Collective durotaxis of cohesive cell clusters on a stiffness gradient

THE EUROPEAN PHYSICAL JOURNAL E Eur. Phys. J. E (2022)45:7 https://doi.org/10.1140/epje/s10189-021-00150-6 Regular Article Collective durotaxis of cohesive cell clusters on a stiffness gradient Irina Pi-Jaumà1,2 , Ricard Alert3,4,5,6 , and Jaume Casademunt1,2,a 1 Departament de Fı́sica de la Matèria Condensada, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain Universitat de Barcelona Institut of Complex Systems (UBICS), 08028 Barcelona, Spain 3 Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA 4 Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544, USA 5 Max Planck Institute for the Physics of Complex Systems, Nöthnitzerst. 38, 01187 Dresden, Germany 6 Center for Systems Biology Dresden, Pfotenhauerst. 108, 01307 Dresden, Germany 2 Received 2 July 2021 / Accepted 15 November 2021 © The Author(s) 2022 Abstract Many types of motile cells perform durotaxis, namely directed migration following gradients of substrate stiffness. Recent experiments have revealed that cell monolayers can migrate toward stiffer regions even when individual cells do not—a phenomenon known as collective durotaxis. Here, we address the spontaneous motion of finite cohesive cell monolayers on a stiffness gradient. We theoretically analyze a continuum active polar fluid model that has been tested in recent wetting assays of epithelial tissues and includes two types of active forces (cell–substrate traction and cell–cell contractility). The competition between the two active forces determines whether a cell monolayer spreads or contracts. Here, we show that this model generically predicts collective durotaxis, and that it features a variety of dynamical regimes as a result of the interplay between the spreading state and the global propagation, including sequential contraction and spreading of the monolayer as it moves toward higher stiffness. We solve the model exactly in some relevant cases, which provides both physical insights into the mechanisms of tissue durotaxis and spreading as well as a variety of predictions that could guide the design of future experiments. 1 Introduction The organized motion of cohesive groups of cells, usually referred to as collective cell migration, plays a key role in many instances of morphogenesis, tissue regeneration, and cancer invasion [1–6]. The mechanisms by which cells coordinate their motion are diverse and often not fully understood. Recent work has shown that groups of cells may respond to external stimuli as a whole, that is, in the form of collectively organized directed motion, in ways similar to what single cells do. Such collective migration can arise in response to a variety of external stimuli such as gradients in either chemical concentrations or in the stiffness of the environment, which, respectively, lead to collective chemotaxis [7] and durotaxis. We are interested in the phenomenon of durotaxis, which refers to the directed motion of cells along stiffness gradients of the extracellular matrix, typically toward stiffer regions. This is a well-known phenomenon for single-cell migration [8], which is rather common in many types of cells and has important implications for cancer invasion. More recently, durotaxis has been reported also for collective cell migration [9,10]. a e-mail: author) (corresponding 0123456789().: V,-vol Remarkably, large cell monolayers can perform durotaxis collectively even when their constituent cells do not [9], and in some cases, there is an optimal intermediate stiffness for tissue spreading [11,12]. Collective durotaxis has been theoretically described both via hybrid computational models [13–16] and via a continuum active polar fluid model [17] that generalized previous work on tissue wetting [18]. This continuum model was solved numerically to reveal two possible mechanisms of collective durotaxis [17]. Here, we extend the work in Ref. [17] to provide a more comprehensive classification of the dynamical regimes of the model in terms of physical parameters. Remarkably, we solve the model analytically in some simple but relevant situations, allowing for a better grasp of the physical mechanisms at play. As shown in Ref. [18], the model predictions can be fitted to experimental data to infer physical parameters that are often elusive to direct measurement. The model describes cell monolayers moving on a substrate as a quasi-two-dimensional viscous fluid with two types of active forces: cell–substrate traction and cell–cell contractility. The competition between both active forces was shown to give rise to the socalled active wetting transition, whereby a tissue either spreads or retracts depending on its size [18]. The same model also predicted a fingering instability of the lead- 123 7 Page 2 of 15 ing edge of the tissue [19]. In addition to the active forces, the model also features two passive forces: an effective viscosity, which arises from cell–cell adhesion, and a friction force due to cell–substrate interactions. All these forces are treated in a coarse-grained way at the supracellular scale. The rationale of the approach is to identify the dynamical behaviors of cell monolayers that are of mechanical origin, explicitly excluding any signaling effects that cannot be encoded in the mechanical parameters of the model. To what extent such purely mechanical approach may succeed as a first step to account for the observed phenomenology is an interesting open question that might be settled by future experiments. 2 Hydrodynamic model Our model stems from a hydrodynamic approach to cell tissues, a strategy that has proven useful when tissues are organized at a supracellular scale, such that information at the cellular scale is not relevant [20–24]. This is the case in many examples of collective cell migration, where coarse-grained fields such as velocity, cell density, and polarization are treated as smooth fields varying on scales larger than the cell size. Continuum field theories based on linear irreversible thermodynamics, often called active gels theories, were first devised to account for active matter at the cellular scale, such as the cytoskeleton [25–28], but have more recently been extended to multicellular scales [29]. The basic idea is that tissues can be modeled to some extent as continuous active materials, in such a way that the biological properties are encoded in a series of physical parameters, including passive ones such as viscosity or friction, and active ones such as contractility or traction. These parameters will in general be time and space dependent to account for the biological regulation of the cell properties and interactions. For instance, in a simple model for the spreading of epithelial monolayers [30], it was shown that their effective viscosity increases with time as they become thinner due to the spreading. This type of approach is useful to identify activity-driven hydrodynamic instabilitie (...truncated)