The complexity of biological systems creates challenges for fully understanding their behaviour. This is particularly true for cell migration which requires the co-ordinated activity of hundreds of individual components within cells. Mathematical modelling can help understand these complex systems by breaking the system into discrete steps which can then be interrogated in silico . In this review, we highlight scenarios in cell migration where mathematical modelling can be applied and discuss what types of modelling are most suited. Almost any aspect of cell migration is amenable to mathematical modelling from the modelling of intracellular processes such as chemokine receptor signalling and actin filament branching to larger scale processes such as the movement of individual cells or populations of cells through their environment. Two common ways of approaching this modelling are the use of models based on differential equations or agent-based modelling. The application of both these approaches to cell migration are discussed with specific examples along with common software tools to facilitate the process for non-mathematicians. We also highlight the challenges of modelling cell migration and the need for rigorous experimental work to effectively parameterise a model.

Migration of epithelial cells is fundamental to multiple developmental processes, epithelial tissue morphogenesis and maintenance, wound healing and metastasis. While migrating epithelial cells utilize the basic acto-myosin based machinery as do other non-epithelial cells, they are distinguished by their copious keratin intermediate filament (KF) cytoskeleton, which comprises differentially expressed members of two large multigene families and presents highly complex patterns of post-translational modification. We will discuss how the unique mechanophysical and biochemical properties conferred by the different keratin isotypes and their modifications serve as finely tunable modulators of epithelial cell migration. We will furthermore argue that KFs together with their associated desmosomal cell–cell junctions and hemidesmosomal cell–extracellular matrix (ECM) adhesions serve as important counterbalances to the contractile acto-myosin apparatus either allowing and optimizing directed cell migration or preventing it. The differential keratin expression in leaders and followers of collectively migrating epithelial cell sheets provides a compelling example of isotype-specific keratin functions. Taken together, we conclude that the expression levels and specific combination of keratins impinge on cell migration by conferring biomechanical properties on any given epithelial cell affecting cytoplasmic viscoelasticity and adhesion to neighboring cells and the ECM.

The ability of cells to migrate is a fundamental physiological process involved in embryonic development, tissue homeostasis, immune surveillance and wound healing. In order for cells to migrate, they must interact with their environment using adhesion receptors, such as integrins, and form specialized adhesion complexes that mediate responses to different extracellular cues. In this review, we discuss the role of integrin adhesion complexes (IACs) in cell migration, highlighting the layers of regulation that are involved, including intracellular signalling cascades, mechanosensing and reciprocal feedback to the extracellular environment. We also discuss the role of IACs in extracellular matrix remodeling and how they impact upon cell migration.

Directed cell migration is critical for embryogenesis and organ development, wound healing and the immune response. Microtubules are dynamic polymers that control directional migration through a number of coordinated processes: microtubules are the tracks for long-distance intracellular transport, crucial for delivery of new membrane components and signalling molecules to the leading edge of a migrating cell and the recycling of adhesion receptors. Microtubules act as force generators and compressive elements to support sustained cell protrusions. The assembly and disassembly of microtubules is coupled to Rho GTPase signalling, thereby controlling actin polymerisation, myosin-driven contractility and the turnover of cellular adhesions locally. Cross-talk of actin and microtubule dynamics is mediated through a number of common binding proteins and regulators. Furthermore, cortical microtubule capture sites are physically linked to focal adhesions, facilitating the delivery of secretory vesicles and efficient cross-talk. Here we summarise the diverse functions of microtubules during cell migration, aiming to show how they contribute to the spatially and temporally coordinated sequence of events that permit efficient, directional and persistent migration.

Migration of cells is required in multiple tissue-level processes, such as in inflammation or cancer metastasis. Endocytosis is an extremely regulated cellular process by which cells uptake extracellular molecules or internalise cell surface receptors. While the role of endocytosis of focal adhesions (FA) and plasma membrane (PM) turnover at the leading edge of migratory cells is wide known, the contribution of endocytic proteins per se in migration has been frequently disregarded. In this review, we describe the novel functions of the most well-known endocytic proteins in cancer cell migration, focusing on clathrin, caveolin, flotillins and GRAF1. In addition, we highlight the relevance of the macropinocytic pathway in amoeboid-like cell migration.

Cell migration is a fundamental biological process involved in tissue formation and homeostasis. The correct polarization of motile cells is critical to ensure directed movement, and is orchestrated by many intrinsic and extrinsic factors. Of these, the subcellular distribution of mRNAs and the consequent spatial control of translation are key modulators of cell polarity. mRNA transport is dependent on cis -regulatory elements within transcripts, which are recognized by trans -acting proteins that ensure the efficient delivery of certain messages to the leading edge of migrating cells. At their destination, translation of localized mRNAs then participates in regional cellular responses underlying cell motility. In this review, we summarize the key findings that established mRNA targetting as a critical driver of cell migration and how the characterization of polarized mRNAs in motile cells has been expanded from just a few species to hundreds of transcripts. We also describe the molecular control of mRNA trafficking, subsequent mechanisms of local protein synthesis and how these ultimately regulate cell polarity during migration.