A dissipative mass conserved reaction–diffusion system reveals switching between coexisting polar and oscillatory cell motility states Open Access

Eukaryotic cells display distinct migration modes. While some migratory modes are tightly linked to specific functions and cell types, also migratory plasticity may occur, where different modes of locomotion are observed within the same cell type. It remains unclear, however, whether transitions between the migratory modes require changes in external conditions, or whether the different modes are coexisting states that emerge from the underlying signaling network. This theoretical and experimental study provides a distinct approach for addressing the question of internally coexisting dynamical structures in cell motility and presents the first potential mechanism of this essential aspect of cortical pattern formation.

Motile eukaryotic cells play a key role in many important biological processes, such as early embryonic development or functions of the immune system.^1,2 The mechanical forces that drive their locomotion are generated by the assembly of filamentous actin (F-actin) that causes protrusion of the cell edge.³ The cytoskeletal dynamics is governed by upstream signaling pathways and provides the basis not only for motility but also for other central cellular functions, such as nutrient uptake and cell division.^4–10 Also numerous pathological phenomena are related to defects in F-actin regulation, among these, aberrant cell motility in metastatic cancer.¹¹ 

Actin-driven cell motility relies on self-organized space-time patterns that emerge from the underlying regulatory dynamics.^12–25 Central players in this regulatory network are small GTPases that interact rapidly to form a biochemical “pre-pattern” inside the cell. Small GTPases have been intensely studied experimentally^26,27 as well as theoretically over the past few decades.^24,28–35 While small GTPases may locally promote nucleation and polymerization of F-actin, they also interact with F-actin via feedback loops. This gives rise to a variety of space-time patterns broadly known as “actin waves,”^36–41 a prominent example of pattern formation at the subcellular level, see Refs. 42 and 43 and the references therein.

Eukaryotic cells can move in many different ways, ranging from disordered (random-like walks) to highly persistent migration.^1,44 The different modes of locomotion are related to distinct cell shapes and cytoskeletal arrangements, such as the flat, extended, and stable leading edge (lamellipodium) of fish keratocytes or the small, compact, and short-lived membrane protrusions (pseudopodia) of amoeboid cells. While some migratory modes are tightly linked to specific functions and cell types, migratory plasticity can also occur, where distinct modes of locomotion occur within the same cell type. For example, during cancer progression, metastatic cells undergo transitions between different migratory modes, such as amoeboid and mesenchymal motility.^45,46

One of the common model organisms to study basic mechanisms of switching between different migratory modes is the social amoeba Dictyostelium discoideum (D. discoideum). Besides their common amoeboid motility, D. discoideum cells may also exhibit a keratocyte-like (so-called “fan-shaped”) mode of locomotion. While amoeboid motility is characterized by the formation of highly dynamic, localized pseudopodia, resulting in erratic, random displacements, fan-shaped cells move in a highly persistent fashion and show a stable cell shape that is elongated perpendicular to their direction of motion.⁴⁷ Fan-shaped D. discoideum cells were observed as a consequence of genetic mutations^47,48 and specific developmental conditions.⁴⁹ Their persistent forward motion is driven by a ring-shaped actin wave that covers most of their ventral membrane.⁵⁰ Actin waves of this type have been thoroughly characterized in D. discoideum.^51–53 They serve as precursors of endocytic cups and their formation is controlled by small GTPase signaling.^7,39 Thus, the fan-shaped mode was also directly induced in D. discoideum cells by synthetically increasing their level of small GTPase activity.^41,54 However, spontaneous switching within the same cell was also observed.⁵⁵ 

The above examples of switching between distinct motility modes can be explained in different ways, e.g., by invoking changes in connectivity of the underlying regulatory network or by changes in system parameters. In this study, we focus on a third possibility, where distinct motility modes coexist in the same regulatory systems and for a fixed set of parameters, so that external perturbations merely trigger switches between them. We first devise and analyze a simplified reaction–diffusion model with mass conservation, mimicking the coupled GTPase and F-actin dynamics to reveal a bistability region, where polarized and oscillatory states generically coexist. Second, we demonstrate why, upon a large perturbation, the polarized state may become unstable, resulting in a transition to oscillatory dynamics, associated with transitions between two coexisting migratory modes. Finally, we show an experimental demonstration of a fan-shaped D. discoideum cell that, upon collision with a rigid boundary, becomes unstable and undergoes a transition to a disordered non-polar mode of migration.

An important feature in (1) is the basal rate of activation $b>0$⁠; mathematically, this term excludes the existence of trivial solutions. In addition, $γ$ is the rate of auto-activation (positive feedback of active GTPase to its own activation rate), $s$ is the F-actin dependent inactivation rate, $η$ is the F-actin time scale parameter, $p 0$ is the F-actin basal growth rate, $p 1$ is the GTPase dependent F-actin assembly rate, and $D F≪D<1$ are the diffusion coefficients of F-actin and active GTPase, respectively. In what follows, we use $s$ as a control parameter while keeping all other parameters fixed. We note that a more detailed mathematical analysis of (1) is presented in Ref. 65.

We start by numerically computing the uniform steady states $P ∗=( u ∗, v ∗, F ∗)$ of (1), which result in up to three biologically relevant solutions, $P 0 , 1 , 2 ∗>0$⁠, forming an inverse “S” form, as shown in Fig. 1(b). Linear stability analysis in 1D of $P ∗$ to infinitesimal perturbations⁶⁶ leads to solutions $P(x,t)− P ∗∝exp⁡(σt+iqx)$⁠, where $σ$ is the growth rate of wavenumber $q$⁠, resulting in three dispersion relations for $σ(q;s)$⁠. We find that, while $P 0 ∗$ is linearly stable, the state $P 2 ∗$ exhibits a simultaneous long-wavelength and finite wavenumber Hopf instability at $(s,b)=( s c, b c)≈(0.409,0.067)$⁠, as shown in the inset of Fig. 1(b); such simultaneous instabilities are also known as codimension-2 bifurcations. The former instability occurs around $q=0$ and leads to the bifurcation of steady states in subcritical direction (toward a stable portion of $P 2 ∗$⁠, $s< s c$⁠). The latter gives rise to both traveling and standing waves (TWs and SWs with $q c≈2.031$⁠, respectively) that bifurcate supercritically (toward the unstable portion of $P 2 ∗$⁠, $s> s c$⁠) and where TWs are linearly stable (here, SWs are ignored and for further details we refer to Ref. 65). We used the package AUTO⁶⁸ to compute the bifurcating branches and solutions; linear stability of nonuniform states is obtained via the eigs function in MATLAB.

In our context of cells, whose size is finite, we set the domain length $L=3 λ c$⁠, where $λ c=2π/ q c≈3.093$ and $q c≈2.031$ is the critical wavenumber at the onset of the finite wavenumber Hopf instability (⁠ $s= s c$⁠). In Fig. 1(b), we show that the WP solutions, $P WP 3 λ c$⁠, bifurcate subcritically from $s≳ s c$ and fold to the right $( at s= s S N − W P )$⁠, where their profile resembles a hole-like state, as shown in Fig. 1(c). Then, the branch continues to the right, folds again to the left \big(at $s= s SN + WP$\big), and terminates near the fold of $P 0 ∗$ at an additional long-wavelength instability. At $s= s SN + WP$⁠, the profile resembles a peak-like solution, whereas between the folds, the profiles correspond to mesa states, as shown in Fig. 1(c) by selected intermediate profiles. The mesa states are “box-like patterns that join two flat regions of space with sharp transition layers”;⁶⁹ however, the flat tops and bottoms of the mesas here are not steady uniform solutions of (1). Mesa solutions were first reported in dissipative models^69,70 and are also known as “wave-pinning solutions.”³¹ In our model, high values of these plateaus correspond to high concentrations of F-actin and its regulator at the “front edge” of a cell.^31,34,62,71

To compute the bifurcating TW branch, we employ a comoving frame $ξ=x−ct$⁠, where $c$ is the speed and at the onset (⁠ $s= s c$⁠) is given by the phase speed $c c= Imσ/ q c≈0.191$⁠. The TWs branch, $P TW λ c$⁠, is supercritical, i.e., in the direction of increasing $s$ and after a fold at $s SN TW$⁠, it ends in a parity-breaking bifurcation on the $P W P λ c$ branch (not shown here) with the corresponding length $L= λ c$ [“” in Fig. 1(b)], for details see Ref. 65. Figure 1(b) shows the coexisting branch of TWs and, moreover, the bistability region, where the steady mesa states and TWs are linearly stable. Next, we address the role of large amplitude perturbations within this bistable regime, as in the case of collisions.

Bistability and the role of large perturbations—In the case of motile cells, the steady mesa states correspond to strongly polarized F-actin distributions and, therefore, can be taken as representations of fan-shaped cells with highly polarized, directed migration. Let us consider a polarized cell moving toward a wall and, upon collision, extending laterally, as schematically shown in Fig. 2(a). One of the possible and simplest nonlinear (large) perturbations of the mesa state is that it may become wider. However, due to mass conservation (here $M=2$⁠), the mesa state cannot maintain its amplitude, i.e., its peak value will decrease at the expense of widening.⁷³ To ensure the mass constraint, we first fixed the value of $s$ for the “pre-perturbed state” on the stable portion of $PWP3λc$ [“” in Fig. 1(c)]. Then, we select a state with a different value of $s$ [“” in Fig. 1(c)], representing a 21% increase in the width of the top plateau (for the same $M$ value). We note that collisions comprise many other perturbations (e.g., parameter variations in space^43,74) that are likely to enhance the instability of the polarized state (mesa states). However, such perturbations are model-dependent and, thus, beyond the scope of our simplified framework Eq. (1).

Solving Eq. (1) via direct numerical integration on domain length $L=3 λ c$⁠, we show that the “perturbed” initial state develops oscillations leading to outward propagating TWs, as shown in Fig. 2(b). However, the form of the oscillations is of lesser importance, as they lead to disordered dynamics in higher spatial dimensions due to modulational instabilities^66,75 (e.g., Eckhaus–Benjamin–Feir instability), even in the absence of stochastic contributions or shape deformations.^{24,48,49,60,76–81} This means that, in general, oscillations and disordered wave patterns belong to the same universality class, i.e., the latter are aperiodic patterns of the former type.^{24,41,49,60,82,} As an example, a snapshot of the resulting dynamics on an ellipsoid surface, corresponding to a larger domain, is displayed in Fig. 2(c). The differences in colors correspond to different actin concentrations, representing intracellular forces that drive the formation of membrane protrusions and, thus, mimic the fluctuating directions of motion in the non-polarized mode of motility, see Fig. 3. In the context of cell motility, disordered TWs result in ripples and irregular protrusions along a cell edge, generating mechanical forces in various directions and, thus, leading to disordered motility behavior.

Wall collisions of fan-shaped cells—Inspired by our theoretical findings, we sought a simple experimental example of coexisting modes of migration, and how cells switch between them. Here, we show that directed and irregular migratory modes of D. discoideum before and after a wall collision provide such an example. We recall that a wall collision serves as a strong perturbation that could trigger transitions between motility modes.

We recorded the migration of D. discoideum cells in PDMS-based microfluidic chambers of different geometry. For our imaging experiments, we used a non-axenic knockout cell line (DdB wildtype background) that exhibited increased small GTPase activity,⁸³ resulting in the abundant formation of actin waves and an increased ratio of fan-shaped cells.^6,,55 The cell line furthermore expressed green and red fluorescent fusion proteins (LifeAct-GFP and PH $CRAC$-mCherry) allowing for fluorescence imaging of actin and PIP $3$ dynamics, respectively, for details see supplementary material.

Our recordings showed repeated interactions of migrating cells with the sidewalls of the microfluidic chamber. While in most cases cells maintained their mode of migration, we also observed instances, where, upon hitting the boundary, a fan shaped cell lost its polarity and switched to a disordered mode of locomotion based on small and dynamically changing pseudopodia. An example is displayed in Fig. 3, where a fan-shaped cell approaches a corner in the PDMS sidewall of the chamber from the bottom right. The transition in the migratory mode upon collision with the sidewall is reflected in a change of cell shape from the stable elongated fan to an irregular morphology with small protrusions around the cell border. A color coded temporal sequence of cell contours before, during, and after the collision can be seen in Fig. 3(a), where examples of a polarized cell before the collision and a non-polarized cell after the collision are highlighted as bold contours. They correspond to the fluorescence images displayed in the two panels in Fig. 3(b). The loss of polarity upon collision with the sidewall is also illustrated in a kymograph representation of the local motion of the cell border, shown in Fig. 3(c). During the fan-shaped motion, a stable protruding cell front and a retracting back (0 to $π$ and $π$ to $2π$⁠, respectively) can be seen in the kymograph until the collision at around $t=400$ s. After the collision, polarity is lost and protrusive and retractive activities are distributed all around the cell border.

Discussion—We uncover a mechanism for the transition from polarized to oscillatory dynamics, following the analysis of a mass-conserving reaction–diffusion system that agrees with experimental recordings of collisions of D. discoideum cells with solid boundaries (Fig. 3). In this mechanism, mass conservation is a key component, as similar behavior cannot arise in reaction–diffusion models without this feature (Fig. 1). While polarized F-actin distributions are associated with fan-shaped motility, the oscillatory states represent F-actin dynamics that result in non-polarized disordered motility (Fig. 2). Moreover, our theory is also consistent with the so-called “crawling” and “ruffling,” corresponding to TWs that move along the edge of adherent cells.^37,38,84 Taken together, our results suggest that GTPase signaling coupled to F-actin feedback^41,85,86 may account for coexisting migratory modes in eukaryotic cells, an essential prerequisite to understanding and potentially controlling their motility, paving the way toward novel functionalities and applications.