---

Active phase separation and protein aggregation

• Diffusion in singularly perturbed domains is a powerful analytical method for solving two-dimensional (2D) and three-dimensional (3D) diffusion problems in domains containing small holes or traps within the domain interior.
• The analysis proceeds by partitioning the domain into a set of inner regions around each hole, and an outer region where mean-field interactions occur. Asymptotically matching the inner and outer stationary solutions generates an asymptotic expansion of the particle concentration that depends on the positions of the holes and the geometry of the domain.
• The goal of this research topic is to apply the analysis of diffusion in singularly perturbed domains to two forms of self-organization in biological cells: active phase separation and protein aggregation.
• Active liquid-liquid phase separation is thought to be the mechanism underlying the formation of biological condensates, which are protein rich membrane-less structures in the cytoplasm and nucleus of cells. In this case the domain holes correspond to droplets in the high density phase.
• Protein aggregation underlies the formation of large-scale molecular assemblies such as lipid rafts and protein-rich postsynaptic domains within the cell membranes. In this case the domain holes represent protein clusters.

---

Encounter-based models of diffusion in partially absorbing domains

• Encounter-based methods provide a general probabilistic framework for modelling diffusion-mediated absorption phenomena on the surface or interior of some domain. An absorption event occurs when the particle contact time with the absorption domain exceeds a random threshold.
• If the probability distribution of the latter is an exponential function, then one obtains absorption at a constant rate, whereas a non-exponential distribution signifies a time-dependent absorption process with memory.
• In the case of a partially absorbing interior (surface), the contact time is given by a Brownian functional known as the occupation time (boundary local time).
• The goal of this research topic is to apply encounter-based methods to a wide range of diffusive and active transport processes, including search processes with stochastic resetting, anomalous diffusion, active run-and-tumble particles, and interacting particle systems.

---

Stochastic models of diffusion through semi-permeable membranes

• Diffusion through semipermeable interfaces has a wide range of applications, including molecular transport through biological membranes, reverse osmosis, synaptic receptor trafficking, and drug delivery.
• Snapping out Brownian motion (BM) is a probabilistic model of interfacial diffusion, which sews together successive rounds of partially reflecting BMs that are restricted to either side (U or V) of a semipermeable interface S.
• Each round is killed (absorbed) at the interface when its Brownian local time exceeds a random threshold. A new round is then immediately started in U with probability p or V with probability 1-p. If p ≠ 1/2 (directed transport), then there is a discontinuity in the chemical potential across the interface.
• The probability density for snapping out BM satisfies a renewal equation that relates the full density to the probability densities of partially reflected BM on either side of the interface. In the case of an exponentially distributed local time threshold, the solution of the renewal equation satisfies the classical diffusion equation for a semi-permeable membrane with constant permeability.
• On the other hand, if the threshold distribution is non-exponential, then the resulting permeability is a time-dependent function that tends to be heavy-tailed.
• The goal of this research topic is to develop the theory and applications of snapping out BM. Current projects include the following: (a) Analyzing the stochastic thermodynamics of diffusion across a semipermeable interface; (b) Connecting the effective permeability with biophysical models of semipermeable membranes. (c) Comparing semipermeable interfaces with actively switching interfaces, and the boundaries of biological condensates; (d) Investigating the screening effects of semipermeable interfaces on stochastic resetting.

---

Cytoneme-based morphogenesis and viral spread

• Morphogen protein gradients play an essential role in the spatial regulation of patterning during embryonic development. The most commonly accepted mechanism of protein gradient formation involves the diffusion and degradation of morphogens from a localized source.
• Recently, an alternative mechanism has been proposed, which is based on cell-to-cell transport via thin, actin-rich cellular extensions known as cytonemes. It has been hypothesized that cytonemes find their targets via a random search process based on alternating periods of retraction and growth, perhaps mediated by some chemoattractant.
• There is also experimental evidence showing that cytonemes can be hijacked by viral particles, thus providing a longer range substrate for cell-to-cell viral transmission.
• We have developed a search-and-capture model of cytoneme-based morphogenesis, in which nucleating cytonemes from a source cell dynamically grow and shrink until making contact with a target cell and delivering a burst of morphogen. Renewal theory can be used to calculate the splitting probabilities and conditional mean first passage times (MFPTs) for the cytoneme to be captured by a given target cell.
• It can also be shown that multiple rounds of search-and-capture, morphogen delivery, cytoneme retraction and nucleation events lead to the formation of a morphogen gradient. This can be achieved by formulating the morphogen bursting model as a G/M/infinity queuing process.
• The goal of this research topic is the construction and analysis of mathematical models of cytoneme-based transport processes. Current projects include the following: (a) Modeling the accumulation of target cell morphogens as a G/M/1 queue. (b) Incorporating the effects of cytoneme bending in search processes; (c) Developing a model of cytoneme-mediated within-host viral spread.

---

Probabilistic formulations of stochastic resetting

• Stochastic resetting is a mechanism whereby a system is returned to its initial state at a random sequence of times that is typically generated by a Poisson process with constant rate r.
• Within the specific application domains of animal movement and cell biology, stochastic resetting has been posited as a mechanism for reducing the expected time to find a hidden target within some large search domain.
• In the case of a foraging animal, the target represents a local region of resources such as food or shelter, and resetting mimics the observed tendency for an animal to return to its home base in order to rest or resupply. The stochastic process cycles between a diffusive search phase, a ballistic return phase, and a refractory phase (non-instantaneous resetting)
• Two examples of search processes with non-instantaneous resetting at the cellular level are motor-based vesicular transport and cytoneme-based morphogen transport.
• One of the limitations of a purely diffusive process as a stochastic search mechanism is that the mean first passage time (MFPT) to find a target diverges as the size of the search domain goes to infinity. The introduction of a stochastic resetting protocol can support a finite MFPT that has a unique minimum as a function of the resetting rate.
• An alternative physical realisation of non-instantaneous resetting is to consider diffusion in an external trapping or confining potential that is intermittently switched on and off. During an ON phase, a diffusing particle tends to move towards the minimum of the potential, which thus plays an analogous role to the reset position in perfect resetting. This return phase is clearly of finite duration. Moreover, once the particle reaches a neighbourhood of the minimum, it tends to remain there until switching to an OFF state, which is analogous to a refractory phase.
• The goal of this research topic is to develop probabilistic formulations of instantaneous stochastic resetting and diffusion in an intermittent potential. These can be represented by a jump-diffusion process and a switching diffusion (hybrid stochastic) process, respectively. Possible applications include stochastic thermodynamics with resetting, interacting particle systems with resetting, and stochastic resetting in partially absorbing media.

---

Stochastically switching (hybrid) systems

• There are many processes in cell biology that can be modelled in terms of an actively switching particle. The continuous degrees of freedom of the particle evolve according to a hybrid stochastic differential equation (hSDE) whose drift term depends on a discrete internal or environmental state N(t) that switches according to a continuous time Markov chain.
• Examples include Brownian motion in a randomly switching environment, membrane voltage fluctuations in neurons, protein synthesis in gene networks, bacterial run-and-tumble motion, and motor-driven intracellular transport.
• Mathematically speaking, at the single particle level, the analysis of an hSDE holds irrespective of whether N(t) is interpreted as a discrete internal state (intrinsic switching) or an external environmental state (extrinsic switching). However, for a population of actively switching particles, the two scenarios differ significantly, even in the absence of particle-particle interactions. In particular, a common randomly switching environment induces statistical correlations between the particles.
• The goal of this research project is to analyze populations of particles with intrinsic or environmental switching using a combination of mean field theory and path integral methods