Substrate curvature and stiffness equivalence in governing cell mechanosensing

Abstract

Mechanosensing enables cells to perceive and interpret their mechanical microenvironment, including forces, stiffness and topography. Although focal adhesions (FAs) are central to this process, their structural adaptation to mechanical stimuli remains poorly understood. Here, we uncover FA tilting - the inclination of the FA plane relative to the substrate - as a mechanically regulated architectural feature. Using reverse cell imprinting and atomic force microscopy, we reveal a strong inverse correlation between FA tilting angle and substrate stiffness. A two-dimensional clutch model shows that tilting emerges from force distribution across the FA–substrate interface and contributes to cell mechanosensing. By engineering rigid substrates with defined curvatures, we impose specific tilting angles independently of stiffness and modulate the cellular mechanostate, revealing a curvature–stiffness mechanical equivalence principle. This enables the construction of a correlation map linking curvature values to equivalent stiffness levels.

Together, our results identify FA tilting as a geometrical and mechanical transducer and a powerful design parameter for instructive biomaterials in physio-pathological tissue engineering.

This study reveals how cells translate substrate stiffness and curvature into biological responses through focal adhesion tilting, uncovering a mechanical equivalence that opens new routes for designing cell-instructive biomaterials.

Main

Mechanosensing, the ability of cells to detect and respond to the physical properties of their environment, is a crucial process that underpins many biological phenomena, including tissue development, regeneration, and the progression of diseases, such as cancer^1–4. At the foundation of this process are focal adhesions (FAs), multi-protein complexes that form at the interface between the cell and the extracellular matrix (ECM). They act as molecular hubs, converting external mechanical cues into biochemical signals. These signals regulate cellular processes, including migration, proliferation, apoptosis, and reprogramming^5–9. Mechanosensing involves a dynamic interplay between FAs, the actin cytoskeleton (CSK), and the nucleus. The CSK defines the mechanical identity of the cell, regulating the overall mechanical stress exerted on the nuclear envelope on one side while generating tension on FAs on the other^10–12. FAs act as mechanical adaptive joints, with their dynamics finely regulated by the balance between intracellular and extracellular forces¹³. The force balance enables cells to adapt to mechanical stimuli by altering FA composition, morphology, and signaling. These changes reorganize the CSK and trigger downstream pathways such as Hippo and Wnt, ultimately altering gene expression and cell behavior^14–17.

Among the diverse mechanical signals sensed by cells through FAs and CSK, ECM stiffness has emerged as a particularly influential factor. Over the past two decades, stiffness has been shown to regulate a wide range of cellular processes from stem cell differentiation to fibrosis and cancer progression^18–21. Stiffness modulates nearly every level of cellular organization from FA maturation to CSK contractility and nuclear architecture. On stiff substrates, cells typically form larger FAs, spread more extensively and assemble robust, contractile actomyosin networks^15,22. These mechanical adaptations are coupled with biochemical changes, including the nuclear translocation of YAP/TAZ, which activate gene expression programs associated with proliferation and differentiation²³. These processes are driven by forces transmitted across the integrin–CSK interface, arising from intracellular contractility and ECM properties, which dynamically regulate FA organization and mechanosensitive signaling^24,25. Conversely, on soft substrates, cells form smaller FAs, a less organized actin CSK, and exhibit impaired activation of mechanical signaling. This results in diminished cytoskeletal tension and reduced adhesion to the substrate.

These stiffness-dependent changes are closely linked to the structural and compositional remodeling of FAs. While much attention has focused on in-plane changes in FA size, composition, and turnover, it remains unclear whether out-of-plane architectural adaptations also contribute to how cells sense and respond to matrix mechanics. Variations in matrix rigidity modulate the recruitment and mechanical loading of key FA components, ultimately tuning the strength and dynamics of cell–ECM coupling. Super-resolution microscopy studies have revealed a defined nanoscale vertical organization of FA components within the ~200 nm thickness of the adhesion complex²⁶. Paxillin is predominantly localized in the proximal layer of the adhesion, close to the plasma membrane, where it serves as a platform for signaling and adaptor proteins involved in integrin-mediated mechanotransduction. It is one of the earliest proteins recruited during FA assembly and participates in the coordination of downstream signaling pathways by interacting with focal adhesion kinase (FAK), Src family kinases, and other regulatory effectors^27,28. Among other core FA components, talin plays a central mechanosensitive role, linking integrins to the actin CSK and undergoing force-induced unfolding that exposes binding sites for proteins such as vinculin²⁹. These force-dependent conformational changes contribute to a feedback loop in which mechanical loading drives protein recruitment and structural reorganization, ultimately modulating the mechanical and signaling output of the adhesion^29,30. These molecular rearrangements influence not only how forces are transmitted through the CSK but also how cells migrate across different mechanical landscapes. Migration tends to slow on stiff substrates, where FAs are more stable and to accelerate on softer matrices, where FAs are more dynamic and undergo faster turnover^6,31.

Understanding how these dynamic adhesions mediate the transmission of intracellular forces to the ECM has been the subject of extensive theoretical modeling. One of the most widely adopted frameworks is the motor–clutch model, which conceptualizes how actin retrograde flow and traction forces are regulated through stochastic engagement and disengagement of mechanical linkages, referred to as “clutches”, that connect the CSK to the ECM. Physically, the system is represented as a combination of elastic springs, with the substrate modeled as a single linear spring which reflects its effective stiffness, and the molecular clutches represented as multiple identical springs arranged in parallel³². Myosin-driven actin retrograde flow applies force to this network, progressively stretching the engaged clutches and deforming the substrate. As the applied force increases, individual clutch elements may stochastically unbind once their force threshold is exceeded. This collective behavior gives rise to emergent phenomena, including load-dependent FA binding and the generation of traction forces that scale with the mechanical properties of the substrate. Although originally formulated to describe cellular responses to matrix stiffness, the motor–clutch model has since been adapted to incorporate additional mechanical features of the microenvironment, such as substrate viscoelasticity, strain energy and ligand spacing, as well as intracellular response mechanisms like molecular reinforcement via talin unfolding^33–36. These extensions highlight the versatility of the framework in capturing the interplay between extracellular cues and intracellular force-regulation.

Yet, despite this progress, a crucial question remains unresolved: how do FAs adapt their three-dimensional architecture to modulate force transmission under different mechanical conditions? Existing models focus on molecular composition and in-plane dynamics, overlooking potential out-of-plane structural adaptations that may encode mechanical information. In this work, we uncover a previously unrecognized feature of FAs—tilting—that emerges as a direct geometric response to substrate stiffness and curvature. We propose that tilting is not a passive deformation, but a tunable, force-sensitive element that adds an additional dimension to FA-mediated mechanosensing.

Using reverse cell imprinting (RCI) combined with high-resolution atomic force microscopy (AFM), we analysed FAs architecture across substrates of varying stiffness. We discovered a systematic inclination of the FA plane relative to the substrate, a phenomenon we define as FA tilting. Importantly, tilting angle inversely correlated with stiffness, supporting its role as a geometric readout of substrate mechanics. This structural adaptation emerged as a geometric readout of substrate mechanics. Notably, tilting was not a random feature but showed a strong inverse correlation with substrate stiffness, pointing to a functional role in cellular mechanosensing. Specifically, soft substrates were associated with increased tilting and shorter FAs, whereas stiff substrates led to reduced tilting and more elongated FAs. To gain mechanistic insight into this phenomenon, we developed a two-dimensional clutch model that captures how differences in force distribution at the FA–substrate interface give rise to distinct tilting angles on soft versus stiff materials. The model resolves the local force balance and successfully reproduces the inverse relationship between substrate stiffness and tilting angle, supporting the view that tilting is not a passive consequence but a mechanical adaptation to the interplay between tangential and normal forces. In this context, tilting, a force-dependent reorientation of the adhesion plane, acts as a finely tuned sensor that modulates the balance of strain forces between the substrate and FA-associated proteins. By adjusting the distribution of these forces, tilting may contribute to how cells interpret the mechanical properties of their environment.

In addition to stiffness, geometry, particularly substrate curvature, has emerged as a potent mechanical signal. Substrate curvature has been shown to modulate a variety of cellular behaviors, including stem cell differentiation, neuronal guidance, and cancer cell motility, in ways that often mirror stiffness-dependent responses^37–42. These observations underscore the mechanosensitive nature of curvature across multiple cell types. The underlying mechanisms by which geometric cues influence cytoskeletal organization and adhesion remain poorly investigated. In this study, we demonstrate that by forcing cells to adopt specific FA tilting angles on engineered rigid substrates with controlled curvature, we can effectively modulate their mechanostate. To systematically explore this effect, we generated a library of curvature-defined substrates to impose varying degrees of tilting and assess the resulting cellular responses. These platforms revealed that tilting was associated with modulation of multiple mechanosensitive outputs, including FA length, cell spreading, CSK stiffness (Young’s modulus), nuclear deformation, and YAP localization. Strikingly, we identified a mechanical equivalence between cells cultured on rigid curved substrates with imposed tilting angles and those on flat, soft substrates. This led to the development of a mechanical correlation map linking specific curvature values to equivalent stiffness levels. These findings provide insight into how cells achieve mechanosensitive precision across diverse mechanical environments, with potential applications for cell engineering and programming, tissue engineering and disease modeling.

Results

Cell mechanosensing is governed by focal adhesion tilting

Advanced imaging techniques have unveiled the complex nanoscale organization of FAs, including their vertical organization and lateral polarity^26,43. Yet, how this architecture adapts to mechanical cues remains largely unexplored. Motivated by the hypothesis that spatial geometry may influence force transmission, we investigated whether structural rearrangements within FAs contribute to mechanosensing. To this end, NIH/3T3 fibroblasts were cultured on polydimethylsiloxane (PDMS) substrates with stiffness values of 1000, 200, and 36 kPa. Substrates were functionalized with RGD peptides to promote cell adhesion, using an optimized protocol to ensure uniform peptide coverage and consistent cell attachment across the different conditions. The RCI method was employed to evaluate the cellular footprint, while high-resolution AFM imaging enabled the reconstruction of FA spatial configuration. This approach preserved the integrity of basally associated proteins, allowing combined AFM and confocal imaging with nanometric resolution along the z-axis (Fig. 1A, B, and Supplementary Fig. 1). The interaction between cells and substrates was characterized by mapping the topography of FAs on cells adhered to substrates with different stiffness. By overlapping confocal and AFM data, we quantify the length and z-coordinates of the FAs on these substrates. Immunostaining revealed that FAs length, defined by major axis, increased with substrate stiffness (Fig. 1C). This observation is consistent with previous findings, showing that stiffer substrates promote enhanced adhesion maturation and assembly^44,45. The z-coordinates of FAs were assessed at their extremities corresponding to the start and end points, highlighting distinct inclination respect to the substrates (Supplementary Figs. 1 and 2). Specifically, we observed a consistent elevation at the distal end of each FA relative to its proximal end, indicating a measurable tilting. This tilting aligned with the traction forces exerted along the corresponding stress fibers (SFs), and the extent of tilting was influenced by the mechanical properties of the substrate. FA tilting arises from the combined action of tangential and normal stress components at adhesion sites, which generate a rotational moment. The presence of such torques at FAs has been reported in a previous study⁴⁶. Here, we investigate their role in driving the structural reorientation of the adhesion complex. Quantitative analysis revealed a stiffness-dependent reduction in FA tilting. Median inclination angles decreased from approximately 2.5° on 36 kPa substrates to 1.5° on 200 kPa and 0.8° on 1000 kPa substrates, corresponding to an approximately threefold reduction in tilt on the stiffest condition (Fig. 1C). The observed changes in FA tilting suggest that substrate stiffness influences the mechanical loading experienced by scaffolding proteins. On softer substrates, larger tilting angles could reduce force transmission along the FA axis, potentially limiting protein extension and adhesion growth. Conversely, smaller tilting angles on stiffer substrates may enhance tangential force, favoring adhesion maturation.

Fig. 1. Mechanistic insights into focal adhesion tilting and its impact on cell mechanosensing.

A Correlative paxillin immunostaining and AFM imaging enables FA structures to be associated with their corresponding height profiles. B Three-dimensional AFM topography reveals a height gradient along individual FAs, providing a measure of their inclination relative to the substrate. C Quantification of FA tilting angles and lengths on substrates with different stiffnesses shows that FAs display larger tilting angles and shorter lengths on softer substrates, whereas stiffer substrates promote smaller tilting angles and longer adhesions (n = 2, 3, 3 biological replicates, N_FA = 363, 291, 365 for 36,200 and 1000 kPa substrates, respectively). D Schematic of the two-dimensional clutch model, in which the substrate is represented by tangential and normal springs and engaged clutches transmit actin-generated forces to the substrate. Created in BioRender. Imparato, G. (2026) https://BioRender.com/rrvr5ku. E The model predicts that the number of engaged clutches increases with substrate stiffness, while actin retrograde flow decreases. F Simulations show that the FA tilting angle decreases and the traction force increases with substrate stiffness. Data in (E, F) are presented as time-weighted means ± time-weighted standard deviation, calculated over 100,000 stochastic simulation events. G Schematic representation of stiffness-dependent FA organization, showing larger tilting angles and altered cytoskeletal organization on softer substrates, and reduced tilting on stiffer substrates. Created in BioRender. Panzetta, V. (2026) https://BioRender.com/u5xc6e. H Proposed mechanical mechanism by which FA tilting increases the transduction angle, reducing the force component aligned with the adhesion axis and limiting adhesion maturation. Conversely, reduced tilting favors efficient force transmission. Created in BioRender. Panzetta, V. (2026) https://BioRender.com/bssrguo. I SEM image of fibroblasts cultured on a concave curvature-controlled PDMS substrate (κ = − 1/297 µm⁻¹). J FA length increases with decreasing substrate curvature, indicating a relationship between surface geometry and adhesion maturation (n = 2 biological replicates; N_FA = 684, 647, 614, 450 for 297 concave, 470 concave, 535 concave, and 835 concave, respectively). Boxplots indicate the interquartile range, central lines the median, and whiskers extend to 1.5× the interquartile range. Statistical significance was determined using the Shapiro–Wilk normality test, followed by a Kruskal−Wallis test and by predefined pairwise two-sided Mann–Whitney U tests. *p < 0.05; **p < 0.01; and ***p < 0.001.

To gain mechanistic insight into how FA tilting arises and contributes to mechanosensing, we developed a two-dimensional clutch model that focuses on force-dependent clutch engagement and rupture dynamics, capturing the interplay between intracellular forces, substrate compliance, and FA geometry. The model simulates the dynamic interactions between actin filaments, molecular clutches (representing FAs), and the ECM, allowing predictions of how FAs respond to varying mechanical properties of their environment. The two-dimensional clutch model we developed, illustrated in Fig. 1D, builds upon traditional one-dimensional frameworks but incorporating both normal and tangential elastic responses of the substrate. This two-dimensional refinement enables the model to capture features of ECM deformation that are overlooked in canonical 1D approaches, thereby providing a more accurate and spatially resolved understanding of the mechanosensing process. By schematizing the substrate using three orthogonal springs (Supplementary Fig. 3 and Supplementary Table 1), the model reveals how traction forces applied by the cell generate force fields with opposing signs: tensile forces dominate at the FA far edges, while compressive forces emerge beneath at the FA proximal edge⁴⁷. In this framework, FA tilting is not prescribed a priori but emerges from the mechanical coupling between actomyosin-generated forces and the elastic response of the substrate. Specifically, the interaction between tangential and normal force components transmitted through the deformable substrate generates an asymmetric deformation field at the adhesion–substrate interface, which manifests as a tilted FA configuration. Importantly, the present formulation does not include rotational degrees of freedom or explicit torque balance within the adhesion complex. Therefore, FA tilting should be interpreted as an emergent geometric consequence of asymmetric force transmission through the deformable substrate rather than as the result of a direct moment balance acting on a spatially resolved adhesion structure. In the discrete formulation, spring stiffness was linked to the macroscopic substrate modulus to ensure consistency with the mechanical properties of the PDMS substrates used experimentally (Supplementary Note 1 and Supplementary Figs. 4–6). Analysis of the model dynamics further reveals two characteristic regimes of clutch behavior (Supplementary Note 2 and Supplementary Fig. 7). When the number of available clutches is low, the system operates in a frictional slippage regime characterized by rapid binding–unbinding events that limit force buildup. As the number of engaged clutches increases, the system transitions toward a load-and-release regime in which forces accumulate before the release of a large fraction of bound clutches, rather than complete collective detachment, enabling more efficient transmission of traction forces to the substrate (Supplementary Fig. 7). The model should be interpreted as a minimal mechanical framework and as one possible physical explanation for the emergence of FA tilting, rather than as a complete or unique description of adhesion architecture. In particular, the current formulation represents the FA–substrate coupling as a lumped mechanical element, whereas real FAs are spatially extended structures and their molecular components can deform under shear. Furthermore, the clutch binding and rupture kinetics are described using a simplified non-equilibrium formulation that does not explicitly enforce detailed balance when force-free fibers are allowed to move. While more elaborate kinetic schemes could be considered, this simplification is not expected to qualitatively affect the predicted dependence of FA tilting on substrate stiffness. However, this framework recapitulates important hallmark behaviors of mechanosensing. As shown in Fig. 1E, the model accurately predicts that the average number of engaged clutches increases with substrate stiffness. It is important to note that it is associated with the optimal number of clutches that minimizes actin retrograde flow and represents the equilibrium condition for the cell (Supplementary Fig. 8). This result aligns with experimental findings, which demonstrate that FAs mature more effectively on stiffer substrates, enabling them to support greater mechanical forces and exhibit increased stability. Moreover, the model successfully replicates the relationship between traction force and substrate stiffness: on stiffer substrates, clutches engage more robustly, resulting in higher traction forces, while softer substrates support fewer engaged clutches, reducing the traction forces generated by the cell (Fig. 1F)^36,44,48. Strikingly, the simulations mirror our experimental findings, showing that increasing stiffness leads to a reduction in FA inclination respect to the substrate (Fig. 1F). These simulations are consistent with our experimental observations and support the interpretation that FA tilting emerges from the interplay between tangential and normal forces at adhesion sites. On softer substrates, higher tilting angles arise from the greater deformability of the material, which reduces the tangential component of the transmitted forces and is associated with smaller FAs. Conversely, stiffer substrates exhibit lower tilting angles, increasing the tangential force component and correlating with larger FAs. This shift in tilting is more than a structural adaptation; it provides a crucial insight into how FAs sense mechanical changes. The pronounced tilt on softer substrates suggests less efficient force transmission due to deformed adhesions being unable to anchor as firmly. Conversely, the reduced tilt on stiffer substrates indicates more efficient and direct force transmission. This tilting mechanism introduces a perspective on cellular mechanosensing, revealing that FAs not only react to mechanical forces but also adjust their geometry to interpret the mechanical properties of their environment (Fig. 1G).

A more refined mechanical interpretation of our results suggests that the tilting angle is tightly linked to another geometric feature that shapes cellular mechanosensing, although this parameter was not directly quantified in the present study. We refer to this as the transduction angle, defined as the angle between the SF and the FA plane (Fig. 1H). This angle governs the transmission of forces along the CSK–FA–ECM axis. Classic one-dimensional clutch models typically assume that force vectors are aligned with the adhesion axis and therefore do not explicitly account for the angular geometry of force transmission. Recent theoretical developments have begun to consider the role of force directionality, showing that oblique pulling can impair adhesion self-stabilization⁴⁹. In this context, our two-dimensional framework provides a complementary mechanical interpretation in which substrate-induced tilting modulates the effective angle of force transmission. On soft substrates, increased tilting rotates the adhesion plane relative to the incoming SF, effectively widening the transduction angle and reducing the component of force transmitted along the FA axis. Conversely, on stiffer substrates, reduced tilting narrows this angle, improves mechanical efficiency and promotes adhesion stabilization. Additional torque-generating mechanisms may contribute to tilting, including the internal organization of adhesion molecules, shear-induced deformation of clutch components, actin filament orientation, and mechanical interactions between adhesion proteins and cytoskeletal elements. Incorporating explicit torque balance, distributed FA architecture, and molecular-scale shear deformation would refine the quantitative description of the clutch model.

To directly test whether cytoskeletal forces regulate FA tilting, we modulated actomyosin contractility both experimentally and in silico. In the clutch model, increasing cytoskeletal contractility by raising the number of active myosin motors (n_m) led to higher traction forces transmitted through engaged clutches and to larger predicted tilting angles (Supplementary Note 3, Supplementary Fig. 9, and Supplementary Table 2). Consistently, actomyosin contractility was increased experimentally using lysophosphatidic acid (LPA) in NIH/3T3 cells cultured on 200 kPa substrates. Quantification of FA inclination using RCI revealed a significant increase in tilting angle following LPA treatment, while substrate stiffness and surface chemistry remained unchanged (Supplementary Note 3 and Supplementary Fig. 9).

Together, these results indicate that FA tilting is regulated by intracellular contractile forces and modulates the effective angle of force transmission at adhesion sites. We therefore asked whether this geometrical parameter could be externally imposed and used to modulate mechanosensitive responses. To this end, we engineered curvature-controlled substrates specifically designed to induce defined tilting angles in FAs and thereby isolate the effect of tilting from other variables, such as stiffness (Fig. 1I). These substrates featured concave topographies with precisely controlled radii of curvature. The platforms were fabricated as arrays of microlenses with diameters of 250 µm and curvature values (κ) ranging from 1/297 to 1/835 µm⁻¹. All substrates were made from PDMS with a stiffness of 1000 kPa, ensuring that any observed effects were attributable to topography rather than material compliance. To exclude potential artifacts arising from substrate geometry, we performed control experiments to assess both surface topography and ligand distribution. AFM measurements revealed homogeneous nanoscale roughness across both concave and flat substrates (Supplementary Note 4, Supplementary Figs. 10, 11, and Supplementary Tables 3, 4). In parallel, fluorescence mapping using aminofluorescein, coupled through the same Sulfo-SANPAH chemistry used for RGD functionalization, confirmed that ligand distribution was spatially uniform and independent of substrate curvature (Supplementary Note 5, Supplementary Fig. 12, and Supplementary Table 5). The curvature of the concave surfaces was designed to reproduce the tilting behavior naturally observed on soft substrates, thereby mimicking the geometric cue that modulates FA dynamics and the associated distribution of traction forces. We systematically analyzed FA length as a readout of adhesion maturation and mechanosensing in agreement with previous studies showing that FA elongation correlates with efficient force transmission and mechanotransductive signaling. We observed that FA length progressively decreased with increasing substrate curvature. Specifically, cells on highly concave surfaces (i.e., $\kappa =-\frac{1}{297}\mu {\mathrm{m}}^{-1}$) exhibited significantly shorter adhesions, while cells on flatter substrates (i.e., $\kappa =-\frac{1}{835}\mu {\mathrm{m}}^{-1}$) formed longer, more elongated FAs (Fig. 1J). This trend supports the idea that curvature-induced tilting modulates the magnitude of intracellular forces, with greater tilting disrupting effective load transfer and limiting FA growth. Several previous studies have reported that substrate curvature influences cell mechanics, including CSK architecture, nuclear orientation, and traction force generation^37,41,50,51. Our results suggest that a key element in this process is the tilting of the FA plane, which alters the transduction angle and redistributes mechanical forces within the adhesion complex. By quantifying this tilting and linking it to FA geometry, we provide a mechanistic interpretation that may help to unify and extend previous observations on curvature-guided cellular responses. These experiments establish a causal framework linking substrate mechanics, FA geometry, and cellular response. On flat substrates, reduced stiffness increases FA tilting through enhanced substrate deformation under cellular forces. Conversely, curvature-controlled substrates allow us to impose defined tilting angles on mechanically rigid materials without altering substrate stiffness. Remarkably, curvature-induced tilting produces cellular responses similar to those observed on soft substrates. Together, these observations suggest that FA tilting acts as a geometrical mediator through which both stiffness and curvature regulate cellular mechanosensing.

Substrate curvature mimics stiffness to control adhesion and migration

Having established FA tilting as a mechanosensing mechanism, we next aimed to exploit this principle for the rational design of cell-instructive materials capable of modulating mechanical cell identity. Because curvature allows us to impose defined FA tilting angles independently of substrate stiffness, this system provides a unique framework to test whether adhesion tilting itself can reproduce cellular responses typically associated with softer substrates. To this aim, we extended our curvature-controlled platform to include convex topographies (i.e., $\kappa =+\frac{1}{297}{\mu \mathrm{m}}^{-1}$), complementing the previously tested concave geometries. This addition enabled us to explore how directional curvature, at constant material stiffness, affects cellular behavior. First, we assessed the cell spreading area, a critical indicator of how cells interact with their microenvironment and adapt to varying mechanical cues^52,53. The spreading area reflects the extent of cellular adhesion and force transmission to the substrate, providing insights into the cell’s ability to sense and respond to its physical surroundings. An increase in basal cell area is observed when transitioning from concave to flat substrates. This trend becomes even more pronounced as the curvature shifts from a more accentuated concave profile to a convex profile (Fig. 2A, B, Supplementary Fig. 13, and Supplementary Table 6 for detailed p-values). Specifically, compared to the control condition on the flat rigid substrate (1000 kPa), a reduction of approximately 56% in cell spreading area on the 297 µm concave substrates, and a reduction of about 33% on the 535 µm concave substrates have been observed. Conversely, the 297 µm convex substrate shows a slight increase in spreading area of around 6%. The exploitation of convex topography optimizes the mechanical forces experienced by FAs, promoting a well-organized cytoskeletal arrangement and facilitating increased cellular spreading compared to the planar configuration. The quantification of the spreading area across substrates with varying curvatures highlights how curvature-induced spatial cues influence cellular adaptation and mechanosensing. The different adhesive capacity of the cells is also evident in the FIB-SEM images, which show variations in the adhesion sites between the cell and the substrate (Supplementary Figs. 14 and 15). This parameter not only reflects the extent of cell–substrate interactions but also establishes a link between substrate topography, cellular behavior, and FA tilting dynamics, providing further evidence of the mechanosensing role of FA tilt. Moreover, we assessed FAs under these different conditions. Importantly, while previous analyses had focused on curvature-dependent differences, we now directly compared curved substrates to flat ones of identical stiffness to isolate the effect of curvature. Despite all substrates being fabricated from the same material (PDMS, 1000 kPa), we observed a marked reduction in FA length on concave surfaces compared to flat controls (Fig. 2A–C and Supplementary Fig. 13), consistent with the predicted effect of increased imposed tilting disrupting force transmission efficiency. Conversely, on convex surfaces, FA length did not significantly exceed that observed on flat substrates, despite the increase in cell spreading area. This plateau effect suggests that FA elongation may reach a limit beyond which further increases in CSK forces do not translate into additional growth⁵⁴. To assess whether these curvature-dependent responses were cell-type specific, we performed the same analysis on mesenchymal stem cells (mMSCs). Notably, mMSCs displayed trends closely resembling those observed in NIH/3T3 fibroblasts, with reduced spreading and shorter FAs on concave substrates and increased spreading on the flat configuration with the same stiffness (Fig. 2E–G, Supplementary Fig. 13, and Supplementary Table 9). These results indicate that curvature-driven modulation of adhesion mechanics is not restricted to fibroblasts but represents a more general mechanosensing response across different cell types. In order to interpret the impact of curvature in terms of equivalent mechanical input, we next compared these results to cells cultured on flat PDMS substrates with varying stiffness. Remarkably, the FA size obtained on the substrates with the most pronounced concavity ($\kappa =-\frac{1}{297}{\mu \mathrm{m}}^{-1},1000\mathrm{kPa}$), were statistically indistinguishable from those observed on flat substrates with a stiffness of two orders of magnitude lower ($8\mathrm{kPa}$) for both cell lines. Expanding this concept, we found that the FA size on concave substrates with a curvature $\kappa =-\frac{1}{535}{\mu \mathrm{m}}^{-1}\left(1000\mathrm{kPa}\right)$ mirrored that on flat substrates with a stiffness of 120 kPa (Supplementary Tables 7 and 10 for detailed p-values). This equivalence allowed us to construct a mechanical correlation matrix that links specific tilting angles with their corresponding stiffness values. These findings demonstrate that substrate curvature can effectively replicate the mechanical properties of softer or stiffer environments, underscoring its potential as a tool for precisely modulating the cellular microenvironment. This curvature–stiffness relationship provides a powerful framework for manipulating cell behavior and advancing our understanding of mechanotransduction.

Fig. 2. Substrate curvature mimics stiffness to control adhesion and migration.

A Representative images and migration trajectories of NIH/3T3 fibroblasts on flat substrates (8, 120, and 1000 kPa) and curved substrates (±1/297 and −1/535 µm⁻¹). Actin is green, and paxillin-labelled FAs are white. B NIH/3T3 basal cell area across substrate conditions (n = 2 biological replicates; N_cell = 47, 44, 73, 45, 47, 45 for 8 kPa, 297 concave, 120 kPa, 535 concave, 1000 kPa and 297 convex, respectively). C NIH/3T3 FA lengths. Concave substrates produce shorter FAs, with −1/297 and −1/535 µm⁻¹ conditions resembling 8 and 120 kPa flat substrates, respectively (n = 2 biological replicates; N_cell ≥15 per condition and N_FA = 535, 682, 615, 612, 1350, 496 for 8 kPa, 297 concave, 120 kPa, 535 concave, 1000 kPa and 297 convex, respectively). D Left, schematic of FA tilting-mediated migration: larger tilting reduces adhesion stability and favors motility. Created in BioRender. Imparato, G. (2026) https://BioRender.com/fo0lmz. Right, NIH/3T3 motility across substrate conditions (n = 2 biological replicates; N_cell = 44, 47, 50, 43, 35, 39 for 8 kPa, 297 concave, 120 kPa, 535 concave, 1000 kPa and 297 convex, respectively). E Representative actin and paxillin images of mMSCs under corresponding conditions. F mMSC basal cell area (n = 2 biological replicates; N_cell = 45, 49, 48, 52, 56 for 8 kPa, 297 concave, 120 kPa, 535 concave and 1000 kPa, respectively). G mMSC FA lengths, showing smaller FAs on concave than corresponding flat substrates (n = 2 biological replicates; N_cell ≥15 per condition and N_FA = 366, 354, 506, 560, 785 for 8 kPa, 297 concave, 120 kPa, 535 concave and 1000 kPa, respectively). Boxplots indicate the interquartile range, central lines the median, and whiskers extend to 1.5× the interquartile range. Raincloud plots show the data distribution, with white dots indicating the median and individual dots representing the measured data points. Statistical significance was determined using the Shapiro–Wilk normality test, followed by a Kruskal−Wallis test and by predefined pairwise two-sided Mann–Whitney U tests. In the graphs, p-values are indicated as follows: * denotes comparisons of 120 kPa or 1000 kPa with respect to 8 kPa; # denotes comparisons of 535 concave or 297 convex substrates with respect to 297 concave substrates; $ denotes comparisons of 1000 kPa with respect to 120 kPa; § denotes comparisons of 297 convex with respect to 535 concave substrates. Differences between substrate pairs (8 kPa vs 297 concave, 120 kPa vs 535 concave, and 1000 kPa vs 297 convex) are indicated directly within the plots when present. ***, ###, §§§ p < 0.001, **, ##, §§§ p < 0.01, $ p < 0.05. Detailed p-values are available in the Supplementary Tables 6–10.

Coherently with the results gathered on cell spreading area and FA length, FA tilting appears to play a significant role in regulating cell migration. Time-lapse video analysis of NIH/3T3 cells cultured on substrates with varying curvature revealed a clear inverse correlation between curvature and cell motility rate (Fig. 2D). Cells exhibited significantly higher motility on concave surfaces compared to convex ones. These findings are in line with previous reports showing increased motility on softer or curved substrates^41,55,56. However, we reveal a continuous and quantifiable trend in cell motility across a range of defined curvatures, demonstrating that curvature can act as a predictable and engineerable parameter for guiding cell behavior. This observation reinforces the curvature–stiffness equivalence discussed above and further supports the role of FA tilting as a key mediator of mechanosensing (Supplementary Table 8 for detailed p-values). In analogy with soft substrates, where enhanced motility is commonly attributed to reduced adhesion stability, the increased migration observed on highly curved concave surfaces likely reflects the presence of shorter, less stable FAs. By contrast, long and stable FAs, typically formed on stiffer or convex substrates, are associated with reduced motility, in line with their higher mechanical resistance⁵⁷. Interestingly, we also observed increased directional persistence on convex substrates, suggesting that ECM curvature may influence not only migration speed but also guidance (Supplementary Fig. 16). Together, these results underscore the dynamic interplay between adhesion strength and cellular migration and highlight FA tilting as a tunable feature for modulating mechanosensing in engineered microenvironments.

Substrate curvature reprograms cell mechanical identity

FA length, number and orientation collectively define the intensity and directionality of cytoskeletal forces, thereby shaping the mechanical identity of CSK^44,45,58. Acting as a load-bearing network, the CSK plays a central role in transducing mechanical signals throughout the cells⁵⁹. Its ability to transmit signals efficiently stems from the numerous signaling molecules that bind to the CSK components, which are activated by mechanical stimuli^60,61. Consequently, altering the mechanical identity of CSK can lead to a shift in signal transduction and ultimately influence cell decisions. We have previously demonstrated that a change in CSK mechanical structure induced by patterning cell morpho-physical cues can result in robust cellular instructions through mechanoregulation^62–65. Accordingly, we hypothesized that a similar outcome could be achieved by regulating the FA tilting pattern. To test this hypothesis, we employed AFM to examine how alterations in FA tilting angle influenced the mechanical properties of the CSK (Fig. 3A). We note that AFM-derived Young’s modulus should be interpreted as a comparative readout of the cellular mechanical state rather than as a direct measurement of traction forces. As expected, a significant increase in the average CSK Young’s modulus was observed as the substrate curvature transitioned from negative (concave) to positive (convex) values (Fig. 3B). Despite employing the same rigid material, inducing a tilted configuration in FAs enables us to effectively mimic the mechanical effects of a compliant material on cell behavior. Indeed, cell mechanical properties (i.e., Young’s modulus) exhibit a reduction of approximately 77% on the 297 µm concave substrate and approximately 10% on the 535 µm concave substrate compared to the flat substrate (Fig. 3C and Supplementary Table 11 for detailed p-values). These findings successfully establish the equivalence between the curved rigid substrate $\left(\kappa =-\frac{1}{297}\mu {\mathrm{m}}^{-1},1000\mathrm{kPa}\right)$ and a soft flat substrate ($8\mathrm{kPa}$). For the substrate with a curvature $\kappa =-\frac{1}{535}\mu {\mathrm{m}}^{-1}\left(1000\mathrm{kPa}\right),$ the equivalent stiffness lies within the range of $10$ to$120\mathrm{kPa}$. On a convex substrate, FA extension leads to a stiffer CSK compared to a flat substrate, with an increase of approximately 13% in Young’s modulus. Importantly, comparable curvature-dependent changes in cytoskeletal stiffness were also observed in mMSCs, which exhibited reduced Young’s modulus on concave substrates compared to the corresponding flat configuration (Fig. 3D), confirming that curvature-driven modulation of cellular mechanical identity is conserved across cell types. Consistent with the behavior observed in NIH/3T3 cells, concave substrates induced a significant softening of the CSK, yielding Young’s modulus values comparable to those measured on flat substrates of lower stiffness. In particular, the Young’s modulus measured on concave substrates with curvature $\kappa =-\frac{1}{297}{\mu \mathrm{m}}^{-1}\left(1000\mathrm{kPa}\right)$ was comparable to that observed on flat substrates with a stiffness of 8 kPa (Supplementary Table 12). Overall, these findings highlight the tight coupling between cell adhesion, spreading, cytoskeletal organization, and mechanical properties, emphasizing the capacity of substrate curvature to modulate cellular mechanosensing. By directly altering the FA tilting angle, we demonstrate the ability to reprogram the mechanical identity of the CSK, thereby confirming FA tilting as a regulator of cellular mechanotransduction.

Fig. 3. Substrate curvature reprograms cell mechanical identity.

A Schematic illustrating the influence of substrate curvature on cell mechanics. Created in BioRender. Imparato, G. (2026) https://BioRender.com/v5l2zi. B Representative AFM deflection curves and topographical images of NIH/3T3 cells on concave, planar, and convex substrates. C Box plot illustrating the measured Young’s modulus of NIH/3T3 cells across different substrate configurations. Cell stiffness increases from concave to planar and convex substrates, with cells on −1/297 µm⁻¹ concave substrates showing a Young’s modulus comparable to those on 8 kPa flat substrates. Additionally, equivalence in Young’s modulus for cells cultured on concave substrates with a curvature of −1/535 µm⁻¹ (1000 kPa) lies within the range of 10 to 120 kPa (n = 2 biological replicates, N_modulus = 192, 362, 448, 310, 320, 352 for 8 kPa, 297 concave, 120 kPa, 535 concave, 1000 kPa and 297 convex, respectively). D Quantification of the Young’s modulus for mMSCs cultured on the same substrates reveals trends consistent with those observed for NIH/3T3 cells, confirming that curvature-dependent modulation of cellular mechanical properties is conserved across cell types, including a similar matching between concave rigid substrates and softer flat environments (n = 2 biological replicates, N_modulus = 443, 512, 544, 512, 512 for 8 kPa, 297 concave, 120 kPa, 535 concave and 1000 kPa, respectively). Boxplots indicate the interquartile range, central lines the median, and whiskers extend to 1.5× the interquartile range. Statistical significance was determined using the Shapiro–Wilk normality test, followed by a Kruskal−Wallis test and by predefined pairwise two-sided Mann–Whitney U tests. In the graphs, p-values are indicated as follows: * denotes comparisons of 120 kPa or 1000 kPa with respect to 8 kPa; # denotes comparisons of 535 concave or 297 convex substrates with respect to 297 concave substrates; $ denotes comparisons of 1000 kPa with respect to 120 kPa; § denotes comparisons of 297 convex with respect to 535 concave substrates. Differences between substrate pairs (8 kPa vs 297 concave, 120 kPa vs 535 concave, and 1000 kPa vs 297 convex) are indicated directly within the plots when present. ***, ###, §§§ p < 0.001, **, ##, §§§ p < 0.01, $ p < 0.05. Detailed p-values are available in the Supplementary Tables 11 and 12.

FA tilting coordinates nuclear shape, chromatin, and YAP signaling

Having demonstrated that changes in FA tilting angle directly impact the mechanical structure of the CSK, we aimed to explore whether variations in FA geometry correlate with nuclear deformation and downstream mechanotransduction responses. The CSK establishes a mechanical connection with the nucleus via the LINC complex (linker of nucleoskeleton to cytoskeleton)⁶⁶. This linkage enables the nucleus to function as a central player in cellular mechanosensing, transducing external mechanical stimuli into intracellular response⁶⁷. Nuclear deformations are governed by the dynamic interplay between CSK–nucleus connection, particularly the actomyosin and the vimentin networks, the structural integrity of the nuclear lamina and the chromatin organization^68–70. Since FA tilting influences the intensity and distribution of CSK-borne forces, it may also affect nuclear deformation. Indeed, significant alterations in nuclear shape were observed on curved substrates, as evidence by 3D reconstruction images generated by using Imaris software (Fig. 4). Fibroblasts cultured on concave substrates exhibited a more spherical nuclear morphology, with a significantly increased nuclear height compared to those on flat substrates (Fig. 4A, B, and Supplementary Table 13 for detailed p-values). These changes are closely associated with varying forces exerted on the nucleus. Substrate curvature acts as a regulator, modulating the contractile forces generated by the actin SFs and potentially influencing the compressive forces exerted by microtubules. By fine-tuning FA length through substrate tilting, the distribution and magnitude of cytoskeletal forces within the cell can be actively remodeled. Conversely, cells cultured on convex substrates displayed flattened and elongated nuclei. This deformation may be attributed to strong push-forces exerted by individual fibres of the perinuclear actin cap, leading to indentation of the nuclear membrane and the formation of grooves on its surface (Fig. 4A)⁷¹. These findings emphasize the role of substrate curvature in shaping the direction and intensity of cytoskeletal forces, which in turn significantly influence nuclear morphology (Fig. 4B). This reinforces the hypothesis that FA tilting contributes to cellular decision-making processes through mechanotransduction. To further explore this phenomenon, we analysed the level of chromatin condensation in correlation with substrate curvature. Trends in chromatin condensation mirrored nuclear morphology data, showing reduced chromatin compaction as the substrate transitioned from concave to convex curvature (Fig. 4C and Supplementary Table 14 for detailed p-values). This highlights a mechanistic link between substrate curvature and chromatin condensation state. Additionally, we investigated the expression and localization of Yes-associated protein (YAP), a critical mechanotransduction regulator. Our findings revealed that substrate curvature serves as a potent modulator of YAP nuclear translocation, complementing the well-established role of substrate stiffness⁷². Specifically, the nuclear-to-cytoplasmic (N/C) ratio of YAP increased on convex substrates compared to both concave and flat ones (Fig. 4D, Supplementary Fig. 13, and Supplementary Table 15 for detailed p-values). Importantly, similar curvature-dependent modulation of YAP localization was also observed in mMSCs, which exhibited increased YAP N/C ratio on flat rigid substrate and reduced nuclear localization on concave geometries (Fig. 4E, Supplementary Fig. 13, and Supplementary Table 16 for detailed p-values). In agreement with the observations in NIH/3T3 cells, concave substrates reproduced YAP nuclear levels comparable to those measured on softer flat substrates, further supporting the curvature–stiffness equivalence across different cell types. This observation aligns with prior studies demonstrating that increased substrate stiffness promotes YAP translocation to the nucleus^23,73. In our study, convex curvature mimicked the effects of stiffer substrates, amplifying YAP nuclear localization. In this context, substrate curvature appears capable of modulating similar mechanosensitive outputs. These findings suggest that substrate curvature can impact not only nuclear shape but also chromatin organization and YAP signaling, which are key regulators of gene transcription and mechanotransduction pathways^23,74. The relationship between chromatin condensation, YAP translocation, and the cell’s mechanical state is mediated by multiple pathways and factors, such as histone modifications, cellular contractility, and nuclear deformation^74,75. While our findings do not establish a direct causal link between FA tilting and nuclear signaling, they indicate that variations in adhesion geometry and cytoskeletal organization correlate with nuclear mechanotransduction responses within the broader ECM–FA–CSK–nucleus mechanical axis^76,77. By modulating the transmission of mechanical signals from the extracellular environment through FAs and the CSK to the nucleus, FA tilting may represent the geometrical adaptation necessary to the cellular response to mechanical environments. This interconnected system orchestrates nuclear deformation, chromatin organization, and mechanotransduction signaling pathways, emphasizing the importance of substrate curvature in influencing cellular responses and decision-making processes.

Fig. 4. Focal adhesion tilting: shaping nuclear morphology, chromatin state, and YAP signaling in mechanotransduction.

A Mid-plane FIB-SEM cross-sections of NIH/3T3 fibroblasts cultured on 297 concave, flat, and 297 convex substrates. B Representative 3D reconstructions showing nuclear shape in NIH/3T3 fibroblasts cultured on substrates with varying stiffness (8, 120, and 1000 kPa) and curvature (297 concave, 535 concave, and 297 convex). Nuclei are more spherical and taller on concave or softer substrates and become progressively flatter on planar, convex or stiffer substrates (n = 2 biological replicates, N_cell = 19, 9, 26, 13, 10, 28 for 8 kPa, 297 concave, 120 kPa, 535 concave, 1000 kPa and 297 convex, respectively). C Representative Hoechst-stained nuclei and quantification of chromatin condensation parameter (CCP). Chromatin condensation appears to decrease from concave to convex substrates and from softer to stiffer flat substrates (n = 2 biological replicates, N_cell = 12, 10, 13, 10, 11, 10 for 8 kPa, 297 concave, 120 kPa, 535 concave, 1000 kPa and 297 convex, respectively). D Representative immunofluorescence images and quantification of the YAP N/C ratio in NIH/3T3 cells. YAP images were normalized to the mean intensity of the corresponding cell. Nuclear YAP localization increases with substrate stiffness and from concave to convex configurations, with concave substrates reproducing responses observed on softer flat substrates (n = 2 biological replicates, N_cell = 14, 11, 14, 10, 11, 10 for 8 kPa, 297 concave, 120 kPa, 535 concave, 1000 kPa and 297 convex, respectively). E Representative images and quantification of the YAP N/C ratio in mMSCs cultured under the same substrate conditions. mMSCs show similar curvature- and stiffness-dependent responses, supporting curvature–stiffness matching across cell types (n = 2 biological replicates, N_cell = 23, 20, 22, 19, 21 for 8 kPa, 297 concave, 120 kPa, 535 concave and 1000 kPa, respectively). Boxplots indicate the interquartile range, central lines the median, and whiskers extend to 1.5× the interquartile range. Statistical significance was determined using the Shapiro–Wilk normality test, followed by a Kruskal−Wallis test and by predefined pairwise two-sided Mann–Whitney U tests. In the graphs, p-values are indicated as follows: * denotes comparisons of 120 kPa or 1000 kPa with respect to 8 kPa; # denotes comparisons of 535 concave or 297 convex substrates with respect to 297 concave substrates; $ denotes comparisons of 1000 kPa with respect to 120 kPa; § denotes comparisons of 297 convex with respect to 535 concave substrates. Differences between substrate pairs (8 kPa vs 297 concave, 120 kPa vs 535 concave, and 1000 kPa vs 297 convex) are indicated directly within the plots when present. ***, ###, §§§ p < 0.001, **, ##, §§§ p < 0.01, $ p < 0.05. Detailed p-values are available in Supplementary Tables 13–16.

Discussion

Our results suggest a mechanistic sequence linking substrate mechanics to cellular response. On flat substrates, reduced stiffness increases FA tilting as a consequence of enhanced substrate deformation under cellular forces. Furthermore, curvature-controlled substrates impose defined tilting angles while maintaining constant material stiffness. The observation that curvature-induced tilting reproduces the cellular responses observed on soft substrates supports the view that FA tilting acts as a geometrical mediator of mechanotransduction. Our findings indicate that FA tilting regulates the spatial configuration of proteins within FA, thereby affecting the transmission of mechanical forces from the ECM through the CSK to the nucleus. A strong correlation was observed between FA inclination and their size, suggesting that tilting directly influences FA assembly and disassembly. Furthermore, FA inclination correlates with substrate deformability, highlighting a unified mechanosensing process mediated by the geometrical organization of adhesion proteins. Specifically, substrate deformation drives FA tilting, which could in turn alter the spatial organization and mechanical state of proteins within FAs, thereby governing cellular responses to external stimuli. Our findings extend the current understanding of mechanosensing by demonstrating that FA tilting unifies the cellular interpretation of mechanical (stiffness) and topographical (curvature) signals. A substrate stiffness-curvature equivalence principle was established, unifying cell response to both mechanical and curvature cues. Cells cultured on rigid concave substrates (k = 1/297 µm⁻¹, E = 1000 kPa) exhibit mechanical properties and behavior akin to those cultured on soft flat substrates (8 kPa). This equivalence highlights FA tilting as a key mechanotransduction mechanism, offering the potential to map substrate curvature to equivalent stiffness values. Importantly, we show that FA tilting influences nuclear morphology, chromatin condensation, and YAP signaling, suggesting that its effect might extend beyond short-term responses like migration and spreading to long-term outcomes, such as functionality and differentiation. Importantly, the biological relevance of curvature-dependent mechanosensing extends beyond the specific experimental system used here. Curved microenvironments are widespread in vivo, where cells interact with folded epithelia, tubular structures, vascular walls, glandular architectures, and curved extracellular matrices. Previous studies have shown that curved substrates influence stem cell fate decisions and spatial tissue organization and homeostasis^50,51,78. Within this broader framework, our results suggest that FA tilting provides a local adhesion-scale mechanism through which substrate curvature can be coupled to cytoskeletal force transmission and downstream mechanosensitive outputs. These findings suggest a potential strategy for controlling cellular behavior by leveraging FA tilting to create cell-instructive materials. Such an approach has far-reaching implications for tissue engineering, regenerative medicine, and disease modeling, enabling unprecedented control over both immediate and long-term cellular outcomes.

Methods

PDMS substrates preparation for cell cultures

PDMS substrates with varying stiffnesses and curvatures were prepared using Sylgard 184 and Sylgard 527 silicone elastomer kits (Dow Corning). Two different ratios of elastomer to curing agent were used for Sylgard 184, 1:10 and 1:30, resulting in Young’s moduli of 1 MPa and 200 kPa, respectively. Sylgard 527 was mixed in a 1:1 ratio as per the manufacturer’s instructions. By blending Sylgard 184 (1:10) with Sylgard 527 in specific proportions, additional stiffness levels were achieved, yielding substrates with moduli of 8 kPa (1:40 ratio), 36 kPa (1:20), and 120 kPa (1:5 ratio). The mechanical properties of the PDMS substrates were characterized using small-amplitude oscillatory shear tests performed with an Anton Paar shear-controlled rheometer, equipped with a 25 mm diameter parallel plate geometry. These tests provided measurements of the linear elastic properties of the substrates, confirming that all the substrates behaved as predominantly linear elastic materials across the range of tested moduli (Supplementary Note 6 and Supplementary Fig. 17). This allowed us to exclude the effects of viscoelasticity on cellular responses, ensuring that cellular behavior was solely influenced by substrate stiffness and topography rather than time-dependent viscoelastic effects. Concave surfaces were formed by replica moulding against microlens arrays (Suss Microoptics) used as masters. In contrast, convex substrates were produced through a double casting method involving plasma treatment, followed by an ethanol treatment to prevent adherence during the casting of the second PDMS layer⁷⁹. The substrates were sterilized by overnight incubation in a penicillin-streptomycin solution, followed by a 1-h UV treatment. Surface modification for enhanced cell adhesion was carried out using Arg-Gly-Asp (RGD) peptides. The PDMS surfaces were first treated with a bifunctional photolinker, N-sulphosuccinimidyl-6-(4′-azido-2′-nitrophenylamino)hexanoate (sulpho-SANPAH 0.5 mg mL⁻¹ in deionized water, Thermo Fisher Scientific), activated by UV light (50 W, 365 nm, Ted Pella Inc) for 20 min serving as a cross-linking agent for peptides immobilization. Following PBS washes, the substrates were incubated overnight at 4 °C with a 1 mM RGD solution in 50 mM carbonate buffer (pH 8.5). Unreacted NHS groups were blocked using 0.2 mM ethanolamine (Sigma) at 4 °C for 30 min^45,80.

Cell culture and immunofluorescence

Mouse embryo fibroblasts (NIH/3T3, 93061524) were cultured under standard conditions at 37 °C in a humidified atmosphere with 5% CO₂. NIH/3T3 were maintained in high glucose Dulbecco’s Modified Eagle Medium (DMEM, Sigma), supplemented with 10% calf serum, 2 mM L-glutamine, 1000 U/L penicillin, and 100 mg/L streptomycin (Sigma). D1 ORL UVA bone marrow-derived murine mesenchymal stem cells (mMSC, 12424 ATCC) were cultured in DMEM with 2 mM L-glutamine, supplemented with 10% foetal bovine serum, penicillin (100 U mL⁻¹) and streptomycin (100 mg mL⁻¹) (Sigma). The medium was refreshed every 48–72 h, and cells were passed at 80% confluence using trypsin-EDTA. For all experiments, cells were seeded onto PDMS substrates and incubated for 24 h prior to fixation.

Fixation and permeabilization

Cells were fixed in 4% paraformaldehyde for 15 min at room temperature to preserve cellular structures. Following fixation, cells were permeabilized with 0.1% Triton X−100 for 15 min to allow antibody access to intracellular targets. Non-specific binding was blocked by incubating the samples in 3% bovine serum albumin (BSA) in phosphate-buffered saline (PBS) for 1.5 h.

Immunostaining procedure

Primary antibodies were used to target specific cellular components. Anti-paxillin monoclonal antibody (ab32084 Abcam, 1:250 dilution) was employed to visualize FAs, as paxillin is a key marker of these adhesion sites. Anti-YAP1 polyclonal antibody (PA1-46189 Thermo Fisher Scientific, 1:250 dilution) was utilized to assess nuclear mechanotransduction pathways, given YAP1’s role as a mechanosensitive transcriptional regulator. Paxillin antibody was incubated for 1.5 h at room temperature. Conversely, anti-YAP1 polyclonal antibody was incubated overnight at 4 °C. After three washes with PBS 1×, the samples were incubated for 1 h with species-specific secondary antibodies conjugated with Alexa Fluor 546 and Alexa Fluor 488 (Invitrogen), diluted in 3% BSA/PBS 1× solution. Nuclei were stained with Hoechst 33342 (Invitrogen) at a concentration of 1 µg/ml for 15 min. To visualize actin filaments, cells were stained with Alexa Fluor 488 phalloidin (Invitrogen, A12379) at a dilution of 1:200 in PBS 1× for 1 h.

Imaging and analysis

Immunofluorescence imaging was performed using both a Zeiss LSM 700 and Leica STED-SP5 confocal microscope. High-resolution images of FAs, nuclear morphology, and cytoskeletal organization were acquired using 63× oil immersion and 40× water immersion objectives. Additionally, the cell adhesion area and migration patterns were analysed using a 10× objective, which provided a broader field of view for capturing larger regions of interest (ROI). FAs were quantified using ImageJ/Fiji software, with a custom routine developed to eliminate background noise and enhance signal detection. Specifically, uneven background fluorescence was corrected using a rolling-ball background subtraction. The images were then smoothed using a median filter to reduce pixel-level noise while preserving the sharp boundaries of adhesion structures. Bright outlier pixels, typically arising from detector noise or isolated fluorescence spikes, were removed using the Remove Outliers function. Finally, a despeckle filter was applied to further suppress residual high-frequency noise. This preprocessing sequence improves the signal-to-noise ratio and enhances elongated high-intensity structures corresponding to FAs, facilitating their subsequent segmentation and quantitative analysis. Cell spreading area was also quantified using ImageJ, with automated routines ensuring consistent measurements across samples. Nuclear morphology was reconstructed in three dimensions using the Surface function in Imaris software (version 10.1.1, Bitplane). Confocal z-stacks of nuclear staining were imported into Imaris, and nuclei were segmented by generating 3D surfaces based on fluorescence intensity thresholds. YAP nuclear localization was assessed by measuring the ratio of nuclear to cytoplasmic fluorescence intensity (ImageJ), providing insights into the activation of mechanotransduction pathways. The routines are available upon request from the corresponding author. Chromatin condensation levels were quantified by calculating the CCP from confocal images of cell nuclei⁸¹. Briefly, images were first pre-processed to convert the nuclear regions into intensity matrices. Edge detection was then performed using the Sobel algorithm, which calculates the gradient magnitude at each pixel to identify the edges of chromatin structures. The resulting edge map was subjected to thresholding and skeletonization to reduce the edges to a single-pixel width. The CCP was calculated by dividing the number of detected edges by the cross-sectional area of the nucleus, providing a normalized measure of chromatin condensation.

Reverse cell imprinting (RCI) method

After 24 h from culture, cells were washed twice in PBS, fixed for 20 min in 4% paraformaldehyde (Sigma), washed twice in PBS, rinsed with 70 and 100% ethanol for 30 s each and then dried in a flow of nitrogen. A droplet of UV optical adhesive (NOA63, Norland) was distributed on the sample and pressed with a coverslip. The sample was exposed to UV light (365 nm, 22 mW/cm², Ted Pella Inc) for 5 min to solidify the adhesive. Then, the coverslip-attached adhesive was gently peeled off the PDMS substrates, exposing the backside of the cell^82,83. Direct measurements of FA tilting were performed on mechanically stable PDMS substrates (200 and 1000 kPa). Softer substrates were not suitable for RCI measurements because very compliant materials deform during the inversion and imprinting steps, compromising the nanoscale reconstruction of the cell–substrate interface.

Atomic force microscopy (AFM)

To scan the backside of cells, we used a commercial AFM (JPK Nanowizard II AFM, JPK Instruments) to image the cell basal side mounted on a 700 LSM confocal microscope (Carl Zeiss). Fixed cells were scanned in liquid in contact mode using triangular shaped, reflective gold-coated silicon nitride cantilevers (MSCT, Bruker) or with rectangular silicon cantilevers (FESP-V2, Bruker). Before inversion, cells were stained for paxillin. Fluorescence images used for correlation with AFM measurements were acquired after inversion using a Zeiss 40× water-immersion objective. The mechanical properties of cells, instead, have been tested using an AFM tip featuring a 6 µm diameter polystyrene bead. An area of 2 × 2 µm² has been covered employing an indentation depth of 100 nm and a velocity of 2 μm/s. The indentation points were selected in the perinuclear cytoplasmic region while explicitly excluding the nuclear area. The nucleus was identified optically before indentation, and force curves acquired directly above the nucleus were not included in the analysis. Measurements were also avoided at the extreme cell periphery, where reduced cell thickness and local substrate effects may influence the mechanical readout. This strategy allowed us to probe a cytoplasmic region representative of the cellular mechanical state while minimizing direct contributions from nuclear stiffness, nuclear compression, and substrate-related artifacts. For experiments on curved substrates, AFM indentations were performed only on cells located within the central region of each curvature-defined microlens. The center of the concave or convex feature was identified from the optical reference, and a circular ROI extending up to approximately 100 µm from this center was used for analysis. Cells located near the microlens edge, at the transition between curved and flat regions, or partially spanning different curvature zones were excluded. This precaution was adopted to minimize the local surface inclination relative to the AFM indentation axis. When the surface is inclined, the applied indentation force is not perfectly aligned with the surface normal, which can lead to a slight underestimation of the elastic modulus derived from the force–indentation curves. Considering the smallest radius of curvature used in this study (R = 297 µm), the maximum expected inclination within the probed region results in a deviation of approximately 6% in the normal force component (worst-case estimate corresponding to an indentation point located ~100 µm from the center of the concave substrate, i.e., radial distance r ≈ 100 µm), representing a conservative upper bound that does not affect the interpretation of the comparative mechanical measurements.

FA tilting measurements

The AFM system used in this study (combined AFM–optical platform) includes an integrated motorized stage and dedicated software that performs automatic spatial calibration between the optical and AFM reference frames. This calibration relies on coordinate transformation between the optical image coordinate system and the scanner coordinate system, enabling direct overlay of confocal and AFM topography images within the same field of view, as reported in correlative AFM–optical microscopy workflows^83. However, when high-magnification objectives are used (40× in the present case), the reduced field of view can make the overlay between optical and AFM datasets more sensitive to small positioning variations. Therefore, following the automatic calibration, a manual fine alignment step was performed. AFM and confocal images were overlaid, and clearly identifiable morphological landmarks (cell perimeter and membrane contours; Supplementary Fig. 1) were used to refine the alignment. A rigid translation was applied until pixel-level correspondence of the cell borders was achieved.

To obtain a statistically robust dataset, a semi-automated procedure for FA inclination analysis was implemented. AFM topography images were exported from the JPK software as height matrices $z\left(x,y\right)$ consisting of $N\times M$ data points (1024 × 1024 pixels over a 100 µm scan area), where each matrix element corresponds to the surface height at the spatial coordinate $\left(x,y\right)$.

Paxillin fluorescence images were first rescaled to match the AFM image dimensions, then thresholded to reduce background fluorescence and converted to binary masks using ImageJ. For tilting analysis, only FAs with lengths between 1 and 5 µm were considered. This range was selected to exclude nascent adhesions (<1 µm), and fibrillar adhesions (>5 µm), which are primarily involved in matrix remodeling⁸⁴. In addition, FAs displaying tilting angles greater than 10° were excluded from the analysis, as these are typically located in the perinuclear region where the weight of the nucleus can introduce inclination unrelated to substrate mechanics. each identified FA, the coordinates of its two extreme points (distal and proximal ends) were determined from the binary mask. These two points define the principal adhesion axis in the x-y plane.

The corresponding height values at these two positions, $z_{distal}$ and $z_{proximal}$, were extracted from the AFM topographic matrix. The FA inclination angle $\theta$ was then calculated as:

$$\theta =\arctan \left(\frac{\mathrm{\Delta }z}{L_{FA}}\right),$$ 1

where $\mathrm{\Delta }z=z_{proximal}-z_{distal}$ and $L_{FA}$ is the projected adhesion length in the x-y plane between the two extreme points.

Because AFM imaging was performed on inverted cells (i.e., imaging the basal side in reversed configuration), the measured height gradient appears reversed relative to the native cellular orientation. After accounting for this inversion, FA height profiles consistently decreased from the proximal to the distal end in the original cellular configuration, consistent with the direction of traction forces transmitted along the associated stress fibres.

FIB-SEM

Focused ion beam scanning electron microscopy (FIBSEM) was the approach used to evaluate cell-material interaction. Cells were fixed in 2.5% glutaraldehyde (Electron Microscopy Sciences) in 0.1 M sodium cacodylate buffer (Electron Microscopy Sciences) overnight at 4 °C to preserve structures. After rinse in sodium cacodylate buffer, samples were incubated in a 20 mM glycine solution (Sigma) at 4 °C for 20 min to crosslink the reactive groups. To increase the contrast and preserve the ultrastructure during FIBSEM imaging, the ROTO (reduced osmium-thiocarbohydrazide-osmium) protocol was performed. Briefly samples were post fixed in a solution of 2% osmium tetroxide/1% potassium ferrocyanide (Electron Microscopy Sciences) in 0.1 M sodium cacodylate buffer for 1 h at 4 °C in the dark (RO step), then washed three times in distilled water and incubated in 1% thiocarbohydrazide aqueous solution (Electron Microscopy Sciences) for 20 min at room temperature in the dark (T step). Finally, cells were incubated in 1% osmium tetroxide aqueous solution (O step) for 30 min at room temperature in the dark before the overnight incubation with 1% uranyl acetate solution. To increase the membrane contrast, cells were incubated in 0,15% tannic acid (Sigma) for 3 min, then rinsed three times in distilled water before dehydration. The dehydration was performed in ascendent series of ethanol (30%, 50%, 70%, 95%, and 100%), each step for 10 min at 4 °C, except for absolute ethanol in which three steps were performed at room temperature. Samples started the embedding in low viscosity resin (Electron Microscopy Sciences) according to the ultra-thin plasticization protocol (UTP)^85,86. For this purpose, samples were embedded in a mix of absolute ethanol/resin with different ratios (2/1 for 3 h; 1/1 overnight; 1/2 for 3 h; absolute resin overnight and over day), then put in a vertical position for 3 h to remove the excess of resin before polymerization in oven at 70 °C for 24 h. Samples were mounted on 12 mm aluminum pin stub (Ted-Pella) with silver paste (RS company), coated with 20 nm of gold by using of sputter coating (HR208_Cressington) and loaded in the FIBSEM (Helios CX5_Thermo Scientific) chamber for milling. The cell surface was scanned with 3 kV electron beam to identify a ROI. One micrometer Pt layer was deposited in the selected ROI, and the ion milling was carried out at a 30 kV with a current between 0.23 and 0.43 nA. After selection of ROI all milling and imaging parameters were set (z-depth, dwell time) and the Auto Slice and View (Thermoscientific software) performed the automatic slice and acquisition to make a final AVI of the selected area. The acquisition was performed by Through Lens Detector (TLD) in Backscattered electron mode with dynamic focus mode in a range of magnification between 5 KX and 35 KX.

The motor clutch model

The current implementation extends a previously established molecular-clutch framework by transitioning from a one-dimensional to a two-dimensional formulation^87,88. This extension enables a more realistic representation of intracellular force transmission and ECM interactions by accounting for spatial force distribution and FA dynamics. A schematic representation of the physical model and its working principle are provided in Supplementary Fig. 3A, B, and Supplementary Movie 1. In particular, the substrate is modeled using three orthogonal elastic springs with stiffness $k_{S,T}^{left}$, $k_{S,T}^{right}$, and $k_{S,N}$, capturing both tangential and normal responses to CSK-applied forces. To ensure consistency with the original 1D formulation, spring constants were selected such that the equivalent tangential and normal stiffness matched the target substrate stiffness. A dedicated static sensitivity analysis was performed to quantify geometry-induced effects arising from variations in spring rest lengths (100–10,000 nm); deviations from the reference 1D stiffness remained limited across stiffness regimes and were minimized when all rest lengths were set to 1 μm (Supplementary Note 1 and Supplementary Figs. 4–6). Accordingly, identical rest lengths of 1 μm were adopted in all simulations.

To account for the fact that the overall rigidity of the system depends on $k_{S,T}^{left}$, $k_{S,T}^{right}$, and $k_{S,N}$, these values were used to compute an effective network constant (Supplementary Note 1), tuned to span the range of experimentally tested substrate stiffnesses.

In the dynamic simulations, the model describes $N_c$ stochastic molecular clutches connecting the actin cytoskeleton to the deformable substrate through the junction node $S$, whose equilibrium position is ${\mathbf{x}}_{\mathit{S}}={\left(x_S,y_S\right)}^T$(see Supplementary Fig. 3). Note that vector quantities are denoted in boldface, whereas scalar components are indicated in italic.

Each clutch binds with rate $K_{on}$ and unbinds with a force-dependent rate $K_{off,i}^{*}$:

$$\frac{{dP}_{b,i}}{dt}=\left(1-P_{b,i}\right)K_{on}-P_{b,i}{K}_{off,i}^{*}$$ 2

Once engaged, each clutch, modeled as a linear springs of stiffness $k_{c,i}$, develops tension as it is stretched by F-actin retrograde flow and the force-dependent off-rate follows Bell’s law:

$$K_{off,i}^{*}=K_{off}\exp \left(\frac{∣{\mathit{F}}_{c,i}∣}{F_b}\right)$$ 3

where $∣{\mathit{F}}_{c,i}∣$ is the magnitude of the axial force in the $i$-th clutch and $F_b$ is the characteristic rupture force.

Myosin motors ($n_m$) generate contractile forces that drive rearward actin motion. In the absence of load, the actin filament moves at the unloaded velocity $v_u$. Under mechanical resistance, the retrograde velocity decreases according to a linearized Hill-type force–velocity relationship:

$$v_f=v_u\left(1-\frac{F_{S,x}}{n_m{F}_m}\right)$$ 4

where $F_m$ is the single-motor stall force and $F_{S,x}$ is the magnitude of the horizontal component of the substrate reaction force acting on node $\mathrm{S}$,

$$F_{S,x}=F_{T,left,x}+F_{T,right,x}+F_{N,x}$$ 5

Thus, only the force component opposing actin motion contributes to the reduction of sliding velocity. In the present model, we do not explicitly resolve whole-cell-scale variations in stress-fiber orientation from central to peripheral FAs. The model assumes that the pulling direction is aligned with the principal FA axis and therefore focuses on the local mechanical coupling between clutch forces and substrate deformation.

The components of velocity of the $i$-th clutch attachment point are defined as a weighted average between the substrate velocity and the actin retrograde velocity, depending on its binding state. Along the horizontal direction

$$\frac{d{x}_{c,i}}{dt}=\left(1-P_{b,i}\right)\frac{d{x}_S}{dt}+P_{b,i}{v}_f$$ 6

Along the vertical direction, where actin retrograde flow has no component

$$\frac{d{y}_{c,i}}{dt}=\left(1-P_{b,i}\right)\frac{d{y}_S}{dt}+P_{b,i}\left(0\right)$$ 7

At each stochastic event, the junction node $S=\left(x_S,y_S\right)$ is assumed to reach quasi-static mechanical equilibrium. The axial forces generated by the engaged clutches are transmitted to the deformable substrate and must be balanced by the elastic reaction of the substrate springs.

Mechanical equilibrium is therefore obtained by enforcing force balance at node $S$, where the sum of the substrate spring forces and the clutch forces vanishes. Solving this nonlinear system yields the instantaneous equilibrium position $\left(x_S,y_S\right)$, at which substrate deformation compensates for the tensile forces developed by the engaged clutches.

The force balance is

$${\mathit{F}}_{T,left}\left({\mathbf{x}}_S\right)+{\mathit{F}}_{T,right}\left({\mathbf{x}}_S\right)+{\mathit{F}}_N\left({\mathbf{x}}_S\right)+{\sum }_{i\in ϵ\left(t\right)}{\mathit{F}}_{C,i}\left({\mathbf{x}}_S,{\mathbf{x}}_{C,i}\right)=\mathbf{0}$$ 8

where $\mathcal{E}\left(t\right)$ denotes the set of engaged clutches at time $t$. In the dynamic simulations, clutch forces are computed individually and summed only over the engaged clutches, in contrast to the equivalent stiffness formulation used in the static analysis.

The substrate spring forces are given by

$${\mathit{F}}_{T,left}=k_{S,T}^{left}\left(1-\frac{{\ell }_{L,0}}{{\ell }_L}\right){\mathit{d}}_L$$ 9

$${\mathit{F}}_{T,right}=k_{S,T}^{right}\left(1-\frac{{\ell }_{R,0}}{{\ell }_R}\right){\mathit{d}}_R$$ 10

$${\mathit{F}}_N=k_{S,N}\left(1-\frac{{\ell }_{N,0}}{{\ell }_N}\right){\mathit{d}}_N$$ 11

The vectors ${\mathbf{d}}_i$, current lengths ${\ell }_i$, and the explicit expressions of the substrate spring forces are defined as described in Supplementary Fig. 3. For completeness, the clutch force acting on node $S$ is written as

$${\mathit{F}}_{C,i}=k_{ci}\left(1-\frac{{\ell }_{C,0}}{{\ell }_{C,i}}\right){\mathit{d}}_{C,i}$$ 12

where

$${\mathit{d}}_{C,i}={\mathbf{x}}_S-{\mathbf{x}}_{C,i},{\ell }_{C,i}=\parallel d_{C,i}\parallel$$ 13

The coupled stochastic–mechanical system was solved using a next-reaction variant of the Gillespie stochastic algorithm implemented within a Monte Carlo framework in MATLAB (M R2022B; R2023A). Specifically, for each possible clutch engagement or disengagement event, a candidate event time was randomly generated according to:

$$t_{\frac{eng}{diseng},i}=-\ln \left({random}_i\right)/k_i$$ 14

where ${random}_i$ is a random number between 0 and 1 and $k_i$ is the kinetic rate for a clutch to engage ($K_{on}$) or disengage ($K_{off,i}^{*}$). The event associated with the shortest candidate time was selected and executed, and the simulation clock was advanced accordingly.

At each stochastic event, mechanical equilibrium was enforced by solving the nonlinear force-balance equations. The number of simulated events was capped at 100,000 per simulation. A complete list of model parameters is provided in Supplementary Table 1. To evaluate the robustness of the model predictions, sensitivity analyses were performed by varying the clutch kinetic parameters $k_{on}$ and $k_{off}$ over physiologically relevant ranges⁸⁷.

(Supplementary Tables 2, and 17, 18). The dependence of FA tilting on substrate stiffness remained qualitatively unchanged.

Statistical analysis

Statistical distributions for each experimental condition were visualized using box plots, bar plots or rain plots which display individual data points. The boxplot represents the first and third quartiles, with the median indicated within the box. The whisker length is set to 1.5 times the interquartile range. Independent experiments were performed in which a substrate platform was fabricated and seeded with cells from different passages. Each replicate therefore represents an independent experiment incorporating both substrate-to-substrate variability and biological variability associated with cell culture. The number of independent experiments is indicated as n, whereas N denotes the total number of analysed data points pooled across experiments. The subscript of N indicates the nature of the data point; for example, N_FA denotes the number of FAs. Exact values of n and N are reported in the corresponding figure legends. Statistical analyses were performed using the online software Statistics Kingdom. Outliers were excluded. Data normality was assessed using the Shapiro–Wilk test, with a threshold of p < 0.05 indicating a deviation from normality. For normally distributed data, statistical comparisons were made using Student’s unpaired t-test, while the nonparametric Kruskal–Wallis test was applied for data that did not meet normality assumptions followed by predefined pairwise two-sided Mann–Whitney U tests between biologically motivated experimental conditions. Statistical significance is indicated as follows: *p < 0.05; **p < 0.01; and ***p < 0.001. Detailed p-values for all comparisons are reported in the Supplementary Tables 6−16 to ensure clarity in the main figures.

Reporting summary

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

Author contributions

Conceptualization: S.F. and P.A.N. Methodology: C.F., V.P., V.M., R.M. and S.S. Investigation: C.F. and V.P. Visualization: C.F. Supervision: V.P., S.F. and P.A.N. Writing (original draft): C.F. Writing (review & editing): C.F., V.P., V.M., R.M., S.S., S.F. and P.A.N.

Peer review

Peer review information

Nature Communications thanks Ansgar Petersen, Laurent Pieuchot, who co-reviewed with Ismail Tahmaz and the other anonymous reviewer for their contribution to the peer review of this work. A peer review file is available.

Funding

This work was financially supported by the Italian Ministry of University and Research (MUR) through the project FIT4MEDROB: Fit for Medical Robotics – PNRR MUR (PNC0000007) and the project Space It Up (Contract No. 2024-5-E.0, CUP No. I53D24000060005).

Data availability

All data used in the analysis are available in the main text, in the Supplementary materials, in the Source data file, on Zenodo at https://doi.org/10.5281/zenodo.21640658 or upon request to the corresponding authors. Source data are provided with the paper.

Code availability

The code used in the analysis is available in Supplementary Note 7 and on Zenodo at https://doi.org/10.5281/zenodo.21600489.

Competing interests

The authors declare no competing interests.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the author(s) used ChatGPT from OpenAI in order to revise English in some sentences. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-76866-w.