Diffusive queues with finite resources as a model for cell signalling

Cellular responses to environmental stimuli are often initiated by ligand-receptor binding. Our analysis of live-cell immune signaling data reveals that such ligand-binding events are processed by the cell in a manner similar to customers and servers within queueing theory. Classical queuing theory assumes instantaneous service once a server is available, but in biochemical networks agents must diffuse within range of one another before interactions can occur. Based on this, we developed a "diffusive queuing model" in which K active "server" molecules diffuse with coefficient D and interact with "customers" (ligand-bound receptors) only upon close approach. This makes the service rate depend on the diffusive coefficient D, the "service time" that a customer occupies a server for, and the time-evolving spatial distribution of agents. We identify two regimes governed by D: (i) a diffusion-limited regime at low D, where spatial segregation produces long-tailed waiting-time distributions and dynamical heterogeneity, and (ii) a well-mixed regime at high D, where waiting times show divergences characteristic of queuing theory. Our work reveals how diffusivity and shared-resource depletion can control information processing in biochemical signaling. We present analytical results corroborated with stochastic simulations, and suggest implications for signalling architectures in cells.