Mechanically primed cells transfer memory to fibrous matrices for invasion across environments of distinct stiffness and dimensionality

INTRODUCTION

In fundamental biological processes of development, disease, and regeneration, cells move to and from mechanically distinct environments. For example, neural crest cells move throughout the embryo to lay the foundation for structurally complex organs (Shellard et al., 2018), and cancer cells escape stiff tumors and invade through softer healthy tissue to initiate metastasis (Paszek et al., 2005; Acerbi et al., 2015). In any given environment, cells sense and respond to stiffness of their extracellular matrix (ECM) through actin–myosin force generation and focal adhesion signaling (Pelham and Wang, 1997; Li et al., 2017). Accordingly, epithelial cells are known to migrate faster on stiffer two-dimensional (2D) surfaces (Ng et al., 2012). When such epithelial cell collectives encounter mechanical interfaces between stiff and soft matrices, greater cellular forces on stiffer ECM pull the whole cell colony to migrate toward the stiffer region—a process termed collective durotaxis (Sunyer et al., 2016). However, it remains unknown how grouped cells negotiate such mechanically dissimilar interfaces in fibrous three-dimensional (3D) environments. If the durotaxis model, previously understood on 2D surfaces, holds true in 3D, cells moving from a stiff fibrotic–like environment into softer healthy-like tissues would potentially slow down due to a net force balance toward the stiffer side, which remains untested thus far.

Further complicating this cross-environment mechanical heterogeneity, cells not only sense their current environment but also remember their past environments by retaining activated expression of mechanosensitive genes. For example, stem cells have been shown to store a mechanical memory of their past stiffness on flat substrates via a RUNX2 and yes-associated protein (YAP) signaling axis (Li et al., 2017). This cellular mechanical memory has been described through synergistic feedback between protein-level mechanotransduction, mechanosensitive transcriptional activity, and epigenetic remodeling (Price et al., 2021). In the context of collective cell migration, as epithelial cells move from stiff to soft 2D surfaces, adequate stiff priming enhances their migration on future soft surfaces due to a YAP-based mechanical memory (Nasrollahi et al., 2017). However, it is not yet clear whether such memory of cellular mechanical priming can be imparted to 3D collagen microenvironments and whether this remodeled collagen can alter future cell invasion strategies into new, mechanically distinct environments (Figure 1A).

Unlike 2D elastic substrates (Wozniak et al., 2003; Li et al., 2017; Nasrollahi et al., 2017), 3D matrices composed of collagen fibers undergo plastic remodeling orchestrated by cellular adhesions and forces (Cukierman et al., 2001; Ilina et al., 2020). Conversely, collagen deformation, alignment, and degradation can effectuate different modes of cell invasion (Provenzano et al., 2008; Kraning-Rush et al., 2013). Alternatively, when the collagen fiber architecture is remodeled beforehand, cells exploit matrix contact guidance mediated by cellular mechanoactivation and efficient alignment of forces, which in turn lower the energy barrier required for cell migration through 3D microenvironments (Kim et al., 2015; Seo et al., 2020). Although the effect of collagen structure on cellular response is well appreciated, it remains unknown whether past mechanical priming of cells can enable different migration strategies for cell invasion into future collagen environments. Because mechanically “stiff-primed” cells remain mechanoactivated after leaving their priming environment (Nasrollahi et al., 2017), it is possible that the enhanced forces of stiff-primed cells may result in greater collagen remodeling, and this remodeled collagen architecture could become advantageous for impending cell invasion. In this scenario, cell invasion outcomes would depend not only on the current collagen structure but also on the environmental memory of cells. Here, we investigate these questions through spatiotemporal invasion measurements of primed cells implanted in collagen and in silico modeling with an energy-minimization model that integrates primed mechanoactivation, immediate mechanosensing, and collagen remodeling within a lattice-based framework for cell invasion.

RESULTS

Past priming of cells on stiff substrates increases collagen remodeling

To combine two mechanically distinct environments for cellular mechanical priming and future invasion within one assay, we developed a dual-matrix scaffold with synthetic hydrogels and 3D collagen type 1 (Figure 1B). We first cultured MCF10A human breast epithelial cells on soft (0.08 kPa) or stiff (16 kPa) hydrogel substrates, which led to greater expression of RhoA and Rac1 in stiff-primed cells (Figure 1C). Next, we implanted these hydrogel disks with adhered cells within 3D collagen (2.3 mg/ml) to allow spontaneous invasion of soft- or stiff-primed cells into a new and mechanically different environment without intermediate detachment (see Materials and Methods; Figure 1B). After 3 days of invasion through collagen (Figure 1D; Supplemental Figure S1A), we found that cells previously stiff primed had more aligned F-actin fibers (Figure 1E; Supplemental Figure S1B) compared with soft-primed cells. We also found greater collagen fiber alignment (defined as coherency) around invaded cells that migrated from stiff-priming substrates (Figure 1F; Supplemental Figure S1C). To compare forces generated by differentially primed cells, we embedded fluorescent beads in collagen, trypsinized cells after 3 days of primed invasion to release cellular tension, and tracked bead displacements, which are expected to proportionally increase with forces originally generated by the invading cells. We found that stiff-primed cells led to greater collagen deformation in the 3D space around the invasive front that expands approximately 300 µm from the edge of the gel with a depth of deformation of close to 10–20 µm (Figure 1G). Scanning electron microscopy (SEM) showed more collagen bundling and accumulation around stiff-primed invading cells (Figure 1H). Although both soft- and stiff-primed cells resided in the same collagen condition for ∼3 days postpriming, cells that were previously primed on a stiff surface increased collagen remodeling in terms of both structural modifications and tension stored in collagen fibers (Figure 1).

Persistent invasion of stiff-primed cells is preceded by collagen deformation at the invasive front

Given the known importance of cell–matrix engagement in cell migration, the differential collagen remodeling caused by soft- versus stiff-primed cells (Figure 1) could also result in different outcomes for 3D cell invasion. In our measurements, stiff priming led to almost double the invaded distance and a higher number of invaded cells, compared with soft priming (Figure 2, A–C). Notably, these differences in the number of invaded cells grew larger as the invasive front moved farther away from the priming environment (Figure 2C), which could indicate that the influence of priming-dependent collagen remodeling on cell invasion is potentially compounded over time. We confirmed a similar advantage of stiff priming for three additional cell types—human breast cancer cell lines MDA-MB-231, MCF-7, and malignant conversion of MCF10A—by overexpressing ErbB2 (Lu et al., 2009), a known epithelial-mesenchymal transition (EMT)-related oncogene in breast tumors (Supplemental Figure S2, A–D). When tracked over time and distance from the polyacrylamide (PA) gel, stiff-primed cells pull and accumulate collagen behind the invasive front (Supplemental Movie S1; Figure 2D). These high-density collagen “anchors” remain stable over the duration of observed invasion. After the initial ∼12 h period of rapid matrix remodeling, the collagen deformation rate is reduced, which could indicate establishment of pretension within the collagen matrix (Figure 2E). As a result, the invasion speed of stiff-primed cells almost doubled compared with that of their soft-primed counterparts. By plotting spatial maps of collagen bead density over time (Figure 2D), we show that regions of high-density collagen accumulation are larger and more pronounced in the case of stiff-primed cells and once formed, these regions do not dissolve over time. Here, the collagen deformation rate around soft-primed cells remained too low (Figure 2E; Supplemental Movie S2) to cause collagen accumulation or tension generation (Figure 2D).

To better define the role of mechanical priming in subsequent invasion across an interface, we considered several alternate hypotheses and scenarios. We note that in the chosen design for cross-environment invasion, cells need to be continuously adhered to an ECM. After priming, when we trypsinized and implanted cell colonies by themselves, that is, without the attached PA gel, we found that such trypsinization was too stressful for cells to form epithelial colonies, and this process erased any mechanical priming (unpublished data; similar observations were made in an earlier study with stem cells [Yang et al., 2014]). We did not pursue this method because, unlike this trypsinization procedure, cells within the body traverse across environments without clipping all their adhesions. Cell densities behind the invasive front were similar in the soft- and stiff-primed conditions after 5 days of culture (Supplemental Figure S1, D and E). We also considered and discounted another possibility, that cells invading through collagen could laterally sense the PA gel, which is left hundreds of microns behind the invasive front where key measurements are made. If such a tug-of-war with the past PA gel were to occur, according to the known long-distance durotaxis on substrates of gradient stiffness (Sunyer et al., 2016), the stiff PA gel left behind would exert stronger forces compared with the softer PA or 3D collagen. As a result, there will be a net backward force pointing toward the stiff PA gel, which in turn would slow down cell invasion across the stiff-to-collagen interface. Instead, we see the opposite phenotype, in which cells coming from the stiff ECM invade faster, indicating an active process of forward migration. Finally, to show the functional importance of priming in active cell invasion, we repeated experiments in which cells were minimally primed before their 3D invasion. Here, MCF10A cells were seeded on soft and stiff PA gels, allowed to adhere for 6 h (to ensure robust adhesion and handling of samples), and immediately implanted into 3D collagen matrices. In this “minimal priming” experiment, we found that the cell invasion distance did not change between the invading cells from soft or stiff PA gels (Supplemental Figure S4). Thus, the enhanced invasion of cells stiff primed for 3 days, shown in Figure 2, cannot be explained purely by interface effects (from stiff PA to softer collagen); instead, extended priming is required for advantageous collagen remodeling and cell invasion.

Integrating primed cell mechanoactivation, matrix remodeling, and direct mechanosensing in an energy-minimization model explains cross-environment cell invasion

Previously, mechanosensitive cell migration has been broadly described in terms of a dynamic spatiotemporal coordination between adhesions with the current matrix, active intracellular contractility, bulk cell mechanics, and front–rear polarity of protrusions (Lauffenburger and Horwitz, 1996; Danuser et al., 2013). These basic principles have been incorporated into motor-clutch models, coupling adhesions with forces, to explain stiffness-sensitive cell migration and collective durotaxis (Chan and Odde, 2008; Sunyer et al., 2016). Similar principles of force balance have been used to understand cell migration and invasion in 3D environments (Zaman et al., 2005). These models have not only provided fundamental insights into how the intricate coupling of subcellular and extracellular processes come together to generate cell motion but have also enabled new predictions, for example, biphasic and stick-slip cell migration (Pathak and Kumar, 2011; Bangasser et al., 2017). However, the role of past cellular priming in current cell migration remains undocumented in previous models, which in turn limits our mechanistic understanding of cross-environment cell invasion and further predictions in this scenario, where both present and past environments could affect cell migration. Here, we develop a minimalist model of primed cell invasion with these goals: to incorporate the idea of priming and matrix feedback into cell migration; to understand whether our experimental results can be explained by a consistent set of physical rules; to inform future experiments via predictions; and to expand the basic framework of cell migration by including both past and present mechanosensing.

Based on our experimental results thus far, cellular forces due to past stiff priming perform collagen remodeling (measured as accumulation, alignment, and tension) that helps propel future cell invasion through 3D collagen matrices. Without such remodeling, when cells enter a 3D matrix, they first experience a resistance (γ) due to matrix density that raises the energy barrier required for cell migration (Figure 3A). To enable forward migration, cells must overcome this resistance via collagen remodeling (α) performed by cellular forces, which is assumed to be directly driven by cellular mechanosensing (φ) (Figure 3A). Combining these factors, cell migration becomes more probable when the energy barrier due to ECM resistance (γ) can be lowered by collagen remodeling (α). On the basis of these basic principles of force-based collagen remodeling, we define a resultant potential for effective cell protrusions as –ΔH_protrusions = μ (Figure 3B; Eqs. 2 and 5), which is defined as a function of φ, α, and γ wherein the negative sign represents lowering of the energy barrier. On the basis of our experiments, we posit that collagen remodeling results from both primed mechanoactivation (ψ) from the past ECM and direct mechanosensing (φ) of the current matrix. While conventional models of cell invasion rely on direct ECM mechanosensing, this simple energy-minimization criterion could potentially capture our experimental observations of primed cellular mechanoactivation followed by collagen remodeling and future cell invasion. In the absence of this priming-dependent collagen remodeling, cells would succumb to bulk resistance from the newly encountered collagen matrix, as they move across environments. In our model, primed mechanoactivation decays slowly, consistent with previous studies (Zaman et al., 2005; Nasrollahi et al., 2017), and direct mechanosensing (φ) adapts quickly to the new environment. Using a lattice-based framework, we simulate primed cell invasion based on an energy balance of direct mechanosensing (φ), resistance (γ), and priming-dependent collagen remodeling (α) that together cause a net migration potential (μ) (see Materials and Methods for details).

As we begin to understand how these newly unveiled processes of priming-dependent matrix remodeling regulate cell invasion, we first consider classic models of cell movement (Lauffenburger and Horwitz, 1996) and assess which minimal modifications will be needed to explain the observed memory-dependent cell invasion across environments (Figure 3B). According to conventional models (DiMilla et al., 1991; Lauffenburger and Horwitz, 1996), a migrating cell extends protrusions, forms new adhesions, and breaks rear adhesions via contractile forces. Each of these steps in migration has corresponding energetic costs. Here, we define the total energy of the system (H) as the sum of the energies associated with adhesions (H_adhesions), bulk mechanical properties of the cell (H_bulk), and the net frontward protrusions described above (H_protrusions) (Figure 3B). Cell movement occurs such that the net energy of the system is minimized, ΔH < 0 (Figure 3B). While the terms defining adhesions and bulk mechanics are somewhat similar to those of previous models (as described in Materials and Methods), the priming-dependent matrix remodeling and its effect on the cell migration potential is an addition. Specifically, the stiff-primed cellular mechanoactivation causes collagen remodeling, which in turn reduces the net protrusion potential and thus lowers the net ΔH. Proportionally, this reduction in energy cost increases the probability of moving the corresponding cellular lattice points. By contrast, reduced collagen remodeling due to comparatively lower levels of soft-primed mechanoactivation keeps the energy barrier high, thus making it less favorable to move lattice points (Figure 3B).

To explain and understand experimental results, we simulated the spatial and temporal evolution of stiff-primed (φ = ψ = 1) or soft-primed (φ = ψ = 0.3) cell colonies migrating into a collagen matrix (ρ = 2.3 mg/ml). As cells enter collagen, the mechanosensing signal (φ) quickly adapts to the current soft signal received from the new collagen environment, and the net migration potential (μ) drops because of the ECM resistance (γ) (Figure 3C, top row). However, the primed mechanoactivation signal (ψ) is modeled to remain high for ∼12 h, allowing collagen remodeling (α) to rise (Figure 3C, top row) and accumulating collagen at the invasive front similar to experiments (defined by high-intensity regions of α; Supplemental Figure S3A). Collagen remodeling (α) reduces the preexisting matrix resistance to invasion and causes a rise in net migration potential (μ). After 12 h, although primed mechanoactivation (ψ) adapts to new soft collagen, matrix remodeling (α) continues to oppose ECM resistance (γ) to maintain high net migration potential (μ), which results in cell invasion going forward (Figure 3C, top row; Supplemental Movie S3). In comparison, soft primed cells do not carry high enough levels of primed mechanoactivation that could be used for collagen remodeling, and thus they cannot overcome the ECM resistance (γ). As a result, net migration potential (μ) does not rise, leading to a low level of invasion (Figure 3C, bottom row; Supplemental Movie S3). Thus, the ability of primed cellular mechanoactivation to remodel collagen before or during cell migration into the new environment is a crucial step in the cross-environment collective cell invasion into collagen.

Loss of stable cellular priming obviates memory-dependent matrix remodeling and cell invasion

Our model integrates priming kinetics, direct mechanosensing of immediate matrix, and collagen remodeling with conventional models of mechanosensitive cell migration (Lauffenburger and Horwitz, 1996; Ahmadzadeh et al., 2017). Using this model, we next sought to further understand the importance of past cellular priming in future collagen remodeling and the ensued invasion. To this end, we performed simulations where a stiff-primed cell state could not be stably sustained. In our model, the influence of primed cellular mechanoactivation on collagen remodeling and invasion is regulated by a slow decay of primed mechanoactivation and quick adaptation to current environment. Specifically, the rate of decay of primed mechanoactivation is captured by the parameter ζ, whose higher values ensure more stable priming (Figure 4A). We simulated cells primed for 5 days on stiff substrates followed by invasion into collagen, but their primed mechanoactivation was allowed to deplete rapidly (ζ = 50) (Figure 4, A–C). Our simulations show that such rapid depletion of priming (ψ) does not allow enough collagen remodeling (α), whichis required to overcome the ECM resistance (Figure 4, B and C). As a result, cells devoid of past priming invaded collagen at a slower speed compared with those with stable priming, despite having the same levels of prior stiff priming in both cases (Figure 4, B and C).

To test our model prediction of memory-independent invasion in the case of unstable priming, we depleted YAP, a known regulator of cellular mechanical memory (Yang et al., 2014; Nasrollahi et al., 2017), in MCF10A cells and repeated experiments of primed-cell invasion. Here, MCF10A-shYAP cells were stiff or soft primed and allowed to invade 3D collagen (as described above; Figures 1 and 2). We found that there was no substantial difference between soft- and stiff-primed cells in terms of alignment of F-actin fibers and collagen fibers (Figure 4, D–F). As cells invade, the resulting collagen deformation is priming independent, somewhat temporally uncoordinated, and smaller compared with that in the control case (Figure 4H). Consistent with our model predictions of priming-dependent collagen remodeling before cell invasion, a lower level of collagen deformation by shYAP cells coincided with their slower invasion speeds (Figure 4I), regardless of soft or stiff priming. We also found that myosin inhibition via blebbistatin treatment decreased all collagen deformation and cellular invasion. However, some invasion differences between soft- and stiff-primed cells persisted despite actin–myosin inhibition of contractility via blebbistatin, which indicates that, unlike YAP depletion, myosin inhibition does not erase past priming and simply reduces cell invasion for any previous priming (Supplemental Figure S4, C and D). To quantify the effect of past priming on future cell invasion, we calculated a “priming advantage,” defined as the difference in the invasion speeds of soft- and stiff-primed cells. While this priming advantage increased over time for control cells, it significantly decreased in the case of shYAP (Figure 4J). Overall, our model and the experimental validation with shYAP show that the loss of stable primed mechanoactivation in cells eliminates differences in collagen remodeling, which in turn results in priming-independent cell invasion.

Priming advantage for cell invasion persists in dense or stiffened collagen

Because dense or cross-linked collagen matrices are known to restrict cell invasion (Wolf et al., 2007; Ilina et al., 2020), we asked whether prior mechanical priming of cells could prove advantageous to cells when they encounter such challenging environments (Figure 5A). We repeated experiments in two collagen compositions: first, higher collagen density of 3.1 mg/ml and second, with cross-linking to increase stiffness but maintain density by mixing 2.3 mg/ml collagen with the photo-cross-linker riboflavin followed by UV exposure (Grunert et al., 2015). We measured stiffness of collagen gels using atomic force microscopy (AFM) and found that increasing collagen density to 3.1 mg/ml almost doubled the stiffness to an average of ∼1 kPa compared with ∼0.5 kPa for a control collagen density of 2.3 mg/ml (Figure 5B). Photo-cross-linked collagen led to an even greater increase to ∼2.4 kPa average stiffness. To understand how differentially primed cells behave in these collagen matrices, we allowed soft- and stiff-primed cells to invade, imaged collagen fibers (Figure 5C), and found higher levels of cell invasion and fiber alignment (Figure 5D; Supplemental Figure S5, A and B) by stiff-primed cells. Compared to the control collagen concentration, higher density and cross-linked collagen matrices led to overall slower invasion and lower collagen deformation (Figure 5, E and F, left column). Even so, the stiff-primed cells generated a higher collagen deformation rate (Figure 5E, right column; Supplemental Figure S6) and resulted in faster cell invasion compared with the soft-priming case (Figure 5F, right column). Although the cell invasion rate was reduced in these challenging matrices of higher density and cross-linking, the priming advantage toward invasion was preserved (Figure 5G). Through collagen bead intensity kymographs (Figure 5H; Supplemental Figure S6), we show that stiff-primed cells generated a higher level of collagen accumulation in both dense and cross-linked collagen matrices. To test whether our computational model can capture these experimental findings of priming advantage in various collagen compositions, we repeated simulations with a higher collagen density (ρ = 3.1 mg/ml). To define stiffened and cross-linked collagen in the model, we used ρ = 2.3 mg/ml, = ρ/4, γ₀ = 0.06—these parameters represent increased collagen stiffness and bulk resistance, which is physically consistent with the corresponding matrix in experiments (Figure 5B) that resulted in thicker fibers (Supplemental Figure S5C) and reduced pore size (Supplemental Figure S5D). Consistent with experiments, although collagen deformation and invasion rates were reduced in dense and cross-linked collagen, stiff-primed cells continued to perform better than soft-primed cells (Figure 5I). Overall, our experiments and simulations show that prior mechanical priming continues to be advantageous in denser and stiffened collagen matrices due to priming-mediated collagen remodeling.

Unstable collagen remodeling disrupts the connection between past cellular priming and future invasion

According to our model and results thus far, priming-dependent cellular forces pull on collagen fibers to enable cell invasion. This collagen remodeling needs to be temporally stable such that cells can continue to move as they go from one environment to another (Figure 6A). To test the effect of collagen remodeling kinetics on invasion, we first used our model to vary rate constant r_α, which determines the speed of collagen remodeling (Figure 6B). According to our simulation, smaller values of r_α led to smaller amounts of remodeled collagen (α) by stiff-primed cells (Figure 6C). If matrix remodeling is not performed before the primed cellular mechanoactivation dissipates, the past environment has no way of impacting future cell invasion. Because remodeled collagen is required to reduce the energy barrier for cell invasion, lower levels of collagen remodeling (due to lower values of r_α) led to slower cell invasion (Figure 6C). To experimentally test the role of stable collagen remodeling in primed invasion, we used β-aminopropionitrile (BAPN) to inhibit the lysyl oxidase (LOX) that cross-links collagen. After soft or stiff priming of cells, BAPN was added, and cells were allowed to invade into collagen (Figure 6D; Supplemental Figure S7A). Through collagen fiber imaging and analysis, we found that collagen fiber alignment was rendered independent of past soft or stiff priming (Figure 6E; Supplemental Figure S7B). In the case of stiff-primed cells, LOX inhibition reduced collagen fiber bundling compared with that of the control (untreated) case (Figure 6F). These structural analyses of collagen fibers (Figure 6, E and F) indicate that LOX inhibition reduces collagen remodeling in either of these two cases of mechanical priming of cells (Supplemental Figure S7C). When tracked over time (∼2 days), both collagen deformation and invasion rates of stiff-primed cells remained low (Figure 6G), which is consistent with modeling predictions.

Spatial propagation of collagen remodeling is required for priming-dependent invasion

As we have shown, the ability of cells to sustain priming-dependent forces and remodel collagen is important for invasion (Figures 1–4). We next asked whether cellular forces are necessary only near the invasive front, or whether they need to be propagated over long distances into the matrix to enable collagen remodeling (Figure 7A). In our model, after primed cells reach collagen, they remodel their surrounding collagen and the resulting field of lowered energy must spatially propagate through the collagen matrix to enable sustained cell invasion. To test this effect of spatial propagation of collagen remodeling, we repeated simulations with a reduced distance of propagation of collagen remodeling α (parameter n = 1; Figure 7B). Here, a higher value of n caused a reduction in net accumulation of remodeled collagen (α). In this case, because an invasive front of stiff-primed cells is unable to cause large-scale changes in the matrix, the net cell invasion rate was reduced (Figure 7C). To implement an analogous spatial disruption of force-based collagen remodeling in experiments, after the stiff-primed cells invaded into the collagen and at the onset of invasion, we performed laser-based ablation of collagen fibers ahead of the invasive front. As shown in a representative image (Figure 7D), collagen fibers in contact with invading cells are intact; however, they are disconnected from the rest of the collagen matrix. In this case, overall collagen deformation was substantially reduced within a few hours after laser ablation of fibers and remained in a low net deformation state over time (Figure 7E; Supplemental Figure S8). This depleted effect of ablated matrix indicates the importance of long-distance force propagation within collagen for its bulk remodeling. Similarly, although the invasion rate of stiff-primed cells started at a high level, it subsided within 6–8 h and remained low over 2 d of tracking (Figure 7F). In sum, as the stability of primed mechanoactivation was important for cells (Figure 4), stable remodeling and long-distance force propagation are important in collagen matrices for priming-dependent matrix remodeling and subsequent cell invasion.

DISCUSSION

Collagen remodeling by cell-generated forces and reinforcement of cellular mechanoresponse by matrix mechanics have been known and studied for several decades. To enable force-based migration strategies, cells sense their environment and accordingly up-regulate actin–myosin contractility and Rho signaling, which in turn allows them to either squeeze through narrow pores or align collagen fibers to make paths in 3D environments (Beadle et al., 2008; Friedl and Wolf, 2010). Recent studies have revealed cells’ ability to store mechanical memory of their past environments, which has been explained in the form of persistently high levels of YAP, a mechanosensory protein, after stiff priming of cells (Yang et al., 2014; Nasrollahi et al., 2017). Here, we asked whether a similar mechanical priming could sustain cellular forces long enough to remodel 3D fibrous matrices and thus influence cell invasion across environments of different stiffness and dimensionality, which has not been studied before. To address this question, we developed a novel hydrogel–collagen device to combine mechanical priming and cross-environment invasion. Using this system, we show that cells previously primed by stiff environments continue to invade through softer 3D collagen matrices—a multifaceted process mediated by collagen fiber remodeling.

Considering previous studies on cellular mechanical memory (Yang et al., 2014; Nasrollahi et al., 2017), it is not surprising that past priming on stiff substrates enhances future cell migration. Adding to the emerging idea of stored cellular memory (Lele et al., 2019; Gonzales et al., 2021), our study reveals that mechanical priming of cells by their past environment not only affects cellular response but also remodels the extracellular collagen microenvironment, which in turn regulates future cell invasion. By mapping collagen deformation and cell invasion over time and space, we showed that much of the collagen remodeling happens before the outward cell invasion (Figure 2). During this process, cells accumulate “anchors” of high collagen density behind the invasive front and use aligned collagen fibers to invade. In these somewhat sequential steps, collagen remodeling and cell invasion are temporally staggered processes—high cellular forces carried over due to past mechanical priming are first employed in remodeling the matrix, and then cells exploit this remodeled matrix to further invade. To test alternate possibilities that our observations could be due to some mechanical artifacts of the cross-environment interface between stiff hydrogel and soft collagen, we used YAP-depleted cells (Figure 4) or those minimally primed cells (Supplemental Figure S4), both of which do not undergo priming-dependent invasion despite crossing the same soft or stiff interfaces as the wild-type cells. These results build the case that an intricate coupling of both present and past cellular states along with collagen remodeling gives rise to memory-dependent cell invasion across distinct environments.

Based on these ideas of cell and matrix memory dependence, our model combines individual influences of direct mechanosensing and past priming of differential kinetics. Through iterative model predictions and experimental validation, we show that priming-based cellular forces and matrix remodeling are such intimately coupled processes that any disruptions in collagen cross-linking or force propagation disrupt the connection between past cellular priming and future invasion. These basic mechanisms of priming-dependent matrix remodeling and invasion hold true for multiple cell types and collagen compositions. Overall, our results show that past “stiff priming” provides cells a previously unappreciated advantage in navigating challenging new environments.

In cell migration on flat surfaces with a stiffness gradient, the process of durotaxis allows preferential cell migration toward stiffer regions, due to subcellular and intercellular force propagation (Sunyer et al., 2016; Shellard and Mayor, 2021), which also occurs in vivo along cell-generated stiffness gradients (Klein, 2015). Here, we have shown that the fibrous microenvironment does not act merely as a passive provider of mechanical cues; instead, differentially primed cells can actively remodel the matrix for persistent invasion into new environments. Thus, when the new environment is fibrous and plastic, albeit soft, cells are not bound by the durotaxis model, because the matrix not only governs cellular mechanosensing but is also remodeled by primed cell mechanoactivation, causing persistent cell invasion despite the preexisting stiff-to-soft gradient. We speculate that such a process could potentially be described in terms of rudimentary steps in classic neurological memoryencoding, storage, and retrieval (Klein, 2015). Consistent with this analogy, our mathematical model is indeed built around three key steps: encoding prior mechanical priming of cells in the form of sustained mechanoactivation, storing the primed cellular state into the matrix via force-based fiber remodeling, and retrieving matrix remodeling for enhanced cell invasion through 3D collagen.

Although stem cell reprogramming and mechanical memory have been shown previously (Yang et al., 2014; Li et al., 2017; Gonzales et al., 2021), our findings reveal that such cellular memory could also differentially modify the extracellular matrix. In stem cells, cardiac cells, and breast cancer cells, mechanical priming can result in chromatin remodeling (Killaars et al., 2019; Stowers et al., 2019; Seelbinder et al., 2021), causing epigenetic modifications and potentially mechanical memory. On the basis of our findings, we speculate that while the biochemically stored cellular memory may be depleted due to changing transcriptional and epigenetic landscapes, the potential matrix remodeling or memory is mechanical in nature and thus may have long-lasting effects on cellular response. Our experimental and mathematical models combining cellular priming and matrix remodeling broaden the previously understood processes of collective cell invasion through complex fibrous environments. This memory-dependent cell–matrix coupling could apply to wide-ranging biological processes wherever cells remodel their environments over time and distance, for example, fibrosis, cancer, and aging.

MATERIALS AND METHODS

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Cell culture and reagents

MCF10A (American Type Culture Collection), nontumorigenic human breast epithelial cells, MCF10A with anti-YAP short hairpin RNA (shRNA) (shYAP) (courtesy of G.D. Longmore, Washington University), and mutant MCF10A cells overexpressing ErbB2 (kindly provided by Dihua Yu), were cultured in DMEM/F12 (GE Healthcare Life Sciences), supplemented with 5% horse serum (Invitrogen), 20 ng/ml epidermal growth factor (EGF; Miltenyi Biotec), 0.5 mg/ml hydrocortisone (Sigma-Aldrich), 10 µg/ml insulin (Sigma-Aldrich), 100 ng/ml cholera toxin (Sigma-Aldrich), and 0.2 % Normocin (Invitrogen). Media was changed every 3 d while cells were in culture. For YAP depletion, the healthy MCF10A cell line was transfected using a lentiviral pFLRU vector containing anti-YAP shRNA, developed and verified in our previous work (Nasrollahi et al., 2017).

Polyacrylamide gel preparation

Glass coverslips of 5 mm diameter (Thermo Fisher Scientific) and glass slides were prepared for gel adhesion. Coverslips were activated by plasma cleaning and a 10 min treatment with bind-silane solution, composed of 94.7% ethanol, 5% acetic acid, and 0.3% bind-silane (GE Healthcare Life Sciences). Following treatment, coverslips were washed with ethanol and air-dried. While waiting for the coverslips to dry, glass slides were treated with Sigmacoat solution (Millipore) to create a hydrophobic surface. Polyacrylamide gels with distinct stiffness were fabricated through step-by-step polymerization of the PA solution. Briefly, precursor solutions combining acrylamide (A; Bio-Rad), bis-acrylamide (B; Bio-Rad), and ultrapure water were mixed for respective contents of 3, 0.05, and 96.95% (∼0.08 kPa stiffness) and 12, 0.15, and 87.85% (∼16 kPa stiffness). These solutions were degassed by N2 injection. Then, volumes of ammonium persulfate (APS) and N, N, N′, N′-tetramethylethylene diamine (TEMED) were added to yield concentrations in the final gel solution of 0.5 and 0.05%, respectively. To form gels, this solution was sandwiched between the treated glass slides and 5 mm coverslips. These constructs were placed in a degasser for 15 min to polymerize and then were submerged in Dulbecco’s phosphate-buffered saline (DPBS) (Life Technologies) for 30 min. Coverslips were lifted, and formed gels were placed for 1 h under UV for sterilization. To facilitate collagen conjugation, gels were functionalized with a 0.5 mg/ml solution of sulfosuccinimidyl 6-(4′-azido-2′-nitrophenylamino) hexanoate (Sulfo-SANPAH) (Thermo Fisher Scientific) prepared in 50 mM HEPES buffer (Santa Cruz Biotechnologies). To adhere Sulfo-SANPAH to PA gel surfaces and provide binding sites for collagen, Sulfo-SANPAH–coated gels were exposed to 365 nm UV for 10 min. Gels were then washed twice before coating with 0.05 mg/ml collagen type I (rat tail; Santa Cruz Biotechnologies). Finally, to achieve sufficient collagen adhesion, gels were incubated in this solution overnight at 4°C.

Mechanical priming of cells and collagen gel fabrication

To mechanically prime cell colonies before their invasion, cells were plated on PA gels of different stiffnesses (∼0.08 and ∼16 kPa). Cells in culture flasks were detached using 0.25 mg/ml trypsin (Life Technologies). PA gels were air-dried for 10 min and then droplets containing 10,000 cells were seeded in the form of monolayers on the gel surface. Monolayers were left to grow on soft or stiff gels for 5 d at 37°C in 5% CO₂ to ensure their long-term priming by defined ECM stiffness. After this priming period, collagen gels were prepared. Collagen type I solution (rat tail; 4.0 mg/ml; Advanced Biomatrix) was diluted in cold media to yield a working collagen concentration of 2.3 mg/ml. This solution was adjusted to pH 7.5–7.8 using 1 M NaOH. Once this pH was reached, a sonicated solution containing 1 µm of fluorescent beads (Thermo Fisher) was added to achieve a final concentration of 4 µl of fluorescent beads per milliliter of solution. This solution, containing collagen and beads, was carefully mixed to prevent bubbles and then was pipetted into the wells of a chilled glass-bottom 24-well plate (Fisher Scientific). PA gels containing primed cells were carefully placed on top of this collagen solution, PA gel side down. The plate was then incubated at 37°C and 5% CO₂ for 45 min to facilitate cross-linking. After cross-linking this lower collagen gel layer, another layer of collagen solution with fluorescent beads was added on top of the PA gel. To facilitate cross-linking of this upper collagen layer, gels were incubated for another 45 min. Finally, media was added, with care taken not to disrupt the collagen gels. Completed gel constructs were incubated at 37°C and 5% CO₂ for 3 d to test the duration of the mechanical memory phenotype that we might observe. Note that in this setup, cells are not trypsinized after priming, before implanting into collagen. It was necessary to devise a system in which cells are primed and then allowed to invade without detachment, because trypsinization is a harsh treatment that resets cellular state and obviates memory, as shown previously (Yang et al., 2014). Moreover, chemical detachment of cells while crossing the interface between two environments is unrealistic.

Immunofluorescence imaging of cells

After 3 d of invasion, constructs were prepared for immunofluorescence imaging. Cells were fixed with 4% paraformaldehyde for 15 min, washed with DPBS, and then permeabilized with a 0.3% solution of Triton X-100 (Santa Cruz) for 10 min. Nonspecific binding was blocked by 2% bovine serum albumin (BSA) in DPBS overnight at 4°C, followed by two washes with DPBS. To visualize nuclei, cells were incubated with Hoechst 33258 (1:50; Thermo Fisher) for 30 min at room temperature and then washed with DPBS. For actin visualization, gels were incubated with phalloidin (1:125; Invitrogen) for 35 min and washed with DPBS. Stained gels were stored at 4°C until imaging. Fluorescent images were recorded using a laser-scanning confocal microscope (Zeiss LSM 730; Carl Zeiss Micro Imaging, Germany) at 10, 20, and 40× objectives, and z-stacks were acquired at 5 µm intervals. Experiments were performed in triplicate and quadruples per well. Meanwhile, laser intensity and exposure time were kept constant to enable quantitative analysis across samples. To prevent potential biasing, the images used for analysis were randomly selected from six to seven fields of view for each condition.

Time-lapse microscopy

Live-cell imaging to visualize collective cell migration and fluorescent bead movement was done using a Zeiss Cell Observer microscope (10× objective; Carl Zeiss Microscopy) equipped with an incubation chamber. In each experiment, phase-contrast and tracking of 1 µm orange (540/560) fluorescent bead images were acquired to facilitate analysis of cell invasion and bead displacement, respectively. Images were collected every 1 h for 48–72 h, and cells were maintained at 37°C and 5% CO₂ for the duration of time-lapse imaging.

Modulation of collagen density and collagen cross-linking

To determine how primed cells interact with different collagen densities and fiber structure, collagen gels of three different modalities, 2.3 mg/ml, 2.3 mg/ml + cross-linked, and 3.1 mg/ml, were constructed, using the same procedure as described above. Collagen cross-linking was achieved by the addition of riboflavin 5′ phosphate sodium slat hydrate (Sigma Aldrich) to the media–collagen solution (final concentration 0.5 mM). This collagen solution was then prepared as previously described to create the two fully cross-linked layers. After media was added, the constructs were exposed to 365 nm UV for 15 s to activate riboflavin and cross-link collagen fibers to create bundle and potentially thicker fibers in the gel.

Spatial and temporal analyses of cell invasion

Imaris software (Bitplane) was used to create a 3D reconstruction of the invading cells in collagen utilizing confocal images and stacks. A custom batch process was developed to reconstruct cell nuclei from the Hoechst signal, record cell coordinates in the 3D space, and count the invaded cells at a given distance. A custom MATLAB code was written to calculate the distance vector of the cells in relation to the edge of the gel. To determine the rate of invasion of the primed cells into the collagen, rectangular regions of interest (ROIs) of dimensions 387 µm × 213 µm were selected within collagen and cell migration was analyzed from time-lapse videos. A custom ImageJ (https://imagej.net/Fiji) macro was developed to identify invading cells using a Gaussian blur. From this macro, the area occupied by the invading cells was extracted and Microsoft Excel was used to calculate the difference between the area occupied by invaded cells at time t and the area at time t = 0, and this difference was normalized by the width of the rectangular ROI to yield invasion distance (µm) per unit hour. To determine the advantage of stiff-primed cells over soft-primed cells, we also calculated the difference in their invasion rates, termed here as the “priming advantage.”

Collagen deformation and accumulation analyses

The ROI approach described above was also used to track the collagen deformation rate over time. To track the fluorescent beads, particle image velocimetry (PIV) analysis was done to calculate spatiotemporal profiles of velocity magnitudes through the PIVlab package in MATLAB (Thielicke and Stamhuis, 2014). For each condition, PIV was performed by using three passes of 64-, 32-, and 16-pixel windows to obtain the velocity field (v_i). These values were exported as MATLAB workspace. A custom MATLAB code was used to calculate the average velocity of the beads over time to determine the amount of deformation of the collagen gel due to primed cells invasion. Collagen accumulation can be quantified by the changes to the bead fluorescent intensity because the beads are attached to collagen fibers. The bead fluorescence intensity in collagen gels was used to create kymographs by calculating the bead fluorescence intensity in five regions along the length of the rectangular ROI. The regions were defined by how close they were to the edge of the PA gel. Using ImageJ, the intensities in each region were extracted and then averaged among samples and regions. Normalization of bead intensity across different regions was done against intensity at t = 0. Using these values, a kymograph was plotted to display temporal and spatial changes in bead fluorescence intensity.

Collagen relaxation analysis

To assess net tension in bulk collagen stored by the primed cells after they invaded for 3 d, trypsinization was used to cause cellular detachment from collagen to quantify differences in collagen displacements. For collagen displacement, fluorescent beads (1 µm) were embedded in 2.3 mg/ml collagen. MCF10As were then allowed to invade for 3 days and then trypsin was added, causing cells to detach from the collagen. The tracking of the bead displacement was done using confocal microscopy. Fluorescent beads were imaged at the same laser power and exposure. These images were processed through PIV to get vector components of bead movement, which were processed in MATLAB to calculate the bead displacement in the X and Z planes and plot heatmaps of bead displacement, superimposed with locations of cells (as black circles) present in collagen before trypsinization.

Collagen visualization and alignment analyses

To visualize collagen structure and fiber orientation, reflectance was used through a small wide wavelength that passes through the sample and then captures the backscattering of light passing through the collagen matrix, allowing us to visualize collagen fiber structures. Reflection images of collagen were taken using a 20× objective, 5-μm z-stacks, and consistent laser power. To analyze the orientation of the collagen, we used the Z-projection of images and utilized the OrientationJ Fiji plug-in in combination with a batch ImageJ macro to calculated fiber orientation angles. This is paired with an R script that normalizes the angle distribution between –90°and 90° (Franco-Barraza et al., 2016; Kaur et al., 2019). Similar analysis was done for the images with F-actin fibers.

Fiber coherency

To determine the level of coherency of collagen or F-actin fibers, the OrientationJ plug-in in ImageJ was used. Nine to 10 ROIs were randomly selected, and the coherency value was calculated using collagen reflectance or F-actin images, as applicable. Coherency values were averaged across samples for each condition.

Collagen bundling analysis

To quantify collagen bundling after 3 d of priming, we extracted ROIs of similar sizes and developed a custom macro in ImageJ to determine the degree of fiber bundling seen in reflectance images. To analyze collagen bundling, first the images were processed using ImageJ in these steps: mean intensity within a 960 µm × 960 µm ROI is measured; this mean is subtracted from the ROI; images are converted to binary; the final raw intensity yields the degree of bundling of collagen fibers. To normalize bundling across images, raw intensity values were divided by 255 to get the number of pixels, which were divided by the area of the ROI.

Measuring pore size in collagen

Utilizing reflectance images of each collagen density, an ImageJ macro was developed to extract the Feret diameter from a stack of images. The macro selects the stack placing a Gaussian blur on the image. Then, the threshold is adjusted to highlight the empty spaces found between fibers; these spaces represent pores within the collagen matrix. This threshold value is applied to create a binary image and Feret diameter, representing pore size as measured for all Z-slices.

Pharmacological inhibition studies

To understand the effects of contractility and collagen cross-linking, inhibitors were used during live imaging to quantify their effect. For LOX inhibition, 10 mM BAPN (Sigma Aldrich), was added 10 min before live imaging. Similarly, we used 20 µM blebbistatin (Sigma Aldrich), a known myosin-II inhibitor, 10 min before live imaging to test whether contractility is enough to maintain the priming phenotype that we observe.

Quantitative PCR (qPCR) of primed cells

MCF10a cells were primed for 5 days on stiff and soft PA gels. The RNA was isolated from the cells using Qiagen’s Rneasy kit following manufacturer instructions. The RNA quality and quantity were measured using Nano dropper (Thermo Fisher) and then standardized to 200 µg/ml and converted to cDNA using the C1000 Thermocycler (Bio-Rad). A volume of cDNA was added to each well of the 96-well plate that contained 10 μl of a solution of TaqMan Fast Advanced Master Mix (Applied Biosystems), nuclease-free water (Invitrogen), and TaqMan primers (Thermo Fisher). We used B2M (Assay ID: Hs00187842) as our housekeeping gene and ran reactions for Rac1 (Assay ID: Hs01902432) and RhoA (Assay ID: Hs00357608) using QuantStudio real-time PCR in triplicates for an N = 6. Expressions of each gene were normalized to the housekeeping gene. The relative expressions were calculated from the comparison of the difference in C_t between target genes and the reference gene. This fold change was done in comparison of stiff to soft. For a list and sequences of the oligonucleotides used, see Table 1.

Laser ablation of collagen

To ablate collagen fibers, we used an Andor Micropoint Laser Ablation system connected to the Zeiss Cell Observer system and controlled with Andor i8 software (Andor). A straight line was ablated in the collagen before invasion began, creating a cut approximately 150 µm away, parallel from the edge of the gel. The cut was made on the Z-plane where the cells appeared in focus. Bead tracking for laser ablation was done using PIV. Fixed imaging of the cut collagen and invaded cells was performed as described above.

Mechanical characterization of collagen stiffness for different concentrations

To quantify Young’s modulus of collagen of different compositions, AFM was performed. Collagen stiffness was measured using a Bruker BioScope Resolve Bio-atomic force microscope (AFM; Bruker). AFM probes were custom-made by Nanoscope to have a 4.5 µm bead attached to the cantilever with a nominal stiffness, 0.01 N/m. Elastic moduli were analyzed from collected force curves using the Hertz model (MacKay and Kumar, 2013).

Scanning electron microscopy

Scanning electron microcopy was performed to visualize the collagen alignment and cells as they invade into the collagen. The in vitro PA–collagen gel system with invaded cells, after 5 d of priming and 3 d of invasion described above, was fixed using 2.5% glutaraldehyde that had been mixed with a solution of 2% paraformaldehyde in 0.15 M cacodylate buffer at a pH of 7.4 with 2 mM calcium chloride. Once warmed, cell media was removed, and a fixative was added and incubated at 35°C and 5% CO₂ for 15 min. Samples were removed from the incubator and placed in a shaker overnight at room temperature. Images were acquired using a high-resolution SEM (Zeiss Merlin FE-SEM) at the Washington University Center for Cellular Imaging.

Statistics and reproducibility

All data represent at least four replicates from separate experiments. All bar graphs are presented as mean ± SEM. All box-and-whisker plots represent mid-line as the mean and whiskers as data within 90-10% range. Similar sample sizes and statistical tests were used for each experiment, and these are indicated in the figure legends. Statistical significance was calculated mostly with tests for pairwise comparisons and one-way analysis of variance, unless specified otherwise. All statistical analyses were performed in Prism.f

COMPUTATIONAL MODEL

Overview: modeling memory-dependent invasion

Our experimental findings connect primed cellular mechanoactivation to direct cellular mechanosensing along with active ECM remodeling feedback. Thus, several intercoupled cellular and extracellular processes, some mechanical and some biochemical, give rise to memory-dependent cell invasion. Here, we develop a theoretical model and a computational framework to better understand these complex processes, which cannot be explained by existing models of cell migration that largely account for mechanosensing in current mechanical environments but not in past environments.

Cell migration as an energy-minimization problem

We recall that cell migration has traditionally been understood as three basic steps (Lauffenburger and Horwitz, 1996): 1) extension of frontward protrusions attempting to propel the cell forward; 2) formation of cell–ECM focal adhesions that provide traction and resistance against motion; 3) generation of cytoskeletal contractile forces that break adhesions to reduce resistance and enable net cell translocation. How these three processes generate cell migration can be interpreted as an energy-minimization problem. From a resting state, biochemical signaling for actin polymerization gives rise to protrusions, and this protrusive energy is minimized by anchoring protrusions via adhesion formation. Mechanotransductive signaling gives rise to actin–myosin contractile forces that pull on focal adhesions. As a result, there is potential energy stored in stretched focal adhesions, which can be minimized by breaking adhesions. As such, the forward translocation of the cell can be understood as minimizing energy fluctuations due to dynamic evolution of protrusions, contraction, and adhesions. Cell migration would not occur if energy did not rise due to protrusions or contraction-based stretching of adhesions. We formulate a net energy balance (Figure 3B) as

Here, H_protrusions is the energy associated with stable protrusions, H_cytoskeleton is the energy associated with cell body properties like supracellular contractility and the cell rigidity that helps maintain cell shape, and H_adhesions is the energy associated with cell contacts (combined cell–cell and cell–ECM adhesions in this case). In 3D environments, cells experience resistance not only from adhesions but also from drag due to bulk crowding of ECM proteins. For net cell movement to occur, the change in net energy ΔH should be less than zero. Unlike 2D substrates, cells can align and remodel 3D fibrous matrices (Seo et al., 2020), which is an additional energy term that is not accounted for in the energy consideration described above (Eq. 1). We propose that collagen remodeling acts to lower the energy barrier for invasion (Figure 3B). We implement this process through ΔH_protrusions, because net stable protrusions must overcome ECM barriers in 3D, written as

This formulation is analogous to previous models of 3D cell invasion that calculate net cell movement as a sum of forces due to contractility, protrusion, and drag (Zaman et al., 2005). Here, our key addition is the active feedback between contractile forces and ECM resistance via force-based ECM remodeling. In Eq. 2, the direct mechanosensing signal from the ECM (φ), collagen remodeling (α), and resistance via ECM (γ) together result in net protrusive potential μ, which lowers ΔH_protrusions in the net energy balance (Eq. 1). This conceptual framework of energy minimization causing cell invasion in 3D environments is somewhat independent of computational methods used to visualize and describe cells, that is, in principle, it can be implemented in agent-based, element-based, or lattice-based methodologies. Here, we utilize the Cellular Potts method (Graner and Glazier, 1992; Swat et al., 2012) to implement this model due to its several advantages—computational efficiency, description of the ECM as a dynamic energy field, and relatively straightforward cell–cell and cell–ECM interactions.

Defining adhesion and bulk cell energies

The Cellular Potts model (Graner and Glazier, 1992) represents a space as a discrete collection of lattice points—pixels on a 2D grid or voxels in a 3D grid. We model a collection of biological cells by attaching to each lattice point (i; j) of a square lattice a label σ_ij, which identifies the corresponding cell, and a label τ(σ_ij), which identifies cell type. Adjacent lattice sites are defined to lie within the same cell if they have the same value of σ_ij. The system evolves by the random movement of individual pixels that move according to transition probabilities from Monte Carlo simulations based on the energy criterion described above (Swat et al., 2012). At each Monte Carlo step (MSC), two neighboring pixels are chosen randomly, with one as source pixel and the other as target pixel. If both pixels belong to the same cell (i.e., σ[source] = σ[target]), then no changes are made to the lattice. Otherwise, the source pixel attempts to occupy the target pixel based on the Monte Carlo acceptance probability, which is calculated from the difference in the total system energy. The total system energy associated with the configuration, before and after the move, is defined as per Eq. 1. We provide specific definitions for each term in Eq. 1, as follows:

Here, J_{τ(σ(i)),τ(σ(j))} represents the contact energy for the two cell types in contact and (τ(σ(i)),τ(σ(j))) represents the contribution from the total energy due to cell–cell adhesions.

In H_cytoskeleton, two terms represent contributions from bulk elasticity of the cell and cell-surface contractility, respectively. K_a and K_p are constants for bulk elasticity and contractility, respectively. V₀ and A₀ are target volume and surface area that the cell has in isolation. After calculating the energy of the system before (H_i) and after (H_f), the copy attempt will always be successful if H_f < H_i H_f < H_i, that is, ΔH < 0. If ΔH ≥ 0, the copy attempt is accepted with a probability of e−ΔH/T, T = 20. Higher values of T would tend to accept more unfavorable copy attempts.

Defining protrusive potential to model directed migration into collagen

Earlier, we described H_protrusions as the net protrusive energy for cell migration. For net motion at any given time step, ΔH_protrusions must be less than zero. In other words, −ΔH_protrusions = μ > 0 will propel migration into the collagen; if μ ≤ 0, there is no bias due to protrusions to migrate in a preferred direction. As cells enter a 3D matrix of normalized density , resistance (γ) due to mat© must be overcome to enable invasion, as the cells sense this new environment and adapt to it (φ). We propose that cells overcome this resistance via collagen remodeling (α). Thus, net protrusions μ is defined as

Here, φ is the direct mechanosensing signal from the current ECM. We note that φ is an abstract quantity that isolates signal due to direct mechanosensing alone, which cannot be measured, because the measurable cell state (μ) is the net sum of its real context—a combination of primed mechanoactivation (ψ), ECM resistance, and remodeling. With increasing collagen density, the pore size in collagen matrices is reduced, which increases resistance. Simultaneously, the rise in ligand density has been shown to give rise to biphasic cell migration (DiMilla et al., 1991; Zaman et al., 2005). Thus, we model the ECM resistance γ proportional to square of density (ρ): γ = kρ², where k = 3 × 10⁻² ml²mg⁻² is the proportionality constant. The term (discussed later) in Eq. 5 represents collagen remodeling (Wolf et al., 2007), which lowers the energy barrier posed by resistance γ. Here, φ and α evolve over time according to changes in the ECM and cell state.

Memory-dependent collagen remodeling via primed mechanoactivation (ψ) and direct mechanosensing (φ)

As mechanically primed cells enter collagen, they apply forces on collagen fibers. Cellular forces are understood to arise from the net mechanoactivation state of the cell. According to previous models of cell migration, rapid mechanotransduction due to focal adhesion signaling enables direct mechanosensing of current ECM conditions (stiffness, architecture). On the basis of our experimental findings and previous results of cell state regulation from mechanical memory (Nasrollahi et al., 2017), we update this model by formulating collagen remodeling α resulting from a combination of primed mechanoactivation from past ECM ψ and direct mechanosensing due to the current ECM φ (Figure 3):

where r_α = 10⁻² MCS⁻¹ = 1.7 × 10⁻⁴ s⁻¹. The first term represents the contribution to remodeling from two mechanoactivation signals–primed mechanoactivation ψ of the past ECM and direct mechanosensing φ of the current ECM (Figure 3A). Mechanically, it should be easier to pull and remodel softer ECM compared with rigid ECM; thus g(ρ) represents the bulk stiffness of the collagen matrix and is defined as g(ρ) = (ρ/3.7)². As the invading cells pull on collagen and remodel it, depending on collagen properties, the remodeling is transmitted to the rest of the ECM (Hall et al., 2016). On the basis of the continuum model of collagen ECM for a single cell, we model that remodeling (α_ECM) in the ECM at a point away from the invading front is given by α_ECM = α_LE/(x_ECM – x_LE)ⁿ, where α_LE is the average collagen remodeling at the leading-edge front, whose x coordinate is at x_LE, and x_ECM is the x coordinate of the point in the ECM where we are trying to calculate the transmitted collagen remodeling. Here, n = 0.2 represents how far the collagen remodeling is transmitted. Higher values of n reduce how far collagen remodeling is transmitted. ECM tension is an important process in the transmission of collagen remodeling, and we calculate the normalized net tension in the ECM α by summing α over the entire simulation space (normalized by the simulation space area). This represents the average tension buildup in the ECM as cells remodel it.

Differential kinetics of primed mechanoactivation and direct mechanosensing

As noted above, the mechanosensing signal φ represents direct sensing of the current ECM and rapid mechanotransductive response that would predict quick adaptation, written as (Figure 3A).

Here r_φ = 0.5 MCS^–1 = 8.7 × 10^–3 s^–1. is the mechanosensitive signal due to collagen stiffness and pore size, both of which affect cell migration in 3D (Beadle et al., 2008; Friedl and Wolf, 2010) . As collagen density increases, the mechanosensing signal increases and eventually saturates (Saez et al., 2005; Califano and Reinhart-King, 2010; Han et al., 2012).

In contrast to direct mechanosensing, primed mechanoactivation ψ is slow to decay and adapt to the new environment, consistent with previous experimental findings of YAP (Yang et al., 2014; Nasrollahi et al., 2017) such that nuclear YAP localization continues to retain primed mechanoactivation, written as (Figure 3A)

Here, the governing rate r_ψ for primed mechanoactivation is modeled as a switch-like system such that the rate is low (∼0) in the beginning when cells start invading into collagen and approaches r_φ (the same as that for φ in Eq. 7) in about 2 d:

where ζ = 3000 MCS∼48 h. Lower values of ζ reduce memory by enhancing the adaptation of ψ to the current collagen matrix.

Simulation details

We utilize CompuCell3D to solve the system of differential equations described above over a defined field (150 pixels × 75 pixels, 1 pixel = 2 μm). At each time step, φ, ψ, μ, and γ are solved for each cell, whereas α is also solved over space. The total simulation time is 3000 MCS, which is equivalent to 48 h. At t = 0 MCS, φ = ψ = 1 for stiff-primed cells and φ = ψ = 0.3 for soft-primed cells, with α = 0 everywhere. Cell monolayer is defined as a flat collection of cells of diameter 10 μm. Here, we assume that the collagen matrix is in a 3D space and show simulation for a given 2D plane of cell invasion.

Data and materials availability

All data are available in the main text or in the Supplemental Materials.

FOOTNOTES

ACKNOWLEDGMENTS

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