104 research outputs found
The noisy edge of traveling waves
Traveling waves are ubiquitous in nature and control the speed of many
important dynamical processes, including chemical reactions, epidemic
outbreaks, and biological evolution. Despite their fundamental role in complex
systems, traveling waves remain elusive because they are often dominated by
rare fluctuations in the wave tip, which have defied any rigorous analysis so
far. Here, we show that by adjusting nonlinear model details, noisy traveling
waves can be solved exactly. The moment equations of these tuned models are
closed and have a simple analytical structure resembling the deterministic
approximation supplemented by a nonlocal cutoff term. The peculiar form of the
cutoff shapes the noisy edge of traveling waves and is critical for the correct
prediction of the wave speed and its fluctuations. Our approach is illustrated
and benchmarked using the example of fitness waves arising in simple models of
microbial evolution, which are highly sensitive to number fluctuations. We
demonstrate explicitly how these models can be tuned to account for finite
population sizes and determine how quickly populations adapt as a function of
population size and mutation rates. More generally, our method is shown to
apply to a broad class of models, in which number fluctuations are generated by
branching processes. Because of this versatility, the method of model tuning
may serve as a promising route toward unraveling universal properties of
complex discrete particle systems.Comment: For supplementary material and published open access article, see
http://www.pnas.org/content/108/5/1783.abstract?sid=693e63f3-fd1a-407a-983e-c521efc6c8c5
See also Commentary Article by D. S. Fisher,
http://www.pnas.org/content/108/7/2633.extrac
Life at the front of an expanding population
Recent microbial experiments suggest that enhanced genetic drift at the
frontier of a two-dimensional range expansion can cause genetic sectoring
patterns with fractal domain boundaries. Here, we propose and analyze a simple
model of asexual biological evolution at expanding frontiers to explain these
neutral patterns and predict the effect of natural selection. Our model
attributes the observed gradual decrease in the number of sectors at the
leading edge to an unbiased random walk of sector boundaries. Natural selection
introduces a deterministic bias in the wandering of domain boundaries that
renders beneficial mutations more likely to escape genetic drift and become
established in a sector. We find that the opening angle of those sectors and
the rate at which they become established depend sensitively on the selective
advantage of the mutants. Deleterious mutations, on the other hand, are not
able to establish a sector permanently. They can, however, temporarily "surf"
on the population front, and thereby reach unusual high frequencies. As a
consequence, expanding frontiers are susceptible to deleterious mutations as
revealed by the high fraction of mutants at mutation-selection balance.
Numerically, we also determine the condition at which the wild type is lost in
favor of deleterious mutants (genetic meltdown) at a growing front. Our
prediction for this error threshold differs qualitatively from existing
well-mixed theories, and sets tight constraints on sustainable mutation rates
for populations that undergo frequent range expansions.Comment: Updat
Noise Driven Evolutionary Waves
Adaptation in spatially extended populations entails the propagation of evolutionary novelties across habitat ranges. Driven by natural selection, beneficial mutations sweep through the population in a “wave of advance”. The standard model for these traveling waves, due to R. Fisher and A. Kolmogorov, plays an important role in many scientific areas besides evolution, such as ecology, epidemiology, chemical kinetics, and recently even in particle physics. Here, we extend the Fisher–Kolmogorov model to account for mutations that confer an increase in the density of the population, for instance as a result of an improved metabolic efficiency. We show that these mutations invade by the action of random genetic drift, even if the mutations are slightly deleterious. The ensuing class of noise-driven waves are characterized by a wave speed that decreases with increasing population sizes, contrary to conventional Fisher–Kolmogorov waves. When a trade-off exists between density and growth rate, an evolutionary optimal population density can be predicted. Our simulations and analytical results show that genetic drift in conjunction with spatial structure promotes the economical use of limited resources. The simplicity of our model, which lacks any complex interactions between individuals, suggests that noise-induced pattern formation may arise in many complex biological systems including evolution
Tension dynamics in semiflexible polymers. II. Scaling solutions and applications
In part I O. Hallatschek , preceding paper, Phys. Rev. E 75, 031905 (2007)] of this contribution, a systematic coarse-grained description of the dynamics of a weakly bending semiflexible polymer was developed. Here, we discuss analytical solutions of the established deterministic partial integro-differential equation for the spatiotemporal relaxation of the backbone tension. For prototypal experimental situations, such as the sudden application or release of a strong external pulling force, it is demonstrated that the tensile dynamics reflects the self-affine conformational fluctuation spectrum in a variety of intermediate asymptotic power laws. Detailed and explicit analytical predictions for the tension propagation and relaxation and corresponding results for common observables, such as the end-to-end distance, are obtained
Coupling of transverse and longitudinal response in stiff polymers
The time-dependent transverse response of stiff polymers, represented as
weakly-bending wormlike chains (WLCs), is well-understood on the linear level,
where transverse degrees of freedom evolve independently from the longitudinal
ones. We show that, beyond a characteristic time scale, the nonlinear coupling
of transverse and longitudinal motion in an inextensible WLC significantly
weakens the polymer response compared to the widely used linear response
predictions. The corresponding feedback mechanism is rationalized by scaling
arguments and quantified by a multiple scale approach that exploits an inherent
separation of transverse and longitudinal correlation length scales. Crossover
scaling laws and exact analytical and numerical solutions for characteristic
response quantities are derived for different experimentally relevant setups.
Our findings are applicable to cytoskeletal filaments as well as DNA under
tension.Comment: 4 pages, 3 figures, 1 table; final versio
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