Beyond stiffness: interfacial slippage and active cellular matrix remodelling in focal adhesion dynamics

Abstract

Focal adhesion (FA) dynamics and cell migration depend sensitively on the mechanical properties of the extracellular matrix, yet substrate stiffness alone cannot account for the distinct behaviours observed on natural versus artificial materials. In this review, we synthesize experimental findings on cells interacting with collagen I matrices and viscoelastic hydrogels, and identify the key physical mechanisms that govern adhesion stability, turnover, and migratory efficiency. Collagen I substrates are characterized by structural anisotropy, spatial heterogeneity, and a broad spectrum of relaxation times, in contrast to the simplified and often isotropic response of artificial hydrogels. Based on these observations, we highlight the central role of interfacial slippage at the FA–substrate biointerface enabled by matrix remodelling and multi-timescale viscoelastic dissipation. We further discuss how cell-generated forces can induce collagen reorganization and surface tension gradients, potentially driving Marangoni-like flows that contribute to dynamic matrix redistribution. These processes collectively maintain adhesions in a state between stabilization and turnover, favourable for persistent migration. Finally, we consider the interplay between these mechanical mechanisms and mechanosensitive signalling, particularly involving Piezo1 ion channels. This perspective emphasizes that effective cell–matrix coupling arises from the ability of the substrate to dissipate and reorganize distribution of mechanical energy, rather than from stiffness alone.

Introduction

Collective cell migration is a fundamental process underlying a wide range of physiological and pathological phenomena. It drives morphogenesis, where coordinated cell movements shape tissues and organs (Barriga and Mayor 2019; Shellard and Mayor 2020; Pajic-Lijakovic et al. 2026). In adult organisms, it is essential for wound healing, enabling epithelial sheets to close gaps and restore barrier function. Conversely, perturbed regulation of collective migration is a hallmark of cancer progression, where malignant cells invade surrounding tissues and disseminate to form metastases (Clark and Vignjevic 2015; Millar et al. 2017; Riehl et al. 2021). Importantly, migrating cells do not maintain a fixed adhesive state, but dynamically adapt cell–cell and cell–matrix adhesions in response to mechanical loading and environmental resistance. In this sense, collective migration can be viewed through a “mechanostat-like” framework, analogous to the concept originally introduced by Harold Frost, in which biological systems continuously regulate their mechanical state to maintain functional force balance (Frost 1987).

The efficiency and speed of collective migration are governed by the dynamics of FAs (Hynes 2002; Oliva et al. 2025; Ciccone et al. 2025; Sharma et al. 2026). These specialized multi-protein complexes serve as mechanical linkages between the intracellular actomyosin cytoskeleton and the extracellular matrix (ECM). Rather than acting as purely motile entities, cells continuously explore and adapt to their mechanical environment through dynamic regulation of adhesions, force transmission, and mechanosensitive signalling. Consequently, a defining feature of cell–matrix interactions is the coexistence of multiple intrinsic timescales. On short timescales (seconds to tens of seconds), actomyosin contractility generates fluctuating forces that are transmitted to the substrate through molecular clutches composed of integrins, adaptor proteins such as talin and vinculin, and actin filaments. The molecular clutch framework (Chan and Odde 2008) provides a conceptual basis for understanding how force transmission depends on substrate mechanics and clutch kinetics.

On longer timescales (minutes to hours), FAs undergo continuous remodelling, including changes in clutch binding, adhesion size, mechanosensitive signalling and structural mechano-adaptation. This remodeling involves pathways such as the activation of mechanosensitive ion channels, including Piezo1, and associated calcium influx, which modulate actomyosin contractility and FA remodeling (Pthak et al., 2014; Yao et al. 2022; Oliva et al., 2025). Thus, FA dynamics emerges from the interplay between rapid mechanical fluctuations and slower biochemical adaptation depending on substrate viscoelasticity. Despite substantial progress, there remains a need for an interpretative framework capable of unifying recent findings on multiscale FA rearrangements with the emerging roles of substrate viscoelasticity, interacting timescales, and interfacial mechanical effects.

In an effort to fill this gap, we adopt a recently-proposed conceptual framework in which long-term FA dynamics is interpreted through active wetting and de-wetting (Bruinsma and Sackmann, 2001). Wetting and de-wetting have been discussed in the context of collective migration of epithelial monolayers and rearrangement of epithelial clusters on substrate matrices and more generally offer a framework for understanding dynamic cell–substrate interactions during mechanical exploration and adaptation. (Pérez-González et al. 2019; Pajic-Lijakovic and Milivojevic 2023a). In this picture, FAs behave as active interfacial domains whose spreading is governed by a balance between adhesion energy and cohesion energy per unit area (de Gennes, 1985). When adhesion dominates, the FA undergoes wetting and expands; when this balance is reversed, de-wetting occurs, leading to shrinkage and eventual detachment. Importantly, this balance is not static but is dynamically regulated by the mechanical response of the substrate. The viscoelasticity and interfacial slip of the matrix play central roles in modulating this energy balance. In particular, collagen I networks are a primary structural component of the ECM in connective tissues, providing a prototype example of a matrix exhibiting persistent interfacial slip, which effectively reduces adhesion energy at the rear of the cell (Pelham and Wang 1997). Various factors influence FA slippage on collagen I substrates such as: (i) the nature of the integrin-ligand bonds (Yao et al. 2022), (ii) enzymatic degradation of collagen I network (Narasimhan et al., 2025), and (iii) disruption of the network caused by cell tractions (Pajic-Lijakovic and Milivojevic 2023b). These processes control de-wetting, enabling rear detachment and preventing mechanical arrest. In contrast, substrates with limited slip promote excessive adhesion stabilization, leading to more stable, less dynamically remodelling adhesions. We will discuss here the impact of slippage effects and the viscoelastic response of collagen I substrates on FA wetting and de-wetting. In the following, we compare the rheological behaviour of physically cross-linked collagen I substrates and artificial viscoelastic substrates.

Artificial hydrogels—such as PEG-based systems, micro-patterned polyacrylamide (PAAm), alginate gels, and their composites—have provided powerful reductionist platforms to investigate cellular mechanosensing (Oliva et al., 2025; Ciccone et al. 2025; Sharma et al. 2026). These systems have been particularly useful in isolating the role of specific mechanical parameters, often yielding responses dominated by a single or narrowly distributed set of relaxation modes. Such studies have clarified how FA dynamics depend on well-defined relationships between substrate stiffness and characteristic relaxation times. Building on these insights, it is useful to consider more complex substrates, such as collagen I networks, which exhibit a broad and hierarchical spectrum of relaxation and retardation times arising from their fibrous architecture and dynamic crosslinking (Pajic-Lijakovic et al. 2022). This richer dynamical behaviour enables a more intricate coupling between cellular activity and matrix response. In this context, the mechanical behaviour of FAs on simplified substrates is often governed by a relationship between substrate stiffness and a dominant relaxation timescale. While recent studies on artificial substrates like PEG and PAAm demonstrate that FA surface area is highly dependent on such relationships (Oliva et al., 2025; Ciccone et al. 2025), FAs on collagen I can, in contrast, move beyond this “stiffness trap” through the interplay of multiple relaxation modes (Pajic-Lijakovic et al. 2022).From a physical perspective, an effective substrate for FA-mediated migration must reconcile competing mechanical requirements across timescales: (i) elastic resistance at short times to support force transmission, (ii) viscoelastic dissipation at intermediate times to prevent excessive stress accumulation, and (iii) sufficient long-time compliance or interfacial mobility to enable adhesion turnover and cell translocation (Pajic-Lijakovic et al. 2022). More generally, these requirements reflect the need for cells to dynamically regulate their mechanical coupling to the substrate, maintaining a balance between stable attachment and adaptive remodeling. Substrates that exhibit multi-scale viscoelasticity and allow interfacial mobility can support such behaviour. Collagen I networks provide a particularly well-characterised example of this class of materials, owing to their hierarchical architecture and dynamic interfacial properties. In this work, we use collagen I as a model system to explore how these physical features contribute to mechanically adaptive cell–matrix interactions. Using a multi-regime Zener-type viscoelastic framework, we analyze how the spectrum of relaxation and retardation times regulates FA stability and turnover. In this context, FAs can remain mechanically engaged during short-time contractile events while retaining their ability to undergo the long-term wetting and de-wetting transitions required for migration.

The remainder of the review develops this conceptual framework in a stepwise manner. We first examine how the hierarchical architecture of collagen I networks gives rise to a broad spectrum of viscoelastic relaxation and retardation times, enabling both elastic force transmission and progressive energy dissipation during cell–matrix interactions. We then discuss how this multiscale viscoelastic response promotes interfacial slippage between FAs and the substrate, thereby modulating the effective adhesion energy and establishing front–rear asymmetry in focal adhesion wetting and de-wetting. Finally, we describe how these mechanically-regulated interfacial processes govern focal adhesion turnover, Piezo1-mediated mechanosensing, and the adaptive balance between stable attachment and efficient collective cell migration. Accordingly, we argue that efficient collective migration emerges from an intrinsic front–rear asymmetry within ventral FAs, in which adhesion growth and stabilization at the leading edge are balanced by slip-mediated, force-assisted de-wetting and disassembly at the trailing edge. This dynamic asymmetry provides the mechanistic link between collagen I viscoelasticity, interfacial mechanics, and adaptive mechanosensing during collective migration.

Phenomenological description of coupling systems

A deeper understanding of FA remodelling requires consideration of the coupled interactions between the ventral actomyosin cytoskeleton, the ensemble of molecular clutches constituting the FA, and the viscoelastic substrate matrix. Ventral actomyosin stress fibers generate oscillatory forces due to active myosin-driven contractions that are transmitted through FAs to the substrate. These cell-generated tractions induce structural remodelling of the substrate, resulting in both energy storage and dissipation. A fraction of the energy released by the substrate is transmitted back toward the FA and cytoskeleton, thereby closing the mechanical feedback loop. These three subsystems are effectively coupled in series (Chan and Odde 2008) implying that the force is conserved across them in the absence of slippage at the FA–substrate biointerface. However, slip introduces additional energy dissipation and partially relaxes this force continuity, enabling dynamic redistribution of stresses between the FA and the substrate. Before analysing this coupled system in detail, we introduce a phenomenological description of each component.

Ventral actomyosin stress fibers

Ventral stress fibers are composed of bundles of actin filaments and non-muscle myosin II filaments, which are interconnected and cross-linked by α-actinin and various actin-binding proteins (Tojkander et al. 2012; Hohmann and Dehghani 2019). This structural interconnection of stress fibers facilitates the sliding of myosin along the actin filaments. Mature ventral stress fibers are made up of 10–30 actin filaments (Rajan et al. 2023). These fibers are anchored at both ends to focal adhesions (FAs), thereby linking adjacent FAs across the ventral surface of the cell (Vallenius 2013). As a result, in the ventral cytoskeletal network, FAs serve as mechanical junctions that integrate the contractile stress fiber bundles into a cohesive, load-bearing system (Hohmann and Dehghani 2019). The behaviour of stress fibers resembles that of rigid chains, consistent with the observation that the persistence length of filaments is considerably greater than their contour length. Stress fibers are interconnected and stabilized by actin-binding proteins. Additionally, the diameter, surface density, and interconnectivity of stress fibers can vary across the ventral region of the cytoskeleton. The organisation and homogeneity of the ventral cytoskeleton significantly influence the dynamics of FAs.

Structural organisation of focal adhesions on the ventral cytoskeleton

A single FA is composed of thousands of molecular clutches that mechanically couple the actomyosin cytoskeleton to the substrate matrix (Chan and Odde 2008). However, only a small fraction of these clutches are active at any given time. The density of active clutches depends on substrate stiffness because stiffer substrates promote focal adhesion growth, with typical FA areas ranging from 50 to 100 $\:{{\upmu\:}\text{m}}^{2}$ (Swaminathan and Waterman 2016). Each FA contains hundreds to thousands of talin molecules per $\:{{\upmu\:}\text{m}}^{2}$ (Lemke et al. 2019).

Although talin possesses 11 potential vinculin-binding sites (VBSs), only a limited number are occupied simultaneously, resulting in an average of approximately two vinculin molecules bound per talin in mature FAs (Case et al. 2015). Vinculin functions as a molecular “lock” that stabilizes talin in its unfolded state by preventing refolding of the talin rod domain (Yao et al. 2014). As long as vinculin remains bound to VBSs, talin is mechanically constrained from returning to its native helical bundle conformation. When mechanical tension is reduced—for example, upon myosin inhibition—the vinculin–talin complex remains stable for about 30 min before vinculin dissociates (Carisey et al. 2013). Following dissociation, the talin rod rapidly refolds to its native conformation within less than one minute under low-force conditions. A schematic representation of an ensemble of molecular clutches representing part of an FA is shown in Fig. 1:

Fig. 1

A schematic representation of an ensemble of molecular clutches forming part of an FA, integrated within a group of FAs, which are linked to actomyosin stress fibers and the substrate. Focal adhesions constitute a sophisticated protein network capable of dynamic remodelling in conjunction with the ventral cytoskeleton and the substrate matrix

Each individual clutch consists of a single integrin heterodimer bound to an extracellular ligand and mechanically linked intracellularly to a single talin molecule (Bachir et al. 2014). Integrins are organized into nano-scale clusters that connect extracellular ligands to talin and other FA proteins. Nevertheless, only a small subset of integrins actively bears mechanical load. Additional key components of the clutch include: (i) kindlin, which is essential for integrin activation; (ii) α-actinin, which provides an auxiliary linkage between talin and actin; (iii) focal adhesion kinase (FAK), which serves as a signalling regulator; (iv) paxillin, a major scaffold protein; and (v) zyxin, which is recruited under high tension and contributes to the reinforcement and repair of the actin–clutch interface under mechanical stress (Moser et al. 2009; Lele et al. 2006; Theodosiou et al. 2016). The clutch activity and stability of an FA depend on the viscoelasticity of substrate matrices.

Collagen I substrates vs. artificial viscoelastic hydrogel substrates

Collagen networks consist of triple-helical strands of collagen polypeptides that self-assemble into larger fibers and form cross-linked, overlapping network structures (Shoulders and Raines 2009). The key characteristics of physically cross-linked collagen I substrates include: (i) anisotropic rearrangement of collagen fibers, (ii) inhomogeneous fiber distributions driven by cell-generated tractions, and (iii) a mechanical response characterized by a broad spectrum of relaxation and retardation times under external loading. Anisotropy—characteristic of fibrilar collagen—provides a geometric form of bioactivity. Aligned fibers act as physical “tracks” that instruct cells to polarize and migrate efficiently along a specific axis. Standard artificial hydrogels such as: PEG-based hydrogels, micro-patterned PAAm, alginate hydrogels, and their composites are typically isotropic (uniform in all directions), providing a uniform density of ligands. While this makes experiments workable, it lacks the “bioactive complexity” of collagen, which is naturally inhomogeneous. These variations in ligand density and pore size in collagen create micro-gradients that can drive more complex cellular responses, such as branching or specialized protrusion formation.

The structural dynamics of collagen I networks span multiple time scales, ranging from milliseconds to hours (Pryse et al. 2003; Gautieri et al. 2012; Nam et al. 2016; Pajic-Lijakovic et al. 2022) as shown schematically in Fig. 2:

Fig. 2

A schematic representation of multi-scale remodelling of collagen I substrate caused by cell traction. Time scales from nanoseconds to milliseconds are associated with intra-filament interactions governed by individual fiber conformations, whereas time scales on the order of seconds reflect inter-fiber interactions and the organization of fibers within mesoscopic domains. Processes occurring over minutes correspond to sliding between network domains, while longer timescales, from tens of minutes to hours, are associated with large-scale rearrangement of these domains

Time scales from nanoseconds to milliseconds are associated with intra-filament interactions governed by individual fiber conformations, whereas time scales on the order of seconds reflect inter-fiber interactions and the organization of fibers within mesoscopic domains. Processes occurring over minutes correspond to sliding between network domains, while longer timescales, from tens of minutes to hours, are linked to large-scale rearrangement of these domains (Pryse et al. 2003; Legant et al. 2009; Gautieri et al. 2012; Nam et al. 2016).

Viscoelasticity plays a central role in enabling matrix remodelling by allowing time-dependent stress relaxation. In contrast to purely elastic substrates, which resist deformation with a nearly time-independent response, viscoelastic matrices can dissipate energy and progressively reorganise under cell-generated forces. This behaviour enables cells to restructure the surrounding network, modify local ligand distribution, and adapt their mechanical coupling to the substrate. The broad spectrum of relaxation and retardation times in collagen I substrates arises from their hierarchical architecture, spanning molecular to macroscopic scales (Pryse et al. 2003; Pajic-Lijakovic et al. 2022). Unlike simpler artificial hydrogels engineered with a narrow range of relaxation times, collagen dissipates energy through multiple distinct microstructural processes. During migration, cells exert traction forces on substrates in the range of approximately 10–100 nN (Hall et al. 2016; Steinwachs et al. 2016; Emon et al. 2021), which are several orders of magnitude larger than the ~ 20 pN required to disrupt individual electrostatic and hydrophobic bonds within collagen I networks (Nam et al. 2016). Although forces at the single-bond level are small, the collective transmission of cellular traction through FAs can rupture multiple intermolecular bonds and crosslinks within the network. Indeed, forces of only a few nNs are sufficient to stretch individual collagen fibrils to strains of up to ~ 20% (Gautieri et al. 2012), placing them within a nonlinear, and potentially damage-inducing, mechanical regime. Moreover, cellular traction forces are generally insufficient to disrupt the underlying network structures of artificial hydrogels.

FA morphology and dynamics depend sensitively on the mechanical properties of the substrate. On fibrous and remodeling matrices such as collagen I, FAs are typically smaller, more elongated, and more dynamic (Wang et al. 2003; Subramanian et al. 2025), particularly when aligned with underlying fibers, which can promote quasi-one-dimensional force transmission (Friedl and Wolf 2003, 2010). On more homogeneous viscoelastic substrates, such as PEG or polyacrylamide gels, FAs tend to be larger, less elongated, and more stable (Chaudhuri et al. 2016). These differences can be interpreted as reflecting distinct regimes of cell–substrate coupling, associated with differences in structural organisation and the spectrum of relaxation times. In fibrous matrices, cells interact with a hierarchy of timescales, enabling dynamic adjustment of force transmission, whereas more uniform substrates tend to favour more stabilised adhesion states. In this context, cells can effectively engage with different timescales of the substrate, with rapid relaxation limiting tension build-up and slower relaxation supporting sustained force transmission along aligned structures.We next consider the the influence of the surface properties and viscoelastic nature of collagen I substrates and FAs on the maturation and disassembly of FAs.

Surface characteristics of focal adhesion and substrates: wetting and de-wetting of FAs

The mismatch between adhesion and cohesion energy per unit surface influences FA wetting (extension) and de-wetting (compression) (Bruinsma and Sackmann, 2001; Nicolas et al. 2004). These processes are governed by a dynamic spreading factor $\:{S}_{FA}\left(r,t\right)$. The spreading factor can be formulated as: $\:{S}_{FA}={e}_{a}-{e}_{coh}$ (where $\:{e}_{a}$ is the FA-substrate adhesion energy per unit surface and $\:{e}_{coh}$ is the cohesion energy of FA complex). The adhesion energy can be expressed as: $\:{e}_{a}\left(r,t\right)={\rho\:}_{I}{E}_{bI}$ (where $\:{\rho\:}_{I}$ is the surface density of integrin molecules and $\:{E}_{bI}$ is the integrin-ligand bond energy). The surface density $\:{\rho\:}_{I}$ is inhomogeneous per single FA (Kanchanawong et al. 2010). The adhesion energy $\:{e}_{a}$ can be estimated for the surface density of integrin in the range of $\:500-1500\:\frac{\text{m}\text{o}\text{l}\text{e}\text{c}\text{u}\text{l}\text{e}\text{s}}{{{\upmu\:}\text{m}}^{2}}$ (Kanchanawong et al. 2010) and the bond energy for higher affinity integrin-ligand complexes in the range of $\:20-25\:{k}_{B}T$ (Burgos-Bravo et al. 2018), such that $\:{e}_{a}\sim1x{10}^{-1}\:\frac{\text{m}\text{N}}{\text{m}}$.

The cohesion energy of the FA complex $\:{e}_{coh}$ can be interpreted in a coarse-grained, thermodynamic sense in terms of an effective surface tension of the FA and expressed as: : $\:{e}_{coh}=2{\gamma\:}_{FA}$. The surface tension represents the surface energy per unit surface area. Stretching and tilting of molecular clutches during FA growth induces lateral binding of vinculin molecules, which enhance the cohesion of the FA network. When $\:{S}_{FA}>0$, an FA undergoes wetting (extension), which facilitates the maturation of the FA. In this context, FA extension encourages the recruitment of new active clutches to preserve its structural integrity. The stretching and tilting of clutches attract vinculin to unfold talin and strengthen the internal protein network, thereby enhancing the cohesion energy of the FA. An increase in cohesion energy in comparison to adhesion energy results in a reduction of the spreading factor. When the spreading factor is equal to zero ($\:{S}_{FA}=0$), the FA attains a matured, quasi-steady state. In this case, $\:{\gamma\:}_{FA}$ can be estimated as: $\:{\gamma\:}_{FA}\sim\frac{1}{2}\langle{e}_{a}\left(r,\tau\:\right)\rangle$, which is equal to $\:{\gamma\:}_{FA}\sim0.5x{10}^{-1}\:\frac{\text{m}\text{N}}{\text{m}}$.

For a comprehensive analysis, it is imperative to examine the surface tension of FAs, the surface tension of the collagen substrate, and the interfacial tension between the FA and the substrate. After this, it is necessary to characterize the extension (wetting) and retraction (de-wetting) of FAs on both homogeneous, isotropic artificial substrates and inhomogeneous and anisotropic collagen I substrates.

Surface tension of FA

The surface tension of an FA in contact with a liquid medium $\:{\gamma\:}_{FA}\left(r,t\right)$ represents a measure of FA cohesion. It is a space-time-dependent physical parameter. Because the remodelling of FA is highly dissipative active systems, our formulation utilizes an effective thermodynamic framework mapped onto a non-equilibrium steady states. Surface tension is typically considered a state variable and can be thermodynamically articulated via the Virial equation, which depends on the surface packing density of the system’s components (Myers and Freed 1993). As a result, the dynamic surface tension of the FA complex, denoted as $\:{\gamma\:}_{FA}$, is influenced by: (i) the surface packing density of clutches ρ(r, t) and (ii) the concentration of intracellular calcium [$\:{C}_{a}^{2+}]$ (labelled as $\:C$), which has a feedback impact on the interactions between clutches. Consequently, coarse-grained interactions between active molecular clutches, which are in a non-equilibrium steady state can be described by the second virial coefficient. The FA surface tension $\:{\gamma\:}_{FA}={\gamma\:}_{FA}\left(\rho\:,C\right)$ can be formulated as:

$$\:\frac{{{\gamma\:}_{\:}}_{FA}\left(r,t\right)}{{e}_{eff}\rho\:}=1-{B}_{2}\rho\:\left(r,t\right)$$

(1)

where $\:{e}_{eff}\:$ is the effective binding energy per single active clutch, and $\:{B}_{2}={B}_{2}\left(C\right)$ signifies the second virial coefficient, which is contingent upon the intracellular calcium level. This coefficient can be articulated as: $\:{B}_{2}\left(C\right)=-\frac{1}{2}{\int\:}_{0}^{\infty\:}{d}^{2}{r}_{12}{\int\:}_{0}^{2\pi\:}\left({e}^{-\frac{U\left({r}_{12},{\theta\:}_{12},C\right)\:}{{k}_{B}T}}-1\right)d{\theta\:}_{12}$. Here, $\:{r}_{12}$ indicates the distance between two clutches, $\:{\theta\:}_{12}$ refers to the orientation angle between neighboring clutches, and $\:U\left({r}_{12},{\theta\:}_{12},C\right)$ represents the pair interaction potential between clutches, encompassing both positional interactions $\:U\left({r}_{12}\right)$ and orientational interactions $\:U\left({\theta\:}_{12},C\right)$. The positional interactions may exhibit attractive or repulsive characteristics, which can be modeled by the Lennard-Jones potential $\:U\left({r}_{12},C\right)=4 \epsilon \left(C\right)\left[{\left(\frac{d}{{r}_{12}}\right)}^{12}-{\left(\frac{d}{{r}_{12}}\right)}^{6}\right]$ (where $\:\epsilon$ is the strength of the interactions and $\:d$ is the distance at which the potential $\:U\left({r}_{12},C\right)$ is zero). The orientational potential can be formulated as: $\:U\left({\theta\:}_{12},C\right)=\frac{1}{2}{k}_{\theta\:}\left(C\right){{\varDelta\:\theta\:}_{12}}^{2}$ (where $\:{k}_{\theta\:}$ is the torsional spring constant of a single clutch). During the extension of the FA, the potential $\:U\left({r}_{12},C\right)$ is attractive. In this scenario, clutches tilt while maintaining alignment such that $\:{\varDelta\:\theta\:}_{12}\to\:0$ leading to the orientational potential $\:U\left({\theta\:}_{12},C\right)\to\:0$ approaching zero as well. Conversely, during the compression of the FA, the alignment of clutches is perturbed, resulting in an increase in the orientational potential $\:U\left({\theta\:}_{12},C\right)$, while the positional potential $\:U\left({r}_{12}\right)$ becomes repulsive. As a result, the virial coefficient $\:{B}_{2}$ adheres to the condition: $\:{B}_{2}^{exten}<{B}_{2}^{compress}$. An increase in the packing density of active clutches, coupled with a reduction in the second virial coefficient $\:{B}_{2}$, leads to an increase in the surface tension $\:{\gamma\:}_{FA}$, as well as an increase in cohesion energy relative to adhesion energy, which in turn results in a decrease in the spreading factor during FA wetting. When the spreading factor becomes negative, the FA undergoes compression, which intensifies the interactions among clutches, thereby increasing the second virial coefficient $\:{B}_{2}$ and triggering the disassembly of an FA during de-wetting. An increase in the intracellular calcium concentration, resulting from the concurrent activation of Piezo1 channels, may lead to a reduction in the virial coefficient $\:{B}_{2}$. This reduction subsequently contributes to a decrease in the virial coefficient $\:{B}_{2}$ and an enhancement in the cohesion of an FA (Oliva et al., 2025). Furthermore, a reduction in the lifetime and stability of the FAs results in a decrease in the surface tension $\:{\gamma\:}_{FA}$.

The surface tension $\:{\gamma\:}_{FA}$ is inhomogeneously distributed along the FA. The surface density of clutches is greater in the peripheral region of biological cells compared to the core region (Kumar et al. 2016). This observation aligns with the fact that clutches in the peripheral region undergo greater mechanical stress, which activates a larger number of clutches in order to maintain the FA’s structural integrity. Clutches situated in the peripheral region are more stretched and tilted, resulting in the binding of vinculin to active sites of unfolded talin molecules, thereby enhancing both the cohesion of FAs and the surface tension $\:{\gamma\:}_{FA}$. Consequently, the gradient of the surface tension $\:\overrightarrow{\nabla\:}{\gamma\:}_{FA}$ influences the rearrangement of clutches and generates mechanical stress in FAs during cell tractions.

In addition to FA surface tension, the surface tension of the collagen I substrate—and its spatial distribution—regulate FA wetting and de-wetting both directly, through the spreading factor, and indirectly, by inducing slip along the biointerface.

The surface characteristics of the collagen substrate matrix

Cellular traction forces (~ 10–100 nN) greatly exceed single-bond strengths (~ 20 pN) in collagen I networks, so force transmission through FAs act collectively to rupture multiple bonds and crosslinks (Nam et al. 2016; Pajic-Lijakovic et al. 2022). This indicates that cells have the ability to cause considerable changes in both the volume and surface structure of collagen I gel, which in turn affects the persistence of migration (Clark et al. 2022). Changes in the volumetric arrangement of the collagen I matrix, induced by movement of free, already disrupted collagen fibers leads to an increase in matrix stiffness, whereas alterations in the surface arrangement affect the surface tension of the matrix. The volumetric and surface rearrangements of collagen I gel are interconnected, as described by the Young-Laplace equation (Pajic-Lijakovic and Milivojevic 2023b).

Clark et al. (2022) pointed out that the migration of small cell clusters made by 2–50 cells on a collagen I substrate establishes a concentration gradient of collagen fibers. The resultant distribution of collagen concentration around the cell cluster is asymmetric such that the collagen concentration near the cluster front region is $\:\sim30\%$ lower than that near the cluster rear (Clark et al. 2022). It means that a single FA can experience the concentration gradient of collagen fibers induced by cell tractions.

Surface tension of substrate matrix

The surface tension represents the surface free energy per unit surface area. The establishment of a surface concentration gradient of non-gelled collagen film results in a gradient of collagen surface tension. Kezwon and Wojciechowski (2014) pointed out that an increase in collagen concentration from 1 mg/ml to 4 mg/ml (in experiments without cells) reduces the collagen I surface tension $\:{\gamma\:}_{m}$ from 62 mN/m to 57 mN/m at 21 $\:{}^{o}C$.

The establishment of a concentration gradient of collagen fibers triggers two mechanisms responsible for the movement of collagen fibers. One is the Marangoni flow, which relies on the gradient of surface tension $\:\overrightarrow{\nabla\:}{\gamma\:}_{m}$, while the other is diffusion. The gradient of surface tension $\:\overrightarrow{\nabla\:}{\gamma\:}_{m}$ is expected to promote movement of collagen fibers from the regions of lower surface tension (and higher collagen concentration) to the regions of higher surface tension and (lower collagen concentration). The diffusion of collagen fibers acts in the same direction from the region of higher concentration to the regions of lower concentration of free fibers. The interplay between these two mechanisms can be discussed in the context of the dimensionless Marangoni number expressed as: $\:{M}_{a}=\frac{\lceil\frac{d{\gamma\:}_{m}}{d{c}_{f}}\rceil\varDelta\:{c}_{f}L}{{\eta\:}_{m}{D}_{m}}$ (where $\:\varDelta\:{c}_{f}$ is the difference in collagen surface concentration, $\:{\eta\:}_{m}$ is the viscosity of dense collagen solution, $\:{D}_{m}$ is the diffusion coefficient of collagen fibers, and $\:L$ is the characteristic length scale over which the collagen concentration gradient exists). The $\:{M}_{a}$ number can be estimated for: (i) the concentration of collagen of $\:2\:\frac{\text{m}\text{g}}{\text{m}\text{l}}$ and an increase in concentration $\:\varDelta\:{c}_{f}$ of 30% (Clark et al. 2022), (ii) $\:\frac{d{\gamma\:}_{m}}{d{c}_{f}}\sim1.67\:\frac{\text{N}{\text{m}}^{2}}{\text{k}\text{g}}$(Kezwon and Wojciechowski 2014), (iii) the characteristic length of $\:L=10\:{\upmu\:}\text{m}$ (Clark et al. 2022), (iv) the viscosity of dense collagen solution $\:{\eta\:}_{m}\sim10\:\text{P}\text{a}\text{s}$ (Frantz et al. 2010), and (v) diffusion constant $\:{D}_{m}\sim1X{10}^{-11}\:\frac{{\text{m}}^{2}}{\text{s}}$ (Kihara et al. 2013). The calculated $\:{M}_{a}$ number is $\:{M}_{a}\sim{10}^{5}$. This estimation suggests that the Marangoni flow may dominantly influence the transport of free, already-disrupted fibers. The evolution of collagen concentration can be expressed as:

$$\:\frac{\partial\:{c}_{f}}{\partial\:t}={\mu\:}_{M}\overrightarrow{\nabla\:}\left[{c}_{f}\overrightarrow{\nabla\:}{\gamma\:}_{m}\left({c}_{f}\right)\right]$$

(2)

where the gradient of substrate surface tension is equal to: $\:\overrightarrow{\nabla\:}{\gamma\:}_{m}=\frac{\partial\:{\gamma\:}_{m}}{\partial\:{c}_{f}}\overrightarrow{\nabla\:}{c}_{f}$ and $\:{\mu\:}_{M}$ is the Marangoni coefficient, which quantifies the mobility of collagen fibers. Within this framework, Marangoni-like redistribution may generate a positive feedback loop for migration. The Marangoni force acting in the direction of migration is one of the reasons why collagen I is superior to artificial hydrogels. It transforms the substrate from a passive obstacle into an active propulsion system that specifically facilitates the long-term processes of wetting and de-wetting. The surface tension of the substrate, accompanied by the surface tension of an FA contributes, to the FA-substrate interfacial tension.

FA-substrate interfacial tension

The surface tension of FA, surface tension of substrate, and adhesion energy between them contribute to the FA-substrate interfacial tension $\:{\gamma\:}_{FA-m}$. The interfacial tension can be expressed at the mesoscopic scale through a coarse-graining procedure as:

$$\:{\gamma\:}_{FA-m}={\gamma\:}_{FA}+{\gamma\:}_{m}-{e}_{a}$$

(3)

An increase in the adhesion between integrins and ligands by increasing the surface density of these bonds results in a reduction of interfacial tension. The gradient of interfacial tension is generated by: (i) the inhomogeneous distribution of surface density of molecular clutches and the interactions between clutches, (ii) the movement of collagen fibers driven by the Marangoni effect, and (iii) the inhomogeneous distribution of the surface density of integrin-ligand bonds. This can be represented as: $\:\overrightarrow{\nabla\:}{\gamma\:}_{FA-m}=\overrightarrow{\nabla\:}{\gamma\:}_{FA}+\overrightarrow{\nabla\:}{\gamma\:}_{m}-\overrightarrow{\nabla\:}{e}_{a}$. The interfacial tension and interfacial tension gradient can induce slip effects along the biointerface, which have a feedback impact on the coupling between FA and substrate and, on that basis, the FA wetting/de-wetting.

Slippage acts as a dynamic thermostat (a mechanostat) for the spreading factor by keeping the spreading factor near zero for a long time. Without slipping, the adhesion energy $\:{e}_{a}$ would climb rapidly as the cell pulls on the substrate, quickly overwhelming the cohesion $\:{e}_{coh}$ and forcing the spreading factor $\:{S}_{FA}$ to a high positive value. On collagen I substrates, the slipping along the biointerface prevents this “energy accumulation” by constantly dissipating the work done by the actin cytoskeleton. By the Marangoni flow the displacement of FA $\:{\overrightarrow{u}}_{FA}\left(r,t\right)$ is prevented of reaching its “maximal elastic value.” Both adhesion and cohesion energy per unit surface depend on the distribution and activity of mechanosensitive Piezo1 channels.

The distribution of Piezo1 channels around FA in epithelial cells on collagen I vs. artificial substrates

Piezo1 participates in a bidirectional mechanochemical feedback loop with actomyosin contractility that regulates FA rearrangement in a threshold-dependent manner. Contractile forces generated by the actomyosin cytoskeleton activate Piezo1 channels and induce a Ca²⁺ influx that, at moderate levels, promotes activation and recruitment of molecular clutches (e.g., integrins, talin, vinculin), thereby enhancing force transmission and stabilizing adhesions. This Ca²⁺ signalling also reinforces contractility (via MLCK/ROCK pathways), forming a short-time positive feedback. However, when Ca²⁺ levels become sufficiently elevated or sustained, calpain is activated, triggering proteolytic cleavage of adhesion components and accelerating clutch disengagement and FA turnover. As adhesions remodel, traction forces are redistributed, which alters the local mechanical input to Piezo1 and introduces a delayed negative feedback. Together, this interplay produces spatially heterogeneous, pulsatile Piezo1 activity and dynamically reorganizes FAs, enabling cells to continuously tune adhesion strength and cytoskeletal coupling to their mechanical environment.

Piezo1 molecules cluster around FAs in non-transformed cells such as epithelial cells and fibroblasts solely when there is sufficient sustained actomyosin contractility (Yao et al. 2022). Slippage phenomena at the FA-substrate biointerface, particularly evident in collagen I substrates, may diminish the intensity of the traction force. Under these conditions, Piezo1 channels were observed to exhibit a more uniform distribution across the plasma membrane of non-transformed cells (Yao et al. 2022). Conversely, actomyosin contractility is enhanced and slippage effects are less pronounced in fibronectin-coated glass substrates, where Piezo1 channels tend to cluster near the ventral FAs.

The primary physical mechanism that initiates the movement of Piezo1 channels towards FAs is the gradient of membrane surface tension, whereas diffusion facilitates the uniform distribution of these channels across the membrane (Pajic-Lijakovic et al. 2026). Within this conceptual framework, the redistribution can be interpreted as a membrane-scale analogue of a Marangoni-like process. The gradient is predominantly produced due to the formation of local curvatures surrounding FAs. Such local curvature induces conformational alterations in the actin-cortex and lipid rearrangement, resulting in a slight expansion of the membrane surface area and, consequently, an increase in membrane surface tension (Pajic-Lijakovic et al. 2026). The curvatures surrounding FAs are influenced by the size, stability, and contractility of the cell. Given that FAs are smaller and exhibit greater dynamism when cells are cultured on collagen I substrates, these FAs generate smaller curvatures and a reduced gradient of membrane surface tension, which is insufficient to guide the migration of Piezo1 channels towards FAs. It is important to emphasize that this mechanism describes a proposed membrane-intrinsic lateral transport process, whereby Piezo1 channels redistribute along the membrane in response to surface tension gradients generated by local curvature heterogeneities and coupled interactions between the lipid bilayer and the cortical cytoskeleton. Diffusion describes stochastic lateral motion of membrane proteins within the membrane plane, acting in opposition to energy-gradient–biased redistribution. In contrast, the Marangoni effect in the FA–substrate system is described at a coarse-grained level in terms of an effective interfacial tension gradient emerging from FA–ECM interactions (Sect. FA-substrate interfacial tension).

Upon further reflection, it is essential to highlight the influence exerted by FA slipping on collagen I substrates in relation to the FA-substrate coupling, which subsequently affects the remodelling of FAs.

Impacts of the dynamics along the FA-substrate biointerface on the coupling between FA and substrate

During FA remodelling, interfacial slipping between the FA and the collagen I substrate produces spatially heterogeneous stress transmission, leading to an imbalance between the mechanical stresses within the FA and those supported by the substrate. This stress discontinuity can be represented as a traction force density difference at the FA–matrix biointerface $\:\varDelta\:{\overrightarrow{T}}_{F}$, which can be expressed through an effective interfacial force balance:

$$\:\varDelta\:{\overrightarrow{T}}_{F}=\left({\stackrel{\sim}{\sigma\:}}_{FA}-{\stackrel{\sim}{\sigma\:}}_{m}\right)\cdot\:\overrightarrow{n}$$

(4)

where $\:{\stackrel{\sim}{\sigma\:}}_{FA}$ is the FA stress, $\:{\stackrel{\sim}{\sigma\:}}_{m}$ is the substrate stress, and $\:\overrightarrow{n}$ is the unit vector perpendicular to the biointerface. This stress imbalance directly correlates with the interfacial forces:

$$\:\varDelta\:{\overrightarrow{T}}_{F}={\gamma\:}_{FA-m}\kappa\:\cdot\:\overrightarrow{n}+\overrightarrow{\nabla\:}{\gamma\:}_{FA-m}$$

(5)

where $\:\kappa\:$ is the curvature equal to $\:\kappa\:=\left(\nabla\:\cdot\:\overrightarrow{m}\right)$, and $\:\overrightarrow{m}$ is the normal vector within the biointerface. The traction force density difference $\:\varDelta\:{\overrightarrow{T}}_{F}$ represents the mechanical imbalance at the biointerface, which triggers the biointerface displacement $\:{\overrightarrow{u}}_{slip}$. When the surface effects along the biointerface are neglected $\:\varDelta\:{\overrightarrow{T}}_{F}=0$ and the traction force density is the same within FA and substrate, i.e., $\:{\overrightarrow{T}}_{F}={\stackrel{\sim}{\sigma\:}}_{FA}\cdot\:\overrightarrow{n}={\stackrel{\sim}{\sigma\:}}_{m}\cdot\:\overrightarrow{n}$. However, when the surface effects are included, the difference of traction force density $\:\varDelta\:{\overrightarrow{T}}_{F}\ne\:0$. A schematic presentation of interplay between dynamics along the biointerface responsible for FA slippage and the stress imbalance is shown in Fig. 3:

Fig. 3

Schematic representation of the physical mechanisms governing focal adhesion (FA) dynamics at the FA–matrix biointerface. Spatial variations in the effective interfacial tension and its gradients generate Marangoni-like interfacial forces that induce FA slippage relative to the underlying collagen I substrate. The resulting slippage produces a mechanical stress imbalance between the FA and the substrate, which is manifested as a traction force density difference across the biointerface. This interplay between interfacial forces, FA slippage, and stress transmission regulates asymmetric FA wetting and de-wetting during cell migration on collagen I substrates

Two limiting regimes can be considered depending on the characteristic timescale of the applied traction and the relaxation dynamics of the biointerface. At shorter timescales or higher frequencies, the relaxation processes of the biointerface cannot fully develop, and the response is predominantly elastic: $\:\varDelta\:{\overrightarrow{T}}_{F}={k}_{int}{\overrightarrow{u}}_{int}$ (where $\:{k}_{int}$ is the spring constant of the biointerface per unit surface area of FA and $\:{\overrightarrow{u}}_{int}$ is the displacement of the biointerface). On longer timescales or at lower frequencies, relaxation processes, molecular bond rearrangement, matrix remodelling, and interfacial slippage become dominant, resulting in a dissipative response: ., $\:\varDelta\:{\overrightarrow{T}}_{F}=\xi\:{\overrightarrow{v}}_{int}$ (where ξ is the friction coefficient per unit surface area, and $\:{\overrightarrow{v}}_{int}=\frac{{d\overrightarrow{u}}_{int}}{dt}$ is the slip velocity). The displacement $\:{\overrightarrow{u}}_{int}$ is equal to: $\:{\overrightarrow{u}}_{int}\left(r,t\right)={\overrightarrow{u}}_{a}-\sum\:_{j}^{}{\overrightarrow{u}}_{mj}-{\overrightarrow{u}}_{FA}$ (where $\:{\overrightarrow{u}}_{a}$ is the displacement of actomyosin stress fibers and $\:{\overrightarrow{u}}_{mj}$ is the substrate displacement for the corresponding j-th relaxation mode).

The traction forces transmitted to FAs $\:{\overrightarrow{F}}_{FA}\left(r,t\right)={\overrightarrow{T}}_{F}\varDelta\:A$ and the traction force within the substrate $\:{\overrightarrow{F}}_{m}\left(r,t\right)={\overrightarrow{T}}_{m}\varDelta\:A$ (where $\:\varDelta\:A\left(r,t\right)$ is the increment of FA surface area) satisfies the force balance:

$$\:\overrightarrow{F}\left(r,t\right)={\overrightarrow{F}}_{FA}={\overrightarrow{F}}_{m}+{\overrightarrow{F}}_{int}$$

(6)

where $\:\overrightarrow{F}\left(r,t\right)$ is the force exerted by actomyosin equal to: $\:\overrightarrow{F}\left(r,t\right)={\overrightarrow{F}}_{0}\left(r,t\right)\text{sin}\left({\omega\:}_{a}\left(r\right)t+\varphi\:\left(r,t\right)\right)$, $\:{\omega\:}_{a}$ is the angular velocity of actomyosin oscillations equal to: $\:{\omega\:}_{a}=\frac{2\pi\:}{T}$, $\:T$ is the period of actomyosin oscillations equal to $\:T=30-90\:\text{s}$ (Banerjee et al. 2017), $\:{\overrightarrow{F}}_{0}\left(r,t\right)$ is the force amplitude that changes slowly with time, i.e., $\:\frac{d{\overrightarrow{F}}_{0}}{dt}\ll\:{\overrightarrow{F}}_{0}{\omega\:}_{a}$, $\:\varphi\:\left(r,t\right)$ is the lag phase between the force $\:\overrightarrow{F}$ and the displacement $\:{\overrightarrow{u}}_{a}\left(r,t\right)$, which is also a slow-changing variable, while $\:{\overrightarrow{F}}_{int}$ is the force along the biointerface equal to $\:{\overrightarrow{F}}_{int}=\varDelta\:{\overrightarrow{T}}_{F}\varDelta\:A$ (where $\:\varDelta\:A$ is part of the FA surface area). The displacement of actomyosin is equal to: $\:{\overrightarrow{u}}_{a}\left(r,t\right)={\overrightarrow{u}}_{a0}\left(r,t\right)\text{s}\text{i}\text{n}\left({\omega\:}_{a}\left(r\right)t\right)$, such that $\:{\overrightarrow{u}}_{a0}$ changes slowly with time, i.e., $\:\frac{d{\overrightarrow{u}}_{a0}}{dt}\ll\:{\overrightarrow{u}}_{a0}{\omega\:}_{a}$. The coupling between FAs and the substrate depends on their viscoelasticity. The viscoelasticity will be discussed based on the proposed constitutive models that point to mechanisms of energy storage and dissipation during FA-substrate coupling.

Viscoelasticity of the collagen I substrate and FAs

In the following, we consider how these properties can be captured through appropriate constitutive descriptions of the main interacting components (the substrate and the focal adhesions), focusing on their mesoscopic rheological behaviour.

Several key aspects of collagen network viscoelasticity influence cell–ECM interactions, including stress relaxation under sustained deformation, the presence of multiple relaxation modes, the accumulation of residual stresses during repeated loading, and nonlinear effects such as strain stiffening (Pryse et al. 2003; Nam et al. 2016; Pajic-Lijakovic et al. 2022). These behaviours arise from the underlying filament architecture, including conformational flexibility, fibers alignment, extension/compression, and cross-linking density. Additionally, they are influenced by the magnitude of strain and strain rate (Ban et al. 2018; Jansen et al. 2018).

The rheological properties of FAs exhibit complexity and are determined by the surface packing density of molecular clutches ρ(r, t), the cumulative effects of their stretching and compression, and the interactions that occur between them, which are dependent on the timescale (Ladoux and Mège 2017). These clutches experience elastic stretching, while the packing density remains nearly constant over seconds (Chan and Odde 2008). Nevertheless, the long-term remodelling of FAs involves: (i) the recruitment of new active clutches, their stretching, and tilting during the wetting process, and (ii) significant interactions among clutches and their turnover during de-wetting, which leads to both energy storage and energy dissipation that occur over minutes. The inhomogeneous distribution of the surface packing density of clutches causes a non-uniform viscoelastic response throughout the FA. The resulting mechanisms of energy storage and dissipation over these timescales should be captured by a viscoelastic constitutive model capable of describing both the short-time elastic response and the longer-term, approximately linear viscoelastic behaviour of the system.

Constitutive model for collagen I substrates

Cell tractions lead to a non-uniform distribution of collagen concentration across substrates (Clark et al. 2022). The substrates exhibit a distribution of relaxation times (Pryse et al. 2003; Pajic-Lijakovic et al. 2022) and the traction force $\:{\overrightarrow{F}}_{m}$ can be expressed as a Prony series in the time domain from seconds to minutes, i.e., $\:{\overrightarrow{F}}_{m}\left(r,t\right)=\sum\:_{j}^{}{\overrightarrow{F}}_{mj}$. The force contributions can be described by a Zener model suitable for viscoelastic solids. In this approach, the model parameters are: (i) functions of the spatial coordinate, reflecting the local collagen density and (ii) change over a long-time. The corresponding constitutive model for the j-th relaxation mode can be expressed as:

$$\begin{aligned}&\:{\overrightarrow{F}}_{mj}\left(r,t\right)+{\tau\:}_{1mj}\left(r,t\right)\frac{d{\overrightarrow{F}}_{mj}}{dt}={k}_{mj}\left(r,t\right){\overrightarrow{u}}_{mj}\\&+{k}_{mj}\left(r,t\right){\tau\:}_{2mj}\left(r,\tau\:\right)\frac{d{\overrightarrow{\varvec{u}}}_{\varvec{m}\varvec{j}}}{dt}\end{aligned}$$

(7)

where $\:{\overrightarrow{F}}_{mj}\left(r,t\right)$ is the j-th contribution to the stress of collagen substrate, $\:{\overrightarrow{u}}_{mj}$ is the substrate displacement for the j-th relaxation mode, $\:{\tau\:}_{1mj}$ and $\:{\tau\:}_{2mj}$ are the j-th relaxation and retardation times that satisfy the condition that $\:{\tau\:}_{1mj}<{\tau\:}_{2mj}$ (Lin 2025), respectively, and $\:{k}_{mj}\left(r,t\right)$ is the equilibrium spring constant. Multiple modes in collagen relaxation ensure that, as the FA transitions from elastic to viscous behaviour, there is always a substrate mode that matches its current state.

The complex spring constant $\:{K}_{mj}^{*}$ can be obtained after Fourier transform of Eq. (7), $\:{\overrightarrow{F}}_{mj}\left(r,\omega\:\right)={K}_{mj}^{*}\left(r,\omega\:\right){\overrightarrow{u}}_{mj}\left(r,\omega\:\right)$, such that the complex spring constant of the j-th mode $\:{K}_{mj}^{*}\left(r,\omega\:\right)$ is equal to: $\:{K}_{mj}^{*}\left(r,\omega\:\right)={k}_{mj}\frac{1+i{\tau\:}_{2mj}\omega\:}{1+i{\tau\:}_{1mj}\omega\:}$). The complex modulus of $\:{K}_{mj}^{*}$ is equal to $\:{K}_{mj}^{*}={K}_{mj}^{{\prime\:}}+i{K}_{mj}^{{\prime\:}{\prime\:}}$ (where $\:i=\sqrt{-1}$, $\:{K}_{mj}^{{\prime\:}}$ and $\:{K}_{mj}^{{\prime\:}{\prime\:}}$ quantify energy storage and dissipation within the substrate for the j-th mode of relaxation, respectively). The phase lag between the force $\:{\overrightarrow{F}}_{mj}$ and displacement $\:{\overrightarrow{u}}_{mj}$ is expressed as: $\:{\theta\:}_{mj}=\frac{{K}_{mj}^{{\prime\:}{\prime\:}}}{{K}_{mj}^{{\prime\:}}}$ and can be written as: $\:{\theta\:}_{mj}=\text{a}\text{r}\text{c}\text{t}\text{a}\text{n}\omega\:\frac{1}{{\tau\:}_{1mj}}-\text{a}\text{r}\text{c}\text{t}\text{a}\text{n}\omega\:\frac{1}{{\tau\:}_{2mj}}$. Although the constitutive model yields complex moduli over a wide frequency range, the actomyosin oscillations introduce a characteristic frequency $\:{\omega\:}_{a}$, such that the effective governing transmission of the mechanical response is primarily determined by the values of the moduli evaluated at $\:{\omega\:}_{a}$.

If the frequency of actomyosin oscillations $\:{\omega\:}_{a}$ satisfies the condition: $\:{\tau\:}_{1mj}{\omega\:}_{a},{\tau\:}_{2mj}{\omega\:}_{a}\sim1$, the phase lag $\:{\theta\:}_{mj}$ can be large, up to $\:{90}^{\text{o}}$. It creates a massive mismatch between the displacements $\:{\overrightarrow{u}}_{a\:}\left(r,t\right)$ and $\:{\overrightarrow{u}}_{mj\:}\left(r,t\right)$ forcing the displacement of FA $\:{\overrightarrow{u}}_{FA\:}\left(r,t\right)$ to spike to accommodate the difference, which will be discussed in Sect.  Force balances.

Constitutive model for FA

FAs consist of interconnected protein networks that maintain structural integrity throughout maturation and collectively exhibit viscoelastic solid-like behavior. The Zener model, which is commonly applied to viscoelastic solids, can be utilized for the characterization of FA viscoelastic properties. The corresponding model parameters such as FA stiffness, as well as relaxation and retardation times depend on the surface packing density of engaged clutches, thereby linking microscopic FA structural changes to the mesoscopic rheological response. Consequently, FAs exhibit a range of relaxation and retardation times, which corresponds to minutes. The Zener constitutive model can be expressed as:

$$\begin{aligned}\:{\overrightarrow{F}}_{FA}\left(r,t\right)+{\tau\:}_{1FA}\left(\rho\:\right)\frac{d{\overrightarrow{F}}_{FA}}{dt}={k}_{FA}\left(\rho\:\right){\overrightarrow{u}}_{FA}\\+{k}_{FA}\left(\rho\:\right){\tau\:}_{2FA}\left(\rho\:\right)\frac{d{\overrightarrow{u}}_{FA}}{dt}\end{aligned}$$

(8)

where $\:{\tau\:}_{1FA}\left(\rho\:\right)$ and $\:{\tau\:}_{2FA}\left(\rho\:\right)$ are the relaxation and retardation times of FA, respectively, $\:{k}_{FA}\left(\rho\:\right)$ is the equilibrium spring constant of an FA equal to: $\:{k}_{FA}\left(\rho\:\right)=\rho\:\varDelta\:A{k}_{c}$, $\:{k}_{c}$ is the average spring constant of a single clutch, $\:{\overrightarrow{u}}_{FA}$ is the FA displacement field, and $\:\rho\:\left(r,t\right)$ is the surface packing density of molecular clutches. On short timescales, the surface packing density $\:\rho\:\left(r,t\right)\approx\:const$ and the relaxation and retardation times satisfy the condition that: $\:{\tau\:}_{1FA}\left(\rho\:\right)$, $\:{\tau\:}_{2FA}\left(\rho\:\right)$ $\:\gg\:\frac{1}{\omega\:}$ (where $\:\omega\:$ is the angular velocity of actomyosin oscillations). Consequently, FA behaves elastically at a timescale of seconds, i.e. $\:{\overrightarrow{F}}_{FA}\left(r,t\right)\approx\:{k}_{FA}\frac{{\tau\:}_{2FA}}{{\tau\:}_{1FA}}\left(\rho\:\right){\overrightarrow{u}}_{FA}$ (where $\:{k}_{FA0}$ is the initial spring constant of an FA equal to $\:{k}_{FA0}={k}_{FA}\frac{{\tau\:}_{2FA}}{{\tau\:}_{1FA}}$) (Chan and Odde 2008). This provides the immediate mechanical grip needed for the molecular clutch to engage. Without this initial elastic “snap,” the cell would never be able to transmit the force required to remodel the collagen or to initiate the Marangoni flow.

The complex spring constant $\:{K}_{FA}^{*}$ can be obtained after Fourier transformation of Eq. (8), $\:{\overrightarrow{F}}_{FA}\left(r,\omega\:\right)={K}_{FA}^{*}\left(r,\omega\:\right){\overrightarrow{u}}_{FA}\left(r,\omega\:\right)$, such that $\:{K}_{FA}^{*}\left(r,\omega\:\right)={k}_{FA}\frac{1+i{\tau\:}_{2FA}\omega\:}{1+i{\tau\:}_{1FA}\omega\:}$. The complex modulus of an FA $\:{K}_{FA}^{*}$ is equal to $\:{K}_{FA}^{*}={K}_{FA}^{{\prime\:}}+i{K}_{FA}^{{\prime\:}{\prime\:}}$ (where $\:{K}_{FA}^{{\prime\:}}$ and $\:{K}_{FA}^{{\prime\:}{\prime\:}}$ quantify energy storage and dissipation within an FA, respectively). The phase lag between the force $\:{\overrightarrow{F}}_{FA}$ and displacement $\:{\overrightarrow{u}}_{FA}$ is expressed as: $\:{\theta\:}_{FA}=\frac{{K}_{FA}^{{\prime\:}{\prime\:}}}{{K}_{FA}^{{\prime\:}}}$. At short times the FA behaves elastically, i.e., $\:{K}_{FA}^{{\prime\:}}\to\:{k}_{FA}$ and $\:{K}_{FA}^{{\prime\:}{\prime\:}}\to\:0$. Energy storage and dissipation during coupling of the actomyosin-FA-substrate system can be expressed by effective storage and loss moduli, which include the contributions of all interconnected systems. Energy storage and dissipation during FA-substrate coupling can be extracted from the constitutive models in the form of a complex effective spring constant.

Energy storage and dissipation during coupling of the actomyosin-FA-substrate system

Actomyosin stress fibers exert work on the FA and substrate. A part of this energy is storage within the FA, biointerface, and substrate, while the other part is dissipated within these systems. The ratio between energy storage and dissipation, depending on the viscoelasticity of these elements and the way of their coupling, can be expressed in the form of their complex moduli.

The complex spring constant, depends on the complex spring constants of the FA $\:{K}_{FA}^{*}\left(r,\omega\:\right)$, substrate $\:{K}_{m}^{*}\left(r,\omega\:\right)={\sum\:}_{\varvec{j}}^{}{K}_{mj}^{*}\left(r,\omega\:\right)$, and the biointerface between them $\:{K}_{int}^{*}\left(r,\omega\:\right)$. The complex spring constant of the biointerface $\:{K}_{int}^{*}\left(r,\omega\:\right)$ can be expressed as: (i) $\:{K}_{int}^{*}\left(r,\omega\:\right){=k}_{int}$, which reflects elastic behaviour, such that $\:{K}_{int}^{{\prime\:}}\left(r,\omega\:\right){=k}_{int}$ and $\:{K}_{slip}^{{\prime\:}{\prime\:}}\left(r,\omega\:\right)=0$ for higher frequencies $\:\omega\:\to\:\infty\:$ and (ii) $\:{K}_{int}^{*}\left(r,\omega\:\right)=i\xi\:\omega\:$, which reflects viscous behaviour, such that $\:{K}_{int}^{{\prime\:}}\left(r,\omega\:\right)=0$ and $\:{K}_{int}^{{\prime\:}{\prime\:}}\left(r,\omega\:\right)=\xi\:\omega\:$ for lower frequencies (and longer times). The effective complex spring constant $\:{K}_{eff}^{*}\left(r,\omega\:\right)$ encodes the full viscoelastic spectrum of the coupled FA–substrate system, while the actomyosin oscillations select an effective characteristic frequency $\:{\omega\:}_{a}$, which emerges from the interplay between active contractility and the mechanical response of the coupled system. Thus, measurable quantities such as force amplitude $\:{\overrightarrow{F}}_{0}$, phase lag $\:\varphi\:$, and work done by actomyosin per cycle are determined by evaluating $\:{K}_{eff}^{*}$ at $\:{\omega\:}_{a}$. For series coupling between these interconnected systems, the effective complex modulus $\:{K}_{eff}^{*}\left(r,{\omega\:}_{a}\right)=\frac{\overrightarrow{F}\left(r,{\omega\:}_{a}\right)}{{\overrightarrow{u}}_{a}\left(r,{\omega\:}_{a}\right)}$ can be expressed as:

$$\:\frac{1}{{K}_{eff}^{\text{*}}\left(r,{\omega\:}_{a}\right)}=\frac{1}{{K}_{FA}^{\text{*}}}+\frac{1}{{K}_{m}^{\text{*}}+{K}_{int}^{\text{*}}}$$

(9)

The storage and loss effective spring constants for the whole system $\:{K}_{eff}^{{\prime\:}}$ and $\:{K}_{eff}^{{\prime\:}{\prime\:}}$, respectively, as a measure of energy storage and energy dissipation, during coupling between these systems can be expressed from the effective spring constant $\:{K}_{eff}^{*}\left(r,\omega\:\right)$ while the phase lag between the force $\:\overrightarrow{F}$ and the actomyosin displacement $\:{\overrightarrow{u}}_{a}$ is equal to: $\:\varphi\:\left(r,{\omega\:}_{a}\right)=arctan\frac{{K}_{eff}^{{\prime\:}{\prime\:}}}{{K}_{eff}^{{\prime\:}}}$ and $\:\text{s}\text{i}\text{n}\varphi\:=\frac{{K}_{eff}^{{\prime\:}{\prime\:}}}{\left|{K}_{eff}^{*}\right|}$.

The work done by the actomyosin over a single oscillation, i.e. the total energy input from the cytoskeleton, can be expressed as:

$$\:W\left(r,t\right)={\int\:}_{t}^{t+T}F\left(r,t^{\prime\:}\right)\frac{d{u}_{a}\left(r,{t}^{{\prime\:}}\right)}{d{t}^{{\prime\:}}}d{t}^{{\prime\:}}$$

(10)

where the work $\:W\left(r,t\right)=\pi\:{F}_{0}{u}_{a0}\text{s}\text{i}\text{n}\varphi\:$ represents a long-time evolution of the total energy input per cycle, capturing the slow changes in the system amplitudes. The force amplitude $\:{\overrightarrow{F}}_{0}\left(r,{\omega\:}_{a}\right)$ depends on the actin displacement amplitude $\:{\overrightarrow{u}}_{a0}\left(r,{\omega\:}_{a}\right)$ and the modulus of the complex spring constant $\:\left|{K}_{eff}^{*}\:\right|$ and can be expressed as: $\:{\overrightarrow{F}}_{0}\left(r,{\omega\:}_{a}\right)={\overrightarrow{u}}_{a0}\sqrt[]{{K}_{eff}^{{\prime\:}2}+{K}_{eff}^{{\prime\:}{\prime\:}2}}$. Consequently, the work done by the actomyosin ventral stress fibers is equal to: $\:W\left(r,{\omega\:}_{a}\right)=\pi\:{\overrightarrow{u}}_{a0}^{2}{K}_{eff}^{{\prime\:}{\prime\:}}$ (where $\begin{aligned}\:&{K}_{eff}^{{\prime\:}{\prime\:}}=\\& \frac{\frac{{K}_{FA}^{{\prime\:}{\prime\:}}}{{K}_{FA}^{{\prime\:}2}+{K}_{FA}^{{\prime\:}{\prime\:}2}}+\frac{{K}_{m+int}^{{\prime\:}{\prime\:}}}{{K}_{m+int}^{{\prime\:}2}+{K}_{m+int}^{{\prime\:}{\prime\:}2}}}{{\left[\frac{{K}_{FA}^{{\prime\:}}}{{K}_{FA}^{{\prime\:}2}+{K}_{FA}^{{\prime\:}{\prime\:}2}}+\frac{{K}_{m+int}^{{\prime\:}}}{{K}_{m+int}^{{\prime\:}2}+{K}_{m+int}^{{\prime\:}{\prime\:}2}}\right]}^{2}+{\left[\frac{{K}_{FA}^{{\prime\:}{\prime\:}}}{{K}_{FA}^{{\prime\:}2}+{K}_{FA}^{{\prime\:}{\prime\:}2}}+\frac{{K}_{m+int}^{{\prime\:}{\prime\:}}}{{K}_{m+int}^{{\prime\:}2}+{K}_{m+int}^{{\prime\:}{\prime\:}2}}\right]}^{2}}\end{aligned}$ and $\:{K}_{m+int}^{{\prime\:}}={K}_{m}^{{\prime\:}}+{K}_{int}^{{\prime\:}}$ and $\:{K}_{m+int}^{{\prime\:}{\prime\:}}={K}_{m}^{{\prime\:}{\prime\:}}+{K}_{int}^{{\prime\:}{\prime\:}}$). This work transmitted to the FA and substrate represents dissipation for the actin cytoskeleton itself.

The frequency range, which ensures maximum transfer of energy from the cytoskeleton can be obtained from the condition $\:\frac{dW\left(r,{\omega\:}_{a}\right)}{d{\omega\:}_{a}}=0$. The corresponding frequency range depends on the set of physical parameters, i.e., $\:{\omega\:}_{a\:max}\left(r\right)=$${\omega\:}_{a\:max}\left(\rho\:,{k}_{FA},\:{\tau\:}_{1FA},{\tau\:}_{2FA};\sum\:_{j}^{}{k}_{mj},{\tau\:}_{1mj},{\tau\:}_{2mj};{k}_{int},\xi\:\right)$ . The viscoelastic properties of the FA and substrates, along with the slipping of the FA, significantly affect the FA’s wetting and de-wetting processes. Furthermore, it is essential to examine the primary characteristics of FA wetting and de-wetting on collagen I substrates as well as on artificial viscoelastic substrates.

Wetting and de-wetting of FA on collagen I vs. artificial substrate matrices: a comparative analysis

Wetting and de-wetting of FAs on collagen I vs. artificial substrates vary in their rate and uniformity. On homogeneous, isotropic artificial substrates, wetting and de-wetting are uniform and rapid, whereas on collagen I, they are non-uniform and gradual (Wang et al. 2003). Because artificial gels are homogeneous, the spreading factor is identical in all directions. The FA wets symmetrically, expanding at the same rate at its front and rear. This symmetry leads to comparable anchoring at the front and rear, so that de-wetting requires overcoming a more uniformly stabilised adhesion state. Wetting and de-wetting are generally more spatially heterogeneous on collagen I substrates, reflecting the underlying structural and dynamical complexity of the matrix. Processes such as fiber redistribution and interfacial stress gradients can contribute to breaking the front–rear symmetry observed on more homogeneous substrates. As a result, wetting is often more pronounced at the front of the adhesion, while the rear may exhibit reduced wetting or enhanced de-wetting, consistent with the direction of cell-generated forces. Rather than defining sharply distinct regimes, these behaviours are better viewed as part of a continuum of possible responses, shaped by the local balance between adhesion, cohesion, and substrate remodelling across multiple timescales. A schematic illustration summarising these trends for both collagen I and artificial substrates is presented Fig. 3:

Fig. 4

Schematic representation of the main characteristics of the wetting and de-wetting of an FA on both collagen I substrate and an artificial substrate. The upper scheme illustrates the mechanical coupling between the FA, cytoskeleton, substrate, and directiom of cell migration. The purple arrow indicates the latter direction, whereas the blue arrow represents FA slipping relative to the substrate during migration. FA wetting and de-wetting are symmetric on artificial substrates, while they become asymmetric on collagen I substrates due to directional FA slipping. This slipping promotes FA wetting at the front side of the adhesion while reducing wetting at the rear side. Conversely, slipping suppresses de-wetting at the front side and enhances de-wetting at the rear side. The lower scheme illustrates three regimes of FA behaviour on collagen I substrates: (i) wetting, (ii) partial de-wetting, and (iii) de-wetting. In contrast, artificial viscoelastic substrates exhibit symmetric FA remodelling characterized by two regimes: (i) wetting and (ii) de-wetting. Green arrows indicate adhesive interactions and FA spreading (wetting), whereas red arrows indicate FA contraction and detachment processes associated with de-wetting. The asymmetry of FA remodeling on collagen I substrates arises from the coupling between directional slipping, substrate–adhesion interactions, and cytoskeletal forces

When the substrate is described by a single set of Zener parameters, it exhibits a single dominant relaxation timescale and rapidly approaches its asymptotic elastic regime once this timescale has elapsed. Beyond this point, the material effectively behaves as a purely elastic spring. This abrupt transition leads to a rapid increase in focal adhesion–substrate adhesion energy, promoting fast and effectively irreversible FA maturation. In contrast, collagen I substrates possess a broad spectrum of relaxation modes, which introduces a progressive relaxation response across multiple timescales. As FAs expand, they continuously couple to increasingly slower relaxation modes of the matrix. This distributed viscoelastic response, together with slippage at the FA–substrate interface, suppresses rapid stress accumulation and maintains the effective spreading factor near zero over extended time intervals (on the order of minutes). As a result, force transmission is regulated by the hierarchical relaxation structure of the substrate, which acts as a time-dependent mechanical filter. On more homogeneous and isotropic substrates, the rear of the focal adhesion tends to become uniformly wetted and stabilised, leading to the formation of relatively strong, mature adhesions. Under these conditions, de-wetting requires large rupture forces to overcome multiple engaged clutches, resulting in pronounced stick–slip dynamics. In contrast, focal adhesions on collagen I substrates exhibit qualitatively different de-wetting behaviour. Due to sustained stress relaxation and interfacial dissipation, rear wetting remains dynamically regulated, preventing full mechanical solidification of the adhesion zone and maintaining a more fluid-like, continuously remodeling state. De-wetting occurs as a smooth compression over minutes, triggered easily as adhesion energy $\:{e}_{a}$ drops below the vinculin-mediated cohesion $\:{e}_{coh}$. Consequently, these differences in rate and uniformity are the direct result of the sum of Zener elements $\:\sum\:_{j}^{}{\overrightarrow{F}}_{mj}$ and the Marangoni flow in collagen I substrates caused by cell tractions. In order to relate the rate of FA wetting/de-wetting with FA slipping along the biointerface, it is necessary to formulate proper force and mass balances of the FA and collagen I substrate.

Force and mass balances for wetting/de-wetting of FA

The rate of change in the spreading factor $\:{S}_{FA}$ depends on the rates of change of the adhesion energy and cohesion energy $\:\frac{d{e}_{a}}{dt}$ and $\:\frac{d{e}_{coh}}{dt}$. The adhesion energy can be expressed at a mesoscopic level as: $\:{e}_{a}\left(r,t\right)=\frac{1}{2}{k}_{FA}{\overrightarrow{u}}_{FA}^{2}$, while the cohesion energy has been expressed as: $\:{e}_{coh}\left(r,t\right)=2{\gamma\:}_{FA}$. The rate of change in the adhesion energy depends on: (i) the rate of change in the FA displacement field and (ii) the FA displacement field itself, i.e. $\:\frac{d{e}_{a}}{dt}\sim{\overrightarrow{u}}_{FA}\frac{d{\overrightarrow{u}}_{FA}}{dt}$. The change in FA displacement depends on: (i) the viscoelasticity of the collagen I substrate and (ii) the FA-substrate interfacial tension and interfacial tension gradient. The rate of change in cohesion energy depends on the change in the surface packing density of active molecular clutches, i.e., $\:\frac{d{e}_{coh}}{dt}\sim\frac{d\rho\:}{dt}$. Changes in the displacement field $\:\frac{d{\overrightarrow{u}}_{FA}}{dt}$ and surface packing density of active clutches $\:\frac{d\rho\:}{dt}$ will be discussed based on force and mass balances.

Force balances

The change of the FA displacement field $\:{\overrightarrow{u}}_{FA}$ has a feedback impact on the stability of FA and can be expressed as:

$$\begin{aligned}\:\frac{d{\overrightarrow{u}}_{FA}\left(r,t\right)}{dt}={\omega\:}_{a}{\overrightarrow{u}}_{a0\:}\left(r,t\right)\text{c}\text{o}\text{s}\left({\omega\:}_{a}\left(r\right)t\right)\\-\sum\:_{j}^{}\frac{d{\overrightarrow{u}}_{mj}\left(r,t\right)}{dt}-\frac{d{\overrightarrow{u}}_{int}\left(r,t\right)}{dt}\end{aligned}$$

(11)

where the rate $\:\frac{d{\overrightarrow{u}}_{int}\left(r,t\right)}{dt}$ is equal to $\:\frac{d{\overrightarrow{u}}_{int}\left(r,t\right)}{dt}=\frac{1}{{k}_{int}}\frac{d{\overrightarrow{F}}_{int}}{dt}$ for the short-time regime and $\:\frac{d{\overrightarrow{u}}_{int}\left(r,t\right)}{dt}=\frac{1}{\xi\:}{\overrightarrow{F}}_{int}$ for the long-time regime. The spectrum of rates $\:\sum\:_{j}^{}\frac{d{\overrightarrow{u}}_{mj}\left(r,t\right)}{dt}$ depends on the viscoelasticity and surface characteristics of collagen I gel and can be formulated based on the force balance as:

$$\:\sum\:_{j}^{}\frac{d{\overrightarrow{u}}_{mj}\left(r,t\right)}{dt}={\mu\:}_{m}\left[\overrightarrow{\nabla\:}{\gamma\:}_{m}\varDelta\:A+\sum\:_{j}^{}{\overrightarrow{F}}_{mj}\right]$$

(12)

where $\:{\mu\:}_{m}$ is the mobility of collagen fibers. As the FA attempts to wet the surface, the long-term Zener modes allow the collagen fibers to reorganize over minutes. In the long-time regime, interfacial slippage acts as an effective regulatory mechanism that maintains the spreading factor close to zero by continuously dissipating energy.

The summary of spectrum $\:\sum\:_{j}^{}\frac{d{\overrightarrow{u}}_{mj}\left(r,t\right)}{dt}$ effects can be considered within three time-regimes: (i) a short-time regime for $\:{\tau\:}_{1mj},{\tau\:}_{2mj}\ll\:\frac{1}{{\omega\:}_{a}}$, (ii) a $\:{\tau\:}_{1mj},{\tau\:}_{2mj}\sim\frac{1}{{\omega\:}_{a}}$ middle-time regime, and (iii) a long-time regime for $\:{\tau\:}_{1mj},{\tau\:}_{2mj}\gg\:\frac{1}{{\omega\:}_{a}}$. The phase-lag between the force $\:{\overrightarrow{F}}_{mj}$ and displacement $\:{\overrightarrow{u}}_{mj}$ is: (i) $\:{\theta\:}_{mj}\sim0$ in the short-time regime, (ii) $\:{\theta\:}_{mj}\sim\text{m}\text{a}\text{x}\text{i}\text{m}\text{u}\text{m}$ in the middle-time regime, and (iii) $\:{\theta\:}_{mj}\sim0$ in the long-time regime. In the short-time regime, the rates $\:\frac{d{\overrightarrow{u}}_{int}\left(r,t\right)}{dt}$ and $\:\sum\:_{j}^{}\frac{d{\overrightarrow{u}}_{mj}\left(r,t\right)}{dt}$ are low and elastic. However, in the long-time regime, the rate $\:\frac{d{\overrightarrow{u}}_{int}\left(r,t\right)}{dt}$ accounts for the viscous slip effect and the rate $\:\sum\:_{j}^{}\frac{d{\overrightarrow{u}}_{mj}\left(r,t\right)}{dt}$ depends on the Marangoni flow of collagen fibers and longer modes of viscoelasticity (Eq. 12). The transition of the FA from elastic to dissipative is associated with a smooth, long-term “maturation” process of the FA (an increase in $\:{e}_{coh}$), rather than a sudden “freezing” of the biointerface. The benefit of the spectrum is that the total substrate velocity $\:\sum\:_{j}^{}\frac{d{\overrightarrow{u}}_{mj}\left(r,t\right)}{dt}$ can “match” the actin drive within slow and fast regimes. The substrate relaxation modes within the middle regime are critical for actin substrate coupling and can produce a viscoelastic resonance effect.

Consequently the sum of substrate displacements $\:{\sum\:}_{j}^{}{\overrightarrow{u}}_{mj}\left(r,t\right)$ acts as mechanical damper. It balances the peak displacements of $\:{\overrightarrow{u}}_{FA}\left(r,t\right),$ keeping the spreading factor $\:{S}_{FA}$ near zero and facilitating the “slow-down” wetting and controlled maturation of the FA in the long-time regime, i.e., $\:\frac{d{e}_{a}}{dt}\sim\frac{d{e}_{coh}}{dt}$. In contrast to a collagen I substrate, $\:{\overrightarrow{u}}_{FA}\left(r,t\right)$ oscillates with higher amplitude on an artificial substrates with single relaxation and retardation times because the substrate cannot adapt to the actin frequency. In light of the fact that the rate at which cohesion energy changes is contingent upon the variation in surface density of active molecular clutches, it is crucial to establish the mass balance for these active molecular clutches.

Mass balances of molecular clutches

The surface packing density of molecular clutches is: (i) constant in the short-time regime (regime i), i.e., $\:\rho\:\left(r,t\right)\approx\:const$, (ii) starts to increase in the middle time regime and (iii) continues to increase during wetting or decrease during de-wetting depending on the magnitude of spreading factor. A change in the surface packing density depends on the relationship between recruitment rate, disengagement rate, and the lateral movement of molecular clutches. These changes in the density of active clutches directly influence the cohesion of an FA. The mass balance of clutches can be expressed as:

$$\:\frac{\partial\:\rho\:\left(r,t\right)}{\partial\:t}={k}_{on}\left({\rho\:}_{max}-\rho\:\right)-{k}_{off}\rho\:-\overrightarrow{\nabla\:}\left({\overrightarrow{v}}_{r}\rho\:\right)$$

(13)

where $\:{k}_{on}$ is the recruitment rate constant for active clutches, which depends on the spreading factor $\:{S}_{FA}$ and can be expressed as: $\:{k}_{on}={k}_{on}^{0}\left(1+\alpha\:{S}_{FA}\right)$, $\:{k}_{on}^{0}$ is the baseline binding rate, $\:\alpha\:$ is a sensitivity coefficient, $\:{k}_{off}\left({B}_{2}\right)$ is the rate at which active clutches disengage or become inactive on collagen I substrates which depends on the second virial coefficient $\:{B}_{2}$, $\:\rho\:$ is the surface packing density of clutches, $\:{\rho\:}_{max}$ is the maximum surface density of clutches in a matured FA. An increase in spreading factor induces the stretching and tilting of molecular clutches, which speeds up the recruitment of new active clutches. An increase in the positional and orientational contributions to the interaction potential between clutches, as well as the second virial coefficient caused by FA de-wetting, causes an increase in the disengagement rate of clutches. The relative velocity between the cytoskeleton and substrate is equal to: $\:{\overrightarrow{v}}_{r}=\frac{d{\overrightarrow{u}}_{a}\left(r,t\right)}{dt}+{\sum\:}_{j}^{}\frac{d{\overrightarrow{u}}_{m}\left(r,t\right)}{dt}+\frac{d{\overrightarrow{u}}_{int}\left(r,t\right)}{dt}$. The rate $\:\frac{d{\overrightarrow{u}}_{a}\left(r,t\right)}{dt}$ opposes cell migration, whereas the rate $\:\frac{d{\overrightarrow{u}}_{m}\left(r,t\right)}{dt}$ aligns with the direction of cell migration. The gradient term $\:\overrightarrow{\nabla\:}\left({\overrightarrow{v}}_{r}\rho\:\right)$ acts as a mechanical drainage for FAs. As the cell pulls collagen fibers to the back, the Marangoni effect creates a forward-movement of the fibers. The high velocity $\:{\overrightarrow{v}}_{r}$ ensures that energy is dissipated as frictional flow rather than being stored as elastic strain. Because the long-term modes stay in phase with the actomyosin oscillations, the “summation” of $\:{\overrightarrow{v}}_{r}$ is coherent rather than fluctuating randomly. Consequently, the FA experiences reduced mechanical noise arising from destructive interference between competing relaxation modes. This allows the clutches to stay engaged while moving quickly, facilitating the “slow-down” wetting where the FA expands over minutes without locking. Considering these factors, it is essential to highlight the advantages of collagen I substrates for FA dynamics, particularly regarding effective cell migration, in comparison to artificial viscoelastic substrates.

Collagen I substrates as a model system for adaptive FA dynamics

Substrates that exhibit a hierarchical spectrum of relaxation times can enable focal adhesions (FAs) to display both stability and mobility, supporting a dynamic balance between anchoring and reorganisation. Collagen I networks provide a well-characterised example of such behaviour, owing to their multi-scale viscoelasticity and interfacial properties. In this context, interfacial slippage can arise from gradients in FA–substrate interactions. These gradients may reflect the inhomogeneous distribution of several coupled factors, including: (i) the surface packing density of active clutches (Kumar et al. 2016), (ii) the local concentration of collagen fibres, which can be modified by cell-generated tractions (Clark et al. 2022; Pajic-Lijakovic et al. 2022), and (iii) variations in integrin–ligand adhesion energy (Yao et al. 2022).

. All gradients that contribute to the gradient of interfacial tension, i.e., $\:\overrightarrow{\nabla\:}{\gamma\:}_{FA}$, $\:\overrightarrow{\nabla\:}{\gamma\:}_{m}$, and $\:\overrightarrow{\nabla\:}{e}_{a}$ are aligned in the direction of cell migration. Consequently, the slipping of an FA is induced by the Marangoni flow of active clutches driven by the gradient of interfacial tension $\:\overrightarrow{\nabla\:}{\gamma\:}_{FA-m}$ during wetting and the Marangoni flow of collagen fibers driven by the gradient of substrate surface tension $\:\overrightarrow{\nabla\:}{\gamma\:}_{m}$ as shown schematically in Fig. 5:

Fig. 5

Interplay between physical parameters responsible for asymmetric FA wetting and de-wetting on collagen I substrates. The spreading factor of FAs is determined by the balance between integrin–ligand adhesion energy per unit FA surface area and the cohesion energy of the FA protein network. An increase in adhesion energy relative to cohesion energy promotes FA wetting and spreading, whereas a decrease in this balance favors FA de-wetting and shrinkage. Interfacial slippage at the FA–substrate interface modulates this balance by altering effective adhesion energy and contributes to asymmetric wetting/de-wetting dynamics. Marangoni-like flows of collagen fibers and active clutches are proposed mechanisms that may contribute to interfacial redistribution and slippage-mediated FA remodelling. The arrows indicate conceptual relationships between the physical processes involved in FA adaptation rather than direct causal relationships established for all biological systems

The phenomenon of FA ageing, which is especially noticeable on collagen I substrates, leads to a gradual and prolonged wetting process. FA stability and lifetime depend sensitively on the mechanical properties of the underlying substrate. On more homogeneous substrates, adhesions tend to become more uniformly stabilised, leading to longer lifetimes and reduced dynamical turnover. In contrast, on heterogeneous and multi-timescale substrates such as collagen I, FAs often remain more dynamically regulated, with ongoing processes of growth and disassembly occurring over comparable timescales. These differences can be understood in terms of the underlying relaxation behaviour of the substrate. Systems characterised by a single dominant relaxation timescale tend to promote rapid adhesion stabilisation, resulting in long-lived, relatively immobile adhesions. By contrast, substrates exhibiting a broad spectrum of relaxation modes enable continuous adjustment of the adhesion state, supporting a balance between mechanical engagement and turnover. In this sense, collagen I provides a well-characterised example of a system in which multi-timescale dynamics can sustain adaptive FA behaviour, without requiring a transition to either fully stable or fully transient states. In the following, we discuss how such behaviour can be interpreted in terms of distinct viscoelastic regimes that emerge from the underlying spectrum of relaxation processes. The coupling between FA and collagen I substrates can be discussed within three time-regimes (Figure 5):

• Regime (i) Short-Time Elastic Anchor ($\:{\tau\:}_{1mj},{\tau\:}_{2mj}\ll\:\frac{1}{{\omega\:}_{a}})$: Because both the FA and the substrate are in their elastic limit, they are perfectly synchronized. The phase lag is $\:{\theta\:}_{mj}\sim0$. This minimizes $\:{\overrightarrow{u}}_{m}$ providing the traction needed to stretch talin and begin the maturation process. The surface density of active clutches is constant.

• Regime (ii) The damping regime ($\:{\tau\:}_{1mj},{\tau\:}_{2mj}\sim\frac{1}{{\omega\:}_{a}}$): As we discussed, the collagen spectrum smooths this resonance. The phase lag $\:{\theta\:}_{mj}$ is maximal. The FA and substrate show the transition from elastic to dissipative behaviour. It prevents a single-mode “mechanical shock” from rupturing the bonds, allowing the energy of the actomyosin oscillation to be dissipated across the network hierarchy rather than being concentrated into a destructive slip. The surface density of active clutches starts increasing.

• Regime (iii) Long-Time Dissipative Flow ($\:{\tau\:}_{1mj},{\tau\:}_{2mj}\gg\:\frac{1}{{\omega\:}_{a}}$): The phase lag is $\:{\theta\:}_{mj}\sim0$ (the relaxed state) and the FA and substrate behave as viscous ( i.e., slippage and flow). Because the biointerface becomes dissipative here, it allows for the “slippage along the biointerface” that limits the accumulation of adhesion energy $\:{e}_{a}$. The relaxation of collagen I substrates in multiple modes, clutch slippage, and bond turnover prevent excessive elastic energy storage in adhesions. Instead of $\:{e}_{a}$ quickly jumping to a high value, it increases gradually or saturates. By yielding over minutes, it ensures that the spreading factor $\:{S}_{FA}$ stays near zero. The FA “oozes” forward rather than locking, which is exactly why the cell can maintain fluid migration over long periods. The surface density of active clutches increases/decreases during FA wetting/de-wetting.

In contrast to collagen I substrates, artificial substrates, characterized by a single or narrowly distributed set of relaxation modes, often fail to support stable adhesion dynamics because they lack sufficient separation between relaxation regimes. As a result, they either relax too rapidly, leading to loss of effective mechanical anchoring, or relax too slowly, resulting in excessive stress accumulation and impaired adaptive remodeling. A single set of relaxation and retardation times, characteristic for artificial substrates, leads to a rapid increase in $\:{e}_{a}$ that exceeds $\:{e}_{coh}$ forcing the FA into an “over-wetted” state. The FA matures into a massive, immobile plaque that cannot easily transition back to a de-wetting state. However, the multiple relaxation modes in collagen I substrates ensure that as $\:{e}_{coh}$ increases (due to vinculin recruitment and higher $\:{\gamma\:}_{FA}$) the substrate continues to “yield”. This “near-equilibrium” state is what makes collagen I more suitable for migration; the cell can flip from wetting to de-wetting with very small changes in internal tension.

The complex coupling dynamics between FAs and collagen I substrates are closely associated with Marangoni-like transport mechanisms operating across multiple spatial scales. These mechanisms govern: (i) migration of free collagen fibers along the substrate surface driven by gradients in substrate surface tension ( Sect.  Surface tension of substrate matrix), (ii) slipping of FAs induced by gradients in effective FA–substrate interfacial tension (Sect.  FA-substrate interfacial tension), and (iii) lateral redistribution of mechanosensitive Piezo1 channels along the cell membrane driven by gradients in membrane surface tension ( Sect.  The distribution of Piezo1 channels around FA in epithelial cells on collagen I vs. artificial substrates). The distribution and activity of Piezo1 channels exert feedback effects on calcium influx and, consequently, on the rearrangement of molecular clutches within FAs.

Conclusion

The behaviour of focal adhesions (FAs) on collagen I substrates cannot be understood solely in terms of stiffness or even viscoelasticity; rather, it emerges from a coupled system of multi-timescale matrix relaxation, active surface remodelling, and interfacial slippage. Unlike artificial hydrogels, collagen I exhibits a hierarchical structure that enables continuous reorganization under cell-generated forces. These forces not only deform the bulk matrix, but also generate surface concentration gradients of collagen fibers, which in turn establish surface tension gradients driving Marangoni flows. As a result, the substrate behaves as an active, evolving interface rather than a passive mechanical support.

Within this framework, slippage along the FA–substrate biointerface represents the central regulatory mechanism. It arises from the combined effects of collagen matrix yielding, bond rupture and reformation dynamics, and Marangoni-driven fiber transport: it functions as an effective dynamic regulator that continuously dissipates the mechanical work generated by the actomyosin cytoskeleton. This prevents excessive accumulation of adhesion energy and maintains the spreading factor near zero, allowing FAs to persist in a state of near-equilibrium between wetting and de-wetting. Consequently, the FAs remain sufficiently stable to transmit forces, while retaining the mobility required for sustained migration. In contrast, simplified artificial substrates with limited hierarchical relaxation and reduced active surface remodelling capacity generally exhibit lower interfacial slippage, leading to rapid energy accumulation, over-maturation of adhesions, and loss of migratory adaptability.

The regulation of this mechanically permissive state is further reinforced by the spatial organization and activity of Piezo1 channels. On viscoelastic artificial substrates, sustained traction forces promote Piezo1 clustering at FAs, amplifying local Ca²⁺ signalling and stabilizing adhesions through positive feedback with contractility. However, on collagen I substrates, interfacial slippage and force redistribution reduce sustained traction peaks, resulting in a more uniform and dynamically fluctuating Piezo1 activity. This prevents excessive adhesion stabilization and promotes turnover through a balance of Ca²⁺-mediated reinforcement and calpain-driven disassembly.

Together, these results establish a unified physical picture in which collagen I substrates function as adaptive, self-organizing systems, where bulk viscoelasticity, surface tension gradients, and interfacial slippage are tightly coupled. This coupling maintains cells in a near-critical mechanical regime, where small variations in contractility are sufficient to switch between adhesion growth and disassembly. Such a regime is essential for efficient and persistent migration and highlights that the key determinant of FA dynamics is not stiffness alone, but the ability of the substrate to dissipate, redistribute, and actively reorganize mechanical energy at the biointerface.

It should be noted that the artificial substrates discussed here refer primarily to simplified and relatively homogeneous hydrogel systems with controlled ligand presentation and mechanical properties. More complex hydrogel platforms incorporating spatial heterogeneity, dynamic remodelling, or tunable viscoelasticity may reproduce additional features of native extracellular matrices and could therefore exhibit more diverse regulation of focal adhesion dynamics. The present comparison therefore specifically emphasizes the role of the hierarchical fibrillar organization and active interfacial remodelling characteristic of collagen I in generating asymmetric wetting–de-wetting behaviour.