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Intracellular protein patterns govern essential cellular functions by dynamically redistributing proteins between membrane-bound and cytosolic states, conserving their total numbers. This review presents a theoretical framework for…

We consider a reaction-diffusion system with discontinuous reaction terms modeled by non-ideal relays. The system is motivated by an epigenetic population model of evolution of two-phenotype bacteria which switch phenotype in response to…

Crystallization of supersaturated liquids usually starts by heterogeneous nucleation. Mounting evidence shows that even homogeneous nucleation in simple liquids takes place in two steps; first a dense amorphous precursor forms, and the…

Formation of spatial patterns of cells is a recurring theme in biology and often depends on regulated cell motility. Motility of M. xanthus depends on two motility machineries: the S-engine and A-engine. Moving M. xanthus cells can organize…

Biological systems excel at building spatial structures on scales ranging from nanometers to kilometers and exhibit temporal patterning from milliseconds to years. One approach that nature has taken to accomplish this relies on the…

We present an analytically solvable model for self-assembly of a molecular complex on a filament. The process is driven by a seed molecule that undergoes facilitated diffusion, which is a search strategy that combines diffusion in…

The different cell types in a living organism acquire their identity through the process of cell differentiation in which the multipotent progenitor cells differentiate into distinct cell types. Experimental evidence and analysis of…

Many essential cellular processes, including cell division and the establishment of cell polarity during embryogenesis, are regulated by pattern-forming proteins. These proteins often need to bind to a substrate, such as the cell membrane,…

The proper functioning of multicellular organisms requires the robust establishment of precise proportions between distinct cell-types. This developmental differentiation process typically involves intracellular regulatory and stochastic…

One of the most interesting phenomena in the soft-matter realm consists in the spontaneous formation of super-molecular structures (microphases) in condition of thermodynamic equilibrium. A simple mechanism responsible for this…

We numerically examine a binary system of particles with repulsive interactions, where one species is driven by a rotating drive and the other is subjected either to a constant drive in a fixed direction or to a rotating drive that is out…

Growth patterns generated by filamentous organisms (e.g. actinomycetes and fungi) involve spatial and temporal dynamics at different length scales. Several mathematical models have been proposed in the last thirty years to address these…

Bacteria possess diverse mechanisms to regulate their motility in response to environmental and physiological signals, enabling them to navigate complex habitats and adapt their behavior. Among these mechanisms, interspecies recognition…

Vegetation in semi-arid environments self-organizes into striking spatial patterns -- bands, spots, labyrinths, and gaps -- with characteristic wavelengths on the order of tens to hundreds of meters. Existing reaction-diffusion models…

We demonstrate that active rotations in chemically signalling particles, such as autochemotactic {\it E. coli} close to walls, create a route for pattern formation based on a nonlinear yet deterministic instability mechanism. For slow…

We study diffusion-driven pattern-formation in networks of networks, a class of multilayer systems, where different layers have the same topology, but different internal dynamics. Agents are assumed to disperse within a layer by undergoing…

Coupling diffusion process of signaling molecules with nonlinear interactions of intracellular processes and cellular growth/transformation leads to a system of reaction-diffusion equations coupled with ordinary differential equations…

Living cells exhibit a complex organization comprising numerous compartments, among which are RNA- and protein-rich membraneless, liquid-like organelles known as biomolecular condensates. Energy-consuming processes regulate their formation…

Active stresses in biological cells and tissues drive many developmental processes. However, increasing experimental evidence suggests that additional mechanical interactions with surrounding material can play a crucial role in guiding…

Mechanisms of pattern formation---of which the Turing instability is an archetype---constitute an important class of dynamical processes occurring in biological, ecological and chemical systems. Recently, it has been shown that the Turing…