In applications spaning from image analysis and speech recognition, to energy dissipation in turbulence and time-to failure of fatigued materials, researchers and engineers want to calculate how often a stochastic observable crosses a specific level,
such as zero. At first glance this problem looks simple, but it is in fact theoretically very challenging. And therefore, few exact results exist. One exception is the celebrated Rice formula that gives the mean number of zero-crossings in a fixed time interval of a zero-mean Gaussian stationary processes. In this study we use the so-called Independent Interval Approximation to go beyond Rices result and derive analytic expressions for all higher-order zero-crossing cumulants and moments. Our results agrees well with simulations for the non-Markovian autoregressive model.
30% of the DNA in E. coli bacteria is covered by proteins. Such high degree of crowding affect the dynamics of generic biological processes (e.g. gene regulation, DNA repair, protein diffusion etc.) in ways that are not yet fully understood. In this
paper, we theoretically address the diffusion constant of a tracer particle in a one dimensional system surrounded by impenetrable crowder particles. While the tracer particle always stays on the lattice, crowder particles may unbind to a surrounding bulk and rebind at another or the same location. In this scenario we determine how the long time diffusion constant ${cal D}$ (after many unbinding events) depends on (i) the unbinding rate of crowder particles $k_{rm off}$, and (ii) crowder particle line density $rho$, from simulations (Gillespie algorithm) and analytical calculations. For small $k_{rm off}$, we find ${cal D}sim k_{rm off}/rho^2$ when crowder particles are immobile on the line, and ${cal D}sim sqrt{D k_{rm off}}/rho$ when they are diffusing; $D$ is the free particle diffusion constant. For large $k_{rm off}$, we find agreement with mean-field results which do not depend on $k_{rm off}$. From literature values of $k_{rm off}$ and $D$, we show that the small $k_{rm off}$-limit is relevant for in vivo protein diffusion on a crowded DNA. Our results applies to single-molecule tracking experiments.
