Single-cell chiral symmetry breaking under confinement

Abstract

Single cells confined by the extracellular matrix can exhibit persistent rotational motion, yet the physical mechanisms underlying this chiral symmetry breaking remain unclear. Here, we address this gap with a cellular phase field model that couples cell deformation, cell polarization governed by stochastic excitable dynamics, and confinement. We identify the confinement strength as a bifurcation parameter determining three regimes: strong confinement prevents rotation through spatial constraints, intermediate confinement induces stochastic transitions between chiral and non-chiral states, and weak confinement allows persistent rotational motion. For the intermediate regime, we develop a semi-Markovian renewal process framework that characterizes the stochastic dynamics through dwell time statistics, transition probabilities and first-passage times. For the weak confinement regime, we reveal that a mechanochemical feedback enables coherent rotations despite internal noise through the reduction of local excitability mediated by mechanical contraction. We formalize this feedback analytically using Kramers escape theory. Experiments on epithelial MCF10A cells in Matrigel validate predictions for the weak confinement regime. Our results establish a theoretical approach for understanding single-cell chiral symmetry breaking under confinement, with implications for controlling single-cell dynamics by tuning extracellular matrix properties.

I. INTRODUCTION

Chiral symmetry breaking, the spontaneous establishment of a preferred handedness, is a robust emergent phenomenon in macroscopic active systems. Its diverse dynamical consequences have been reported in fish schools [1], ant mills [2], Dictyostelium discoideum cell clusters [3], starfish embryos [4], pancreatic spheres [5], mosh pits [6], and in confined multicellular structures [7–9]. In these systems, chirality typically emerges from interactions among many constituents, but also from their respective interactions with the surroundings. In the case of cells in physiologically relevant microenvironments, e.g. the extracellular matrix (ECM), cell-ECM interactions start to play a fundamental role (see [10] and references therein). The ECM locally confines cells, reducing the available space for protrusion growth and migration. This confinement has the ability of forcing multiple cells to self-organize into chiral multi-cellular aggregates [7, 9], and this phenomenon is observed in other types of biological confinement as well, including micropatterned substrates [11–13].

Chirality can also arise at the level of a single constituent, when that unit is an out-of-equilibrium system. Eukaryotic cells provide an excellent example, because they exhibit a wide and diverse range of subcellular processes, including membrane deformations and membrane-bound biochemical signaling, that break temporal and spatial symmetries [14–16]. Two relevant processes in single-cell chirality are cell polarization [17, 18] and protrusion creation triggered by F-actin polymerization [19–21]. In both, the appearance of well-defined temporal and spatial scales is governed by excitable wave-like dynamics of membrane-bound proteins at the cell cortex [22–27], which are maintained despite intrinsic noise [25, 28, 29]. Experiments have shown that transient rotational protrusions in single cells arise before eventually transitioning to a fully polarized mesenchymal mode of migration on 2D substrates [30, 31]. Moreover, experiments have demonstrated the possibility of maintaining a coherently rotating cell (MCF10A) in Matrigel [7], yet the physical mechanisms that originate and sustain such behavior have not been fully addressed. Intriguingly, single-cell rotation appears to be variable even within the same cell type under similar confinement conditions: while some MCF10A cells rotate [7], others do not [32]. This variability extends across cell types, as a recent study has reported the absence of rotational motion in single MDCK cells in Matrigel [33], suggesting that the phenomenon may involve intrinsic cellular dynamics with both deterministic and stochastic components.

A simple theoretical framework that integrates cell-ECM interactions, cell mechanics, and signaling dynamics to understand when and why single-cells under confinement rotate is currently unavailable. This framework needs to address key questions: What are the minimal mechanochemical ingredients governing the transition from a non-chiral to chiral cells? What mechanisms maintain rotational coherence against intrinsic noise?

Here we present a 3D phase field model, which couples cell membrane deformation, a polarization mechanism dictated by stochastic excitable dynamics, and homogeneous ECM confinement that robustly show the spontaneous chiral symmetry breaking of single cells. The confinement does not impose the chirality but rather establishes the necessary conditions to observe the spontaneous emergence of cell rotations. We find that the persistence of chiral cells depend on the level of confinement: under strong confinement, cell dynamics is governed by disorganized short-lived protrusions and the cell remains static, while under weak confinement, the cell is able to establish coherent rotations associated with long-lived protrusions. The model also predicts an intermediate stochastic regime, governed by intermittent events of rotational coherence.

To describe the stochastic regime, we coarse-grain a two-dimensional representation of the excitable dynamics in our model into a discrete set of states and quantify the dynamics via mean dwell times, transition probabilities, and mean first passage times. We show in this discrete representation that transitions between non-chiral and chiral cells can be captured by a semi-Markov renewal process, providing a compact data-driven approach to analyze stochastic switching in excitable cellular dynamics. For the weak confinement regime, we identify a mechanochemical feedback mechanism: strong contractility induced by large protrusions reduces cortical excitability and, thus, suppresses the noise-triggered nucleation of new protrusions. We formalize this mechanism analytically based on Kramers escape theory: the persistent chiral motion can be understood as a state trapped in a potential well with a contractility-driven barrier.

Finally, we conduct multiple experimental realizations of single epithelial MCF10A cells seeded in Matrigel across a range of concentrations (30%−100%). Comparisons with our theory suggest that MCF10A in Matrigel exhibit dynamics consistent with the weak confinement regime. Therefore, our work provides a theoretical framework for understanding single-cell chiral symmetry breaking under confinement, experimental support for the weak confinement regime, and a foundation for future experimental studies exploring the stochastic and strong confinement regimes.

II. RESULTS

A. Rotational dynamics under confinement

The phase field approach is a flexible continuum method that can couple boundary deformations to the spatiotemporal evolution of bulk scalar fields. It has successfully captured cell morphodynamics [34], cell-substrate adhesion [35], cell interactions [12], cell plasticity [20, 36], and collective cell migration [37]. Specifically, we consider a scalar field ${\phi }_c(\mathbf{\text{r}},t)$ to represent the cell position: ${\phi }_c=1$ inside the cell, ${\phi }_c=0$ outside the cell, and ${\phi }_c=1/2$ corresponding to the interface. A second static phase field, ${\phi }_M(\mathbf{\text{r}})$, represents the confining ECM (see Fig. S1A). The cell interacts with the ECM through a repulsive force ${\mathbf{\text{F}}}_r=-{\eta }_{cM}^r{\phi }_M\stackrel{\mathbf{ˆ}}{\mathbf{\text{n}}}$, where ${\eta }_{cM}^r$ is a parameter that quantifies confinement and $\stackrel{\mathbf{ˆ}}{\mathbf{\text{n}}}=-\nabla {\phi }_c/\left|\nabla {\phi }_c\right|$ is the outward unit normal on the cell’s interface. An additional friction force mediates the interaction between the cell and the ECM, ${\mathbf{\text{F}}}_f=-\xi {\phi }_M\mathbf{\text{v}}$, where $\xi$ is a friction coefficient and $\mathbf{\text{v}}$ is the velocity of the cell interface. This friction term phenomenologically encompasses hydrodynamic drag, viscous dissipation from ECM deformation, and dissipation from integrin-based adhesive interactions. Cell size, $V_c$, is maintained by a size-restoring pressure force: ${\mathbf{\text{F}}}_v=-{\eta }_v\left(\int {\phi }_{c}d\mathbf{\text{r}}-V_c\right)\stackrel{\mathbf{ˆ}}{\mathbf{\text{n}}}$, where ${\eta }_v$ controls the strength of this constraint. This force produces nonlocal mechanical coupling: local deformations induce compensating membrane changes throughout the cell. Membrane tension, ${\mathbf{\text{F}}}_t$, is modeled through an interfacial energy with surface surface tension coefficient $\gamma$, which resists membrane deformations (see Supplemental Material and Fig. S1B for details [38]). The cell is driven out-of-equilibrium with a protrusion force ${\mathbf{\text{F}}}_p={\eta }_{p}A\stackrel{\mathbf{ˆ}}{\mathbf{\text{n}}}$ [12, 34], parameterized by ${\eta }_p$, where the activator $A$ is the mean-field representation of a membrane-bound protein that controls protrusion growth. The activator is coupled to the evolution of a slow membrane-bound inhibitor $R$, and the two-variable system obeys stochastic excitable dynamics:

$$\begin{gathered} {\partial }_t\left(A{\Omega }_c\right)={\Omega }_c\left(𝒜(A,R)+\sqrt{\Gamma }{\gamma }_1(\mathbf{\text{r}},t)\right)+\nabla \left(D_A{\Omega }_c\nabla A\right) \\ {\partial }_t\left(R{\Omega }_c\right)={\Omega }_cℛ\left(A,R\right)+\nabla \left(D_R{\Omega }_c\nabla R\right). \end{gathered}$$ (1)

The functionals $𝒜$ and $ℛ$ include commonly used positive and negative feedbacks (detailed in the Supplemental Material [38]) between the activator and inhibitor species [20, 25, 36]. The excitable dynamics follows from a single intersection in the phase space, defined by the homogeneous solution $\left\{A_o,R_o\right\}$, between the linear and cubic nullcline of the activator-inhibitor system (Fig. S1C). The functional ${\Omega }_c={\Omega }_c\left({\phi }_c\right)$ is an indicator of the cell position (see Supplemental Material for details [38]), and it couples cell membrane deformations to the signaling dynamics. Both species diffuse with diffusivities $D_A$ and $D_R$, respectively. Stochasticity is introduced as Gaussian noise ${\gamma }_1(\mathbf{\text{r}},t)$ with amplitude $\Gamma$, a common approximation for systems where fluctuations emerge from many independent subcellular processes [25, 28, 29, 39, 40]. Cell motion is governed by the advection equation

$${\partial }_t{\phi }_c=-\mathbf{\text{v}}\cdot \overrightarrow{\nabla }{\phi }_c,$$ (2)

where the velocity is obtained from a force balance at the cell interface that includes all forces described above and is evaluated in an overdamped limit. Consistent with this approximation, we assume that translational cell dynamics being much slower than cell membrane relaxation [41].

We numerically integrate the phase-field model (Eq. 2) from a spherical initial condition corresponding to the resting state $\left\{A_s,R_s\right\}$ (parameters in Table S1; numerical details in the Supplemental Material [38]). Varying the confinement parameter ${\eta }_r^{cM}$ produces three qualitatively distinct single-cell dynamical regimes (Videos S1–S3). Parameter values are presented unitless in the main text for readability; the associated physical units are provided in Table S1 (Supplemental Material [38]). Under strong confinement $\left({\eta }_{cM}^r\gtrsim 7\right)$, the cell is jammed with an approximately spherical shape, a noisy activator background, and a centroid that remains stationary on average (Fig. 1Ai–ii). To visualize activator dynamics, we analyze its behavior on an arbitrary plane $\left\{p_1^{\prime },p_2^{\prime }\right\}$, which passes through the centroid. This shows that short-lived and small-amplitude protrusions appear, but are not able to produce organized dynamics (Video S4 and Fig. 1Aiii).

FIG. 1.

Chiral symmetry breaking under confinement in the 3D phase field model. (A) Cell under high confinement conditions $\left({\eta }_{cM}^r=32\right)$, (B) cell in the stochastic regime with ${\eta }_{cM}^r=5.3$, and (C) deterministic rotating cell $\left({\eta }_{cM}^r=4\right)$. For all the dynamical regimes, the panels show: (i) cell shape $\left({\phi }_c=1/2\right)$ and normalized activator field $\bar{A}=A/\text{max}(A)$, (ii) trajectories of the centroid of the cells over $60T$, and (iii) activator distributions on planes $\left\{p_1^{\prime },p_2^{\prime }\right\}$ (A and B) or $\left\{p_1,p_2\right\}$ (C). The red (*) symbols in (Aiii) indicate short-lived protrusions. In (Biii) and (Ciii), the blue arrows highlight the coherent rotational motion and the blue triangles indicate the position of the rotating protrusions. The trajectories in (Biii) illustrate the connection between cell centroid motion and cell shape deformation during the loss of rotational coherence. The time variables $T$ (ii-iii) and $t_o$ (iii) correspond to the period of the rotational motion at ${\eta }_{cM}^r=3$ and an arbitrary initial time, respectively.

At intermediate confinement $\left(5\lesssim {\eta }_{cM}^r\lesssim 7\right)$, the single-cell dynamics becomes complex: sufficiently large protrusions form and trigger rich spatiotemporal activator patterns, including spiral waves (Video S2). Fig. 1Bi illustrates the case ${\eta }_{cM}^r=5.3$, where a transient cell shape and activator field are depicted. The activator shows a spiral wave fragment (left) and the early stage of a protrusion (right). The protrusion, correlated with a local accumulation of the activator, interacts with the ECM and is propagated as a target wave along the surface. Stochastic fluctuations or heterogeneities in refractoriness can break this pulse (see Figs. S2 and S3 in the Supplemental Material [38]), leading to the formation of spiral waves. The resulting centroid trajectory contains localized periods of coherent rotations (shaded red) with the motion governed by two spiral wave fragments synchronously rotating along the cell surface (Video S2 and Fig. 1Bii). The rotational coherence can be lost after some time (shaded pink). Choosing a visualization plane that encompasses the centroid positions during the coherent rotation reveals a single “pulse” that circulates along the periphery until its disappearance (Video S5 and Fig. 1Biii). Eventually, the centroid trajectory exhibits coherent rotations again, but on a different plane (Videos S2 and S5).

For weak confinement $\left({\eta }_{cM}^r\lesssim 5\right)$, the cell enters a permanently rotating state after an initial transient (Fig.1C and Video S3). The shape adopts a symmetric two-lobe geometry correlated with a frustrated “figure of eight” activator pattern (Fig.1Ci and Fig. S4A in the Supplemental Material [38]). The centroid trajectory is almost fully bounded to the symmetry plane defined by the two-lobe geometry, $\left\{p_1,p_2\right\}$, where a permanently rotating pulse is observed (Video S6 and Fig. 1Cii–iii). Since the rotational period of the pulse is established by the activator-inhibitor dynamics (Eq. 1), we can define a unique time scale, chosen here to be the period of the rotating state for ${\eta }_{cM}^r=3:T$. We note that this rotational period can be redefined by rescaling time and the right-hand side in Eqs. (1) and (2). Additional centroid trajectories in the confinement range $2\le {\eta }_{cM}^r\le 32$ are shown in Fig. S5 (Supplemental Material [38]).

Finally, for confinement values below ${\eta }_{cM}^r\approx 0.3$, a single protrusion quickly polarizes the cell, inducing a permanent unidirectional motion rather than rotational dynamics. This unconfined regime is characterized by effective cell penetration into the ECM, where confinement no longer governs the dynamics, and thus, we restrict our study to ${\eta }_{cM}^r\gtrsim 0.3$. Together, these results demonstrate that signatures of rotation are observed within a finite confinement window and the maintenance of such dynamics is stochastic or purely deterministic depending on the value of ${\eta }_{cM}^r$.

B. Discrete state representation captures stochastic transitions

To systematically investigate the emergence and maintenance of rotating dynamics, and motivated by the simpler picture of pulse-dynamics in a plane (Fig. 1), we next reduce our 3D model to a 2D phase field model (see Section V in the Supplemental Material for details [38]). Importantly, in this reduced description, coherent and non-coherent rotations are still observed. We note that while the three confinement regimes persist in 2D, the range of the intermediate regime is larger, $5\lesssim {\eta }_{cM}^r\lesssim 52$, presumably because rotational waves nucleate more readily in the 2D case, where waves propagate along a 1D boundary rather than a 2D surface. Our simulations reveal ten possible 2D activator patterns across confinements, $𝒮={\left\{{𝒮}_i\right\}}_{i=1,..,10}$, shown in Fig. 2A (see also Video S7). We take advantage of the small discrete number of possible states, and decide to study the continuous 2D dynamics from a simpler, discrete perspective. This discrete description allows us to use established concepts in stochastic processes and statistical physics (see below), captures the switch between chiral and non-chiral behaviors while removing irrelevant microscopic details, and places confined cell dynamics in a broader theoretical framework. In this framework, the dynamics is governed by quasi-deterministic and stochastic transitions between the states, and their respective waiting times before transitioning (dwell times) ${\tilde{\tau }}_D$, where the superscript indicates normalization by the rotational period $T$.

FIG. 2.

Emergence and maintenance of rotating states in $𝒮$-space. (A) Normalized activator fields from 2D simulations of the phase field model illustrating the ten states ${𝒮}_i$ considered in the analysis. The triangles indicate rotating protrusions while the half ring illustrate non-rotating protrusions. (B) Example of a trajectory of the low-dimensional model highlighting the birth and death of a rotating state: ${𝒮}_{\mathit{\text{rot}}}$. The time has been normalized by the period of the rotating state: $\tilde{t}=t/T$. (C) Survival probabilities on a semi-log scale for the resting state $\left({𝒮}_1\right)$, rotating state $\left({𝒮}_{\mathit{\text{rot}}}\right)$, two chiral pulse state $\left({𝒮}_6\right)$ and three chiral pulse state $\left({𝒮}_7\right)$ for different degrees of confinement. The dashed lines indicate exponential decay for the corresponding cases. (D) Mean dwell times of the discrete states for different confinement values. The horizontal dashed line highlights one rotation period, and the vertical dashed line is ${\eta }_{cM}^r=5$. Bootstrap standard errors (500 trajectory resamples) of $\mathbb{E}\left[{\tilde{\tau }}_D\right]$ are indicated by solid red lines. (E) Transition probabilities for two different confinement values. The transitions involving ${𝒮}_{\mathit{\text{rot}}}$ are dashed. (F) Mean first passage time from the resting to the rotating state and vice-versa, computed using the 2D phase field model (squares) and the semi-Markov renewal process (triangles) as a function of ${\eta }_{cM}^r$. The orange whiskers correspond to the ${10}^{\mathit{\text{th}}}$ and ${90}^{\mathit{\text{th}}}$ percentiles of the first-passage time distributions. The orange (x) symbols indicates the lower bound (maximum first passage time observed) when a ${90}^{\mathit{\text{th}}}$ cannot be estimated (Supplemental Material [38]). Bootstrap standard errors (500 trajectory resamples) of both $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)\right]$ and $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_1\right)\right]$ are indicated by solid red lines.

The states are defined as follows: ${𝒮}_1$ is the resting fixed point of the excitable system (Fig. S1C), ${𝒮}_2$ is a non-chiral protrusion, and ${𝒮}_3$ (or ${𝒮}_{\mathit{\text{rot}}}$) is a rotational protrusion of either handedness, corresponding to the coherent rotational motion in the system. The states ${𝒮}_4,{𝒮}_5,{𝒮}_6$ are two-protrusion configurations: one chiral and one non-chiral, two non-chiral, and two chiral of opposite handedness, respectively. The last four, ${𝒮}_7-{𝒮}_{10}$, are three-protrusion states: (i) three chiral protrusions, with two sharing the same handedness; (ii) two chiral protrusions of opposite handedness plus one non-chiral; (iii) two non-chiral and one chiral; and (iv) three non-chiral, respectively.

The numerical procedure to coarse-grain the 2D data into $𝒮$-space is based on counting activator pulses on the cell periphery, assigning an angular displacement to distinguish between chiral and non-chiral pulses, and defining a time scale separation to account for extreme events of short duration (Figs. S6, S7, and S8 in the Supplemental Material [38]). A key simplification involves collapsing the states $\left\{{𝒮}_8,{𝒮}_9,{𝒮}_{10}\right\}$ into a single degenerated state, defined as ${𝒮}_{deg}$ (or the new ${𝒮}_8$). This is a remedy for the otherwise difficult task of introducing a time scale separation for these three states, since their durations are governed by the same underlying mechanism, a single, non-chiral pulse dynamics. Fig. 2B shows a sample trajectory in $𝒮$-space for ${\eta }_{cM}^r=7$, highlighting the creation $\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)$ and destruction $\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_1\right)$ of a coherent rotation (see exemplary trajectories for other confinements in Fig. S7 in the Supplemental Material [38]). Additionally, this trajectory exemplifies some of the quasi-deterministic transitions: division of a non-chiral protrusion into two counter-propagating pulses $({𝒮}_2\to {𝒮}_6)$, and merging of the two counter-propagating pulses into a non-chiral pulse and its posterior disappearance $\left({𝒮}_6\to {𝒮}_2\to {𝒮}_1\right)$.

We exploit the discrete-state description to extract robust statistics (Fig. S9 in the Supplemental Material [38]) and to quantify the dynamics of each state across different confinements, using trajectories of duration (13T). While this duration captures multiple transitions when sufficiently far from the critical confinement $\left({\eta }_{cM}^r\approx 5\right)$, close to it, trajectories can become right-censored. We consider these censored trajectories in our analysis (see Supplemental Material [38]), as they provide valuable information near the critical confinement ${\eta }_{cM}^r\approx 5$, where the dynamics exhibit longer episodes of rotation. A useful statistical measure to distinguish between different states and confinement regimes is the survival probability of a state, $\mathit{\text{Pr}}\left\{{\tilde{\tau }}_D>\tilde{t}\right\}$ (Fig. 2C). In the weak and intermediate confinement regimes, the survival curves show that the system spends most of the time in the rotating state, while visits to other states are shorter than one rotation (for survival curves of additional states, see Fig. S10 in the Supplemental Material [38]). In the regime of permanent rotations (${\eta }_{cM}^r=2$ in Fig. 2C), no survival curve is computed for state ${𝒮}_{\mathit{\text{rot}}}$ as all the trajectories are right-censored, i.e., once the system enters ${𝒮}_{\mathit{\text{rot}}}$, it never leaves within the trajectory duration. The survival curves for the resting and the two-chiral pulse state exhibit quasi-deterministic waiting times (plateaus) before transitioning. The plateau in ${𝒮}_1$ persists for all confinements values, pointing to a deterministic time scale imposed by the excitable activator-inhibitor dynamics, the refractory period. The plateau of around half a rotation in ${𝒮}_6$ corresponds to the creation and annihilation of the two pulses with opposite handedness, i.e., the time spent in ${𝒮}_6$ within the passage ${𝒮}_1\to {𝒮}_6\to {𝒮}_1$, which represents the low-dimensional version of target wave propagation in 3D. Noisy activations increasingly perturb this deterministic motion as confinement increases, and the plateaus are replaced by decaying segments (Fig. S10 [38]). The survival curves for the state ${𝒮}_4$ display short plateaus when ${\eta }_{cM}^r\le 7$ while for ${𝒮}_5$ there is only a short plateau when ${\eta }_{cM}^r=7$ (Fig. S10 [38]). This is likely associated with the timescale of appearance and disappearance of a non-chiral pulse when another pulse is present.

The decay of the survival curves give further insights into the processes underlying the stochastic dynamics. Most states have a complex decay law, which do not follow a single exponential (see also Fig. S10), indicating the presence of multiple timescales rather than simple memory-less dynamics [42]. However, state ${𝒮}_6$, after its initial plateau, and state ${𝒮}_{\mathit{\text{rot}}}$ display a clear exponential decay across confinements, indicative of memory-less behavior. As confinement weakens toward ${\eta }_{cM}\simeq 5$, the survival curve of the rotating state exhibits a pronounced slowdown. At ${\eta }_{cM}=7$ and ${\eta }_{cM}=12$, the survival probability has not decayed to zero within the observation window (Fig. 2C, $S_{\text{rot}}$), corresponding to a considerable fraction of right-censored trajectories (59% and 18%; Fig. S9, respectively).

We quantify state-dependent persistence by estimating the mean dwell time $\mathbb{E}\left[{\tilde{\tau }}_D\right]$ for all states in confinement conditions where it is well-defined (Fig. 2D and Section VI in the Supplemental Material [38]). Notice that because some trajectories are right-censored, $\mathbb{E}\left[{\tilde{\tau }}_D\right]$ is a lower bound on the true mean. Nevertheless, this measurement clearly illustrates that the residence time in the rotating state increases as confinement is relaxed, and this motion becomes permanent when ${\eta }_{cM}^r\lesssim 5$ (no exit from ${𝒮}_{\mathit{\text{rot}}}$). We have verified that the persistence in ${𝒮}_{\mathit{\text{rot}}}$ is robust against variations in the amplitude of the noise (Fig. S11 in the Supplemental Material [38]).

The dwell time statistics can be complemented with the calculation of the transition probabilities $P\left({𝒮}_j\mid {𝒮}_i\right)$ between states (details in Section VI in the Supplemental Material [38]). The transition probability graphs in Fig. 2E illustrate several characteristics of the pulse-dynamics. First, not all states are directly connected to each other due to the underlying deterministic dynamics. For example, the transitions ${𝒮}_2\leftrightarrow {𝒮}_{\mathit{\text{rot}}}$ and ${𝒮}_4\leftrightarrow {𝒮}_6$ are forbidden because of the topological reciprocal transformation between a non-chiral pulse and two pulses of opposite chirality. Second, some direct transitions are suppressed while others become more likely when the confinement is weakened (see also Fig. S12 in the Supplemental Material [38]). Specifically, when the confinement decreases from ${\eta }_{cM}^r=32$ to ${\eta }_{cM}^r=7$, the direct passages ${𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}$ and ${𝒮}_{\mathit{\text{deg}}}\to {𝒮}_{\mathit{\text{rot}}}$ disappear, and the quasi-deterministic transition probabilities ${𝒮}_6\to {𝒮}_2,{𝒮}_4\to {𝒮}_3$ become larger.

The transition probabilities and dwell time statistics characterize only the local behavior of the discrete dynamics. To have a global picture of the process, we study first-passage times $\tilde{\tau }\left({𝒮}_i\to {𝒮}_j\right)$, from a starting state ${𝒮}_i$ to a target state ${𝒮}_j$, focusing on the constructive and destructive pathways that drive the resting state to rotate $\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)$ and return to the resting state $\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_1\right)$. The left panel of Fig. 2F shows the mean first-passage time $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)\right]$ of creation, which is shorter than four rotation periods for all confinement parameters and follows a non-monotonic trend. It also reveals that, in the weak confinement regime $({\eta }_{cM}^r=2)$, the establishment of permanent rotations is relatively fast. The right panel displays the destructive mean first-passage time $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_1\right)\right]$, exhibiting an increasing trend as confinement weakens. Notice that for confinements ${\eta }_{cM}^r\ge 17,\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_1\right)\right]$ is approximately four times larger than the corresponding mean dwell time in ${𝒮}_{\mathit{\text{rot}}}$ (Fig. 2D). This observation, combined with short mean dwell times in states ${𝒮}_{i\ne rot}$, suggests that the destruction path is on average an ensemble of multiple visits to ${𝒮}_{\mathit{\text{rot}}}$. This behavior is determined by $P\left({𝒮}_4\mid {𝒮}_{\mathit{\text{rot}}}\right)$, the only possible exit from ${𝒮}_{\mathit{\text{rot}}}$ (aside from a small transition probability to ${𝒮}_1$ at ${\eta }_{cM}^r=32$), which then triggers the quasi-deterministic sequence: ${𝒮}_{\mathit{\text{rot}}}\to {𝒮}_4\to {𝒮}_7\to {𝒮}_4\to {𝒮}_{\mathit{\text{rot}}}$. This sequence becomes increasingly deterministic as confinement weakens, and when ${\eta }_{cM}^r\lesssim 17$, the right-censoring in the distribution of the first passage times $\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_1\right)$ is larger than 77% (Fig. S13 in the Supplemental Material [38]). In these cases, indicated by shaded regions in Fig. 2F and Fig. S14 in the Supplemental Material [38], we are not able to compute the median (${50}^{\mathit{\text{th}}}$ percentile) of the distribution and we do not calculate the corresponding mean first passage times.

To test whether computable mean first-passage times can be inferred from the local dynamics, we adopt a data-driven semi-Markov renewal process (see Section VII in the Supplemental Material [38]). This framework accounts for arbitrary, state-dependent dwell time distributions, as several distributions deviate from exponential behavior (Fig. 2C and Fig. S10 [38]), and is characterized by the following modeled mean first passage times (renewal equation) [43]:

$$\mathbb{E}\left[\tilde{\tau }\left({𝒮}_i\to {𝒮}_j\right)\right]=\mathbb{E}{\left[{\tilde{\tau }}_D\right]}_i+\sum _{k\ne j}P\left({𝒮}_k\mid {𝒮}_i\right)\mathbb{E}\left[\tilde{\tau }\left({𝒮}_k\to {𝒮}_j\right)\right],$$ (3)

where $\mathbb{E}{\left[{\tilde{\tau }}_D\right]}_i$ indicates the mean dwell time in state ${𝒮}_i$. Using Eq. (3), we successfully capture the constructive and destructive paths, as well as the other mean first-passage times from ${𝒮}_{\mathit{\text{rot}}}$ (Fig. S14 in the Supplemental Material [38]) across almost all confinement values, despite right-censoring in the distribution. The differences between the empirical and modeled mean first-passage times are smaller than $2T$, except for the cases $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_2\right)\right],\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_7\right)\right]$ and $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_{\mathit{\text{deg}}}\right)\right]$ at ${\eta }_{cM}^r=12$, where is around $5T$. We note that when the right-censoring in the first passage time distributions is less than 25%, the match between the empirical and modeled expectation values is perfect (e.g. ${𝒮}_{\mathit{\text{rot}}}\to {𝒮}_4$; Fig. S14 in the Supplemental Material [38]).

C. Mechanochemical feedback stabilizes persistent rotation

Three features suggest that the stochastic behavior of the excitable activator-inhibitor dynamics is controlled by confinement: 1) the persistent coherent rotations for sufficiently weak confinements $\left({\eta }_{cM}^r\le 5\right)$, 2) the increase in the probability of quasi-deterministic transitions when weakening the confinement, and 3) the dominance of a single sequence of states close to ${\eta }_{cM}^r=5$. Although stochastic activations can orchestrate the constructive path, they become suppressed once cell rotation is established in weak confinement regimes. This hints to a mechanochemical feedback, which decreases the ability to nucleate new activator pulses (or protrusions) when the rotating pulse is present. To probe this feedback, we focus on the influence of the rotating protrusion on the rest of the cell membrane. Specifically, we concentrate on studying locally the opposite side of the cell, which we term the back (Fig. 3A), and define the variable ${\phi }_b$: a zero-dimensional representation of the cell phase field at the back (Section IX in the Supplemental Material [38]). The protrusion deformation induces a back contraction $\left({\partial }_t{\phi }_b\approx v_n\left|{\partial }_n{\phi }_b\right|<0\right)$ via membrane tension and the size-restoring force, where the subscript $n$ indicates the normal direction at the interface. Thus, larger protrusions, allowed under weak confinement and characterized by a broader activator pattern—quantified by $\int {\phi }_{c}Ad\mathbf{\text{r}}/\int {\phi }_{c}d\mathbf{\text{r}}$—trigger stronger back contractions (Fig. 3A). Importantly, these contractions in the weak confinement regime are strong enough to rotate the entire cell and not just the activator pulse (Fig. S15 in the Supplemental Material [38]). The strength of the contractile response can be tuned by the nonlocal mechanical force: reducing ${\eta }_v$ decreases the back contraction magnitude across confinements (Fig. S16 in the Supplemental Material [38]). In addition, reductions in ${\eta }_v$ affect the coherent rotational motion of the cell, as reflected by shorter mean dwell times in ${𝒮}_{\mathit{\text{rot}}}$, even when the confinement is weak (Fig. 3B). These observations demonstrate that strong nonlocal mechanical contraction is essential for persistent rotation.

FIG. 3.

Back contraction regulates rotational persistence via mechanochemical feedback. (A) Confinement dependence of the mechanical feedback term ${\partial }_t{\phi }_b$, calculated at the back of the cell in noiseless conditions. The insets show some of the spatial patterns of the normalized activator $\bar{A}$, and a plot illustrating protrusion enlargement as confinement is weakened. (B) Phase diagram in ${\eta }_{cM}^r-{\bar{\eta }}_v$ space showing the mean dwell time in the rotating state. For display purposes we have introduced a logarithmic scale, ${\bar{\eta }}_v=\mathit{\text{log}}\left({\eta }_v\right)+d$ with $d=4.6$. The blue triangles indicate measurable mean dwell times at ${\eta }_{cM}^r=2$ (exit from ${𝒮}_{\mathit{\text{rot}}}$ is possible). (C) Free energy $𝒰$ in the quasi-steady approximation for two different confinements. The arrows represent the potential barrier $\Delta 𝒰$ and the potential width $\Delta A$. The circles are the relevant minima and maxima of $𝒰;A_b^s$ and $A_b^u$, respectively. (D) Variations of the potential width (left panel) and of the potential barrier (right panel) as a function of the confinement. (E) Growth of the mean escape time in the quasi-steady approximation as confinement is weakened. The algebraic fit is ${\widetilde{𝒯}}_o-a{\left({\eta }_{cM}^r-{\hat{\eta }}_{cM}^r\right)}^c$ with fitting parameters $a=0.001$ and $c=2.93$. At ${\hat{\eta }}_{cM}^r=2.42$ the crossing at $A_b^u$ ceases to exist, and ${\widetilde{𝒯}}_o=0.07$ is the normalized mean escape time at at a baseline confinement ${\eta }_{cM}^r=32$. (F) Divergence of $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_4\right)\right]$ in the 2D model, close to the critical confinement ${\eta }_{cM}^r\approx 5$. The algebraic fit is $\mathbb{E}{\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_4\right)\right]}_o-a{\left({\eta }_{cM}^r-{\bar{\eta }}_{cM}^r\right)}^c$ with fitting parameters $a=20.93$ and $c=1.25$. At ${\bar{\eta }}_{cM}^r=4.8$, no exits from ${𝒮}_{\mathit{\text{rot}}}$ are observed after $100T$. The baseline $\mathbb{E}{\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_4\right)\right]}_o=0.24$ is measured at ${\eta }_{cM}^r=32$. The orange whiskers correspond to the ${10}^{th}$ and ${90}^{th}$ percentiles of the first-passage time distributions. The orange (x) symbols indicates the lower bound (maximum first passage time observed) when a ${90}^{\mathit{\text{th}}}$ cannot be estimated (Supplemental Material [38]).

For this back contraction to enhance coherent rotations, it must modulate the activator-inhibitor dynamics. A back contraction of sufficiently large magnitude affects the local activator-inhibitor $\left\{A_b,R_b\right\}$ excitable dynamics through the coupling with the phase field dynamics (Eq. 1), and thus with cell mechanical deformations. Specifically, the local dynamics of the activator and the inhibitor is modified by the linear terms $-2{\partial }_t{\phi }_b{A}_b$ and $-2{\partial }_t{\phi }_b{R}_b$ (Section IX in the Supplemental Material [38]), reducing the stochastic excitable dynamics at the back:

$$\begin{gathered} {\partial }_t{A}_b\approx 𝒜\left(A_b,R_b\right)+\sqrt{\Gamma }{\gamma }_1(\mathbf{\text{r}},t)\delta \left(\mathbf{\text{r}}-{\mathbf{\text{r}}}_b\right)-2{\partial }_t{\phi }_b{A}_b \\ {\partial }_t{R}_b\approx ℛ\left(A_b,R_b\right)-2{\partial }_t{\phi }_b{R}_b, \end{gathered}$$ (4)

where ${\mathbf{\text{r}}}_b$ indicates the position of the back. The mechanical corrections modify the local excitable dynamics by increasing the distance between the fixed point (or resting state) $\left\{A_b^s,R_b^s\right\}$ and the minimum of the cubic nullcline, raising the effective excitability threshold (Fig. S17 in the Supplemental Material [38]). This increase of the threshold limits the activation of new pulses, i.e., $P\left({𝒮}_4\mid {𝒮}_{\mathit{\text{rot}}}\right)\to 0$, trapping the system in a persistent rotating state. Below a critical confinement, ${\eta }_{cM}^r\approx 2.3$, the fixed point at the back disappears and the zero-dimensional picture of the back of the cell being influenced by an isolated rotating pulse is not valid anymore. The slight confinement difference between the disappearance of the fixed point (at the cell back) and the emergence of the persistent coherent rotation $\left({\eta }_{cM}^r\approx 5\right)$ is likely due to finite-size nucleation effects in 2D or 3D geometry.

To formally address the excitability reduction and the transition to persistent coherent rotations, we adopt a quasi-steady approximation of the back activator-inhibitor dynamics: The slow inhibitor is frozen to its fixed point value $R_b=R_b^s$, and the fast-scale dynamics for $A_b$ drives the system. In this approximation, the back activator-inhibitor system can be written as an overdamped Langevin equation for $A_b$ under a potential $𝒰\left(A_b,{\eta }_{cM}^r\right)$ (Sextion X in the Supplemental Material [38] and Fig. 3C):

$$d{A}_b=-\frac{d𝒰\left(A_b,{\eta }_{cM}^r\right)}{d{A}_b}dt+\sqrt{\Gamma }d{W}_t,$$ (5)

where $W_t$ is a Wiener process [44] and the confinement dependence of the potential comes from the back contraction $\left|{\partial }_t{\phi }_b\right|$. As confinement weakens, both the potential well width $\Delta A=A_b^u-A_b^s$ (with $A_b^u$ denoting the crossing of $R=R_b^s$ with the unstable branch of the nullcline; Fig. S1C) and the barrier height $\Delta 𝒰$ increase (Fig. 3D). The growth of $\Delta A$ indicates that the distance from the fixed point to the excitability threshold becomes larger, i.e., the cell becomes less excitable. From a stochastic perspective, the increase of $\Delta 𝒰$ maps to the overdamped Kramers escape problem [45]; a large potential barrier abolishes escapes at finite times.

Transitions out from the rotating state correspond to escapes from the potential well $𝒰\left(A_b,{\eta }_{cM}^r\right)$, allowing nucleation of new protrusions; ${𝒮}_{\mathit{\text{rot}}}\to {𝒮}_4$. Therefore, to bridge the statistical analysis of discrete states with the zero-dimensional approach, we can quantify the stability of the rotating state by estimating the mean escape time from the potential well as a function of confinement, $\widetilde{𝒯}(𝒰)$. Introducing the corresponding Fokker-Planck equation for Eq. (5) and following standard analytical calculations [44], we obtain (Section X in the Supplemental Material [38]):

$$\widetilde{𝒯}(𝒰)=2\pi \frac{e^{2\Delta 𝒰/\Gamma }}{\sqrt{\left.{\left.\frac{d^2𝒰}{d{A}_b^2}\right|}_{A_b^s}{\left|\frac{d^2𝒰}{d{A}_b^2}\right|}_{A_b^u}\right|}}.$$ (6)

This mean escape time diverges algebraically as confinement decreases (Fig. 3E), indicating that the rotating state becomes increasingly stable. The singularity is marginally shifted from ${\eta }_{cM}^r\approx 2.3$, and is related to the extinction of the crossing at the unstable branch $A_b^u$. This mean escape time, valid in the zero-dimensional approach, serves as a mechanistic proxy for the empirical mean first passage time $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_4\right)\right]$, conditioned to the escape being possible. Consistently, $\mathbb{E}\left[\tilde{\tau }\left({𝒮}_{\mathit{\text{rot}}}\to {𝒮}_4\right)\right]$ also shows an algebraic diverging behavior near the transition to the regime of permanent rotation (Fig. 3F).

D. Experiments validate weak confinement regime predictions

Our theory provides a possible resolution to the apparent conflict among experimental observations of single MCF10A cell chirality in Matrigel: sometimes these cells rotate [7], and sometimes they do not [32]. Specifically, our modeling framework suggests that the breaking of chiral symmetry is stochastic and depends on the level of confinement. Therefore, an experiment with sufficient statistics across different confining conditions should reveal the probabilistic behavior of single-cell chiral motion. We test this prediction by seeding multiple MCF10A cells in Matrigel at different concentrations (30%−100%) and tracking their behavior by imaging F-actin (LifeAct-GFP) and the nucleus (H2B-mCherry) over a maximum of 16 h (see Sextion XI in the Supplemental Material [38]), prior to cell division (18–20 h).

In some cells, we observe distinctive F-actin configurations consistent with states ${𝒮}_1,{𝒮}_2,{𝒮}_{\mathit{\text{rot}}},{𝒮}_5$ and ${𝒮}_6$ (Fig. 4A, and Videos S8–S10 in the Supplemental Material [38]). The most frequently observed experimental state is ${𝒮}_1$. A plausible reason for not observing ${𝒮}_7$ and ${𝒮}_8$ is that resolving three F-actin patches requires a cell size that is large relative to the patch size. While individual F-actin patches can be identified manually, systematic quantification of their dynamics, similar to Fig. 2, is challenging due to difficulties in robustly thresholding the images to distinguish states, as well as the limited observation windows imposed by cell division. We therefore adopt a simpler binary classification, labeling each experimental realization as either exhibiting cell rotation or not (see Section XI in the Supplemental Material [38]).

FIG. 4.

Chiral motion of MCF10A cells in Matrigel. (A) Temporal snapshots of F-actin distributions corresponding to distinct $𝒮$-states observed in 70% and 100% Matrigel. (B) Nucleus trajectory of the rotating cell, ${𝒮}_{\mathit{\text{rot}}}$, shown in (A). (C) Nucleus trajectory of a non-rotating cell in 100% Matrigel. (D-E) Absorption probability of the passage from a non-chiral to chiral state $({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}})$ for different Matrigel concentrations in the experiments and for different ${\eta }_{cM}^r$ values within the weak confinement regime in the model, respectively. Bars in (D) indicate standard deviation.

Rotating cells exhibited two key features consistent with the weak confinement regime of our model $\left({\eta }_{cM}^r\lesssim 5\right)$. First, the nucleus, a proxy of the cell centroid, translated during the rotational motion (Fig. 4B), demonstrating that the entire cell moves instead of a protrusion orbiting around the surface of an almost static cell. Second, rotating cells displayed a distinct F-actin patch on their surface (Fig. 4A and Video S9), likely corresponding to the large protrusion that drives rotation in the simulations. In contrast, non-rotating cells showed no rotational motion over the entire window of observation, and in several cases cells drift across the ECM (Fig. 4C and Video S11). This translational motion likely arises from cells pushing and deforming the matrix while being still confined, a mechanism not captured by the static ECM in our model.

The analysis of more than 10 cells at each Matrigel concentration revealed that rotation does not emerge deterministically but rather occurs with a probability that depends on the concentration (Fig. 4D). In our framework, this corresponds to the absorption probability of the constructive path $P_{\mathit{\text{abs}}}\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)$ : the probability of transitioning from the testing state to the rotating state over a finite observation window. We note that at 30% Matrigel cells sink to the bottom, pointing to a minimal confinement to have chances of observing rotations. While the physical mechanisms differ, this breakdown of confinement parallels the unconfined limit in the model $\left({\eta }_{cM}^r\lesssim 0.3\right)$. To compare the stochastic nature of the emergence of rotations between experiments and theory, we compute $P_{\mathit{\text{abs}}}\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)$ in the model across weak confinement values $({\eta }_{cM}^r\lesssim 5)$. The theory predicts that $P_{\mathit{\text{abs}}}\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)=1$ if the observation window accounts for many rotation periods $T$ (see the exemplary trajectories in Fig. S7 for ${\eta }_{cM}^r=2$). However, in the experiment, the observation window is restricted to approximately $2T$, with $T=9.7\pm 1.2\text{h}$. This biological constraint dramatically affects $P_{\mathit{\text{abs}}}\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)$, as cells may not transition to a chiral state before division. Using an observation window of $2T$ to compute $P_{\mathit{\text{abs}}}\left({𝒮}_1\to {𝒮}_{\mathit{\text{rot}}}\right)$, across different weak confinements, yields qualitative agreement with experiments (Fig. 4E). The experimental decay in the probability of observing rotations at the lowest Matrigel concentration where cells are still confined (50%) possibly reflects insufficient confinement, which slows protrusion retraction, and thus the birth of rotational waves.

III. DISCUSSION

A. Spontaneous chiral symmetry breaking

One of the key findings of our work is that single-cell chirality arises spontaneously from an intrinsically non-chiral model. The homogeneous confinement does not impose chirality but rather establishes geometric and mechanical conditions (e.g. protrusion size) where spontaneous symmetry breaking becomes possible. While previous experimental works demonstrated cell chirality in single and multiple cells [7, 33, 46], they addressed this emergent phenomenon as a consequence of an explicit symmetry breaking either inside or outside the cell. There is theoretical work approaching spontaneous chiral symmetry breaking in polarized epithelial cells, but only from a multicellular perspective [47]. Our results suggest that there is no necessity of pre-existing chirality at the molecular level for the rotations to emerge, but that nonlinear coupling between excitable dynamics, confinement and cell deformations at the macroscopic level is sufficient. This positions single-cell chirality within a general physics framework of symmetry-breaking in out-of-equilibrium systems [48, 49].

B. Physical interpretation of cell confinement

In our model, confinement is prescribed by a local isotropic force acting at the cell-ECM interface. This confining effect can be understood through a simplified one-dimensional version of Eq. (2), in which we find that, at first order approximation, the local normal velocity at the cell interface obeys $v_n=\left(B{\eta }_p-C{\eta }_{cM}^r\right)/\xi$, where $B$ and $C$ are positive constants (Section XII in the Supplemental Material [38]). This simple relationship aligns with the findings presented in the text: Strong confinement can result in negative values for this normal velocity and thus suppresses the growth of local deformations. In other words, large values of ${\eta }_{cM}^r$ impose fast protrusion retraction that overcomes protrusion extension. The analogy between the degree of confinement and the protrusion retraction timescale suggests that the confinement modeled in this work can be interpreted as reflecting ECM stiffness [50], where the strong (weak) confinement regime could represent a stiff (soft) environment. This interpretation can be directly connected to the results in Figs. 4D and 4E, where variations in Matrigel concentration, with reduced concentrations corresponding to decreased stiffness [51], are mapped to variations in ${\eta }_{cM}^r$. Reductions in Matrigel concentration have also been linked to a decrease in ECM ligand density, which weakens cell-ECM adhesive interactions [52]. This lowers resistance to cell membrane deformation, effectively reducing cell confinement and suggesting that cell-ECM adhesion provides an additional mechanism for controlling it, which could be tested in an extension of our model.

The notion of confinement may also be extended beyond cell-ECM interactions. For example, a cell surrounded by neighboring cells could be sufficient to establish the necessary confinement for cells to display chiral symmetry breaking. In fact, during Drosophila oogenesis, there is a stage where nurse cells—surrounded by either other nurse cells, the oocyte, or follicle cells—dump cytoplasm into the oocyte while sustaining rotating cortical myosin waves on their surfaces [53].

In our experimental study, we considered epithelial MCF10A cells because they do not remodel the ECM as much as other cells. However, some cells have the ability to significantly remodel the ECM, enabling them to control their own confinement. This is the case for MDA-MB-231 cells seeded in collagen type I, which release matrix metalloproteinases to degrade their surroundings and make room for cells to come after cell division [9]. From the modeling perspective of a single cell, the degradation could be understood as a reduction in effective confinement due to extra free space (Section XII in the Supplemental Material [38]). Beyond such active remodeling, the effective confinement can also change over time through ECM stress relaxation [54]. Future studies could include matrix degradation, stress relaxation dynamics, and exploring how cell-ECM interactions are modified after ECM remodeling [55, 56].

C. Physical interpretation of volume/area conservation and nonlocal mechanics

Our phase field model includes a size-restoring force, conserving the volume (in 3D) or the area (in 2D). Although the motivation of this force relies on the argument of the incompressibility of the cytoplasm, volume control in real cells involves multiple processes such as osmotic pressure, ion pumps, and tension generated by the membrane and the actomyosin cortex [57–59]. The nonlocal character of ${\mathbf{\text{F}}}_v$, where local deformations (protrusions) induce immediate long-range contractions elsewhere, mirrors the stress transmission effect of the actomyosin cortex [60]. Thus, the simple nonlocal force—widely used in phase field modeling—incorporates not only the phenomenology of volume conservation, but also other processes involving cell contraction. As we showed in Fig.3B, small ${\eta }_v$ values do not sustain rotations even for weak confinements. This could be connected to experimental observations where inhibition of actomyosin network regulators (Myosin II, myosin light-chain kinase and Rho-associated kinase) led to loss of rotational coherence [7]. In addition, the relevance of cell contractility for persistent rotational motion (Fig. 3B) may be related to experimental findings in endothelial cell doublets seeded on micropatterns, where the cell with higher contractility imposes the chirality of the doublet [61].

D. Semi-Markov renewal process for excitable cellular dynamics

The semi-Markovian approach we developed for the stochastic regime represents a broadly applicable technique for studying excitable systems with complex spatiotemporal dynamics [62]. This statistical framework is particularly useful for systems where the states of the system have characteristic timescales (refractory periods, slow wave dynamics) and experience noise-driven transitions. The interplay between deterministic and stochastic elements naturally map into semi-Markov dynamics. Our statistical approach could be extended to other biological systems—governed by excitable responses and membrane deformations—exhibiting stochastic switching between dynamical behaviors. For example, F-actin wave dynamics in Dictyostelium discoideum cells on substrates display birth-death dynamics [63], offering a well-known experimental system to extend the coarse-grained technique. We highlight that our method is data-driven so given the enough spatiotemporal resolution it should be easily applied to experimental data.

E. Robustness of chiral motion

The model introduced here requires fixing 19 parameters (see Table S1), which control cell mechanics and wave dynamics within the cell. It is thus important to assess how the parameter choices influence the emergence and maintenance of chiral motion in single-cells.

The parameters governing the local activator-inhibitor dynamics establish either an excitable or an oscillatory regime, depending at which point the linear and cubic nullcline intersect (fixed point). Extensive experimental evidence shows that cortical actin waves in migrating and mechanically constrained cells predominantly exhibit excitable properties, including threshold responses and refractory periods [22–24, 26, 64]. Relevant to our case, excitable waves have also been reported under geometric or adhesive confinement [65, 66], suggesting that excitable dynamics is the prevailing behavior of actin waves across diverse biological contexts. For this reason, we developed our theoretical framework considering the excitable regime of Eq. (1), but additional simulations in the oscillatory regime also show the emergence and maintenance of rotating states (see Fig. S18 and Section XIII in the Supplemental Material [38]).

As illustrated in Fig. S17 [38], the level of excitability depends on the distance $𝒟$ from the fixed point to the minimum of the cubic nullcline. Therefore, to robustly obtain coherent rotations for which surpassing the excitability threshold is mandatory, the parameters controlling the local dynamics in Eq. (1) must satisfy the necessary condition of a sufficiently small $𝒟$ (specifically, $𝒟\lesssim 0.32$), enabling activator nucleations (Fig. S19 in the Supplemental Material [38]).

The possibility of nucleating an activator wave, either in the excitable or in the oscillatory regime, is not only dictated by the local nonlinear dynamics of Eq. (1), but also depends on the length scales of the spatially extended system, which are set by the diffusion coefficients $D_A$ and $D_R$. Larger activator length scales (larger $D_A$), relative to the cell size, make the nucleation of activator waves more difficult, which in turn will decrease the probability of observing rotating waves where multiple nucleations are necessary. Moreover, fluctuations mediate the emergence of rotational motion, and their appearance also depend on the activator length scale being sufficiently small. Therefore, given a cell size—here motivated by MCF10A cells (diameter $~15\mu \text{m}$)-and the parameters in Table S1, the ratio $D_A/D_R$ must be smaller than 0.7 to robustly observe cell rotations (Fig. S20 in the Supplemental Material [38]).

The effects of the parameters directly influencing cell mechanics are easier to grasp compared to the parameters involved in the activator-inhibitor dynamics. For example, if the friction force dominates over all others (large $\xi$) rotational coherence is impaired due to the inability of the cell to produce motion (Fig. S21 in the Supplemental Material [38]). Likewise, decreasing the protrusive force relative to the other forces (small ${\eta }_p$) leads to the loss of rotations due to the inability of the system to generate a persistently polarized protrusion [20, 35].

While our model qualitatively captures the chiral motion of MCF10A cells within Matrigel (Figs. 4D and 4E), systematic optimization of the model parameters discussed in this subsection could potentially improve the quantitative agreement.

IV. CONCLUSIONS

In summary, we demonstrate that confinement by the ECM shapes single-cell chiral dynamics through a rich interplay between mechanics, coarse-grained chemical signaling and stochasticity. Our theoretical framework identifies three dynamical regimes controlled by confinement strength. In the intermediate, or stochastic, regime, a data-driven analysis establishes that a semi-Markov renewal process governs the emergence and maintenance of the rotational motion. Under weak confinement, a mechanochemical feedback stabilizes persistent chiral states. This feedback operates in the form of a nonlocal contraction, activated in the presence of the rotating protrusion, lowering the cortical excitability. Experiments on MCF10A cells in Matrigel validate theoretical predictions for the weak confinement regime. Our work opens avenues for understanding and potentially controlling single-cell dynamics under confined conditions, and suggests immediate follow-ups. In particular, promising directions include new experiments that test the theoretically predicted transition from persistent to stochastic behavior, and investigations into the biological role of single-cell rotation and its relationship to multicellular rotating structures formed after cell division.