Haiqin
Wang
^{ab},
Tiezheng
Qian
^{c} and
Xinpeng
Xu
*^{ab}
^{a}Technion – Israel Institute of Technology, Haifa, 32000, Israel
^{b}Physics Program, Guangdong Technion – Israel Institute of Technology, Shantou, Guangdong 515063, China. E-mail: xu.xinpeng@gtiit.edu.cn
^{c}Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

Received
21st November 2020
, Accepted 4th January 2021

First published on 5th January 2021

Onsagers variational principle (OVP) was originally proposed by Lars Onsager in 1931 [L. Onsager, Phys. Rev., 1931, 37, 405]. This fundamental principle provides a very powerful tool for formulating thermodynamically consistent models. It can also be employed to find approximate solutions, especially in the study of soft matter dynamics. In this work, OVP is extended and applied to the dynamic modeling of active soft matter such as suspensions of bacteria and aggregates of animal cells. We first extend the general formulation of OVP to active matter dynamics where active forces are included as external non-conservative forces. We then use OVP to analyze the directional motion of individual active units: a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer. Next we use OVP to formulate a diffuse-interface model for an active polar droplet on a solid substrate. In addition to the generalized hydrodynamic equations for active polar fluids in the bulk region, we have also derived thermodynamically consistent boundary conditions. Finally, we consider the dynamics of a thin active polar droplet under the lubrication approximation. We use OVP to derive a generalized thin film equation and then employ OVP as an approximation tool to find the spreading laws for the thin active polar droplet. By incorporating the activity of biological systems into OVP, we develop a general approach to construct thermodynamically consistent models for better understanding the emergent behaviors of individual animal cells and cell aggregates or tissues.

A distinctive feature of active matter is that the system is locally driven out of equilibrium by active units at the length scale of a constituent component.^{1–3,27} This is distinct from those nonequilibrium systems that are driven at the system boundaries.^{28} The presence of self-propelled units in active matter breaks the detailed balance and time-reversal symmetry (TRS),^{1,3,5,8} resulting in a wealth of intriguing macroscopic structures and behaviors, such as spontaneous flows,^{3,4} motility-induced phase separation,^{3,6,8} unusual mechanical and rheological properties,^{2,10} wave propagation and sustained oscillations even in the absence of inertia,^{29–31}etc. One of the most interesting questions in the nonequilibrium dynamics of active matter is how the local driving forces operating at the small scale of individual active unit can produce the observable macroscopic emergent phenomena at the large scale of the whole system. Answering this question will not only shed new light on the fundamental statistical mechanics,^{3,6,8,26} but also deepen our understanding of biological processes,^{1–4} and help design new generations of biomimetic active materials that balance structural flexibility and stability.^{1,3,5}

The study of active matter can be brought into the framework of condensed matter physics based on the consideration that the collective behaviors of active matter emerge from the interactions among the constituent self-propelling units and the dissipation mechanisms operating inside the system. In particular, soft condensed matter physics provides many useful model systems for reference to active matter,^{2,5}e.g. the wetting of substrates by liquid droplets,^{32–34} the dynamics of colloid suspensions,^{35,36} the dynamics of nematic liquid crystals,^{37} the dynamics and rheology of polymer gels,^{38,39} and the phase segregation of surfactants,^{40,41}etc. The major challenge is to couple these model systems with active and molecularly specific processes, such as the active force generation by self-propelling units, and the binding and unbinding of transmembrane adhesion receptors on solid substrates.^{2,5} Over the last decade, several soft matter systems have been revisited with a focus on this point of view.^{2} The physical understanding for the emergent structures and behaviors of active soft matter has been rapidly growing,^{1–10} with particular attention paid to dry active matter,^{3,18} active polar fluids,^{42} active nematics,^{43} active gels,^{44} and active membranes.^{45}

Theoretically there have been two major approaches to the study of active soft matter: particle-based models^{3,6–8,26,46} and continuum phenomenological models.^{1,2,4–6,14,15,43} In particle-based models, the active units are usually modeled as self-propelled particles with fixed or variable speed and random orientation moving in an inert background, following the seminal work of Vicsek et al.^{47} It provides a straightforward approach to the study of active soft matter with an emphasis on the order and fluctuations rather than the forces and mechanics.^{3,7,8,26} In continuum phenomenological models, active units are represented by a smooth density field rather than individually resolved particles. A continuum model for active soft matter is usually constructed by modifying the dynamic model of a proper reference soft matter system.^{1,2,4–6,43} This is typically accomplished by adding a minimal set of extra terms that cannot be derived from any free energy or dissipation functions. This is an effective way to introduce the activity and break the TRS such as active forces, active fluxes, and active chemical potentials.^{1,43,48,49} However, there is another more systematic way of including activity by introducing the mechanochemical coupling between passive dissipative processes and some relevant biochemical reactions^{1,4} in Onsager's framework of irreversible thermodynamics.^{50,51} The two theoretical approaches are complementary. The particle-based approach involves only a small number of parameters for each active unit, and therefore the theoretical predictions can be readily compared with experiments for some model active systems such as self-propelled colloids.^{52} However, the model for interacting self-propelled particles sometimes oversimplifies the problem, and hence may lose some generality and applicability of its conclusions when applied to real systems, especially in vivo biological systems.^{1,2,4,43} By contrast, the formulation of phenomenological models is based upon symmetry consideration, conservation laws of mass, momentum, and angular momentum, and laws of thermodynamics. This gives the continuum approach a large range of applicability and generality when applied to real biological processes.^{1,2,4,43} In this work, we focus on the continuum phenomenological models and show that Onsagers variational principle, which has been widely used in the study of soft matter dynamics,^{53} can be extended for the study of active soft matter.

Onsager's variational principle (OVP) was originally proposed by Lars Onsager in his seminar papers in 1931.^{54,55} He showed that for irreversible processes in a near-equilibrium thermodynamic system, the thermodynamic fluxes can be written as linear combinations of conjugate thermodynamic forces, and the proportionality coefficient matrix must be positive-definite and symmetric according to Onsager's reciprocal relations (ORR). The ORR lay the foundation for the theoretical framework of linear irreversible thermodynamics. In the end of his paper, Onsager proposed OVP as a variational principle that is equivalent to the linear force–flux relations in describing dissipative dynamics. In addition, OVP can be regarded as an extension of “the principle of the least dissipation of energy” proposed by Lord Rayleigh.^{56} For isothermal systems, OVP takes a simple form as follows.^{53,57,58} The irreversible processes described by the thermodynamic fluxes follow the dynamic path that minimizes the function:

() = Φ() + Ḟ(;α). | (1) |

OVP can be used to derive many transport equations for soft matter dynamics,^{53,62}e.g., the Stokes equation for incompressible low-Reynolds-number flows,^{58} the diffusion equation,^{53,62} the reaction-diffusion equations for (low molecular weight) multi-component solutions,^{51,64,65} the thin film evolution equations,^{63,66–68} the phase field model for two-phase hydrodynamics,^{58,69} the electrorheological hydrodynamic equation,^{70} the two-fluid model for the phase separation dynamics in colloids and polymers,^{53,62,63} the dynamics of polymer gels,^{39,63} the dynamic equations of lipid membrane,^{71–73}etc. Moreover, OVP also provides a very convenient way to derive thermodynamically consistent boundary conditions that supplement the transport equations in the bulk region.^{58,69} Examples include the generalized Navier boundary condition (GNBC) for the contact line hydrodynamics,^{58} the generalized nemato-hydrodynamic boundary conditions for liquid crystals,^{74} and the boundary conditions for block copolymer solution films,^{75}etc. In addition, Doi and his collaborators have recently proposed that OVP can be used as a direct variational method to find approximate solutions for complex soft matter dynamics.^{63,68,76} This approximation method has been successfully used to study the evolution of droplets and thin films,^{63,67,77,78} the dynamics of the beads-on-string structure of viscoelastic polymer filaments,^{63,79} the sedimentation in colloidal suspensions,^{80} and the translocation of a vesicle through a narrow hole,^{81}etc. Although OVP has been widely applied with great successes in the study of inert soft matter dynamics, it is rarely used in the study of active soft matter dynamics.^{65,82–84} In the present work, we will show that OVP can be readily extended to include biochemical activity and conveniently applied to study the emergent structures and behaviors of active soft matter. OVP can not only be applied to formulate thermodynamically consistent models, but also be used to generate approximate solutions for the complex dynamics of active soft matter.

This paper is organized as follows. In Section 2, a brief review of OVP is provided, and a simple extension of OVP is presented for applications to active matter, in which the active forces are treated as non-conservative forces that cannot be derived from any free energy and dissipation functions. In the next three sections, we apply OVP and its extended form to three representative active matter problems motivated by the biology of bacteria and animal cells. In Section 3, we present the first application of OVP to the directional motion of an individual active unit, e.g., a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer moving in a viscous fluid. In Section 4, we consider the two-phase hydrodynamics for active polar droplets. We use OVP to formulate a diffuse-interface model for an active polar droplet on a solid substrate. This hydrodynamic model is thermodynamically consistent and consists of hydrodynamic equations in the bulk region and boundary conditions at the solid surface. In Section 5, we consider a thin active polar droplet moving on a solid substrate in two dimensions. Under the lubrication approximation, we firstly apply OVP to derive the classical thin film equation that has been obtained previously. We then use OVP as an approximation tool to find the scaling laws for the spreading of a thin active droplet in the respective limits of negligible activity and strong activity. In Section 6, we summarize our major results, make some general remarks, and envision a few potential applications of OVP to more realistic biological problems.

(2) |

〈f_{ri}(t)〉 = 0, 〈f_{ri}(t)f_{rj}(t′)〉 = 2ζ_{ij}k_{B}Tδ(t − t′), | (3) |

(4) |

From the Langevin eqn (2), we calculate the transition probability^{60}P(α′,t + dt|α,t) from the state α at time t to α′ at t + dt:

(5) |

(6) |

(;α) = Φ(,) + Ḟ(;α) − Ẇ_{a}(;α) | (7) |

(8) |

We would like to give some remarks on OVP as follows.

(i) A term appearing in the Onsager–Machlup function in eqn (6) does not contribute to the Rayleighian because we are considering the most probable state α′ at an immediate future time close to t, which is to be determined from the prescribed state α and hence the prescribed forces f(α,t) and Ψ(f(α,t),f(α,t)).

(ii) The active matter may also be subject to the influence of some external forces that do not arise locally from the consumption of chemical energy of the system. This can be treated by subtracting the work power Ẇ_{ext}(;α) done by the external forces from the Rayleighian in eqn (7):

(;α) = Φ(,) + Ḟ(;α) − Ẇ_{a}(;α) − Ẇ_{ext}(;α). | (9) |

(iii) In a continuum model of active matter, the set of slow variables α can represent the field variables both in the bulk and at the boundary. Then eqn (8) derived from OVP gives the dynamic equations both in the bulk and at the boundary, with the latter becoming dynamic boundary conditions. Furthermore, if there are external forces applied at the system boundary, then their contributions may be described by Ẇ_{ext} in the Rayleighian in eqn (9).

(iv) It is important to note that OVP is a local principle that can be used to find the most probable state only in the immediate future (without additional constraints). To locate the most probable paths that can take the system to the far future under various constraints,^{60,87} we can divide the long time interval (say, t − t_{0}) into N_{t} sub-intervals with dt = (t − t_{0})/N_{t}, then

(10) |

(11) |

(12) |

Variational principles have been proposed in various fields of physics and several variational methods have been developed accordingly to find approximation solutions such as Ritz method and the least-squares method.^{89} Recently, Doi developed a Ritz-type variational method based on OVP^{76} by assuming that state variables α(t) is a certain function of a small number of parameters denoted by a = (a_{1},a_{2},…), i.e., α(t) = α(a(t)). Then the rate of the state variables can be written as

(13) |

Similarly, direct variational method can also be developed based on the OMVP to approximate the long time kinetic paths or states of the system.^{68} We consider certain kinetic path which involves a parameter set α(a(t)). The best guess for the actual path is the path which gives the smallest value of the Onsager–Machlup functional [α(a(t))] with respect to the parameter functions a(t). This variational method is similar to the least-square method but with a target function that is more physically meaningful based on physical principles.^{68,89}

These direct variational methods are useful particularly when we have an idea for the probable kinetic path and can write down the functions α(a(t)). It has been applied successfully to many problems in soft matter dynamics. In this work, we will show that these approximation methods can also be used to study the dynamics of active soft matter.

• Scalar formulation. The variational principles involve only physical quantities that can be defined without reference to a particular set of generalized coordinates, namely the dissipation function, free energy, and active work power. This formulation is therefore automatically invariant with respect to the choice of coordinates for the system, which allows us a great flexibility in choosing state variables and rates.

• Thermodynamic consistency. The variational principles incorporate the intrinsic structure of Onsagers theory of non-equilibrium thermodynamics clearly. They provide compact invariant ways of obtaining thermodynamically-consistent dynamic equation systems where the pairs of rates and forces are obtained automatically.

• Direct variational approximation tools. The direct Ritz-type variational method of finding approximation solutions for the system dynamics bypasses the derivation of the Euler–Lagrange equations and goes directly from a variational statement of the problem to the solution of the Euler–Lagrange equations. This approximation method helps to pick up the most important dynamic behaviors and to simplify the calculations significantly from complicated partial differential equation systems to simple ordinary differential equations. In addition, the direct least-square-type variational method based on OMVP of minimizing the Onsager–Machlup integral further optimizes the search for more realistic kinetic paths and provides a new method of studying long-time steady-state dynamics of the system.

The above variational principles have been successfully applied to the diffusion in electrolyte solutions by Onsager himself in 1940s,^{57} and more recently applied to various soft matter systems such as multiphase flows,^{58,69} electrorheological fluids,^{70} colloid suspensions,^{80} polymer solutions,^{53,62,63} polymer gels,^{39,63} liquid crystals,^{62,74} vesicles,^{81} membranes^{71–73} and so on. This indicates that OVP is an important principle in soft matter dynamics.^{62,63,68} In this work, we present its applications to active soft matter dynamics that is mostly motivated by biological applications.

Before ending this section, we would like to summarize the general steps for applying OVP to the dynamics of active soft matter for which the dynamic equations are not yet known or still controversial.^{62,63}

(i) Choose a set of coarse-grained, slow variables, α ≡ {α_{1},α_{2},…}, to describe the time evolution of the macroscopic state of the system.

(ii) Construct the free energy function, F(α), and calculate the rate of change of the free energy, Ḟ(;α).

(iii) Construct the dissipation function, Φ(,), which is quadratic in the rates/fluxes .

(iv) Find the work power done by the active forces, Ẇ_{a}(;α), based on the specific activity considered, and find the work power done by some other external forces, Ẇ_{ext}(;α). The external forces are usually applied at the system boundary and do not arise locally from the consumption of chemical energy of the system.

(v) Minimize the Rayleighian in eqn (9): (;α) = Φ(,) + Ḟ(;α) − Ẇ_{a}(;α) − Ẇ_{ext}(;α), with respect to the rates/fluxes . Note that some additional constraints on the system dynamics may need to be imposed by using Lagrange multipliers.

Furthermore, if we have an idea about the most probable kinetic path, then we can write down the slow variables α = α(a(t)) as functions of a small number of parameters, a = (a_{1},a_{2},…). We can follow the above steps and obtain the Rayleighian as a function of ȧ and a as (ȧ;a). The minimization of with respect to ȧ will then provide an approximate description for the active matter dynamics directly.

Fig. 1 Directional motion of individual active units. (a) A myosin motor catalyzes ATP hydrolysis and converts the released chemical energy into its directional motion on an actin filament^{94} toward the plus (barbed) end. Meanwhile, an external force, f_{ext}, is applied on the myosin through optical trapping of a nano-bead attached to the motor. (b) A toy two-sphere microswimmer swims in viscous fluids. The body length, 2(t), of the microswimmer oscillates cyclically and the friction coefficient, ζ, is asymmetric for forward motion (with smaller friction) and backward motion (with larger friction), as shown in eqn (23). The front-back asymmetry in friction is indicated by inclined thorns on the microspheres. |

ATP ⇌ ADP + P_{i}. | (14) |

In this subsection, we use OVP to formulate a thermodynamic description in the linear (near-equilibrium) regime for the directional motion of a translationary molecular motor along a polar filament against an external force, as shown in Fig. 1a. This Onsager-type description is pioneered by Kedem & Caplan^{91} and extended by Chen & Hill.^{92} We take our thermodynamic system to include the molecular motor and the surrounding solution of ATP, ADP, and P_{i}. The system is coupled to a heat reservoir and a work reservoir, which can apply external forces on the motor, for example, by optical tweezers, that is, optical trapping of a nano-probe attached to the motor (see Fig. 1a). The states of the thermodynamic system can be described by the average motor position x, the polarization vector p (describing the polarity of the filament and assuming to point from minus end to plus end), and the average number N_{α} of chemical components involved in ATP hydrolysis with α = ATP, ADP, and P_{i}.

The reaction free energy for ATP hydrolysis takes the form of

_{r} = _{r}(N_{ATP},N_{ADP},N_{Pi}), | (15) |

(16) |

(17) |

The irreversible dynamics of the thermodynamic motor/filament system is characterized by two rates: the reaction rate r and the average motor velocity v = ẋ. In the linear response regime close to equilibrium, the dissipation function is a quadratic function of the rates given by

(18) |

Ẇ_{ext} = f_{ext}·v. | (19) |

Using the Rayleighian in the presence of the external force and eqn (16)–(19), we minimize with respect to r and v and obtain

Δμ = Λr − λv, | (20a) |

f_{ext} = −λr + ζv. | (20b) |

To be specific, for the in vivo motion of myosin motors along actin filaments, ATP is usually in excess with a constant Δμ > 0 and myosin motor always move towards to the plus end of the actin filament. Therefore, the load-free motor velocity (for f_{ext} = 0) must be positive (and hence λ > 0 and 0 < q < 1). In this case, we can identify the following four regimes from eqn (20):

(i) For f_{ext} > 0, we have v > 0 and r > 0 (hence f_{ext}v > 0 and rΔμ > 0), the external force pulls the motor to the plus end. Meanwhile, the excess ATP hydrolyzes and the chemical energy is consumed to drive the motor along the same direction to the plus end.

(ii) For small negative force f_{ext} < 0 and f_{ext} > f_{stall,v}, we still have v > 0 and r > 0 (hence f_{ext}v < 0 and rΔμ > 0), the excess ATP hydrolyzes and the released chemical energy is converted into mechanical work. Here f_{stall,v} = −λΔμ/Λ < 0 is called stall force of myosin motion and when f_{ext} = f_{stall,v}, the motor is stationary with v = 0.

(iii) For f_{stall,r} < f_{ext} < f_{stall,v}, the moving direction of the motor is reversed with v < 0 and hence f_{ext}v > 0. That is, the external force is doing positive work on the motor moving along the plus end. However, we still have r > 0 and hence rΔμ > 0. That is, the excess ATP hydrolyzes and the released chemical energy also drives the motion the motor towards the same plus end. Here f_{stall,r} = −Δμζ/λ < 0 is called stall force of ATP hydrolysis and when f_{ext} = f_{stall,r}, the ATP hydrolysis is inhibited with r = 0.

(iv) For f_{ext} < f_{stall,r} < 0, we have v < 0 and r < 0, hence f_{ext}v > 0 and rΔμ < 0. That is, the external force is doing positive work on the motor and produces ATP that is already in excess; the system then works as an ATP pump.

Therefore, the motor/filament system is a reversible machine: it can not only convert chemical energy into mechanical work, but can also convert mechanical work into chemical energy. In this work, we are particularly interested in the regime (ii), in which ATP hydrolysis occurs spontaneously and the released chemical energy is used to drive the system out of equilibrium continuously.

Now let's consider a practical limit at which the ATP hydrolysis rate r (or equivalently the active force f_{a} = λr) is taken as a given positive parameter that measures the activity of the system. That is, the effect of mechanical forces on the ATP hydrolysis is neglected and the rate of ATP hydrolysis is determined dominantly by the chemical affinity as r ≈ Λ^{−1}Δμ > 0. This leads to a reduced description in which the position of the molecular motor in directional motion becomes the only state variable, while the amounts of the reactants and products in the ATP hydrolysis are no longer involved. As a result, the Rayleighian reduces to its extended form in eqn (9) as

(21) |

Finally, we would like to point out that the results obtained here from thermodynamic description for the motion of molecular motors on polar filaments are completely independent of any underlying microscopic mechanisms. However, the above linear-response theory applies only to the linear regime near equilibrium where Δμ/k_{B}T ≪ 1 and f_{ext}ξ/k_{B}T ≪ 1 with ξ being the typical molecular size of relevant proteins. In real life, molecular motors mostly operate far from equilibrium (with Δμ ∼ 10k_{B}T) and the velocity v(f_{ext},Δμ) and the rate of ATP consumption r(f_{ext},Δμ) are in general highly nonlinear. Therefore, more specific models such as a minimal two-states model for molecular motors should be constructed to arrive at a more comprehensive understanding of the specificity and robustness of the directional motion of motors in highly fluctuating environment.^{95}

Specifically, to show how periodic shape changes can generate directional self-propulsion, here we consider a toy microswimmer that is composed of two microspheres^{99} as shown in Fig. 1b. Let x_{1} and x_{2} denote the coordinates of these two microspheres. Then the directional motion and the shape changes of the microswimmer can be described by the temporal evolution of the center-of-mass position and the half-body-length of the swimmer , which are taken as the two slow variables. The toy microswimmer can actively change its shape by periodically changing its body length (or the distance between the two spheres) 2(t) as

(t) = _{0} + asin(ωt), | (22) |

The toy microswimmer subjected to the periodic shape oscillations can achieve directional motion only when there exists some mechanisms that break the front-back symmetry. Here we consider an asymmetry in the viscous friction, defined by f_{v} = −ζ(ẋ)ẋ, in which the friction coefficient ζ(ẋ) depends on the moving direction of each microsphere according to:

(23) |

(24) |

In most microswimmers, it is natural to assume that there is a clear separation of time scales between their shape oscillations and the directional motion. The directional motion of the microswimmer is usually much slower than its shape oscillations, i.e., _{0}/v ≫ T_{}. We can, therefore, integrate out the relatively fast varying variable, the half-body-length (t), in one cycle of shape oscillation and arrive at a time-averaged dissipation function of the slow variable, the center-of-mass velocity v, by as

(25) |

(26) |

v = f_{a,eff}/ζ_{eff}, | (27) |

We would like to give some remarks on the directional motion of the toy two-sphere microswimmer as follows.

(i) Most dynamic behaviors of biological systems show strong nonlinearity. For example, in the toy two-sphere microswimmer, the swimmer migration velocity shows highly nonlinear dependence on the active force or the active shape-changing velocity of the swimmer. However, in many cases, OVP can still be employed if we expand the set of slow state variables properly.^{63} For example, here our set of slow variables includes not only the center-of-mass position of the swimmer but also its fast-changing body length.

(ii) The dissipation function is non-zero even for symmetric microswimmers (with ζ = ζ_{−} = ζ_{+}) when there is no average directional motion (i.e., v = 0): . It arises in the symmetric microswimmer from the viscous dissipation due to the fast shape-oscillation in viscous fluids.

(iii) Similar to the walk of molecular motors in the previous example, the active shape changes of microswimmers are also driven by spontaneous ATP hydrolysis. Then the irreversible dynamics of the microswimmer should be characterized by the rate of ATP hydrolysis r in addition to the sphere velocities, ẋ_{1} and ẋ_{2}. The rate of the change of free energy is given in eqn (16) by . The dissipation function is given, to the leading order in the rates, by

(28) |

(29) |

f_{a} − ζ_{1}ẋ_{1} = 0, −f_{a} − ζ_{2}ẋ_{2} = 0, | (30) |

(31) |

Finally, we would like to point out that in our toy two-sphere microswimmer, the two necessary conditions for a steady directional motion are the active shape oscillations as the energy input and the frictional asymmetry that breaks the front-back symmetry. Similarly, for a long thin swimming micro-filament, the hydrodynamic friction is anisotropic: it experiences less friction when moving along its axis than perpendicular to it. In this case, a cyclic beat pattern on the filament will be able to drive directional propulsion in a similar manner as in the above one-dimensional toy microswimmer‡.

Fig. 2 Active stresses generated by active units: (a) extensile force dipoles generated by the bacterial microswimmer, and (b) contractile force dipoles generated by the actomyosin filament. |

In this subsection, we present an alternative derivation of the generalized hydrodynamic model for an active polar fluid that is regarded as a reactive fluid involving the ATP hydrolysis/synthesis. In the next subsection, we will show that in the same active polar fluid, if the effects of polarization and flow on the ATP hydrolysis are negligible, then the rate of ATP hydrolysis becomes a constant simply determined by the preset constant chemical affinity Δμ. The activities, driven by the spontaneous ATP hydrolysis, are then represented by local external non-conservative force fields that are added as the active terms to the dynamic model of a passive polar fluid.

To be specific, here we use OVP to derive a diffuse-interface model for a droplet of active polar fluids moving on a solid substrate, as schematically shown in Fig. 3a. The states of such an active polar droplet can be described by the following slow field variables: the scalar composition field ϕ(r,t) (distinguishing the coexisting passive isotropic phase from the active polar phase), the polarization vector field p(r,t) (describing the average orientation of active polar agents), the average fluid velocity field v(r,t), and the density field n_{α}(r,t) of chemical components involved in ATP hydrolysis (eqn (14)) with α = ATP, ADP, and P_{i}. For an active polar fluid that is confined between solid substrates or flows at the solid surfaces, the total free energy includes four contributions, [ϕ,p,n_{α}] = _{ϕ} + _{p} + _{r} + _{s}, as respectively given by

(32a) |

(32b) |

(32c) |

(32d) |

The composition variable, ϕ, is a conserved phase parameter and its dynamics follows the following conservation equation

∂_{t}ϕ = −∇·(ϕv + J). | (33) |

∂_{t}n_{α} = −∇·(n_{α}v) + rν_{α}. | (34) |

Using eqn (33) and (34) and the definition of ṗ, we obtain the change rate of free energy from eqn (32) as

(35) |

σ^{e} = −I − K_{ϕ}∇ϕ∇ϕ − K_{p}∇p_{k}∇p_{k} − cp∇ϕ, | (36) |

(37) |

(38) |

L ≡ a_{s} + K_{ϕ}·∇ϕ + c·p, | (39) |

(40) |

H ≡ b_{s}p· + K_{p}·∇p. | (41) |

The energy dissipation function Φ is a quadratic function of three dissipative rates: the shear rate , the rate of change of polarization ṗ, and the rate of ATP hydrolysis r. These rates have the same time parity and from symmetry considerations, Φ can be written into the following invariant scalar form as^{62}

(42) |

P ≡ ṗ − ω × p = ṗ + Ω·p, | (43) |

Then the Rayleighian is given by

(44) |

−∇p + ∇·(σ^{e} + σ^{v} + σ^{a}) = 0, | (45a) |

h = γ_{1}P + γ_{2}p· − h^{a}, | (45b) |

Δμ = −λp·P − pp: + Λr, | (45c) |

σ^{v} = α_{1}(:pp)pp + α_{2}pP + α_{3}Pp + α_{4} + α_{5}pp· + α_{6}·pp, | (46a) |

σ^{a} = −rpp, | (46b) |

h^{a} ≡ λrp. | (47) |

Furthermore, from the minimization of [v,J,ṗ,r], we can also obtain the thermodynamically-consistent boundary conditions at the solid surfaces that supplements the dynamic equations in the bulk fluids:

∂_{t}ϕ + v_{τ}·∇_{τ}ϕ = −ΓL(ϕ,p), | (48a) |

(48b) |

·v = 0, ·J = 0, | (48c) |

H ≡ b_{s}p· + K_{p}·∇p = 0. | (48d) |

The phenomenological coefficients in the above equation systems are functions of the frictional coefficients β_{i} in the dissipation function Φ in eqn (42) as^{62}

(49) |

(50) |

(51) |

(52) |

σ^{a} = −ζpp, | (53a) |

h^{a} = ζ_{h}p, | (53b) |

We would like to further point out that eqn (52) indicates that in the limit of constant reaction rate of ATP hydrolysis, the ATP-induced activity in the active polar fluid can be regarded simply as some local non-conservative fields applied externally on the passive polar fluid. The active characteristic of these external fields is reflected in the fact that the active stress σ^{a} and the active molecular field h^{a} both depend on the local state variable (the polarization), p. Furthermore, these active fields driven by spontaneous ATP hydrolysis break the time-reversal symmetry of the polar fluids.

It follows that according to the general form of eqn (7), the Rayleighian in eqn (44) can be rewritten as

(54) |

(55) |

Minimization of gives the following simplified dynamic equation system:

(i) The dynamic equations for v: the incompressibility condition ∇·v = 0, the generalized Stokes' equation in eqn (45a) with the stress tensors σ^{e}, σ^{v}, and σ^{a}, given in eqn (36), (46a), and (53a), respectively;

(ii) The dynamic equation for ϕ: the conservation equation in eqn (33) with J = −M∇μ_{ϕ};

(iii) The dynamic equations for p:

P = γ_{1}^{−1}h − γ_{1}^{−1}γ_{2}p· + γ_{1}^{−1}h^{a}, | (56) |

(iv) The boundary conditions in eqn (48) still apply to the present case.

We would like to point out that in Section 4.1, a complete model is constructed to incorporate the chemical reaction and explicitly describe the mechanochemical coupling. In this description, the time-reversal symmetry (TRS) is preserved, and so is Onsagers reciprocal relation (ORR) for mechanochemical coupling. In Section 4.2, the limit of constant reaction rate is taken, and a simplified model is obtained from the complete one in Section 4.1. In this limit, the TRS is lost, and so is ORR for mechanochemical coupling. However, the Parodi relation for the cross coupling in the passive polar fluid is still preserved.

Finally, we note that the dynamic equation system for two-phase active polar flows on solid substrates is similar to that for two-phase passive polar flows on solid substrates, but is supplemented by some extra active terms (here the active stress σ^{a} and the active molecular field h^{a}) that break the TRS. This type of diffuse-interface model has been solved numerically as a minimal model for cell motility.^{49,104}

(i) The active fluid is incompressible, satisfying the incompressibility condition, ∇·v = ∂_{x}u + ∂_{z}w = 0, from which we get the local mass conservation equation for the evolution of film height as

(57) |

(ii) The lubrication approximation^{66,77,112,113} is applied to the thin-film dynamics of the active polar droplet on the solid substrate. In the long-wave limit, the characteristic film thickness h_{0} is much smaller than the length scale R_{0} for variations in the x direction, i.e., h_{0}/R_{0} ≪ 1. It follows that the film thickness varies slowly in space with |∂_{x}h| ≪ 1. Given h_{0}/R_{0} ≪ 1 and ∇·v = 0, we obtain that the flow velocity v is approximately along the x direction with w ≪ u.

(iii) The equilibrium contact angle of the droplet θ_{e} is very small such that Young's equation is approximated as

(58) |

(iv) We only consider droplet dynamics with left-right symmetry and the droplet shape is mainly determined by its interfacial energy and the effects of nematic elastic energy can be neglected. This arises when the characteristic thickness of the droplet is much larger than h_{K} ∼ K_{p}/γ with K_{p} being the elastic constant defined in eqn (32b). Then the total energy functional of the droplet is given by

(59) |

(60) |

(v) The active filaments inside the droplet lack a head–tail polarity, that is, p and −p are equivalent, but they can show average nematic alignment. Furthermore, in the case of thin active droplets, the z dependence of p = (p_{x},p_{z}) is determined by the equilibrium equations,

∂_{z}^{2}p_{x} = ∂_{z}^{2}p_{z} = 0, | (61) |

(vi) We assume the planar anchoring conditions at any bounding surfaces with which the active filaments are in contact, that is, the polarization vector p is parallel to the tangent direction of all the bounding surfaces. Here we then have: p = at z = 0 and p = ≈ (1,∂_{x}h) at the free surface z = h, in which is the unit vector along the x-direction and is the unit tangent vector of the free surface of the droplet. This anchoring boundary condition is mainly motivated by the stress-fiber structure in adherent cells^{2} and by the experimental observations^{114} on thin films of amoeboid cells, in which the cells lie in the plane of the glass slide on which they spread and form nematic liquid-crystal structures. The planar anchoring conditions at the free surface have been employed in many previous works.^{98,102,105,106} In contrast, in our diffuse-interface model of active polar droplets in Section 4, we have assumed planar anchoring condition at the solid surface but perpendicular anchoring condition at the free interface, which mimics the orientation of actin filaments in the lamellipodium of migrating cells. Such anchoring boundary conditions have also been used in many previous works.^{43,49,101,104}

Then based on the assumptions in (v) and the planar anchoring conditions in (vi), we can solve the polarization vector p from eqn (61) independently of the flow velocity v for a given drop profile h(x,t) and obtain^{102,105}

p_{x} ≈ 1, p_{z} ≈ (z/h)∂_{x}h. | (62) |

(vii) We consider only the dynamic limit of active polar fluids discussed in Section 4.2, at which the effects of polarization and flow on the ATP hydrolysis are negligibly small, and the rate r ≈ Λ^{−1}Δμ is a constant parameter. In this case, the active stress σ^{a} = −ζpp in eqn (53a) is only a function of the local polarization vector p and it breaks the TRS, driving the system out of equilibrium locally. The work power done by σ^{a} on the thin droplet is approximated to the leading order as

(63) |

(viii) Using the lubrication approximation, the dissipation functional is given to the leading order by^{66,77,112,113}

(64) |

From the above discussions, we then obtain the Rayleighian as

(65) |

(66a) |

∂_{z}p = 0, | (66b) |

p(x,h,t) = p_{0} − γ∂_{x}^{2}h, | (67a) |

η∂_{z}u|_{z=h} = 0, | (67b) |

(68) |

As in the classical problems of thin film fluids,^{66,107,115} the solution of the above closed equation system gives a parabolic profile for u(x,z,t) in the form of

(69) |

(70) |

(71) |

We would like to comment and compare our model for the thin active droplets on solid substrates with other models in the literature^{101,102,106} as follows.

(i) In comparison to the thin-film model by Joanny & Ramaswamy,^{102} we have neglected the effect of nematic elastic energy in determining the droplet shape. As a result, our thin film equation is a limiting case of their model when the droplet thickness is much larger than h_{K} ∼ K_{p}/γ as discussed above near eqn (59). However, if h_{K} is not very small, our boundary condition (68) in the vicinity of the contact line with h_{K} > h ∼ 0 may be problematic and elastic contributions have to be included.

(ii) In the thin-film model by Loisy et al.,^{106} the flows inside the active droplets are induced by the winding of the polarization field. This winding introduces a dramatic change in the orientation of the polarization p along the thickness z-direction: , in which θ is the angle of p relative to the x-axis and ω is an integer winding number that counts the number of quarter turns of p across the drop height. In comparison, in the present work, we have not considered such internal polarization winding and the orientation of p varies along z as θ ∼ p_{z}/p_{x} ≈ (z/h)∂_{x}h (see eqn (62)). Such difference in the variation of p orientation leads to the difference in the final form of thin film equation between the present work and that by Loisy et al.^{106}

(iii) In the recent work by Trinschek et al.,^{101} the authors have proposed a more complete model for active polar droplets, which is similar to our model presented in Section 4 but have introduced one additional active contribution from the treadmilling or self-propulsion of active units in the direction of their polarization. Using their more complete free energy, we can still apply our variational approach and the lubrication approximation to study the more complicated thin film dynamics by following similar methods that we have done for thin films of binary mixtures before.^{66}

We assume that the height profile of the droplet h(x,t) is given by a parabolic function

(72a) |

(72b) |

To achieve an approximate description of the droplet dynamics, the time-dependent parameters θ(t), R(t), and u_{0}(x,t) must be determined by OVP. However, note that these parameters are not independent. Firstly, from the conservation of the droplet area (or mass), A_{0}, that is, , we have

(73) |

(74) |

(75) |

Substituting the droplet profile h(x,t) in eqn (72a) into eqn (59), we obtain the total free energy

(76) |

(77) |

(78) |

(79) |

Then from eqn (77)–(79), we obtain the Rayleighian . Minimizing with respect to Ṙ gives the following evolution equation

(80) |

(81) |

(82) |

Particularly, for k_{act} ≪ 1, the first term on the right-hand side of eqn (80) can be ignored, and the evolution equation for R becomes

(83) |

(84) |

(85) |

(86) |

Note that as mentioned by Joanny & Ramaswamy,^{102} the effects of activity on the droplet spreading enter at the same order in gradients as those of gravity, but with a different dependence on the film height. Furthermore, similar dynamic equation as eqn (80) for thin droplets on solid substrates has been obtained in a very different scenario where evaporation occurs at the free surface of the droplet.^{77} The effects of activity on the droplet spreading enter at the same order as those of evaporation, but with a different dependence on the droplet radius or contact length.

In addition, the formulation and calculations presented here can be readily extended to the thin-film dynamics of three-dimensional droplets on solid substrates, particularly for the spreading dynamics of a droplet with cylindrical symmetry. Furthermore, the effects of nematic energy on the spreading dynamics can also be considered by including nematic elastic energy,^{102} which takes the simple form of .

The first application of OVP presented in Section 3 is about the directional motion of individual active units: a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer moving in a viscous fluid. In the motor/filament system, we consider the mechanochemical cross-coupling which indicates that the system is a reversible machine: it can not only convert chemical energy into mechanical work, but can also convert mechanical work into chemical energy. In the toy microswimmer, we show how directional self-propulsion can be generated by cyclic body-shape oscillations together with front-back asymmetry in hydrodynamic friction. It is shown that mechanochemical cross-coupling in biological systems can be considered in Onsager's framework of non-equilibrium thermodynamics. Activity and the broken time reversal symmetry in active matter are basically resulted from the persistent consumption and conversion of chemical energy, released during spontaneous ATP hydrolysis, into motion or mechanical work.

The second application presented in Section 4 is about the two-phase hydrodynamics for a droplet of active polar fluids, which is composed of suspending contractile or extensile active units such as bacteria, actomyosin units, and animal cells. We use OVP to formulate a diffuse-interface model for an active polar droplet moving on a solid substrate. This hydrodynamic model is thermodynamically consistent in both hydrodynamic equations in the bulk fluid and matching boundary conditions at the solid surface.

The third application presented in Section 5 is about the motion of a thin active polar droplet on a solid substrate in two dimensions. Using the lubrication approximation, we firstly apply OVP to derive the classical thin film equation that has been obtained previously. We then use OVP as an approximation tool to find two scaling laws for the spreading (or dewetting) of the thin active droplet in the respective limits of negligible activity and strong activity. It is interesting to note that the reduced equation obtained for the spreading (or dewetting) dynamics of thin active droplets takes a similar form as that has been obtained previously for the dewetting dynamics of an evaporating droplet on solid substrates.

Below we make a few general remarks and outlook.

(i) Near-equilibrium assumption of OVP. OVP is proposed in Onsager's linear-response framework of non-equilibrium thermodynamics, which is based on the near-equilibrium assumption. However, biological systems are usually far away from equilibrium. Therefore, the validity and the range of the OVP applications should be and can only be justified by solving real biological problems and comparing with quantitative experiments.^{1,4}

(ii) Relationships between OVP/OMVP and other approaches in nonequilibrium thermodynamics. Following the pioneering works of Onsager, there have been various approaches developed for nonequilibrium thermodynamics. In particular, there have been various variational principles formulated for the study of irreversible processes.^{116} More recently, the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) formalism has been proposed as an extension of Hamiltonian's formalism of classical mechanics to nonequilibrium thermodynamic systems with both reversible and irreversible dynamics.^{117} However, a general discussion on the relationships among the various approaches is beyond the scope of this work.

(iii) Applications of Onsager–Machlup variational principle (OMVP). In the end of Section 2.1, we have mentioned that Onsager and Machlup^{60} introduced OMVP in their study of the statistical fluctuations of kinetic paths in the framework of Langevin equation. They have shown that the most probable kinetic path over a certain long-time period is determined by the minimization of a time integral, i.e., the Onsager–Machlup integral in eqn (12). Recently, Doi et al.^{68} proposed that OMVP can be used to approximate the long-time dynamics of nonequilibrium systems. However, in this work, we have not given applications of using OMVP to find approximate solutions for long-time behaviors such as steady states. We mention two potential applications of OMVP as follows: the steady-state for the wave propagation and sustained oscillations observed in migrating cells;^{29–31} the steady-state (spontaneous) retraction dynamics of an injured axon^{118} or a laser-cutting stress-fiber bundle in adherent cells.^{119}

(iv) Applications of OVP and OMVP to more specific biological problems. The applications considered in this work are mostly toy models or simplified models of mostly theoretical interests. We are now trying to apply the extended form of OVP to more specific biological problems such as cell spreading, cell curvotaxis, wound closure, tissue folding, and so on. However, in these real systems, we usually need to involve many more complex active processes^{16} in addition to active forces and cyclic body-shape oscillations, such as tensional homeostasis,^{2} cell division and apoptosis,^{13} topological cell rearrangements,^{16} memory effects.^{1,17}

In summary, the variational method proposed in this work about incorporating biochemical activity into OVP will help to construct thermodynamically-consistent models and to find approximate dynamic solutions in active soft matter. Particularly, this will help to deepen our understanding of the emergent structure and dynamic behaviors of real in vivo biological systems such as bacteria suspensions, individual animal cells and cell aggregates (or tissues).^{1,2,13}

(i) Fluxes are chosen to be r, P, (or the flow velocity v) and the corresponding forces are Δμ, h, σ^{v}, respectively, as taken in Section 4 of this work. Here all the three fluxes have the same time parity.

(ii) Fluxes are chosen to be r, P, σ^{v} (the momentum flux) and the corresponding forces are Δμ, h, , respectively, as taken by Marchetti et al.^{1} Here the time parity of the flux σ^{v} is different from the other two fluxes r and P. Furthermore, note that in this case, the new pair of flux σ^{v} and force is a swap of the pair of flux and force σ^{v} in the first choice (i).

The symmetry of Onsager matrix coupling fluxes and forces depends on the time parity (i.e., the time-reversal signature) of the fluxes:^{1,50} the Onsager coupling matrix is symmetric for fluxes of the same time parity and is antisymmetric for fluxes of opposite time parity. Therefore, for the first choice (i) with fluxes of the same time parity, the Onsager coupling matrix is symmetric. In comparison, for the second choice (ii), the cross-coupling coefficients between the flux σ^{v} and the flux r or between the flux σ^{v} and the flux P are both antisymmetric as shown in Marchetti et al.^{1}

Given the non-unique choice of flux–force pairs, we will make some general discussions in this Appendix about the consequences of different choices of thermodynamic fluxes and forces.

(A1) |

If alternatively, we swap X_{2} with _{2}, that is, we choose the fluxes to be _{1} and X_{2}, and the corresponding forces are X_{1} and _{2}. Then the time parities of the two fluxes _{1} and X_{2} are now different. In this case, the linear force–flux relations become

(A2) |

(A3) |

X_{i} ≡ − ∂/∂α_{i}, Y_{i} ≡ − ∂/∂β_{i} | (A4) |

X = X^{d} + X^{r}, Y = Y^{d} + Y^{r}. | (A5) |

(A6) |

(A7) |

The dissipation function is given by

(A8) |

(A9) |

(A10) |

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## Footnotes |

† It is interesting to mention that in active matter, the presence of active forces and the breakdown of TRS can significantly change the statistical behaviors of the system such as the distribution of α and barrier crossing kinetics. The dynamics of downhill and uphill processes are very different even for passive systems, and hence must be treated using their respective variational approaches. In this work, we are interested in the active dynamics that would reduce to downhill processes if the activity vanishes. However, the fluctuation effects can still be investigated in a variational framework.^{46,86} |

‡ Private communications with M. Doi. |

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