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Modeling the lamellipodium of motile cells

Many cells move over surfaces during embryonic development, immune response, wound healing or cancer metastasis by protruding flat lamellipodia into the direction of migration. Although the phenomenon of cell motility has been fascinating scientists for many years, we are only beginning to gain a quantitative understanding. Many issues remain unresolved. Open questions are for example: Why and how do lamellipodia initially form? Why do some cells exhibit stable lamellipodia while others show protruding and retracting lamellipodia or form a lamellipodium that subsequently collapses? Which changes occur inside the lamellipodium during phases of protrusion and retraction? Is the activation of signaling molecules necessary? The force-velocity relation of keratocytes has been measured experimentally. What is the reason for its unexpected concave shape? Why do moving cells first react very sensitively to tiny forces but can then still move at forces that are orders of magnitude larger? How large are the forces that cells can exert on surrounding cells in tissue and what determines the strength of lamellipodia? What are the mechanical properties of the lamellipodium and which structures bring them about? In this thesis, a mathematical model is developed that contributes substantially to answering those questions. It describes the formation of lamellipodia, their stability, strength and leading edge dynamics. Two regions of the lamellipodium are distinguished in the model. The bulk of the lamellipodium contains a dense cross-linked actin network called actin gel. The newly polymerized tips of the actin filaments at the leading edge constitute a highly dynamic boundary layer called semiflexible region (SR). The forces that single treadmilling actin filaments in the SR exert on the leading edge membrane are calculated, and the balance of forces with viscous and external forces determines the velocity of the leading edge. Specifically, the previously published filament model is supplemented by including retrograde flow in the actin gel, and a feasible description of a variable filament density in the SR due to nucleation of new and capping and severing of existing filaments. With several simplifications, the model is represented by a system of ordinary differential equations for a one dimensional cross-section through the thin lamellipodium. A stability analysis of the model identifies three qualitatively different parameter regions: with stationary filament density and leading edge motion, oscillatory motion, or a filament density of zero and no leading edge motion. That defines criteria for the existence of stable lamellipodia and allows for describing different cell types. Zero filament density means no stable lamellipodium can exist. However, due to excitability, lamellipodia can still form transiently. The measured subsequent protrusions and retractions of lamellipodia in epithelial cells are very well reproduced by assuming random nucleation of single short filaments from the actin cortex in the excitable regime. The modeling results support ideas on lamellipodium formation, the formation of actin arcs in the lamellum, and show that in principle no signaling is necessary for cycles of protrusion and retraction. Furthermore, the model results are fitted to the force-velocity relation of keratocytes, which has been measured by placing a flexible scanning force microscopy (SFM)-cantilever into the cell’s path of migration. Due to the good agreement between experiment and simulations, a mechanism leading to the characteristic features of the force-velocity relation is suggested. Moreover, properties of the structure of the stable keratocyte lamellipodium, like the length of actin filaments at