Matching material and cellular timescales maximizes cell spreading on viscoelastic substrates

Ze Gong, Spencer E. Szczesny, Steven R. Caliari, Elisabeth E. Charrier, Ovijit Chaudhuri, Xuan Cao, Yuan Lin, Robert L. Mauck, Paul A. Janmey, Jason A. Burdick, and Vivek B. Shenoy
PNAS March 20, 2018 115 (12) E2686-E2695; first published March 5, 2018 https://doi.org/10.1073/pnas.1716620115

  1. Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved February 5, 2018 (received for review September 21, 2017)

Significance

It is well known that cell proliferation, differentiation, and migration depend strongly on the mechanical stiffness of the extracellular matrix (ECM). Natural ECMs also exhibit dissipative (i.e., plastic, viscoelastic) properties, which can modulate cellular behavior. However, to fully utilize this information in bioengineering applications, a systematic understanding of the role of substrate viscosity on cell function is needed. Using combined theoretical and experimental approaches, we demonstrated that viscous dissipation can be as important as elasticity in determining cell response. Specifically, we found that intermediate viscosity maximizes cell spreading on soft substrates, while cell spreading is independent of viscosity on stiff substrates. This information can now be used to design dissipative biomaterials for optimal control of cell behavior.

Abstract

Recent evidence has shown that, in addition to rigidity, the viscous response of the extracellular matrix (ECM) significantly affects the behavior and function of cells. However, the mechanism behind such mechanosensitivity toward viscoelasticity remains unclear. In this study, we systematically examined the dynamics of motor clutches (i.e., focal adhesions) formed between the cell and a viscoelastic substrate using analytical methods and direct Monte Carlo simulation. Interestingly, we observe that, for low ECM rigidity, maximum cell spreading is achieved at an optimal level of viscosity in which the substrate relaxation time falls between the timescale for clutch binding and its characteristic binding lifetime. That is, viscosity serves to stiffen soft substrates on a timescale faster than the clutch off-rate, which enhances cell−ECM adhesion and cell spreading. On the other hand, for substrates that are stiff, our model predicts that viscosity will not influence cell spreading, since the bound clutches are saturated by the elevated stiffness. The model was tested and validated using experimental measurements on three different material systems and explained the different observed effects of viscosity on each substrate. By capturing the mechanism by which substrate viscoelasticity affects cell spreading across a wide range of material parameters, our analytical model provides a useful tool for designing biomaterials that optimize cellular adhesion and mechanosensing.

Mounting evidence has demonstrated that cells can sense and react to the physical properties of the extracellular matrix (ECM), an ability that plays a key role in processes such as cell migration (1⇓–3), spreading (4, 5), and proliferation (6⇓–8). It is commonly believed that focal adhesions (FAs), which anchor the cell to the ECM as well as serving as hubs for the exchange of biological and mechanical stimuli (9, 10), are responsible for such mechanosensitivity of cells.

It has been recognized that cells probe the stiffness of their surroundings by gauging the resistance of FAs to actin retrograde flow generated by intracellular myosin contractions (11⇓⇓–14). FAs, acting like molecular clutches, alter the movement of actin intracellular structures by providing a tunable connection to the ECM (13, 15). Based on this picture, the well-known motor-clutch model (15⇓–17) was developed, which predicted a biphasic dependence of cell adhesion traction (and consequently cell spreading) on ECM rigidity. However, recent experiments have found a monotonic increase of cell spreading with the ECM rigidity (18, 19), which can be attributed to reinforcement mechanisms including, for example, the activation of adhesion proteins under high surrounding stiffness/load level or the recruitment of integrins into the FAs (20, 21).

Beyond substrate rigidity, most natural ECM materials such as collagen, fibrin (22), and tissues (23⇓–25) are viscoelastic in nature and exhibit a strong frequency-dependent mechanical response. Interestingly, it was reported recently that cell spreading can be enhanced by stress relaxation of a cell culture substrate (e.g., alginate, polyacrylamide), which was dependent on the elastic modulus of the substrate (4, 26). This was explained by local remodeling (leading to increased ligand density) of the matrix during deformation (4), which corresponds to a plastic rather than viscous response. On the other hand, our own experiments (to be discussed in Model Predicts Cells Spreading for Several Different Cell Types Across a Wide Range of Elastic and Viscous Substrate Properties) suggest that viscosity has a negligible effect on how cells spread. However, it is unclear how a purely viscoelastic substrate can have different effects on cell spreading, largely due to the lack of a theoretical model capable of revealing the physical mechanisms governing the cellular response to viscoelasticity.

To address this critical issue, we systematically examined how cell spreading is influenced by the viscoelastic components of the ECM via an analytical mean-field analysis and direct Monte Carlo simulations. Specifically, by treating the ECM as a standard linear viscoelastic solid, we found that an intermediate level of viscosity can promote cell spreading when the ECM rigidity is relatively low, reflecting the fact that the substrate relaxation time under such circumstance falls between the timescale for clutch binding and its characteristic binding lifetime. That is, viscosity serves to stiffen soft substrates, which enhances cell−ECM adhesion and cell spreading. As for high rigidity, the large tension borne by the clutches triggers an increase in their binding rates as well as an increase in the integrin density (clutch reinforcement), which saturates the cell response to substrate stiffness and eliminates any role of viscosity. These data are presented in heat maps of spreading response in the parameter space spanned by the substrate and cellular timescales.

Our theoretical predictions that incorporate the viscoelastic relaxation time spectrum analysis are in excellent agreement with previous observations as well as providing the basis for understanding our own experiments where ECMs with tunable viscoelastic properties were fabricated by two different methods. By capturing the mechanism by which substrate viscoelasticity affects cell spreading across a wide range of material parameters, our analytical model provides a useful tool for designing biomaterials that optimize cellular adhesion and mechanosensing.

Model Description

The canonical motor-clutch model (15), as schematically illustrated in Fig. 1A, was adopted to study the impact of viscoelasticity on spreading. In this model, myosin motors pull the actin filament bundle toward the cell center, generating retrograde flow of actin. On the other hand, the molecular bonds/clutches, connecting the F-actin with the substrate, were assumed to be able to randomly break or reengage with a dissociation or association rate of roff,i or ron, respectively. Engagement of the clutches leads to slowing down of the retrograde flow, allowing the polymerization at the leading edge to push the cell membrane forward, resulting in the spreading of the cell. The master equation describing the evolution of the state of each clutch can be written asdPb,idt=(1−Pb,i)ron−Pb,i roff,i,[1]where Pb,i represents the probability for clutch i to remain engaged. As the F-actin slides, a bound clutch (treated as a linear spring with stiffness, kc) undergoes stretching and hence generates a force Fc,i resisting the retrograde flow of actin. If the displacements of the filament end and substrate end of the clutch are denoted as xc,i and xs, then the force can be expressed as Fc,i= kc(xc,i−xs). Using a Bell model (27), the dissociation rate is expected to increase exponentially with Fc,i; that is, roff,i=roff0⋅exp(Fc,i/Fb), where Fb is a characteristic force and roff0 represents the breaking rate of clutches in the absence of any force. Recent experiments demonstrated that FAs grow on stiff substrates through the recruitment of integrins to the adhesion complex via talin unfolding (20, 21). Following the model developed by Elosegui-Artola et al. (21) that describes such FA reinforcement, the association rate ron is replaced by ron0(1+α(Fa−Fcr)) once the average clutch force Fa has surpassed a force threshold Fcr (refer to Supporting Information for more details). A value of 0.2 per piconewton was chosen for α based on fits of cellular spreading on purely elastic substrates (Fig. S7).

Fig. 1.
Fig. 1.

Modified motor-clutch model investigating FA and cell spreading behavior on viscoelastic substrates. (A) Schematic of the motor-clutch model of a cell attached to a viscoelastic substrate represented as a standard linear solid. Myosin motors pull the actin bundle toward cell center at a retrograde flow velocity, Vf. Clutches connect the actin bundle to the substrate based on the reaction rates ron and roff and resist the retrograde flow. Greater resistance to the retrograde flow increases cell spreading due to polymerization of the actin bundle. Three timescales govern the model behavior: clutch binding timescale (τb=1/ron), FA lifetime scale (τl=Fs/v0kl), and substrate relaxation timescale (τs=η/ka). (B) Representative cell spreading speed as a function of substrate stiffness, for a purely elastic substrate, predicted by our model. The red curve corresponds to the case where clutch is reinforced by force over a threshold level, while the green curve represents the scenario without any reinforcement. The black dashed line indicates the critical stiffness above which clutch reinforcement begins to take effect. (C and D) Results for a typical set of parameters (ka=1 pN/nm,kl=0.1 pN/nm,η=1pN⋅s/nm) show that the (C) fraction of engaged clutches and (D) retrograde flow (ODE, red line) and the KMC method (blue line) are almost identical. Insets in C show whether the clutches are attached or detached. The dash-dotted line in D indicates the mean spreading speed over one cycle of clutch engagment and detachment.

While many viscoelastic substrates possess multiple relaxation times, we chose to represent the substrate as a standard linear viscoelastic material for two main reasons (Fig. 1A). First, this model captures the essential effects of the competition between the viscoelastic timescale of the substrate and the intrinsic cellular timescales and allows us to identify the impact of different mechanical parameters (long-term and additional stiffness). Second, by fitting our experimental relaxation data to a Prony series (i.e., the generalized Maxwell model with multiple timescales) and carrying out simulations of spreading using this model (Simulation with Multiple Relaxation Timescales and Fig. S8), we found that only one relaxation time significantly affects the spreading behavior. This single relaxation time constant was determined using the time spectrum analysis, which is described in Results. The force−displacement relation of standard linear viscoelastic model is given by(ka+kl)ηdxsdt+kaklxs=kaFs+ηdFsdt.[2]Here ka and kl are the additional and long-term stiffness, respectively, of substrate, while η represents the viscosity for dashpot (Fig. 1A). The total load transmitted to the substrate Fs is the sum of all forces sustained by bounded clutches; that is,Fs=kc∑i=1nc(xc,i−xs).[3]On the other hand, the substrate force can be related to the retrograde flow speed, Vf, through the Hill relation (15)Vf=vu⋅[1−FsNmFm],[4]where Nm is the number of myosin motors, Fm refers to the characteristic stalling force, and vu represents the maximum retrograde flow velocity. For a clutch that remains engaged, its filament end moves with the F-actin. In comparison, since a broken clutch carries zero load and moves with the substrate, we can effectively set xc,i=xs. Consequently, on average, the moving velocity of the filament end of a clutch can be expressed asdxc,idt=Pb,iVf+(1−Pb,i)dxsdt.[5]To connect this analysis with cell spreading, we recall that cells spread via lamellipodia, flat sheet-like structures driven by the actin polymerization-induced protrusive forces (pushing against the cell membrane) (28). During spreading, actin monomers flow from the cell center to the edge to sustain continuous polymerization and offset the retrograde flow of F-actin (29⇓–31). Importantly, it has been found that the sum of the retrograde flow velocity and the speed of actin protrusion remains approximately constant (5, 30, 31). Based on these observations, we were able to relate the cell spreading speed, Vs, to the retrograde flow (Vf) and polymerization velocity (Vp, treated as a constant) asVs=Vp−Vf.[6]Here, it should be noted that the spreading velocity Vs is the initial spreading speed when the cells are seeded on the substrates. After this initial phase of polymerization, outward motion of the actin filaments will be countered not only by rearward forces from the myosin motors but also by the resistance of the plasma membrane, eventually leading to a steady-state configuration where the cell spreading area becomes constant (Fig. S2). This steady-state spreading area is a function of the initial spreading velocity, initial spreading radius, and the properties of the cytoskeleton and cell membrane (Initial Spreading Speed Determines the Final Spreading Area). Kinetic Monte Carlo (KMC) simulations show that, given the initial radius and cytoskeleton properties, the steady-state spreading radius increases in an almost linear manner with respect to the initial spreading speed (Fig. S2).

From Eqs. 1–6, the dynamic behavior of the molecular clutches as well as of cell spreading can be simulated by a KMC method (15). In addition, a mean-field approach can also be applied (see Supporting Information for details) to examine the problem in an analytical manner. In this case, the behaviors of all clutches are averaged into a single “ensemble” clutch with effective stiffness kcncPb, where Pb is the fraction of clutches that remain engaged. By solving the corresponding ordinary differential equation (ODE) from the mean-field theory, the temporal evolution of the clutch status can be obtained. The values of parameters are given in Table S1 (more details are provided in General Model Parameters).

Results

Intermediate Viscosity for Soft Substrates Promote Cell Spreading.

First, we consider purely elastic substrates, where our model predicts that cell spreading increases gradually with ECM stiffness and eventually saturates (Fig. 1B, red line). In contrast, if no reinforcement is included in the model, the spreading velocity will actually start to decrease with the ECM rigidity for stiffness higher than ∼1 pN/nm (Fig. 1B, green line). Since our experimental data consistently demonstrated that cells exhibit a monotonic relation between spreading area and elastic stiffness, we focused on the reinforcement case (the case without reinforcement is included in Supporting Information for comparison). For viscoelastic substrates, typical results from our model for single FA life cycle are shown in Fig. 1 C and D. The clutches gradually engage with the sliding filament at the beginning (Fig. 1C) to transmit higher traction forces, leading to an increased substrate displacement and spreading speed (Fig. 1D). However, after passing the binding time tbind (marked in Fig. 1C), bounded clutches start to break as the load level inside keeps increasing. Finally, at t=tlife (i.e., the binding lifetime for a FA marked in Fig. 1D), all clutches break. This produces a dynamic binding cycle that will repeat itself with a period of tlife, which is the same “load and fail” response produced by other motor-clutch models (15, 16). For comparison, representative trajectories from the KMC simulations are also given in Fig. 1 C and D, which are in good agreement with the mean-field analysis. It must be pointed out that the lifetime, tlife, of such load and fail response would gradually increase and even diverge (i.e., stable FAs that cannot break) as the reinforcement takes effect and increases the binding rate (Fig. S1). To characterize the spreading behavior for cells, we use the mean spreading speed, Vs, calculated by averaging the speed over its lifetime (Supporting Information). Given the agreement between the two methods (Fig. S1A), the ODE solution will be applied for all following parametric studies to reveal the ensemble behavior of FAs over their lifetime.

As shown in Fig. 1A, the response of a substrate represented by the standard linear model is characterized by three parameters: the long-term stiffness kl, the additional stiffness ka, and the viscosity η. For a constant applied substrate displacement, ka+kl represents the initial stiffness of the substrate, ka represents the drop in stiffness as the substrate relaxes, and kl represents the equilibrium substrate stiffness. Note that we assumed a typical FA length of 1 µm (11); therefore, a stiffness of 1 pN/nm is equivalent to a modulus of 1 kPa (Extracting Substrate Parameters from Relaxation Test). The heat maps of spreading speed with respect to the entire parameter space (ka,kl, η) are given in Fig. 2A.

Fig. 2.
Fig. 2.

Cell spreading speed is significantly different based on the elastic and viscous parameters (ka,kl, η) of the substrate. (A) Heat maps of spreading speed, Vs, plotted as a function of long-term stiffness, kl, and viscosity, η, from the ODE method. The substrate additional stiffness, ka, increases from 0.1 pN/nm to 10 pN/nm from Left to Right. (B) Mean spreading speed versus viscosity, η, under different values of ka (corresponding to the dashed lines in A) and a fixed kl of 0.1 pN/nm. There is an optimum viscosity that maximizes cell spreading, which becomes more significant as ka increases from 0.1 pN/nm to 0.5 pN/nm. Above ka = 5 pN/nm, the substrate behaves like a Kelvin−Voigt material with a sharp increase in cell spreading with viscosity. (C) The heat map of the optimum viscosity (for inducing maximum spreading speed) plotted as a function of long-term and additional stiffness. The spreading speed is maximized at intermediate viscosity in regime I, while a monotonic increase of the spreading speed with viscosity is observed in regime II. The symbols in C correspond to three cases for different ka in B. Based on typical length of FAs (∼1 μm), a substrate stiffness of 1 pN/nm is equivalent to a modulus of 1 kPa (Supporting Information).

Depending on the elastic parameters, viscosity can influence cell spreading in very different ways. When viscosity is very small (η<10−2 pN⋅s/nm), the response of cells is largely determined by the long-term ECM stiffness, which is similar to the case of purely elastic substrates. However, as viscosity increases, the spreading behavior is highly affected by both the long-term and additional stiffness of the substrate. Specifically, viscosity was found to promote cell spreading for soft substrates (i.e., with relatively low long-term stiffness) as it increases the effective stiffness that cells can sense, leading to maximized cell spreading at intermediate viscosity values (Fig. 2B). In comparison, for substrates with a large long-term stiffness (kl>5 pN/nm), viscosity has negligible effect on cell spreading, as the number of clutches that can be form has reached a plateau due to the elevated ECM rigidity in this case (Fig. 2A). The optimum viscosity (for inducing maximized cell spreading) as a function of both the long-term and additional substrate stiffness is given in Fig. 2C. In regime I, optimal cell spreading is achieved at intermediate levels of viscosity for compliant ECMs, while, in regime II, largest spreading occurs when viscosity is large. A summary of how cell spreading is influenced by the elastic and viscous parameters (at low, medium, and high levels) is provided in Fig. S4. In contrast, when there is no clutch reinforcement, cell spreading is suppressed at high ECM stiffness, which leads to frictional slippage behaviors (15) of FAs (Binding Timescale and the Characteristic Lifetime Dictate the Behavior of FAs). The corresponding phase diagram in this case is also shown in Fig. S3 for comparison.

Spreading Is Optimal When Substrate Relaxation Timescale Is Comparable to Clutch Binding Timescale.

To investigate the optimum in cell spreading that is observed at intermediate viscosities for soft substrates, we analyzed the relevant timescales controlling cell spreading in our model. First, as the clutches attach to the actin filament and stretch, the force they exert on the substrate increases at a rate vukl. Therefore, the characteristic time it takes the clutches to develop sufficient force to stall the motors forces (FmNm) can be identified as τl=FmNm/vukl. It should be noted that the actual lifetime tlife of an FA is proportional to τl (Supporting Information). Similarly, τb=1/ron defines the timescale for the clutches to bind, which is also correlated with the formation time (tbind) of the FAs (Supporting Information). Depending on the value of the two timescales, the FA cluster exhibits different behaviors that eventually lead to different cell spreading (Supporting Information). Finally, the viscoelastic nature of the ECM introduces a characteristic relaxation timescale, τs=η/ka. Essentially, upon loading, the effective substrate stiffness decays exponentially (with the timescale τs) from its initial value of ka+kl to the long-term stiffness, kl.

Plotting the mean spreading speed as a function of the ratio between the relaxation and binding timescales shows that maximal cell spreading occurs at intermediate viscosity values when the substrate relaxation timescale τs is within an order of magnitude of τb (Fig. 3A). This behavior can be explained by examining the cell spreading that occurs during a single FA lifetime in relation with the three pertinent timescales (τb, τl, and τs).

  • i) When τs<τb, the ECM stiffness relaxes rapidly to its long-term value faster than the clutches can fully bind to the substrate. Consequently, the FAs only sense the long-term stiffness, kl, and the effect of viscosity becomes negligible. In this case, a typical clutch behavior in one lifetime cycle is shown by the red curve in Fig. 3C, which corresponds to the red cross in Fig. 3B.

  • ii) When τb<τs<τl, FAs sense a gradual change of stiffness during their formation−disruption cycle. The green curve (Fig. 3C corresponding to η=100  pN⋅s/nm) shows how the cell spreading speed varies with respect to time. Interestingly, a clear inflection can be seen at around 2 s, demonstrating a change in the effective ECM stiffness. Before 2 s, the cell spreading speed is almost same as that of η=102  pN⋅s/nm, suggesting only the initial ECM stiffness is detected by the cell. After the inflection point, cells gradually sense the relaxed stiffness that helps FAs extend their lifetime corresponding to the long-term stiffness. The net outcome is that actin retrograde flow is suppressed at the beginning while a long FA lifetime is maintained, thereby maximizing cell spreading.

  • iii) When τs>τl, the substrate does not relax during the lifetime of the FA. As such, the ECM behaves elastically, with the effective stiffness ka+kl. A representative response of an FA in this regime is shown by the black curve in Fig. 3C, with a sharp increase in the spreading speed at the beginning, yet a very short lifetime.

Fig. 3.
Fig. 3.

Optimal substrate viscosity is determined by the ratio between the substrate relaxation (τs) and clutch binding (τb) timescales. (A) Mean spreading speed plotted as a function of τs/τb for three different values of kl (with fixed initial stiffness ka+kl=1.1 pN/nm) shows that the greatest cell spreading occurs on a substrate where τs is slightly greater than τb. (B) Heat map of spreading speed, Vs, plotted as a function of the long-term stiffness, kl, and viscosity, η, calculated using the ODE method for ka=1 pN/nm. (C) Spreading speed vs. time for three typical values of η corresponding to the cross points in B, with dash-dotted lines representing the time average. The black line represents a highly viscous substrate that relaxes slowly such that the cells only sense the initial substrate stiffness (i.e., τs>τl>τb). The red line represents a substrate with low viscosity that relaxes quickly on which cells mainly sense the long-term stiffness (i.e., τs≪τb). The green line represents the optimal viscosity where the substrate relaxes during the FA lifetime (i.e., τs≈τb). (D) Schematic of three regimes for effect based on relaxation timescales: (I) when τs≪τb, the viscoelastic substrate has the same effect on cell spreading as an elastic substrate with the long-term stiffness kl; (II) when τs≈τb, the maximum spreading is observed on viscoelastic substrates; and (III) when τs τb, the viscoelastic substrate gives the same spreading speed as an elastic substrate with the initial stiffness ka+kl.

Three regimes in Fig. 3D show the distinct regulation effects as the relaxation timescales are varied. Note that this only applies for soft substrates; as the long-term stiffness increases, clutch reinforcement mechanism increases the binding rate, saturates bounded clutches, and weakens the viscosity effects. The end effect is that the substrate properties and clutch binding kinetics produce phenomenological timescales that must be precisely tuned so that substrate viscoelasticity has a significant impact on cell spreading. Specifically, for viscosity-based regulation to be significant, the material relaxation timescale τs should be larger than (or at least comparable to) the characteristic binding time τb of the clutches.

Model Predicts Cells Spreading for Several Different Cell Types Across a Wide Range of Elastic and Viscous Substrate Properties.

To test the predictions of our model, we performed cell spreading experiments on viscoelastic substrates synthesized by two different methods (refer to Figs. 4A and 5A). The measured spreading area and the predicted spreading speed of 3T3 fibroblasts cultured on substrates, fabricated by integrating viscous linear polyacrylamide (5%) in the structure of an elastic cross-linked polyacrylamide network (8%), are shown in Fig. 4D. Compared with a purely elastic substrate with the same long-term stiffness, viscosity significantly increased cell spreading (Fig. 4D). Interestingly, the same experiment using human mesenchymal stem cells (MSCs) conducted on a different type of viscoelastic substrate, fabricated by combining covalent and supramolecular cross-linking, indicated that viscosity has no significant effect (Fig. 5D). To understand the reason behind these differences in cellular response, stress relaxation tests were conducted to characterize the viscoelastic properties of the different substrates (Figs. 4B and 5B). Time spectrum analysis based on a generalized Maxwell model (see Methods and Extracting Substrate Parameters from Relaxation Test) revealed the presence of multiple relaxation timescales for the substrates ( Figs. 4C, Inset and 5C, Inset). Following analysis of the competition between the lifetime, binding, and relaxation timescales, only the relaxation timescales that are longer than the binding timescale (τs≥τb, i.e., the peaks on the right part of dashed line) play a role in viscosity regulation (Figs. 4C and 5C). Although these two substrates share similar long-term moduli, the difference in effective relaxation timescales in the two cases puts them into different regimes in the phase diagram. Specifically, substrates fabricated with viscoelastic polyacrylamide exhibit a prominent relaxation timescale of τs≈1 s (Fig. 4C). The triangle marker in Fig. 4E denotes the position of these viscoelastic material parameters (with the cross marker representing the elastic substrate with the same long-term stiffness) in the heat maps of the predicted spreading speeds. Clearly, cell spreading speed (Fig. 4E) and hence spreading area (Fig. 4D) are in good agreement with experimental results. In contrast, for substrates prepared from covalent and supramolecular cross-linkers, the relaxation time spectrum indicates there are no prominent peaks (significant relaxation timescales) beyond the characteristic binding time (Fig. 5C). By choosing the small peak at τs≈3 s, simulations indicate that the viscosity of these ECMs has a negligible effect on cell spreading (Fig. 5D). Since the additional stiffness is small, when the other small peaks were considered, viscosity still did not influence cell spreading velocity or spreading (Fig. 5 D and E). In fact, this insignificant effect of viscosity on cell spreading was predicted to be independent of the long-term substrate stiffness (Fig. 5E), which was confirmed by our additional experiments on gels that are both softer and stiffer (Fig. S5).

Fig. 4.
Fig. 4.

Model explains that viscosity increases cell spreading for substrates synthesized by combining a network of cross-linked polyacrylamide with linear acrylamide. (A) The method used to prepare viscoelastic gels: combining a network of cross-linked polyacrylamide (elastic) with linear polyacrylamide (viscous). (B) The relaxation stiffness as a function of time for cross-linked polyacrylamide substrate (elastic, blue dots) and cross-linked polyacrylamide and linear acrylamide substrate (viscoelastic, red dots). (Inset) Fitting of the relaxation time spectra (black line) for the normalized stiffness of the viscoelastic substrate (red dot markers). (C) Relaxation spectra of the viscoelastic substrate shows that τ=1 s is the dominant relaxation timescale greater than the binding timescale. (Inset) The generalized Maxwell model with multiple relaxation timescales. (D) For gels synthesized with this method (A), viscosity increases cell spreading 4 h after seeding, where a significant difference (n ≥ 180 cells, **P < 0.01, Student’s test) is observed. Note that cross and triangle symbols show the simulated cell spreading area on elastic and viscoelastic substrate, respectively. (E) The relevant positions are also marked on the heat map of spreading speed for ka=1 pN/nm. The trend predicted by the model agrees with the experimental cell spreading area.

Fig. 5.
Fig. 5.

Model explains that viscosity has no effect on cell spreading for substrates synthesized by combining covalent and supramolecular cross-linkers. (A) The method used to prepare viscoelastic gels: combining covalent and supramolecular cross-linkers (see Methods). (B) The relaxation stiffness as a function of time for covalently cross-linked substrate (elastic, blue dots) and covalent and supramolecular cross-linked substrate (viscoelastic, red dots). (Inset) Fitting of the relaxation time spectra (black line) for the normalized stiffness of viscoelastic substrates (red dot markers). (C) The relaxation spectrum H(τ) was plotted as a function of the relaxation timescale τ, which represents (Inset) the generalized Maxwell model. There is no apparent relaxation timescale that is larger than (or comparable to) the binding timescale (τb ≈ 1 s). (D) For substrates synthesized with the method in A, cell spreading area was similar for elastic and viscoelastic substrates; n≥ 80 cells, and n.s. indicates that no statistically significant difference is observed (using the Student’s test). Note that cross and triangle symbols show the simulated cell spreading area on elastic and viscoelastic substrate, respectively. (E) The relevant positions are also marked on the heat map of spreading speed for ka=0.1 pN/nm. The trend predicted by the model agrees with the experimental cell spreading area.

We also compared our model predictions with previous data of U2OS cells seeded on alginate gels that also demonstrated that viscosity promotes cell spreading on soft substrates while reducing spreading on stiff substrates (4) (Fig. 6 D and E). Time spectrum analysis shows that the effective relaxation timescale of this type of viscoelastic substrate is around 15 s (Fig. S6), indicating that the appearance of viscosity would enhance cell spreading compared with elastic substrate with the same long-term stiffness. However, since the initial stiffness (ka+kl) was kept constant for the elastic and viscoelastic substrates in that study, the phase diagram has to be replotted accordingly (Fig. S6C). The corresponding positions of the experimental configurations are shown on the heat maps for cell spreading in Fig. 6 A–C. Again, by choosing an appropriate steady spreading timescale, the simulation results fit well with the experimental data (Fig. 6D). Note that we also carried out simulations with the generalized Maxwell model that incorporates multiple relaxation timescales to verify the choice of the effective timescale (Fig. S8 and Simulation with Multiple Relaxation Timescales).

Fig. 6.
Fig. 6.

Model explains experimental findings that viscosity increases cell spreading for low initial stiffness and suppresses cell spreading for high initial stiffness. (A–C) Heat maps of spreading speed, Vs, plotted as a function of the long-term stiffness, kl, and viscosity, η, calculated using the ODE method for a fixed initial stiffness of (A) 1.4 kPa, (B) 3.4 kPa, and (C) 9 kPa. Crosses and circles indicate the measured properties of the elastic and viscoelastic substrates used for experiments in ref. 4. (see Supporting Information for more details). The viscoelastic relaxation spectrum for this system is given in Fig. S6. (D) Quantification of the cell spreading area as a function of the initial moduli of cells on elastic (gray) or viscoelastic (maroon) substrates. Reprinted with permission from ref. 4. Data are shown as mean ± SD, and ***P < 0.001 (Student’s t test). (E) Model predictions for cell spreading on elastic (gray) or stress-relaxing (maroon) substrates exhibit a behavior similar to the experimental results, where substrate viscosity leads to an increase in cell spreading at a low initial stiffness and decrease in spreading at high initial stiffness.

At this point, we have shown that different observations of viscoelastic regulation are, in fact, attributed to the difference of substrate parameter values. This difference in the parameters leads to the different lifetime and relaxation timescales, which puts the system in a different region in the phase diagram. It must be pointed out that, although we have validated our model with experimental data based on a wide range of parameter values, a large region parameter space remains for further experimental exploration.

Discussion and Conclusions

In summary, we use both analytical and Monte Carlo methods to simulate FA engagement and cell spreading behavior. Rather than using a stochastic lattice model (4), we combined the motor-clutch model with a simple standard linear viscoelastic substrate to show how different viscoelastic material parameters (long-term stiffness, additional stiffness and viscosity) regulate cell spreading. To be specific, at low long-term and additional stiffness, viscosity promotes cell spreading, and maximum cell spreading is achieved at an intermediate viscosity value. In contrast, viscosity has a negligible effect on spreading when the long-term stiffness is large. This insensitivity of cell spreading on viscosity in this regime stems from the clutch reinforcement effect that leads to a saturation of the bounded clutches that can be formed on stiff ECMs. However, such a reinforcing mechanism (under high clutch load) may not be present in certain cell types, like neurons (32, 33), which eventually leads to suppressed cell spreading when the substrate becomes very rigid. The effect of viscosity for these cell types is presented in Supporting Information.

Timescale analysis revealed that substrate viscoelasticity regulates cell spreading based on the magnitude of the substrate relaxation timescale compared with the timescales of motor clutch binding and FA lifetime. The clutch binding time (τb) is solely due to the stochastic binding nature of FA molecules (e.g., integrins, talin, vinculin) with value of ∼1 s (15, 16). The interaction between the pulling force of myosin motors and the substrate stiffness yields an FA lifetime timescale (τl) in the range of 10−1 s to 103 s. When reinforcement does not take effect, the FA lifetime is proportional to its lifetime timescale, τl (16). The average lifetime is ∼10 s to 100 s depending on the stiffness, which conforms with nascent FAs within the lamellipodium (34). However, once the reinforcement takes effect, the FA lifetime can be in minutes or even longer (divergent case), which is consistent with many experiments showing stable FAs for stiffer substrates (1, 9, 35). The substrate relaxation timescale (τs) would span a wide range of 10−2 s to 102 s, corresponding to the viscoelastic nature of different substrate. The relationship between the three timescales (τl, τb,τs) clearly explains how viscosity regulates cell spreading. The regulation effect based on the timescales analysis is summarized in Fig. 7. Specifically, when τl>τb, cell spreading is greatest when the relaxation timescale τs falls between the binding timescale τb and the lifetime timescale τl (τb<τs<τl); when τl<τb, clutch reinforcement takes effect and leads to saturation of the cell spreading area; in this case, viscosity has no effect on cell spreading. The relaxation timescales that are smaller than binding timescale (τs<τb) have a negligible effect on cell spreading.

Fig. 7.
Fig. 7.

Optimal viscosity minimizes actin retrograde flow while maximizing FA lifetime for soft substrates. Schematic describes effects of substrate viscoelasticity on cellular behavior based on the comparison of the clutch binding timescale, τb, substrate relaxation timescale, τs, and lifetime scale, τl. In the load and fail region (τl>τb), cell spreading is maximized when τl>τs>τb such that the cells experience a substrate with a high initial stiffness that relaxes shortly after initial clutch engagement. This minimizes actin retrograde flow while maintaining a long FA lifetime, thereby maximizing cell spreading. When the substrate viscosity is too low (i.e., τs<τb), cells only sense the long-term stiffness, which provides a long FA lifetime without restricting actin retrograde flow. When the substrate viscosity is too high (i.e., τs>τl), the cells only sense the high initial stiffness that doesn’t diminish during the FA lifetime, causing the clutches to prematurely fail. In comparison, for stiff ECMs, a large number of clutches will be formed due to clutch reinforcement, leading to an increased FA lifetime that limits the retrograde flow and enhances the spreading speed.

We also developed a practical way to determine the effective relaxation timescale of viscoelastic substrate regarding the influence on cell spreading. First, relaxation time spectra (Figs. 4C and 5C and Fig. S6) were obtained from stress relaxation data of viscoelastic substrates. Then, we picked the most significant timescale (i.e., the highest peak for τs≥τb) from relaxation spectrum as the effective timescale. Explicit simulations with multiple timescales confirmed that our choice of the effective timescale faithfully describes the dynamics cell spreading (Fig. S8 and Simulation with Multiple Relaxation Timescales) on viscoelastic substrates. In cases where there are multiple prominent relaxation times beyond the binding timescale, simulations show that the resulting cell spreading is approximately a weighted average of the effect for each timescale.

Previous studies suggested that the effects of substrate viscoelasticity on cell spreading are due to local substrate densification and plastic flow (4). However, these experimental results can be reproduced by our model (Fig. 6), demonstrating that viscoelasticity alone is capable of explaining their findings. It is important to note that our model successfully explains the viscoelastic regulation of cell spreading for three completely distinct types of hydrogels [hyaluronic acid (HA), polyacrylamide, alginate] with different ways of imparting viscoelasticity (supramolecular interactions, semiinterpenetrating network/entanglements, ionic cross-linking), different stiffnesses (range from 10−1 pN/nm to 101 pN/nm), and different cell types (human MSCs, 3T3 fibroblasts, U2OS osteosarcoma line). In addition, our experiments using substrates produced with different fabrication methods (and exhibiting virtually no plasticity) are also consistent with the model predictions. While these results demonstrate that viscoelasticity alone has a strong influence on cell spreading, plasticity and ECM remodeling may be equally important. A more complex model that includes plastic theory should be considered in future work.

In conclusion, we have developed an analytical model that incorporates the viscoelastic relaxation time spectrum and successfully explains the effects of substrate viscoelasticity on cell spreading. Our model can be used as a theoretical tool for further study of the viscoelastic regulation of cell behavior for various cell types and tissues. It can also help predict cell spreading over the full parameter space for viscoelastic substrates, which will enable the rational design of biomaterials. Overall, our work provides both physical insight and a practical method to investigate how cellular and material timescales intersect to regulate the cell behavior.

Methods

HA Modification.

Norbornene-modified HA (NorHA) was synthesized by amidation with 5-norbornene-2-methylamine (TCI America) and (benzotriazol-1-yloxy) Tris-(dimethylamino) phosphonium hexafluorophosphate (BOP; Sigma) under nitrogen at room temperature for 2 h. NorHA was further modified with adamantane groups (AdNorHA) as previously described (36) to permit supramolecular interactions in viscoelastic HA hydrogels via esterification with 1-adamantane acetic acid (TCI America), 4-(dimethylamino)pyridine (Sigma), and ditertbutyl dicarbonate (Boc2O; Sigma) under nitrogen at 45 °C for 20 h. The β-cyclodextrin-modified HA (CDHA) was synthesized as previously described (36) to enable supramolecular interactions with adamantane groups on AdNorHA. Briefly, an amine-functionalized cyclodextrin [6-(6-aminohexyl)amino-6-deoxy-β-cyclodextrin] was synthesized (36) and coupled to carboxylic acids on HA using the BOP-mediated amidation protocol described for NorHA above. All reactions were quenched with cold water, purified via dialysis (Spectra/Por, 6- to 8-kDa molecular weight cutoff) and filtration for 7 d at room temperature, frozen, and lyophilized.

HA Hydrogel Fabrication and Rheological Characterization.

NorHA elastic hydrogels with covalent cross-linking only were fabricated via UV light-mediated thiol−ene addition (37). NorHA hydrogels were photopolymerized (2 mW⋅cm−2) in the presence of 1 mM lithium acylphosphinate (38, 39) photoinitiator for 2 min along with thiolated RGD (1 mM, GCGYGRGDSPG; GenScript) to permit cell attachment and DTT (Sigma) to promote covalent cross-linking. AdNorHA−CDHA viscoelastic hydrogels with both covalent and supramolecular cross-links were fabricated similarly to the NorHA elastic hydrogels, with AdNorHA and CDHA mixed to promote Ad−CD guest−host interactions followed by thiol−ene photopolymerization. All hydrogels were fabricated between untreated and thiolated (37) coverslips (50 μL, 18 × 18 mm) to make films with ∼100-μm thickness. HA hydrogel stress relaxation properties were assessed using an AR2000ex rheometer (TA Instruments) equipped with a cone and plate geometry (1°, 20 mm diameter) at a 10% applied strain for 5 min.

MSC Cell Culture and Spreading Analysis on HA Hydrogels.

Human bone marrow-derived MSCs (Lonza) were used at passage 3 for all experiments. Culture media was composed of α-MEM supplemented with 16.7 vol/vol% FBS (Gibco), 1 vol/vol% penicillin streptomycin (Invitrogen), 1 vol/vol% l-glutamine (Invitrogen), and 1 vol/vol% fungizone amphotericin B (Invitrogen). All polymer, peptide, and cell culture reagents were either sterile-filtered or sterilized via germicidal UV irradiation before cell culture. MSCs were seeded atop hydrogel thin films (18 × 18 mm) in six-well plates at a density of 2 × 103 cells/cm2 and cultured for 2 d. Phase contrast images of MSCs atop HA hydrogels were acquired on a Zeiss Axiovert 200 inverted microscope (Hitech Instruments, Inc.), and spread area was quantified for at least 80 cells per experimental group using National Institutes of Health (NIH) ImageJ.

The 3T3 Fibroblast Cells Culture.

The 3T3 NIH cells (ATCC) were maintained in DMEM (Gibco) supplemented with 10% FBS (ATCC) and 1% penicillin streptomycin (Gibco) at 37 °C and 5% CO2. Cells were suspended with trypsin 0.05% and plated on gels. The projected area of single fibroblasts was manually quantified 4 h after plating, on bright field images taken with a LEICA II inverted microscope with a 10× objective.

Viscoelastic Polyacrylamide Gels Fabrication and Rheological Characterization.

A 5% (wt/vol) acrylamide solution is polymerized without bis-acrylamide. The resulting linear polyacrylamide molecules are entrapped in a network made of 8% (wt/vol) acrylamide and 0.125% (wt/vol) bis-acrylamide. This hydrogel is sterilized by a 15-min exposure to germicidal UV light, and activated by immersion in a sulfosuccinimidyl 6-(4′-azido-2′-nitrophenylamino)hexanoate solution in Hepes 50 mM, pH = 8.2 and exposed to UV for 8 min. The surface of gels is functionalized with a 2-h RT incubation with 50 μg/mL of rat-tail collagen I (Corning) in Hepes 50 mM pH = 8.2. The resulting gel presents collagen I cross-linked to both the linear and the networked polyacrylamide. The rheology of polyacrylamide gels was characterized with a Kinexus rheometer (Malvern) using parallel plate geometry (20 mm in diameter plate). The polyacrylamide gels display the typical behavior of viscoelastic solids as quantified with stress relaxation experiments (strain = 0.1).

Numerical Methods and Time Spectrum Analysis.

We use both KMC based on Gillespie algorithm and analytical solution by solving ODE to simulate the process of cell spreading. The relaxation time spectrum was obtained from the stress relaxation data through the regularized Inverse Laplace Transformation (40). All simulations and analysis were carried out in MATLAB; details can be found in Supporting Information.

Footnotes

  • ↵1To whom correspondence may be addressed. Email: vshenoy{at}seas.upenn.edu or ylin{at}hku.hk.

  • Author contributions: V.B.S. designed research; Z.G., S.R.C., E.E.C., and O.C. performed research; Z.G., S.E.S., X.C., R.L.M., P.A.J., and J.A.B. analyzed data; Z.G., S.E.S., Y.L., and V.B.S. wrote the paper; and S.R.C., E.E.C., and O.C. conducted the experiments.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1716620115/-/DCSupplemental.

Published under the PNAS license.

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Matching material and cellular timescales maximizes cell spreading on viscoelastic substrates
Ze Gong, Spencer E. Szczesny, Steven R. Caliari, Elisabeth E. Charrier, Ovijit Chaudhuri, Xuan Cao, Yuan Lin, Robert L. Mauck, Paul A. Janmey, Jason A. Burdick, Vivek B. Shenoy
Proceedings of the National Academy of Sciences Mar 2018, 115 (12) E2686-E2695; DOI: 10.1073/pnas.1716620115
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