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Register a new userDispersal and organization of polarized cells: non-linear diffusion and cluster formation without adhesion
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Experimental studies of cell motility in culture have shown that under adequate conditions these living organisms possess the ability to organize themselves into complex structures. Such structures may exhibit a synergy that greatly increases their survival rate and facilitate growth or spreading to different tissues. These properties are even more significant for cancer cells and related pathologies. Theoretical studies supported by experimental evidence have also shown that adhesion plays a significant role in cellular organization. Here we show that the directional persistence observed in polarized displacements permits the formation of stable cell aggregates in the absence of adhesion, even in low-density regimes. We introduce a discrete stochastic model for the dispersal of polarized cells with exclusion and derive the hydrodynamic limit. We demonstrate that the persistence coupled with the cell-cell exclusion hinders the cellular motility around other cells, leading to a non-linear diffusion which facilitates their capture into larger aggregates.
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We investigate clustering of malignant glioma cells. emph{In vitro} experiments in collagen gels identified a cell line that formed clusters in a region of low cell density, whereas a very similar cell line (which lacks an important mutation) did not cluster significantly. We hypothesize that the mutation affects the strength of cell-cell adhesion. We investigate this effect in a new experiment, which follows the clustering dynamics of glioma cells on a surface. We interpret our results in terms of a stochastic model and identify two mechanisms of clustering. First, there is a critical value of the strength of adhesion; above the threshold, large clusters grow from a homogeneous suspension of cells; below it, the system remains homogeneous, similarly to the ordinary phase separation. Second, when cells form a cluster, we have evidence that they increase their proliferation rate. We have successfully reproduced the experimental findings and found that both mechanisms are crucial for cluster formation and growth.
Universality in the behavior of complex systems often reveals itself in the form of scale-invariant distributions that are essentially independent of the details of the microscopic dynamics. A representative paradigm of complex behavior in nature is cooperative evolution. The interaction of individuals gives rise to a wide variety of collective phenomena that strongly differ from individual dynamics---such as demographic evolution, cultural and technological development, and economic activity. A striking example of such cooperative phenomena is the formation of urban aggregates which, in turn, can be considered to cooperate in giving rise to nations. We find that population and area distributions of nations follow an inverse power-law behavior, as is known for cities. The exponents, however, are radically different in the two cases ($mu approx 1$ for nations, $mu approx 2$ for cities). We interpret these findings by developing growth models for cities and for nations related to basic properties of partition of the plane. These models allow one to understand the empirical findings without resort to the introduction of complex socio-economic factors.
We study the dynamics of exchange value in a system composed of many interacting agents. The simple model we propose exhibits cooperative emergence and collapse of global value for individual goods. We demonstrate that the demand that drives the value exhibits non Gaussian fat tails and typical fluctuations which grow with time interval with a Hurst exponent of 0.7.
Anomalous diffusion, manifest as a nonlinear temporal evolution of the position mean square displacement, and/or non-Gaussian features of the position statistics, is prevalent in biological transport processes. Likewise, collective behavior is often observed to emerge spontaneously from the mutual interactions between constituent motile units in biological systems. Examples where these phenomena can be observed simultaneously have been identified in recent experiments on bird flocks, fish schools and bacterial swarms. These results pose an intriguing question, which cannot be resolved by existing theories of active matter: How is the collective motion of these systems affected by the anomalous diffusion of the constituent units? Here, we answer this question for a microscopic model of active Levy matter -- a collection of active particles that perform superdiffusion akin to a Levy flight and interact by promoting polar alignment of their orientations. We present in details the derivation of the hydrodynamic equations of motion of the model, obtain from these equations the criteria for a disordered or ordered state, and apply linear stability analysis on these states at the onset of collective motion. Our analysis reveals that the disorder-order phase transition in active Levy matter is critical, in contrast to ordinary active fluids where the phase transition is, instead, first-order. Correspondingly, we estimate the critical exponents of the transition by finite size scaling analysis and use these numerical estimates to relate our findings to known universality classes. These results highlight the novel physics exhibited by active matter integrating both anomalous diffusive single-particle motility and inter-particle interactions.
We are interested in the tail behavior of the pdf of mass density within the one and $d$-dimensional Burgers/adhesion model used, e.g., to model the formation of large-scale structures in the Universe after baryon-photon decoupling. We show that large densities are localized near ``kurtoparabolic singularities residing on space-time manifolds of codimension two ($d le 2$) or higher ($d ge 3$). For smooth initial conditions, such singularities are obtained from the convex hull of the Lagrangian potential (the initial velocity potential minus a parabolic term). The singularities contribute {em hbox{universal} power-law tails} to the density pdf when the initial conditions are random. In one dimension the singularities are preshocks (nascent shocks), whereas in two and three dimensions they persist in time and correspond to boundaries of shocks; in all cases the corresponding density pdf has the exponent -7/2, originally proposed by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for the pdf of velocity gradients in one-dimensional forced Burgers turbulence. We also briefly consider models permitting particle crossings and thus multi-stream solutions, such as the Zeldovich approximation and the (Jeans)--Vlasov--Poisson equation with single-stream initial data: they have singularities of codimension one, yielding power-law tails with exponent -3.
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