Survival Probabilities at Spherical Frontiers

Motivated by tumor growth and spatial population genetics, we study the interplay between evolutionary and spatial dynamics at the surfaces of three-dimensional, spherical range expansions. We consider range expansion radii that grow with an arbitrary power-law in time: $R(t)=R_0(1+t/t^*)^{\Theta}$, where $\Theta$ is a growth exponent, $R_0$ is the initial radius, and $t^*$ is a characteristic time for the growth, to be affected by the inflating geometry. We vary the parameters $t^*$ and $\Theta$ to capture a variety of possible growth regimes. Guided by recent results for two-dimensional inflating range expansions, we identify key dimensionless parameters that describe the survival probability of a mutant cell with a small selective advantage arising at the population frontier. Using analytical techniques, we calculate this probability for arbitrary $\Theta$. We compare our results to simulations of linearly inflating expansions ($\Theta=1$ spherical Fisher-Kolmogorov-Petrovsky-Piscunov waves) and treadmilling populations ($\Theta=0$, with cells in the interior removed by apoptosis or a similar process). We find that mutations at linearly inflating fronts have survival probabilities enhanced by factors of 100 or more relative to mutations at treadmilling population frontiers. We also discuss the special properties of "marginally inflating" $(\Theta=1/2)$ expansions.Comment: 35 pages, 11 figures, revised versio

Energy flux near the junction of two Ising chains at different temperatures

We consider a system in a non-equilibrium steady state by joining two semi-infinite Ising chains coupled to thermal reservoirs with {\em different} temperatures, $T$ and $T^{\prime}$. To compute the energy flux from the hot bath through our system into the cold bath, we exploit Glauber heat-bath dynamics to derive an exact equation for the two-spin correlations, which we solve for $T^{\prime}=\infty$ and arbitrary $T$. We find that, in the $T'=\infty$ sector, the in-flux occurs only at the first spin. In the $T<\infty$ sector (sites $x=1,2,...$), the out-flux shows a non-trivial profile: $F(x)$. Far from the junction of the two chains, $F(x)$ decays as $e^{-x/\xi}$, where $\xi$ is twice the correlation length of the {\em equilibrium} Ising chain. As $T\rightarrow 0$, this decay crosses over to a power law ($x^{-3}$) and resembles a "critical" system. Simulations affirm our analytic results.Comment: 6 pages, 4 figures, submitted to EP