Mean deformation metrics for quantifying 3D cell-matrix interactions without requiring information about matrix material properties

Abstract

Mechanobiology relates cellular processes to mechanical signals, such as determining the effect of variations in matrix stiffness with cell tractions. Cell traction recorded via traction force microscopy (TFM) commonly takes place on materials such as polyacrylamide- and polyethylene glycol-based gels. Such experiments remain limited in physiological relevance because cells natively migrate within complex tissue microenvironments that are spatially heterogeneous and hierarchical. Yet, TFM requires determination of the matrix constitutive law (stress-strain relationship), which is not always readily available. In addition, the currently achievable displacement resolution limits the accuracy of TFM for relatively small cells. To overcome these limitations, and increase the physiological relevance of in vitro experimental design, we present a new approach and a set of associated biomechanical signatures that are based purely on measurements of the matrix's displacements without requiring any knowledge of its constitutive laws. We show that our mean deformation metrics (MDM) approach can provide significant biophysical information without the need to explicitly determine cell tractions. In the process of demonstrating the use of our MDM approach, we succeeded in expanding the capability of our displacement measurement technique such that it can now measure the 3D deformations around relatively small cells (∼10 micrometers), such as neutrophils. Furthermore, we also report previously unseen deformation patterns generated by motile neutrophils in 3D collagen gels.

Keywords: confocal microscopy; large deformations; neutrophil; traction force microscopy.

Fig. 1.

(A–D) Validation of MDM on a spherical cell (gray) undergoing stretch with three stretch ratios ${\lambda }_1,$ ${\lambda }_2,$ ${\lambda }_3$ (A), rotation with angle θ around axis $x_1$ (B), simple shear with magnitude k (C), and an Eshelby transformation strain ${\mathrm{ϵ}}_T$ (D). (A–D, Left) Applied deformation gradient $\mathbf{F}$. (A–D, Center) Vector cone plot of the displacement fields interpolated onto the surface of the sphere for A–D, color-coded by normalized magnitude $\left|\mathit{u}\right|.$ (A–D, Right) Results for the recovered mean deformation gradient $\langle \mathbf{F}\rangle$ for A–D with associated error (ε) (A–C, less than ${10}^{-12}$ because of numerical error; D, less than ${10}^{-5}$ because of small strain assumption). Calculation of full-field matrix displacements using the analytical Eshelby solution for an inclusion undergoing an eigenstrain. (E) Schematic illustrating the canonical Eshelby inclusion problem, where an initially spherical inclusion (gray) in an infinite matrix undergoes an eigenstrain ${\mathrm{ϵ}}_T$ and induces a known analytical displacement field $\mathit{u}$ and confinement strain ${\mathrm{ϵ}}_C$ external to its boundary. (F) Vector cone plot of the solution of the matrix displacements calculated analytically from the inclusion eigenstrain ${\mathrm{ϵ}}_T.$ For visualization purposes, the displacement field is interpolated onto random locations as vector cones color-coded by magnitude $\left|\mathit{u}\right|.$ (G and H) $x_2$ − $x_3$ (G) and $x_1$ − $x_3$ (H) plane views of the solution in F.

Fig. 2.

Cell-induced hydrogel deformations and calculation of surface displacements. (A, Left) Volume rendering of a DiI-stained neutrophil (red) embedded in a fibrillar type I collagen matrix (cyan) imaged with reflectance confocal microscopy. (A, Right) Confocal micrograph of a neutrophil (red) migrating in a 3D collagen matrix containing randomly distributed 0.5-μm fluorescent microspheres (green) used to measure cell-induced displacement fields. (Scale bar, 10 μm.) (B) Vector cone plot of the induced 3D displacement field measured by FIDVC surrounding a chemotactic neutrophil (gray) with volume V and surface $\partial V.$ (Scale bar, 5 μm.) (C) Schematic outlining the interpolation of the 3D displacement field to $\partial V$ to produce the surface displacements $\mathit{u}\mathrm{.}$ The surface normal $\mathit{n}$ (green) is used to calculate the corresponding normal ${\mathit{u}}_{\perp }$ and tangential ${\mathit{u}}_{\parallel }$ component of $\mathit{u}$ with respect to the cell.

Fig. S1.

Planar fMLF gradient diffusion system. (A) Schematic of the 3D diffusion system where the fMLF chemoattractant is introduced into the front well (blue) to induce neutrophil (red) chemotaxis against the planar diffusion gradient. The direction of the diffusion gradient is indicated by the shaded fMLF volume (blue to white), which is aligned with the $x_1$ axis of the imaging coordinate system. (B, Left) Volume rendering of a DiI-stained neutrophil (red) embedded in a fibrillar type I collagen matrix (cyan) imaged with reflectance confocal microscopy. (B, Right) Confocal micrograph of a neutrophil (red) migrating in a corresponding 3D collagen matrix containing randomly distributed 0.5-μm fluorescent microspheres (green) used to measure cell-induced matrix displacements. (Scale bar, 10 μm.) (C) Boxplot of neutrophil speed for all time points and tested conditions: K – N (solid white; $n=30$); T – N (solid gray; $n=48$); K – L (hatched white; $n=44$); and T – L (gray hatched gray; $n=57$). Red lines indicate medians; upper and lower boxes indicate upper and lower quartiles; whiskers indicate maximum and minimum values. $P<0.05$ by Mann–Whitney U test across corresponding chemokinesis and chemotaxis groups (*) and naïve and LPS-activated groups $(\#).$ (D) Plot showing least-squares fit (solid) using Eq. S1 to the experimentally measured (square) normalized rhodamine dye intensity $I/I_0$ at increasing depths (green, blue, red, and gray) in the collagen matrix. (Inset) Confocal micrograph of the intensity of the rhodamine dye diffusion front at the beginning of the experiment. (Scale bar, 200 μm.) (E and F) Normalized chemoattractant concentration $\widehat{C}=c/c_0$ (E) and concentration gradient (F) in the diffusion system as a function of normalized chamber length $L/L_0,$ as predicted by Eq. S1.

Fig. S2.

Measurement of surface displacement gradients exerted by a migrating neutrophil. (A) Contour plot of the displacement gradient Frobenius (L_2,2) norm $\Vert \mathbf{\nabla }\mathit{u}\Vert$ on the surface $\partial V$, color-coded by magnitude. (B and C) Contour plot of the skew norm $\Vert \mathrm{skew}\left(\mathbf{\nabla }\mathit{u}\right)\Vert$ (rotational) and symmetric norm $\Vert \mathrm{sym}\left(\mathbf{\nabla }\mathit{u}\right)\Vert$) (stretch) parts of $\mathbf{\nabla }\mathit{u}$, color-coded by magnitude. (Scale bar, 5 μm.) (D–F) Mollweide projection of corresponding surface data in A–C onto a 2D plane. Data interior to the solid circle (white) are located on the front hemisphere ($x_1>$ cell centroid on $\partial V$), whereas data outside lie on the back hemisphere ($x_1<$ cell centroid on $\partial V$) of the cell. (G and H) Boxplot of the maximum value of $\mathbf{\nabla }\mathit{u}$ on $\partial V$ (G) and mean displacement gradient $\Vert \langle \mathbf{\nabla }\mathit{u}\rangle \Vert$ (H) for all time points and tested conditions: K – N (solid white; $n=30$); T – N (solid gray; $n=48$); K – L (hatched white; $n=44$); and T – L (gray hatched gray; $n=57$). Red lines indicate medians; upper and lower boxes indicate upper and lower quartiles; whiskers indicate maximum and minimum values. (I) Boxplot of the three mean principal strain values normalized by its associated Frobenius norm $\langle {\widehat{E}}_i\rangle ,$ calculated from the mean Lagrangian strain tensor $\langle \mathbf{E}\rangle .$ Negative values of $\langle {\widehat{E}}_i\rangle$ signify mean contraction, whereas positive values of $\langle {\widehat{E}}_i\rangle$ are mean expansions of the cell along the associated principal axis. $P<0.05$ by Mann–Whitney U test across corresponding chemokinesis and chemotaxis groups (*), naïve and LPS activated groups (#), and ${\widehat{E}}_1$, ${\widehat{E}}_2$, ${\widehat{E}}_3$ (+).

Fig. 3.

MDM for neutrophils calculated from the mean deformation gradient tensor $\langle \mathbf{F}\rangle .$ (A) Boxplot of the mean volume change ratio $\langle J\rangle$ of the cell (dotted line, constant volume), defined as the determinant of $\langle \mathbf{F}\rangle$ for all time points of the tested conditions: K – N, chemokinesis and naïve (solid white; $n=30$); T – N, chemotaxis and naïve (solid gray; $n=48$); K – L, chemokinesis and LPS-activated (hatched white; $n=44$); and T – L, chemotaxis and LPS-activated (hatched gray; $n=57$). Red lines indicate medians; upper and lower boxes indicate upper and lower quartiles; whiskers indicate maximum and minimum values. (B) Boxplot of the three mean contractility values $\langle {\lambda }_i\rangle$ calculated from the right Cauchy–Green stretch tensor $\langle \mathbf{U}\rangle .$ Values of $\langle {\lambda }_i\rangle$ below 1 signify mean contraction, whereas values of $\langle {\lambda }_i\rangle$ above 1 are mean expansion of the cell along the associated principal axis. (C) Schematic illustrating minimum $\langle {\lambda }_1\rangle$ and maximum $\langle {\lambda }_3\rangle$ principal stretches exerted by the cell with corresponding eigenvectors $\langle {\mathit{N}}_1\rangle$ and $\langle {\mathit{N}}_3\rangle .$ (D and E) Schematic illustrating mean cell rotation angle $\langle \theta \rangle$ (D) and corresponding boxplot for all time points and conditions (E). (F) Integrated $\langle \theta \rangle$ for each cell over its migration time t for all tested conditions: K – N (solid red; $n=4$ cells); T – N (solid blue; $n=3$ cells); K – L (dashed magenta; $n=5$ cells); and T – L (dashed cyan; $n=3$ cells). The middle line indicates the mean; the shaded area indicates the SD of integrals evaluated at the specified time point. $P<0.05$ by Mann–Whitney U test across corresponding chemokinesis and chemotaxis groups (*), naïve and LPS-activated groups (#), and across $\langle {\lambda }_i\rangle$ (+) in B.

Fig. 4.

Measurement of 3D surface displacements exerted by a chemotactic neutrophil and its associated homolographic 2D projections. (A) Vector cone plot of the 3D displacement field interpolated onto the surface $\partial V$ (gray) of the neutrophil, color-coded by magnitude $\left|\mathit{u}\right|.$ (B) Elevation data of the Earth projected onto $\partial V$ to illustrate the spatial position of surface data. (C) Contour plot of the normal component of $\mathit{u}$ $\mathrm{(}{\mathit{u}}_{\perp })$ with respect to $\partial V,$ color-coded by direction and magnitude (red is outward and blue is inward normal surface displacement). (D) Streamline plot of the tangential displacement component ${\mathit{u}}_{\parallel }$ along $\partial V,$ color-coded by magnitude with arrowheads pointing along the vector field. (Scale bar, 5 μm.) (E) Mollweide mapping of the spherical projection of Earth’s elevation data in B to allow for user-friendly visualization of all 3D data onto a plane. Dashed grid lines represent spacing of 45° along latitude and longitude lines. Data interior to the solid white circle are located on the front hemisphere ($x_1$ > cell centroid on $\partial V$), whereas data outside lie on the back hemisphere $(x_1$ < cell centroid on $\partial V)$ of the cell. (F and G) Mapped contour plot of ${\mathit{u}}_{\perp }$ in C and streamline plot of ${\mathit{u}}_{\parallel }$ in D using the Mollweide projection in E. (H and I) Magnified view of F and G highlighting sink-like (green triangle) (Top), source-like (green star) (Middle), and saddle-like (green plus symbol) (Bottom) features of ${\mathit{u}}_{\parallel }$ and ${\mathit{u}}_{\perp }.$

Fig. S3.

Calculation of surface displacements using the analytical solution of Eshelby outside an inclusion undergoing an eigenstrain ${\mathbf{ϵ}}_T.$ (A) Cone plot of the displacement field interpolated onto the surface $\partial V$) (gray) of the inclusion, color-coded by magnitude $\left|\mathit{u}\right|.$ (B) Contour plot of the normal displacement component ${\mathit{u}}_{\perp }$ with respect to $\partial V,$ color-coded by direction and magnitude (red is outward and blue is inward normal surface displacement). (C) Streamline plot of the tangential displacement component ${\mathit{u}}_{\parallel }$ along $\partial V,$ color-coded by magnitude and arrowheads are pointed along the vector field. (D) Contour plot of the displacement gradient Frobenius (L_2,2) norm $\Vert \mathbf{\nabla }\mathit{u}\Vert$ on $\partial V,$ color-coded by magnitude. (E–G) Mollweide projection of corresponding surface data in B–D onto a 2D plane. Data interior to the solid circle (white) are located on the front hemisphere ($x_1>$ inclusion centroid on $\partial V$), whereas data outside lie on the back hemisphere $(x_1<$ inclusion centroid on $\partial V)$ of the inclusion.

Fig. S5.

Comparison of the biophysical metrics provided by the MDM approach and TFM.

Fig. S4.

Calculation of mean deformation error associated with estimating the cell boundary. (A) Actual cell boundary $\partial V,$ minimum enclosing ellipsoid $\partial V_E,$ and minimum enclosing sphere $\partial V_S.$ (B) Error e in the norm of the mean deformation gradient tensor averaged over all cells and conditions $(n=179)$ of the cell boundary and minimum circumscribed ellipsoid and sphere. Horizontal lines indicate the 5th, 25th, 50th (dotted), 75th, and 95th percentiles. The notch indicates SD.