A minimal cell model for lamellipodia-based cellular dynamics and migration

Eukaryotic cells intrinsically change their shape by deforming their membrane through local variations in the membrane composition and by reorganizing their underlying cytoskeleton. The many molecular components of the cell membrane and cytoskeleton involved in the cellular shape changes and motility have been, and are still being actively, exposed. Despite the extensive knowledge of the components of this system, an open question is to understand how the observed cellular shape dynamics emerge from the huge molecular complexity. This open challenge gives motivation for constructing simplified theoretical models that describe cellular shape dynamics using a general and simplified set of components. We review here one such theoretical approach.

Cells in our bodies have a myriad of different shapes depending on their function (Frey and Idema, 2021), from cells in the gut that are covered with microvilli (short protrusions) (Sauvanet et al., 2015) to the highly branched neurons. These shapes all involve deforming the flexible cell membrane into a variety of archetypal forms, including cylindrical protrusions, such as filopodia (Mattila Pieta and Lappalainen, 2008), sheet-like extensions, such as lamellipodia (Innocenti, 2018), and cup-like invaginations, such as those occurring during endocytosis and phagocytosis (Flannagan et al., 2012), to name just a few common examples. These shapes can be rather static and maintain their form over long timescales (years), such as those of the stereocilia of the hair-cells in the inner ear (Naoz et al., 2008; Orly et al., 2015), whereas many are highly dynamic, forming and disassembling over timescales of minutes. Such dynamic shapes appear, for example, during development when cells differentiate, during cell motility (Bodor et al., 2020) and throughout the normal function of differentiated cells. One principal mechanism that allows cells to deform the membrane and achieve the desired shape is their cytoskeleton, mainly based on actin filaments (Pollard and Cooper, 2009; Salbreux et al., 2012). Polymerizing actin filaments allow cells to produce protrusive forces that push the bilayer membrane outwards (Mogilner and Oster, 2003; Liu et al., 2008), while the same actin network allows the cells to contract their membrane through the recruitment of myosin-II molecular motors (Koenderink and Paluch, 2018). An outstanding question in this field is how cells control the actin cytoskeleton in space and time so that polymerization and contractility self-organize where and when they should.

The underlying elements of the actin cytoskeleton that direct its polymerization to the cell membrane have been, and are still being, intensively explored. They include several families of actin polymerization promoter proteins that can form membrane-bound complexes (Welch and Mullins, 2002). Examples of such membrane-bound complexes include the Scar/WAVE complex (Machesky et al., 1999) and its binding partners, such as IRSp53 (also known as BAIAP2; Pipathsouk et al., 2021) and Abl (ABL1 in mammals; Zhu and Bhat, 2011) at the leading edge of lamellipodia. Typically, these actin polymerization promoters can be in an active or inactive form, with the activation occurring when they are part of the membrane-bound complex. Clearly this activation ‘switch’ allows the cells to control where actin polymerization occurs, by forming the active complexes (where actin polymerization occurs) in localized regions on the cell membrane. What determines the localization of these activated complexes on the cell membrane? One type of control can be exerted by external signaling, for instance chemokine molecules binding to membrane receptors could trigger a signaling cascade that locally activates membrane-bound actin polymerization promoters, inducing the polymerization of actin near the cell membrane (cortical actin). However, many cellular shapes form spontaneously, in the absence of external trigger or guidance.

One way to describe the dynamics of actin polymerization on the cell membrane, and how it forms dynamic patterns, has been in the form of reaction–diffusion (RD) equations. In this framework, several key components of the cortical actin network are described explicitly, including actin polymerization promoters, inhibitors of actin polymerization and both the polymerized and monomeric actin. The level of detail varies between these models, given that including the full complexity of the actin cytoskeleton and its associated proteins would make the models intractable (Mori et al., 2008; Bhattacharya et al., 2020; Driscoll et al., 2012; Bernitt et al., 2017). Nevertheless, these models have been successful, especially with respect to describing patterns of cortical actin polymerization and membrane composition such as actin waves propagating on the cell membrane, which also trigger lamellipodia protrusions (Taniguchi et al., 2013; Flemming et al., 2020). Most of these approaches so far have not included the deformation of the membrane as part of modeling the formation of spatiotemporal actin polymerization patterns. This is a valid approach for phenomena involving only small membrane deformations, such as those occurring on the basal surface of adhered cells. When membrane deformations are described by the RD framework, owing to the active forces (Campbell and Bagchi, 2018; Saito and Sawai, 2021), the membrane is usually described by a flexible substrate for the RD dynamics, but there is no explicit feedback whereby the membrane shape affects the chemical reactions that drive the RD dynamics (Ben Isaac et al., 2013). Explicit feedback between the membrane shape and the RD pattern formation has been considered in the absence of membrane deformations by active (actin-polymerization induced) forces (Wu and Liu, 2021; Tamemoto and Noguchi, 2020, 2021).

A few theoretical studies have addressed the three-dimensional shapes of isolated spreading (Serpelloni et al., 2021) and motile cells on surfaces of different curvatures (Link et al., 2023). One model contains a detailed description of the cellular mechanics and is based on the assumption of a central role for the nuclear dynamics and deformations in controlling the cell migration on the curved surface (Vassaux et al., 2019). A similar approach of modelling cell migration, where the process is dominated by the coupling between the nucleus and random peripheral protrusions (He and Jiang, 2017), produced migration patterns that were in qualitative agreement with observations (Song et al., 2015). Another model provides a simpler and more general description of three-dimensional cell migration in terms of an active fluid (Winkler et al., 2019), but its predictions were not systematically compared to experiments. A similar model was proposed to describe ameboid cells moving along ridges, guided by a reaction-diffusion mechanism adapted from macropinocytic cup formation (Honda et al., 2021).

We review here another approach, where the membrane dynamics is driven by a population of curved membrane proteins, which recruit actin polymerization that exerts active, protrusive forces on the membrane. By ‘curved membrane proteins’ we mean in general a membrane-bound protein complex that has an intrinsic shape (Zimmerberg and Kozlov, 2006). Examples of intrinsically curved membrane-bound proteins include the BAR family (Liu et al., 2015). When such protein complexes form on the membrane, they tend to bend the membrane to conform to their intrinsic shape (McMahon and Boucrot, 2015; McMahon and Gallop, 2005; Alimohamadi and Rangamani, 2018).

In this model, active protrusive forces are exerted by the actin cytoskeleton (Mesarec et al., 2021), where the curved protein complexes (CMCs) direct the actin to polymerize and grow against the membrane at their locations. The CMC might contain proteins that act as nucleators of actin polymerization, such as WAVE proteins or WASp (also known as Was) (Takenawa and Miki, 2001; Pollitt and Insall, 2009; Stradal et al., 2004). There is evidence that curved membrane proteins form complexes with actin nucleators, especially at the leading edge of membrane protrusions, such as lamellipodia (Pipathsouk et al., 2021; Begemann et al., 2019). When actin polymerization occurs near the membrane it exerts a pressure that pushes the membrane outwards, and the actin network backwards (retrograde flow) (Mogilner and Oster, 1996; Upadhyaya et al., 2003; Giardini et al., 2003; Carlsson, 2018). This is termed an ‘active’ force, to denote that it originates from processes that consume ATP and are therefore indicative that the living cell is not in thermal equilibrium. These forces convert chemical energy into mechanical work.

The combination of the intrinsic curvature of membrane-bound proteins and the protrusive forces of the cytoskeleton deforms the membrane and give rise to the emergence of different spontaneous shapes (Gov, 2018; Fošnarič et al., 2019). We demonstrate that this coupling between curvature and cytoskeletal forces constitutes a ‘minimal cell’ model, which exhibits several fundamental cellular behaviors that resemble cell spreading (Sadhu et al., 2021, 2023a), phagocytosis (Sadhu et al., 2022) and cell migration. Although the numerical simulations allow us to calculate complex membrane dynamics and shapes in three-dimensional space, which are not easily (or at all) amenable to analytic description, the basic modeling scheme is very simple with respect to the number of components. In this framework, we do not attempt to describe the dynamics of actin polymerization at the level of the individual filaments, which can be cross-linked and undergo depolymerization. Models with a similar level of complexity of description have been proposed in other studies (Hu and Papoian, 2010). Although these models contain more realistic details of the actin dynamics, they are computationally very demanding, making the description of large membrane shapes difficult to achieve. In addition, their complexity renders understanding of the resulting dynamics very difficult.

We aim to keep our theoretical model simple, so that we can systematically explore, as well as gain deeper understanding of, the shapes and dynamics that our proposed coupling produces. We therefore avoid description in terms of specific proteins and instead denote their basic properties that are essential for our model, such as the intrinsic curvature of membrane proteins (or protein–lipid complexes) and the strength of the protrusive force actin polymerization exerts at their location on the membrane. The active force is exerted in our model at the locations of the curved membrane proteins in the direction of the locale outwards, normal to the membrane surface. This form of protrusive force, which acts as a local pressure field on the membrane, is most natural for describing branched actin networks, as opposed to highly oriented actin bundles (such as inside filopodia), and we therefore limit the discussion in this Review to cellular shapes driven by such networks. We start by introducing the theoretical model, before presenting a series of examples where we demonstrate how this model can help explain different forms of cellular dynamics, firstly, the spreading of a model minimal cell on a flat adhesive substrate, where we obtain different shapes and the emergence of a motile, polarized phenotype that resembles lamellipodia-driven cells. We then explore how the model minimal cell engulfs rigid adhesive particles, resembling phagocytosis, and demonstrate that its protrusions spontaneously coil on adhesive fibers, similar to the behavior of cellular protrusions. Finally, we utilize the model minimal cell to explore curvotaxis, the response of cell migration to curved surfaces (Schamberger et al., 2023; Pawluchin and Galic, 2022), which suggests that physical principles give rise to universal rules of cell migration on curved substrates.

Our theoretical model is based on a coarse-grained continuum model (Fošnarič et al., 2019; Sadhu et al., 2021; Drab et al., 2023), where the flexible membrane of a vesicle is described as a two-dimensional surface that is built of triangles. The vertices of this triangulated surface are displaced by random Monte Carlo moves, driving the dynamics of the membrane (see Box 1 and figure therein). During the simulation, each displacement of the vertices is accepted if the total energy of the system decreases owing to this displacement (the energy terms are explained below). If the energy is increased by the displacement, the move is accepted according to the probability function that describes thermal fluctuations that drive such transient increases in the energy of the system. The use of a coarse-grained model means that we deal with length scales where the continuum description of the membrane is valid (i.e. larger than tens of nanometers), and do not include details of the molecular scale. In addition, we do not describe in this model the fluid flows that develop around (and in) the membrane as it moves and deforms. Such flows exert drag on the membrane and set the timescale for membrane shape deformations, and we therefore do describe the real timescale for the shape dynamics that we calculate. This means that the model describes the most energetically favorable shape changes that affect the dynamics, without the correct absolute timescales. We are able to compare relative times of shape changes by comparing their average number of Monte Carlo steps.

Box 1. Model equations of the energy terms

We provide here a list of the energy terms that are calculated at each Monte Carlo displacement of the nodes of the triangulated membrane (see middle section of figure). The moves are accepted or rejected depending on the energy change due to this local displacement of a node, such that a move that decreases the energy is always accepted, while a move that increases the energy is accepted according to the probability given by the thermal distribution of states (Drab et al., 2023). In addition to the energy terms that we describe below, bond-flip events are implemented during the simulation (see right section of figure), which are accepted as long as the edge length is within some length bounds that maintain the topological stability of the calculation (Fošnarič et al., 2019). These bond flips result in effective fluidization of the triangulated surface, thereby facilitating the diffusion of the CMCs on the vesicle surface.

Bending energy

The lipid bilayer membrane minimizes its energy when the lipid head groups and the fatty tails have an optimal packing. This molecular arrangement endows the membrane with a preferred intrinsic (spontaneous) curvature (see left section of figure). Unless the membrane composition has a large asymmetry between the two bilayer leaflets, this intrinsic curvature is close to zero, and the membrane prefers to be flat. However, there are specific membrane-bound proteins that have a non-zero preferred curvature. We consider that these protein complexes are more rigid than the bilayer, and therefore the energy of the membrane can be written as a discretized version of the Helfrich form (Helfrich, 1974):

[1]

with κ the bending modulus, h_i the mean curvature and c_0,i the spontaneous curvature of each node i, associated with area element A_i. The spontaneous curvature of the bare membrane nodes is usually taken to be zero (c_0,i=0), and for vertices that contain CMC, we will use here a high intrinsic convex curvature, (protruding outwards), where l_min is the minimal allowed length of the edges in the triangulated surface (the length-scale in the problem). Eqn [1] has a minimum value of zero, when the local mean curvature of the membrane fits with the local intrinsic curvature of each of the vertices on the surface. When this mismatch is larger, a bending energy cost incurs.

Protein–protein binding energy

We consider that CMCs can bind to each other to form clusters and aggregates on the membrane (see left section of figure and Fig. 1). This is implemented by a binding energy between neighboring vertices on the triangulated surface of:

[2]

with the binding energy between nearest-neighbor (nn) CMC (ρ_j=1 for CMC, and ρ_j=0 for bare membrane vertex) of strength w>0 per CMC–CMC bond.

Membrane–substrate adhesion energy

Given that cells can adhere to external surfaces, we consider an adhesion energy term. For example, in the case of a flat adhesive surface, this amounts to (Sadhu et al., 2021):

[3]

with energy E_ad per each adhered node, which is within a distance Δz of the substrate, where the rigid surface is located at z=z₀, and all the membrane vertices that are within a distance of Δz=l_min from this surface contribute an adhesion energy per unit area of − E_ad. In addition, the membrane is prevented from moving across the external surface, which acts as a rigid barrier.

Membrane tension energy

We can also calculate the explicit membrane tension using the following energy (Graziano et al., 2019):

[4]

where σ the effective membrane tension, and a_j,a₀ are the area and target area of each triangle.

In most of the simulations, we used a simpler condition that constrains the area changes per triangle, by setting minimal and maximal lengths per edge l_min, l_max=1.7l_min. This condition prevents pathological deformations and efficiently implements area conservation.

Active force that represents the force exerted by actin polymerization

The active force, that pushes the membrane towards the outwards normal (⁠⁠) at each node that contains a CMC (see right section of figure), is written as an additional energy (work) term:

[5]

such that the change in the energy of the system due to a small displacement of node i along the direction of the outwards normal ⁠, is set by the force of strength F. This term is inherently non-equilibrium in the sense that it has no lower bound.

Figure adapted from Gov (2018) with permission from the Royal Society of Chemistry and from Schamberger et al. (2023) with permission from Elsevier.

The energy terms that we consider in the model and their mathematical implementation are outlined below and in Box 1. We include the minimal energy terms, that necessarily exist in this system.

The first term accounts for the energy cost of bending the membrane. The lipid bilayer membrane minimizes its energy when the lipid head groups and the fatty tails have an optimal packing. This molecular arrangement endows the membrane with a preferred intrinsic (spontaneous) curvature. Unless the membrane composition has a large asymmetry between the two bilayer leaflets, this intrinsic curvature is close to zero, and the membrane prefers to be flat (Deserno, 2015; Safran, 2003). However, there are specific membrane-bound proteins that have a non-zero preferred curvature, and the membrane–protein complex minimizes its energy when its curved. The bending energy term is always positive (Helfrich, 1974).

We also consider that CMCs can bind to each other to form clusters and aggregates on the membrane. This binding is described as a negative energy term, which is therefore maximized when the CMC binds to form large clusters. This process, together with the spontaneous curvature of the CMC, can describe the spontaneous aggregation of the CMC to form small hemi-spherical buds, which can further aggregate to form ‘pearled’ clusters (Fig. 1). The buds have the exact spontaneous curvature of the CMC, thereby minimizing their bending energy, while their protein–protein binding energy stabilizes them against thermal fluctuations that act to break them apart. The formation of the pearled clusters maximizes this protein–protein binding energy by minimizing the number of isolated clusters. Such shapes might be observed in living cells for clusters of curved membrane proteins and nano-domains. Note that the membrane tension is not explicitly considered in most of our simulations, and we use a simpler condition that constrains the area changes per triangle, by setting minimal and maximal lengths per edge. This condition prevents pathological deformations and efficiently implements area conservation.

Given that cells can adhere to external surfaces, we consider an adhesion energy term. This energy represents the binding of the membrane to an external surface. It is therefore implemented as a negative energy increment whenever a vertex of the membrane is within a close distance to the surface. More complex adhesion rules can be implemented, but the simplest is to assume that the adhesion strength is uniform over all the membrane that is in proximity to the surface (Sadhu et al., 2021). In addition, the membrane is prevented from moving across the external surface, which acts as a rigid barrier. The final energy term that we consider is a representation of the active force exerted by actin polymerization.

The active force, which pushes the membrane towards the outwards normal at each vertex that contains a CMC, represents the pressure that acts on the membrane owing to actin polymerization. This term is inherently non-equilibrium in the sense that it has no lower bound.

Actin polymerization is converted into an efficient protrusive force on the membrane (Mogilner and Oster, 1996) when the retrograde flow of the actin filaments experiences an effective friction with the substrate (Craig et al., 2015), mediated by specialized adhesion molecules (Gardel et al., 2010). We do not describe these adhesion molecules explicitly in our description, and simply allow each CMC to exert the same active force on the membrane, assuming that this friction affects all regions of the membrane equally. In addition, these adhesion molecules, and their binding–unbinding dynamics affects the movement of the cell membrane over the adhesive surface (Sackmann and Smith, 2014), exerting an effective friction that balanced the traction forces and determines the migration speed of the cell (DiMilla et al., 1991). We do not explicitly describe the dynamics of these adhesion molecules in our model.

Therefore, this model has only a few components. In the next sections, we present the different membrane shapes and dynamics that emerge in this model when the CMC vertices exert active protrusive forces, which represent the recruitment of actin polymerization, as well as adhesion to an external substrate.

Many cell types are observed to spread and adhere to external rigid surfaces (Döbereiner et al., 2004; Cavalcanti-Adam et al., 2007). Such cells can either spread and remain stationary and adherent, or they become polarized and migrate on the surface. This process often involves the formation of a thin sheet-like protrusion around the cell edge, called a lamellipodium, which is driven by the formation of a branched actin network (Blanchoin et al., 2014; Le Clainche and Carlier, 2008), with the actin polymerization promoter, such as the WAVE complex, localized along the highly curved leading edge of the lamellipodium (Fritz-Laylin et al., 2017; Bieling and Rottner, 2023). As we show below, our model of a minimal cell describes a process that closely resembles the observed cell spreading by lamellipodia-like protrusions (Cuvelier et al., 2007; Xiong et al., 2010; Döbereiner et al., 2004).

When we let our minimal cell vesicle adhere and spread over a flat rigid surface, of uniform adhesiveness, we first find that this process is strongly affected by the presence of passive CMCs (i.e. in the absence of actin-driven forces; Fig. 2A). For the vesicle to spread, the adhesion energy gain has to offset the bending energy cost of forming a highly curved rim along the cell edge. The bending energy can prevent spreading on weakly adhered substrates; these are relevant to cells: in the absence of actin cytoskeleton activity, the bare cell membrane–substrate adhesion is usually weak and is insufficient to drive spreading over the surface (Guo et al., 2017). Given that cells need to be able to detach, move and remodel their adhesions, they do not form extremely strong, ‘super-glue’-like interactions with the substrate. We find that the highly curved, convex CMCs spontaneously aggregate along the curved rim of the spreading vesicle, thereby reducing the bending energy cost that acts to resist the spreading. Although this mechanism can be utilized by cells to facilitate spreading on weakly adhered substrates, it involves a large concentration of CMCs on the cell membrane.

When the CMCs recruit the protrusive force of actin polymerization, we find that they self-organize to form large aggregates along the cell rim, which drive robust spreading on the substrate, even at low CMC concentrations (Fig. 2B). The mechanism for this robust spreading is the positive feedback between the aggregation of the CMCs along the highly curved rim and the protrusive forces that they recruit, which push the membrane outwards and maintain the high curvature. The resulting adhered shape resembles adherent cells that are either round, with lamellipodia all along their edge (Szewczyk et al., 2013; Li et al., 2015), or elongated with mainly two competing lamellipodia protrusions at opposite ends of the cell. Note that adhered cells form stress fibers, which rely on and exert contractile forces on the cell membrane and external substrate (Schwarz and Safran, 2013). We do not include stress fibers in our current model. Our model captures the dynamics of the spreading observed in cells, including the ruffles that form at the leading edge of the lamellipodia during the spreading process (Safran, 2003; Helfrich, 1974; Mogilner and Oster, 1996).

The elongated shapes arise in our model when the density of CMCs is so low that there are not sufficient CMCs to complete a continuous ring-like cluster along the leading edge of a circular spread vesicle (Fig. 2B). In this regime, we find that for intermediate actin force strength and strong adhesion, a polarized and motile phenotype emerges spontaneously in our model (Fig. 2C). This motile vesicle has a single, crescent cluster of CMCs along its leading edge, where the actin-driven forces are applied and maintain a sharp edge. The back of the vesicle is rounded, minimizing the bending energy of the membrane. We find that this motile vesicle resembles the shape of polarized motile cells, such as keratocytes. However, it is a rather fragile object – when our motile vesicle hits a barrier, or even due to spontaneous fluctuations, the leading-edge cluster of CMCs can break apart, and an immotile, elongated shape emerges (Fig. 2D). Real cells are often observed to undergo similar events of leading-edge splitting (Andrew and Insall, 2007), but they have internal mechanisms to re-establish polarity, which we do not have in our current model (Maiuri et al., 2015).

Despite the simplicity of our model, it can be used to rationalize some puzzling experimental observations. For example, in the study of Spence et al. (2012) a type of mammalian breast cancer cells was found to be not very persistent and often had elongated shapes with multiple and competing leading edges. Upon inactivation of some of the actin polymerization nucleators, these cells became crescent-shaped and persistent, with a single leading edge (Döbereiner et al., 2004). This puzzling observation can be explained with our model (Fig. 2C), in that for high actin-driven forces, elongated (non-motile) shapes dominate in our model, which correspond to the regular form of these cancer cells. When the magnitude of the active forces is reduced in the model, a motile vesicle appears, corresponding to the motile cells that appeared for lower values of the actin polymerization activity.

Our model shows that the shape of a lamellipodium, in both adherent and motile cells, can spontaneously arise due to the coupling between convex CMCs and the protrusive forces of actin polymerization. This coupling leads to a self-organization process, maintaining the aggregation of CMCs along the lamellipodia-containing leading edge, giving rise to its distinct sheet-like morphology. This mechanism of lamellipodia formation, which we argue arises purely from a minimization of the system's energy and work, is controlled in the cell by signaling networks. These biochemical controls allow the cell to spatially and temporally fine tune its response to different external stimuli.

Having established that our minimal cell model can recapitulate the spontaneous formation of lamellipodia during cell spreading, as well as the emergence of a motile phenotype, on a flat substrate, we next discuss how this model might explain the interactions of cells with curved surfaces.

An in vivo example where a cell engages with a curved substrate, is during phagocytosis, the engulfment of a rigid particle by a cell (Flannagan et al., 2012; Cannon and Swanson, 1992; Kumari et al., 2010). Here, the cell membrane adheres to the particle and then spreads its membrane over it until complete engulfment occurs. During this process, there is a competition between the gain in adhesion energy and the cost of bending the membrane during engulfment. Similar to the case of spreading over a flat substrate (Fig. 2A), we find that the engulfment process can be facilitated by the presence of passive CMCs, which spontaneously aggregate along the highly curved leading edge (Fig. 3A,C) (Sadhu et al., 2021, 2022). This aggregation lowers the bending energy, such that the gain in adhesion energy overcomes the bending energy cost and the engulfment can be completed.

Phagocytosis is, however, an active process that is known to involve actin-mediated forces that push the engulfing membrane forward (Mylvaganam et al., 2021). From numerous studies, we know that phagocytosis involves highly complex and dynamic rearrangements of the cytoskeleton, membrane shape deformations and protein aggregations (Niedergang and Chavrier, 2004). At present, there is no complete theoretical understanding of the dynamics of the self-organization of the membrane and the actin cytoskeleton, including the active forces it exerts during the engulfment process (Richards and Endres, 2017). However, we show that this problem can be addressed using our theoretical model (Sadhu et al., 2022).

When CMCs recruit the protrusive actin-mediated forces, their aggregation along the leading edge is more robust, driving complete engulfment at lower density of CMCs compared to passive CMCs (Fig. 3A,B) and for non-spherical particles (Sadhu et al., 2022), which present a higher energy barrier for engulfment. In addition to lowering the bending energy barrier for engulfment, the actin-driven forces also contribute directly to the engulfment by pushing the leading edge of the membrane over the surface of the particle. The model predicts that this directed force is effective in driving the engulfment even when the cluster of CMCs at the leading edge does not form a complete ring but is composed of either a single partial arc or fragmented arcs (Fig. 3D). In these cases, a complete CMC ring usually forms at the later stages, as the engulfment progresses towards completion. These features, which appear in our model of fragmented actin clusters at the leading edge of the phagocytic cup, a complete actin ring as engulfment nears completion and final dispersal of the actin clustering after engulfment, have also been observed in recent experiments in cells using high-resolution imaging (Vorselen et al., 2021) (Fig. 3E).

Finally, as the engulfment is completed, the model predicts that the cluster of CMCs disperses spontaneously, as the narrow neck of membrane (our model does not allow fusion or fission of the membrane) does not have the mean curvature the CMCs require (Fig. 3D). The model therefore demonstrates that a complex biological process such a phagocytosis can be driven and coordinated using physical principles of minimization of energy and effective work. In the cell, these physical mechanisms are controlled by additional layers of biological signals, which determine when and where they are activated.

In the next two sections, we explore the prediction of our model for cells that spread over extended curved surfaces.

Cellular protrusions play important roles in exploring and sensing the extracellular environment during cell spreading and adhesion, cell migration and cell–cell interactions (Le Clainche and Carlier, 2008; Caswell and Zech, 2018). Lamellipodia protrusions enable cells to adhere and spread on fiber-like surfaces (Callens et al., 2020; Assoian et al., 2019; Koons et al., 2017), such as the fibers of the extracellular matrix (ECM) (Clark et al., 1982), as well as cylindrical protrusions of other cells, such as glial cells spreading over neighboring axonal extensions (Stadelmann et al., 2019; Djannatian et al., 2019). In vitro studies of cellular spreading and migration on fibers have shown the organization of different cell types on these fibers (Bade et al., 2017; Svitkina et al., 1995; Hwang et al., 2009; Meehan and Nain, 2014; Kennedy et al., 2017; Mukherjee et al., 2019; Guetta-Terrier et al., 2015), with cellular shape and motility found to depend on the curvature (diameter) of the fibers. These experiments have found indications for leading-edge cellular protrusions coiling (wrapping) around extracellular fibers, for instance in metastatic cancer cells (breast and ovarian), as well as in several other cell types (fibroblasts, epithelial and endothelial) (Koons et al., 2017; Mukherjee et al., 2019; Guetta-Terrier et al., 2015). However, the mechanisms that drive the tendency of the leading edge of cellular protrusions to rotate while cells are spreading on fibers is not understood at present.

Therefore, we have attempted to explain this behavior using our model, starting with an elongated, adherent vesicle that has two leading edges on opposite ends (Fig. 4A) (Sadhu et al., 2023a). This configuration, which forms spontaneously in our model (Fig. 2B), resembles protrusions extended by cells on fibers (Koons et al., 2017; Mukherjee et al., 2019; Guetta-Terrier et al., 2015), with each protrusion having a lamellipodia-like leading edge, which in our model is composed of and driven by a cluster of CMCs. We find in our model that these protrusions spontaneously prefer to reorient along the circumferential direction, which gives rise to coiling-like motion (Fig. 4B) (Sadhu et al., 2023a). The origin of this preference is minimization of (mainly) bending and adhesion energy (Sadhu et al., 2023a). Using this insight, we used the model to predict behavior on fibers with a non-circular cross-section with sharp edges; here, the bending energy cost of coiling will prevail over the adhesion energy, and coiling or wrapping will be inhibited (Fig. 4C,D). This prediction was verified with experiments of cells spreading on fibers flattened to form ribbons with very narrow edges (Sadhu et al., 2023a), where the cellular protrusions do not coil around the fiber. As discussed in the previous examples above, despite the fact that a cell exhibits a highly complex ruffling dynamics at the leading edge, a simple model based on few physical principles can explain the tendency for coiling of the leading edge of a protrusion.

When a model vesicle spreads on a fiber, it can form a single leading edge and obtain the motile phenotype that we observed on the flat surface. The dynamics of such motile vesicles on curved surfaces is explored in the next section.

Cells often migrate on curved surfaces inside the body, such as curved tissues, blood vessels, fibers of the extracellular matrix or cylindrical protrusions of other cells. Recent in vitro experiments provide clear evidence that motile cells are affected by the curvature of the substrate on which they migrate (Vassaux et al., 2019; Song et al., 2015; Assoian et al., 2019; Werner et al., 2020, 2018; Driscoll et al., 2014; Sanz-Herrera et al., 2009), preferring certain curvatures to others, a process termed curvotaxis. However, the origin and underlying mechanism of this curvature sensitivity are not well understood.

We focus here on two simple types of curved surfaces – a flat surface with a sinusoidal height undulation along one direction and a cylindrical fiber (as in the previous section). On both these surfaces, we can calculate the dynamic behavior of the model motile vesicle and compare it to the experimental observations (Sadhu et al., 2023b preprint).

A sinusoidal geometry has been experimentally investigated in several studies (Song et al., 2015; Pieuchot et al., 2018). For instance, the migration of T-lymphocytes has been studied on a surface with a unidirectional sinusoidal (wavy) height undulation (Song et al., 2015). Here, the cells were found to move axially (along the pattern) when inside the grooves (minima) of the surface topography, avoiding migration on the ridges (maxima) by crossing the ridges orthogonally. Indeed, our motile model vesicle exhibits this exact same qualitative behavior when it is small compared to the undulation wavelength (Fig. 5A,B) (Song et al., 2015; Sadhu et al., 2023b preprint). In contrast, when the undulation wavelength is smaller and the cell spans more than one groove or ridge, the vesicle can maintain stable motility orthogonal to the undulation pattern (Fig. 5C), while undergoing periodic changes in its speed.

Note that adherent cells, which are dominated by stress fibers and are weakly motile (such as fibroblasts), have been found to settle in the concave grooves or adhere aligned to the undulation axis (both on grooves and ridges) (Werner et al., 2018, 2019). In many adherent cells, their direction of migration and axis of cellular elongation is determined by the competition between the bending energy of the stress fibers, deformation of the nucleus and the contractile forces applied by the stress fibers (Biton and Safran, 2009; Werner et al., 2020; Sanz-Herrera et al., 2009). These components are not included in our current model.

The agreement between the model and experiments on sinusoidal surfaces indicates that this form of curvotaxis can arise from only a few physical parameters. However, a sinusoidal surface has both positive and negative mean curvatures, which complicates the analysis of the resulting motion in terms of energy minimization. We therefore investigate next the migration over a cylindrical fiber, a surface with a uniform mean curvature.

To simulate migration on a fiber, we started with our motile vesicle initially aligned along the axis of the fiber (Sadhu et al., 2023b preprint). The vesicle spontaneously rotates to align its motion along the circumferential direction (Fig. 5D) and continues to rotate around the fiber axis. The main energies that drive this reorientation, and stabilize the circumferential orientation, are the bending and adhesion energies. This is the same energy minimization process that gives rise to the coiling of elongated adherent vesicles (Fig. 4A) and can be used to explain the coiling dynamics of cellular protrusions on fibers. The increased adhesion in the circumferential orientation arises from the ability of the leading-edge cluster of CMCs to stretch the vesicle very effectively along the zero-curvature axial direction in this configuration.

Independently of our model, this predicted tendency of migrating cells to rotate around fibers has already been demonstrated experimentally in a PhD thesis (Blum, 2015) for the orientation of Dictyostelium discoideum on a cylindrical fiber, clearly exhibiting the tendency of the cells to prefer the circumferential orientation (Fig. 5E,F) (Bade et al., 2017).

Taken together, we demonstrate here that a minimal cell model of a motile cell, based on only a few parameters and energy terms, is able to describe and explain several curvotaxis features of lamellipodia-driven cell migration on curved adhesive substrates. The curvotaxis features explained by the model, such as the tendency of motile cells to migrate aligned within grooves, avoid ridges and rotate around fibers, all arise owing to minimization of the adhesion and bending energies of the vesicle. Real cells contain numerous additional layers of complexity that our simple model does not, such as the effects of contractility, stress fibers and internal organelles (mainly the nucleus), which can all affect migration on curved substrates. Nevertheless, the agreement between the predictions of the model and the observations of curvotaxis in different types of migrating cells suggests that the simple energetic considerations in our model might indeed drive universal curvotactic features in lamellipodia-based motile cells.

These results from modeling a minimal cell demonstrate that complex cellular behavior might have underlying physical underpinnings, where energy and effective work minimization are the driving principle. As discussed above, a wide range of cellular shape dynamics and migration patterns can be obtained from a very simple model containing only a few components owing to the strong feedback between the intrinsic shape of the CMC and the active forces that deform the membrane. Our model shows that curved membrane complexes that recruit the forces exerted by actin polymerization constitute a versatile mechanism for spontaneous pattern formation on the cellular level, which are able to drive numerous cellular shapes and dynamics that correspond to observed cellular behavior (Graziano et al., 2019). This physical mechanism is regulated in the cell, in space and time, by additional layers of biological complexity and signaling networks.

This approach also demonstrates the power of simple physical models in exposing very general mechanisms for the self-organization of the cytoskeleton, which are not cell type specific, but are often obscured by biochemical details. Gaining such general understanding greatly advances our ability to control and predict the dynamics of cells over a wide range of cell types and cellular dynamics. Such models can also motivate experiments aimed at recreating a similar minimal cell in vitro (Streicher et al., 2009). A recent extension of the work reviewed here demonstrates that the minimal cell model might explain the responses of lamellipodia-driven motile cells to shear flow (Sadhukhan et al., 2023). Future extensions of the work presented here will need to explore additional factors, such as the effects of CMCs with anisotropic shapes (Pipathsouk et al., 2021; Kabaso et al., 2012; Mesarec, et al., 2023) and mixtures of CMCs of several types of curvatures (both convex and concave) (Ravid et al., 2023). We reviewed here modeling cellular shapes driven by branched-actin networks, such as lamellipodia and ruffles, but this framework can also be extended to describe the effects of actin filament bundling, which occurs inside finger-like filopodia protrusions (Ravid et al., 2023).

N.S.G. is the incumbent of the Lee and William Abramowitz Professorial Chair of Biophysics.