Mechanical Restrictions on Biological Responses by Adherent Cells within Collagen Gels

Abstract

Cell-seeded collagen and fibrin gels represent excellent assays for studying interactions between adherent interstitial cells and the three-dimensional extracellular matrix in which they reside. Over one hundred papers have employed the free-floating collagen gel assay alone since its introduction in 1979 and much has been learned about mechanobiological responses of diverse types of cells. Yet, given that mechanobiology is the study of biological responses by cells to mechanical stimuli that must respect the basic laws of mechanics, we must quantify better the mechanical conditions that are imposed on or arise in cell-seeded gels. In this paper, we suggest that cell responses and associated changes in matrix organization within the classical free-floating gel assay are highly restricted by the mechanics. In particular, many salient but heretofore unexplained or misinterpreted observations in free-floating gels can be understood in terms of apparent cell-mediated residual stress fields that satisfy quasi-static equilibria and continuity of tractions. There is a continuing need, therefore, to bring together the allied fields of mechanobiology and biomechanics as we continue to elucidate cellular function within both native connective tissues and tissue equivalents that are used in basic scientific investigations or regenerative medicine.

1. INTRODUCTION

Interactions between adherent cells and extracellular matrix are fundamental to diverse tissue-level processes, including morphogenesis, development, adaptation, disease progression, and wound healing (Tomasek et al., 2002; Humphrey, 2008; Chiquet et al., 2009; Hinz, 2010; Schwartz et al., 2010; Dufort et al. 2011). Not only does the extracellular matrix (often consisting primarily of elastic fibers, collagens, and proteoglycans) endow tissues with important mechanical properties, it sequesters diverse biomolecules (e.g., growth factors, cytokines, and proteinases) and its mechanical state (e.g., stiffness, stress, or strain) can contribute both directly and indirectly to the regulation of cell proliferation, migration, differentiation, and even contraction. For example, mechanical stress can participate in the release or activation of transforming growth factor-beta within the extracellular matrix and cells can participate mechanically in this process (Wipff et al., 2007). There are many reasons, therefore, to understand better the mechanics of cell – matrix interactions and attendant effects on the mechanobiology and pathobiology.

Considerable insight has been gleaned by observing the behavior of cells plated on substrates consisting of, or structures functionalized with, different extracellular matrix proteins (Discher et al., 2005; Chen, 2008), yet adherent interstitial cells typically behave very differently on 2-D surfaces than within 3-D matrix (Cukierman et al., 2001; Chiquet et al., 2009). For this reason, cell-seeded collagen and fibrin gels (or, tissue equivalents) have become particularly useful model systems (Tomasek et al., 2002; Grinnell, 2003). Notwithstanding tremendous understanding gained by studying cellular responses in such constructs, there has been little attention devoted to the associated mechanics. This situation needs to be addressed since mechanobiology seeks to quantify cellular responses to changes in mechanical stimuli, and it is only via mechanics that we can quantify how these stimuli change as a function of changes in the geometry and mechanical properties of the tissue or tissue equivalent as well as the changes in the applied loads that act on them. In this paper, we present mathematical arguments relevant to one of the most commonly used assays for studying cell – matrix interactions, cell-induced contraction of thin, untethered, circular collagen gels. It is shown that, for the class of deformations and constitutive behaviors considered, equilibrium and boundary conditions impose strong mathematical restrictions on otherwise physically unrestricted (free floating) constructs that implicate cell behaviors consistent with the many reported experimental observations.

2. METHODS

2.1 Theoretical Framework

Newton’s second and third laws can be stated simply: the rate of change of linear momentum must balance the sum of all forces acting on a body relative to an appropriate reference frame (2^nd law) and for every action, there must be an equal and opposite reaction (3^rd law). Due largely to extensions by L. Euler, A. Cauchy, and other savants, the modern statements of these two laws for continua are divt + ρb = ρa and T⁽ⁿ⁾ = −T⁽⁻ⁿ⁾, where t is the Cauchy (true) stress tensor, ρ the mass density, b the body force vector, a the acceleration vector, and T⁽ⁿ⁾ the traction vector (having units of stress)¹ that associates with an outward unit normal vector n on a differential area of interest via the relation T⁽ⁿ⁾ = t · n (Humphrey, 2002). Effects of gravity have been reported to be negligible in thin collagen gels (Costa et al., 2003) and it is intuitive that inertial effects are negligible in cell-mediated remodeling processes that occur over periods of hours to days (Humphrey and Rajagopal, 2002). Hence, we assume that salient aspects of the cell-mediated contraction of an untethered collagen gel can be studied as a series of quasi-static equilibria in the absence of body forces.

Although we will allow material properties to vary with radial location, respecting axisymmetry and assuming no variations in the z-direction because of the thinness of the gel (on the order of 0.05 to 0.1 cm thick relative to diameters ranging from 0.5 to 2.2 cm), equilibrium and boundary conditions reduce to (Simon and Humphrey, 2012)

$$\frac{\partial t_{\mathit{rr}}}{\partial r}+\frac{1}{r}(t_{\mathit{rr}}-t_{\mathit{\vartheta \vartheta }})=0,t_{\mathit{zz}}(z=\pm h/2)=0,t_{\mathit{rr}}(r=r_o)=0,$$ (1)

for all r ∈ [0, r_o], ϑ ∈ [0, 2π], and z ∈ [−h / 2, h / 2] where r_o is the current outer radius and h is the associated thickness. That is, we must satisfy equilibrium and traction continuity at all (r, ϑ, z), including traction continuity on the traction-free upper and lower surfaces as well as around the periphery of the untethered circular gel.

Cell-mediated contraction of these gels results in large deformations (cf. Dallon and Ehrlich, 2008), but there is no evidence of dominant viscoelastic effects over the long time scales of interest. Hence, we employ methods of nonlinear elasticity to study the boundary value problem of interest, including a traditional semi-inverse approach wherein one prescribes the deformation (motivated by laboratory observations) and then computes the associated stresses that satisfy equations 1. Restricting our attention to a sub-class of gels that are initially circular and that remain circular and uniformly thin, which is supported by empirical observations under certain conditions, consider homogeneous axisymmetric finite deformations whereby material particles originally at (R, Θ, Z) are mapped to (r, ϑ, z) according to

$$r=\mathit{\lambda R},\mathit{\vartheta }=\mathrm{\Theta },z=\mathrm{\Lambda }Z,$$ (2)

where λ and Λ are radial and axial stretch ratios, respectively (with λ < 1 for contraction of the gel). Associated physical components of the deformation gradient tensor F are (Humphrey, 2002)

$$\mathbf{F}=\left[\begin{array}{ccc}\frac{\partial r}{\partial R}& \frac{\partial r}{R\partial \mathrm{\Theta }}& \frac{\partial r}{\partial Z}\\ \frac{r\partial \mathit{\vartheta }}{\partial R}& \frac{r\partial \mathit{\vartheta }}{R\partial \mathrm{\Theta }}& \frac{r\partial \mathit{\vartheta }}{\partial Z}\\ \frac{\partial z}{\partial R}& \frac{\partial z}{R\partial \mathrm{\Theta }}& \frac{\partial z}{\partial Z}\end{array}\right]=\left[\begin{array}{ccc}\lambda & 0& 0\\ 0& \lambda & 0\\ 0& 0& \mathrm{\Lambda }\end{array}\right].$$ (3)

Because fluid is exuded from the gel as it contracts (cf. Bell et al., 1979), we assume an overall compressible behavior. Hence, the standard measure of volume changes, J ≡ det F = λ² Λ, need not equal unity as in incompressible problems. Note, too, that based on preliminary calculations, we neglected the small, near hydrostatic pressure that is exerted on these gels while immersed in culture medium.

Unfortunately, there has been little careful work on the likely evolving mechanical properties of free-floating collagen or fibrin gels. Nevertheless, it appears that these gels initially exhibit a low tensile stiffness (~2 to 24 kPa) and that they are initially isotropic (Roeder et al., 2004; Marnezana et al., 2006); the compressive stiffness is expected to be even lower (~0.015 to 0.075 kPa) due to the high porosity and permeability (Barocas et al., 1995; Harley et al., 2007). Hence, we assume that these gels can be characterized initially by a Blatz-Ko constitutive model (cf. Horgan, 1996; Humphrey, 2002). In addition, however, observations suggest that both the cells and matrix can slowly align parallel to the outer edge (Dallon and Ehrlich, 2008; Costa et al., 2003), hence inducing a potentially radially varying transverse-isotropy (with a preferred circumferential direction). We thus consider an “extended Blatz-Ko” constitutive behavior of the form

$$W({\mathit{II}}_C,{\mathit{III}}_C,{\mathit{IV}}_C)={\phi }_{\mathit{iso}}(r)\frac{\mu }{2}(\frac{{\mathit{II}}_C}{{\mathit{III}}_C}+2{\mathit{III}}_C^{1/2}-5)+{\phi }_{\mathit{aniso}}(r)\frac{c}{4}{({\mathit{IV}}_C-1)}^2,$$ (4)

at each time during contraction, where φ_iso (r) ∈ [0, 1] and φ_aniso (r) ∈ [0, 1] are “mass fractions” that capture the isotropic and potentially radially varying transversely isotropic contributions, respectively. Moreover, φ_iso (r) + φ_aniso (r) = 1 for all r ∈ [0, r_o], μ and c are material parameters that model the matrix stiffness (having units of kPa) at any particular time, and the coordinate invariant measures of deformation are

$$2{\mathit{II}}_C={(\mathit{tr}\mathbf{C})}^2-\mathit{tr}{\mathbf{C}}^2,{\mathit{III}}_C=\det \mathbf{C},{\mathit{IV}}_C=\mathbf{M}·\mathbf{CM},$$ (5)

with C = F^T F the right Cauchy-Green tensor and M a unit vector that denotes the preferred (transversely isotropic) direction in the reference configuration. Note that αm = FM, where α is the stretch ratio of a line element originally oriented in direction M, which has direction m after deforming. We consider only a circumferential transverse isotropy, hence for the prescribed deformations m ≡ e_ϑ.

It can be shown that the Cauchy stress response, with t = 2F (∂W / ∂C) F^T / det F, for the extended Blatz-Ko material defined by equation 4 is

$$\mathbf{t}={\phi }_{\mathit{iso}}(r)\frac{\mu }{{\mathit{III}}_C^{1/2}}({\mathit{III}}_C^{1/2}\mathbf{I}-{\mathbf{B}}^{-1})+{\phi }_{\mathit{aniso}}(r)\frac{c}{{\mathit{III}}_C^{1/2}}({\mathit{IV}}_C^2-{\mathit{IV}}_C){\mathbf{e}}_{\mathit{\vartheta }}\otimes {\mathbf{e}}_{\mathit{\vartheta }},$$ (6)

where I is the identity tensor, B = FF^T is the left Cauchy-Green tensor, and ⊗ denotes the tensor product. Finally, to generalize equation 6 to account for possible radial variations in cell-induced contraction, we introduce isotropic (t_a I) and transversely-isotropic (t_ce_ϑ ⊗ e_ϑ) active contributions to the Cauchy stress whereby

$$\mathbf{t}={\phi }_{\mathit{iso}}(r)\left(\frac{\mu }{{\mathit{III}}_C^{1/2}}({\mathit{III}}_C^{1/2}\mathbf{I}-{\mathbf{B}}^{-\mathbf{1}})+t_a\mathbf{I}\right)+{\phi }_{\mathit{aniso}}(r)\left(\frac{c}{{\mathit{III}}_C^{1/2}}({\mathit{IV}}_C^2-{\mathit{IV}}_C+t_c\right){\mathbf{e}}_{\mathit{\vartheta }}\otimes {\mathbf{e}}_{\mathit{\vartheta }}.$$ (7)

That is, given our current lack of understanding of cellular contractile behavior in tissue equivalents, we do not specify the specific functional forms a priori.

2.2 Preparation of Cell-seeded Collagen Gels

NIH/3T3 fibroblasts (ATCC) were maintained in Dulbecco’s modified Eagle’s medium (DMEM), 10% calf serum, and 1% antibiotic-antimycotic (Invitrogen) in an incubator at 37°C and 5% CO₂. They were passaged at 70-80% confluence to obtain cells from passages 4 to 6 for seeding the collagen gels, which were prepared using a protocol modified from Hu et al. (2009) and Ehrlich and Rittenberg (2000). Briefly, 50,000 cells were suspended in 554 μL of a seeding culture medium consisting of DMEM, 10% fetal bovine serum (HyClone), 10% porcine serum (Invitrogen), penicillin G (100 U/mL, Sigma), ascorbic acid (50 μg/mL, Sigma), CuSO₄ (3 ng/mL, Sigma), proline (50 μg/mL, Sigma), alanine (20 μg/mL, Sigma), glycine (50 μg/mL, Sigma), HEPES (0.01M, Invitrogen), basic fibroblastic growth factor (10 ng/mL, R&D Systems), and platelet-derived growth factor-BB (10 ng/mL, R&D Systems) – see Niklason et al. (2011). Alternatively, some cells were suspended in medium that also contained transforming growth factor-β1 (10 ng/mL, Sigma).

These suspensions of cells were mixed with a solution containing 146 μL of concentrated type I rat tail collagen (8.58 mg/mL, BD Biosciences), 200 μL of 5X DMEM, and 100 μL of a 10X reconstitution buffer (0.1N NaOH and 20M HEPES) (cf. Hu et al., 2009) and then cast in a 12-well plate (each well being 2.2 cm in diameter). The final solution of 1 mL thus contained 50,000 cells and 1.25 mg of collagen. The 12-well plate was placed in the incubator for 30 min for gelation. Gels were then released from the wells by suspending them in their seeding medium, transferred to 6.0 cm diameter dishes, and maintained for up to 7 days in the incubator. Culture medium was replaced at 1, 3, and 5 days. Images of the resulting free-floating gels were acquired, using a Motic DMW143 Digital Stereo Microscope, every 4 hours from 0 to 16, every 12 hours from 24 to 120 (5 days), and at day 7, with all times recorded following release of the gel from the 12-well plate. Changes in radius were measured easily at each time whereas changes in thickness required destructive tests. Hence, one gel at each of the 14 times was used to determine thickness. Briefly, each of these gels was fixed in a 10% formalin solution to prevent changes in dimensions while sectioning the gel to obtain a 1 to 2 mm wide by ~5 mm long strip. Each of these strips was then imaged using the stereo microscope to record the thickness along the axial direction of each strip.

The state of stress in the gels was assessed qualitatively using additional gels cast either with or without fibroblasts. First, to assess the possible presence of a residual stress field (i.e., tensile circumferential stresses at the outer edge and compressive circumferential stresses in the inner region), a razor blade was used to introduce a radial cut from the center to the outer edge of the gels and any subsequent opening angles were recorded using the stereo microscope. Second, to assess the possible presence of a nearly equibiaxial compressive stress field in the central region of the gels, a 1 mm diameter biopsy punch (Miltex, with outer diameter of 1.05 mm) was used to excise a circular portion from the gel at an off-center location, and the dimensions of the newly introduced hole were measured and compared with the dimensions of the biopsy punch.

2.3 Histology

The remaining intact gels at 0, 5, and 7 days were fixed in a 10% formalin solution for 24 hours at 4°C, then rinsed 3 times in phosphate-buffered saline and preserved in a 70% ethanol solution. Preserved gels were embedded in paraffin and sectioned at 5 to 8 μm. Half of the sections were stained with picrosirius red (PSR) to examine collagen fiber orientation while the others were immunohistochemically stained for α-smooth muscle actin (αSMA) and counter-stained with hematoxylin to examine cell phenotype and orientation, respectively. Images were acquired at 200X or 300X magnification using an Olympus BX51 microscope with the PSR section examined under circularly polarized light to show birefringent collagen. Collagen alignment was assessed via the power spectrum of a 2D discrete Fourier transform (2D-DFT) as described previously (Bayan et al., 2009). Briefly, PSR images were converted to grayscale and cropped to 256 × 256 pixel regions of interest. These images were processed using a custom MATLAB code that calculated the 2D-DFT, shifted the low frequency components to the center of the image, calculated the power spectrum as the squared amplitude of the 2D-DFT, and rotated it 90 degrees to align the data with the initial image and to map the results into polar coordinates (ρ, θ). A line average was taken along ρ for each value of θ to determine the spatial frequency-averaged intensity as a function of θ, that is I(θ), with the angle associated with the maximum intensity identified as the dominant angle. I(θ) was normalized for each image by the total intensity and shifted so that the dominant angle was at 0 degrees with ±90 degrees plotted on either side. Randomly oriented fibers thus resulted in random power spectrum distributions while highly organized and aligned fibers exhibited a well-defined peak at the corresponding dominant angle.

3. RESULTS

3.1 Theoretical Findings

For the deformation given by equation 2, the non-trivial components of Cauchy stress are

$$t_{\mathit{rr}}={\phi }_{\mathit{iso}}(r)\left(\mu (1-\frac{1}{{\lambda }^4\mathrm{\Lambda }})+t_a\right),$$ (8)

$$t_{\mathit{\vartheta \vartheta }}={\phi }_{\mathit{iso}}(r)\left(\mu (1-\frac{1}{{\lambda }^4\mathrm{\Lambda }})+t_a\right)+{\phi }_{\mathit{aniso}}(r)\left(\frac{c}{\mathrm{\Lambda }}({\lambda }^2-1)+t_c\right),$$ (9)

$$t_{\mathit{zz}}={\phi }_{\mathit{iso}}(r)\left(\mu (1-\frac{1}{{\lambda }^2{\mathrm{\Lambda }}^3})+t_a\right).$$ (10)

Note that the traction-free boundary condition at the periphery (equation 13) is satisfied identically for all functions φ_iso(r = r_o) = 0, which we treat as a restriction on the allowable forms for the radially varying “mass” fractions that dictate the material symmetry. Conversely, the traction-free boundary condition on the upper and lower surfaces (equation 12) requires

$$t_a=-\mu \left(1-\frac{1}{{\lambda }^2{\mathrm{\Lambda }}^3}\right)\forall r\in [0,r_o],$$ (11)

which reveals that the cell-mediated isotropic contraction stress must remain proportional to the (possibly evolving) isotropic matrix stiffness. Let us now consider two special cases defined by particular variations in material symmetry.

First, consider a linear variation whereby φ_aniso = r / r_o and thus φ_iso = 1 − r / r_o. In other words, let the material exhibit an isotropic response at the center (r = 0) but a strongly transversely isotropic (circumferential) response at the periphery (r = r_o); at increasing radial positions between the center and edge there is an increasing transverse isotropy. Clearly, this choice of mass fractions satisfies the traction-free condition at the periphery. Radial equilibrium (equation 11) thus requires, at all radial locations,

$$-\mu \left(\frac{1}{{\lambda }^2{\mathrm{\Lambda }}^3}-\frac{1}{{\lambda }^4\mathrm{\Lambda }}\right)=\frac{c}{\mathrm{\Lambda }}({\lambda }^2-1)+t_c\to t_c=-\mu \left(\frac{1}{{\lambda }^2{\mathrm{\Lambda }}^3}-\frac{1}{{\lambda }^4\mathrm{\Lambda }}\right)-\frac{c}{\mathrm{\Lambda }}({\lambda }^2-1),$$ (12)

with λ < 1 for contraction of the gel. Note, again, that the cell-mediated contraction stress must be proportional to matrix stiffness. The final state of plane stress thus becomes

$$t_{\mathit{rr}}(r)=\mu \left(1-\frac{r}{r_o}\right)\left(\frac{1}{{\lambda }^2{\mathrm{\Lambda }}^3}-\frac{1}{{\lambda }^4\mathrm{\Lambda }}\right),t_{\mathit{\vartheta \vartheta }}(r)=\mu \left(1-\frac{2r}{r_o}\right)\left(\frac{1}{{\lambda }^2{\mathrm{\Lambda }}^3}-\frac{1}{{\lambda }^4\mathrm{\Lambda }}\right),$$ (13)

which can be written equivalently as

$$t_{\mathit{rr}}(r)=\mu \left(1-\frac{r}{r_o}\right)\left(\frac{{\lambda }^4}{J^3}-\frac{1}{{\lambda }^{2}J}\right),t_{\mathit{\vartheta \vartheta }}(r)=\mu \left(1-\frac{2r}{r_o}\right)\left(\frac{{\lambda }^4}{J^3}-\frac{1}{{\lambda }^{2}J}\right),$$ (14)

where J = det F = λ² Λ is the aforementioned measure of volume changes. It can be shown numerically, as would be expected in this traction-free problem, that one obtains a “residual stress type” distribution of stresses wherein regions of compression balance those of tension (cf. Simon and Humphrey, 2012). This type of stress field is also seen in the second case, which is considered in more detail below.

Second, consider an exponential radial variation in material symmetry, specifically φ_aniso = exp[n (r / r_o − 1)] and thus φ_iso = 1 − exp[n (r / r_o −1)]. Albeit involving slightly more algebra, one finds a result similar to that for the linear radial variation in symmetry. Radial equilibrium requires

$$t_c=-\left(\frac{r}{r_o}\right)\mathit{n\mu }\left(\frac{1}{{\lambda }^2{\mathrm{\Lambda }}^3}-\frac{1}{{\lambda }^4\mathrm{\Lambda }}\right)-\frac{c}{\mathrm{\Lambda }}({\lambda }^2-1),$$ (15)

which reveals that the anisotropic contribution of active cell stress must now vary radially as well. The overall plane state of stress can thus be written

$$t_{\mathit{rr}}(r)=\mu (1-e^{n(r/r_o-1)})\left(\frac{{\lambda }^4}{J^3}-\frac{1}{{\lambda }^{2}J}\right),t_{\mathit{\vartheta \vartheta }}(r)=\mu (1-(1+\mathit{nr}/r_o)e^{n(r/r_o-1)})\left(\frac{{\lambda }^4}{J^3}-\frac{1}{{\lambda }^{2}J}\right).$$ (16)

Figures 1 to 4 show illustrative parametric studies for n = 10, which pushes the transversely isotropic contributions to matrix stiffness and cell contractility towards the periphery (see below). Specifically, Figure 1 shows radial distributions of stress, normalized with respect to the possibly evolving material parameter μ, for multiple degrees of contraction (λ = 0.6, 0.5, 0.4) and a representative fixed value of J = 0.25 (i.e., a 75% loss of volume upon contraction). Figure 2 similarly shows similar radial distributions for a 50% contraction (λ = 0.5) for multiple possible values of volume change (J = 0.75, 0.50, 0.25). Although these results – particularly, the existence of a self-equilibrating residual stress field – are qualitatively similar to those for the case of linear radial variations in material properties (not shown), focusing the anisotropy near the periphery creates stronger gradients in the distribution of stress.

Figure 1.

Predicted spatial distributions of radial (top) and circumferential (bottom) Cauchy stress, normalized by the isotropic matrix stiffness μ, in a model fibroblast-seeded thin collagen gel for an exponentially varying radial anisotropy (equation 16 with n = 10) and different degrees of contraction (λ = 0.6, 0.5, and 0.4) for a fixed degree of volume change (J = 0.25). Note that the value of μ would be expected to evolve with increased contraction and that changes in volume should parallel changes in contraction; the present model allows both. Results are intended for illustrative purposes only, not to describe a particular experiment. The schema in the bottom panel shows the components of stress at a generic radial location.

Figure 4.

Predicted distributions of the transversely-isotropic cellular stress (equation 15) for different values of Γ = c / μ and a fixed contraction (λ = 0.5) and volume change (J = 0.25). Note that the results are not sensitive to the ratio Γ = c / μ.

Figure 2.

Similar to Figure 1 except for different degrees of volume change (J = 0.75, 0.50, and 0.25) at a fixed degree of contraction (λ = 0.5). See equation 16.

If we introduce the nondimensional parameter Γ = c / μ, then distributions of cell-induced active stress can also be normalized by the Blatz-Ko modulus μ and plotted as a general function of radius. Figure 3 shows results for a situation similar to that in Figure 1 (i.e., J = 0.25 and multiple degrees of overall contraction of the gel), except for levels of isotropic and oriented cell contractility that yield Γ = c / μ =1. Figure 4 shows results for Γ = c / μ =1, 5, 25, 125, with λ = 0.5 and J = 0.25, which reveals that such results are not very sensitive to this parameter.

Figure 3.

Predicted spatial distributions of the isotropic cell stress given by equation 11 (top) and the transversely-isotropic (circumferential) cell stress given by equation 15 (bottom). Both stresses are normalized by the isotropic matrix stiffness μ, with Γ = c / μ =1, and are shown for different degrees of contraction (λ = 0.6, 0.5, and 0.4) for a fixed degree of volume change (J = 0.25).

3.2 Experimental Findings

When cultured for 5 days, fibroblast-seeded collagen gels had a final normalized outer radius of 0.34±0.04 (mean ± SD), that is, a mean radial stretch ratio λ = 0.34 (cf. Figure 5 Panel A, middle column). PSR-stained images of the gels revealed a dramatic change in organization of the collagen fibers from day 0 to day 5. Collagen fibers in the central region were randomly oriented at both day 0 and day 5 (Figure 5 Panel A, left column), but matrix compaction was qualitatively greater at day 5. In contrast, there was a stark difference in the organization of collagen fibers at the outer edge at day 5 (Figure 5 Panel A, right column). Whereas the fibers were randomly oriented at the periphery at day 0, similar to the fibers within the central region, they showed a high degree of preferential alignment parallel to the outer edge (i.e., circumferential direction) at day 5 (Figure 5 Panel B). That is, there appeared to be an abrupt transition from a random to a preferred orientation near the periphery. Moreover, the collagen fibers near the periphery appeared red under polarized light, which suggested qualitatively either more compacted and/or thicker fibers.

Figure 5.

(Panel A) Circularly polarized light images of picrosirius red-stained sections (left and right columns) from gels at day 0 (top row) and day 5 (bottom row); the middle column shows the entire gel as viewed via the stereo microscope. Note further that the left column shows collagen in the central region of the gels, which compacts but retains its isotropic character at day 5. In contrast, the right column shows collagen at the periphery of the gels, which appears both compacted and aligned parallel to the edge at day 5. White scale bar is 5 mm and yellow scale bar is 50 μm. (Panel B) Assessment of collagen fiber orientation, using a 2D-DFT Power Spectrum, in the central region (left) and at the outer edge (right) of a cell-seeded gel at day 5. The results indicate a lack of preferred direction in the central region, but a high degree of circumferential alignment (shifted to angle 0) of fibers at the outer edge. For visual simplicity, the power spectrums were plotted for ±90 degrees about the measured dominant angle, which was shifted to 0 degrees.

The radial-cut tests were consistent with a cell-mediated development of a residual stress field, characterized by self-equilibrating tensile circumferential stresses at the periphery and compressive circumferential stresses within the central region of the gels (Figure 6, bottom). The associated “opening angle” was 20.60±6.42 degrees for days 1 through 5 (n = 3 per day), with a value of 21.90±1.9 degrees on day 1 consistent with the dramatic contraction achieved during the first day as noted below. In contrast, the 0 day gels with cells showed no considerable opening angle, with measured values of 2.39±1.18 degrees likely reflecting the width of the razor blade. Finally, gels without cells also exhibited no substantial opening angle, with values of 2.36±1.31 degrees measured at days 0 through 5 (again likely due to the thickness of the razor blade; Figure 6, top).

Figure 6.

Collagen gels at day 2 either without (top) or with (bottom) fibroblasts shown 15 minutes after introducing a radial cut. Whereas the cell-free gels showed little to no appreciable opening angle, the cell-seeded gels exhibited considerable opening angles at all times examined (days 1 through 5) except day 0. Such opening angles are consistent with the radial cut relieving residual stresses, particularly tensile circumferential stresses at the periphery and compressive circumferential stresses in the central region. The scale bar is 5 mm.

The off-center biopsy punch tests were similarly consistent with the development of equibiaxial compressive stresses in the central region of the cell-seeded gels. Notwithstanding some undulations at the inner edge, possibly due to local buckling phenomena, the circularly punched holes remained nearly circular at days 1 to 5 and had measured diameters of 0.82±0.06 mm (Figure 7, bottom). In contrast, punched holes in cell-seeded gels at day 0 retained diameters of 1.00±0.03 mm. In gels without cells, the punched holes also maintained diameters similar to the outer diameter of the biopsy punch (1.03±0.06 mm vs. 1.05 mm) at days 0 to 5 (Figure 7, top).

Figure 7.

Collagen gels at day 2 either without (top) or with (bottom) fibroblasts shown 15 minutes after introducing a circular hole in the central region. Whereas holes in the cell-free gels maintained dimensions similar to the biopsy punch, those in the cell-seeded gels closed in at all times examined (days 1 through 5) except day 0. Reduction in hole size while retaining a roughly circular shape is consistent with a nearly equibiaxial compressive stress field in the central region. The scale bar is 0.5 mm.

Sections stained with hematoxylin and antibodies against αSMA revealed an increase in cell density at the periphery of the 5 day cell-seeded gels, with a high degree of orientation parallel to the edge similar to that for the collagen (compare Figures 8-A and 8-B). Strong expression of αSMA (brown) was found in most of the oriented cells, which suggested a phenotypic change from fibroblasts to myofibroblasts. Day 7 gels showed trends similar to the day 5 gels except that changes near the periphery were more dramatic (Figure 8-C). As noted in Methods, some gels were cultured in the presence of exogenous TGF-β1. These gels compacted to a higher degree (λ~0.15), but did not maintain an overall flat, disk-shaped geometry. As seen in Figure 8-D, the cells were even more highly aligned and densely packed at the periphery. The majority of cells at this edge were also positive for αSMA as were many of the cells in the central region.

Figure 8.

Sections of collagen gels immunostained for αSMA and stained with hematoxylin. Brown indicates positive staining for αSMA and purple the cell nuclei. (A) Cells in the central region of a day 5 gel. (B) Cells near the periphery of a day 5 gel. (C) Cells near the periphery of a day 7 gel. (D) Cells near the periphery of a day 5 gel cultured with exogenous TGF-β1. Scale bars are 50 μm.

4. DISCUSSION

The circular, free-floating, fibroblast-populated collagen lattice studied theoretically herein was introduced experimentally over thirty years ago by Bell et al. (1979). Many investigators point to this work as possibly the beginning of tissue engineering. Regardless, free-floating gels have since been the focus of over one hundred studies of cell – matrix interactions (cf. Dallon and Ehrlich, 2008). It has been shown, for example, that the rate and extent of cell-mediated gel contraction depends on the initial density of the collagen, the initial density of the cells, and the availability of particular growth factors. It has also been shown that fibroblasts can contract gels consisting of type III collagen more rapidly and to a greater extent than those consisting of type I collagen (Ehrlich, 1988) and, although exposed to the same biochemical milieu, fibroblasts from different tissues contract gels to different extents (Dahlmann-Noor et al., 2007). These and similar findings remind us that specific results depend on the particular experimental conditions. Nevertheless, many general findings have surfaced. Of particular interest herein, the initial remodeling and compaction of collagen is accomplished by fibroblasts repeatedly extending podial protrusions into the gel, attaching to the collagen via appropriate integrins, and retracting the collagen via actinomyosin based mechanisms (Ehrlich and Rittenberg, 2000); similar observations have also been reported by others in complementary experiments (Roy et al., 1999; Meshel et al., 2005; Dahlmann-Noor et al., 2007). After a gel contracts significantly under typical conditions, fibroblasts within the outermost region begin to differentiate into myofibroblasts (Ehrlich, 1988), a process that is typically characterized by the addition of alpha-smooth muscle actin to the stress fibers, the maturation of focal adhesions into larger “super” focal adhesions, and the ability to sustain a larger contractility (cf. Hinz, 2010). Note, therefore, that increased matrix stiffness, tensile mechanical stress, and the presence of cytokines that are typically present in culture medium containing serum (e.g., TGF-β1), collectively represent a strong stimulus for fibroblasts to differentiate into myofibroblasts. One would expect, therefore, that significant tensile stresses develop in the outer region of the contracted gel as cells reorient and compact the collagen, which in turn could increase the stiffness locally and anisotropically. Indeed, it has also been reported that cell density tends to be higher in this outer region, with the cells preferentially aligned parallel to the free boundary and the collagen aligned with the cells (Ehrlich, 1988; Dallon and Ehrlich, 2008). These localized effects at the boundary can be particularly dramatic in gels having an initially high density (> 6 × 10⁵ cells/ml) of cells (Ehrlich and Rittenberg, 2000). Conversely, it appears that cells within the central region of the gel retain their fibroblast phenotype, but experience increased apoptosis. Because normal levels of mechanical stress are thought to diminish apoptosis (Bride et al., 2004), the relative lower rate of apoptosis in the outermost region further supports the possibility of increased tensile stress in that region.

Notwithstanding such observations, many investigators have suggested that free-floating fibroblast-populated collagen gels remain stress free. For example, Grinnell (2003) wrote, “if the matrix is floating … the matrix does not develop tension; that is, it remains mechanically unloaded”. Bride et al. (2004) said it this way, “In contracting free floating collagen lattices, the tension is distributed isotropically and the extracellular matrix remains mechanically relaxed during contraction of the gel,” whereas Tomasek et al. (2002) wrote, “Mechanical tension cannot develop in untethered lattices.” More recently, John et al. (2010) used a thermoelastic analog (finite element) model to simulate a uniform contraction of a homogeneous, isotropic gel and similarly concluded that stresses remain minimal in free-floating gels.

Assuming that the collagenous matrix and active cell stress are both homogeneous and sotropic (i.e., φ_iso (r) = 1 with t_a ≠ 0 but t_c = 0), our theoretical results (not shown) confirm a possible non-trivial solution that allows compaction of a gel via a uniform zero-stress field wherein the active (tensile) cell stress balances everywhere the (compressive) stress in the matrix. This solution requires equal radial and axial deformations (i.e., λ = Λ), however. Because our measurements of radial contraction and axial thinning revealed that λ ≠ Λ at every time considered (Figure 9), this special solution appears to be admissible mathematically, but not relevant experimentally. In other words, we submit that free-floating collagen gels are not stress-free as suggested previously.

Figure 9.

Experimentally measured mean changes in gel thickness (open diamonds), radial contraction (open circles), and overall volume (x) over 5 days. Note the strongly compressible behavior and that changes in radial contraction and thickness were not the same (i.e., that λ ≠ Λ), which excludes the mathematically admissible solution of a uniform stress-free gel (see text).

Using incompressible finite strain elasticity to model the initiation of contraction in free-floating gels, we previously showed that it is essential to account for the reported observations of radial variations in cell phenotype and cell-mediated remodeling of matrix in any associated mechanical analysis (Simon and Humphrey, 2012). Herein, we extended our prior results by allowing the gel to be compressible (cf. Figure 9). We again found that a residual-type plane stress field is admitted mathematically whereby the outermost region experiences a tensile circumferential stress and the innermost region experiences a compressive circumferential stress; the radial stress was found to be everywhere compressive except at the periphery where it must vanish to satisfy a traction-free boundary condition (cf. Figures 1-4). This non-uniform stress field appears to be consistent with both our general understanding of the cell mechanobiology (cf. Chiquet et al., 2009; Dufort et al., 2010; Hinz, 2010; Schwartz, 2010) and diverse observations from free-floating gel experiments (Ehrlich, 1988; Grinnell, 2003; Dallon and Ehrlich, 2008). Note, therefore, that cells in solution are initially stress-free when introduced within the initially stress-free gels. Moreover, substantial evidence now suggests that normally adherent cells seek to achieve and maintain a target level of stress (Humphrey, 2008), which in collagen gels appears to translate into a target level of tissue-level stress – a process that has been referred to as “tensional homeostasis” (Brown et al., 1998). For example, fibroblasts in tethered collagen gels tend to develop and maintain stresses on the order of 4.5 kPa (Kolodney and Wysolmerski, 1992). Interestingly, stresses at focal adhesions in fibroblasts similarly tend to be maintained at ~3 to 5.5 kPa (Balaban et al., 2001), which is consistent from the perspective of traction continuity. We suggest, therefore, that the contraction of a free-floating gel by seeded cells is an attempt at tensional homeostasis. Because of the lack of tethering, however, the cells can only generate a residual-type stress field while satisfying restrictions imposed by equilibrium and boundary conditions. The associated tensile circumferential stresses in the outermost portion of the circular gel appear to be more favorable mechanobiologically for the fibroblasts, thus facilitating their differentiation into myofibroblasts as observed by many. Because the fibroblasts in the central region appear to experience a small, nearly equibiaxial compressive stress, such a state of stress may encourage increased apoptosis, which has also been observed. That is, in contrast with many prior suggestions that free-floating gels cannot develop any stress, our identification of a residual-type stress field is consistent with prior observations of myofibroblasts and aligned collagen near the periphery and apoptotic fibroblasts and non-aligned collagen in the central region.

The present histological results (Figures 5 and 8) suggest that the transition from an isotropic central region to an anisotropic outer region may be abrupt, with circumferential alignment of cells and fibers being highly localized to the outer edge of the gel. This steep gradient can be accounted for by our model by adjusting the parameter n in the exponential function for the “symmetry mass fractions,” namely φ_iso and φ_aniso. Small values for n (~5) result in a gradual transition from an isotropic to anisotropic matrix whereas larger values of n can create a steep gradient that results in a highly localized anisotropic state at the outer edge (Figure 10) that still satisfies both equilibrium and boundary conditions.

Figure 10.

Parameterization of n for the radially varying functions for material symmetry, φ_iso (top) and φ_aniso (bottom). Increasing values of n localize the anisotropy closer to the traction-free edge while satisfying equilibrium and boundary conditions, thus showing that changes from isotropy to transverse-isotropy can be both abrupt and highly localized to the outer edge.

Referring to early mechanical models of interactions between cells and the substrates on which they are cultured, Discher et al. (2005) suggested that “A major challenge in all such modeling is to clarify the principal enigma: how contractive traction forces exerted by a cell tend to increase with stiffness of the cell’s substrate.” Although we did not address this particular experimental set-up herein, we found that, for the class of deformations and constitutive behaviors considered, forcing cells to respect the mechanics revealed that the level of contraction must be proportional to the matrix stiffness to satisfy equilibrium and boundary conditions (cf. equations 11 and 15 and Figures 3 and 4). Indeed, mathematical restrictions resulting from the axisymmetry, thinness, and traction-free surfaces enabled a complete closed-form solution without prescribing the specific functional form for the active stresses a priori. It appears, therefore, that the present solution, which suggests the development of a cell-mediated residual stress field, is consistent with both general observations and experiment-specific observations, including new findings herein that radially-cut gels open up as would be expected with the relief of residual stresses and that interior punched-holes close nearly uniformly as would be expected for a local equibiaxial compressive stress field (Figures 6 and 7).

In summary, most investigators agree that mechanics plays a central role in dictating cell behaviors in tissues and tissue equivalents, yet most prior papers on cell-seeded gels have speculated on roles of the mechanics rather than considering the mechanics explicitly (cf. Winer et al., 2009). Not only can such speculations be misleading, the mechanics can often expand and enrich our understanding. Hence, we agree with Tomasek et al. (2002) that the “inescapable mechanical laws that are integral to the regulation of myofibroblasts” and other mechano-sensitive cells should be considered when interpreting mechanobiological responses in both tissue equivalents and native tissues.

Highlights.

• Residual-type stress fields can exist in free-floating collagen gels

• Regional anisotropy can develop naturally in free-floating collagen gels

• Cell tractions must be proportional to matrix stiffness in free-floating collagen gels

• Cell phenotype can be dictated, in part, by equilibrium and boundary conditions.