The crawling motility of many eukaryotic cells is driven by filamentous actin (F-actin), and regulated by a network of signaling proteins and lipids (including small GTPases). The tangle of positive and negative feedback loops gives rise to various experimentally observed dynamic patterns (``actin waves''). Here we consider a recent prototypical model for actin waves in which F-actin exerts negative feedback onto a GTPase. Guided by recent numerical PDE bifurcation analysis in Hughes (2025) and Hughes et al (2026), we explore cell shapes and motility associated with polar, oscillatory, and traveling waves solutions of a mass-conserved partial differential equation (PDE) model. We use Morpheus (cellular Potts) simulations to investigate the implications of such regimes of behavior on the shapes and motion of cells, and on transitions between modes of behavior. The model demonstrates various cell states, including resting (spatially uniform GTPase), polar cells (static ``zones'' of GTPase), and traveling waves along the cell edge. In some parameter regimes, such states can coexist, so that cells can transition from one behavior to another in response to noisy stimuli.
Understanding eukaryotic cell motility continues to be one of the grand challenges of modern cell biology, engendering both experimental and theoretical research. Experimentally, many complex cell motility modes have been observed in a wide variety of cell types [7,37,40,52]. Commonly, cells experience directed cell motion, which is key in many biological processes including cancer metastasis [47] and wound healing [55,58]. Other modes of motility can also occur such as random amoeboid-type motion, and cell turning and ruffling. Among the fundamental questions to be addressed, we ask how cell motility is regulated and what causes cells to change motility modes.
While new mechanisms powering cell motility are continually being discovered, the leading role of filamantous actin (F-actin) is well-recognized in basic directed migration of neutrophils, keratocytes, social amoeba (Dictyostelum discoideum) and other cell types. F-actin is a key player in cell motility because the “barbed ends” of these filaments, at which actin monomers assemble, accumulate near the front edge of a cell and cause the edge to protrude outwards. However, it is not always clear what interactions between F-actin and its regulators are at the core of the complex motility machinery. Our interest here is in accounting for basic cell motility patterns such as directed motion, turning, ruffling, and random motility. We consider a small subset of the central signaling network, consisting of a small GTPase (such as Rho, Rac, or Cdc42) (Figure 1a) and the interaction with F-actin.
GTPases act as molecular switches, with the active form bound to the membrane and the inactive in the cell interior (cytosol). The activation and inactivation of these GTPases are on the scale of seconds/minutes, while any loss or production is on a much longer scale [34]. This justifies the common modeling assumption that the total quantity of a given GTPase is conserved over the timescale of interest for motile cells (minutes). The conversion from inactive to active form is governed by GEFs (guanine nucleotide exchange factors) and the reverse by GAPs (GTPaseactivating proteins). The patterns induced by these Rho-GTPases dictate the distribution of F-actin.
Recent experimental evidence shows that F-actin is not only governed by Rho GTPases but, in certain cells, it can also affect the activity of the GTPase. Feedback between GTPases and F-actin results in a variety of experimentally observable dynamic patterns, commonly denoted “actin waves” [4,6,7,9,10,12,17,19,20,31,39,36,54,59]. For example, F-actin can promote GTPase inactivation by recruiting GAPs [8] (Figure 1b). This negative feedback loop leads to many interesting actin wave structures and forms the basis of the actin waves modeling in this paper. Experimental work shows a rich structure of waves along the cell edge, including oscillatory waves [20], traveling waves along the cell front [7] and waves on the cell membrane [60,38]. Thus, these wave-like dynamics appear in many different settings, although the proposed mechanisms governing these patterns vary. A key motivation of our work is to explore how these actin waves affect cell motility.
Models describing actin waves tend to be complex, often relying on numerical simulations alone to understand model behavior. Such studies favor model details over unraveling generic model structure. It can be challenging to fully understand the results. Using simulations alone can also miss important parameter regimes where there is coexistence of different behaviors or transitions from one behavior to another. In this work, we leverage a previous study that analyzed a simple reaction-diffusion (RD) model for actin waves [27]. There, numerical bifurcation analysis was used to identify a mathematical mechanism for coexistence of polar patterns (with a clear front and back) and multi-peak traveling waves suspected to lead to directed cell motion and cell ruffling, respectively. In particular, a parameter regime was identified where such patterns are stable and coexist. Other works in which numerical bifurcation analysis has been used to study actin waves models include [10,61]. Here, our key motivation is to then determine what types of cell shapes and motility occur based on the structures identified by [27].
The intracellular dynamics of actin waves only provides part of the picture at the cell-scale. To understand how actin waves affect cell shape and motility, the dynamics must be coupled to a model for cell deformations. Common techniques for capturing deformable cell domains include Lagrangian marker point, level set methods [43], phase field methods [46,15,5], finite elements [16,18], and Metropolis-based methods including the cellular Potts model (CPM) [3]. Additional references using such methods can be found in the reviews by [1,11,13,14,24,48,50]. The goal of the present study is to provide a simple, reproducible connection between the reactiondiffusion d
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