0 stresses, viscosity η = Eτ and an active stress of magnitude cess. Mechanical events during cell division are active and a σ0. typically contribute to anisotropic active stresses [24]. Many tissues are sheet-like epithelia. These are two An important feature of tissues is cell division and cell dimensional cell layers which play a key role in many devel- death. Cell death often occurs via a regulated process called opmental processes [91, 92]. An advantage of the study of apoptosis. A key characteristics of tissues is the number den- epithelia is that the cell shape can often be directly observed. sity n(x) of cells which obeys the balance equation The local average cell shape can be characterized by a trace- c ∂tn + ∂α(nv )=(kd ka)n, (117) less symmetric cell shape tensor Q˜ which in two dimen- α − αβ sions is a 2 2 matrix. If both the cell shape tensor and the where kd and ka denote the average division and apoptosis × c rates in local volume elements. Here, we have introduced the tissue deformation rate vαβ are observed, this provides infor- cell velocity vc . mation about the decomposition of tissue shear deformations α c The material properties of a tissue can be elastic or visco- v˜αβ in contributions from cell shape changes and from cell elastic. Cell turnover and cell divisions can fluidify an elastic rearrangements [23, 28–30, 93] tissue [24]. In order to discuss these properties, we start from DQ˜ αβ a tissue characterized by compression and shear elastic mod- v˜c = + R˜ . (123) αβ Dt αβ uli χ and E, respectively. Using equation (117), the isotropic stress then obeys Here, Rαβ is a traceless symmetric tensor that captures the contributions of tissue shear stemming from cell rearrange- dσ c ments. Using this decomposition in a continuum approach, χ(vγγ (kd ka)), (118) dt − − one can start from a constitutive equation describing elastic where d/dt denotes a convected time derivative. Typically, cell mechanics division and apoptosis rates are regulated by growth factors and related chemical signals. However in general, they also σ˜αβ = 2EQ˜ αβ + ζ∆µ˜qαβ, (124) depend on local stresses and in particular the local cell pres where ζ∆µq˜αβ is an active stress or equivalently a sponta- sure. At homeostatic conditions, kd ka = 0. If such homeo- − neous cell elongation. We then also require a constitutive static condition exist for some pressure Ph, we can expand the division and apoptosis rates to lowest order in the vicinity of material equation for cell rearrangements. Using the simplest the homeostatic state: choice ˜ 1 ˜ 1 Rαβ = τa− Qαβ + γ∆µ˜qαβ, (125) kd ka η¯− (Ph + σ). (119) − where τa is a timescale and γ a phenomenological coefficient, We then obtain a constitutive equation for the isotropic stress we again obtain equation (122). However, in tissues we have

16 Rep. Prog. Phys. 81 (2018) 076601 Review features of a system far from equilibrium. Furthermore, the the possibility to directly measure the cell shape Q˜ αβ which is related to the elastic deformation as separate observables. approach of irreversible thermodynamics discussed here is In typical Maxwell gels this variable is not easily accessible. based on linear response theory and expresses dynamic equations to first order in the thermodynamic forces that drive the Furthermore, the constitutive equation for R˜ in typical αβ dynamics. tissues can be more complex than equation (125) and may Biological systems typically differ from such ideal regimes involve further relaxation processes and memory effects. In in several ways. Cells are mesoscopic systems and a large such cases the tissue rheology becomes complex and can length scale limit can often not be taken. Filaments of the exhibit unconventional features [29, 30]. cytoskeleton can have lengths of a micrometer and more which implies that a continuum limit may emerge only on 6. Discussion scales beyond the cell size. Biological systems also exhibit very important dynamics on intermediate time scales, for We have reviewed a generic approach to active matter based example the process of cell division. Events that take place on on fundamental concepts of irreversible thermodynamics. fixed time scales do not correspond to hydrodynamic modes. Irreversible thermodynamics captures the dynamics of macro- Also, biological systems are examples of matter that is driven scopic systems in the vicinity of thermodynamic equilibrium far from thermodynamic equilibrium. In a typical cell many when only a few degrees of freedom are driven away from variables are dynamic and are kept away from thermodynamic equilibrium. This corresponds to a local equilibrium approx equilibrium, and many of these variables do not correspond to imation. The constitutive equations of active matter which classical hydrodynamic modes. This is particularly clear for we discussed are based on linear Onsager relations which are dynamic cellular processes such as cell polarization, cell divi- rigorously valid in a regime close to equilibrium. In the limit sion and cellular signalling. Hence, to capture important bio- of large length scales, slow hydrodynamic modes govern the logical phenomena a theory based on the generic properties of dynamics of spatially extended systems. Such hydrodynamic hydrodynamic modes needs to be complemented by models modes emerge as a result of conservation laws for mass, for non-hydrodynamic variables. momentum and energy. Furthermore, broken symmetries that Furthermore, some dynamic processes in living cells are are for example associated with polar or nematic order can expected to operate further away from equilibrium that can be also give rise to hydrodynamic modes. Hydrodynamic modes accurately captured by linear response theory. However local are generic and independent of the microscopic details that equilibrium concepts do still have much value when applied underlie a particular system. Therefore, systems that can be to reconstituted cellular systems or when used inside cells. In observed over long times and in the limit of large system sizes fact, many degrees of freedom are equilibrated on time scales exhibit robust behaviours governed by such a hydrodynamic relevant to cellular dynamics. For example, a local temper theory [32, 94]. ature remains a key concept, and temperatures as well as heat The irreversible thermodynamics of hydrodynamic systems produced in biological systems and organisms can be meas- provides a powerful framework for the study of active mat- ured [95, 96]. Key features of many collective processes, such ter. Hydrodynamic modes represent a small number of slow as the assembly of molecular compartments by phase separa- degrees of freedom of the system that are well captured by tion, can in many cases be described using irreversible ther- irreversible thermodynamics. Because hydrodynamic modes modynamics and local equilibrium approximations [97, 98]. are slow, a large number of faster relaxing modes become To capture the essence of the physics of biological dynamics equilibrated when considering dynamics on large lengths and self-organization, it is therefore important for a given bio- scales and in a continuum limit. Thus the local equilibrium logical process to identify the relevant dynamic variables. The approximation becomes valid. Irreversible thermodynamics number of such relevant variables can be significantly larger therefore captures the behaviours of hydrodynamic modes than the number of conventional hydrodynamic modes, but accurately. In real systems one is often also interested in the this number will still be orders of magnitudes smaller than the dynamics of collective modes that relax more rapidly. In such total number of microscopic degrees of freedom. As a result, cases the hydrodynamic approach can be extended to faster models for biological dynamics are often not fully generic but relaxing modes, however in such cases the local equilibrium can depend in important ways on details of biological inter- condition is not always a good approximation. In addition, a action networks and signalling pathways. To conclude, the continuum limit may not be always appropriate to capture the framework described in this review provides a systematic and physics of mesoscopic systems. Noise and fluctuations can be physically coherent approach to non-equilibrium dynamics systematically introduced to the irreversible thermodynam- of active systems that obeys important conservation laws and ics approach to capture fluctuations on mesoscopic scales. basic physical principles. It also provides the correct equilib- However, only in the vicinity of equilibrium is there a rigor- rium physics in the absence of active processes. ous procedure to specify the properties of such noise via the An alternative approach to develop a hydrodynamic the- fluctuation–dissipation theorem. Thus, the conceptual frame- ory of of active matter is to directly provide an expansion of work discussed here has a number of limitations in particular dynamic variables writing all terms allowed by the symmetries when it is applied to active biological systems. One important of a given system [15, 17]. Such a general theory must still be point is that irreversible thermodynamics is valid in the vicin- based on fundamental conservation laws but may not make ity of a thermodynamic equilibrium and may not capture some use of equilibrium thermodynamics. An advantage of such an

17 Rep. Prog. Phys. 81 (2018) 076601 Review approach is that it is not restricted to the regime in the vicin- epithelial spreading and epiboly progression in the develop- ity of a thermodynamic equilibrium. This advantage, however, ing zebrafish embryo [72]. In these cases, active gel theory comes at the cost that there is no longer a systematic procedure was used to characterize the force-balance that underlie the that helps to identify relevant variables that govern the dynam- associated actomyosin dynamics. The structure of myosin in ics, apart from the slow modes stemming from conservation a highly contractile cortex is known to be non-homogeneous, laws and symmetries. Furthermore, a theory constructed far myosin is concentrated into aggregates called myosin foci from equilibrium does not necessarily provide the correct lim- that undergo a pulsatile activity, with new ones forming and its when approaching equilibrium. In general, theories devel- old foci disappearing on a time scale of one minute [83, 84]. oped using more general approaches and theories based on Such processes result from the dynamic interplay of chemical irreversible thermodynamics share many features in common oscillations and active gel dynamics in a mechano-chemical and often exhibit the same physics. An important example is process [84]. Generally, cell dynamics typically emerges from the active stress, which is a central concept to active matter mechano-chemical processes that combine active gels with that was introduced independently using different approaches chemical patterns [81]. [11, 15]. The active matter equations discussed in this review Similar principles operate at the scale of tissues. Chemical can lead to the spontaneous formation of complex structures. patterns emerge from signalling processes and such chemical These include spontaneous flows, topological defects, and patterns can influence mechanical events. A reaction–diffusion spatiotemporal structures such as traveling waves [11, 76, 99]. system consisting of several specific signalling molecules Numerical simulations of active nematic fluids have shown drives digit formation in the mouse [109]. However, if and that low Reynolds number turbulence can arise in an active how growth and deformation feeds back on the pattern form- system [100–102]. ing process in this system is not yet known. Interestingly, the The hydrodynamic theory of active matter outlined here process of positioning skin follicles, the structures of the skin generalizes the classic hydrodynamic theories of passive liq- that grow hairs in humans and feathers in chicken, is depend- uid crystals [6, 34–39] to the case where chemical processes ent on a patterning process where mechanical and regulatory maintain such liquid crystals away from equilibrium. In the pathways cannot be separated. The dermis of the skin, which biological context, active materials are often based on the is a tissue layer that is below the epidermis, appears to undergo cytoskeleton where molecular motors interact with filaments a contractile instability, which causes compression and mecha- and form an active gel. An important property of biological otransduction of β-catenin in the epidermis and hair follicle matter is that it is fundamentally chiral. This results from the specification [110]. We think that theoretical approaches based chirality of biomolecules. Active chiral processes mediated on continuum theories of active matter will play an increasing by motor molecules and chiral filaments contribute to move- role to shed further light on mechanochemical patterning pro- ments and flows that are generated in cells and in tissues. The cesses in biology from the scales of cells to tissues. resulting chirality of cellular processes in turn plays a key role for the breaking of left right symmetry in cells and in devel- – Acknowledgments oping organisms [103–106]. Equations for the hydrodynamics of active matter have We thank Abhik Basu, Andrew Callan-Jones, Jaume been expressed here in cartesian coordinates. Morphogenesis Casademunt, Erwin Frey, Jean-Francois Joanny, Karsten Kruse, in biology can also occur through deformation of thin layers Tom Lubensky, Cristina Marchetti, Jacques Prost, Sriram of active material, such as the cell cortex or two-dimensional Ramaswamy, Ken Sekimoto and John Toner for interesting tissues called epithelia. In that case, the shape of the system is discussions and stimulating interactions. GS was supported by well-described by approximating it by a moving surface. The the Francis Crick Institute which receives its core funding from hydrodynamics of active matter can be generalized to con- Cancer Research UK (FC001317), the UK Medical Research sider flows on a deformable manifold. A new feature is that Council (FC001317), and the Wellcome Trust (FC001317). active internal bending moments generated in the thin layer play a role, in addition to active stresses [107]. The approach to investigate the irreversible thermodynam- Appendix A. Irreversible thermodynamics ics of hydrodynamic systems has been powerful for shedding of multicomponent fluids light on the physical basis of biological processes ranging from the intracellular activities to the dynamics of tissues. A.1. Thermodynamic potentials and thermodynamic variables In the one cell-stage Caenorhabditis elegans embryo, flows that serve to polarize the zygote where shown to be driven by We briefly review the irreversible thermodynamics of mul- inhomogeneities in the distribution of myosin motor proteins ticomponent fluids [33]. We first consider the equilibrium [53, 64]. These flows serve to trigger the formation of a cell thermodynamics of a multicomponent fluid with local con- polarity pattern, by transporting markers of polarity along the centrations ni = Ni/V of molecular species i and mass density N cell surface [75, 108]. Cell division occurs via active contrac- ρ = i=0 mini in a volume element of volume V. The center tion in a contractile ring. Active gel can capture the mechanics of mass velocity is given by of the cell cortex and the resulting cell shape changes [71]. 1 At the multicellular scale, actomyosin flows into a supra- v = m N vi , α M i i α (A.1) cellular ring-like structure generate the forces that drive i

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N i A.2. Translation and rotation invariance where M = i=0 Nimi is the total mass and vα the local average velocity of molecules of type i. The total momentum of We consider the change in free energy of a finite volume the volume element is G = M , the momentum density is α vα element under infinitesimal translations xα = xα + δxα, gα = ρvα. with constant translation vector δxα. Defining translations As a result of Galilei invariance of the microscopic equa- as n (x )=n (x ), we have δn (x )=n (x ) n (x ), or i α i α i α i α − i α tions of motion, the thermodynamics of a local element of δni(xα)=ni(xα δxα) ni(xα). This argument applies to volume V can always be studied in the the center of mass all variables. The− free energy− can then be written as reference frame in which vα = 0, the momentum vanishes 3 tot ∂f0 G = 0 and the free energy is denoted F0. The free energy δF = d x(vαδgα + µ δni sδT)+ dSα( f δxα + δni), α i − ∂(∂ n ) with centre of mass motion then has the form α i (A.10) G2 where the second term takes into account that the boundary F(T, V, N , G )= α + F (T, V, N ), (A.2) i α 2M 0 i is displaced with the system. The change δF given by equation (A.10) must vanish in the absence of external forces where N denotes the number of particles of species i and T i because of translation invariance. Using δn = δx ∂ n , denotes temperature. Similarly, we define the free energy den- i − α α i 2 δgα = δxβ∂βgα and δT = δxα∂αT , we have sity f (T, ni, gα)=gα/2ρ + f0(T, ni), with f0 = F0/V. We − − then have 3 tot δF = d x (∂βvα)gα +(∂βµi )ni + s∂βT δxβ ∂F ∂f vα = = . (A.3) tot ∂f0 ∂Gα ∂gα + dS ( f v g µ n )δ ∂ n δx . V,Ni,T ni,T α γ γ i i αβ β i β − − − ∂(∂αni) We define the following thermodynamic quantities: (A.11) We identify the equilibrium stress, ∂F0 (0) P0 = = f0 + µi ni (A.4) − ∂V − e ∂f0 T,Ni i σαβ =(f0 µini)δαβ ∂αni. (A.12) − − ∂(∂βni) (0) ∂F0 ∂f0 Translational invariance then imposes the Gibbs–Duhem µi = = (A.5) ∂Ni ∂ni relation T,V,Nj T,nj ∂ σe =(∂ µ )n + s∂ T β αβ α i i α (A.13) ∂F0 ∂f0 − S = = V , (A.6) − ∂T − ∂T that ensures that δF = 0 for pure translations. The Gibbs– V,Ni ni Duhem relation shows that the equilibrium stress is divergence (0) where P0 is the hydrostatic pressure and µ i are the chemical free at thermodynamic equilibrium when ∂αµi and ∂αT van- potentials of a homogeneous system. ish. However, in an out of equilibrium situation, local equilib- Inhomogeneous systems which are locally at equilibrium rium stresses are not force balanced. In addition to translation can be captured using a free energy density f0, where we also invariance, F also obeys rotation invariance. Rotation invari- include contributions of concentration gradients to the free ance of the free energy implies that the equilibrium stress is e e energy. The free energy reads symmetric, σαβ = σβα and the equilibrium angular momentum flux Me and the antisymmetric part of the equilibrium 2 αβγ 3 gα e,a F = d x + f (T, n , ∂ n ) , (A.7) stress σαβ vanish. 2ρ 0 i α i 2 where f (T, ni, ∂αni, gα)=gα/2ρ + f0(T, ni, ∂αni) is the total A.3. Rate of free energy changes free energy density. In inhomogeneous systems, we generalise the definition of the chemical potential as µi = δF0/δni and We now express the balance of free energy in several steps. thus write The change of free energy in a volume of fixed shape (non- moving boundary) can be expressed as ∂f0 ∂f0 µi = ∂α . (A.8) ∂n − ∂(∂ n ) dF 3 1 2 tot 1 2 i α i = d x ∂t ρv + µ + miv ∂tni s∂tT dt 2 i 2 α − We also introduce the total chemical potential µtot = δF/δn i ∂f with + dS 0 ∂ n α ∂(∂ n ) t i α i 1 3 tot ∂f0 tot 2 = d x vα∂tgα + µi ∂tni s∂tT + dSα ∂tni µi = mivα + µi. (A.9) − ∂(∂ n ) −2 α i In the case of inhomogeneous systems, we also need to = d3x(v ∂ σtot + µtot(r ∂ Ji ) s∂ T) α β αβ i i − α α − t generalise the concept of equilibrium pressure which becomes ∂f0 an equilibrium stress. This equilibrium stress follows from + dSα ∂tni. (A.14) ∂(∂αni) translation invariance considerations.

19 Rep. Prog. Phys. 81 (2018) 076601 Review

We can now to use partial integrations in order to create s where σαβ =(σαβ + σβα)/2, vαβ =(∂αvβ + ∂βvα)/2 and boundary contributions that correspond to the total work per- ωαβ =(∂αvβ ∂βvα)/2. Note that a contribution corre formed at the surface. We first write sponding to the− mechanical work of torques occurs at the dF boundary. = d3x( (∂ v )σtot + µtotr +(ji + n v )∂ µtot s∂ T) dt − β α αβ i i α i α α i − t tot tot i ∂f0 A.5. Balance of free energy, entropy and energy + dSα vβσβα µi Jα + ∂tni . (A.15) − ∂(∂αni) The surface term of equation (A.19), defines the free energy Using the Gibbs Duhem relation (A.13) we then have f f – flux Jα = jα + f vα with

dF 3 tot tot i tot e = d x( (∂βvα)σαβ + µi ri + jα∂αµi vα∂βσαβ f i 1 π ∂f0 dni dt − − jα = vβσβα + µijα Mβγαωβγ . (A.20) − − 2 − ∂(∂αni) dt vαgβ(∂αvβ) svα∂αT s∂tT) − − − From the bulk term, we obtain the rate of free energy produc- tot 1 2 i ∂f0 + dSα vβσ (µi miv )( j + nivα)+ ∂tni βα − − 2 α ∂(∂ n ) tion which reads α i 3 tot e = d x( ∂βvα(σαβ + vαgβ σαβ) s e 1 d i dT − − θf = vαβ(σαβ σαβ) Mαβγ ∂γ ωαβ + µiri + jα∂αµi s . − − − 2 − dt + µ r + ji ∂ µ s(∂ T + v ∂ T)) i i α α i − t α α (A.21) tot e 1 2 + dSα vβ(σ σ )+vα( ρv The flux of free energy is related to the heat flux as βα − βα 2 Q e f e jα = jα jα, where jα is the energy flux in the barycentric i ∂f0 − µini) µijα + ∂tni . (A.16) reference frame. The source of free energy θf can be related − − ∂(∂αni) to the entropy production rate θ per unit volume. Using Using the definition of the equilibrium stress (A.12) in the e f s equations (2) with (14), (15) and Jα = Jα + TJα, we have surface terms, we finally obtain ∂ (Ts)+∂ (TJs )= θ and jQ = Tjs t α α − f α α dF dT s 3 e i Tθ = θf Jα∂αT s∂tT = d x( ∂βvα(σαβ σαβ)+µiri + jα∂αµi s ) − − − dt − − − dt (A.22) Q ∂αT dT = θf j s . i ∂f0 dni − − α T − dt + dSα vβσβα vαf µijα + , − − ∂(∂αni) dt (A.17) We thus obtain the entropy production rate per unit volume tot θ with where σαβ = σαβ + vαgβ and we have introduced the convected time derivative d/dt = ∂t + vγ ∂γ. The final expression M given in (A.17) has a surface contribution to free energy changes. Tθ = v (σs σe )+ rI∆µI αβ αβ − αβ This surface term, except for the free energy convection f v , I=1 − α corresponds to the work performed at the boundary of the sys- N 1 ∂ T tem. The bulk terms reflect the local sources of free energy. ji ∂ µ¯ + Md ∂ ω jQ α , (A.23) − α α i 2 αβγ γ αβ − α T i=1 A.4. Angular momentum conservation I I where ∆µ = i ai µ¯i. Furthermore, we have expressed the From rotational invariance of the free energy it follows that dissipation rate in terms of the chemical potential differences both the antisymmetric part of the equilibrium stress and the µ¯i = µi µ0mi/m0 relative to the solvent i = 0. The entropy e,a − 0 equilibrium angular momentum flux vanish: σαβ = 0 and density s = ∂f /∂T obeys the balance equation (14) with e d,a d − Mαβγ = 0. However, deviatoric parts σαβ and Mαβγ could entropy source θ and entropy flux in principle exist in nonequilibrium conditions. For the simple Q s jα case where the density of intrinsic angular momentum lα can Jα = svα + . (A.24) be neglected, angular momentum conservation then implies T The energy is conserved such that the energy density d d,a ∂γ M = 2σ . (A.18) αβγ αβ ∂f e = f T 0 (A.25) This allows us to eliminate the antisymmetric stress from − ∂T equation (A.17). We then have with energy flux

dF 3 s e 1 d e f Q = d x( vαβ(σ σ ) M ∂γ ωαβ + µiri J = ev + j + j (A.26) dt − αβ − αβ − 2 αβγ α α α α dT obeys the conservation law (2). + ji ∂ µ s ) α α i − dt 1 A.6. Dynamic equations + dS v σ v f µ ji + Mπ ω α β βα − α − i α 2 βγα βγ We now use the quantities introduced so far to express dynamic ∂f dn + 0 i , (A.19) equations. We obtain a dynamic equation for the temperature ∂(∂αni) dt 20 Rep. Prog. Phys. 81 (2018) 076601 Review

using energy conservation. The conservation equation (2) for diffusive fluxes are characterised by the coefficients Γi. The the energy density e = f T∂f0/∂T implies chemical reaction kinetics is described by the symmetric − matrix MIJ of coefficients that specify the reaction rates as ∂e ∂e ∂e ∂e e I ∂tT + ∂tni + ∂t∂αni + ∂tgα = ∂αJα. well as a coupling γ to the divergence of the flow field which ∂T ∂ni ∂(∂αni) ∂gα − (A.27) captures pressure generation by chemical reactions. The coefficient κ¯ describes heat conductivity. Interestingly, there can Note that because of Galilei invariance ∂e/∂g = v . We α α exist a dissipative coupling κ generating a contribution to the then have a antisymmetric stress σαβ =(κ/2)∂γ ∂γ ωαβ, which in simple ∂e ∂e f Q fluids is irrelevant in the hydrodynamic limit and thus usu- c∂tT = ∂tni ∂t∂αni vα∂tgα ∂α(evα + jα + jα) −∂ni − ∂(∂αni) − − ally neglected. Using σtot = σ ρv v , ∆µI = aIµ¯ (A.28) αβ αβ − α β i i i and µ¯i = µi µ0mi/m0, equations (A.29) and (A.30) provide where c = T∂2f /∂T2 is the specific heat. − − 0 a dynamic description of the multicomponent fluid if the free Given a state of the system described by the functions ni(r), energy density f is given. T(r) and vα(r), the time derivatives of these state variables can be determined if in addition to the free energy f the quanti tot i Q ties σ = σ ρv v , j , ri and j are known at time t: αβ αβ − α β α α Appendix B. Translation and rotation invariance 1 ∂ v = v ∂ v + ρ− ∂ σ of the free energy in polar and nematic systems t α − β β α β αβ ∂ n = ∂ (n v + ji )+r t i − α i α α i B.1. Translation invariance 1 ∂e i ∂ T = c− (∂ (n v + j ) r ) t ∂n α i α α − i We first consider a polar system with free energy density i 2 f = gα/(2ρ)+f0( pα, ∂βpα, ni). An infinitesimal transla- ∂e i + ∂α(∂β(nivβ + jβ) ri) tion of a volume V to V by a small and constant transla- ∂(∂αni) − tion vector δxα can be described by the displaced positions tot f Q xα = xα + δxα. To linear order in the displacement vector, vα∂βσαβ ∂α(evα + jα + jα) . − − (A.29) 3 3 the change in free energy δF = d xf d xf can be V − V expressed as tot i Q The thermodynamic fluxes σαβ, jα, ri and jα in general are functions (or functionals) of ni, T and vα which obey transla- 3 tot tion and rotation invariance. These functions define themat δF = d x(vαδgα + µi δni hαδpα)+ dSα( f δxα V − erial properties and are called constitutive material relations. ∂f0 Linear response theory can be used to express these consti- + δpβ). (B.1) ∂(∂αpβ) tutive relations to linear order in the vicinity of a thermodynamic equilibrium. Here, the first term corresponds to f f due to the displacement of the fields in the original volume.− The second term A.7. Linear response theory captures the contributions to δF at the translated boundary The description of the system is completed by adding con- when shifting from V to V . i I Expressing translations of the concentration fields with stitutive relations to express the quantities vαβ, jα, r Q constant displacement vector δxα as ni (xα )=ni(xα), and jα in terms of their conjugate thermodynamic forces e I we have δni(xα)=ni (xα) ni(xα). Therefore δni(xα)= σαβ σαβ, ∂αµ¯i, ∆µ and ∂αT using linear response theory. − − n (x δx ) n (x ) δx ∂ n . Similar transfor- To linear order, we write i α − α − i α − α α i mations hold for the other fields: δg δx ∂ g and α − β β α 1 δpβ δxα∂αpβ. The free energy change due to a transla- σαβ = 2η(vαβ vγγδαβ)+¯ηvγγδαβ − 3 tion −is then given by 1 + γI∆µIδ + σe + ∂ Mπ αβ αβ 2 γ αβγ i 3 tot j = Γ ∂ µ¯ Γ ∂ T δF = d x (∂βvα)gα +(∂βµ )ni + hα∂βpα δxβ α − ij α j − i α i I IJ J I r = M ∆µ γ vγγ ∂ − tot f0 Q + dSα ( f vγ gγ µi ni)δαβ ∂βpγ δxβ. = κ∂¯ Γ ∂ µ¯ − − − ∂(∂αpγ ) jα αT i α i π − − (B.2) Mαβγ = κ∂γ ωαβ. (A.30) From the surface contribution we identify the equilibrium Here we have introduced dissipative coefficients and have stress (21). Translational invariance δF = 0 implies the neglected higher order terms. The bulk and shear viscosity are Gibbs–Duhem relation (22). denoted η¯ and η. Diffusion coefficients and the cross coupling For the case of a system with nematic order Qαβ and 2 of diffusion fluxes is described by the symmetric matrix Γij. f = gα/(2ρ)+f0(Qαβ, ∂γ Qαβ, ni), one finds using the same Thermophoretic effects that couple temperature gradients to arguments the equilibrium stress

21 Rep. Prog. Phys. 81 (2018) 076601 Review Appendix C. Irreversible thermodynamics with e ∂f0 intrinsic angular momentum σαβ =(f0 µini)δαβ ∂αQγν. (B.3) − − ∂(∂βQγν) C.1. Local thermodynamics with angular momentum The Gibbs Duhem relation is given by – We can also take angular momentum conservation explicitly into e account following [41]. We introduce the the angular momen- ∂βσαβ =(∂αµi)ni + Hγν∂αQγν. (B.4) − tum Lαβ of each volume element V. In the ensemble where total momentum and angular momentum are given, we can define the free energy F(V, N , T, G , L )=Vf (n , T, g , l ) B.2. Rotation invariance i α αβ i α αβ where lαβ denotes the free energy density. We have We consider the behavior of the free energy under infinitesi- 1 ∂F ∂f mal rotations described by δxα = αβγ θβxγ with infinitesimal Ωαβ = = . (C.1) 2 ∂Lαβ ∂lαβ rotation pseudo vector θβ. The polarity field then transform as V,Ni,T,Gα ni,T,gα δpα = δxβ∂βpα + αβγ θβpγ and a similar expression holds The factor 1/2 stems from the relation . − Ωαβ Lαβ = 2ΩαLα for δgα. Scalar fields transform as δni = δxβ∂βni. Using Using Gallilei invariance, the free energy can be written in − equation (B.1), we find the form

2 3 Gα δF = d x [(∂ µ )n + h (∂ p )δx h θ p ] F(V, Ni, T, Gα, Lαβ)= + F0(V, Ni, T, Lαβ). (C.2) β i i α β α β − α αβγ β γ 2M ∂f0 Similarly, we write f (n , T, g , l )=g2 /(2ρ)+f (n , T, l ), + dSα(( f0 µini)δxα (δxβ∂βpγ γβδθβpδ)) i α αβ α 0 i αβ − − ∂(∂αpγ ) − with f0 = F0/V and ρ = i mini. We define the following = d3x ( ∂ σe ) θ x h θ p thermodynamic quantities: − α βα βγδ γ δ − β βγδ γ δ ∂F 1 ∂f0 0 + dS (σe θ x + θ p ) P0 = = f0 + µini + Ωαβlαβ (C.3) α βα βγδ γ δ ∂(∂ p ) γβδ β δ − ∂V − 2 α γ T,Ni,Lαβ i 3 e ∂f0 = d x (σβα hβpα)βγαθγ + dSδ pαβγαθγ , − ∂(∂δpβ) ∂F ∂f 0 0 (B.5) µi = = (C.4) ∂Ni ∂ni T,V,Nj,Lαβ T,lαβ where we have used the Gibbs–Duhem relation (22). From the condition that δF in equation (B.5) must vanish for small rota- ∂F0 ∂f0 S = = V (C.5) tions θβ, we obtain the relation (25) with the definition (24). − ∂T − ∂ni V,Ni,Lαβ ni,lαβ For a nematic system, the order parameter transforms as δQαβ = δxγ ∂γ Qαβ + αγν θγ Qνβ + βγνθγ Qαν under where P0 is hydrostatic pressure and µi denotes chemical rotations.− We then find potential.

e e,a ∂γ Mαβγ = 2σ + 2(Qαν Hβν QβνHαν ) (B.6) C.2. Rate of free energy change in a polar system with spin αβ − with The thermodynamics of an isothermal polar system with spin can be described by the free energy density ∂f0 ∂f0 2 e f = gα/(2ρ)+f0(lαβ, pα, ∂αpβ, ni). Following the derivation Mαβγ = 2( Qαν Qβν). (B.7) ∂(∂γ Qβν) − ∂(∂γ Qαν ) given in equation (26), we have

dF 1 ∂f = d3x v ∂ g + Ω ∂ l + µtot∂ n h ∂ p + dS 0 ∂ p dt α t α 2 αβ t αβ i t i − α t α α ∂(∂ p ) t β α β 1 ∂f = d3x(v ∂ σtot + Ω (∂ Mπ 2σa )+µtot(r ∂ Ji ) h ∂ p )+ dS 0 ∂ p α β αβ 2 αβ γ αβγ − αβ i i − α α − α t α α ∂(∂ p ) t β α β 1 = d3x( (∂ v )σtot + Ω (∂ Mπ 2σa )+µtotr +(ji + n v )∂ µtot h ∂ p ) − β α αβ 2 αβ γ αβγ − αβ i i α i α α i − α t α tot tot i ∂f0 + dSα vβσβα µi Jα + ∂tpβ , (C.6) − ∂(∂αpβ)

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where h = δF /δp , µ = ∂f /∂n and Ω = 2∂f /∂l . where we have used the convected time derivative α − 0 α i 0 i αβ 0 αβ Note that because f0 contains contributions of kinetic energy d/dt = ∂t + vγ ∂γ. Since a system with finite but constant Ωαβ associated with internal angular momentum, the field hα can be at equilibrium, we rewrite this as defined here contains contributions that depend on spin [41]. dF 3 s e,s The equilibrium stress now reads = d x( vαβ(σ σ ) dt − αβ − αβ e ∂f0 a e,a 1 d σαβ =(f0 µini lγ Ωγ )δαβ ∂αpγ (C.7) +(ωαβ Ωαβ)(σαβ σαβ pαhβ ∂γ Mαβγ ) − − − ∂(∂βpγ ) − − − − 2 1 Dp (∂ ω )Md + µ r + ji ∂ µ h α ) and the Gibbs–Duhem relation becomes − 2 γ αβ αβγ i i α α i − α Dt 1 i 1 π e + dSα vβσβα vαf µij + ωβγ(M + vαlβγ) ∂βσαβ = (∂αΩβγ)lβγ +(∂αµi)ni + hβ∂αpβ. (C.8) − − α 2 βγα − 2 ∂f0 Dpβ At equilibrium with ∂αΩβγ = 0, ∂αµi = 0 and hα = 0, the + , (C.12) equilibrium stress satisfies the force balance ∂ σe = 0. ∂(∂αpβ) Dt β αβ Rotational invariance now implies the rotational Gibbs Duhem – tot d π e relation where σαβ = σαβ + vαgβ, Mαβγ = Mαβγ Mαβγ + vγ lαβ. We finally identify the free energy flux − e e,a ∂γ Mαβγ = 2σ +(pαhβ pβhα)+(Ωαlβ Ωβlα). (C.9) αβ − − f 1 π i ∂f0 Dpβ jα = vβσβα ωβγ(Mβγα + vαlβγ)+µijα . At a thermodynamic equilibrium, hα vanishes and − − 2 − ∂(∂αpβ) Dt Ω l Ω l = 0. In this case ∂ Me = 2σe,a becomes the (C.13) α β − β α γ αβγ αβ equilibrium torque balance. Indeed, from equations (6) and The entropy production rate is (C.9) it follows that at equilibrium ∂ l =Ω l Ω l . For t αβ α β − β α an equilibrium state, ∂tlα = 0. Therefore Ωαlβ Ωβlα must 1 − Tθ = v σs,d + (Ω ω )σa,d + (∂ ω )Md vanish for a global thermodynamic equilibrium. In the context αβ αβ αβ − αβ αβ 2 γ αβ αβγ of a spinning top, the same condition implies that a top spins M N Dp in a steady state around a fixed axis [111]. Only in this case + rI∆µI ji ∂ µ¯ + h α (C.14) − α α i α Dt does a thermodynamic equilibrium exist. I=1 i=1 Using the Gibbs–Duhem relation (C.8) and the relation (C.9) we have with the deviatoric part of the antisymmetric stress

dF 1 = d3x( (∂ v )σtot + Ω (∂ Mπ 2σa ) dt − β α αβ 2 αβ γ αβγ − αβ 1 + µtotr + ji ∂ µtot v ∂ σe v g (∂ v ) h ∂ p v h ∂ p v (∂ Ω )l ) i i α α i − α β αβ − α β α β − α t α − α β α β − 2 α α βγ βγ tot 1 2 i ∂f0 + dSα vβσ (µi miv )( j + nivα)+ ∂tpβ βα − − 2 α ∂(∂ p ) α β = d3x( ∂ v (σtot + v g σe )+ω h p − β α αβ α β − αβ αβ α β 1 + Ω (∂ (Mπ Me ) 2(σa σe,a )+(p h p h )+(Ω l Ω l )) 2 αβ γ αβγ − αβγ − αβ − αβ α β − β α α β − β α 1 + µ r + ji ∂ µ h ∂ p v h ∂ p h ω p v l ∂ Ω ) i i α α i − α t α − α β α β − α αβ β − 2 α βγ α βγ tot e 1 2 i ∂f0 + dSα vβ(σβα σβα)+vα( ρv µini) µijα + ∂tpβ . (C.10) − 2 − − ∂(∂αpβ) Rewriting the expressions and using equation (29) and the definition of the Ericksen stress C.7( ) in the surface terms, we obtain dF = d3x( v (σtot,s + v g σe,s )+(ω Ω )(σa σe,a p h ) dt − αβ αβ α β − αβ αβ − αβ αβ − αβ − α β 1 1 Dpα + Ω ∂ (Mπ Me ) v (∂ Ω )l + µ r + ji ∂ µ h ) 2 αβ γ αβγ − αβγ − 2 γ γ αβ αβ i i α α i − α Dt tot i ∂f0 dpβ + dSα vβ(σβα + vβgα) vα( f lγ Ωγ ) µijα + , (C.11) − − − ∂(∂αpβ) dt

23 Rep. Prog. Phys. 81 (2018) 076601 Review

1 1 last two terms in equation (C.21) are active terms proportional σa,d = σa σe,a ( p h p h ) ∂ Md . (C.15) αβ αβ − αβ − 2 α β − β α − 2 γ αβγ to gradients of the polarity field. Such terms in general also exist when including all terms allowed by symmetry in the Angular momentum conservation (6) then implies a dynamic constitutive equation (80). However in equation (80) these equation for the intrinsic angular momentum terms were not written for simplicity because they are unim- a,d portant in the hydrodynamic limit. This shows that active ∂tlαβ+∂γ (vγ lαβ)= 2σαβ + (Ωαlβ Ωβlα). (C.16) − − terms that contribute to the antisymmetric stress and to angu- Note that the intrinsic angular momentum does not introduce lar momentum flux effectively renormalize active terms in the a separate hydrodynamic variable because the balance equa- redefined symmetric stress and the physics does not change. tion has a source term. Therefore, the intrinsic rotation rate Ω I However, the physics described by the active coefficients ζa, relaxes on non-hydrodynamic time scales [41]. which generate relative internal rotations described by equation (86) is missing when using symmetric stress tensors only. C.3. Symmetrization of the stress tensor The reason is that after redefining the stress to be symmetric, information about the difference of rotation rates Ω ω αβ − αβ The antisymmetric part of the stress can be eliminated by the is lost. Note that while Ωαβ ωαβ vanishes in passive fluids in following redefinition of stress and momentum density [39] the hydrodynamic limit, it can− be finite at long times in active 1 chiral fluids where it is thus a relevant variable. In this case a g = gα + ∂βlαβ (C.17) redefinition of the stress tensor to be symmetric misses physics α 2 of the system even in the hydrodynamic limit. l αβ = 0 (C.18)

1 1 ORCID iDs σtot = σtot σa + ∂ Mπ + ∂ Mπ (C.19) αβ αβ − αβ 2 γ αγβ 2 γ βγα Frank Jülicher https://orcid.org/0000-0003-4731-9185 π Guillaume Salbreux https://orcid.org/0000-0001-7041-1292 Mαβγ = 0. (C.20) The redefined stress tensor σ is symmetric and obeys the αβ References tot momentum conservation equation ∂tgα = ∂βσαβ as well as the angular momentum conservation (4). Note that the physics [1] Bray D 2000 Cell Movements: from Molecules to Motility does not change under this redefinition which amounts to a use (New York: Garland Science) of different variables. The pairs of conjugate thermodynamic [2] Alberts B, Johnson A, Lewis J, Morgan D, Raff M, forces and fluxes remain the same but have to be expressed in Roberts M and Walter P 2014 Molecular Biology of the Cell the new variables. For passive systems, the redefined quanti (New York: Garland Science) ties approach the original ones in the hydrodynamic limit. [3] Howard J 2001 Mechanics of Motor Proteins and the Cytoskeleton (Sunderland, MA: Sinauer) One can then express in the hydrodynamic limit the thermo- [4] Salbreux G, Charras G and Paluch E 2012 Actin cortex dynamic fluxes and forces directly in the redefined variables. mechanics and cellular morphogenesis Trends Cell Biol. This suggests that in the hydrodynamic limit the thermody- 22 536–45 namic flux σa,d is not significant, and the corresponding dis- [5] Doubrovinski K and Kruse K 2007 Self-organization of αβ treadmilling filaments Phys. Rev. Lett. 99 228104 sipative and reactive couplings do not matter. However, while [6] de Gennes P-G and Prost J 1993 The Physics of Liquid this is true for passive systems, relevant active chiral terms can Crystals (Oxford: Oxford University Press) be missed by the symmetrisation of the stress tensor. [7] Toner J and Tu Y 1995 Long-range order in a two-dimensional To illustrate these points, we briefly discuss the effects of dynamical xy model: how birds fly together Phys. Rev. Lett. 75 4326 the redefinition (C.17) (C.20) of the stress tensor in the case – [8] Toner J and Tu Y 1998 Flocks, herds and schools: A of active chiral systems discussed in section 4.4. We consider quantitative theory of flocking Phys. Rev. E 58 4828 for simplicity the limit when the intrinsic angular momentum [9] Simha R A and Ramaswamy S 2002 Hydrodynamic density can be neglected, lαβ 0. Using equations (80) and fluctuations and instabilities in ordered suspensions of self- (88) the redefined dissipative part of the traceless symmetric propelled particles Phys. Rev. Lett. 89 058101 [10] Ramaswamy S, Simha R A and Toner J 2003 Active nematics stress reads on a substrate: giant number fluctuations and long-time tails Europhys. Lett. 62 196 d,s ν1 2 I I σ˜αβ = 2ηv˜αβ + ( pαhβ + pβhα pγ hγ δαβ)+ζ ∆µ q˜αβ [11] Kruse K, Joanny J-F, Jülicher F, Prost J and Sekimoto K 2004 2 − 3 Asters, vortices and rotating spirals in active gels of polar 1 I,eff I I filaments Phys. Rev. Lett. 92 078101 + (ζ2 + ζ3)∆µ (αγν ∂γ q˜νβ + βγν∂γ q˜να) 2 [12] Toner J, Tu Y and Ramaswamy S 2005 Hydrodynamics and 1 2 phases of flocks Ann. Phys. 318 170 ζI ∆µI(∂ p + ∂ p ∂ p δ ) , (C.21) − 2 4 β α α β − 3 γ γ αβ [13] Alt W and Dembo M 1999 Cytoplasm dynamics and cell motion: two-phase flow models Math. Biosci. 156 207–28 eff eff [14] Ramaswamy S, Toner J and Prost J 2000 Nonequilibrium where for simplicity, we have set κ =¯η =¯ν2 =¯ν3 =¯ν4 = 0 I fluctuations, traveling waves and instabilities in active and have omitted terms proportional to gradients of ∆µ . The membranes Phys. Rev. Lett. 84 3494

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Frank Jülicher studied Physics at the University of Stuttgart and RWTH Aachen University. He did his PhD work at the Institute for Solid State Research at the Research Center Jülich and received his PhD in 1994 from the University of Cologne. His postdoctoral work at the Simon Fraser University in Vancouver and the Institut Curie and ESPCI in Paris was followed by a CNRS research position at the Institut Curie in Paris in 1998. Since 2002, he is a Director at the Max Planck Institute for the Physics of Complex Systems in Dresden and a professor of biophysics at the Technical University of Dresden. His research interests are theoretical approaches to active matter and the spatiotemporal organization of cells and tissues.

Stephan Grill studied physics at the University of Heidelberg, did his PhD work at the European Molecular Biology Laboratory (EMBL), and received his PhD in 2002 from the University of Munich. His postdoctoral work at the University of California in Berkeley and at the Lawrence Berkeley National Laboratory was followed by a research group leader position, jointly at the Max Planck Institute for the Physics of Complex Systems and the Max Planck Institute for Molecular Cell Biology and Genetics in Dresden. Since 2013, he is a full professor a the Biotechnology Center (BIOTEC) of the Technical University of Dresden. His research combines experimental and theoretical approaches in the areas of cellular and molecular biophysics.

Guillaume Salbreux did his PhD in the Institut Curie in Paris. He then moved for a postdoctoral work to the University of Michigan. From 2011–2015 he headed a research group at the Max Planck Institute for the Physics of Complex Systems. Since 2015 he is a group leader in the Francis Crick Institute in London. He is interested in the physics of biological systems, morphogenesis and biological self-organisation.