A Two-State Random Walk Model of Sperm Search on Confined Domains

Mammalian fertilization depends on sperm successfully navigating a spatially and chemically complex microenvironment in the female reproductive tract. This process is often conceptualized as a competitive race, but is better understood as a collective random search. Sperm within an ejaculate exhibit a diverse distribution of motility patterns, with some moving in relatively straight lines and others following tightly turning trajectories. Here, we present a two-state random walk model in which sperm switch from high-persistence-length to low-persistence-length motility modes. In reproductive biology, such a switch is often recognized as "hyperactivation". We study a circularly symmetric setup with sperm emerging at the center and searching a finite-area disk. We explore the implications of switching on search efficiency. The first proposed model describes an adaptive search strategy in which sperm achieve improved spatial coverage without cell-to-cell or environment-to-cell communication. The second model that we study adds a small amount of environment-to-cell communication. The models resemble macroscopic search-and-rescue tactics, but without organization or networked communication. Our findings provide a quantitative framework linking sperm motility patterns to efficient search strategies, offering insights into sperm physiology and the stochastic search dynamics of self-propelled particles.