Cellular Microbiaxial Stretching to Measure a Single-Cell Strain Energy Density Function

Abstract

The stress in a cell due to extracellular mechanical stimulus is determined by its mechanical properties, and the structural organization of many adherent cells suggests that their properties are anisotropic. This anisotropy may significantly influence the cells' mechanotransductive response to complex loads, and has important implications for development of accurate models of tissue biomechanics. Standard methods for measuring cellular mechanics report linear moduli that cannot capture large-deformation anisotropic properties, which in a continuum mechanics framework are best described by a strain energy density function (SED). In tissues, the SED is most robustly measured using biaxial testing. Here, we describe a cellular microbiaxial stretching (CμBS) method that modifies this tissue-scale approach to measure the anisotropic elastic behavior of individual vascular smooth muscle cells (VSMCs) with nativelike cytoarchitecture. Using CμBS, we reveal that VSMCs are highly anisotropic under large deformations. We then characterize a Holzapfel-Gasser-Ogden type SED for individual VSMCs and find that architecture-dependent properties of the cells can be robustly described using a formulation solely based on the organization of their actin cytoskeleton. These results suggest that cellular anisotropy should be considered when developing biomechanical models, and could play an important role in cellular mechano-adaptation.

Fig. 1

Fabrication of substrate and cellular microbiaxial stretching device: (a) schematic representation of substrate fabrication process, (b) schematic representation of the C$\mu$BS device. Inset: cell substrate, (c) grip strain versus measured substrate strain under applied uniaxial grip strain (n = 10), and (d) grip strain versus measured substrate strain under equibiaxial grip strain. Error bars: standard deviation (n = 10).

Fig. 2

Cell stretching and stress measurement: (a) uniaxial and biaxial stretching protocols, (b) protocol to determine substrate displacements used to calculate cell-induced substrate traction force, (c) schematic for calculating first Piola–Kirchhoff stress from measured substrate traction force, and (d) flowchart describing C$\mu$BS microscopy technique and cell stress calculation

Fig. 3

Repeated cell stretching and hysteresis: (a) representative images of a single AR4 VSMC during one cycle of loading and unloading. Left columns: brightfield image of cell. Right columns: cell-induced bead displacement field. (b) Total traction force generated by AR4 cells undergoing loading and unloading cycles during repeated uniaxial-A stretch (n = 9). (c) Total traction force exerted by AR4 cells during cyclic loading over four sequential stretches (n = 5). (d) Normalized cycle-to-cycle total traction force relative to the first stretching cycle for cells exposed to four sequential uniaxial-A stretches. All error bars: standard deviation.

Fig. 4

Mechanical anisotropy in biaxially stretched micropatterned VSMCs. (a), (c), and (e) Representative cell-induced displacement fields for unstretched and 16% strain AR4 VSMCs undergoing (a) uniaxial-A (n = 10), (c) uniaxial-T (n = 13), and (e) equibiaxial stretch (n = 9). (b), (d), and (f) First Piola–Kirchhoff stresses in AR4 cells during (b) uniaxial-A (*, * = significant from 0%, p < 0.05), (d) uniaxial-T (* = significant from 0%, 4%, 8%, p < 0.05), and (f) equibiaxial stretching (* = significant from 0%, p < 0.05). All error bars: standard deviation.

Fig. 5

Cell shape influences mechanical properties. (a) Brightfield images of micropatterned cells with identical adhesive area, but varied aspect ratios (1:1 (AR1), 2:1 (AR2), 4:1 (AR4), 8:1 (AR8)). Scale bar: 20 μm. (b) Measured cell cross-sectional areas from average cell thickness maps. (c) First Piola–Kirchhoff stresses for all active cells during uniaxial-A, uniaxial-T, and equibiaxial stretching. Error bars: standard deviation. Uniaxial-A: AR1 (n = 10), AR2 (n = 10), AR4 (n = 10), AR8 (n = 9). Uniaxial-T: AR1 (n = 10), AR2 (n = 11), AR4 (n = 13), AR8 (n = 9). Equibiaxial: AR1 (n = 10), AR2 (n = 10), AR4 (n = 9), AR8 (n = 10). (* = P_x significant from P_y at same strain, p < 0.05).

Fig. 6

Cytoskeletal structure influences mechanical properties. (a) Representative immunofluorescent images of F-actin and microtubules in representative micropatterned cells for each aspect ratio. Top: dimethyl sulfoxide control. Middle: nocodazole treated. Bottom: cytochalasin D treated. (b) Microtubule filament orientation. Measured from n = 10 cells. (c) Actin filament orientation. Measured from n = 10 cells. (d), (f), and (h) Axial first Piola–Kirchhoff stress ($P_x$) in VSMCs during uniaxial stretch in axial direction (d) control cells (n = 10). (f) Nocodazole treated (n = 10). (h) Cytochalasin D treated (n = 6). (e), (g), and (i) Transverse first Piola–Kirchhoff stress ($P_y$) in VSMCs during uniaxial stretch in the transverse direction (e) control cells (n = 13). (g) Nocodazole treated (n = 4). (i) Cytochalasin D treated (n = 6). All error bars: standard deviation. Note: data staggered about strain values to prevent overlapping data. (d), (f), and (h) Y-axis scaled to maximum of P_x. (e), (g), and (i) Y-axis scaled to maximum of P_y. (*, *, *, * = significant from control at same strain with same AR p < 0.05 for respective aspect ratios).

Fig. 7

Mechanical models using actin organization-based SED recapitulate experimental results. (a) AR4 experimental data used to determine SED parameters and planar biaxial model fit. (b) AR1, AR2, and AR8 experimental data and planar biaxial model prediction. Error bars: standard deviation.

Fig. 8

FE model for validating cell stretching experiment. (a) Quarter symmetry cell and substrate model generated in COMSOL of AR4 cell undergoing prescribed uniaxial-axial, uniaxial-transverse, and equibiaxial stretch. (b) and (c) Comparison of model and experimental cell induced substrate displacements during (b) uniaxial-axial, (c) uniaxial-transverse, and (d) equibiaxial stretch.

Fig. 9

Quarter-symmetry FE model-predicted substrate displacement compared to the mean experimental substrate displacements. All stretched images represent 16% strain cases.