Active glasses

TL;DR

Active glasses are a new class of non-equilibrium soft matter combining active particle motion with glass-like behavior, with recent advances in experiments, simulations, and theories revealing their complex dynamics and open questions.

Contribution

This review synthesizes recent experimental, numerical, and theoretical developments in understanding active glassy matter, highlighting its unique non-equilibrium properties.

Findings

Active glasses exhibit dynamical slowdown and heterogeneity.

Violation of the Stokes-Einstein relation observed in active glasses.

Open questions include the nature of aging and fragility in active glassy systems.

Abstract

Active glassy matter has recently emerged as a novel class of non-equilibrium soft matter, combining energy-driven, active particle movement with dense and disordered glass-like behavior. Here we review the state-of-the-art in this field from an experimental, numerical, and theoretical perspective. We consider both non-living and living active glassy systems, and discuss how several hallmarks of glassy dynamics (dynamical slowdown, fragility, dynamical heterogeneity, violation of the Stokes-Einstein relation, and aging) are manifested in such materials. We start by reviewing the recent experimental evidence in this area of research, followed by an overview of the main numerical simulation studies and physical theories of active glassy matter. We conclude by outlining several open questions and possible directions for future work.

Full text

I Introduction

I.1 Active and glassy matter

During the last decade, active matter has emerged as a new and rapidly expanding research area within the field of condensed matter science Marchetti et al. (2013); Ramaswamy (2010); Bechinger et al. (2016). The term ’active matter’ refers to materials whose constituent particles (or ’agents’) are capable of converting energy into some form of autonomous motion. In general, the energy can either be stored within the particles themselves or supplied externally, e.g. by introducing a chemical fuel or an external electromagnetic field. The type of active particle motion may occur in the translational, vibrational, or rotational degrees of freedom, or a combination thereof. Examples of active matter in the natural world are abundant, ranging from macroscopic organisms such as flying birds and swimming fish to the microscopic realm of motile bacteria and cells, down to the subcellular level of e.g. the cytoskeleton and molecular motor proteins. In all these cases, ATP is the main fuel source. From the synthetic side, active systems are now also available across many length scales; examples of such man-made structures include electrically-driven robots, granular particles on a vibrating table Narayan et al. (2007), catalytic Howse et al. (2007) and light-activated Palacci et al. (2013) colloids, metal-capped colloids in near-critical mixtures Buttinoni et al. (2013), swimming oil droplets Maass et al. (2016), catalytic stomatocyte nanoparticles Wilson et al. (2012), and artificial molecular motors Koumura et al. (1999); Kelly et al. (1999). Importantly, since each agent in an active-matter system is constantly consuming energy to generate its own movement, the material is said to be out of thermodynamic equilibrium at the single-particle level. This is to be contrasted with many other methods to bring a material out of equilibrium, such as a sudden quench of a thermodynamic control parameter (e.g. temperature or density) or the application of an external force field (e.g. shear); these latter protocols do not act on the scale of individual particles, but rather on the material as a whole or on the boundaries.

One of the central goals in active-matter physics research is to explore and understand how the intrinsic non-equilibrium nature of active particles can give rise to complex, collective, and novel self-organizing behavior that is absent in the passive counterpart. Let us first briefly consider the case of a single, non-interacting active Brownian particle. Both theory and experiment show that such an active colloid will undergo normal diffusion just as a conventional ’passive’ Brownian particle, at least at time scales comparable to (or larger than) the particle’s rotational diffusion time Howse et al. (2007). In this diffusive regime, the only effect of the self-propelled motion is that the effective diffusion constant will be larger than in the passive case (with an enhancement term proportional to the self-propulsion speed squared Howse et al. (2007))–a result that is often interpreted as an effectively higher temperature $T_{eff}$. Thus, in this example, an active material appears to be rather similar to an effective equilibrium system.

The physics can change profoundly, however, when active particles become governed by mutual (two- or many-body) interactions. Since the pioneering 1995 study by Vicsek and co-workers–which provides a minimal model for flocking behavior due to aligning interactions Vicsek et al. (1995)–, numerous novel phenomena in interacting active-matter systems have been discovered, including Motility-Induced Phase Separation Redner et al. (2013); Cates and Tailleur (2015), active turbulence Wensink et al. (2012); Rabani et al. (2013); Giomi (2015); Wu et al. (2017), active nematic liquid-crystalline behavior Keber et al. (2014); Doostmohammadi et al. (2018), spontaneous compartmentalization Spellings et al. (2015); Grauer et al. (2018), and synthetic quorum sensing Bäuerle et al. (2018). These findings not only call for new developments in fundamental non-equilibrium condensed matter physics, but they also offer new possibilities in biology and materials science. Indeed, as will be discussed later in this review, the framework of active matter provides a unique angle of approach to describe the complex phenomenology of living systems from a novel statistical-physics-based point of view. Furthermore, the incorporation of activity in synthetic systems offers unprecedented possibilities to create functional materials with ’smart’ and life-like properties that would be unattainable in thermodynamic equilibrium. Overall, the field of active matter thus offers an exciting new paradigm with relevance in both the fundamental and applied sciences Marchetti et al. (2013); Ramaswamy (2010); Bechinger et al. (2016); Trepat and Sahai (2018); Needleman and Dogic (2017).

A different branch of condensed matter physics concerns the study of glasses Binder and Kob (2011); Berthier and Biroli (2011); Royall and Williams (2015); Berthier and Ediger (2016). Glassy materials exhibit solid-like behavior but, unlike crystalline solids, they lack any long-range structural order. The study of such disordered solids has a long history, with the first man-made glasses dating back to ca. 3500 BC Beck (1934); Moorey (1999). It is now widely accepted that any material can, in principle, exist in a glassy state; indeed, our modern society makes use of a wide variety of amorphous solids, including organic, inorganic, polymeric, metallic, and colloidal glasses. The most common method of producing a glass is to supercool or compress a liquid until the viscosity $\eta$ (or structural relaxation time $\tau$) exceeds a certain threshold value; if crystallization is avoided, the resulting material can then be regarded as an amorphous solid on any practical time scale. The temperature or density at which the viscosity reaches the solidification threshold (typically defined as $10^{12}$ Pa.s) is known as the glass transition. At this transition, the supercooled liquid is said to have fallen out of equilibrium into a non-ergodic glassy state. Curiously, while the experimental process of glass formation has been known for centuries, it is governed by a multitude of complex phenomena that remain notoriously poorly understood to this day Berthier and Biroli (2011). In fact, the nature of the glassy state and the glass transition has been called ”the deepest and most interesting unsolved problem in solid state theory” Anderson (1995), and in 2005 the journal Science declared it one of the ”most compelling puzzles and questions facing scientists today” Sci (2005). There are several excellent reviews which detail the experimental phenomenology and current theoretical understanding of glassy materials Berthier and Biroli (2011); Royall and Williams (2015); Tarjus (2011); Biroli and Garrahan (2013); Langer (2014); for the purpose of this paper, we briefly summarize the main hallmarks of vitrification below.

I.2 Hallmarks of glassy dynamics

Among the many complex phenomena associated with glass formation, we address five of them in this review: i) dramatic dynamical slowdown, ii) fragility, iii) dynamical heterogeneity, iv) violation of the Stokes-Einstein relation, and v) aging (see Fig. 1). The first aspect is arguably the most striking hallmark of vitrification: as a liquid is supercooled toward the glass transition, its viscosity or relaxation increases by many orders of magnitude upon only a mild decrease in temperature. At the same time, however, this spectacular dynamical slowdown is accompanied by only subtle changes in the microstructure of the material Royall and Williams (2015). Indeed, the structure of a supercooled liquid or glass is almost indistinguishable from that of an ordinary liquid (as quantified by e.g. the static structure factor Hansen and McDonald (2013)). It is this apparent disconnect between structural and dynamical properties that lies at the heart of the glass transition problem: there is still no theory to accurately and rationally link the microstructure of a supercooled liquid to its quantitative relaxation dynamics. At a qualitative level, however, the dynamical slowdown has been successfully explained by theories such as Mode-Coupling Theory Gotze (2009) in terms of the so-called cage effect Kob (2002); Weeks and Weitz (2002). This effect signifies that, as the density increases or temperature decreases, particles become trapped in transient cages formed by their neighboring particles. The hindered particle motion of a caged particle in turn also facilitates the effective caging of its neighbors, culminating into a highly non-linear dynamical slowdown upon only a small change in density or temperature.

The second aspect, fragility, refers to the fact that not all materials vitrify in the same manner Angell (1995); Tarjus and Alba-Simionesco (2014). So-called ’strong’ glass formers solidify rather gradually, exhibiting an Arrhenius-type growth of the viscosity upon supercooling, while ’fragile’ materials vitrify more abruptly in a super-Arrhenius fashion. Many materials fall in between these two extremes, and in fact there are numerous examples of systems that also exhibit a fragile-to-strong crossover Elmatad (2011). Although there is consensus that network-forming materials, such as silica, tend to behave as strong glass formers, and that materials dominated by isotropic particle interactions, such as colloidal hard spheres, are generally more fragile Ozawa et al. (2016), a microscopic framework to predict the fragility for a given material composition and microstructure is still lacking Kelton (2017). It is widely believed that a resolution to this problem, i.e. obtaining a detailed understanding of the microstructural origins of fragility, will be key in ultimately achieving a universal description of the glass transition.

The third aspect, dynamical heterogeneity, is a fairly recent addition to the phenomenology of glassy dynamics Ediger (2000); Berthier (2011); Berthier et al. (2011). Dynamical heterogeneity signifies that the structural relaxation dynamics of a supercooled liquid does not proceed uniformly across the entire material, but rather in groups of collectively rearranging particles while the rest of the system remains temporarily frozen. The appearance of such mobile clusters fluctuates both in space and in time (at a fixed supercooled temperature and density), and the cluster size tends to grow upon approaching the glass transition. In a qualitative sense, this can be understood as a consequence of caging, requiring an increasingly large collective effort to mobilize particles as the density increases. The notion of cooperatively rearranging domains dates back to the 1965 work of Adam and Gibbs Adam and Gibbs (1965), but it was not until the late 1990s that dynamical heterogeneity was firmly established in simulation Hurley and Harrowell (1995); Kob et al. (1997) and experiment Kegel and van Blaaderen (2000); Weeks et al. (2000). While it is now widely believed that the size of dynamically heterogeneous regions represents an important dynamical length scale in the vitrification process, it is not yet established whether this length scale will ultimately diverge at a finite temperature to signal a critical phenomenon Berthier et al. (2005); Albert et al. (2016).

A somewhat related aspect of glassy dynamics is violation of the Stokes-Einstein relation Hodgdon and Stillinger (1993); Tarjus and Kivelson (1995); Shi et al. (2013); Charbonneau et al. (2014, 2018). This equation states that the viscosity $\eta$ (or relaxation time $\tau$) and diffusion constant $D$ are related to the temperature $T$ as $\eta D/T=constant$. In ordinary equilibrium liquids, the Stokes-Einstein relation is generally obeyed; in most supercooled liquids, however, the viscosity increase is stronger than the diffusion-constant decrease, and such Stokes-Einstein violation becomes more pronounced as the glass transition temperature is approached. The intuitive explanation for this decoupling is that diffusion is governed by the fastest particles, whereas structural relaxation is dominated by the slowest ones Ediger (2000). Note that the simultaneous existence of both fast and slow particle populations is also the key aspect of dynamically heterogeneous behavior, and hence it is generally assumed that Stokes-Einstein violation in glassy liquids is essentially a manifestation of dynamical heterogeneity Zangi and Kaufman (2007); Sengupta et al. (2013). While it has been suggested that such phenomenology may arise from critical dynamical fluctuations Biroli and Bouchaud (2007), it has also been proposed that non-critical hopping processes, i.e. the effective escape of particles from their local cages, predominantly underlie the observed viscosity-diffusion decoupling Charbonneau et al. (2014). Overall, a general first-principles framework to accurately predict both the degree of Stokes-Einstein violation and dynamical heterogeneity for a glass-forming material is still missing.

The fifth aspect, aging, refers to the fact that the behavior of a material can exhibit an explicit dependence on its age (at a fixed temperature and density). That is, the structural and dynamical properties may slowly evolve as the material becomes older Struik (1977); Berthier and Biroli (2009); Micoulaut (2016). Aging is generically understood as a non-equilibrium phenomenon which signals a material’s gradual approach toward an underlying equilibrium state. A convenient paradigm to describe this behavior is the so-called energy-landscape picture Stillinger (1995); Debenedetti and Stillinger (2001); Heuer (2008); Raza et al. (2015), which represents the system’s total (free) energy as a highly rugged surface in a high-dimensional configuration space (see Fig. 2). The process of aging then corresponds to the system’s progressive exploration of deeper energy minima on this surface Kob and Barrat (1997); Kob et al. (2000); Rehwald et al. (2010). In supercooled liquids, aging effects are typically only observed after a (small) temperature quench; if the liquid is supercooled sufficiently slowly, it will behave as an ordinary equilibrium liquid in the sense that, e.g., ergodicity and the fluctuation-dissipation theorem hold. In the glass state, however, the relaxation time that is needed to reach (quasi-)equilibrium will exceed–by definition–any practical time scale, implying that the supercooled liquid has fallen out of equilibrium. Physical aging is therefore typically observed within the glassy phase, and is a manifestation of the material’s tendency to reach a lower energy state Struik (1977). Ultimately, this aging behavior will bring the material into a deeply supercooled (quasi-equilibrium) liquid phase at a temperature below the original glass transition temperature. It remains to be established whether the equilibrium supercooled branch will eventually terminate at a low but finite temperature (the so-called Kauzmann temperature $T_{K}$ Kauzmann (1948)); this point would then correspond to the lowest theoretically possible glass transition temperature–a temperature at which the configurational entropy of the glass should rigorously vanish. The difficulty in testing this hypothesis is that it would require, in principle, infinitely slow supercooling rates and/or exceptionally long aging and equilibration times below the operational glass transition temperature Royall et al. (2018); such time scales inherently exceed the time scale of any practical simulation or experiment. For a very recent and comprehensive discussion on the configurational entropy of glass-forming liquids, we refer the reader to Ref. Berthier et al. (2019a).

I.3 Active glassy matter

On top of the already rich physics of both glassy and active matter, recent years have witnessed the emergence of a new subfield that lies at the interface of these two classes of materials: active glassy matter. Such materials are comprised of energy-driven motile particles that collectively exhibit dense and disordered glass-like behavior. Hence, active glassy matter combines multiple distinct non-equilibrium properties in a single material, offering an exciting playground for the discovery of fundamentally new physics. In fact, there is now a growing realization that active glassy physics is manifested in many biological systems, and that glassy behavior in living cells may even carry a biological function. In the context of materials science, the combination of activity and glassiness may provide new possibilities to create ’smart’ amorphous materials with life-like and adaptive functionalities. For example, considering that a particle’s self-propelled motion can be activated or de-activated by external cues, an active glassy material may be (locally) fluidized and re-solidified in an externally controllable manner. This in turn holds application potential for, e.g., shape-shifting and phase-changing materials, soft robotics, switchable sensors, self-healing glasses, and on-demand storage and release functionalities. Although the experimental realization of such synthetic active glasses is still in its infancy, the theoretical study of active glassy matter has already seen exciting progress in recent years. This review aims to provide an overview of this emergent field, highlighting the experimental evidence and current theoretical understanding of active glass-like behavior in both artificial and biological systems. We will pay special attention to the manifestation of the five main hallmarks of glassy dynamics in non-equilibrium active matter, and discuss whether the underlying physics of glassy phenomenology may be similar or distinct in passive and active materials.

Before we end this section, let us make a general remark on the distinction between glassy and jamming physics. Both phenomena describe a transition between a fluid-like and solid-like state, and the terminology is often used interchangeably to describe the emergence of rigidity in disordered materials. Here we use the convention Parisi and Zamponi (2010); Berthier et al. (2019b) that the glass transition is a dynamical transition, corresponding to full kinetic arrest and loss of ergodicity. Broadly speaking, this phenomenon arises from a competition between particle crowding (due to e.g. increased density and caging) and the particles’ ability to move (due to e.g. thermal fluctuations, Brownian motion, or self-propelled active motion). Jamming, on the other hand, is interpreted as a geometric transition that is governed by an increase in the particle connectivity or the number of direct particle contacts van Hecke (2009). More generally, the jamming transition occurs in the absence of any kind of (thermal, Brownian, or active) dynamics, and is defined as a zero-temperature and zero-activity limit of the glass transition. Indeed it was recently argued by Berthier, Flenner, and Szamel that strictly speaking, therefore, active particles can undergo jamming only when they are not active Berthier et al. (2019b). Throughout this paper, we will thus refer to aspects such as the dynamical slowdown as glassy, rather than jamming, phenomena.

This paper is organized as follows. In Sec. II, we first review the experimental evidence for glassy behavior in active matter. We start with very recent experiments on synthetic systems, followed by a description of the observed intra- and intercellular glassy dynamics in living cells, respectively. Section III focuses on ’numerical experiments’, i.e., computer simulation studies of active glassy matter. Here again we first consider non-living systems and subsequently turn to the modeling of living cell tissues. Section IV is devoted to recently developed theories of active glassy matter, including active versions of spin-glassy theory, Mode-Coupling Theory, and Random First Order Transition Theory. In all cases, we focus mainly on the manifestation of the five hallmarks of glassy dynamics (Sec. I.2) in active materials. We summarize our key findings in Sec. V and conclude with an overview of open questions and possible directions for future research.

II Experimental studies of active glassy matter

II.1 Synthetic active glasses

We begin by briefly reviewing the experimental work on artificial active glassy matter. From the synthetic materials side, it has thus far proven challenging to achieve self-propelled particles at sufficiently high densities such that glassy dynamics can be observed, and hence the number of experimental studies is still very limited. Indeed, the first experiment on active colloidal glasses was presented only very recently by Klongvessa et al. Klongvessa et al. (2019a, b) in a sedimentation experiment of peroxide-driven colloids. By analyzing the dynamics in a two-dimensional layer as a function of density and activity (measured in terms of an effective temperature $T_{eff}$), it was found that increasing the density always leads to slower structural relaxation, i.e. more glassy behavior. Interestingly, however, they also observed a non-monotonic dependence of the relaxation time on $T_{eff}$: when the passive system becomes weakly active, the dynamics slows down, but as the activity further increases the dynamics speeds up. This non-monotonic effect was only observed in the high-density regime, i.e. at densities where the passive material behaves as a glass. Thus, it appears that the role of activity in dense amorphous colloidal systems cannot be mapped onto an effectively passive system in a simple manner.

In addition to dense active colloids, we also mention another promising class of synthetic non-equilibrium materials that may offer an experimental realization of active glassy matter, namely driven granulates. Such systems can be realized experimentally by placing granular particles on, e.g., an air-fluidized or vibrating bed. It is hoped that future experimental research in this direction will shed more light on the non-trivial effects of non-equilibrium self-propulsion in model glass-forming systems. In particular, such studies should allow for better physical insight that may be less straightforward to achieve in more complex systems such as living cells, and may ultimately pave the way toward the development of synthetic materials with new adaptive functionalities.

II.2 Living active glasses

II.2.1 Intracellular dynamics

Let us now discuss the experimental evidence for active glassy behavior in living systems. One of the oldest demonstrations of active-glass formation is cryopreservation of cells, which amounts to the rapid freezing of living cell tissue at cryogenic temperatures Pegg (2007). During this process, water inside and outside the cells vitrifies into amorphous ice to effectively solidify the entire tissue, thereby ”stopping biological time” and rendering the material suitable for long-term storage. The active components of the cell, such as the cytoskeleton, thus become arrested within a water-rich glassy matrix. It is perhaps interesting to note that, while the apparent lack of microstructural changes upon supercooling makes glass formation one of the most notorious problems in condensed matter physics, it is precisely this aspect that allows cells to conveniently preserve their delicate internal structure as they undergo solidification. Importantly, as in the case of ordinary glass formation, it is crucial that crystallization is avoided during the supercooling process. For living cells, the formation of ice crystals will in fact cause irreparable damage to the tissue, and hence so-called cryoprotectants are commonly added to inhibit water crystallization during cryogenic cell treatment Akiyama et al. (2019).

The explicit link between intracellular cytoplasmic properties and the main hallmarks of glassy physics–including the dynamical slowdown and fragility–was first recognized in 2009 by Fredberg, Weitz, and co-workers Zhou et al. (2009). They found that eukaryotic cells under osmotic stress exhibit an orders-of-magnitude increase of the cytoplasmic viscosity with increasing density. The observed dynamical slowdown conforms to an Arrhenius-type behavior, making the cell’s fragility reminiscent of strong glass formers such as silica or soft deformable spheres Mattsson et al. (2009). It was also discovered, however, that ATP depletion significantly modulates the glass transition behavior of the cell, indicating that non-equilibrium activity plays a non-trivial role in the cellular glassy dynamics.

A 2013 study by Parry et al. Parry et al. (2014) showed that the bacterial cytoplasm also exhibits numerous glass-like properties. Unlike eukaryotic cells, the bacterial cell interior lacks cytoskeletal motor proteins, and the transport properties within the bacterial cytoplasm are therefore thought to be governed primarily by crowding effects. Parry et al. found that the cytoplasm of metabolically inactive bacteria is glass-like in the sense that the transport dynamics becomes extremely slow and dynamically heterogeneous. This constitutes arguably the first demonstration of glassy dynamical heterogeneity at the intracellular level. Interestingly, the study also revealed that metabolic activity fluidizes the cytoplasm, implying that non-equilibrium active processes offer a means to control the bacteria’s glassiness. Indeed, it was hypothesized that the cell’s ability to reversibly switch from a fluid-like to more dormant, glass-like state may enable bacteria to survive in a nutrient-poor environment.

The 2017 work by Nishizawa et al. Nishizawa et al. (2017) provided more insight into the fragility of the intracellular cytoplasm. They studied both eukaryotic and prokaryotic cells, and compared the glassy dynamics of both living samples and in vitro models from which the metabolic components and cytoskeletons were removed. Interestingly, it was found that the viscosity increase of the in vitro cytoplasms upon crowding conforms to a fragile, super-Arrhenius pattern; metabolically active living cells, on the other hand, exhibited a strong, Arrhenius-type growth of the viscosity. This confirms that activity can lead to a qualitatively different behavior of transport properties within cells; in particular, the intracellular fragility of the living cell appears to be fundamentally distinct from its non-living counterpart. A recent simulation study by Oyama et al. Oyama et al. (2019) rationalized the experimental observations of Parry et al. Parry et al. (2014) and Nishizawa et al. Nishizawa et al. (2017) in terms of an ATP-driven conformational change of proteins within the cytoplasm, thus offering a minimal model in which activity leads to volume fluctuations of particles, rather than self-propelled motion. These simulations revealed that only a small change in protein volume is sufficient to fluidize the glassy state and affect the cytoplasmic fragility.

Recent work suggests that more complex organisms also employ glass-like behavior as a biological self-protection mechanism. A 2017 study by Boothby et al. Boothby et al. (2017) found that tardigrades–multicellular micro-organisms which can survive under extremely harsh conditions–undergo intracellular vitrification as they are confronted with dehydration. The process relies on the vitrification of intrinsically disordered proteins within the tardigrade cytoplasm; the resulting glassy mixture may subsequently act as a protective matrix for other dehydration-sensitive cellular components. Since the vitrification process is reversible by rehydration, it is plausible that this liquid-glass-like transition indeed acts as an effective survival strategy to support the extreme resilience of tardigrades.

II.2.2 Intercellular dynamics

There is now a growing body of literature which suggests that glassy physics is also manifested at the intercellular level in e.g. amorphous confluent cell sheets (i.e. cell layers with a packing fraction of unity). One of the first quantitative studies on glassy collective cell dynamics is the 2011 work by Angelini et al. Angelini et al. (2011). They demonstrated that the relaxation dynamics of two-dimensional epithelial Madin-Darby Canine Kidney (MDCK) layers slows down as the cell density increases, and that the cells’ self-diffusion coefficient grows in a non-Arrhenius fashion with increasing density. The corresponding fragility index is comparable to that of a moderately fragile glass former. It was also found that the glassy cell dynamics is manifestly heterogeneous, and that the estimated correlation length associated with this dynamical heterogeneity increases with density. Finally, since the density increase essentially corresponds to a maturation of the cell layer, the observed glassy behavior may be interpreted as a sign of aging. While at a phenomenological level these features show a clear resemblance with ordinary glassy materials, the relaxation dynamics in confluent cell layers has, of course, a fundamentally different origin which stems from the self-generated active cell motion. Moreover, it was noted that cell layers are governed by additional processes such as cell division, apoptosis, and proliferation which are rigorously absent in, e.g., colloidal glass-forming systems.

In 2015, Garcia et al. Garcia et al. (2015) studied the dynamics of a different two-dimensional epithelial layer comprised of human bronchial epithelial cells (HBEC). As in the study of Angelini and co-workers Angelini et al. (2011), they found evidence for dynamical heterogeneity and a significant dynamical slowdown with increasing cell density. Interestingly, they also extracted a dynamical correlation length associated with heterogeneous relaxation that changes non-monotonically with the overall cell density. Furthermore, it was found that the cells exhibit non-trivial correlations in their instantaneous velocities Garcia et al. (2015). This finding is a striking departure from conventional equilibrium physics: in non-active and non-driven systems, the velocities of different particles are always uncorrelated Szamel et al. (2015); Berthier et al. (2019b). Curiously, the correlation length associated with the cells’ velocities also revealed a non-monotonic dependence on cell density, which in turn was correlated with the length scale of dynamical heterogeneities Garcia et al. (2015). Thus, the underlying physics of the observed cellular glassy dynamics, in particular the emergence of dynamical heterogeneity, may be fundamentally distinct from the non-active equilibrium case. Garcia et al. also concluded that the cellular monolayer exhibits features of aging which are not merely due to a gradual increase in cell density, but rather arise from the maturation of cell-cell and cell-substrate contacts Garcia et al. (2015). This result again constitutes a unique aspect of living cells that has no counterpart in ordinary glassy materials.

The relevance of collective cellular glassiness for pathological conditions such as asthma was first realized in 2015 by Park et al. Park et al. (2015). By studying confluent HBEC layers, they found that the cell dynamics of non-asthmatic (i.e. healthy) donors underwent a continuous transition from a mobile, fluid-like state to a quiescent, glass-like state as the cell layer matured over time. Conversely, cells from asthmatic donors remained significantly more mobile and exhibited a delayed glass-like transition. They also found that both cell cultures were dynamically heterogeneous, with maximum correlated cluster sizes on the order of 20 cells. Importantly, they identified a dimensionless metric for the average cell shape–specifically, the average ratio $\bar{p}$ between the cell perimeter and the square root of the cell area–as a structural signature of glassiness. More explicitly, a $\bar{p}$ value above 3.81 corresponds to fluid-like behavior, while $\bar{p}<3.81$ indicates rigidity of cell layer (also see Sec. III.2.2). Note that the cell-shape index $\bar{p}$ is inherently a many-body property; this must be contrasted with conventional two-particle structural quantities of glass-forming liquids such as the radial distribution function or static structure factor Hansen and McDonald (2013). It remains to be established whether two-body liquid-state-theory concepts of (active or passive) particles will also be meaningful to describe the properties of confluent and deformable living cells. One of the first steps in this direction was recently taken by Giavazzi et al. Giavazzi et al. (2017) to relate the static structure factor of cell nuclei to the flocking behavior of an epithelial monolayer.

It was recently recognized that glassy dynamics is also manifested in three-dimensional cellular collectives, with relevance in both healthy and diseased tissues. Indeed, processes such as wound healing, embryonic development, and cancer are all governed by transitions between migratory (’liquid-like’) and stationary (’glass-like’) cellular states Szabó et al. (2010); Kabla (2012); Park et al. (2016); Hakim and Silberzan (2017); Oswald et al. (2017); Malinverno et al. (2017); Palamidessi et al. (2018). For example, in the context of embryonic development, Schötz et al. Schötz et al. (2013) analyzed individual cell tracks of three-dimensional tissue explants from zebrafish embryos. They found that the cell dynamics inside the tissue exhibits subdiffusive and caging behavior which, based on a minimal model, they attributed to enhanced cell-cell adhesion and decreased active force generation. Another potentially important role of glassy dynamics lies in the pathology of cancer: works by e.g. Friedl and co-workers Haeger et al. (2014), Park et al. Park et al. (2016), Oswald et al. Oswald et al. (2017), and Palamidessi et al. Palamidessi et al. (2018) hypothesize that cancer metastasis is governed by cell ’unjamming’ behavior that allows clusters of cells to mobilize and ultimately escape the solid primary tumor. It is not implausible that glassy dynamical heterogeneities may play a role in this process Park et al. (2016); Giuliano et al. (2018). Recent studies by Malinverno et al. Malinverno et al. (2017) and Palamidessi et al. Palamidessi et al. (2018) have further identified a molecular pathway toward fluidization of two- and three-dimensional kinetically arrested cellular collectives, revealing that overexpression of a protein called RAB5A is sufficient to initiate collective cell motion. However, as in many other cell studies, biological factors such as varying cell-cell adhesive contacts, genetic heterogeneities, the extracellular matrix environment, and the cells’ persistent active motion must also be considered to ultimately understand the fate of a cellular collective; this is again a fundamental difference with the description of ’simple’ glassy liquids.

II.2.3 Summarizing remarks

In summary, there is now compelling evidence that living active systems exhibit several hallmarks of glassy dynamics, both at the intra- and intercellular level, both in two and three dimensions, and both in healthy and diseased tissue. Importantly, this manifestation of glassiness may also carry a biological function, e.g. by enabling individual cells to switch into a dormant glassy state under unfavorable environmental conditions, and by allowing collectives of cells to switch between quiescent and migratory behavior to facilitate multicellular processes such as wound healing and tissue development. Conversely, the apparent lack of cellular glassiness may underlie pathological conditions such as asthma and cancer metastasis. Furthermore, due to the cells’ innate activity, and the fact that such activity can influence the glassy dynamics, it is plausible that nature also employs activity as a means to control the cellular glassiness. For example, in collective cell movement, it might be possible that the fragility of a cell sheet–i.e. the abruptness with which a migrating cell layer comes to a halt–is tuned and controlled by the cells’ own self-propulsion forces. In the context of cancer, the possible active inhibition of dynamically heterogeneous regions within the primary tumor may potentially suppress the emergence of metastasizing cell clusters. Future work will hopefully shed more light on these speculative but highly interesting directions of research.

III Numerical simulation studies of active glassy matter

In addition to experiments, computer simulations constitute a powerful complementary approach to study complex physical phenomena such as glass formation. These ”numerical experiments” allow for controllable tuning of the relevant parameter space whilst providing detailed particle-resolved information on the emergent structural and dynamical properties. It is therefore no surprise that simulations are now also widely employed to elucidate the behavior of active glassy matter. In this section we review some of the main simulation studies in this field, both for synthetic and living model systems.

III.1 Synthetic active glasses

For the study of synthetic active glassy materials, there now exist two popular classes of simulation models: Active Brownian Particles (ABPs) Romanczuk et al. (2012) and Active Ornstein-Uhlenbeck Particles (AOUPs) (see Fig. 3) Szamel (2014); Maggi et al. (2015). Aside from particle interactions, the translational motion of ABPs is governed by thermal Brownian motion and a constant self-propulsion speed; the directionality of the self-propelled motion is controlled by Brownian rotational diffusion. The active motion of AOUPs, on the other hand, is governed by the Ornstein-Uhlenbeck process, which provides a stochastic element to both the magnitude and direction of the self-propulsion forces. This culminates into a persistent random walk that is characterized by a persistence time–which describes the duration of persistent active motion–, and an effective temperature which quantifies the strength of the active forces. For so-called athermal AOUPs, the effective temperature provides the only source of motion (aside from particle-particle interaction forces) Szamel et al. (2015); for thermal AOUPs, the translational motion is additionally governed by Brownian fluctuations Feng and Hou (2017). Both ABP and AOUP systems are typically considered in the overdamped limit, i.e. neglecting inertial effects, but underdamped dynamics may in principle also be included (see, e.g., Cugliandolo et al. (2017)). In the limit of vanishing self-propulsion speed or vanishing persistence time, respectively, both the ABP and AOUP model reduce to a passive Brownian system.

III.1.1 Dynamical slowdown, fragility, and dynamical heterogeneity

The 2013 study by Ni, Cohen Stuart, and Dijkstra Ni et al. (2013) constitutes the first active-matter simulation of arguably the simplest structural glass former, colloidal hard spheres. They considered a dense system of ABPs in three dimensions, and made several interesting observations regarding their glassy dynamics. It was found that the structural relaxation time decreases monotonically with increasing activity, implying that enhanced self-propulsion leads to the effective breaking of cages and more liquid-like behavior. In fact, they found that increased activity can ultimately push the glass transition density toward random close packing, i.e. the highest possible density for disordered spheres. A 2014 study by Wysocki et al. Wysocki et al. (2014) on three-dimensional soft active particles also reported liquid-like behavior up to densities near random close packing. In terms of the fragility, Ni et al. Ni et al. (2013) found that a higher self-propulsion speed makes the system more fragile, corresponding to a more abrupt vitrification process upon compression. Features of dynamical heterogeneity were also observed; at a fixed density, however, enhanced activity was found to decrease the cooperative motion, concomitant to the overall decrease in relaxation time. Finally, Ni and co-workers analyzed the changes in microstructure at the level of the static structure factor. They found that increasing the self-propulsion strength leads to a smaller average nearest-neighbor distance but a broader distribution of interparticle distances. They thus concluded that the structure of an equilibrium hard-sphere glass is different from a non-equilibrium active hard-sphere system at the same packing fraction. However, it was hypothesized that the microstructures should become similar in the jamming limit at random close packing. A later study by De Macedo Biniossek et al. de Macedo Biniossek et al. (2018) on two-dimensional ABP hard disks confirmed that activity induces non-trivial microstructural changes, which they attributed to a competition between activity-induced fluidization and enhanced structural order.

Around the same time as the study by Ni et al., Berthier employed Berthier (2014) Monte Carlo simulations to explore the glassy dynamics of two-dimensional AOUP-like hard disks. While the governing equations for this system are somewhat different from AOUPs, the particles also undergo a persistent random walk in the dilute limit. Berthier found signatures of caging and ultimate kinetic arrest as the particle density increases, as well as dynamically heterogeneous correlated motion, reminiscent of conventional glass phenomenology. However, the competition between caging behavior and active persistent motion led to quantitative differences with respect to the equilibrium case. Similar to the study of Ni et al. Ni et al. (2013), increased activity was found to monotonically shift the glass transition density to a higher value, implying that non-equilibrium self-propulsion indeed opens up new relaxation pathways that are closed in equilibrium. It was suggested that this might be due to an ’effective’ attractive force that emerges as active particles propel into each other, making the observed behavior of repulsive active disks somewhat similar to the equilibrium behavior of adhesive hard spheres. Indeed, it is well established that the addition of short-ranged attractions (through e.g. depletion interactions) can fluidize a passive hard-sphere glass Pham et al. (2002); the work of Berthier Berthier (2014) suggests that activity may possibly play a similar role as effective depletion.

Later simulations by Szamel, Flenner, and Berthier Szamel et al. (2015); Flenner et al. (2016) provided a more detailed picture of the glassy properties of (athermal) AOUP systems. They focused on an active version of the three-dimensional Kob-Andersen Lennard-Jones mixture–a well-studied benchmark system for passive glass formation. It was found that, contrary to the behavior of ABPs, active OUPs can become both more liquid-like and more glass-like depending on the nature of active motion. Specifically, for small persistence times at a fixed effective temperature, the active fluid relaxes faster than a passive reference system, but for large persistence times the active material relaxes more slowly. Interestingly, this non-monotonic dependence of the relaxation time was accompanied by a monotonic growth of the main peak of the radial distribution function, implying that the structure-dynamics link in active glassy OUPs is even more complex than in equilibrium supercooled liquids. To account for the non-monotonic behavior, they identified another important feature of the glassy dynamics which is unique to non-equilibrium systems, namely non-zero correlations among instantaneous particle velocities. The existence of such non-trivial velocity correlations was already noted by Garcia et al. Garcia et al. (2015) in an experimental study on confluent cell layers, but Szamel, Flenner, and Berthier were the first to recognize their importance in a combined numerical and theoretical study of an active model glass former. More specifically, Szamel et al. Szamel et al. (2015) argued that the non-monotonic change in the AOUP relaxation time with increased activity is due to a competition between increasing velocity correlations (which speed up the dynamics) and increasing structural correlations (which slow down the dynamics). It was also found that the fragility increases with increasing persistence time Flenner et al. (2016), a result that is consistent with the ABP results of Ni et al. Ni et al. (2013). Finally, signatures of dynamical heterogeneity and Stokes-Einstein violation were also observed. After rescaling all data with respect to the high-temperature behavior of the active liquid, both dynamically heterogenous motion and Stokes-Einstein decoupling were found to be rather similar to the equilibrium behavior in non-active supercooled liquids Flenner et al. (2016).

A follow-up study by Berthier, Flenner, and Szamel on repulsive (Weeks-Chandler-Andersen) AOUPs Berthier et al. (2017) suggested a more subtle scenario of the interplay between glassy relaxation dynamics, microstructure, and activity-induced velocity correlations. More specifically, they found that the second peak of the radial distribution function changes less trivially than the first peak with increased activity, and it was argued that these microstructural differences predominantly underlie the observed non-monotonic changes in the relaxation time. Interestingly, a careful numerical analysis showed that the velocity correlations grow only monotonically with the persistence time. It was therefore concluded that, while emergent velocity correlations accompany the non-equilibrium glass transition, and presumably affect the quantitative dynamics, they are not the main factor responsible for the initial acceleration and subsequent slowdown of AOUP glassy dynamics upon departing from equilibrium. It still remains to be established whether this finding is universal or unique to the specific AOUP model, and whether these insights may also be applicable to the glassy behavior of confluent active cells.

A 2016 study by Mandal et al. Mandal et al. (2016) focused on the glassy dynamics of mixtures of active and passive particles. They considered a three-dimensional Kob-Andersen binary mixture in which only the minority (20%) species undergo active persistent motion; the direction of the active motion was discretized in 8 directions. It was found that increasing the magnitude of the self-propulsion force shifts the glass transition monotonically to a lower temperature, akin to the monotonic increase of the glass transition density in ABP hard spheres Ni et al. (2013). Furthermore, they observed an activity-dependent change in the fragility of the system, including a fragile-to-strong crossover at sufficiently high self-propulsion strengths. In contrast to the ABP results of Ni et al. and the AOUP simulations of Berthier et al., however, the binary material was found to become less fragile with increased activity. While the interaction potentials, material composition, and active equations of motion differ among these studies, the qualitative difference in fragility is striking. It must be noted, however, that Mandal et al. Mandal et al. (2016) considered the fragility as a function of the temperature $T$, whereas Berthier, Flenner, and Szamel Flenner et al. (2016); Berthier et al. (2017) used the effective temperature $T_{eff}$ as the control parameter. The latter accounts for the total amount of injected energy (i.e., energy of both thermal and active origin), and $T_{eff}$ may therefore be a more suitable control parameter when the activity strength exceeds the typical thermal energy $k_{B}T$. Future work should clarify whether the effect of non-equilibrium activity on the fragility can be captured in a single unifying description; thus far, however, it appears that the microscopic details of the active material also play a non-trivial role.

III.1.2 Aging dynamics

Finally, let us discuss the fifth important aspect of glassy dynamics, namely non-equilibrium aging. As noted in Sec. III.1.2, aging in conventional glassy systems is understood as a gradual approach toward a deeper energy minimum in a high-dimensional energy landscape. Active materials, however, are governed by a constant injection and dissipation of energy at the single-particle level, rendering them non-Hamiltonian systems. Hence, the potential (or free) energy is generally not a useful metric to describe the behavior of active matter, and the manifestation of aging in an active glass is a priori far from trivial.

The first numerical evidence that waiting-time-dependent aging dynamics can, indeed, also take place in active glassy systems was presented in 2017 Janssen et al. (2017). This work focused on athermal active repulsive rods with a constant self-propulsion speed; unlike ABP systems, however, the reorientation dynamics in the system arises from collision-induced torques rather than rotational Brownian motion. For sufficiently high densities and sufficiently short rods, it was found that the system freezes into a disordered and kinetically arrested state–i.e., an active glass. By subsequently decreasing and increasing the density in a periodic manner, the athermal active glass could be periodically melted and revitrified, somewhat similar to applying a temperature-cycling protocol for thermal glass formers Zhao et al. (2013); Ketov et al. (2015). Interestingly, upon repeated application of such melting-revitrification cycles, the average number of rearranging particles would decrease progressively, ultimately yielding a configuration in which all relative particle motion has ceased. That is, the system has manifestly aged toward a more stable state that remains solid even at lower densities. It is important to note that the particles’ innate activity is a crucial ingredient for this aging behavior; since the model lacks any thermal noise by construction, the only non-trivial source of motion stems from the activity. Indeed, in the passive reference system, the only form of time-dependent motion would be an instantaneous quench toward the nearest energy minimum.

To rationalize the observed aging behavior in active glasses, a new ’landscape’ paradigm was introduced which focuses not on the total (free) energy, but rather on the mechanical stability of the system Janssen et al. (2017). Let us first recall that aging in passive glass formers is governed by thermal fluctuations which allow the system to surmount local energy barriers and ultimately reach a deep energy minimum (Fig. 2). For non-Hamiltonian active glasses, however, Janssen, Kaiser, and Löwen Janssen et al. (2017) proposed that it is not the energy that is optimized but rather the mechanical stability: as soon as the active athermal material has reached a configuration that is sufficiently stable to survive at the lowest applied densities, it will remain in that configuration indefinitely. Such a state is characterized by a local force balance on all particles; i.e., the intrinsic self-propulsion forces are cancelled against the local particle-particle interactions to effectively stop all relative particle motion. By subsequently plotting all possible global stabilities of the system as a function of configuration space, a rugged high-dimensional ’stability landscape’ emerges (Fig. 4) that is reminiscent of the conventional energy landscape picture of Fig. 2. For the active glassy system studied in Ref. Janssen et al. (2017), the exploration of this stability landscape is not facilitated by the presence of thermal fluctuations but rather by active particle forces, and the observed aging behavior is therefore a purely activity-induced phenomenon. In analogy to work on periodically driven systems Corté et al. (2008), Janssen, Kaiser, and Löwen Janssen et al. (2017) interpreted this novel form of active aging as an irreversible, random self-organization process toward an ’absorbing state’. That is, the active system continues to explore many different configurations until it spontaneously reaches a stable state from which it can no longer escape. We note that in principle there exist many such absorbing states for a given range of densities; at the highest density, all possible absorbing states should correspond to all (active) random close packing configurations.

The role of thermal noise in active glassy aging was also discussed in Ref. Janssen et al. (2017). It was hypothesized that for thermal active glasses every absorbing state on the stability landscape is effectively replaced by a basin of absorbing states, i.e. a set of similar configurations that are separated only by relatively small barriers. However, explicit aging simulations for such thermal active systems have not yet been performed. In future studies, it will be interesting to explore how active physical aging is affected by thermal fluctuations, whether such fluctuations play the same effective role as innate self-propulsion forces in the aging phenomenology of active glasses, and whether e.g. temperature cycling and density cycling will amount to similar aging behavior. Finally, in the context of living systems, a phenomenological link between physical aging of glasses and biological aging of cells was recently established Lou et al. (2017). This raises a highly intriguing question: can our understanding of active glassy physics ultimately also reveal more insight into the long-term aging behavior of living tissues?

III.2 Living active glasses

To simulate the dynamics of living cells, a multitude of coarse-grained computer models have been developed, including simple isotropic and deformable particle models, vertex and Voronoi models, lattice-based cellular Potts models, and models with explicit subcellular elements Camley and Rappel (2017). Broadly speaking, these respective approaches are characterized by an increasing level of complexity but also by a growing computation cost. Hence, there is a priori not a unique model that is ideally suited to describe the relevant physics of living cell systems. Here we focus only on simulation studies that specifically address the glassy dynamics of confluent cell layers; owing to the fact that such simulations require the explicit treatment of a large number of individual cells, these studies have thus far been limited to relatively simple descriptions at the single-cell level. For computational modeling efforts to describe other biological phenomena such as (non-glassy) epithelial morphogenesis, we refer the reader to e.g. Ref. Fletcher et al. (2014) and especially Ref. Camley and Rappel (2017).

III.2.1 Soft-particle models

Arguably the simplest description of a glassy layer of active cells is a soft-particle-based model, in which every cell is represented as a self-propelled soft disk. This approach was first invoked by Henkes, Fily, and Marchetti in a seminal 2011 study on dense active matter Henkes et al. (2011). Their work in fact also precedes any of the synthetic active-glass simulations mentioned in Sec. III.1, and it can therefore be argued that this study constitutes the first numerical realization of active glassy physics. Briefly, Henkes et al. modeled a two-dimensional system of soft, isotropically-interacting repulsive disks that are each equipped with a constant self-propulsion speed. Inspired by recent confluent-cell experiments Angelini et al. (2011, 2010); Trepat et al. (2009); Poujade et al. (2007); Petitjean et al. (2010), they included an internal alignment mechanism such that the self-propulsion direction tends to align with the particle’s instantaneous velocity; in a biological context, this mimics the effect of aligning the cell polarity with its motility. By calculating the state diagram for a circularly confined system, it was found that the particles become kinetically arrested at sufficiently large densities and low self-propulsion speeds, with a weak reentrant behavior at high degrees of activity. We note that, while the authors referred to the arrested dynamics as ’active jamming’, we here follow the convention that this is an active glassy state, owing to the presence of active motion. At the largest self-propulsion speeds studied, all trajectories yielded fluid-like behavior. Within the densely arrested phase, they also observed regular oscillations of the particle displacements around their mean caging positions, which–in analogy to athermal dense packings–they interpreted as low-frequency modes.

We note that the synthetic active-particle simulations discussed in Sec. III.1 may, in principle, also be regarded as a minimal model for living cell tissues, although these studies did not explicitly take into account any cell-specific properties. In particular, the assumption of ABP- or AOUP-like dynamics may conform more accurately to the behavior of artificial active colloids than to densely packed living cells. Conversely, while the study by Henkes et al. assumed an equation of motion that was inspired by biological cell polarization, their work may, of course, also shed light on the behavior of synthetic materials with a similar active alignment mechanism.

III.2.2 Vertex and Voronoi models

A more refined description of glassy cell sheets should also account for the inherent confluence of the layer and the anisotropic shape deformability of the cells. These aspects are naturally incorporated in so-called vertex and Voronoi models, which both represent cells as polygons that collectively tesselate the entire space; as such, they are both confluent by construction. The main difference between these models lies in the choice of the relevant degrees of freedom: in vertex models, the dynamical equations of motion are solved for the polygon vertices, while in Voronoi models the equations of motion apply to the Voronoi centers of each cell (see Fig. 5). An advantage of the vertex model is that it can describe both concave and convex cell shapes, while the Voronoi model is inherently constrained to convex polygons. Nonetheless, it must be noted that a Voronoi tesselation using the cell nuclei as Voronoi centers can already provide a fairly accurate description of the cell shapes observed in experiment Kaliman et al. (2016). For the incorporation of active cell motility, the Voronoi model is generally preferred over a vertex-model simulation, as the Voronoi centers provide a natural means to assign a single active force to every individual cell. Although the idea to represent an epithelial cell layer by polygons was already proposed by Honda in 1978 Honda (1978); Nagai and Honda (2001), it is largely due to the recent pioneering work of Bi, Manning, and co-workers that such modeling approaches are now recognized as a successful tool to describe and understand the glassy dynamics of living cells. We discuss some of these main simulation studies below.

In 2014 and 2015, Bi et al. Bi et al. (2014, 2015) studied the glassy dynamics of a confluent monolayer using a two-dimensional vertex model. Their model assumed that the mechanical energy of each cell is governed by its perimeter $P$ and surface area $A$, effectively accounting for the bulk elasticity of the cell, the active contractility, and a net line tension. The latter arises from a competition between the cortical tension of the active actomyosin layer near the cortex–which tends to minimize the area of cell-cell contact–, and intercellular adhesion forces which maximize the area of cell-cell contact. The cell shape can be non-dimensionalized in terms of the so-called shape index $p=P/\sqrt{A}$; based on their simulations, it was predicted that the confluent cell layer undergoes a fluid-to-solid-like transition at an average value of $\bar{p}=3.81$. This value could also be associated with a change in the energy barrier heights of so-called T-1 transitions– topological transitions which govern cell rearrangements in confluent layers.

The vertex-model-based prediction of a rigidity transition at $\bar{p}=3.81$ was soon found to be in remarkable agreement with experiments of confluent HBEC monolayers, as detailed in the 2015 study by Park et al. Park et al. (2015). In this joint experimental-simulation work, the effect of intrinsic self-propelled cell motion on the transition was also studied. To this end, a small active term was added to the model that, similar to the study by Henkes et al., was set to a constant magnitude and in a direction that tends to align with the cell’s velocity. It was found that increased active forces operate in concert with increased adhesion to fluidize the cellular collective; however, the critical shape-index value at which the transition takes place remains at $\bar{p}=3.81$, regardless of the magnitude of the active fluctuations. It was also noted that increased cell-cell adhesion leads to larger cell perimeters and more liquid-like behavior; paradoxically, higher adhesion in particulate matter generally leads to enhanced gelation and solidifaction, highlighting the fact that adhesive forces may play a fundamentally different role in living and non-living glassy matter. We note, however, that short-ranged attractions in hard spheres can also fluidize a glassy state (and even give rise to reentrant behavior) Pham et al. (2002); this may possibly point toward a non-trivial analogy between passive glass formers and active cellular systems.

A 2016 simulation study by Bi et al. Bi et al. (2016) focused on the glassy dynamics of confluent cell layers using the so-called self-propelled Voronoi model. The mechanical energy of the system was assumed to be the same as in the vertex model, i.e. controlled by the cells’ perimeters and areas. For the self-propulsion forces, an ABP-like equation of motion was invoked in which the activity vector has a constant magnitude $v_{0}$ and a directionality that is governed purely by Brownian rotation. The main result of this study is the state diagram shown in Fig. 6, which indicates regions of parameter space in which the dynamics is glass-like and fluid-like, respectively. Notably, while the degree of glassiness is explicitly and monotonically dependent on both the magnitude of the self-propulsion forces and the preferred perimeter of the cells, the point at which the glass transition takes place always corresponds to an average shape-index value of $\bar{p}=3.81$. The role of the rotational diffusion constant was also explored, revealing a monotonic shift in the glass-transition line. More specifically, a high rotational diffusion constant was found to give rise to largely uncorrelated motion and more solid-like behavior, while a low rotational diffusion constant induces more collective motion and flow within the tissue. In the limit of vanishing self-propulsion speed, all glass-transition lines converge to the point at which the preferred cell perimeter $p_{0}$ equals $\bar{p}=3.81$. A later study by Merkel and Manning Merkel and Manning (2018) revealed that the rigidity transition of the Voronoi model in three dimensions is similarly controlled by the surface-to-volume ratio of the cells. In view of these findings, and their remarkable agreement with earlier vertex-model and experimental studies of glassy cell layers, it appears evident that there is a clear and quantitative correlation between the microstructural cell shapes and the glassy dynamics of a cellular collective. As already noted in Sec. II.2, however, the structural role of the shape index may be unique to confluent deformable cells and need not necessarily apply to isotropic particles undergoing a glass transition, thus possibly constituting a fundamental difference between the glassy physics of living and non-living matter.

A more recent and larger-scale simulation study by Sussman and Merkel Sussman and Merkel (2018) on the self-propelled Voronoi model highlights the difference between glassy and jamming physics in confluent models. In particular, they found that the zero-motility limit ($v_{0}=0$) of the two-dimensional self-propelled Voronoi model is always marginally stable, i.e. the number of particle constraints and the number of degrees of freedom are exactly balanced, regardless of the value of the shape perimeter $p_{0}$. Consequently, although the non-motile Voronoi system undergoes an apparent fluid-to-solid glass transition at $p_{0}=3.81$, the system strictly lacks an unjamming transition. The two-dimensional vertex model Bi et al. (2015) and three-dimensional Voronoi model Merkel and Manning (2018), on the other hand, are always under-constrained and are rigidified by the collective onset of residual stresses. Overall, the work of Sussman and Merkel suggests a scenario in which the dynamical glass transition in the two-dimensional $v_{0}=0$ Voronoi model is governed by the time scale of rotational diffusion rather than by the shape of the cells, and points toward a non-trivial decoupling of glassy and jamming phenomena.

The first simulation study on dynamical heterogeneity and fragility in confluent glassy models was presented in 2018 by Sussman et al. Sussman et al. (2018). They studied the glassy dynamics of a two-dimensional inactive variant of the Voronoi model, in which the translational motion of a cell is governed solely by cell-shape-based interactions and thermal Brownian noise. They found that the dynamics slows down significantly as the temperature is reduced; however, it was noted that the plateauing region in the dynamical scattering function, which is typically associated with caging, is much less pronounced than in a standard glass former. At the corresponding time scales of relaxation, a growing peak of the dynamical susceptibility was observed, indicating marked dynamical heterogeneity and increased cooperative particle motion. Interestingly, in contrast to the self-propelled Voronoi model, they found that the location of the glass transition does not coincide with an average cell shape index of $\bar{p}=3.81$. Thus, for this particular inactive system, the dynamics and microstructural properties are manifestly decoupled. Furthermore, they found an anomalous pattern in the fragility: as the temperature decreases toward the glass transition temperature, the relaxation time grows in a sub-Arrhenius fashion. In terms of the conventional energy-landscape picture, such ’superstrong’ behavior would correspond to a progressive decrease in local energy barriers as the system is cooled. Similar qualitative results were found for the inactive vertex model. It remains to be established whether this unusual fragility is unique to the specific choice of the model, and whether the inclusion of active forces would yield a different phenomenology. It must also be noted that experiments by Angelini et al. on epithelial MDCK layers showed a moderately fragile behavior upon increased cell density Angelini et al. (2011); it is not yet clear how the here reported simulation results can be applied to those experiments, and whether superstrong fragilities indeed may be observed in real biological tissues.

We conclude this section by mentioning several other, recently developed variants of vertex and Voronoi-based simulation models of confluent cells. Giavazzi et al. Giavazzi et al. (2018) studied a self-propelled Voronoi model in which the self-propulsion direction is not governed by Brownian motion, but is rather enslaved to the cell’s own velocity. It was found that, within this model, such polarization alignment yields global migration and can promote both dynamical heterogeneity and solidication. Lång and co-workers Lång et al. (2018) recently developed a new model which combines the active Voronoi model with the Vicsek model, such that every cell’s self-propulsion direction tends to align with the instantaneous velocity of its neighbors. This model predicts large-scale collective motion as the Vicsek-alignment radius reaches a certain threshold, in good agreement with experiments on confluent human keratinocyte cells. On a phenomenological level, the growing Vicsek radius captures the effect of up-regulating cell-cell adhesion by experimentally increasing the calcium concentration; the activity could be controlled experimentally by adding blood serum to the cell layer Lång et al. (2018). Barton et al. Barton et al. (2017) studied an active vertex model which–in addition to cell-specific mechanical properties and cell alignment–also accounts for cell growth, division, and apoptosis. It was found that fluidization in this model can be induced either by increasing the preferred cell perimeter $p_{0}$ or the magnitude of the active driving. Very recently, Czajkowski et al. Czajkowski et al. (2019) also simulated an active vertex model that includes cell motility, cell division, and cell death. They found that a glass-like regime with caging behavior and subdiffusive dynamics can be achieved if the rate of cell cycling is sufficiently low. Finally, a recent series of papers by Ruscher and co-workers Ruscher et al. (2015, 2017, 2018) studied the glassy physics of the so-called Voronoi liquid, which is defined such that each Voronoi position tends to move towards the centroid of the Voronoi cell to which it belongs. Although this model was not motivated by biological experiments, it revealed a multitude of non-trivial glassy features; it will be interesting to explore if and how the Voronoi liquid can be linked to other Voronoi-based models of confluent living cells.

III.2.3 Cellular Potts models

Let us finally mention one other model used for cell-resolved simulations, namely the cellular Potts model. Briefly, this coarse-grained model represents each cell as a domain of connected spins on a lattice; the dynamics of the cell is governed by an effective energy functional which captures all relevant cellular interactions. In 2016, Chiang and Marenduzzo Chiang and Marenduzzo (2016) developed an active variant of the confluent cellular Potts model that is reminiscent of the vertex model of Bi and co-workers Bi et al. (2015); Park et al. (2015). They found that, depending on the cell motility and the interfacial tension between adjacent cells, the model can exhibit several features of glassy dynamics, including very slow relaxation, subdiffusive behavior, and aging. Interestingly, it was found that the fluid-to-solid transition in the cellular Potts model also coincides rather accurately with a specific average shape index $\bar{p}$. This suggests that predictions of the vertex and Voronoi model, as well as observations in experiment, may also be captured within a lattice-based cellular Potts model.

III.2.4 Summarizing remarks

In summary, there now exist numerous simulation models that can provide new insight into the glassy dynamics of densely packed cell layers, in particular on the emergence of a dynamical slowdown and dynamically heterogeneous behavior, as well as on the relevant underlying microstructural changes of the tissue. Thus far, based on several coarse-grained model studies, it appears that a suitable means to capture the structural signature of glassy behavior is encoded in the geometric shape of the cells, although this result need not apply generally to all simulation models of cellular collectives. Furthermore, it is not yet clear whether alternative structural metrics, such as the radial distribution function used in liquid-state theory, may be successfully applied to biological tissues. Aside from the question to what extent the above mentioned simplified cell models can capture all the relevant physics of a realistic biological system, there are also still major unresolved issues relating to the main hallmarks of glassy dynamics in living systems. Notably, the degree of fragility in a confluent cell layer, the manifestation of Stokes-Einstein violation, the physical origins of aging behavior, as well as the possible role of intrinsic cell motility on all these phenomena, still remain to be explored.

IV Theoretical studies of active glassy matter

Theoretical physics offers a third important methodology to study the glassy phenomenology of active matter. During the last decades, a wide variety of theories has been developed for ’passive’ glass formation, including Adam-Gibbs theory Adam and Gibbs (1965), spin-glass theory Castellani and Cavagna (2005), Mode-Coupling Theory (MCT) Gotze (2009), Random First Order Transition Theory (RFOT) Lubchenko and Wolynes (2007), geometric frustration Tarjus et al. (2005), dynamical facilitation Chandler and Garrahan (2010), and kinetically constrained models Cancrini et al. (2009). The differences between these theories essentially stem from their physical point of view, which may be purely thermodynamic, purely dynamic, purely phenomenological, or a combination thereof. For a detailed overview of existing theories of the glass transition, see e.g. Refs. Binder and Kob (2011); Berthier and Biroli (2011); Langer (2014); Royall and Williams (2015). In this section, we review several recent studies which have sought to extend the theoretical description of glass formation to non-equilibrium active systems. In particular, we focus on studies of spin glasses (i.e., disordered spins on a lattice) and active variants of both MCT and RFOT. These approaches have been developed for model glass-forming systems that are governed by a certain effective Hamiltonian (in the case of active spin glasses), or by ABP- and AOUP-like dynamics (in the case of active MCT and RFOT). As such, they do not distinguish a priori between synthetic and living active materials, but rather give general predictions for systems that conform to certain equations of motion.

IV.1 Spin-glass theories

The first theoretical description of active glassy physics is the seminal 2013 study by Berthier and Kurchan Berthier and Kurchan (2013). They studied the mean-field behavior of a non-equilibrium variant of the so-called spherical $p$-spin model; in the equilibrium case, this spin-glass model can be solved exactly and conforms to the RFOT scenario Kirkpatrick and Thirumalai (1987). To make the model active, they included a colored-noise driving term and a dissipative memory term in the effective Hamiltonian. It was found that both colored noise and energy dissipation give rise to a dynamical slowdown that is strongly reminiscent of equilibrium glass formation. The relationship between the magnitude of the driving force and the decrease in the glass transition temperature was found to be roughly linear. Overall they concluded that, although the specific features of the transition may change quantitatively with the nature of the non-thermal driving force, the phenomenology of the slowdown is rather similar to the behavior of thermal glass-forming systems. This suggests that non-equilibrium active glassy systems may, at least in approximate manner, be mapped onto an effective equilibrium material.

In 2014, Pilkiewicz and Eaves Pilkiewicz and Eaves (2014) analyzed the glassy properties of an active generalization of the Sherrington-Kirkpatrick model–another well-known spin-glass model. Specifically, they considered a mixture of passive and active spins, in which the passive spins possess quenched (fixed) disorder while the active spins can undergo annealing (i.e., relaxation of the spin orientation) due to an external driving force. Upon increasing the fraction of active spins, the glass transition temperature was found to shift to lower values in a monotonic manner. However, when a bias was introduced to promote ferromagnetic order, the phase diagram became markedly more complex: for a certain range of the bias parameter, a reentrant behavior was predicted in which the system goes respectively from a paramagnetic (’liquid-like’) phase at high temperatures, to a spin glass at lower temperatures, to a ferromagnetic (’crystal’), and finally to a mixed spin-glass phase with partial order at the lowest possible temperatures. Although the inherent lack of microstructure in spin-glass models renders them, in principle, unsuitable to describe realistic structural glass-formers, it was suggested that the bias parameter may possibly be mapped onto the bulk density of an experimental mixture of active and passive colloids. Thus far, the glassy dynamics of such active-passive mixtures still remains to be explored in experiment.

IV.2 Mode-coupling theories

We now turn to another important class of theories, namely Mode-Coupling Theory Gotze (2009); Reichman and Charbonneau (2005); Janssen (2018). MCT is essentially the only theory of the glass transition that is founded entirely on first principles, aiming to predict the full microscopic glassy dynamics of a material based solely on knowledge of the material’s microstructure. Briefly, the theory starts from the exact equations of motion for a correlated (glassy) liquid, and subsequently applies a series of approximations to obtain a closed equation for the dynamics; this equation can be solved once the static structure factor is provided as structural input. Although MCT in its standard form cannot adequately account for all hallmarks of glassy dynamics, the theory has nonetheless provided a remarkable series of non-trivial predictions for the qualitative and quantitative dynamical slowdown of many glass-forming materials, including the existence of the cage effect, non-trivial scaling laws, non-exponential relaxation, and complex reentrant behavior. Furthermore, proposed modifications of standard MCT may offer more accurate first-principles predictions of, e.g., fragility Janssen et al. (2014); Janssen and Reichman (2015), dynamical heterogeneity Biroli et al. (2006), Stokes-Einstein violation Charbonneau et al. (2018), and aging Latz (2000); Ramírez-González and Medina-Noyola (2010). We refer the reader to Refs. Gotze (2009); Reichman and Charbonneau (2005); Janssen (2018) for a general overview of the technical aspects, successes, failures, and proposed extensions of standard MCT. Here we focus only on new variants of MCT developed specifically for active matter; a review of this emerging field was also recently presented in Ref. Janssen (2018), and we summarize only the key findings below.

Farage and Brader were the first to extend the standard MCT equations to an active-matter system Farage and Brader (2014). In their 2014 paper, they considered the glassy dynamics of hard-sphere ABPs by treating every active particle as an effectively passive colloid but with a higher, activity-controlled diffusion constant. This approach essentially amounts to the explicit removal of all rotational degrees of freedom. Within this approximation, it was found that the increase of particle activity can enhance the breaking of local cages, resulting in a softening and ultimate melting of a passive glass. The location of the glass transition was found to shift monotonically toward higher densities, consistent with the ABP computer simulations of, e.g., Ni, Cohen Stuart, and Dijkstra Ni et al. (2013). A similar MCT approach was later used by Ding et al. Ding et al. (2015) to study the dynamics of mixtures of hard-sphere active and passive particles. They predicted a reentrant behavior in the glassy dynamics: upon increasing the self-propulsion strength of the active fraction at constant density, the system changes first from a glass to a liquid and then to a glass again. It was argued that this reentrance is unique to active-passive mixtures and would disappear in a purely active system. Note, however, that the here reported reentrance is fundamentally distinct from the spin-glass results of Pilkiewicz and Eaves Pilkiewicz and Eaves (2014).

In 2015, Szamel and co-workers presented the first microscopic MCT framework for athermal AOUP systems Szamel et al. (2015); Szamel (2016). By assuming that the particle positions evolve on a time scale much larger than the time scale needed for reorientation of the activity direction, the self-propulsion could be integrated out in an approximation akin to the one invoked by Farage and Brader. An important difference with earlier MCT work, however, is that Szamel’s theory also requires static correlations between particle velocities as input. Contrary to the behavior of ABPs, and in remarkable agreement with AOUP computer simulations, it was predicted that the incorporation of activity can both enhance and suppress glass formation, depending on the persistence time of the active particles. As already noted in Sec. III.1.1, they attributed this non-monotonic behavior to a competition between increasing velocity correlations and increasing structural correlations. For sufficiently large persistence times, it was found that the fitted MCT glass transition temperature increases monotonically with increasing persistence time, suggesting that vitrification occurs more easily as the material becomes more active. An MCT-based scaling analysis for this type of active-matter system was later performed by Nandi and Gov Nandi and Gov (2017).

Feng and Hou subsequently studied a thermal version of the active Ornstein-Uhlenbeck model that also includes thermal translational noise Feng and Hou (2017). Their MCT derivation differs fundamentally from the approach taken by Szamel Szamel (2016), however: it is valid only for sufficiently small persistence times and does not require explicit velocity correlations as input. Instead, their active-MCT dynamics is governed by an averaged diffusion constant and a steady-state structure factor which both depend on the effective temperature, density, and the persistence time of the active particles. They found that the critical glass-transition density shifts monotonically to larger values with increasing magnitude of the self-propulsion force, and that the critical glass-transition temperature increases with the persistence time. These results highlight the fact that different approximations within the theory can lead to fundamentally different results, and that thermal fluctuations may play a non-trivial role in the glassy dynamics of active Ornstein-Uhlenbeck particles. In a more recent study, Feng and Hou also extended their approach to mixtures of active (thermal AOUP) and passive particles Feng and Hou (2018). They found that such mixtures can give rise to reentrant behavior upon increasing the active component fraction, provided that the activity is sufficiently weak and that the active and passive species have different particle sizes. Note that this reentrance effect is distinct from the earlier MCT predictions of Ding et al. Ding et al. (2015).

The 2017 work by Liluashvili, Onody, and Voigtmann constitutes the first active-MCT work in which the translational and rotational degrees of freedom are explicitly coupled and treated on an equal footing Liluashvili et al. (2017). They focused on two-dimensional ABPs and performed a detailed analysis of the theoretical predictions for active hard particles. It was found that increasing the self-propulsion speed generally causes the material to become more liquid-like; this predicted active fluidization effect grows monotonically with increasing activity or inverse rotational diffusion constant, and shifts the glass-transition density to higher values. Interestingly, they also presented a non-trivial glass-transition phase diagram in the three-dimensional parameter space of density, self-propulsion speed, and rotational diffusivity, which is reminiscent of the phase diagram proposed by Bi et al. for the ABP Voronoi model Bi et al. (2016). This thus establishes an intriguing link between the dynamics of living confluent tissues and the first-principles-based statistical physics of glass-forming active liquids. Furthermore, Liluashvili et al. discussed the difference between the monotonic activity dependence of their ABP work and the non-monotonic behavior of athermal AOUPs reported by Szamel; they attributed this difference to the absence of thermal Brownian noise in AOUPs, suggesting that finite thermal diffusive motion generally makes caging less effective.

Very recently, Szamel also presented a new MCT version for thermal ABPs which is based on a similar derivation as his AOUP work, i.e. the rotational degrees of freedom are effectively integrated out Szamel (2019). It was predicted that at short times the self-propulsion always speeds up the relaxation; at long times the relaxation depends explicitly on the steady-state structure factor and the correlation function of steady-state currents. Although numerical results were not presented, it was concluded that the theory should reduce to the standard MCT for passive glass formation in the limit of vanishing self-propulsion. If thermal fluctuations are neglected, the theory becomes equivalent to the athermal AOUP theory Szamel (2016), while in the limit of high rotational diffusivity the theory coincides with the passive MCT at a temperature equal to the effective temperature. It was noted that the presence of non-trivial currents may wash away the ergodicity-breaking transition in active glassy matter; future work will hopefully shed more light on the precise role of activity within this first-principles framework.

Another important line of MCT research concerns the study of glassy dynamics in the limit of infinite spatial dimensions $d$. While this limit is rather abstract from an experimental point of view, the case $d\rightarrow\infty$ holds great relevance in theoretical physics, owing to the fact that it constitutes an exact mean-field limit. Two independent 2010 studies Ikeda and Miyazaki (2010); Schmid and Schilling (2010) have found that, perhaps disappointingly, standard MCT does not become exact for $d\rightarrow\infty$. In a landmark paper from 2016, however, Zamponi and co-workers managed to solve the exact dynamics of glassy hard spheres in infinite dimensions Maimbourg et al. (2016). Interestingly, their final equation is somewhat similar but not identical to MCT. A subsequent 2017 study by Szamel also presented an alternative, physically motivated derivation of this result Szamel (2017). Impressively, Agoritsas, Maimbourg, and Zamponi recently developed a dynamical $d\rightarrow\infty$ mean-field theory Agoritsas et al. (2019a, b) that also applies to non-equilibrium scenarios, including glassy aging and active (Ornstein-Uhlenbeck) dynamics Agoritsas et al. (2019a). Although explicit results for active glasses were not presented, future work will hopefully shed more light on the role of activity in a well-defined mean-field limit. Ultimately, approaches such as a perturbative expansion in $1/d$ may possibly allow one to extend these results to finite-dimensional and experimentally realizable active glassy systems. The reader is also referred to Ref. Agoritsas et al. (2019a) if she is interested in a broader discussion on the relevance of high-dimensional studies in the context of glassy dynamics.

We finally mention another class of non-equilibrium materials that is closely related to active fluids, namely driven granular matter. Between 2010 and 2013, Kranz, Sperl, and Zippelius developed an MCT for driven granulates by modeling the activity as a driving amplitude that gives rise to random particle kicks Kranz et al. (2010); Sperl et al. (2012); Kranz et al. (2013). It was found that the critical glass transition systematically shifts to higher densities as the dissipation due to inelastic particle collisions increases. The properties within the glass phase were also qualitatively and quantitatively affected by the degree of dissipation. As already noted in Sec. II.1, such driven materials may be studied experimentally by, e.g., placing granular particles on an air-fluidized or vibrating table, and it is hoped that experiments in this field will be realized in the near future.

IV.3 Random First Order Transition Theory and the Stokes-Einstein relation

We now discuss two other recent theoretical approaches to describe the glassy physics of active matter, including an active version of RFOT and an analytic treatment of Stokes-Einstein violation in athermal AOUPs. Let us first recall the standard RFOT scenario, which is a popular theoretical framework of the glass transition that is rooted in spin-glass theory Kirkpatrick and Thirumalai (1987); Kirkpatrick and Wolynes (1987). Briefly, RFOT amounts to a combination of the dynamical scenario of MCT at relatively high temperatures and thermodynamic arguments at lower temperatures. In particular, the low-temperature system is described as a mosaic of (Adam-Gibbs-like) amorphous domains with a characteristic configurational entropy. The sizes of these domains are determined by their stability against thermally activated fluctuations and grow progressively upon deeper supercooling. The domains can transform among themselves by overcoming their interfacial free energy due to thermal activation, and the time scale for such rearrangements is defined as the structural relaxation time $\tau$. We refer the reader to Refs. Lubchenko and Wolynes (2007); Kirkpatrick and Thirumalai (2014); Biroli and Bouchaud (2009); Berthier and Biroli (2011); Langer (2014) for a more detailed description of the theory, including its successes and limitations.

In 2018, Nandi et al. Nandi et al. (2018) extended the RFOT framework to non-equilibrium active systems in a phenomenological manner. Specifically, they incorporated the effect of activity in the expression for the configurational entropy; this effect, in turn, was assumed to arise from an activity-induced change in the (mean-field) potential energy, which should be valid only for small deviations from equilibrium. They studied the RFOT dynamics for two different flavors of the active Ornstein-Uhlenbeck process, and concluded that a larger self-propulsion strength always inhibits glassiness. The role of the persistence time, however, was found to be more subtle and dependent on the microscopic details of the activity. Notably, despite the simplifying approximations in their theory, Nandi et al. could account for the seemingly paradoxical simulation results of Flenner et al. Flenner et al. (2016) and Mandal et al. Mandal et al. (2016) regarding the effect of activity on the fragility. Indeed, for the active-particle model used by Mandal et al.  the material becomes stronger as the activity increases, while the athermal AOUP model of Flenner and co-workers yields more fragile behavior as the persistence time grows. The correct prediction of both phenomena (using two different active-RFOT models) highlights the fact that microscopic and material-specific details of the self-propulsion mechanism must be carefully taken into account when addressing glassy dynamics in active systems.

Let us finally return to the topic of Stokes-Einstein violation–an important feature that is observed in most glass-forming materials. Although at a phenomenological level the decoupling of the diffusion constant and the viscosity (or relaxation time) may appear to be similar in passive and active glassy materials–as reported by e.g. Flenner et al. Flenner et al. (2016)–the underlying physics may be fundamentally different. Indeed, in passive equilibrium systems it is rather surprising that the Stokes-Einstein relation breaks down at low temperatures; in non-equilibrium active matter, however, there is a priori no reason to assume that the Stokes-Einstein relation should hold at any temperature. The latter argument stems from the fact that the injection and dissipation of energy in active matter are uncorrelated: injection of energy arises from the conversion of some form of stored energy (either supplied internally or externally), while dissipation is due to friction with the solvent. This is to be contrasted with passive Brownian motion, in which both the movement and dissipative friction of colloidal particles are governed by the same surrounding medium.

In 2016, Fodor et al. theoretically studied this aspect in detail for a system of athermal AOUPs Fodor et al. (2016). They found that for sufficiently small persistence times, the active system behaves as an effective equilibrium system in the sense that detailed balance, time-reversal symmetry, and an effective fluctuation-dissipation theorem hold. As the persistence time increases, however, the material is driven further away from equilibrium, resulting in the breaking of detailed balance and the lack of a generalized Stokes-Einstein relation between damping and fluctuations. Thus, Stokes-Einstein violation should always occur in strongly active (OUP) systems. Although the work of Fodor et al. was not specifically motivated by glassy physics, their work does carry implications for the study of glass formation. In particular, it will be highly interesting to explore whether Stokes-Einstein violation in active glassy matter is also linked to the emergence of dynamical heterogeneity–as is generally believed to be the case for passive glass-forming matter–, and to what extent the microscopic origins of Stokes-Einstein violation in passive glass formers are fundamentally distinct from non-equilibrium active materials.

V Conclusions and outlook

This review has sought to provide an overview of the rapidly growing body of literature on active glassy matter, focusing on the manifestation of the main hallmarks of glassy dynamics in active systems from an experimental, numerical simulation, and theoretical perspective, respectively. Although the field has emerged only in recent years, it is already unambiguously clear that dense active matter–either of a synthetic or biological nature–shares many similarities with the phenomenology of conventional glass-forming materials. We briefly recapitulate the key evidence regarding the five aspects of glassy physics in non-equilibrium active matter below.

1. (i)

Dynamical slowdown. As in equilibrium supercooled liquids, disordered active matter can undergo kinetic arrest at sufficiently high densities. Indeed, this is universally observed in both synthetic model systems and in living cells; for the latter, a dramatic dynamical slowdown has been reported at both the intracellular and intercellular level in various cell types. Depending on the details of the self-propulsion mechanism, however, activity can both promote and inhibit the slowdown. In all cases, and as in conventional glass formers, the orders-of-magnitude growth of the viscosity or relaxation time is accompanied by only subtle changes in the material’s microstructure. For synthetic active model glass formers, small structural changes appear in the radial distribution or static structure factor which are unique to the non-equilibrium material, i.e., they generally cannot be mapped onto the structure of an effective equilibrium system. For living confluent cell layers, the geometric cell shape appears to be an important and unique structural signature of glassiness. Curiously, there is compelling evidence that living cells also employ activity as a means to control their own relaxation time, e.g. by switching from a fluid-like to glass-like dormant state under unfavorable environmental conditions. The apparent lack or delay of a dynamical slowdown in confluent cells has also been associated with pathological conditions such as asthma. 2. (ii)

Fragility. Both synthetic and living active materials can exhibit varying degrees of fragility, ranging from strong Arrhenius-type to fragile super-Arrhenius-type behavior. Depending on the microscopic details of the system, non-equilibrium activity can both increase and decrease the fragility. In living cells, the intracellular cytoplasm appears to switch from fragile to strong as the metabolic activity level increases; active confluent cell layers tend to behave as fragile materials. The biological relevance of fragility in living cells and tissues, as well as the experimental realization of activity-controlled fragility in synthetic materials, still remains to be established. 3. (iii)

Dynamical heterogeneity. As in conventional glass formers, both synthetic and living active materials exhibit growing dynamically heterogeneous behavior upon approaching the glass transition. Thus far, the role of activity appears to amount to a similar phenomenology as in passive equilibrium systems: as the relaxation time increases–either by activity or supercooling–, the dynamics also becomes increasingly more heterogeneous. In a biological context, it has been hypothesized that the emergence of dynamically heterogeneous domains within a primary tumor may partially underlie the formation of metastatic cell clusters. 4. (iv)

Stokes-Einstein violation. The manifestation of Stokes-Einstein violation in active glassy materials has not yet been studied in great detail, but the first results on synthetic model systems suggest that the behavior is rather similar to conventional glass formers. More specifically, at high effective temperatures and low persistence times, the Stokes-Einstein relation appears to hold. By departing from equilibrium through the increase in persistence time, the diffusion constant and relaxation time become more decoupled. Although the microscopic origin of Stokes-Einstein violation in passive supercooled liquids is still debated, and may possibly be associated with the emergence of dynamical heterogeneity, in active matter the origin stems from the inherent lack of a fluctuation-dissipation theorem far away from equilibrium. The latter presumably can occur even in the absence of dynamical heterogeneity. It remains unclear whether Stokes-Einstein violation in active and passive glassy matter may ultimately be captured in a single unifying framework, to what extent dynamical heterogeneity plays a universal underlying role, and to what extent living glassy cells will conform to the same physics as synthetic model systems. 5. (v)

Aging. Owing to the non-Hamiltonian nature of active matter, the conventional interpretation of physical aging as a gradual approach to a deeper energy minimum is a priori inapplicable to active glasses. Thus far, only one simulation study has reported aging in athermal active glassy matter; in this work, the observed aging behavior was interpreted as an activity-driven approach to a more mechanically stable state that effectively acts as an absorbing state. It remains to be established whether, e.g., the presence of thermal fluctuations in active matter will yield a similar phenomenology as aging in passive thermal glasses. In experiments on confluent cell layers, aging was found to be manifested in various physical properties, including the gradual densification of the cell sheet, and the maturation of cell-cell and cell-substrate contacts within the layer. More work is needed to clarify the main fundamental differences in physical aging between active and non-active glassy matter, and between synthetic and living active glasses.

We end by discussing several interesting open questions and possible directions for future research. Regarding the most striking feature of glass formation, i.e. the spectacular slowdown of the dynamics, it remains to be established which microstructural material properties ultimately underlie this behavior. Although the structure-dynamics link has still not been rigorously elucidated for conventional supercooled liquids, it is plausible that this link will be even more complex for non-equilibrium active glassy materials. Indeed, activity is known to affect both the structural and dynamical properties in a non-trivial manner, and the causal relation between these two effects is still unknown. It will therefore be highly interesting to explore whether the presence of self-propulsion forces primarily influences only the structure, only the dynamics, or both. Furthermore, although a glassy slowdown has been widely observed in both living and non-living active systems, the respective microstructures of these materials have thus far been quantified in fundamentally different ways. Future work should clarify whether liquid-state-theory-based concepts, such as the radial distribution function, can also be successfully applied to quantify relevant structural changes in, e.g., confluent cell layers. If so, a theoretical connection may be established between the (first-principles-based) statistical physics of active glassy liquids and biological glassy phenomena. This raises the intriguing prospect that, ultimately, the elusive structure-dynamics link in living and non-living active matter can be captured in a single theoretical framework.

Fragility, i.e. the abruptness with which vitrification takes place, has historically provided an important and unifying concept to classify seemingly disparate classes of (passive) glass-forming systems. Indeed, it is widely believed that a rigorous understanding of its microstructural origins will be paramount to achieve a universal description of glass formation. Since non-equilibrium activity has now been shown to affect the fragility in a non-trivial manner–both in synthetic and in living active systems–, the natural question arises whether the fundamental physical properties underlying a material’s fragility will be similar or distinct in active and passive glasses. As in the case of the dynamical slowdown, this would also carry implications for the development of a general and quantitative framework that can rationally relate structural to dynamical properties, both for living and non-living active systems.

From a more applied point of view, fragility is also known to impact both the processability and functionality of a material; for example, strong glass formers are generally easier to (re-)shape and recycle, while fragile glass formers can be used as, e.g., ultrafast phase-changing materials. We here propose that the complex role of activity on a material’s fragility may also hold relevance in applied materials science and biology. Specifically, we pose the following two questions: i) can the addition of self-propulsion forces be exploited to create novel synthetic active materials with an externally tunable fluidization/solidification response, and ii) do living cells, both at the intra- and intercellular level, invoke activity as a means to control their own fragility? Regarding the latter, it might be possible that the up- or down-regulation of metabolic activity allows cells to control the transport properties within their cytoplasm, and to regulate the abruptness of solidification in collective cell migration processes such as wound healing and embryonic development. More generally, it will be highly interesting to explore if and how a departure from equilibrium will enable novel dynamic control mechanisms in functional glassy matter.

Our current understanding of dynamical heterogeneity, i.e. the increasingly cooperative nature of particle rearrangements near the glass transition, is still far from complete; however, it is broadly accepted that the emergence of dynamically heterogeneous behavior is concomitant to the slowdown in structural relaxation. As already discussed above, the establishment of a framework that can accurately predict the full glassy dynamics, including the degree of dynamical heterogeneity, for a given material composition and structure will constitute a major breakthrough in the field of condensed matter physics. For active materials, it is not yet clear whether the presence of non-equilibrium activity will generally add a fundamentally new ingredient to the physics of dynamical heterogeneity; indeed, the first studies suggest that the phenomenology in passive and active glass-forming materials is rather similar. Nonetheless, it is plausible that active forces, and especially actively aligning interactions, can induce non-trivial spatiotemporal correlations and coordinated movements among particles that would be absent in the passive counterpart. Future work is needed to establish how the manifestation of dynamical heterogeneity, and the possible link with the underlying microstructure, is dependent on the details of the self-propulsion mechanism. As in the case of fragility, it will be interesting to study whether living cells also exploit activity–either directly in the dynamics or indirectly via changes in the microstructure–to effectively control the size and growth of dynamically heterogeneous regions. A speculative but intriguing possibility is that the emergence of such transiently mobile domains within cancerous cell tumors partially contributes to the formation of cell clusters with enhanced mobility, effective initializing the onset of metastasis.

Curiously, the breakdown of the Stokes-Einstein relation, i.e. the decoupling of the temperature-dependent diffusion constant and relaxation time, is currently better understood for active materials than for passive glass-forming materials. The presence of strong non-equilibrium forces gives rise to the breaking of detailed balance and the lack of a fluctuation-dissipation theorem, thus rendering the Stokes-Einstein relation generally inapplicable to active matter. Conversely, for passive supercooled liquids, it has been hypothesized that Stokes-Einstein violation is essentially a manifestation of dynamical heterogeneity; it remains to be established whether an increased violation of the Stokes-Einstein relation in active matter is also generally accompanied by increasingly dynamically heterogeneous behavior, and whether living active cells will also conform to this phenomenology. Even if at a phenomenological level the decoupling of the diffusion constant and relaxation time will be manifestly equivalent, the physical origins of this decoupling need not be the same in equilibrium and far-from-equilibrium systems. Future work will hopefully shed more light on these origins, in particular on the possible links with the underlying microstructure and the degree of dynamical heterogeneity.

Arguably the least studied glassy property in active matter is aging, i.e. the explicit dependence of a material on its age. Although proof-of-principle for this non-equilibrium phenomenon has recently been established for active matter, there is still a multitude of interesting open questions that remain to be answered. For example, can non-equilibrium activity act as an effective proxy for thermal fluctuations in passive aging glasses? Can the notion of a mechanical energy landscape and an absorbing-state formalism be generalized to other systems? How does the applied aging protocol affect the active-aging behavior? Can specific features of physical aging in passive glass formers, such as a logarithmic decay and a time-versus-waiting-time superposition principle, also be observed in active glasses? And to what extent can the aging behavior of synthetic active glasses be translated to the aging of living cells?

Finally, we note another disparate field of research in which the study of active glassy dynamics and aging may potentially find applicability, namely neural networks. For example, in so-called Hopfield networks Hopfield (1982), a memory pattern is encoded as a local energy minimum that acts as an attractor in a rugged energy landscape. Passive spin glasses form a suitable realization of such networks Amit et al. (1985), but it might be possible to extend the scope and functionality of Hopfield networks by adding activity as a new degree of freedom. The question how efficiently different aging protocols drive active (spin) glasses into a basin of attraction may thus also offer new possibilities for the design and implementation of ’active’ neural networks.

Acknowledgements.

It is a pleasure to thank Ludovic Berthier, Grzegorz Szamel, Roberto Cerbino, and Atsushi Ikeda for interesting discussions. Vincent Debets is gratefully acknowledged for his help in preparing Figure 3.