Spatiotemporal control of structure and dynamics in a polar active fluid†

Saptorshi Ghosh ^a, Chaitanya Joshi ^b, Aparna Baskaran *^a and Michael F. Hagan *^a
aMartin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02453, USA. E-mail: aparna@brandeis.edu; hagan@brandeis.edu
^bDepartment of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA

First published on 14th August 2024

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I. Introduction

Active systems are a diverse class of non-equilibrium assemblies composed of anisotropic components that transform stored or ambient energy into motion. Idealized realizations that have contributed to the development and refinement of this conceptual framework include bacterial suspensions,^1–3 minimal systems of purified cytoskeletal proteins,^4–11 synthetic self-propelled colloids,^12–14 swarming bacterial cells,^15–19 and model tissues and cell sheets.^20–32 In these systems, the interplay between internal activity and the interactions among the active agents results in a wide range of emergent collective behaviors.^33–38 These behaviors emerge spontaneously without requiring a central control mechanism. However, there is typically no means to select which behavior emerges or to switch between states, which significantly limits the functionality of active materials.

Recent experimental advances have put the objective of control within our reach. Experiments have demonstrated that light can be used as a control field to assemble self-limited functional structures in active colloids^39,40 and to exert spatiotemporal control of motility induced phase-separation (MIPS).^41,42 By shining sequences of light that vary in space and time on active materials constructed with light-activated motor proteins, researchers can control the average speed of active flows^43,44 and steer defects in active nematics,⁴⁵ and drive the formation and movement of asters in isotropic suspensions.⁴⁶

Theoretical progress toward functionalizing active matter has taken two paths. The first has been to impose spatiotemporal activity patterns and observe their effects on the system dynamics,^45,47–49 thereby identifying easily accessible target states. The second is to use the framework of optimal control^50,51 and optimal transport⁵² to identify spatiotemporal activity patterns that will drive the system to a pre-chosen target dynamics.^47,53,54

The work described in this article belongs in this second class. We study the optimal control theory of the classic active matter theory, a dry 2D active polar fluid, first considered by Toner and Tu.^55,56 We treat the convective speed of the active particles as the control parameter and identify control solutions that drive the system to targeted steady states, including forcing an aster to move to a given spatial location, causing propagating stripes to reorient along a particular direction, and driving a propagating stripe to convert into a stationary aster. In the previous work most closely related to this one, Norton et al.⁵³ obtained a control solution to switch a confined active nematic between two symmetric attractors (changing handedness of a circular flow state). Here, we show that an optimal control framework can solve much more diverse problems, including switching among attractors with very different broken symmetry patterns and driving the system into states which are not stable attractors at a given set of parameters. Further, we analyze the identified optimal solutions to uncover generic control principles that are broadly applicable to active polar fluids.

This paper is laid out as follows. In Section II, we review the hydrodynamic theory and describe the key features of the steady states that arise in the absence of control. Section III describes the method for implementing optimal control theory. In Section IV, we report the results of the control solutions for driving the system to particular steady states or switching between them. Then, we analyze the control solutions in terms of the dynamical equations of motion to identify the essential mechanisms that drive the system into the desired behavior, with the goal of identifying generic principles. We also investigate how robust the optimal control solutions are to errors arising from experimental noise and inaccuracies in model parameters. Finally Section V concludes with a discussion on testing these results in experiments.

II. Model

We consider a macroscopic description of an active polar fluid in 2D, in terms of a conserved density field ρ(r,t) and the polarization field τ(r,t) = ρ(r,t)P(r,t), a vector characterizing orientational order in the fluid. As noted by Toner and Tu,⁵⁵ the order parameter is also a velocity that convects mass in an active fluid. While a number of distinct dynamical equations for these hydrodynamic quantities have been considered in the literature,^57–60 the particular model we study is of the form

∂tρ = −∇·(ωτ − D∇ρ)	(1)

∂tτ + λ1τ·∇τ = −ν(a2(ρ) + a4(ρ)|τ|2)τ − ∇(ωρ) + K∇2τ + λ2τα∇τα + λ3τ∇·τ.	(2)

We briefly discuss the physics it captures and the emergent phenomenology that results from it. Extensive studies can be found in prior work on this model.^61–65

The density dynamics eqn (1) is an advection-diffusion equation where the advective velocity is proportional to the orientational order parameter P(r,t). The dynamics of the orientational order has three features: (i) It has a self-convection term with coefficent λ₁, encoding the absence of Galilean invariance in the ‘dry’ theory.^55,56 (ii) It has terms consistent with model A dynamics that drive the system downhill on a free energy landscape where , encoding the fact that the flocking occurs due to spontaneous symmetry breaking. (iii) It contains a hydrostatic pressure term of the form , where λ₂ encodes the tendency for elongated self-propelled particles to splay due to recollision events.^62,66,67

In this study of spatiotemporal control, we choose a₂(ρ) = (1 − ρ/ρ_c) and a₄(ρ) = (1 + ρ/ρ_c)/ρ², yielding a continuous mean-field transition from an isotropic τ to a homogeneous, polar or a swarming state (|τ| > 0) at the critical density ρ = ρ_c. For simplicity, we choose D = K. Without loss of generality we set the critical density ρ_c = 1. We further reduce the number of independent parameters and set λ₁ = λ₂ = λ₃. We set the unit of time as τ₀ = ν⁻¹, the relaxation time scale of the orientation field, and the unit of length as . The simplified dimensionless equations are:

∂tρ = −∇·(ωτ − ∇ρ)	(3)

∂tτ = −(a2(ρ) + a4(ρ)|τ|2)τ − ∇(ωρ) + ∇2τ + λ(τα∇τα + τ∇·τ − τ·∇τ).	(4)

These reduced equations involve three parameters: (1) The self-propulsion speed or activity, ω, is the main control variable and dictates both density advection and an effective pressure/compressibility; (2) λ captures the interplay between interparticle interactions and activity; and (3) the mean density, ρ₀, which is set by the initial condition. We note that if these equations are derived from a microscopic model of self-propelled particles with aligning interactions, the {λ_i} parameters would be complex functions of multiple parameters from the microscopic model, including the activity strength. To simplify the analysis we have set them all equal to λ and independent of activity. The system behaviors are not qualitatively affected by neglecting this dependence.^62,66,68,69 Additional discussion of these equations and other related models can be found in ref. 62, 66 and 67 and review articles.^37,38,70

The phenomenology of this model is described in ref. 62 and summarized in Fig. 1a. For the purposes of this work, we note that the dynamics of this system admits two inhomogeneous steady states: (i) propagating stripes composed of ordered swarms moving through a disordered background, which are referred to as polar drops elsewhere in the literature,^63,71 when ω/λ ≫ 1, and (ii) a stationary high density aster, again in an isotropic background, when λ/ω ≫ 1. Both of these states arise close to the threshold density for orientational ordering (which we set to ρ_c = 1), and correspond to the system phase separating into a dense ordered phase and a dilute disordered phase. In this study, we fix the homogeneous density at ρ = 1.07 and consider the problem of controlling the inhomogeneous steady states using spatiotemporal patterning of the convective strength ω, referred to as the activity or control parameter in the rest of this paper.

This model has the virtue of mathematical simplicity and a minimal number of parameters, thus enabling physical insight from the optimal control solutions. At the same time, the states we seek to control are directly realizable in experiments (see Section V). Further, in the ESI,† we show that this theory is linearly controllable on short length scales. Thus it is ideal for investigating applications of optimal control in active systems.

III. Optimal control

Spatiotemporal control of the system involves identifying an activity field ω(r,t) in an interval t ∈ [0,t_F] such that the state of the system evolves from some initial condition ρ(r,0), τ(r,0) to a chosen target state ρ*(r,t), τ*(r,t) within the control window [0,t_F]. In the optimal control framework, we identify such a solution by minimizing the scalar objective function

	(5)

subject to the constraints that the dynamical fields ρ(r,t), τ(r,t) obey eqn (3) and (4) at every time point in the control window. We include the activity as ω = ² to constrain the solution to positive activity values, since ω < 0 is unphysical in this system. The term (² − ₀²)² penalizes deviations of the magnitude of the control variable ω from the preferred (i.e. baseline) value of the activity ω₀ = ₀², and thereby restrains the control solution to the vicinity of a chosen value of activity. The terms and promote smoothness of the activity field in space and time. To simplify the presentation of results, we set K = 0 and thus do not penalize time-variations.

The terms (ρ − ρ*)² and (τ − τ*)² measure the deviations from the target state, ρ* and τ*. We constrain our search of optimal state trajectories to those that obey the system dynamics by introducing Lagrange multipliers, η and ν, which are adjoint variables for ρ and τ. These dynamical constraints are enforced in the optimization by adding them to the cost function as

	(6)

The necessary condition for optimality is ,^72,73 so , , , , , , . The first two conditions return eqn (3) and (4) governing ρ and τ. The following two conditions yield the dynamical equations for the adjoint variables η and ν,

∂tη = C(ρ − ρ*) − ∇2η − 2∇·ν + (δρa2(ρ) + δρa4(ρ)|τ|2)(ν·τ).	(7)

∂tν = D(τ − τ*) + (a2(ρ) + a4(ρ)|τ|2)ν + 2a4(ρ)τ(ν·τ) − λ[−τ∇·ν + 2ν∇·τ − ∇(ν·τ) + τ·∇ν − να∇τα] − ∇2ν − 2∇η.	(8)

with boundary conditions at t_F: {η,ν}(r,t_F) = 0, and periodic boundary conditions on the domain. yields an equation to update the control input as,

2A(2 − 02) − 2B∇22 − 2K(d22/dt2) − 2τ·∇η − 2ν·∇ρ = 0.	(9)

We use the direct-adjoint-looping (DAL) method⁷⁴ to minimize the cost function under the constraint that the dynamics satisfies eqn (3) and (4), to yield the optimal schedule of activity in space and time (see ESI† Sections SIV and SV for more details). Specifically, we construct an initial condition by performing a simulation with unperturbed dynamics (eqn (3) and (4)) until reaching steady-state, at a parameter set that leads to a desired initial behavior. We construct a target configuration in the same manner, using a different parameter set that leads to the desired target behavior. We also specify a time duration t_F over which the control protocol will be employed, and an initial trial control protocol. We then perform a series of DAL iterations; in each iteration the system and the adjoint dynamics are integrated from the initial condition for time t_F under the current control protocol, and the cost function (eqn (5)) is computed from the resulting trajectory. The adjoint equations are integrated backwards in time to propagate the residuals. After each backward run, the control protocol is updated via gradient descent, , to minimize the cost function. We employ Armijo backtracking⁷⁵ to adaptively choose the step-size for gradient descent and to ensure convergence of the DAL algorithm. Once the optimal spatiotemporal activity pattern has been computed, performing a forward integration using eqn (3) and (4), the same initial condition, and ω = ² will yield the solution trajectory. We have implemented this entire calculation in the open-source Python finite element method solver FEniCS.⁷⁶ We have provided the associated source code and data here.

IV. Results

Using the optimal control framework described in Section III and the hydrodynamic equations eqn (3) and (4), we have computed spatiotemporal control solutions that steer the system state toward the target configuration, for each of the target behaviors shown in Fig. 1(c)–(e). In this section, we describe these calculations, and physical insights that can be learned by studying the computed control solutions.

A. Aster advection

First, starting in a parameter regime where a stationary aster is stable (ω = 0.05 and λ = 0.8), we seek to advect an aster to a new location. The control problem specifies the initial and final states of the system as well as the elapsed time; that is, the spatial dependence of the density and polarization fields at every point in space at times t = 0 and t = t_F = 2000.

Fig. 2 summarizes the results of this computation. Fig. 2a and b respectively show the time evolution of the system configuration and the applied control field that drives the transformation. At early times, the applied control is strongest at the aster core while it is lowest in front of the aster along the direction we seek to move it. The aster then elongates to assume a comet-like shape, with a denser, polar-ordered head (see snapshots at t = 50, 100), as it advects toward the target location. Note that the activity is largest to the rear of the aster in this region. Thus, the control solution pushes (rather than pulls) the aster.

To quantify how the position and profile of the aster change over the course of advection, we measure its center-of-mass position (x_COM, y_COM) and asphericity. Here, we track the y-coordinate of the center of mass, which is calculated as , and the asphericity is given by the ratio of eigenvalues of shape tensor: , where Latin indices denote grid points and Greek indices correspond to Cartesian coordinates, and r_ij is the distance of the ijth grid point from the center of mass. As shown in Fig. 2c, the control window naturally partitions into two stages. During the initial stage (0 < t ≲ 100) the aster rapidly changes shape into the comet-like configuration, as seen by the decrease in its asphericity, while simultaneously undergoing advection, moving towards the target point. Then, over the remaining long time window (100 ≲ t < t_F) the aster reforms slowly, the asphericity increases back to 1, and it moves the remaining small distance to the target position.

For further description of the aster profile during these stages, we present the angle of the polarization field θ as a function of the azimuthal angle ϕ around the aster core at four time points in Fig. 2d. At the initial time (t = 0) the system is radially symmetric with polarization vectors pointing toward the aster center, while by t = 50 and t = 100 the symmetry of θ in the top [0,π] and bottom [−π,0] quadrants is broken, with more polarization at the bottom points toward the advection direction, and the front-end remains aster-like with a radial configuration. The orientation returns to the aster configuration by t = t_F.

Further, we can understand the control solution physically by noting that the dynamics of τ is such that gradients in the control field ω create a torque on the orientation field, i.e., ∂_tτ ∼ −ρ∇ω or equivalently ∂_tθ ∼ ∇ω × τ. We then calculate the integral of the torque over two domains, Ω_L: [x ∈ {x₀ − 10, x₀}, y ∈ {0, L_y}] on the left and and Ω_R: [x ∈ {x₀, x₀ + 10}, y ∈ {0, L_y}] to the right, with x₀ = 40 in our case (Fig. 2e). When the aster unwinds and advects, the region to the left has a positive torque (countercockwise rotation) and the region on the right experiences negative torque (clockwise rotation), which correspond to the partial unwinding of the aster. This is also illustrated by the snapshot shown for t = 35, where the dashed line (x = 40) represents the axis along the aster's motion. During the subsequent reformation phase, as the aster regains circular symmetry, the profile winds back such that the left/right subdomains experience clockwise/counterclockwise torque respectively. The snapshot shown at t = 1000 illustrates this behavior. Eventually, the system relaxes sufficiently close to an unperturbed aster configuration that the net torque becomes zero.

B. Changing the direction of propagating stripes

Next, starting at a parameter set for which stripes are stable, we obtain an activity profile to change the stripe propagation direction, with initial direction along + and a target direction diagonally oriented along 45°. Note that we obtain similar results for any target orientation, including reversing the stripe direction by 180°. Fig. 4a and b show the system configurations and corresponding control solutions at several time points. At t = 0, the applied activity is strongest at the leading edge of the stripe, and decays over the width of the leading boundary layer (i.e., the region where the polarization changes from isotropic to uniform). These gradients in activity lead to both melting (reduction of the magnitude of polarization) and turning (reorientation of polarization toward the target direction). At the next two time-points (t = 100, 200) the activity has decreased in magnitude, but continues to turn the polarization. By t = 500 the activity is nearly uniform in space and approaching its steady-state value of 0.4. However, some curvature remains near the leading edge of the stripe. The stripe has completely reformed by the last time point (t = 1400). Notably, the timescale for dissolving and reorienting the stripes at this periodic box size 256 × 256 (ESI,† Movie S4) was about 500 in our dimensionless units, which is 2 orders of magnitude lower than obtaining stripes from a random homogenous initial condition in the absence of control.

While a straightforward route to reorienting a strip would be to melt the stripe to an isotropic domain and then have it reform in the new direction, this is not the optimal solution given by the control theory. Starting with the density equation, eqn (3), we investigate the primary forces influencing density evolution during the stripe reorientation process. For this we choose a subdomain 0 < x ≤ 50, 0 < y ≤ 100 within the simulation box of size 256 × 256. We integrate each of the three terms in eqn (3) over the specified subdomain as a function of time (Fig. 4c): −ω∇·τ, which governs the convection of active particles at convection speed ω; −τ·∇ω, which determines the local density dynamics due to gradients in activity; and ∇²ρ, which determines the diffusion due to density gradients. We find that, at all times, the dominant contribution arises from self-propulsion, −ω∇·τ; contributions from gradients in activity and density have negligible contributions. Thus, we conclude that activity gradients are not the driving force for the density dynamics, but rather lead to the torques that reorient the polarization field, as described in our analysis in Section IV A. To quantify the effect of active torque in this case, we illustrate in Fig. 4c that as the difference in orientation between the initial state and target state increases, the active torque also increases, and as the system settles into the target orientation, the active torque goes to 0.

This result can be qualitatively understood as follows. A trajectory which proceeds through an isotropic intermediate state would require large magnitudes and gradients of activity to melt the stripe and then reform it in a new direction. Instead, the optimal solution takes advantage of the fact that a stripe arises through a spontaneous breaking of rotational symmetry, and thus requires only small torques (and correspondingly small orientational biases in gradients of activity) to reorient its direction.

C. Remodeling stripe to aster

So far, we have considered cases where the initial and target states are both steady states of the uncontrolled forward dynamics of our system at specified parameters. To demonstrate the power of the control theory, we start in a parameter regime in which propagating stripes are stable, and obtain an activity profile that drives the system into a stationary aster (not a steady state at these parameters). For the initial condition, we run to steady-state under parameters that lead to propagating stripes, λ = 0.6 and ω = 0.4. To obtain a configuration to specify the target state, we perform an independent simulation in which we obtain a stationary aster steady-state with λ = 0.6 and ω = 0.04. Fig. 5 shows the trajectory and corresponding control solution. We see that at early times the applied activity is strongest at the top- and bottom-edges of the leading boundary layer of the stripe, which bends the polarization vectors toward the core of the target aster. The magnitude of the activity field decreases quickly in time, but the spatial profile remains similar, thus continuing to steer polarization toward the core, and decreasing the net momentum in the + direction. Due to the coupling between ρ and τ (see eqn (3)), the resulting gradients and polarization lead to convection of density toward the core. By t = 1500, there is a density maximum at the core and the system has achieved a radially symmetric state, which leads to a balance of propulsion forces and thus a stationary state.

We note that this control problem also differs from the previous one (reorienting stripes) in that the initial and final state have different symmetries. However, the optimal solution still avoids isotropic intermediate states, since these would require significantly larger magnitudes and gradients of activity.

D. Robustness of the optimal control solution to noise and parameter variations

Since models are never completely accurate and noise is inevitable in any experimental system, we investigated the robustness of our control solution to noise and parameter variations. Because our implementation uses deterministic PDEs, we tested the effects of noise by perturbing the initial condition for the aster translocation problem presented in Fig. 2. Specifically, we added Gaussian noise with a relative magnitude to the values of ρ, τ_x, and τ_y at each pixel, and then integrated the dynamics with the control protocol computed in the absence of noise. Fig. 6 shows the performance of the control solution. We plot the deviation from the trajectory integrated with zero noise and λ = 0.8 (see Fig. 2a):

	(10)

where (ρ*(r,t), τ*(r,t)) is the trajectory in Fig. 2a and the error is normalized by the spatial extent Ω. The inset shows the deviation of the final state from the target , as a function of . We see that noise has a relatively small effect on the performance up to a magnitude of about , after which deviations in the objective function rise dramatically. However, even with and a relatively large value of at t_F, the final state is remarkably close to the target in all qualitative aspects (see Fig. 6c), with a well-formed aster close to the target position. Thus, we conclude that the optimal control solution is robust to noise, at least in the initial condition.

To investigate the effect of model parameter errors on the optimal control solution, we performed trials for the aster advection problem shown in Fig. 2 with varying λ. That is, we used the same initial condition and the optimal protocol computed with λ = 0.8 (i.e. the activity sequence shown in Fig. 2b), but we integrated the equations of motion with different values of λ. As shown in Fig. 7, we see small errors for variations in λ smaller than 10%, beyond which the errors rise significantly. However, similar to the noise results, we observe well-formed asters even for cases with large variations, although their final positions deviate from the target state (Fig. 7c).

V. Discussion and conclusions

We demonstrate an optimal control theory framework that can prescribe activity patterns to guide an active material into desired emergent behaviors, focusing on an active polar fluid as a model system. The capabilities include programmed switching among dynamical attractors with very different dynamics and distinct broken symmetry patterns, and reprogramming the dynamics of existing attractor states. As an example of the former, we identify a spatiotemporal activity pattern that converts a propagating stripes state into a stationary aster. As an example of the latter, we show that a stationary aster can be programmed to self-advect to a new target configuration, either via an arbitrary trajectory or along a prescribed path. Similarly, propagating stripes can be forced to reorient in arbitrary directions. Depending on the choice of terms and weights in the objective function, the spatiotemporal variations of the control inputs can be regulated to limit experimental cost or ensure smooth trajectories.

Further, we show that the optimal control solutions are robust to noise. In particular, perturbing the initial condition by up to 20% leads to minimal quantitative deviations from the target behavior, and the solution remains qualitatively accurate for significantly larger perturbations. Also, we note that additional strategies can be employed for experimental systems where larger noise sources or systematic errors are unavoidable. This includes integrating closed-loop control components. For example, one can observe the current state of the system at regular intervals along a trajectory, and recompute the optimal control solution using the current state as the initial condition. Alternatively, one can add linear feedback terms that analyze deviations from the pre-computed optimal control trajectory.⁷⁸

In addition to directly applying the computed activity protocols, examination of their forms provides both fundamental and practical insights into controlling active materials. In particular, we show how the spatial gradients in the applied activity field lead to localized torques which rotate polarization directions, leading to the programmed reformulation of the pattern of interest (e.g. aster or stripe). Unsurprisingly, the form of the trajectory is different depending on the task being encoded for – changing the broken symmetry state of the system (e.g. stripe-to-aster, Fig. 5) requires very different spatial arrangements of active torques then advection (Fig. 2 and 3) or reorientation (Fig. 4). Notably however, the applied activity field and corresponding trajectory also depend strongly on the time allowed for the transformation. In the example of advecting the aster over a distance many times its size (Fig. 3), the aster mostly retains its form throughout the course of the trajectory when moving at moderate speed, but when forced to complete the journey 4× faster, the applied activity field reshapes the aster into a localized flock or swarm, which reformulates into an aster upon reaching the target position.

The states we seek to control are realizable in experiments. Propagating concentration waves of aligned self-propelled particles have been observed in dense actin motility assays^79–82 and in self-chemotactic bacterial systems.^83–91 Asters are ubiquitous in cell biology in processes such as the formation of the mitotic spindle, oogenesis, and plant cell cytokinesis.^92–99 They can be reliably obtained in in vitro suspensions of cytoskeletal filaments and motor proteins.^{6,46,95,100–110} In such systems, activity can be controlled in space and time by constructing active materials with optogenetic molecular motors. For example, the closely related microtubule-based active nematic system has been modified to contain light-activated kinesins clusters, so that the activity is proportional to the local light intensity.^43–46 The researchers then used a digital light projector to shine programmed spatiotemporal sequences of light on the sample.^43–46 Our optimal control protocol can be used to determine the spatiotemporal light sequence that will drive such a system into a desired state. Due to inevitable accuracies in hydrodynamic descriptions of active matter systems and experimental noise, it is important that we found our optimal control solutions to be robust against noise and parameter inaccuracies (Fig. 6 and 7). Further, optimal control solutions can be corrected by periodically observing the state of the system and applying feedback control. For example, feedback can be implemented by adding linear restoring forces that drive the system toward the computed trajectory, or by recomputing the optimal control solution based on the current state of the system.

The optimal control framework presented here is highly generalizable, and can be readily applied to any system provided there is a means to externally actuate the system and there is a reasonably accurate continuum model. Importantly, the control variable need not be limited to the activity field, since the objective function can be extended to include any property of the material that can be actuated. With the recent success of automated PDE learning tools in discovering continuum models for active systems (e.g. ref. 111–113), applications need not be limited to systems with accurate models already available. Furthermore, since model discovery tools work better when provided with a variety of data, including from non-steady-state observations, we anticipate that combining model discovery tools with optimal control could be a powerful approach to both discover more accurate models and enhance the reliability of the control solutions.

Data availability

The code and data associated with this work are provided in: https://github.com/ghoshsap/optimal-control-activepolarfluid/tree/main.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was primarily supported by the Department of Energy (DOE) DE-SC0022291. Preliminary simulations of the system without control were supported by the National Science Foundation (NSF) DMR-1855914 and the Brandeis Center for Bioinspired Soft Materials, an NSF MRSEC (DMR-2011846). Computing resources were provided by the NSF XSEDE allocation TG-MCB090163 (Stampede and Expanse); the National Energy Research Scientific Computing Center (NERSC), a Department of Energy Office of Science User Facility using NERSC award BES-ERCAP0026774; and the Brandeis HPCC which is partially supported by the NSF through DMR-MRSEC 2011846 and OAC-1920147. AB acknowledges support from NSF – 2202353 and the hospitality of the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452.

References

1. K.-A. Liu and I. Lin, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 86, 011924 CrossRef PubMed.
2. A. Czirók and T. Vicsek, Phys. A, 2000, 281, 17–29 CrossRef.
3. C. Pierce, H. Wijesinghe, E. Mumper, B. Lower, S. Lower and R. Sooryakumar, Phys. Rev. Lett., 2018, 121, 188001 CrossRef PubMed.
4. D. Needleman and Z. Dogic, Nat. Rev. Mater., 2017, 2, 1–14 Search PubMed.
5. G. Sarfati, A. Maitra, R. Voituriez, J.-C. Galas and A. Estevez-Torres, Soft Matter, 2022, 18, 3793–3800 RSC.
6. F. Ndlec, T. Surrey, A. C. Maggs and S. Leibler, Nature, 1997, 389, 305–308 CrossRef.
7. T. Surrey, F. Nédélec, S. Leibler and E. Karsenti, Science, 2001, 292, 1167–1171 CrossRef CAS PubMed.
8. M. L. Gardel, K. E. Kasza, C. P. Brangwynne, J. Liu and D. A. Weitz, Methods Cell Biol., 2008, 89, 487–519 CAS.
9. M. L. Gardel, I. C. Schneider, Y. Aratyn-Schaus and C. M. Waterman, Annu. Rev. Cell Dev. Biol., 2010, 26, 315–333 CrossRef CAS PubMed.
10. M. Dogterom and G. H. Koenderink, Nat. Rev. Mol. Cell Biol., 2019, 20, 38–54 CrossRef CAS.
11. M. Soarese Silva, M. Depken, B. Stuhrmann, M. Korsten, F. C. MacKintosh and G. H. Koenderink, Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 9408–9413 CrossRef PubMed.
12. C. G. Wagner, M. M. Norton, J. S. Park and P. Grover, Phys. Rev. Lett., 2022, 128, 028003 CrossRef CAS.
13. W. Wang, W. Duan, S. Ahmed, A. Sen and T. E. Mallouk, Acc. Chem. Res., 2015, 48, 1938–1946 CrossRef CAS PubMed.
14. J. R. Gomez-Solano, S. Samin, C. Lozano, P. RuedasBatuecas, R. van Roij and C. Bechinger, Sci. Rep., 2017, 7, 14891 CrossRef.
15. O. Hallatschek, S. S. Datta, K. Drescher, J. Dunkel, J. Elgeti, B. Waclaw and N. S. Wingreen, Nat. Rev. Phys., 2023, 1–13 Search PubMed.
16. S. Liu, S. Shankar, M. C. Marchetti and Y. Wu, Nature, 2021, 590, 80–84 CrossRef CAS PubMed.
17. A. E. Hamby, D. K. Vig, S. Safonova and C. W. Wolgemuth, Sci. Adv., 2018, 4, eaau0125 CrossRef CAS PubMed.
18. B. Ni, R. Colin, H. Link, R. G. Endres and V. Sourjik, Proc. Natl. Acad. Sci. U. S. A., 2020, 117, 595–601 CrossRef CAS.
19. E. Lauga and T. R. Powers, Rep. Prog. Phys., 2009, 72, 096601 CrossRef.
20. D. Gonzalez-Rodriguez, K. Guevorkian, S. Douezan and F. Brochard-Wyart, Science, 2012, 338, 910–917 CrossRef CAS PubMed.
21. S. Henkes, K. Kostanjevec, J. M. Collinson, R. Sknepnek and E. Bertin, Nat. Commun., 2020, 11, 1405 CrossRef CAS PubMed.
22. E. Fodor and M. C. Marchetti, Phys. A, 2018, 504, 106–120 CrossRef.
23. J. W. Nichol and A. Khademhosseini, Soft Matter, 2009, 5, 1312–1319 RSC.
24. G. Zhang, R. Mueller, A. Doostmohammadi and J. M. Yeomans, J. R. Soc., Interface, 2020, 17, 20200312 CrossRef.
25. A. Doostmohammadi, S. P. Thampi and J. M. Yeomans, Phys. Rev. Lett., 2016, 117, 048102 CrossRef PubMed.
26. T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeomans and B. Ladoux, Nature, 2017, 544, 212–216 CrossRef PubMed.
27. X. Trepat, M. R. Wasserman, T. E. Angelini, E. Millet, D. A. Weitz, J. P. Butler and J. J. Fredberg, Nat. Phys., 2009, 5, 426–430 Search PubMed.
28. X. Trepat and E. Sahai, Nat. Phys., 2018, 14, 671–682 Search PubMed.
29. G. Duclos, S. Garcia, H. Yevick and P. Silberzan, Soft Matter, 2014, 10, 2346–2353 RSC.
30. C. Blanch-Mercader, V. Yashunsky, S. Garcia, G. Duclos, L. Giomi and P. Silberzan, Phys. Rev. Lett., 2018, 120, 208101 CrossRef PubMed.
31. S. Garcia, E. Hannezo, J. Elgeti, J.-F. Joanny, P. Silberzan and N. S. Gov, Proc. Natl. Acad. Sci. U. S. A., 2015, 112, 15314–15319 CrossRef CAS PubMed.
32. M. Ghibaudo, A. Saez, L. Trichet, A. Xayaphoummine, J. Browaeys, P. Silberzan, A. Buguin and B. Ladoux, Soft Matter, 2008, 4, 1836–1843 RSC.
33. T. Vicsek and A. Zafeiris, Phys. Rep., 2012, 517, 71–140 CrossRef.
34. H. Chaté, Annu. Rev. Condens. Matter Phys., 2020, 11, 189–212 CrossRef.
35. A. Doostmohammadi, J. Ignés-Mullol, J. M. Yeomans and F. Sagués, Nat. Commun., 2018, 9, 1–13 CrossRef CAS PubMed.
36. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe and G. Volpe, Rev. Mod. Phys., 2016, 88, 045006 CrossRef.
37. G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nardini, F. Peruani, H. Löwen, R. Golestanian, U. B. Kaupp and L. Alvarez, et al. , J. Phys.: Condens. Matter, 2020, 32, 193001 CrossRef PubMed.
38. M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao and R. A. Simha, Rev. Mod. Phys., 2013, 85, 1143 CrossRef.
39. A. Aubret, M. Youssef, S. Sacanna and J. Palacci, Nat. Phys., 2018, 14, 1114–1118 Search PubMed.
40. J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine and P. M. Chaikin, Science, 2013, 339, 936–940 CrossRef PubMed.
41. J. Arlt, V. A. Martinez, A. Dawson, T. Pilizota and W. C. Poon, Nat. Commun., 2018, 9, 768 CrossRef.
42. G. Frangipane, D. Dell’Arciprete, S. Petracchini, C. Maggi, F. Saglimbeni, S. Bianchi, G. Vizsnyiczai, M. L. Bernardini and R. Di Leonardo, eLife, 2018, 7, e36608 CrossRef PubMed.
43. L. M. Lemma, M. Varghese, T. D. Ross, M. Thomson, A. Baskaran and Z. Dogic, PNAS Nexus, 2023, 2, pgad130 CrossRef PubMed.
44. Z. Zarei, J. Berezney, A. Hensley, L. Lemma, N. Senbil, Z. Dogic and S. Fraden, Soft Matter, 2023, 19, 6691–6699 RSC.
45. R. Zhang, S. A. Redford, P. V. Ruijgrok, N. Kumar, A. Mozaffari, S. Zemsky, A. R. Dinner, V. Vitelli, Z. Bryant and M. L. Gardel, et al. , Nat. Mater., 2021, 20, 875–882 CrossRef CAS.
46. T. D. Ross, H. J. Lee, Z. Qu, R. A. Banks, R. Phillips and M. Thomson, Nature, 2019, 572, 224–229 CrossRef CAS.
47. S. Shankar, L. V. Scharrer, M. J. Bowick and M. C. Marchetti, Proc. Natl. Acad. Sci. U. S. A., 2024, 121, e2400933121 CrossRef CAS PubMed.
48. M. Nasiri, H. Löwen and B. Liebchen, Europhys. Lett., 2023, 142, 17001 CrossRef.
49. M. Kneević, T. Welker and H. Stark, Sci. Rep., 2022, 12, 19437 CrossRef PubMed.
50. S. L. Brunton and J. N. Kutz, Data-driven science and engineering: Machine learning, dynamical systems, and control, Cambridge University Press, 2022 Search PubMed.
51. G. E. Dullerud and F. Paganini, A course in robust control theory: a convex approach, Springer Science & Business Media, 2013, vol. 36 Search PubMed.
52. C. Villani, et al., Optimal transport: old and new, Springer, 2009, vol. 338 Search PubMed.
53. M. M. Norton, P. Grover, M. F. Hagan and S. Fraden, Phys. Rev. Lett., 2020, 125, 178005 CrossRef CAS.
54. C. Sinigaglia, F. Braghin and M. Serra, Phys. Rev. Lett., 2024, 132, 218302 CrossRef CAS PubMed.
55. J. Toner and Y. Tu, Phys. Rev. Lett., 1995, 75, 4326 CrossRef CAS PubMed.
56. J. Toner and Y. Tu, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 58, 4828 CrossRef CAS.
57. H. Y. Lee and M. Kardar, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2001, 64, 056113 CrossRef CAS PubMed.
58. K. Husain and M. Rao, Phys. Rev. Lett., 2017, 118, 078104 CrossRef PubMed.
59. D. Geyer, A. Morin and D. Bartolo, Nat. Mater., 2018, 17, 789–793 CrossRef CAS.
60. V. M. Worlitzer, G. Ariel, A. Be’er, H. Stark, M. Bär and S. Heidenreich, New J. Phys., 2021, 23, 033012 CrossRef CAS.
61. S. Mishra, A. Baskaran and M. C. Marchetti, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2010, 81, 061916 CrossRef.
62. A. Gopinath, M. F. Hagan, M. C. Marchetti and A. Baskaran, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 85, 061903 CrossRef PubMed.
63. J.-B. Caussin, A. Solon, A. Peshkov, H. Chaté, T. Dauxois, J. Tailleur, V. Vitelli and D. Bartolo, Phys. Rev. Lett., 2014, 112, 148102 CrossRef.
64. H. Reinken, S. H. L. Klapp, M. Bär and S. Heidenreich, Phys. Rev. E, 2018, 97, 022613 CrossRef CAS.
65. W. Ngamsaad and S. Suantai, Phys. Rev. E, 2018, 98, 062618 CrossRef CAS.
66. E. Bertin, M. Droz and G. Grégoire, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2006, 74, 022101 CrossRef.
67. A. Peshkov, I. S. Aranson, E. Bertin, H. Chaté and F. Ginelli, Phys. Rev. Lett., 2012, 109, 268701 CrossRef.
68. A. Baskaran and M. C. Marchetti, Phys. Rev. Lett., 2008, 101, 268101 CrossRef.
69. P. Srivastava, R. Shlomovitz, N. S. Gov and M. Rao, Phys. Rev. Lett., 2013, 110, 168104 CrossRef PubMed.
70. G. I. Menon, Rheology of complex Fluids, 2010, pp. 193–218 Search PubMed.
71. A. P. Solon and J. Tailleur, Phys. Rev. Lett., 2013, 111, 078101 CrossRef PubMed.
72. D. E. Kirk, Optimal control theory: an introduction, Courier Corporation, 2004 Search PubMed.
73. S. Lenhart and J. T. Workman, Optimal control applied to biological models, Chapman and Hall/CRC, 2007 Search PubMed.
74. R. Kerswell, C. C. Pringle and A. Willis, Rep. Prog. Phys., 2014, 77, 085901 CrossRef PubMed.
75. A. Borzí and V. Schulz, Computational optimization of systems governed by partial differential equations, SIAM, 2011 Search PubMed.
76. T. Dupont, J. Hoffman, C. Johnson, R. C. Kirby, M. G. Larson, A. Logg and L. R. Scott, The fenics project, Chalmers Finite Element Centre, Chalmers University of Technology, 2003 Search PubMed.
77. At least up to K = 1, larger values led to convergence problems. We expect that interpolating the target state would lead to a smoother solution profile.
78. J. Bechhoefer, Control Theory for Physicists, Cambridge University Press, 2021 Search PubMed.
79. V. Schaller, C. Weber, C. Semmrich, E. Frey and A. R. Bausch, Nature, 2010, 467, 73–77 CrossRef CAS.
80. L. Huber, R. Suzuki, T. Krüger, E. Frey and A. Bausch, Science, 2018, 361, 255–258 CrossRef CAS.
81. A. Sciortino and A. R. Bausch, Proc. Natl. Acad. Sci. U. S. A., 2021, 118, e2017047118 CrossRef CAS.
82. R. Suzuki and A. R. Bausch, Nat. Commun., 2017, 8, 41 CrossRef.
83. J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 16235–16240 CrossRef PubMed.
84. O. Pohl and H. Stark, Phys. Rev. Lett., 2014, 112, 238303 CrossRef PubMed.
85. T. Bhattacharjee, D. B. Amchin, R. Alert, J. A. Ott and S. S. Datta, eLife, 2022, 11, e71226 CrossRef PubMed.
86. R. Alert, A. Martínez-Calvo and S. S. Datta, Phys. Rev. Lett., 2022, 128, 148101 CrossRef PubMed.
87. A. V. Narla, J. Cremer and T. Hwa, Proc. Natl. Acad. Sci. U. S. A., 2021, 118, e2105138118 CrossRef CAS.
88. J. Cremer, T. Honda, Y. Tang, J. Wong-Ng, M. Vergassola and T. Hwa, Nature, 2019, 575, 658–663 CrossRef CAS.
89. T. Bhattacharjee, D. B. Amchin, J. A. Ott, F. Kratz and S. S. Datta, Biophys. J., 2021, 120, 3483–3497 CrossRef CAS PubMed.
90. M. P. Brenner, L. S. Levitov and E. O. Budrene, Biophys. J., 1998, 74, 1677–1693 CrossRef CAS PubMed.
91. C. Fei, S. Mao, J. Yan, R. Alert, H. A. Stone, B. L. Bassler, N. S. Wingreen and A. Komrlj, Proc. Natl. Acad. Sci. U. S. A., 2020, 117, 7622–7632 CrossRef CAS.
92. T. U. Mayer, T. M. Kapoor, S. J. Haggarty, R. W. King, S. L. Schreiber and T. J. Mitchison, Science, 1999, 286, 971–974 CrossRef CAS PubMed.
93. S. C. Schaffner and J. V. José, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 11166–11171 CrossRef.
94. R. Loughlin, R. Heald and F. Nédélec, J. Cell Biol., 2010, 191, 1239–1249 CrossRef PubMed.
95. M. Shirasu-Hiza, Z. E. Perlman, T. Wittmann, E. Karsenti and T. J. Mitchison, Curr. Biol., 2004, 14, 1941–1945 CrossRef PubMed.
96. S. R. Heidemann and M. W. Kirschner, J. Cell Biol., 1975, 67, 105–117 CrossRef PubMed.
97. S. R. Heidemann and M. W. Kirschner, J. Exp. Zool., 1978, 204, 431–443 CrossRef CAS PubMed.
98. A.-C. Schmit, M. Vantard, J. De Mey and A.-M. Lambert, Plant Cell Rep., 1983, 2, 285–288 CrossRef CAS PubMed.
99. F. Kumagai and S. Hasezawa, Plant Biol., 2001, 3, 4–16 CrossRef.
100. B. Najma, W.-S. Wei, A. Baskaran, P. J. Foster and G. Duclos, Proc. Natl. Acad. Sci. U. S. A., 2024, 121, e2300174121 CrossRef CAS.
101. V. Wollrab, J. M. Belmonte, L. Baldauf, M. Leptin, F. Nédeléc and G. H. Koenderink, J. Cell Sci., 2019, 132, jcs219717 CrossRef CAS.
102. P. J. Foster, S. Fürthauer, M. J. Shelley and D. J. Needleman, eLife, 2015, 4, e10837 CrossRef.
103. A. Colin, P. Singaravelu, M. Théry, L. Blanchoin and Z. Gueroui, Curr. Biol., 2018, 28, 2647–2656 CrossRef CAS PubMed.
104. W. Luo, C.-H. Yu, Z. Z. Lieu, J. Allard, A. Mogilner, M. P. Sheetz and A. D. Bershadsky, J. Cell Biol., 2013, 202, 1057–1073 CrossRef CAS PubMed.
105. S. Stam, S. L. Freedman, S. Banerjee, K. L. Weirich, A. R. Dinner and M. L. Gardel, Proc. Natl. Acad. Sci. U. S. A., 2017, 114, E10037–E10045 CrossRef PubMed.
106. J. Berezney, B. L. Goode, S. Fraden and Z. Dogic, Proc. Natl. Acad. Sci. U. S. A., 2022, 119, e2115895119 CrossRef.
107. T. Thoresen, M. Lenz and M. L. Gardel, Biophys. J., 2011, 100, 2698–2705 CrossRef PubMed.
108. D. V. Köster, K. Husain, E. Iljazi, A. Bhat, P. Bieling, R. D. Mullins, M. Rao and S. Mayor, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, E1645–E1654 CrossRef PubMed.
109. M. Glaser, J. Schnauß, T. Tschirner, B. S. Schmidt, M. Moebius-Winkler, J. A. Käs and D. M. Smith, New J. Phys., 2016, 18, 055001 CrossRef.
110. F. Nédélec, T. Surrey and E. Karsenti, Curr. Opin. Cell Biol., 2003, 15, 118–124 CrossRef.
111. C. Joshi, S. Ray, L. M. Lemma, M. Varghese, G. Sharp, Z. Dogic, A. Baskaran and M. F. Hagan, Phys. Rev. Lett., 2022, 129, 258001 CrossRef CAS PubMed.
112. R. Supekar, B. Song, A. Hastewell, G. P. Choi, A. Mietke and J. Dunkel, Proc. Natl. Acad. Sci. U. S. A., 2023, 120, e2206994120 CrossRef CAS.
113. M. Golden, R. O. Grigoriev, J. Nambisan and A. Fernandez-Nieves, Sci. Adv., 2023, 9, eabq6120 CrossRef CAS PubMed.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4sm00547c

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