Cell reorientation on a cyclically strained substrate

Abstract

Cyclic strain avoidance, the phenomenon of cell and cytoskeleton alignment perpendicular to the direction of cyclic strain of the underlying 2D substrate, is an important characteristic of the adherent cell organization. This alignment has typically been attributed to the stress-fiber reorganization although observations clearly show that stress-fiber reorganization under cyclic loading is closely coupled to cell morphology and reorientation of the cells. Here, we develop a statistical mechanics framework that couples the cytoskeletal stress-fiber organization with cell morphology under imposed cyclic straining and make quantitative comparisons with observations. The framework accurately predicts that cyclic strain avoidance stems primarily from cell reorientation away from the cyclic straining rather than cytoskeletal reorganization within the cell. The reorientation of the cell is a consequence of the cell lowering its free energy by largely avoiding the imposed cyclic straining. Furthermore, we investigate the kinetics of the cyclic strain avoidance mechanism and demonstrate that it emerges primarily due to the rigid body rotation of the cell rather than via a trajectory involving cell straining. Our results provide clear physical insights into the coupled dynamics of cell morphology and stress-fibers, which ultimately leads to cellular organization in cyclically strained tissues.

Keywords: cyclic strain avoidance; fluctuations; homeostasis; stress-fiber alignment.

Fig. 1.

(a) Immunofluorescence images showing actin distributions within U2OS cells subjected to a uniaxial cyclic strain with a stretch amplitude at frequencies f = 0.01 and 1 Hz for around 12 hours. Reproduced from (14). Scale bar 25 μm. (b) Sketch showing a single cell adhered to a substrate subjected to a biaxial cyclic strain in the plane. The cell exchanges high energy nutrients with the nutrient bath. A morphological microstate is defined by the mapping of material points on the cell membrane with material points on the substrate. (c) The 2D approximation of the cells. The components of the cell that are modeled explicitly include an elastic nucleus and cytoplasm as well as the contractile SFs in their polymerized state along with the the unbound components that are free to diffuse within the cytoplasm. (d) The elliptical approximation of the cell as a spatially uniform ellipse on the cyclically loaded substrate. The principal axes 2a₁ and 2a₂, respectively, of the ellipse are labeled along with the definition of the orientation θ of the cell. (e) Sketches to illustrate the orientation δ of an SF relative to the x₁ imposed cyclic strain direction and the orientation ϕ of the SF relative to the major axis of the ellipse.

Fig. 2.

(a) Predictions of the angular distributions of SF concentrations as parameterized by , where δ is the orientation of the SFs with respect to -direction of cyclic stretching. Results are shown for cyclic loading (r = 0) with ε_amp = 0.1 and f = 0.5, 1 Hz, together with the reference case of no imposed cyclic loading (i.e. f = 0). (b) Comparison of the predicted and measured (14) CVs, defined by (12), for selected frequencies f and ε_amp = 0.1. (c) The angular distributions of the SF concentrations as parameterized by within cell with ϕ denoting the orientation of the SFs with respect to the major axis of the ellipse. Predictions are shown for the three straining cases in (a) with the corresponding circular histograms shown as insets. (d) The angular distributions and circular histograms for SFs parameterized by for a circular cell of radius R₀ and subjected to the three straining cases in (a).

Fig. 3.

Probability density functions of the three key morphological observables for loading (r = 0) with a cyclic strain ε_amp = 0.1 and f = 0.5, 1 Hz along with the case of no cyclic strain (f = 0 Hz). (a) Predictions of the cell orientation p(θ) along with comparisons with measurements of Wang et al. (39) for endothelial cells subjected to uniaxial cyclic strain with f = 1 Hz. Corresponding predictions of the (b) normalized cell area and (c) cell aspect ratio p(A_s).

Fig. 4.

Predictions of the normalized free-energy landscapes using axes of the normalized semi-major and semi-minor axes a₁/R₀ and a₂/R₀, respectively, of the ellipse. These landscapes are shown for cyclic strain (r = 0) with ε_amp = 0.1 and f = 1 Hz for cells oriented at (a) θ = 0^○ and 90^○, (b) θ = 45^○ and 135^○, and (c) the reference case of no imposed cyclic strain (i.e., f = 0 Hz). In (c), we only show the landscape for a₁ ≥ a₂, where a₁ is the semi-major axis and there is no θ dependence of the free-energy landscape.

Fig. 5.

Temporal evolution of the probability density functions of (a) the normalized cell area , (b) cell aspect ration p(A_s) and (c) cell orientation p(θ) for cells subjected to cyclic strain (r = 0) with ε_amp = 0.1 and f = 1 Hz. Cell are seeded from suspension onto the cyclically strained substrates at normalized time .

Fig. 6.

(a) Sketches for the two mechanisms via which the cells could orient away from the cyclic straining direction illustrated here for uniaxial cyclic straining (ε₂(t)/ε₁(t) = 0). In the strain mode, cells stretch with negligible cell rotation, while in the rotation mode, cells rotate with negligible morphological changes. The sketch shows the reorientation of the cell by 90^○. To illustrate kinetics of the two processes, we mark a material line (solid black line) corresponding to the minor axis of the initial cell morphology and follow temporal evolution of this material line. In the strain mode, this line does not rotate but stretches to become the major axis, while in the rotation mode, the line rotates by 90^○ but remains the minor axis. (b) Observations of these two modes in fibroblasts seeded on cyclically loaded substrates. Reproduced from (32).

Fig. 7.

Temporal evolution of the probability density function of cell rotation θ_r for cells subjected to cyclic strain (r = 0) with ε_amp = 0.1 and f = 1 Hz. Cells are first allowed to equilibrate on the substrate in the absence of cyclic straining with loading commenced at for cells initially oriented at θ = θ₀. Results are shown for (a) θ₀ = 0^○, (c) θ₀ = 45^○, and (d) θ₀ = 90^○. In (b), we show a comparison between the probability density functions of the cell orientation and the auxiliary rotation at time for the θ₀ = 0^○ case. Note that for the p(θ_r) distributions, we show a range of θ_r in each case such that the integral of p(θ_r) over the range is at least equal to 0.9. Note that the probability distributions are Dirac delta functions at , which, in turn, implies that the modes of probability distributions are large at and hence been cut for clarity.

Fig. 8.

Comparison between measurements and predictions of the temporal evolution of cell orientation with two choices of the damping co-efficient γ/|H_s| = 500 s and 1,000 s. (a) Order parameter S = 〈cos(2θ)〉 is compared with measurements of Jungbauer et al. (41). Cells are randomly oriented at t = 0 and align perpendicular to the strain direction at large times. (b) Evolution of the orientation of a single cell oriented at θ = 0^○ and ≈20^○ at t = 0 in the predictions and measurements (32), respectively. The interquartile of the predictions over the 1,000 Langevin trajectories are indicated in (b) for γ/|H_s| = 500 s.