Glassy adhesion dynamics govern transitions between sub-diffusive and super-diffusive cancer cell migration on viscoelastic substrates

Domain: physics
Published: 2026-01-13
Editorial outcome: major_revision

Abstract
Cell migration is pivotal in cancer metastasis, where cells navigate the extra-cellular matrix (ECM) and invade distant tissues. While the ECM is viscoelastic, exhibiting time-dependent stress relaxation, its influence on cell migration remains poorly understood. Here, we employ an integrated experimental and modeling approach to investigate filopodial cancer cell migration on viscoelastic substrates and uncover a striking transition from sub-diffusive to super-diffusive behavior driven by the substrate’s viscous relaxation timescale. Conventional motor-clutch based migration models fail to capture these anomalous migration modes, as they overlook the complex adhesion dynamics shaped by broad distribution of adhesion lifetimes. To address this, we develop a glassy motor-clutch model that incorporates the rugged energy landscape of adhesion clusters, where multiple metastable states yield long-tailed adhesion timescales. Our model reveals that migration dynamics are governed by the interplay between cellular and substrate timescales: slow-relaxing substrates prolong trapping, leading to sub-diffusion, while fast-relaxing substrates promote larger steps limiting trapping, leading to super-diffusion. Additionally, we uncover the role of actin polymerization and contractility in modulating adhesion dynamics and driving anomalous migration. These findings establish a mechanistic framework linking substrate viscoelasticity to cell motility, with implications for metastasis and cancer progression.

Introduction
Cancer cell migration is a fundamental process in tumor progression and metastasis, facilitating cancer cells to traverse the extracellular matrix (ECM), invade surrounding tissues, and establish secondary tumors. This multifaceted phenomenon is modulated by the mechanical properties of both the cells and their ECM, which provide critical mechanical and biochemical cues1–3. Among these, the viscoelastic characteristics of the ECM, encompassing elasticity (resistance to deformation) and viscosity (energy dissipation), play a pivotal role in influencing migration phenotypes such as amoeboid, mesenchymal, and filopodial migration4,5. The viscoelastic properties of the ECM regulate cell migration through time-dependent stress relaxation and deformation6,7. These dynamic mechanical properties modulate cellular processes, including spreading, division, and motility3,7–9. While the role of elasticity in migration has been extensively investigated, the impact of viscous dissipation and stress relaxation on migration dynamics remains insufficiently explored. This knowledge gap is particularly relevant for understanding anomalous diffusion behaviors frequently observed in active cell migration.

Our previous work7 demonstrated that substrate viscosity and stress relaxation critically influence cell migration dynamics, underscoring the significant regulatory role of ECM viscoelasticity on cellular motility. Nevertheless, this earlier study primarily focused on the phenomenological correlation between viscosity-induced changes in actin cytoskeletal dynamics and overall migration behavior, without providing mechanistic insights into the underlying diffusion mechanism and link to the adhesion level processes. Specifically, we did not investigate how ECM viscoelasticity modulates migration diffusivity characterized through mean squared displacement (MSD), which scales with time as MSD = \mu t^\alpha, where the exponent \alpha characterizes the diffusivity of cell migration, while \mu represents the diffusion constant. Traditional motor-clutch models, while valuable in studying mechanosensitive responses during migration, typically predict changes only in the diffusion constant (\mu) and consistently yield purely diffusive behavior (\alpha = 1) migration10. However, cell migration is inherently an active process that frequently displays anomalous diffusion (\alpha \neq 1), with purely diffusive migration an exception rather than the norm11,12. Super-diffusive motion (\alpha > 1) is linked to persistent, directed migration, while sub-diffusive motion (\alpha < 1) arises from intermittent trapping and saltatory movement.

To uncover the physical mechanisms governing these anomalous cell migration patterns, we develop a glassy motor-clutch model that accounts for the heterogeneous dissociation kinetics of adhesion proteins. The conformational flexibility and mechanical interactions within adhesion complex proteins (e.g., integrins, talin, vinculin), results in a broad distribution of dissociation timescales3,6–9,13. This produces a spectrum of transient and prolonged dissociation events in an adhesion cluster, mimicking the hallmark feature of glassy systems6,14,15. On viscoelastic substrates, these dissociation dynamics are further modulated by fluctuating mechanical loads and the substrate’s time-dependent stress relaxation. Here, we present a physics-based model that unifies sub-diffusive and super-diffusive migration within a single biophysical framework by linking adhesion glassiness to the underlying integrated glassy adhesion dynamics with substrate viscoelasticity, revealing how the interplay between cellular and matrix timescales governs trapping, step sizes, and the transitions between distinct diffusive regimes. We further validate our model through experiments, demonstrating how substrate stress relaxation modulates migration behavior.

Results
Enhanced substrate relaxation promotes the transition from sub-diffusive to super-diffusive migration modes
We first examined how substrate viscoelasticity influences cell migration diffusivity. Prior studies have used tunable viscoelastic hydrogels to investigate migration dynamics7,16. Here, we employed interpenetrating network (IPN) hydrogels composed of alginate and reconstituted basement membrane (rBM), where alginate controls mechanical properties, and rBM provides adhesion sites. By adjusting alginate molecular weight, we systematically modulated substrate stress relaxation timescale while maintaining a constant Young’s modulus (~2 kPa), consistent with soft tissues such as brain and liver6. Human fibrosarcoma HT-1080 cells were cultured on IPNs with low- and high-molecular-weight alginate (Fig. 1A), yielding fast-relaxing (~10 s) and slow-relaxing (~1000 s) substrates, respectively (Fig. 1B). By maintaining a constant Young’s and independently modulating substrate stress relaxation timescale, we isolated specific effect of stress relaxation on migration behavior.

To assess migration phenotypes, we conducted time-lapse microscopy experiments and characterized cell motility using the diffusivity exponent \alpha obtained by fitting MSD/t^\alpha (details of experiments and analysis are provided in the Methods section). For classic Brownian motion \alpha = 1, indicating purely diffusive behavior. However, we observed a striking transition in migration dynamics: cells on fast-relaxing substrates displayed super-diffusive migration (\alpha > 1), while cells on slow-relaxing substrates showed sub-diffusive migration (\alpha < 0.8) (Fig. 1C, D). These findings suggest that substrate stress relaxation independently modulates the transition between super- and sub-diffusive migration.

To further quantify migration persistence, we analyzed two independent metrics: track straightness and velocity autocorrelation (VAC)17,18. Track straightness measures the directionality of a migration path as the ratio of net displacement to total distance traveled, while VAC captures how consistently a cell maintains its direction over time, with slower decay indicating greater persistence. Cells on fast-relaxing substrates exhibit greater track straightness than their slow-relaxing counterparts (Fig. 1E), indicating cells have more directed migration on fast-relaxing substrates. Similarly, VAC analysis revealed a slower decay for cells on fast-relaxing substrates, reflecting prolonged persistence times (Fig. 1F). Despite differences in decay rate, all VAC curves plateaued to zero at long timescales, suggesting that over extended durations, migration ultimately transitions to a diffusive regime. Together, these three independent metrics—MSD scaling, track straightness, and VAC—demonstrate that enhanced substrate stress relaxation promotes persistent, super-diffusive migration, while slow-relaxing substrates favor trapped, sub-diffusive behavior.

Conventional models with a single bond timescale cannot capture anomalous migration modes
We first examined how substrate stress relaxation timescales influence migration modes using a conventional motor–clutch modeling approach. In this model, molecular clutches, which represent adhesion clusters, dynamically link actin filaments to the substrate (Fig. 2A). Clutches are modeled as Hookean springs that randomly engage and disengage, with engagement occurring at a constant association rate and disengagement governed by a force-dependent dissociation rate, as described by Bell’s law (r_off = \tau_off^{-1} e^{F/F_b}). Forces exerted by engaged clutches counteract actomyosin-driven retrograde flow, allowing actin polymerization at the cell front and propelling forward cell migration. The total force transmitted to the substrate thus emerges stochastically from the binding and unbinding dynamics of multiple clutches.

To simulate the impact of stress relaxation timescales, we vary the substrate viscosity (\eta) while maintaining additional stiffness (k_a) and long-term stiffnesses (k_l) constant, mirroring our experimental approach of altering alginate molecular weight ratios to modify hydrogel stress relaxation while keeping elastic properties unchanged. The stress relaxation timescale in the simulations is expressed as \eta/k_a, where a high value of \eta signifies a slow-relaxing substrate and a low value of \eta signifies a fast-relaxing substrate (see Methods for governing equations and simulation details).

Using the conventional motor–clutch model, we found that although cells migrate farther on fast-relaxing substrates (Fig. 1G-i), the model consistently predicts purely diffusive migration (\alpha \approx 1), irrespective of the substrate relaxation timescale (Fig. 1G-ii). Thus, the conventional model fails to capture the experimentally observed transition between sub-diffusive (\alpha < 1) and super-diffusive (\alpha > 1) migration modes. Furthermore, metrics for migration persistence, including track straightness and velocity autocorrelation (VAC), showed no significant differences between slow and fast-relaxing substrates using this conventional approach (Fig. 1G-iii, iv). These observations underscore the missing physics from the conventional models and the need for capturing long-tailed processes in order to capture the anomality in migration modes.

The glassy motor clutch model predicts both super- and sub-diffusive migration modes
To overcome the limitations of the conventional model, we developed a glassy motor–clutch framework (Fig. 2A) that incorporates a broad distribution of adhesion lifetimes. Unlike previous models that assume a single characteristic off-rate (\tau_off), our approach accounts for the rugged energy landscape of adhesion clusters (Fig. 2B), where adhesion proteins undergo stochastic unfolding and unbinding with highly variable kinetics19–21. This behavior is characteristic of glassy systems and gives rise to long-tailed dissociation time distributions (see Methods).

To capture the glassy adhesion dynamics, we adopt a power law distribution of \tau_off, expressed as p(\tau_off) = |(\beta-1)| \tau_{min}^{\beta-1} \tau_{off}^{-\beta}, where \beta and \tau_{min} represent the glass coefficient and characteristic dissociation time constant22,23. This distribution enables us to capture both frequent, short-lived and rare, long-lived adhesion events, which are crucial for modeling the heterogeneity of adhesion cluster behavior. The shape of this distribution is shown in Fig. 2C, and its cumulative form is depicted in Fig. 2D. When \beta < 3 the central limit theorem breaks and the variance of the distribution diverges, leading to non-Gaussian dynamics that are central to reproducing anomalous migration modes (see Methods for details).

Using this glassy model, we simulated cell migration on fast- and slow-relaxing substrates. The model successfully predicts that fast-relaxing substrates promote increased cell migration distance with an MSD power law coefficient (\alpha > 1) leading to super-diffusive migration, whereas cells on slow-relaxing substrates exhibit sub-diffusive migration (\alpha < 1) (Fig. 2F-i, ii). Consequently, enhanced substrate relaxation resulting from decreased viscosity induces a transition from a sub-diffusive to a super-diffusive migration mode. To quantitatively examine this transition, we evaluate migration characteristics through track straightness and VAC. We find that cells on fast-relaxing substrates display greater migration track straightness and a gradually decaying VAC, indicating greater persistency in migration. On the other hand, we find that cells on slow-relaxing substrates display lower migration track straightness and a fast-decaying VAC (Fig. 2F-iii, iv), consistent with experimental observations (Fig. 1), reinforcing a lower persistency in migration behavior. Importantly, both sub- and super-diffusive modes of cell migration are captured only when the glass exponent for the clutch off-rate, \beta, is less than or approximately equal to 3 (Fig. 2E, detailed trajectories on this phase diagram shown in Supplementary Fig. 4). This range of \beta is critical because it allows the distribution of off-rates to exhibit long-tailed behavior, which is a defining characteristic of glassy clutch dynamics. In this regime, the off-rate distribution is heavily skewed toward larger timescales, meaning that adhesion bonds exhibit a broad spectrum of dissociation times, including rare but significantly slow dissociation events. These long tails are essential for representing the diverse relaxation timescales that underlie complex cell migration dynamics on viscoelastic substrates.

Long-tailed distributions are characterized by their influence on the variance of the off-rate constant. As discussed in the last section, for \beta > 3, the variance of the off-rate distribution becomes finite, leading to a more predictable, Gaussian-like behavior. However, for \beta < 3, the variance becomes infinite, which breaks the central limit theorem and leads to a non-Gaussian distribution of dissociation events (see Supplementary Note 1). This variability in the off-rate constant plays a significant role in shaping cell migration trajectories, introducing non-uniformity of each migration step, as observed experimentally. In the following sections, we explore these trajectories in detail, examining how variations in substrate relaxation timescales change their dynamics and eventually contribute to the emergence of sub- and super-diffusive migration modes.

Interplay between cell trapping time and migration step size drives the transition between sub- and super-diffusive migration modes
To gain mechanistic insights into the transition between sub-diffusive and super-diffusive cell migration, we performed a detailed analysis of individual cell trajectories. In the model, cell migration dynamics are governed by both the waiting times between steps and the magnitude of the steps themselves. To better understand these waiting times and step sizes-and how they vary with changes in stress-relaxation, we focus on a simplified one-dimensional model (Fig. 3A). This reduction preserves the essential physics of the system while enabling a clearer, more focused analysis of individual migration trajectories over time.

While the clutches on two ends may initially be balanced (Fig. 3A-i), the stochastic nature of adhesion bond dynamics leads to an imbalance in the number of bound clutches. Consequently, one end harbors fewer bound clutches, causing each bound clutch to bear a greater level of force, as the total force must be equal and opposite on both ends. As the clutches begin to break, the load shared among the bound clutches increases, increasing the likelihood of breakage due to the force dependence of the off rate, r_off. Ultimately, all the clutches on one end undergo catastrophic failure, leading to the disintegration of the entire adhesion cluster (Fig. 3A-ii). The retrograde velocity is greater at the end where more clutches fail (as the lower resistance), which propels the cell toward the opposite end with a velocity of D_s = V_m \Delta t (Fig. 3A-iii). However, before the cell takes its next step in a similar fashion, there is an intermediate period where clutches remain balanced on both ends, as is evident from the distribution of bound clutches (Fig. 3B-i). We term this duration of minimal net displacement the “trapping time” (Fig. 3A-iii, iv). The trapping time is governed by the net dissociation rate r_off, which is dependent on two factors: the glassy dissociation rate constant \tau_off and the clutch forces F_c. These two factors are dictated by the substrate relaxation timescale. Eventually, the cell emerges from the trapped state upon experiencing a force imbalance between the two ends (Fig. 3A-v). Next, we examine cell trajectories on slow- and fast-relaxing substrates, revealing distinct migration patterns (Fig. 3B). On slow-relaxing substrates, cells undergoing sub-diffusive migration display balanced clutches at both ends for extended durations (Fig. 3B: left panel). Consequently, their migration distance profiles exhibit prolonged periods of nearly zero displacement, indicating longer trapping times t_p. The migration velocity curve confirms this trapped state, showing near-zero migration velocity, signifying balanced retrograde flow velocities at both ends. Conversely, on fast-relaxing substrates, the bound clutch probability, migration distance, and velocity plots (Fig. 3B: right panel) illustrate that clutches at the two ends do not remain balanced for prolonged durations, thus limiting the occurrence of extended trapping times.

We then quantified the observed differences in trapping times and step sizes and found that our model predicts significantly longer trapping times on slow-relaxing substrates compared to fast-relaxing substrates (Fig. 3Ei). Further, the model predicts larger migration step-sizes for cells on fast-relaxing substrates in contrast to those on slow-relaxing substrates (Fig. 3Eii). Experiments validated that cells exhibit intermittent trapping between consecutive migration steps, displaying a discontinuous migration pattern (i.e., each migration step is different during the whole trajectory of a cell) (Fig. 3C, D). Quantification of these experimental trapping times confirmed that, on average, cells on slow-relaxing substrates experienced significantly longer trapping times than did their counterparts on fast-relaxing substrates (Fig. 3Fi), in agreement with theoretical predictions. Further experimental quantification of step-sizes validated the presence of larger steps taken by cells on fast-relaxing substrates (Fig. 3Fii).

In addition to differences in the magnitude of trapping times, a defining feature of sub-diffusive migration on slow-relaxing substrates is the emergence of long-tailed trapping-time distributions. Likewise, for cells exhibiting super-diffusive behavior on fast-relaxing substrates, step sizes follow a heavy-tailed distribution, and these statistics are central to the onset of anomalous migration. Sub-diffusive motion is often described using continuous-time random walk (CTRW) or fractional diffusion models11,24,25, in which the waiting time between steps follows a power-law distribution with long tails, unlike standard random walks that assume uniform waiting times. The presence of rare but prolonged trapping events lowers the overall displacement, thereby reducing migration diffusivity. This same mechanism, driven by long-tailed trapping-time statistics, underlies sub-diffusivity in our glassy motor–clutch model.

Similarly, super-diffusive motion is typically represented by the Lévy walk model11,25, where step lengths are drawn from a heavy-tailed distribution to produce occasional long jumps. In our framework, such heavy-tailed step-size behavior is not imposed but emerges naturally from glassy adhesion dynamics and the broad distribution of clutch off-times. These rare, large steps are an intrinsic outcome of the coupling between adhesion kinetics and substrate relaxation. Importantly, CTRW, fractional diffusion, and Lévy walk models are phenomenological: they assume the existence of heavy-tailed statistics but do not explain their physical origin. In contrast, our glassy motor–clutch model is physics-based. The trapping times and step-size distributions arise self-consistently from force transduction and adhesion dynamics that respond to substrate stress relaxation. This mechanistic foundation distinguishes our framework from purely statistical descriptions and reveals how anomalous migration emerges directly from the underlying biophysical processes.

To validate the role of heavy tails for step-size and trap-time distribution as the governing physics controlling diffusivity, we utilize the Kurtosis number (Fig. 3G). The Kurtosis number is a statistical measure that quantifies the “tailed-ness” of a probability distribution compared to a normal distribution. It is calculated using the fourth standardized moment of the distribution and indicates whether the data has heavier, or lighter tails compared to a normal distribution. In this way, the relative contribution of step-size distribution tailed-ness K_step to the trapping time distribution tailed-ness K_trap can be compared, by taking a ratio of their respective Kurtosis numbers. We then use this ratio to compare how viscosity and glass coefficient affect the tailed-ness of both step size and trapping time distributions and consequently migration diffusivity. We note that for low values of glass coefficient \beta, the ratio K_trap/K_step increases as the viscosity increases, and for high values of \beta, the substrate viscosity has negligible effect on the Kurtosis ratio (Fig. 3H). Interestingly, comparing the phase map previously obtained for migration diffusivity (Fig. 2E) with the phase map obtained for the Kurtosis number ratio (Fig. 3H), we find overlapping trends. The region with large K_trap/K_step in Fig. 3H overlaps closely with the sub-diffusive region in Fig. 2E, demonstrating that sub-diffusive migration occurs when the relative contribution of K_trap is large, effectively making K_trap/K_step large. This underscores the importance of a long tail for the trap-time distribution in capturing sub-diffusive migration. Conversely, the region where K_trap/K_step is small in Fig. 3H, we observe a correspondence with the super-diffusive region in Fig. 2E. In this case, the relative contribution of K_step is large, effectively making K_trap/K_step relatively smaller. This observation of increasing tailedness for step-size distribution highlights the significance of long tails in the step-size distribution in generating super-diffusive migration. Both our experiments and simulations for fast- and slow-relaxing substrates confirm these trends (as seen in Fig. 3E, F), where cells on slow-relaxing substrates showing sub-diffusivity have heavy-tailed trapping times while cells on fast-relaxing substrates showing super-diffusivity have heavy-tailed step sizes. Thus, we conclude that migration diffusivity is regulated by the substrate stress relaxation via the competition between trap time, step sizes and their respective distributions.

The inability of conventional motor–clutch models to reproduce these trapping statistics is further demonstrated in Supplementary Figs. 6, 7. As shown in Supplementary Fig. 6, trajectories generated by the standard model with constant clutch off-times produce uniform, Gaussian migration with narrowly distributed trapping durations. Even when model parameters such as clutch number, motor number, or stall duration are systematically varied (Supplementary Fig. 7), the resulting trajectories remain diffusive, showing only changes in average speed or pause length but never exhibiting the long-tailed, non-Gaussian statistics characteristic of anomalous migration. These results confirm that parameter tuning within conventional formulations cannot generate intermittent trapping or anomalous diffusion and that such behaviors emerge only when adhesion glassiness is explicitly incorporated into the model.

Trapping time and step size are regulated by the interplay of substrate relaxation timescales and glassy adhesion dynamics
To understand the mechanisms that regulate trapping times and step sizes in cell migration, we identify three critical timescales governing adhesion dynamics, force generation, and viscoelastic substrate relaxation. These timescales determine whether migration will manifest as sub-diffusive with prolonged trapping periods, or super-diffusive featuring large step sizes.

1. Motor-stall timescale (\tau_l): Myosin motors generate contractile forces as they pull actin filaments against cellular load. This force is regulated by Hill’s relation, where the motor’s velocity declines with increasing load until it reaches a “stall” state. This “stall force” represents the maximum load at which the motor can no longer perform mechanical work against resistance, and at this point, its velocity approaches zero. The timescale for this process is given by Gong et al.3, \tau_l = F_m / (v_u N_m k_l) (~10 s), where F_m represents the individual motor’s stall force, N_m the number of motors, v_u the unloaded motor velocity, and k_l the long-term stiffness. This timescale, \tau_l defines the period required for motors to reach the stall state and it modulates the dynamics of force transmission.

2. Dissociation timescale (\tau_off): Adhesion clusters modeled as molecular clutches serve as the link that transmit actomyosin generated force to the substrate. These clutches can associate (taken as a constant rate) and dissociate with a characteristic dissociation timescale \tau_off at every migration step. We account for the glassy dynamics and power law distribution of the off-rate time constant \tau_off = |(\beta-1)| \tau_{min}^{\beta-1} \tau_{off}^{-\beta} as discussed in the earlier sections (range: ~1–100 s). This timescale regulates the intrinsic tendency of clutches to stay bound and captures the variability in migration at each step.

3. Substrate relaxation timescale (\tau_s): Once the force is transmitted to the substrate through an adhesion cluster, the viscoelastic substrate undergoes stress relaxation. This response is characterized by a timescale, \tau_s = \eta/k_a, where \eta represents the viscosity and k_a is the additional stiffness of the substrate. Stress relaxation reduces the effective force in the clutches over time, thus modulating the adhesion dynamics differently for fast- (\tau_s \approx 10 s) and slow-relaxing substrates (\tau_s \approx 1000 s).

The interplay of these three timescales determines whether cell migration shows sub-diffusive (dominated by prolonged trapped periods) or super-diffusive (characterized by large displacement steps). We outline a few relevant cases of how these timescales specifically affect the propensity for a cell to be trapped or show large migration steps.

For fast-relaxing substrates, the timescale for substrate relaxation is faster than the motor-stall timescale (\tau_s < \tau_l) and we get the following cases:
(i). \tau_off < \tau_s < \tau_l : A low \tau_off means that molecular clutches have an intrinsic nature of quickly disengaging, i.e., they remain engaged for a short time. In the case when (\tau_off < \tau_s) the clutches disengage faster than the substrate can relax to its long-term stiffness k_l. This results in quick clutch failure and lower resistance to retrograde flow. Since the overall retrograde flow is higher, cells take large steps and are unable to remain trapped (Fig. 4B-iv).
(ii). \tau_s < \tau_l < \tau_off : This is possible only for the glassy motor clutch model in the presence of high \tau_off as shown in Fig. 4B. The substrate relaxes to long-term stiffness k_l quickly, and clutches sense a change in stiffness before motors have stalled. The retrograde flow of actin continues to pull on the clutches, resulting in eventual clutch failure (Fig. 4B-iv). In this case, quick relaxation coupled with high retrograde flow causes clutch failure, preventing trapping. Hence, even though clutches are intrinsically stable and can remain engaged for longer, fast substrate relaxation does not let the force build up and resist retrograde flow and hence causes eventual clutch failure (Fig. 4B-iii).

For slow relaxing substrates the substrate takes much longer to relax compared to the motor stall timescale (\tau_s > \tau_l) and we get following cases:
(i). \tau_off < \tau_l < \tau_s : The intrinsic nature of clutches is to disengage quickly due to low \tau_off, and since clutches disengage faster than the substrate relaxes or motors are stalled, the cell cannot remain trapped for longer (akin to Fig. 4B, ii).
(ii). \tau_l < \tau_off < \tau_s : This is only possible for the glassy distribution of \tau_off where we sample high \tau_off values. In this case, clutches remain engaged for long periods because the motors have reached their stall state, intrinsic \tau_off is high enough to provide stability, and the substrate is able to maintain stiffness for long time hence sustaining resistance to retrograde flow. Thus, the cell remains trapped for long periods in this case and undergoes a migration step only after a long time (Fig. 4B-i). We get these long trapping period cases for slow-relaxing substrates when \tau_s is high.

To summarize, for fast-relaxing substrates, since the relaxation time \tau_s is the fastest timescale among the three at play, quick clutch failure is initiated for all cases. This leads to lower retrograde flow resistance, causing cells to take larger migration steps and prevent trapping, leading to super-diffusive migration. Conversely, for slow-relaxing substrates, \tau_s is high and allows motors to stall before the substrate relaxes. In this case, whenever a high \tau_off is sampled, it provides an opportunity for clutches to stay engaged for longer, resulting in cell trapping zones and hence displaying a sub-diffusive migration mode. It is to be noted that cells do not show a uniform migration pattern for every migration step and rather show a saltatory migration pattern. This behavior is effectively captured in our model by the glassy dynamics of adhesion clusters, which introduces a range of dissociation pathways with a probabilistic distribution of the off-rate time constant \tau_off invoked at every migration step. This dynamic interplay of timescales ensure that each migration step is unique, encompassing a spectrum of migration steps and trajectories while predicting the overall migration pattern accurately for both sub- and super-diffusive migration mode. In the next section, we discuss how changes in the active forces generated by actomyosin contractility affect the related timescale and eventually the migration mode since for a cell to show super-diffusive migration, it needs active cellular forces to generate traction and avoid getting trapped.

Glassy motor clutch model reveals super-diffusion necessitates active process and inhibiting actomyosin activity results in sub-diffusive migration
In our model, clutches serve as dynamic linkages between cells and substrates, constraining movement until sufficient traction forces break these connections to enable migration. Actin polymerization and myosin-driven contractility generate the cytoskeletal forces required for this process, relying on the hydrolysis of ATP to ADP with the release of inorganic phosphate (Fig. 5J). Unlike passive Brownian motion, cell migration is an inherently active process, necessitating continuous energy expenditure to produce these dynamic forces11,26. Super-diffusive migration arises when cells sustain contractility, generating forces to overcome trapping and facilitate the cyclic adhesion remodeling essential for persistent motion. To elucidate the role of actomyosin forces in super-diffusive behavior, we manipulated biophysical parameters representing actin polymerization and myosin activity in our model. These computational predictions were experimentally validated by pharmacologically inhibiting actomyosin activity, confirming its critical influence on the observed migration dynamics.

Inhibiting myosin motor activity by reducing the number of myosin motors (parameter N_m) led to a decrease in MSD-\alpha values, indicating a shift from super-diffusive to sub-diffusive migration behaviors on fast-relaxing substrates (Fig. 5A–D). The reduction in number of myosin motors results in decreased actomyosin contractility and the level of force transduction through clutches decreases. Following Bell’s law for adhesion bonds, lower clutch force leads to a lower off-rate r_off, leaving bonds bound for longer durations. As a result, cells experience prolonged trapping times, leading to decreased diffusivity, as explained previously. Therefore, myosin activity inhibition increases trapping time due to decreased contractility, thereby inducing sub-diffusive migration. To validate these findings, we inhibited myosin light chain kinase (-MLCK) and blocked MLC phosphorylation, effectively reducing contractility. Consistently, experimental findings mirrored the model predictions, showing a transition from super-diffusive to sub-diffusive migration on fast-relaxing substrates upon MLCK inhibition (Fig. 5E–H). Furthermore, our model predictions for the effect of MLCK inhibition across a wide range of substrate parameters revealed significant changes in the diffusivity heatmap (Fig. 5I-ii), indicating a predominance of sub-diffusive migration across most parameter regimen.

We similarly simulated actin inhibition by reducing the actin polymerization speed and decreasing the number of myosin motors in the model. Such as for the myosin inhibition study, we observed that actin inhibition resulted in lower clutch force generation and prolonged clutch binding, leading to longer trapping times and sub-diffusive behavior on fast-relaxing substrates, as evidenced by a reduction in the diffusivity exponent (Fig. 5A–D). To validate these findings, we conducted actin polymerization inhibition studies using Latrunculin A (LatA), a potent toxin that disrupts actin filament polymerization by binding to monomeric actin (G-actin), thereby preventing its incorporation into growing actin filaments (F-actin). Consequently, LatA treatment induces the depolymerization of existing actin filaments and inhibits the assembly of new filaments, thereby reducing polymerization at the leading edge. This disruption of the actin network also weakens the scaffolding needed for myosin II to generate tension, leading to a marked decrease in actomyosin contractility27,28. The experimental findings again showed a transition from a super-diffusive to a sub-diffusive regime upon actin inhibition. Furthermore, our model analysis of actin inhibition across a broad parameter range revealed a shift toward predominantly sub-diffusive migration, as depicted in the diffusivity heatmap (Fig. 5I-iii).

Reduction in contractility lowers the motor-stall timescale \tau_l = F_m / (v_u N_m k_l), which is crucial in governing trap time, step sizes, and consequently the migration diffusivity regimes. This causes motors to stall faster, increasing the probability that the dissociation time constant \tau_off sampled and the substrate relaxation timescale \tau_s is higher than the motor-stall timescale \tau_l, resulting in a stable clutch for longer durations. The elongation of stable clutch durations leads to extended cell trapping and the emergence of sub-diffusive behavior, even on fast-relaxing substrates. This shift in migration mode also correlates with reduced migration persistence, as indicated by trends in velocity autocorrelation (VAC) and track straightness under conditions of myosin and actin inhibition (Fig. 5G, H). Sustaining cell migration requires active energy consumption to drive the cytoskeleton machinery for actin polymerization and actomyosin contractility (Fig. 5L). Diffusivity is therefore directly influenced by the energy available to power the migration machinery: higher diffusivity corresponds to greater active energy expenditure. Inhibition of actomyosin activity reduces energy consumption, leading to decreased cell activity and diminished force generation. Consequently, cells are less able to escape trapped zones, increasing trapping times and further reinforcing sub-diffusive migration.

Based on these observations, our glassy motor clutch model provides a framework to describe migration dynamics across regimes. We propose a phase diagram incorporating substrate stress relaxation, system glassiness (quantified by the glass coefficient \beta), and contractility levels (Fig. 5M). The inclusion of glassy dynamics is essential for capturing the anomalous migration behaviors observed experimentally and offers insights into the interplay between substrate properties and active cellular processes.

Methods
Formulation of a glassy motor clutch model for cell migration
We developed a computational model based on the motor-clutch theory3,10,33,34 that employs a 2D framework, featuring independent motor-clutch modules along the X and Y axes (Fig. 2A). In essence, each module consists of molecular clutches representing adhesion clusters and link F-actin filaments to the substrate. The clutches get stretched due to the activity of actomyosin contractility and exhibit random engagement and disengagement with association and dissociation rates r_on and r_off, respectively. The binding of these clutches to actin filaments opposes retrograde flow driven by myosin motors, facilitating polymerization at the protruding end and propelling cell migration. The dynamics of molecular clutches are governed by the master equation:

d/dt P_{b,i} = (1-P_{b,i}) r_{on,i} - P_{b,i} r_{off,i} (1)

Here, P_{b,i} denotes the binding state of the i-th clutch, determined using a Monte Carlo sampling approach. The force exerted by a bound clutch is defined as F_{c,i} = k_c (x_{c,i} - x_s), where k_c represents the molecular clutch stiffness, x_{c,i} denotes the displacement caused by the sliding actin filament, and x_s corresponds to the substrate displacement, which is dependent on its viscoelastic properties. The cumulative force transmitted to the substrate is expressed as the sum of the forces from all the clutches: F_s = \sum k_c (x_{c,i} - x_s).

The extension and dynamics of molecular springs facilitate the transmission of forces between the cell and the substrate, enabling mechanical sensing of the viscoelastic properties of the substrate. We model the substrate as a standard linear solid (SLS) viscoelastic material according to the following equation:

(k_a + k_l) d/dt x_s + k_a k_l x_s = k_a F_s + \eta d/dt F_s (2)

In this equation, one branch of the SLS model comprises a spring of stiffness k_l, which dictates the long-term behavior of the substrate. The other branch consists of two elements in series: a dashpot with viscosity \eta and a spring with stiffness k_a. Upon force application, the initial response is dictated by the deformation of the two springs with stiffness k_a and k_l, representing instantaneous elastic behavior. However, the dashpot viscosity \eta resists deformation over time until the stiffness relaxes to the long-term value k_l.

The relaxation properties of the substrate regulate the transmitted clutch force F_c, thereby influencing the clutch dissociation rate r_off. To model the force-dependent dissociation rate, we employ Bell’s law, where r_off is expressed as:

r_off = \tau_{off}^{-1} exp(F_c / F_{bond}) (3)

In the context of molecular clutches, F_{bond} denotes the characteristic rupture force and the dissociation time constant, \tau_{off}, represents the characteristic timescale at which the clutches disengage. The disengagement of clutches represents the dissociation of the adhesion cluster which consists of unfolding of proteins and subsequent unbinding of protein-protein bonds (detailed further in SI). In conventional motor clutch models, \tau_{off} for adhesion proteins has been considered constant35. However, adhesion complexes consist of multiple proteins, such as integrin, talin, vinculin, and zyxin, each exhibiting distinct and variable folding and unfolding dynamics19–21. Moreover, the conformational energy landscape for protein unfolding is inherently rugged, containing numerous local minima separated by energy barriers36. In such a landscape, some unfolding pathways involve low energy barriers, allowing for rapid transitions, while others encounter high energy barriers, resulting in rare but prolonged unfolding timescales (Fig. 2B). These differences stem from several factors, including the positional flexibility of the protein domains, as well as the stochastic nature of protein-protein interactions within the complex. For instance, talin, a key adhesion protein, displays a highly stochastic unfolding response, manifesting multiple unfolded conformations for the same level of the applied force21. It has additionally been observed that \tau_{off} for certain adhesion proteins, such as zyxin, depends on mechanical fluctuations rather than taking on a fixed, deterministic value20. These observations underscore the non-uniform and stochastic nature of the unfolding and unbinding processes, where external forces and internal fluctuations modulate the overall dissociation dynamics of the adhesion complexes. Given the complexity of the adhesion clusters, consisting of multiple interacting proteins, cannot be adequately described by a constant dissociation time constant, \tau_{off}. Instead, these clusters exhibit glass-like dynamics, characterized by a broad distribution of stochastically emerging relaxation timescales having a range of \tau_{off} values.

To capture the overall glassy behavior of the supramolecular adhesion complex, we adopted a distribution of \tau_{off} assigned to molecular clutches (Fig. 2C). A protein’s transition between conformational states can be expressed using Markov state modeling that reveals a long-tail distribution for the transition path timescales37. Long-tail distributions are particularly suited to representing rare, infrequent events which arise from conformational adaptability and redundancy in unfolding and refolding pathways. As the adhesion cluster stretches due to actomyosin-generated retrograde flow, multiple adhesion proteins undergo unfolding and eventual unbinding. The coupled unfolding and unbinding process are referred to as dissociation in this study and are detailed in the Supplementary Note 2. To account for the heterogeneous transition pathways of both frequent and rare events, we use a power law distribution of \tau_{off} = |(\beta-1)| \tau_{min}^{\beta-1} \tau_{off}^{-\beta}, with \beta and \tau_{min} as the glass coefficient and characteristic dissociation time constant, respectively22,23. This function exhibits a gradual decay from a peak at \tau_{min} for small values of \beta, to a steeper decay for large values of \beta, asymptotically approaching a delta function, thus recovering the conventional model with a single constant time coefficient (Fig. 2C, D). By employing this power law distribution for \tau_{off}, we can tune the molecular clutch dynamics, effectively capturing the glassy nature of adhesion protein mechanics.

Crucially, when \beta \le 3, the power law distribution exhibits infinite variance (derived in the SI), thereby violating the central theorem. This leads to a non-Gaussian long-tailed distribution, distinguishing it from distributions with \beta > 3 that converge to a Gaussian form upon averaging. Such long-tailed behavior is characteristic of glassy dynamics38, a defining feature of complex systems, which include adhesion clusters, where relaxation timescales may span a wider range. The presence of these long tails implies that rare, extreme events significantly influence the system’s behavior, underscoring the need for models that account for this inherent complexity in protein dissociation dynamics. Conventional motor-clutch modules with finite-variance off-rate distribution can be reduced to deterministic modules through mean-field theory3,33, but the glassy model lacks a finite mean off-rate and cannot be reduced to deterministic behavior. This fundamental non-Gaussian character allows the glassy adhesion dynamics to enable the anomalous diffusivity, as shown in SI note 1, and the differences in migration trajectory statistics are shown in Supplementary Figs. 6, 7.

The probability of engaged clutches P_b, and disengaged clutches 1-P_b can be derived by solving the master equation (1) with the off-rate expressed as r_off as described earlier. Engaged clutches have their top end bound to actin filaments and move with the retrograde flow velocity V_r. Disengaged clutches move with the substrate velocity, ensuring that clutch association occurs at zero extension3,33. Therefore, the displacement of each clutch can be expressed as:

d/dt x_{c,i} = P_{b,i} V_r + (1-P_{b,i}) d/dt x_s (4)

The retrograde flow velocity for each clutch module is updated according to Hill’s relation expressed as V_r = V_u (1 - F_c/F_m), where the myosin motor force F_m is determined based on the force equilibrium at the cell center (see SI). The migration velocity V_m in each direction is calculated as the difference in retrograde flow between two protruding ends (Fig. 3A), given by D(t+\delta t) = D(t) + V_m(t) \delta t. By solving the aforementioned system of equations for the motor clutch modules, we can simulate cell migration trajectories and calculate the MSD as

MSD = 1/T \sum (D(t) - D_0)^2 (5)

For Brownian motion, the MSD exhibits a linear dependence with time as MSD = 2Kt, where K denotes the diffusion coefficient. Conversely, anomalous diffusion is characterized by a non-linear dependence, MSD \propto t^\alpha, with \alpha representing the power-law coefficient used to classify the diffusion mode.

Cell culture
Human fibrosarcoma cells HT-1080 (ATCC) were cultured using established protocols. HT-1080 cells were cultured in high-glucose Dulbecco’s Modified Eagles Medium (Gibco) supplemented with 10% fetal bovine serum (FBS) (Hyclone), and 1% Penicillin/Streptomycin (Life Technologies). Cells were cultured in a standard humidified incubator at 37°C in a 5% CO_2 atmosphere. Cells were maintained at sub-confluency and passaged every 2–3 days.

Hydrogel preparation and mechanical Characterization
Interpenetrating (IPN) hydrogels comprised of alginate and reconstituted basement membrane (rBM) were prepared following previously established protocols7. Briefly, low molecular weight (<75 kDa) and high molecular weight alginate (280 kDa) was purchased from Provona UP VL VG NovaMatrix and FMC Biopolymer, Protanal LF 20/40 respectively. Alginate was dialyzed in deionized water for 4 days (MW cutoff of 3500 Da), treated with activated charcoal, sterile-filtered, lyophilized, and then reconstituted to 3.5 wt% in serum-free Dulbecco’s modified Eagle’s medium (DMEM) (Gibco). The alginate was crosslinked using calcium sulfate dihydrate. LMW alginate was used to make fast relaxing substrate while HMW alginate was used to make slow relaxing substrate. The IPNs were formulated as 10 mg/ml alginate – 4.4 mg/ml rBM.

Mechanical characterization of IPNs was performed with a stress-controlled AR2000EX rheometer (TA instruments). IPNs were formulated using luer-lock syringes and directly deposited onto the Peltier plate. A 25 mm flat plate geometry was slowly lowered to make full contact with the gel and a plastic pipette was used to add mineral oil (Sigma) to the edges of the gel to prevent dehydration. During modulus measurement, a time sweep was performed at 1 rad/s, 37°C, and 1% strain for 3 h to allow for the storage and loss moduli to equilibrate. For stress relaxation experiments, the time sweep was followed by applying a constant strain of 5% to the gel, at 37°C, and the resulting stress was recorded over the course of 3 h.

Microscopy
Time-lapse imaging (Leica SP8) was performed for live cell imaging. Live cell conditions were maintained at 37°C and 5% CO_2. R18 membrane dye was reconstituted in DMSO to obtain a 10 mg/ml stock solution. HT-1080 cells were labeled with R18 dye (1:1000 dilution) and the labeled cells were tracked with a 10× NA=0.4 air objective for 24 h. 60 \mu m stack images were acquired every 10 min and imaging parameters were adjusted to minimize photobleaching and avoid cell death.

Inhibition studies
Pharmacological inhibitors were added to cell media 10 min before time-lapse microscopy experiments. The concentrations used for the inhibitors were 100 nM latrunculin A (Tocris Bioscience, actin polymerization inhibitor) and 25 \mu M ML-7 (Tocris, myosin light-chain kinase inhibitor)

Statistics and reproducibility
Measurements were conducted using one to three independent biological replicates from separate experiments. All statistical analyses were carried out in GraphPad Prism (version 9.1.0 for Mac, GraphPad Software). Data are presented as mean values; error bars represent standard deviation. The unpaired two-tailed Student’s t-test was used for data analysis, with p-values adjusted for multiple comparisons when necessary. Further details on the statistical methods are provided in the corresponding figure captions. Immunofluorescence and live-actin imaging were each repeated in three independent experiments.

Cell migration simulation based on the motor–clutch framework
We simulated cell traction and migration dynamics using a stochastic motor–clutch model implemented with a Monte-Carlo update scheme. At each time step, individual clutch binding/unbinding events and motor-driven displacements were sampled probabilistically according to their rate constants, enabling realistic fluctuations in force transmission. All simulations and downstream analyses, including trajectory extraction and force–velocity characterization, were performed in MATLAB (MathWorks)39

Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability
Raw time-lapse images used in this study have been deposited on Figshare40 and are publicly available at (https://doi.org/10.6084/m9.figshare.30763754). The data generated in this study are available in the Source Data file provided with this paper. Source data are provided with this paper.

Code availability
The codes used in this study to simulate glassy motor clutch modules and generate migration trajectories are deposited on Zenodo41 and publicly available at (https://doi.org/10.5281/zenodo.17781658). All the figures were created and assembled using Blender42 and Inkscape43.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this paper, the authors adapted a 2D motor-clutch computational model based on a previous model developed by the Odde lab to investigate the migration behavior of cancer cells on viscoelastic substrates and validated their results with experimental data acquired from human fibrosarcoma HT-1080 on alginate hydrogels with fast and slow relaxing properties. The model is based on a previously published model by the same group (Gong et al., 2018), in which they compared clutch binding with substrate relaxation time scales and found that cell spreading is maximized when the substrate relaxation timescale, τs, is between the clutch binding timescale, τb, and the lifetime scale, τl. In a follow-up study, they further demonstrated that fast-relaxing substrates promote filopodia-mediated cancer cell migration (Adebowale et al.,2021). In the present study, the authors provide both experimental and modeling evidence that cells grown on fast-relaxing substrates exhibit a super-diffusive migration (short pauses, longer steps), while on slow-relaxing substrates the migration is sub-diffusive (longer pauses, small steps) and less straight than on fast-relaxing substrates. Most importantly, they show by modeling that standard adhesion dynamics with a constant value for the clutch dissociation time constant τoff is unable to predict this cell behavior. In contrast, they demonstrate by computational modeling that a more complex, “glassy”, adhesion dynamics following a power law distribution and using a glass coefficient β of less than or equal to three (to account for the heterogenous unbinding kinetics of proteins in focal adhesions) can capture the transition of the migration behavior dependent on the substrate stress relaxation time. Overall, besides the major and minor issues mentioned below, the study is well executed with respect to the modeling and experimental aspects. The manuscript is also well written. The main novelty is the incorporation of the power law distribution for τoff to account for a more complex dissociation behavior involving both protein unfolding and unbinding. The authors use this glassy adhesion dynamics to explain the cell migration behavior on viscoelastic substrates with different substrate relaxation times. Major concerns: 1. In most if not all graphs, especially representing experimental work, it is unclear what one data point represents. Is it a single cell, an average of cells per gel, or a single experiment (multiple gels per experiment)? For example, in Fig. 1C and 1D, is one data point a single cell? 2. Related to the first concern, there is only one data point in Fig. 5B and F for the conditions when actin polymerization and myosin II activity are inhibited in both modeling and experimental studies. This is not acceptable. 3. It is suggested to show higher magnification images to demonstrate the filopodia-type morphology of the cancer cells on the two substrates with different stress relaxation times. Minor concerns: 1. The authors used the standard linear solid model to represent the viscoelastic substrate. Is there any experimental evidence that supports the selection of that model instead of another viscoelastic model? 2. The authors should indicate the different values of β in Fig. 2C and 2D. 3. How do the authors know that the MSD is not affected by the time interval used in the time-lapse imaging? Did they conduct any optimization to define the time interval? 4. What concentration of LatA and MLCK inhibitor was used? 5. The authors should indicate the details of statistical testing of significant differences for all data. 6. Reference 1 information is incomplete. 7. What are the values for τmin in the 3 different graphs of Supplementary Fig. 1? 8. Supplementary Note 3: There is a spelling error on the second line: “…and is based on our previous work” instead of “… and is based on out previous work”", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "Review of “Glassy Adhesion Dynamics Govern Transitions Between Sub-Diffusive and Super-Diffusive Cell Migration on Viscoelastic Substrates”, V. Sharma et al., Apr 2025, Nat. Comm In this manuscript, Sharma et al. investigate the influence of matrix viscoelasticity and the distribution of cell-matrix adhesion lifetimes on cell migration by integrating in vitro experiments with computational modeling. The authors report that human fibrosarcoma cells exhibit super-diffusive migration on fast-relaxing crosslinked alginate hydrogels, while cells on slow- relaxing gels display sub-diffusive behavior. These experimental observations, however, were previously reported by some of the authors in Adebowale et al., Nature Materials, 2021. To interpret these findings, the authors employ a previously developed one-dimensional motor-clutch model and show that assigning a single dissociation rate constant to all clutches results in purely diffusive migration. To recapitulate the experimentally observed anomalous diffusion, the authors introduce a heterogeneous distribution of clutch dissociation rate constants. The computational results are further analyzed by examining clutch cluster failure frequencies and directional step-size displacements under different substrate relaxation conditions. Finally, the authors demonstrate, both experimentally and computationally, that inhibition of actomyosin contractility induces a transition from super-diffusive to sub-diffusive migration. While the manuscript is clearly written and the simulations appear to be carefully implemented, the central finding (that faster matrix stress relaxation enhances cell migration) was already established by the same group and collaborators in Adebowale et al., Nature Materials, 2021, through both experimental data and motor-clutch modeling. Furthermore, the regulatory role of actomyosin contractility in cell migration has been widely explored in the literature. As a result, the present study offers limited novelty and does not meet the threshold for publication in Nature Communications. Major Concerns: 1) A primary concern is the lack of originality. Adebowale et al. (2021) already demonstrated that cells exhibit sub-diffusive and super-diffusive migration depending on substrate relaxation properties and introduced both one- and two-dimensional motor-clutch models that captured the influence of viscoelasticity on cell migration and filopodial dynamics. The current study adds a heterogeneous distribution of clutch dissociation rate constants into the one-dimensional model and analyzes the resulting diffusivity exponent (alpha) under varying conditions. While technically valid, this represents a modest extension that is not essential to capture the observed anomalous diffusion behavior. 2) A greater concern is that the authors claim that the standard motor-clutch model with a single clutch dissociation timescale cannot reproduce sub-diffusive or super-diffusive migration. This claim is incorrect and unsupported. Prior models, including Bangasser et al., Nature Communications, 2017 (in elastic matrices), and Adebowale et al., Nature Materials, 2021 (in viscoelastic matrices), used standard motor-clutch formulations and can successfully capture a range of diffusion behaviors that depend on model parameters. Even a very quick analysis confirms that standard motor-clutch models (e.g. Bangasser et al.) are indeed capable of producing both sub-diffusive and super-diffusive behavior, where it is straightforward to show how the diffusion exponent alpha varies with substrate stiffness for a particular set of parameters. Furthermore, since the authors’ model is fully deterministic, this is likely another reason it fails to capture anomalous diffusion with the authors’ standard formulation. In fact, it is surprising that the authors aim to study subdiffusive and superdiffusive migration using a purely deterministic version of the standard motor clutch model. It is expected that the authors would not observe anomalous diffusion under these conditions, particularly since stochasticity is only introduced later by incorporating a distribution of clutch dissociation rates. Thus, overall, the core mechanistic argument presented in the manuscript is not valid. Overall, the contribution of this work is quite limited. The authors have already studied the effects of viscoelasticity on cell migration, and in this manuscript, the authors simply fit model outputs to extract alpha. Because the model is so minimal, the authors introduce a distribution of clutch dissociation constants, unnecessary for capturing anomalous diffusion, to explain the results. Notably, the authors have not experimentally tested whether such heterogeneity is required. 3) In Gong et al., PNAS (2018), the authors already introduced the three key timescales that govern cell migration in viscoelastic environments, which are repeated in the present study. Therefore, this section does not present novel insights. Moreover, the claim that these three timescales alone determine whether cell migration is subdiffusive or superdiffusive is both unjustified and incorrect. Many additional timescales are known to influence clutch adhesion dynamics and cell migration behavior, including those associated with clutch binding kinetics, clutch rupture forces, the number of clutches in the cell, and substrate stiffness. Reducing the system to just three timescales overlooks these important contributions and misrepresents the complexity of the underlying biophysical processes. 4) A critical issue arises in the formulation of clutch extension dynamics in Equations 1 and 4. In the model, clutch extension is computed based on the relative velocities of actin retrograde flow and substrate displacement. However, the authors do not account for the fact that clutches associate at zero extension, that is, without initial pre-tension, a fundamental assumption in prior models. This oversight compromises the physical accuracy of the model and may explain why the authors fail to capture anomalous migration using a single dissociation timescale. Neglecting this basic mechanical condition can lead to inaccurate force calculations and unrealistic clutch detachment dynamics. This omission is problematic in both elastic and viscoelastic environments, where clutch association and dissociation kinetics as well as substrate deformations directly affect force transmission. The authors should clarify the assumptions underlying Eqs. 1 and 4, assess whether these assumptions are physically realistic, and demonstrate whether the equations can be derived from first principles. 5) The use of a one-dimensional model with fixed protrusions aligned along the x- and y-axes is a limiting simplification. In contrast, the two-dimensional models in prior studies allow protrusions to form, fully retract, and generate forces in arbitrary directions governed by global force balance. This more realistic representation captures key features of dynamic cell morphology and directional migration. By restricting protrusions to orthogonal axes, the present model fails to account for rotation and spatial variability, further reducing its biological relevance. These modeling limitations contribute to erroneous conclusions and weaken the overall impact of the study. Minor concerns: 6) There are also several presentation issues that need attention. The supplemental information, particularly the model description, requires significant revision. Each section should begin with a brief introduction to guide the reader. In Section 3, the origin of the two equations after step 5 is unclear. Specifically, one of the equations simply repeats the same expression on both sides of the equal sign. The second equation equates both sides without explanation, and the inclusion of terms such as Fr and vr multiplied by dt in Equation 2 is not stated. In the supplementary table listing model parameters, references should be more precise. Under references, phrases like \"this article\" are vague and should be replaced with proper citations or explanations of how each parameter was determined, along with the relevant experimental or theoretical basis. The manuscript also contains inconsistencies in figure referencing. Figure 2 is cited before Figure 1. The caption of Figure 1 should define the acronym VAC, even if it is introduced in the main text. 7) It is unclear why the authors simulate actin inhibition by reducing the number of myosin motors. Actin inhibition should, in principle, be modeled by decreasing actin polymerization rates, not motor abundance. The current approach lacks mechanistic justification. 8) The authors reference the Levy walk model as a representative of superdiffusive behavior. However, the authors should verify whether the distribution of step lengths in their model follows a power law distribution, which is a defining feature of Levy walks. Without such evidence, the mention of Levy walks in this context is inappropriate and potentially misleading. 9) The large difference between stimulated and experimental MSD in Figure (i.e. 5A and 5E) should be addressed. OVERALL) Significant revisions to the modeling framework are critically necessary, particularly to clarify the underlying physical assumptions and to ensure a more accurate/realistic representation of model implementation and cell migration dynamics. It is also unclear whether the observed anomalous diffusion patterns extend to other cell types. As it stands, it is the reviewers’ opinion the study does not meet the novelty or rigor expected for publication in Nature Communications and may only be appropriate for consideration in a more specialized or technical journal following substantial revision.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "We would like to thank the reviewers for their thoughtful feedback and critical evaluation. Their comments offered valuable opportunities to clarify the scope, novelty, and conceptual foundations of our work. While most of the feedback did not require substantial changes to the manuscript itself, we have responded in detail to address each concern thoroughly, including additional explanations, references, and clarifications where needed. We believe there may have been a misunderstanding regarding a central claim of our work. As clarified in our responses to Reviewers 3 and 4, our model—unlike prior frameworks—captures both sub-diffusive and super-diffusive migration within a unified mechanism. Crucially, these anomalous behaviors emerge naturally from the glassy adhesion dynamics we introduce, a mechanism essential for reproducing experimental observations, and which previous models lacking glassy dynamics cannot replicate. Most importantly, our model predicts the phenomenon of cell trapping for the first time and explicitly links its regulation to ECM viscoelasticity, which we directly validate through experiments. Collectively, these results represent a novel and significant conceptual advance in the field of cell migration modeling. The additional details and clarifications provided in the revised manuscript and supplementary materials directly address the reviewers’ concerns, and we believe the manuscript is substantially strengthened as a result. (The reviewer comments are in blue, our response in plain text, and the edits made to the manuscript are highlighted in yellow) Reviewer 1 & 2: [Detailed point-by-point response provided in the text] Reviewer 3 & 4: [Detailed point-by-point response provided in the text]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors addressed most of my concerns; however, there are still a few things with respect to the previously mentioned major and minor concerns, that need to be corrected; some may be due to submission errors since not all figures in the revised manuscript seem to be updated. Major concerns: 2. The manuscript file did not contain the updated Figure 5 with the additional replicate data in Fig. 5B and F, as shown in the rebuttal letter. 3. With respect to major concern 3, the authors refer to filopodia images shown in (Adebowale et al., 2021). First, I could not find a Figure S7 in the Supplementary Information showing filopodia images as explained in the rebuttal letter. Second, it is not appropriate to reuse the same images from a previous publication. The authors should show images representing their current data. Lastly, they should check the scale bars of the images shown in the rebuttal letter and in their previous paper. The way they are shown, it is impossible that the magnified inset has a width of almost 5 micrometers based on the 10-micrometer scale bar in the main image. Minor concerns: 5. Whereas the statistical testing information is provided, the legends of Figures 1-3 say “unpaired two tail test”. It probably should say “unpaired two-tailed Student’s t-test”. Please correct.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "Review of Revision for “Glassy Adhesion Dynamics Govern Transitions Between Sub-Diffusive and Super-Diffusive Cell Migration on Viscoelastic Substrates”, V. Sharma et al., Apr 2025, Nat. Comm We thank the authors for their detailed rebuttal. Nevertheless, after carefully reviewing their responses, we remain unconvinced that the manuscript demonstrates a major advance. While the work is technically solid in many ways, it our opinion that the level of originality and mechanistic advance is modest necessary. As such, this is likely a decision for the Editor as our conclusion is that the approach is largely rigorous but the overall advance is modest with regard to the originality needed for publication in Nature Communications. Specific comments: First, the authors claim that their model, through the introduction of “glassy adhesion dynamics,” uniquely captures both sub-diffusive and super-diffusive migration, and that prior models cannot reproduce anomalous diffusion. However, such behaviors can also be captured with conventional motor clutch frameworks. Models by Odde and coworkers, as well as Voituriez and collaborators, have shown that varying the balance between motor generated contractility and clutch engagement naturally yields regimes of trapping: long pauses with small steps and shorter pauses with larger displacements. For example, more sophisticated standard motor clutch models like that of Bangasser et al. are capable of generating a spectrum of migration regimes, including both sub and super diffusive behavior, when model parameter space is examined more broadly. By contrast, the authors’ rebuttal relies on selectively presented MSD curves of that model with unusually large error bars (as in their Fig. R5), and without disclosure of the parameter values used.Therefore, the authors do not provide computational evidence across a broad parameter space, and even worse nor experimental data demonstrating that a heterogeneous distribution of clutch dissociation rates is necessary for anomalous migration. The authors also maintain that their model is the first to predict trapping, defined as clutches remaining engaged at both ends of the cell with minimal displacement. The present study does not establish novelty in this regard. Previous models have shown that conditions of high clutch number and low motor force increase trapping times, previously called stalled regime by the Odde lab and others, with trapping understood as a direct consequence of motor clutch competition. In addition, anomalous pauses and trapping times not only can be modulated by motor-clutch ratios but can also arise from additional mechanisms such as protrusion nucleation dynamics, which are not considered in the present model. Extensive experimental data shows that, when cells do not form clearly defined plasma membrane protrusions, cells typically remain stationary and undergo sub-diffusive behavior at long times. As such, our primary concern in the original review was the limited novelty of the work. Having considered the rebuttal, this concern remains. While we appreciate the authors’ efforts to clarify their perspective and some technical details, their central claims regarding the uniqueness and necessity of glassy adhesion dynamics for anomalous migration, as well as the first-time prediction of trapping, are not supported in the context of prior literature. The present manuscript extends prior motor clutch models by introducing a power law distribution of clutch lifetimes. From a modeling standpoint, this is a modest modification: in real cells, any parameter will follow a distribution. Without new experimental insight or a fundamentally new mechanistic principle, simply changing a model parameter to follow a distribution is not sufficient to advance the field. Consequently, while the manuscript may represent an alternative parameterization and is well conducted, it our opinion that it does not constitute a sufficient conceptual advance for publication in Nature Communications. Minor comment: While on one hand the detailed response from the Authors is appreciated, it also raises concerns. A 17 page response with many rebuttal points being extremely long relative to what is needed to provide a focused response to a Review comment creates concerns that the authors are trying to respond with volume instead of focused and specific responses. More focused responses would improve the impact of the response.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to Reviewers We sincerely thank the reviewers for their continued evaluation of our manuscript. We are grateful that most concerns were resolved in the previous round and appreciate the additional feedback, which has helped us further refine the clarity and presentation of our work. In response to the remaining comments by Reviewers 1 and 2, we added a new experimental dataset with supporting analysis and incorporated figures that were inadvertently omitted from the previous submission. Reviewer 3’s concerns about novelty are now resolved through explicit mathematical proofs and parameter-sweep analyses demonstrating that conventional motor–clutch models cannot reproduce anomalous diffusion. These additions clarify that adhesion glassiness constitutes a genuine conceptual advance that unifies diffusive and anomalous regimes within a single mechanistic framework. Reviewer 4 commended the accessibility of our computational resources, and we have further expanded the codebase to include parameter-sweep capabilities across key model parameters. Together, these updates fully address the reviewers’ feedback and strengthen the manuscript’s clarity, rigor, and impact. The revised version highlights how incorporating glassy adhesion dynamics provides the missing physical mechanism required to explain anomalous cell migration within a unified, physics-based framework. (The reviewer comments are in blue, our response in plain text, and the edits made to the manuscript are highlighted in yellow) [Detailed point-by-point response provided in the text]"}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors have addressed our previous concerns and improved their manuscript.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "While this review still has some issues regarding the novelty of the findings, they are technically strong and will make a considerable contribution to the field. It is my recommendation that the manuscript be accepted for publication.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "We sincerely thank the reviewers for their positive assessment and constructive feedback throughout the review process. We are glad that the revised manuscript has addressed all concerns, and we appreciate their acknowledgment of the improvements made."}]

Decision letter