Continuum Theory of Active Phase Separation in Cellular Aggregates

The yoga of droplets: coalescence in complex fluids

Coalescence of liquid drops is a widely observed phenomenon across a diverse range of natural and industrial processes, ranging from the formation of raindrops and the growth of tumor cells to inkjet printing. The physics involved in this seemingly simple process of coalescence is far from simple, and the rheological complexity of the fluid under investigation makes the problem even more challenging. The present article provides a brief overview of existing knowledge on this topic and discusses the potential avenues for future research. Special attention is given to non-Newtonian fluids and soft materials, which exhibit distinct behaviors compared to Newtonian liquids, revealing fascinating deviations that demand deeper investigation.

Boosting macroscopic diffusion with local resetting

Stochastic interactions generically enhance self-diffusivity in living and biological systems, e.g., optimizing navigation strategies and controlling material properties of cellular tissues and bacterial aggregates. Despite this, the physical mechanisms underlying this nonequilibrium behavior are poorly understood. Here, we introduce a model of interactions between an agent and its environment in the form of a local stochastic resetting mechanism, in which the agent's position is set to the nearest of a predetermined array of sites with a fixed rate. We derive analytic results for the self-diffusion coefficient, showing explicitly that this mechanism enhances diffusivity. Strikingly, we show analytically that this enhancement is optimized by regular arrays of resetting sites. Altogether, our results ultimately provide the conditions for the optimization of the macroscopic transport properties of diffusive systems with local random binding interactions.

Phase ordering in binary mixtures of active nematic fluids

We use a continuum, two-fluid approach to study a mixture of two active nematic fluids. Even in the absence of thermodynamically driven ordering, for mixtures of different activities we observe turbulent microphase separation, where domains form and disintegrate chaotically in an active turbulent background. This is a weak effect if there is no elastic nematic alignment between the two fluid components, but is greatly enhanced in the presence of an elastic alignment or substrate friction. We interpret the results in terms of relative flows between the two species which result from active anchoring at concentration gradients. Our results may have relevance in interpreting epithelial cell sorting and the dynamics of multispecies bacterial colonies.

Interactions between pili affect the outcome of bacterial competition driven by the type VI secretion system

The bacterial type VI secretion system (T6SS) is a widespread, kin-discriminatory weapon capable of shaping microbial communities. Due to the system's dependency on contact, cellular interactions can lead to either competition or kin protection. Cell-to-cell contact is often accomplished via surface-exposed type IV pili (T4Ps). In Vibrio cholerae, these T4Ps facilitate specific interactions when the bacteria colonize natural chitinous surfaces. However, it has remained unclear whether and, if so, how these interactions affect the bacterium's T6SS-mediated killing. In this study, we demonstrate that pilus-mediated interactions can be harnessed by T6SS-equipped V. cholerae to kill non-kin cells under liquid growth conditions. We also show that the naturally occurring diversity of pili determines the likelihood of cell-to-cell contact and, consequently, the extent of T6SS-mediated competition. To determine the factors that enable or hinder the T6SS's targeted reduction of competitors carrying pili, we developed a physics-grounded computational model for autoaggregation. Collectively, our research demonstrates that T4Ps involved in cell-to-cell contact can impose a selective burden when V. cholerae encounters non-kin cells that possess an active T6SS. Additionally, our study underscores the significance of T4P diversity in protecting closely related individuals from T6SS attacks through autoaggregation and spatial segregation.

Collective Cell Radial Ordered Migration in Spatial Confinement

Collective cells, a typical active matter system, exhibit complex coordinated behaviors fundamental for various developmental and physiological processes. The present work discovers a collective radial ordered migration behavior of NIH3T3 fibroblasts that depends on persistent top-down regulation with 2D spatial confinement. Remarkably, individual cells move in a weak-oriented, diffusive-like rather than strong-oriented ballistic manner. Despite this, the collective movement is spatiotemporal heterogeneous and radial ordering at supracellular scale, manifesting as a radial ordered wavefront originated from the boundary and propagated toward the center of pattern. Combining bottom-up cell-to-extracellular matrix (ECM) interaction strategy, numerical simulations based on a developed mechanical model well reproduce and explain above observations. The model further predicts the independence of geometric features on this ordering behavior, which is validated by experiments. These results together indicate such radial ordered collective migration is ascribed to the couple of top-down regulation with spatial restriction and bottom-up cellular endogenous nature.

Mechano-induced homotypic patterned domain formation by monocytes

Matrix stiffness and corresponding mechano-signaling play indispensable roles in cellular phenotypes and functions. How tissue stiffness influences the behavior of monocytes, a major circulating leukocyte of the innate system, and how it may promote the emergence of collective cell behavior is less understood. Here, using tunable collagen-coated hydrogels of physiological stiffness, we show that human primary monocytes undergo a dynamic local phase separation to form highly regular, reversible, multicellular, multi-layered domains on soft matrix. Local activation of the β2 integrin initiates inter-cellular adhesion, while global soluble inhibitory factors maintain the steady state domain pattern over days. Patterned domain formation generated by monocytes is unique among other key immune cells, including macrophages, B cells, T cells, and NK cells. While inhibiting their phagocytic capability, domain formation promotes monocytes' survival. We develop a computational model based on the Cahn-Hilliard equation of phase separation, combined with a Turing mechanism of local activation and global inhibition suggested by our experiments, and provides experimentally validated predictions of the role of seeding density and both chemotactic and random cell migration on domain pattern formation. This work reveals that, unlike active matters, cells can generate complex cell phases by exploiting their mechanosensing abilities and combined short-range interactions and long-range signals to enhance their survival.

Phase Separation Driven by Active Flows

We extend the continuum theories of active nematohydrodynamics to model a two-fluid mixture with separate velocity fields for each fluid component, coupled through a viscous drag. The model is used to study an active nematic fluid mixed with an isotropic fluid. We find microphase separation, and argue that this results from an interplay between active anchoring and active flows driven by concentration gradients. The results may be relevant to cell sorting and the formation of lipid rafts in cell membranes.

Microphase separation of living cells

Self-organization of cells is central to a variety of biological systems and physical concepts of condensed matter have proven instrumental in deciphering some of their properties. Here we show that microphase separation, long studied in polymeric materials and other inert systems, has a natural counterpart in living cells. When placed below a millimetric film of liquid nutritive medium, a quasi two-dimensional, high-density population of Dictyostelium discoideum cells spontaneously assembles into compact domains. Their typical size of 100 μm is governed by a balance between competing interactions: an adhesion acting as a short-range attraction and promoting aggregation, and an effective long-range repulsion stemming from aerotaxis in near anoxic condition. Experimental data, a simple model and cell-based simulations all support this scenario. Our findings establish a generic mechanism for self-organization of living cells and highlight oxygen regulation as an emergent organizing principle for biological matter.

Stress anisotropy in confined populations of growing rods

A central feature of living matter is its ability to grow and multiply. The mechanical activity associated with growth produces both macroscopic flows shaped by confinement, and striking self-organization phenomena, such as orientational order and alignment, which are particularly prominent in populations of rod-shaped bacteria due to their nematic properties. However, how active stresses, passive mechanical interactions and flow-induced effects interact to give rise to the observed global alignment patterns remains elusive. Here, we study in silico colonies of growing rod-shaped particles of different aspect ratios confined in channel-like geometries. A spatially resolved analysis of the stress tensor reveals a strong relationship between near-perfect alignment and an inversion of stress anisotropy for particles with large length-to-width ratios. We show that, in quantitative agreement with an asymptotic theory, strong alignment can lead to a decoupling of active and passive stresses parallel and perpendicular to the direction of growth, respectively. We demonstrate the robustness of these effects in a geometry that provides less restrictive confinement and introduces natural perturbations in alignment. Our results illustrate the complexity arising from the inherent coupling between nematic order and active stresses in growing active matter, which is modulated by geometric and configurational constraints due to confinement.

Unbiased retrieval of frequency-dependent mechanical properties from noisy time-dependent signals

The mechanical response of materials to dynamic loading is often quantified by the frequency-dependent complex modulus. Probing materials directly in the frequency domain faces technical challenges such as a limited range of frequencies, long measurement times, or small sample sizes. Furthermore, many biological samples, such as cells or tissues, can change their properties upon repetitive probing at different frequencies. Therefore, it is common practice to extract the material properties by fitting predefined mechanical models to measurements performed in the time domain. This practice, however, precludes the probing of unique and yet unexplored material properties. In this report, we demonstrate that the frequency-dependent complex modulus can be robustly retrieved in a model-independent manner directly from time-dependent stress-strain measurements. While applying a rolling average eliminates random noise and leads to a reliable complex modulus in the lower frequency range, a Fourier transform with a complex frequency helps to recover the material properties at high frequencies. Finally, by properly designing the probing procedure, the recovery of reliable mechanical properties can be extended to an even wider frequency range. Our approach can be used with many state-of-the-art experimental methods to interrogate the mechanical properties of biological and other complex materials.

Intermittent attractive interactions lead to microphase separation in nonmotile active matter

Nonmotile active matter exhibits a wide range of nonequilibrium collective phenomena yet examples are crucially lacking in the literature. We present a microscopic model inspired by the bacteria Neisseria meningitidis in which diffusive agents feel intermittent attractive forces. Through a formal coarse-graining procedure, we show that this truly scalar model of active matter exhibits the time-reversal-symmetry breaking terms defining the Active Model B+ class. In particular, we confirm the presence of microphase separation by solving the kinetic equations numerically. We show that the switching rate controlling the interactions provides a regulation mechanism tuning the typical cluster size, e.g., in populations of bacteria interacting via type IV pili.

Mathematical and computational models in spheroid-based biofabrication

Ubiquitous in embryonic development, tissue fusion is of interest to tissue engineers who use tissue spheroids or organoids as building blocks of three-dimensional (3D) multicellular constructs. This review presents mathematical models and computer simulations of the fusion of tissue spheroids. The motivation of this study stems from the need to predict the post-printing evolution of 3D bioprinted constructs. First, we provide a brief overview of differential adhesion, the main morphogenetic mechanism involved in post-printing structure formation. It will be shown that clusters of cohesive cells behave as an incompressible viscous fluid on the time scale of hours. The discussion turns then to mathematical models based on the continuum hydrodynamics of highly viscous liquids and on statistical mechanics. Next, we analyze the validity and practical use of computational models of multicellular self-assembly in live constructs created by tissue spheroid bioprinting. Finally, we discuss the perspectives of the field as machine learning starts to reshape experimental design, and modular robotic workstations tend to alleviate the burden of repetitive tasks in biofabrication. STATEMENT OF SIGNIFICANCE: Bioprinted constructs are living systems, which evolve via morphogenetic mechanisms known from developmental biology. This review presents mathematical and computational tools devised for modeling post-printing structure formation. They help achieving a desirable outcome without expensive optimization experiments. While previous reviews mainly focused on assumptions, technical details, strengths, and limitations of computational models of multicellular self-assembly, this article discusses their validity and practical use in biofabrication. It also presents an overview of mathematical models that proved to be useful in the evaluation of experimental data on tissue spheroid fusion, and in the calibration of computational models. Finally, the perspectives of the field are discussed in the advent of robotic biofabrication platforms and bioprinting process optimization by machine learning.

Arrested coalescence of multicellular aggregates

Multicellular aggregates are known to exhibit liquid-like properties. The fusion process of two cell aggregates is commonly studied as the coalescence of two viscous drops. However, tissues are complex materials and can exhibit viscoelastic behaviour. It is known that elastic effects can prevent the complete fusion of two drops, a phenomenon known as arrested coalescence. Here we study this phenomenon in stem cell aggregates and provide a theoretical framework which agrees with the experiments. In addition, agent-based simulations show that active cell fluctuations can control a solid-to-fluid phase transition, revealing that arrested coalescence can be found in the vicinity of an unjamming transition. By analysing the dynamics of the fusion process and combining it with nanoindentation measurements, we obtain the effective viscosity, shear modulus and surface tension of the aggregates. More generally, our work provides a simple, fast and inexpensive method to characterize the mechanical properties of viscoelastic materials.

Phase separation dynamics in deformable droplets

Phase separation can drive spatial organization of multicomponent mixtures. For instance in developing animal embryos, effective phase separation descriptions have been used to account for the spatial organization of different tissue types. Similarly, separation of different tissue types is also observed in stem cell aggregates, where the emergence of a polar organization can mimic early embryonic axis formation. Here, we describe such aggregates as deformable two-phase fluid droplets, which are suspended in a fluid environment (third phase). Using hybrid finite-volume Lattice-Boltzmann simulations, we numerically explore the out-of-equilibrium routes that can lead to the polar equilibrium state of such a droplet. We focus on the interplay between spinodal decomposition and advection with hydrodynamic flows driven by interface tensions, which we characterize by a Peclet number Pe. Consistent with previous work, for large Pe the coarsening process is generally accelerated. However, for intermediate Pe we observe long-lived, strongly elongated droplets, where both phases form an alternating stripe pattern. We show that these "croissant" states are close to mechanical equilibrium and coarsen only slowly through diffusive fluxes in an Ostwald-ripening-like process. Finally, we show that a surface tension asymmetry between both droplet phases leads to transient, rotationally symmetric states whose resolution leads to flows reminiscent of Marangoni flows. Our work highlights the importance of advection for the phase separation process in finite, deformable systems.

Liquid-Liquid Phase Separation: Unraveling the Enigma of Biomolecular Condensates in Microbial Cells

Numerous examples of microbial phase-separated biomolecular condensates have now been identified following advances in fluorescence imaging and single molecule microscopy technologies. The structure, function, and potential applications of these microbial condensates are currently receiving a great deal of attention. By neatly compartmentalizing proteins and their interactors in membrane-less organizations while maintaining free communication between these macromolecules and the external environment, microbial cells are able to achieve enhanced metabolic efficiency. Typically, these condensates also possess the ability to rapidly adapt to internal and external changes. The biological functions of several phase-separated condensates in small bacterial cells show evolutionary convergence with the biological functions of their eukaryotic paralogs. Artificial microbial membrane-less organelles are being constructed with application prospects in biocatalysis, biosynthesis, and biomedicine. In this review, we provide an overview of currently known biomolecular condensates driven by liquid-liquid phase separation (LLPS) in microbial cells, and we elaborate on their biogenesis mechanisms and biological functions. Additionally, we highlight the major challenges and future research prospects in studying microbial LLPS.