Fluctuation-Regularized Front Propagation Dynamics in Reaction-Diffusion Systems

Cohen, Elisheva; Kessler, David A.; Levine, Herbert

doi:10.1103/physrevlett.94.158302

Cited by 17 publications

(9 citation statements)

References 17 publications

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“…The idea is that the mean-field continuous equation (29) fails to describe the behavior of individual cells due to their strong fluctuations at the tip of the front [26]. Therefore, we derive the cut-off continuous approach which describes the system up to a threshold density δ of the order of magnitude of one cell, i.e.…”

Section: Cut-off Mean-field Approximationmentioning

confidence: 99%

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Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion

Hatzikirou

Brusch

Schaller

et al. 2010

Computers & Mathematics with Applications

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a b s t r a c tCancer invasion is the process of cells detaching from a primary tumor and infiltrating the healthy tissue. Cancer invasion has been recognized as a complex system, since a tumor's invasive behavior emerges from the combined effect of tumor cell proliferation, tumor cell migration and cell-microenvironment interactions. Cellular automata (CA) provide simple models of self-organizing complex systems in which collective behavior can emerge out of an ensemble of many interacting ''simple'' components. Here, we introduce a latticegas cellular automaton (LGCA) model of tumor cell proliferation, necrosis and tumor cell migration. The impact of the tumor environment on tumor cells has been investigated in a previous study. Our analysis aims at predicting the velocity of the traveling invasion front, which depends upon fluctuations that arise from the motion of the discrete cells at the front. We find an excellent agreement between the velocities measured in simulations of the LGCA and an analytical estimate derived in the cut-off mean-field approximation via the discrete Lattice Boltzmann equation and its linearization. In particular, we predict the front velocity to scale with the square root of the product of probabilities for mitosis and the migration coefficient. Finally, we calculate the width of the traveling front which is found to be proportional to the front velocity.

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Section: Cut-off Mean-field Approximationmentioning

confidence: 99%

“…In order to improve the mean-field approximation (here we characterize it as ''naive''), we introduce the cut-off meanfield approach [28,29]. The idea is that the mean-field continuous equation (29) fails to describe the behavior of individual cells due to their strong fluctuations at the tip of the front [26].…”

Section: Cut-off Mean-field Approximationmentioning

confidence: 99%

Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion

Hatzikirou

Brusch

Schaller

et al. 2010

Computers & Mathematics with Applications

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“…Noise serves a crucial function in these models as it is required to regularize the wave dynamics. These waves have therefore been called "front-regularized waves" (Cohen et al 2005a). …”

Section: Fluctuations In Dynamics Of Adaptation 1205mentioning

confidence: 99%

Collective Fluctuations in the Dynamics of Adaptation and Other Traveling Waves

Hallatschek

Geyrhofer

2016

Genetics

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The dynamics of adaptation are difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from random genetic drift arising in a vanguard of particularly fit individuals of the population. Several approaches have been developed to analyze the crucial role of genetic drift on the expected dynamics of adaptation, including the mean fitness of the entire population, or the fate of newly arising beneficial deleterious mutations. However, little is known about how genetic drift causes fluctuations to emerge on the population level, where it becomes palpable as variations in the adaptation speed and the fitness distribution. Yet these phenomena control the decay of genetic diversity and variability in evolution experiments and are key to a truly predictive understanding of evolutionary processes. Here, we show that correlations induced by these emergent fluctuations can be computed at any arbitrary order by a suitable choice of a dynamical constraint. The resulting linear equations exhibit fluctuationinduced terms that amplify short-distance correlations and suppress long-distance ones. These terms, which are in general not small, control the decay of genetic diversity and, for wave-tip dominated ("pulled") waves, lead to anticorrelations between the tip of the wave and the lagging bulk of the population. While it is natural to consider the process of adaptation as a branching random walk in fitness space subject to a constraint (due to finite resources), we show that other traveling wave phenomena in ecology and evolution likewise fall into this class of constrained branching random walks. Our methods, therefore, provide a systematic approach toward analyzing fluctuations in a wide range of population biological processes, such as adaptation, genetic meltdown, species invasions, or epidemics.KEYWORDS traveling-wave models; asexual adaptation; Fisher waves; tuned models; stochastic modeling; higher-order correlation functions; exact approaches M ANY important evolutionary and ecological processes rely on the behavior of a small number of individuals that have a large dynamical influence on the population as a whole. This is, perhaps, most obvious in the case of biological adaptation in asexual species, such as microbes (over timescales where horizontal gene transfer is irrelevant). Future generations descend from a small number of currently welladapted individuals. The genetic footprint of the large majority of the population is wiped out over time by the fixation of more fit genotypes. For a large influx of beneficial mutations, these dynamics can be visualized as a traveling wave in fitness space; see Figure 1A. At any time, the currently most fit "pioneer" individuals reside in the small tip of the wave. As time elapses, the wave moves toward higher fitness and the formerly rare most fit individuals dominate the population. By that time, however, a new wave tip of even more fit mutants has formed by mutations and the cycle of transient dominance continues. Evolu...

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“…This value is of crucial importance, as it represents the amount of recombination needed for a finite population to achieve the maximal rate of evolution. Studying this requires inclusion of finite population effects in the evolution equation, for which we employ a heuristic cutoff approach which has been shown to be accurate in a variety of previous investigations [16,17]. In detail, we replace the first part of the mean-field equation (MFE) with the alternate form…”

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confidence: 99%

Recombination Dramatically Speeds Up Evolution of Finite Populations

2005

Self Cite

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We study the role of recombination, in the form of bacterial transformation, in speeding up Darwinian evolution. This is done by adding a new process to a previously studied Markov model of evolution on a smooth fitness landscape; this new process allows alleles to be exchanged with those in the surrounding medium. Our results, both numerical and analytic, indicate that, for a wide range of intermediate population sizes, recombination dramatically speeds up the rate of evolutionary advance.

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Fluctuation-Regularized Front Propagation Dynamics in Reaction-Diffusion Systems

Cited by 17 publications

References 17 publications

Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion

Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion

Collective Fluctuations in the Dynamics of Adaptation and Other Traveling Waves

Recombination Dramatically Speeds Up Evolution of Finite Populations

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