Computational models for active matter

M Reza Shaebani; Adam Wysocki; Roland G Winkler; Gerhard Gompper; Heiko Rieger

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Computational models for active matter

M Reza Shaebani, Adam Wysocki, Roland G Winkler, Gerhard Gompper, Heiko Rieger

arXiv:1910.02528v2Oct 6, 2019cond-mat.softphysics.bio-phphysics.comp-ph

TL;DR

Active-matter modeling must span nonequilibrium, multiscale, nonlinear, and multibody systems while balancing universal descriptions against system-specific details. This review compares microscopic, hydrodynamic, coarse-grained, and continuum approaches, showing how model choice determines which collective behaviors and real-world features can be represented. It also identifies unresolved theoretical and application challenges, including signaling, far-from-equilibrium field theories, and distinguishing generic from specific behavior.

Problem
Active-matter modeling must address systems spanning molecular to animal scales whose out-of-equilibrium, multiscale, nonlinear, and multibody character complicates model selection.
Method
The review compares microscopic dry and fluid models with coarse-grained and continuum approaches, including models for active fluids, cells, and tissues.
Results
The review shows that model complexity and coarse-graining determine which details and collective behaviors can be captured, from generic active-particle phenomena to specific biological features.
Takeaways & Limitations
Simple models emphasize universal properties, whereas detailed models capture specific mechanisms, making the modeling level a delicate choice for real-world applications.
Takeaways & Limitations
Existing field theories rely on expansions appropriate in principle only near equilibrium, while no comprehensive continuum theory yet describes systems far from equilibrium.

Abstract

from arXiv · show

A variety of computational models have been developed to describe active matter at different length and time scales. The diversity of the methods and the challenges in modeling active matter---ranging from molecular motors and cytoskeletal filaments over artificial and biological swimmers on microscopic to groups of animals on macroscopic scales---mainly originate from their out-of-equilibrium character, multiscale nature, nonlinearity, and multibody interactions. In the present review, various modeling approaches and numerical techniques are addressed, compared, and differentiated to illuminate the innovations and current challenges in understanding active matter. The complexity increases from minimal microscopic models of dry active matter toward microscopic models of active matter in fluids. Complementary, coarse-grained descriptions and continuum models are elucidated. Microscopic details are often relevant and strongly affect collective behaviors, which implies that the selection of a proper level of modeling is a delicate choice, with simple models emphasizing universal properties and detailed models capturing specific features. Finally, current approaches to further advance the existing models and techniques to cope with real-world applications, such as complex media and biological environments, are discussed.

DRY ACTIVE MATTER

Dry active-matter models omit momentum conservation, using minimal particle descriptions to study activity-driven collective phenomena and extensions involving shape, architecture, and interactions.

Dry active matter: Dry active systems lack momentum conservation because frictional media absorb momentum or hydrodynamic interactions are less relevant than local interactions and fluctuations.Examples include gliding bacteria, vibrated granular beads, animal flocks, and dense bacterial collections.
Active Brownian particles: The active Brownian particle model describes a self-propelled spherical particle with overdamped translational and rotational Langevin dynamics.Its formulation includes diffusion, active propulsion, interactions or external potentials, and random force and torque terms.
Shape and architecture: Extensions of spherical active-particle models add torques, asymmetric shapes, or polymer-like assemblies, producing altered phase separation, chiral motion, and activity–conformation coupling.These extensions connect particle architecture to emergent collective behavior and polymer dynamics.
Active Brownian particles: Purely repulsive active particles exhibit motility-induced phase separation, wall accumulation, capillary action despite repulsion, and swim pressure.Motility-induced phase separation arises from collision-induced slowing, density increase, and positive feedback in crowded regions.
Active Brownian particles: Spherical active Brownian particles can have a pressure equation of state, whereas elongated nonspherical particles generally do not.In three dimensions, phase-separated high-density spherical particles can also move collectively without an alignment rule.

Active motion with alignment interactions

Alignment-based models explain collective motion through neighbor interactions and noise, while related physical-interaction models can generate coherence without explicit alignment rules.

Active motion with alignment interactions: The Vicsek model updates each particle’s direction by averaging neighboring orientations and adding noise, then advances particles at constant speed.The interaction neighborhood is defined by a circle of radius R, and noise strength is controlled by σ.
Active motion with alignment interactions: Increasing density or decreasing noise produces a continuous transition from disordered to ordered motion in the Vicsek model.The normalized mean velocity m ranges from 0 for random motion to 1 for coherent movement.
Active motion with alignment interactions: Noise, boundary conditions, interaction range and type, and alignment rules influence collective patterns such as bands, rotating chains, and marching groups.The resulting behavior depends on how disorder and interactions are implemented.
Modeling active matter: Figure 1 spans dry-to-wet and microscopic-to-macroscopic active systems, with arrows indicating increasing model generalization or complexity.Its examples include continuum phase separation, active-fluid turbulence, confined polar disks, and a swimming bacterium’s flow field.
Active motion with alignment interactions: Flocking transitions can resemble liquid–gas transitions with traveling ordered bands, whereas the active Ising model exhibits full phase separation.The comparison distinguishes microphase coexistence in the Vicsek model from full phase separation in its lattice counterpart.
Active motion with alignment interactions: Collective motion can arise from inelastic or nematic collisions, short-range interactions, or volume exclusion combined with particle elongation, without explicit polar alignment.These mechanisms provide physical routes to coherent movement in active particle systems.

Continuum models of dry active matter

Continuum models represent active matter through slowly varying fields, using symmetry, conservation laws, and coarse graining to describe collective behavior at larger scales.

Continuum models of dry active matter: Continuum models evolve fields such as number density and velocity to capture collective motion through conservation laws and broken continuous symmetries.They may be derived by coarse graining microscopic models, symmetry arguments, or nonequilibrium thermodynamics near equilibrium.
Dry polar flocks: The Toner–Tu theory describes dry polar flocks with polarization and density fields, including an active flux contribution v0 np.Polar particles distinguish front from rear, and density obeys a continuity equation.
Dry active nematics: Dry active nematics use a symmetric alignment tensor Q and an active density current involving ζ∇·Q.The nematic field represents local alignment while global drift remains zero because of head–tail symmetry.
Dry scalar active matter: Dry scalar active matter retains density as its only slow variable because spherical particles lack alignment interactions and global directional order.Active Model B+ adds time-reversal-breaking chemical-potential and current terms, producing microphase separation and a reverse Ostwald process.
Dry scalar active matter: Active matter with anisotropic interactions can undergo orientational order–disorder transitions, unlike spherical scalar active matter.The distinction follows from whether particle interactions generate polar or nematic orientational structure.

ACTIVE PARTICLES IN FLUIDS

Active particles in fluids require hydrodynamic modeling because fluid-mediated interactions shape propulsion, wall behavior, and collective dynamics. The review connects low-Reynolds-number fluid mechanics, microswimmer flow fields, and hydrodynamic instabilities.

Hydrodynamic interactions: Hydrodynamic interactions are fundamental to most biological and synthetic microswimmers and determine their behavior near walls, in channels, and collectively.For immersed active particles, incorporating solvent dynamics ensures local momentum conservation.
Low-Reynolds-number hydrodynamics: At Re ≲ 10^-3, microswimmer dynamics reduce to reversible Stokes flow with negligible inertial terms.Typical microswimmers have L∼O(µm) and v0∼O(µm/s) in water; consequently, time-reversible strokes cannot propel under the scallop theorem.
Microswimmer flow fields: The Oseen tensor gives the Stokes flow from a point force, while force- and torque-free swimmers are approximated in the far field by dipoles.The dipole strength P=f0L distinguishes pushers (P>0) from pullers (P<0), which have opposite flow directions.
Wall interactions: Dipolar flows generate effective attraction or repulsion and, together with flow-induced torque, attract pushers to walls.Propulsion, slow reorientation, and steric interactions also produce wall attraction independently of swimmer type, while hydrodynamics increases wall detention time.
Collective behavior: Hydrodynamic perturbations destabilize extensile active nematics, whereas long-range vortex flows can synchronize contractions between neighboring cells.Pusher and puller flow fields respectively enhance and reduce a sinusoidal reorientation perturbation in nematic arrangements.

Swimming in viscoelastic fluids

Viscoelastic environments differ from Newtonian fluids by breaking time-reversal symmetry, allowing self-propulsion from time-symmetric internal motion. Their broad range of rheological properties can substantially affect swimming behavior.

Swimming in viscoelastic fluids: Viscoelastic fluids can enable self-propulsion from time-symmetric internal motion by breaking Newtonian time-reversal symmetry.Such environments are commonly polymer solutions spanning properties from shear thinning to viscoelasticity, characterized by storage and loss moduli.
Swimming in viscoelastic fluids: The wide rheological spectrum of polymer solutions means viscoelastic media can affect microswimmer behavior across different material responses.The passage identifies shear thinning, storage modulus, and loss modulus as relevant properties.
Swimming in viscoelastic fluids: Time-symmetric swimming in viscoelastic fluids appears to evade the scallop-theorem constraint applicable to time-reversible Newtonian-fluid motion.The distinction follows from the stated breaking of time-reversal symmetry in viscoelastic environments.

Hydrodynamics: mesoscale simulation techniques

Mesoscale and microswimmer simulations span detailed cell–fluid models, squirmers, and far-field dipoles, trading numerical efficiency against near-field fidelity. Hydrodynamic interactions, swimmer shape, and boundary conditions strongly influence biological and collective behavior.

Mesoscale simulation techniques: Lattice Boltzmann, dissipative particle dynamics, and multiparticle collision dynamics provide alternative mesoscale ways to solve Navier–Stokes equations and generalizations.These methods facilitate simulations of mesoscopic active-matter agents.
Microswimmer representations: Detailed microswimmer models combine cell bodies, flagella, embedding fluids, no-slip boundaries, and momentum-conserving propulsion mechanisms.For example, bacterial swimming can arise from flagellar rotation coupled to counter-rotation of the cell body.
Microswimmer representations: Squirmers coarse-grain microswimmers as colloids with prescribed surface slip velocities and support studies of collective effects in bulk and near surfaces.They represent systems ranging from diffusiophoretic particles to biological cells.
Microswimmer representations: Far-field force-dipole models enable simulations of many microswimmers with minimal numerical effort but omit near-field interactions.This omission matters near surfaces, in thin slits, and for collective behavior in dense systems.
Biological swimmers: Hydrodynamic interactions influence flagellar synchronization, cell–cell scattering, and surface orientation in biological microswimmers.Far-field models predict parallel surface alignment for E. coli and sperm pushers but perpendicular alignment for Chlamydomonas pullers.
Collective behavior: Spherical squirmers form small clusters rather than ABP-like phase separation in thin no-slip films, whereas spheroidal squirmers phase-separate and swarm at relatively small activities.The review attributes these differences jointly to swimmer shape and hydrodynamic effects on orientational dynamics.

Artificial active matter

Artificial active matter spans microswimmers, active fluids, and continuum descriptions that capture propulsion, taxis, alignment, and active stresses. Models range from particle-level mechanisms to coarse-grained equations, with different choices reflecting momentum conservation and constituent symmetry.

Microscopic active agents: Artificial motors propel through phoretic effects including diffusiophoresis, thermophoresis, and electrophoresis.Surface reactions create gradients that drive fluid–particle slippage and propulsion.
Response to external fields – Taxis: Biological microswimmers redirect motion in external fields through taxis, including chemo-, photo-, gravi-, magneto-, and rheo-taxis.Biological chemo- and phototaxis often rely on internal biochemical signalling.
Continuum models for active motion in fluids: Wet active-matter models use conserved quantities and broken-symmetry order parameters to describe momentum-conserving suspensions at large scales.Examples include active gels, wet active nematics, generalized Navier–Stokes equations, and Active Model H.
Continuum models for active motion in fluids: Polar active gels represent orientation with a polarization field and derive dynamics from symmetry, irreversible thermodynamics, or microscopic coarse-graining.Their leading active stress has nematic symmetry, with activity strength ζ>0 for extensile and ζ<0 for contractile particles.
Continuum models for active motion in fluids: Generalized Navier–Stokes models simplify dense active fluids by eliminating either velocity or polarization when concentration is constant.They have been used to study active-fluid rheology and active turbulence.
Continuum models for active motion in fluids: Contractile active stress can produce arrested motility-induced phase separation in scalar active matter.The active contribution is positive for extensile swimmers and negative for contractile swimmers.

CELLS AND TISSUES

Living matter is active across scales, from ATP-consuming molecular machinery and cellular dynamics to tissues, tumors, organs, and groups of animals and humans.

Scales of living matter: Living matter exhibits activity from the protein and cellular scales through multicellular tissues and organisms to groups, swarms, and herds.These different scales necessitate different model approaches.
Protein and cellular scales: At the protein scale, activity includes molecular ATP-consuming machines such as motor proteins, ATPase pumps, protein factories, and ribosomes.
Multicellular and macroscopic scales: At larger biological scales, activity includes cell shape transformation, migration, division, tissue growth, tumors, developing organs, and collective animal or human behavior.

Cytoskeltal filaments and molecular motors

Cytoskeletal activity arises from ATP-dependent filament dynamics and molecular-motor forces, requiring models that span molecular mechanisms, stochastic motors, semiflexible networks, and continuum descriptions.

Cytoskeletal filaments: ATP-dependent polymerization and depolymerization of actin and microtubules generate forces through ratchet mechanisms and cytoskeletal remodeling.
Molecular motors: All-atom molecular dynamics can resolve motor conformational changes, while multiscale simulations address the longer time scales required for motor processes.
Molecular motors: Discrete kinetic and stochastic motor models predict mean velocity and other observables as functions of load force and ATP concentration.
Cytoskeletal networks: Particle-based filament models discretize semiflexible polymers using the wormlike-chain model and simulate their Langevin dynamics.These simulations are challenging for nearly incompressible microtubules.
Active filament networks: Adding molecular motors drives filament networks far from equilibrium and can alter stiffness, amplify stress, or produce contractility.These effects have been studied by introducing force dipoles into extensible wormlike-chain models.

Cell motility models

Cell and tissue models combine cytoskeletal activity, adhesion, growth, mechanics, and changing boundaries across microscopic, lattice-based, and continuum frameworks. Moving interfaces and feedback between growth and stress are central modeling challenges.

Cell motility models: Cell crawling models represent protrusion, front adhesion formation, rear adhesion release, and actomyosin-powered contraction as coordinated migration steps.Actin treadmilling and diffusing nucleators can also drive crawling through polymerization waves without motors.
Continuum descriptions: Continuum cell-migration models use sharp interfaces, level sets, or phase fields to represent moving and deformable cell boundaries.Phase fields distinguish cell interiors from exteriors and can also model multicellular migration.
Microscopic descriptions: Microscopic membrane-and-filament models capture internal fluctuations, persistent or random-walk-like motion, and shape changes under external conditions.
Tissue models: Particle-based tissue models represent deforming, adhering, growing, dividing, and dying cells with volume exclusion and active growth pressure.
Tissue models: Lattice-based tissue models include cellular Potts models and vertex or Voronoi models for cell shape, adhesion, migration, and morphogenesis.
Continuum tissue mechanics: Continuum tissue-growth models couple growth rates to local stress, while hybrid tumor models combine continuum tumor masses with discrete vascular networks.The vascular network represents changing nutrition and oxygen supply.

ANIMAL GROUPS

Models of animal groups use phenomenological, information-processing, and probabilistic approaches to study collective behavior, while the field faces challenges in extending models beyond simple systems and distinguishing generic from system-specific dynamics.

ANIMAL GROUPS: Collective migration models such as the Vicsek model capture prototypical aspects of animal-group behavior, although motion patterns differ substantially across groups.These differences reflect variation in the nature of interactions among animals.
ANIMAL GROUPS: Top-down approaches infer physical interactions from observations, whereas bottom-up approaches model individual information-processing strategies such as delayed signal processing.Collective behavior can also emerge from probabilistic models based on sensed information without predefined social forces or interaction rules.
ANIMAL GROUPS: Quorum sensing facilitates information transfer by coordinating individual activity according to local population density, although only a few individuals may possess required knowledge in some cases.
ANIMAL GROUPS: Collective-behavior studies remain limited to relatively simple systems involving fewer interacting effects, complex environments, external fields, mixtures, and vision-like information exchange.Examples include shape–hydrodynamics interplay, viscoelastic fluids, intricate confinement, gravity, and turbulent flows.
ANIMAL GROUPS: Important modeling gaps include three-dimensional cell motility and biochemical signaling coupled to mechanics in tissues, colonies, wounds, tumors, and animal groups.Signaling has been neglected in most computational active-matter models so far.
ANIMAL GROUPS: The field must distinguish universal properties from specific propulsion or interaction mechanisms that dominate a particular active-matter system.

Computational software packages for simulation of biological systems

The paper identifies computational software packages and methods for simulating biological active matter.

Computational software packages for simulation of biological systems: Supplementary Table S1 lists computational packages for modeling biological active matter.
Computational software packages for simulation of biological systems: The paper also notes particle-based software packages with options for simulating active particle systems.