James Nolen

Cells excel at interpreting noisy chemical gradients during fertilization, development, and immune responses, but the mechanisms underlying this remarkable ability remain poorly understood. Classical models for gradient sensing assume that information is derived solely from ligand-bound receptors, but cells can also utilize information from unbound receptors. Here, we first derive the physical limit for instantaneous gradient detection by a distribution of receptors on a sphere. Surprisingly, we find that a ratiometric biochemical strategy, which compares spatial distributions of active (bound) and inactive (unbound) receptors, can surpass this physical limit. Using theory and simulations, we show this enhancement stems from a local collective memory encoded by downstream proteins that average receptor states sampled over time. Such memory-driven averaging allows cells to reliably interpret gradient direction despite receptor scarcity or noisy conditions, demonstrating how ratiometric sensing mechanisms could broadly enhance cellular responses to directional cues from the environment. Significance Statement Cells detect chemical gradients using receptors on their surface to guide essential processes like fertilization, immune responses, and organismal development. However, chemical environments are often noisy and complex, and it remains unclear how cells can accurately interpret these signals. We show that by using information from inactive as well as active receptors, a cell can average each receptor’s state over time and store that information in a population of downstream signaling proteins near the receptor. This strategy improves gradient sensing accuracy even when receptors are scarce or gradients are noisy, offering a path to improved engineering of microscale biosensors or drug-delivery devices that can orient in complex chemical landscapes.

We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is O ( 1 / ε 2 ) O(1/\varepsilon ^2) , where ε \varepsilon is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map φ \varphi the mixing time becomes O ( | ln ⁡ ε | ) O(\lvert \ln \varepsilon \rvert ) . We also study the dissipation time of this process, and obtain O ( | ln ⁡ ε | ) O(\lvert \ln \varepsilon \rvert ) upper and lower bounds with explicit constants.

We study the asymptotic behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the $k$-th moment dominates the $k$-th power of the first moment for some $k$), while, at distances that grow sub-linearly in time, we show that all the moments converge. A key ingredient in our analysis is a sharp estimate of the transition kernel for the branching process, valid up to linear in time distances from the location of the initial particle.

Accurate detection of extracellular chemical gradients is essential for many cellular behaviors. Gradient sensing is challenging for small cells, which can experience little difference in ligand concentrations on the up-gradient and down-gradient sides of the cell. Nevertheless, the tiny cells of the yeast Saccharomyces cerevisiae reliably decode gradients of extracellular pheromones to find their mates. By imaging the behavior of polarity factors and pheromone receptors, we quantified the accuracy of initial polarization during mating encounters. We found that cells bias the orientation of initial polarity up-gradient, even though they have unevenly distributed receptors. Uneven receptor density means that the gradient of ligand-bound receptors does not accurately reflect the external pheromone gradient. Nevertheless, yeast cells appear to avoid being misled by responding to the fraction of occupied receptors rather than simply the concentration of ligand-bound receptors. Such ratiometric sensing also serves to amplify the gradient of active G protein. However, this process is quite error-prone, and initial errors are corrected during a subsequent indecisive phase in which polarity clusters exhibit erratic mobile behavior.

In this paper we study the asymptotic behavior of Brownian motion in both comb-shaped planar domains, and comb-shaped graphs. We show convergence to a limiting process when both the spacing between the teeth \emph{and} the width of the teeth vanish at the same rate. The limiting process exhibits an anomalous diffusive behavior and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. In the two dimensional setting the main technical step is an oscillation estimate for a Neumann problem, which we prove here using a probabilistic argument. In the one dimensional setting we provide both a direct SDE proof, and a proof using the trapped Brownian motion framework in Ben Arous \etal (Ann.\ Probab.\ '15).