Uniqueness and traveling waves in a cell motility model

Abstract

We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp interface limit equation in the so-called subcritical parameter regime. The proof relies on a Grönwall estimate for a specially chosen weighted L 2 norm.Next, as persistent motion of crawling cells is of central interest to biologists we study the existence of traveling wave solutions. We prove that traveling wave solutions exist in the supercritical parameter regime provided the non-linear term of the sharp interface limit equation possesses certain asymmetry (related, e.g., to myosin contractility).Finally, we numerically investigate traveling wave solutions and simulate their dynamics. Due to non-uniqueness of solutions of the sharp interface limit equation we simulate a related, singularly perturbed PDE system which is uniquely solvable. Our simulations predict instability of traveling wave solutions and capture both bipedal wandering cell motion as well as rotating cell motion; these behaviors qualitatively agree with recent experimental and theoretical findings.2010 Mathematics Subject Classification. 35Q92, 35K55, 65-04. MSM:

Self polarization and traveling wave in a model for cell crawling migration

“…However, there is an important difference: tumor growth naturally involves expanding domain while we consider here incompressible solutions. In the context of the motility of eukaryotic cells on substrates, various free boundary problems have been derived and studied, see [2,3,1,15,4,16]. The models presented in [2,3,1,15,16] present some similarities with our model, but they are obtained as the limit of a second order equation of Allen-Cahn type while we obtain ours as the limit of a fourth order equation of Cahn-Hilliard type.…”

“…In the context of the motility of eukaryotic cells on substrates, various free boundary problems have been derived and studied, see [2,3,1,15,4,16]. The models presented in [2,3,1,15,16] present some similarities with our model, but they are obtained as the limit of a second order equation of Allen-Cahn type while we obtain ours as the limit of a fourth order equation of Cahn-Hilliard type. The model recently proposed in [4] involves a coupled Hele-Shaw/ Keller-Segel type free boundary problem.…”

Minimal model of directed cell motility on patterned substrates

“…In particular, in recent years both free boundary and phase-field models have been extremely successful in replicating, explaining, and predicting experimental results (see, e.g., [13,14,15,16,17,18,19]), see for review [20]. We highlight that recent mathematical analyses have elucidated biological mechanisms [21,22,23,24], and numerical simulations have described a wide range of behaviors ranging from motility initiation via stochastic fluctuations [25] to capturing various modes of motility such as stick-slip and bipedal motions [26,27,28].…”

“…We highlight here that the reduction described above is similar to the one conducted in [28] however we additionally incorporate expansions for the V y component. Moreover the equation for the effective adhesion A 0 is derived using the asymptotic expansion A = A 0 (t)ρ(x, y, t)+δA 1 (x, y, t) which gives rise to the effective coefficients (23).…”

Dynamics of a cell motility model near the sharp interface limit

“…In this work we investigate the minimal phase-field model introduced in [6]. Rigorous analysis of its sharp interface limit dynamics was completed in [15,16], where it was observed that persistent cell motion was not stable. In this work we numerically study the pre-limiting phase-field model near the sharp interface limit, to better understand this lack of persistent motion.…”