Mathematical modeling of eukaryotic cell migration: insights beyond experiments

Abstract

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

Figure 1

Revisiting the Abercrombie model of metazoan cell crawling. Cell migration is divided into discrete steps: (a) protrusion based on actin growth and polymerization force; (b) formation of new adhesions at the front; (c) release and recycling of adhesions at the rear; and finally, (d) actin-myosin-powered contraction of the cytoplasm, resulting in forward translocation of the cell body. We are showing schematically the centrosome and microtubules originating from it, as well as the Golgi complex and Golgi-derived microtubules that play important roles in guiding migration.

Figure 2

Schematics of early models that critically advanced the understanding of mechanisms of cell migration through mathematical modeling. (a) Ratchet model of force generation by actin polymerization (Hill 1981, Peskin et al. 1993). (b) Modeling of F-actin branching, mediated by Arp2/3 complex (black chevrons), capping (black semicircles), disassembly, and G-actin diffusion down its gradient (arrows), added to the polymerization ratchet model, put the dendritic-nucleation model of lamellipodium into a quantitative framework (Carlsson 2001, Mogilner & Edelstein-Keshet 2002). (c) Whole-cell models using discrete mechanical components comprised of springs, dashpots, and actuators (motors) (DiMilla et al. 1991) preempted current whole-cell models. (d) Whole-cell continuum models, such as the two-phase fluid model (Herant & Dembo 2010), integrate the contractile stress, F-actin, and cytosol flow with traction forces and cytoskeletal densities. (e) Early models of chemotaxis and long-term cell migration (Tranquillo et al. 1988).

Figure 3

Ingredients of mathematical models. (a) Physical principles. (Left) At cellular scales, inertia is insignificant, and imbalances of forces quickly dissipate, leading to the force balance principle: The total force on any object or any subcellular region (e.g., dashed box) sums to zero. (Right) Mass conservation. In any region of space (e.g., dashed box), the total amount of a substance can dynamically change only if it is transported out, for example, by diffusion, advection, or active transport, or changes form, for example, by polymerizing or depolymerizing oligomers. (b) The same process or quantity appears differently at different scales. (Top) The adhesion pattern in cells varies spatially. On the scale of the whole cell, adhesion is strong at the front and weak at the rear. At smaller scales, fine structure of individual focal adhesion sites can be resolved. (Bottom) Many cells exhibit cycles of protrusion and retraction at small timescales; however, at larger timescales, the result is smooth net migration forward. (c) Dynamical systems behaviors, including steady states, excitability, and oscillations. Green arrows indicate pulsatile activation of the systems.

Figure 4

Examples of mathematical models and assumptions that the models are built on. (a) Stick-slip adhesion (Chan & Odde 2008). (b) Cell shape, movement, and polarity governed by signaling reaction–diffusion system (Marée et al. 2006).

Figure 5

Mathematical models as data integrators. One role of mathematical modeling is that it allows data on a directly observable variable, such as the F-actin flow field (left), to inform another quantity of interest and ultimately yield biological mechanisms. (a) Wilson et al. (2010) developed a simple mathematical model based on mass conservation. When combined with the cell-scale F-actin flow field, the model predicted a distribution of net F-actin assembly and disassembly that revealed a role for myosin in F-actin disassembly. (b) Ji et al. (2008) developed a model based on force-balance principles, along with other assumptions, such as the short-timescale elasticity of the F-actin network. Micrometer-scale F-actin flow fields then allowed the calculation of intracellular force distributions. Together with further assumptions regarding the orientation of adhesive friction, this permitted elucidation of spatiotemporal distribution forces generated by actomyosin contraction, polymerization at the cell boundary, and adhesions.