A Quantitative Model of Cellular Elasticity Based on Tensegrity

Abstract

A tensegrity structure composed of six struts interconnected with 24 elastic cables is used as a quantitative model of the steady-state elastic response of cells, with the struts and cables representing microtubules and actin filaments, respectively. The model is stretched uniaxially and the Young's modulus (E0) is obtained from the initial slope of the stress versus strain curve of an equivalent continuum. It is found that E0 is directly proportional to the pre-existing tension in the cables (or compression in the struts) and inversely proportional to the cable (or strut) length square. This relationship is used to predict the upper and lower bounds of E0 of cells, assuming that the cable tension equals the yield force of actin (approximately 400 pN) for the upper bound, and that the strut compression equals the critical buckling force of microtubules for the lower bound. The cable (or strut) length is determined from the assumption that model dimensions match the diameter of probes used in standard mechanical tests on cells. Predicted values are compared to reported data for the Young's modulus of various cells. If the probe diameter is greater than or equal to 3 microns, these data are closer to the lower bound than to the upper bound. This, in turn, suggests that microtubules of the CSK carry initial compression that exceeds their critical buckling force (order of 10(0)-10(1) pN), but is much smaller than the yield force of actin. If the probe diameter is less than or equal to 2 microns, experimental data fall outside the region defined by the upper and lower bounds.

Tensegrity I. Cell structure and hierarchical systems biology

“…Largely through the work of Stamenovic and co-workers, this oversimplified micromechanical model continues to be progressively modified and strengthened over time (Coughlin and Stamenovic, 1997;Coughlin and Stamenovic, 1998;Stamenovic and Coughlin, 1999;Stamenovic and Coughlin, 2000;Stamenovic and Ingber, 2002). A more recent formulation of the model includes, for example, semiflexible struts analogous to microtubules, rather than rigid compression struts, and incorporates values for critical features of the individual cytoskeletal filaments (e.g.…”

“…This more refined model is qualitatively and quantitatively superior to that containing rigid struts. Another formulation of the tensegrity model includes intermediate filaments as tension cables that link the cytoskeletal lattice and surface membrane to the cell center (Wang and Stamenovic, 2000 Moreover, all of these tensegrity models yield elastic moduli (stiffness) that are quantitatively similar to those of cultured adherent cells (Stamenovic and Coughlin, 1999;Stamenovic and Coughlin, 2000). Importantly, although models of the cytoskeleton that incorporate only tensile elements (i.e.…”

Tensegrity-based mechanosensing from macro to micro

“…), we developed mathematical models of the cell based on tensegrity starting from first mechanistic principles that provide even more powerful a priori predictions relating to both cell static and dynamic mechanical behavior, which have now been confirmed experimentally in various cell types (Stamenovic et al, 1996;Coughlin and Stamenovic, 1998;Stamenovic and Coughlin, 1999;Stamenovic and Coughlin, 2000;Wang and Stamenovic, 2000;Stamenovic, 2005). Behaviors exhibited by living cells that can be predicted by the tensegrity model include: 1) linear relation between stiffness and applied stress (Wang et al, 1993;Wang and Ingber, 1994), 2) cell mechanics depends on prestress (Lee et al, 1998;Wang and Ingber, 1994), 3) linear relation between stiffness and prestress (Wang et al, 2001;Wang et al, 2002), 4) hysteresivity is independent of prestress (Maksym et al, 2000;Wang et al, 2001); 5) quantitative predictions of cellular elasticity (Stamenovic and Coughlin, 2000), 6) predictions of dynamic mechanical behavior (Sultan et al, 2004), and 7) mechanical contribution of intermediate filaments to cell mechanics.…”

Cell prestress. I. Stiffness and prestress are closely associated in adherent contractile cells

“…Changes of orientation and spacing of the discrete elements represent the central mechanism by which restoring forces arise in stresssupported structures (45,46).…”