Research

Effective pressure in a dense cell monolayer

A system of dividing and growing cells provides an intriguing example of active matter far from equilibrium. Living cells in a dense system are all in contact with each other. The common assumption is that such cells stop dividing due to a lack of space. Recent experimental observations have shown, however, that cells continue dividing for some time, even after a dense cell monolayer is formed. The effective pressure is introduced in order to model the experimentally observed phenomenon in which the average cell size dramatically decreases over time, and cell area distribution becomes narrower. For a non-uniform system, I will consider the cell shift due to the gradient of the effective pressure and examine its effect on the average cell area profiles. Then I will discuss collective cell migration where cells maintain contacts with their neighbors. This migration can be described in terms of a novel front propagation phenomenon; the front speed and the effective pressure profile are found both numerically and analytically.

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E. Khain and J. Straetmans, “Dynamics of an expanding cell monolayer”, Journal of Statistical Physics, accepted (2021).

J. Straetmans and E. Khain, “Modeling Cell Size Dynamics in a Confined Nonuniform Dense Cell Culture”, Journal of Statistical Physics, 176(2), 299-311, (2019).

E. Khain and L. S. Tsimring, “Effective pressure and cell area distribution in a confined monolayer”, Fluid Dynamics Research 50, 051413 (2018).

Levitating granular cluster: typical behavior and noise-induced rare events

In a granular Leidenfrost effect, a solid-like cluster is levitating above the “hot” granular gas. This state was observed experimentally, when granular matter was vertically vibrated in a two-dimensional container. The solid-gas coexistence can be described by using granular hydrodynamics with the properly measured transport coefficients. We performed extensive molecular dynamics simulations of a simple model of inelastic hard spheres driven by a “thermal” bottom wall. Simulations showed that for low wall temperatures, the levitating cluster is stable, while for high wall temperatures, it breaks down, and the hot gas bursts out resembling a volcanic explosion. We found a hysteresis: for a wide range of bottom wall temperatures, both the clustering state and the broken state are stable. However, even if the system is at the (stable) clustering state, a "volcanic explosion" is possible: it is a rare event driven by large fluctuations. We used techniques from the theory of rare events to compute the mean time for cluster breaking to occur; this required the introduction of a two-component reaction coordinate.

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E. Khain, "Thermal conductivity at the high-density limit and the levitating granular cluster", Physical Review E 98, 012903 (2018).

E. Khain and L. M. Sander, "Noise induces rare events in granular media", Physical Review E 94, 032905 (2016).

Migration of adhesive and proliferative cells: front propagation

Consider first a simple discrete model for diffusion and proliferation. Each lattice site can be empty or once occupied. At each time step, a particle is picked at random. Then it can either jump to a neighboring empty site, or proliferate there (a new particle is born). We can ask for is the continuum analog of this model? It was shown that for proliferation rates the propagating fronts in this discrete system can be described by the Fisher-Kolmogorov equation. Suppose now that cells also experience cell-cell adhesion. Here, there are there are two interesting regimes. For subcritical adhesion, there are propagating "pulled" fronts, similarly to those of Fisher-Kolmogorov equation. For supercritical adhesion, there is a nontrivial transient behavior, where density profile exhibits a secondary peak.

In addition to random motion, a cell can perform directed motion in response to a gradient of some chemicals; this phenomenon is called chemotaxis. Chemotaxis can be positive if a cell moves toward a higher concentration of chemoattractants; during negative chemotaxis cells move toward the lower concentrations of chemorepellents. We have recently analyzed the phenomenon of cell migration by deriving a continuum equation for cell density from the underlying microscopic lattice model taking into account both cell-cell adhesion and chemotaxis. The theoretical predictions obtained by solving the resulting system of reaction diffusion equations agree very well with the numerical results of the stochastic hybrid model. We have also shown that when cell chemotaxis is taken into account, the theoretical results agree with the experimental data. We have also investigated the role of hypoxia in migration of brain tumor cells, both theoretically and experimentally. We have shown that hypoxia decreases both the motility of cells and the strength of cell-cell adhesion.

Earlier, we studied possible fingering in the in vitro dynamics of the malignant brain tumor. Experiments with different types of cells showed qualitatively different behavior: one cell line invaded in a spherically symmetric manner, but another gave rise to branches. We formulated a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane diagram in the parameter space were determined. The results were in a qualitative agreement with experimental findings for the two types of cells.

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Clustering of brain tumor cells

It turns out that invasive cells (that detached from the primary tumor and migrated away) have a very low proliferation (division) rate compared to those on the tumor surface, the so-called proliferative cells. This dichotomy between migration and proliferation is sometimes called the ``go or grow" property: that is, invasive cells mostly migrate rather than divide. Therefore, they are almost invisible to standard radiation and chemotherapy treatments, which kill cells that proliferate. The crucial problem in GBM treatment is that invasive cells may eventually switch back to the proliferative phenotype. This switch may occur after a cell has migrated a large distance (up to several centimeters) from the original solid tumor; it gives rise to recurrent tumors. The mechanisms of the phenotypic switch are poorly understood.

An attractive scenario is to see the phenotypic switch as a collective phenomenon. We have proposed that the phenotypic switch is related to the observed clustering of invasive cells. Once such clusters are formed in the invasive region, cells on the surfaces of the clusters can become proliferative again, like the cells on a surface of a primary tumor, thus leading to formation of distant recurrent brain tumors.

To investigate the mechanisms of cell clustering on a substrate, we formulated a discrete stochastic model for cell migration. The model accounts for cells diffusion, proliferation and adhesion. We predicted that cells typically form clusters if the effective strength of cell-cell adhesion exceeds a certain threshold. Another possibility is that recurrent brain tumors can be triggered by a rare event - spontaneous clustering of invasive tumor cells. Once a sufficiently large cluster is formed due to a large fluctuation, cells on the surface of the cluster may become proliferative, triggering rapid tumor growth.

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Fluctuations and stability in front propagation

We investigate the effects of large fluctuations in population dynamics. One example is effect of fluctuations on front propagation in a bistable system. Propagating fronts arising from bistable reaction-diffusion equations are a purely deterministic effect. Stochastic reaction-diffusion processes also show front propagation which coincides with the deterministic effect in the limit of small fluctuations (usually, large populations). However, for larger fluctuations propagation can be affected. We give an example, based on the classic spruce budworm model, where the direction of wave propagation, i.e., the relative stability of two phases, can be reversed by fluctuations.

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Fluctuations in population dynamics

Many populations in nature are fragmented: they consist of local populations occupying separate patches. A local population is prone to extinction due to the shot noise of birth and death processes. A migrating population from another patch can dramatically delay the extinction. What is the optimal migration rate that minimizes the extinction risk of the whole population? Here, we answer this question for a connected network of model habitat patches with different carrying capacities.

We have also considered population dynamics on a network of patches, having the same local dynamics, with different population scales (carrying capacities). It is reasonable to assume that if the patches are coupled by very fast migration the whole system will look like an individual patch with a large effective carrying capacity. This is called a "well-mixed" system. We show that, in general, it is not true that the total population has the same dynamics as each local patch when the migration is fast. Different global dynamics can emerge, and usually must be figured out for each individual case. We give a general condition which must be satisfied for the total population to have the same dynamics as the constituent patches.

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Instabilities and fluid-solid coexistence in dense shear granular flows

It is known that transport coefficients of hard sphere fluid diverge at the density of dense close packing. However, there is recent evidence that the coefficient of shear viscosity diverges at a lower density than other constitutive relations. This may result in a coexistence of "solid-like" and "fluid-like" layers in dense shear flow. The density in "solid-like" layers is higher than the density of viscosity divergence, therefore these layers are at rest or move as a whole.

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Instabilities in driven granular gases

We investigated the long-standing puzzle of phase separation in a granular monolayer vibrated from below. Although this system is three-dimensional, an interesting dynamics occurs mostly in the horizontal plane, perpendicularly to the direction of vibration. Experiments [Olafsen and Urbach, Phys. Rev. Lett. 81, 4369 (1998)] demonstrated that for high amplitude of vibration the system is in the gas-like phase, but when the amplitude becomes smaller than a certain threshold, a phase separation occurs: a solid-like dense condensate of particles forms in the center of the system, surrounded by particles in the gas-like phase. We theoretically explain the experimentally observed coexistence of dilute and dense phases, employing Navier-Stokes granular hydrodynamics. We show that the phase separation is associated with negative compressibility of granular gas.

Another phase-separation instability occurs in a very simple setting: an ensemble of inelastic hard disks driven by a rapidly vibrating side wall in the absence of gravity. This instability is surprisingly similar to the phase-separating instability in the van der Waals gas. Another instability is the thermal granular convection that develops in the same prototypical system, but in the presence of gravity. Convection in a horizontal layer of "classical" fluid heated from below, known as the Rayleigh-Benard convection, is a famous example of pattern formation outside of equilibrium. Understanding the analogous instability in a granular fluid is important for the physics of granular matter. I also investigated the oscillatory instability, where the system is driven by two opposite "thermal" walls at zero gravity. When the inelasticity of particle collisions exceeds a critical value, the "static" clustering state in the middle of the system becomes unstable and develops oscillations. Hydrodynamic predictions have been verified in MD simulations.

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