How cells feel: stochastic model for a molecular mechanosensor

Understanding mechanosensitivity, i.e. how cells sense the stiffness of their environment is very important, yet there is a fundamental difficulty in understanding its mechanism: to measure an elastic modulus one requires two points of application of force - a measuring and a reference point. The cell in contact with substrate has only one (adhesion) point to work with, and thus a new method of measurement needs to be invented. The aim of this theoretical work is to develop a self-consistent physical model for mechanosensitivity, a process by which a cell detects the mechanical stiffness of its environment (e.g. a substrate it is attached to via adhesion points) and generates an appropriate chemical signaling to remodel itself in response to this environment. The model uses the molecular mechanosensing complex of latent TGF-$\beta$ attached to the adhesion point as the biomarker. We show that the underlying Brownian motion in the substrate is the reference element in the measuring process. The model produces the closed expression for the rate of release of active TGF-$\beta$, which depends on the substrate stiffness and the pulling force coming from the cell in a subtle and non-trivial way. It is consistent with basic experimental data showing an increase in signal for stiffer substrates and higher pulling forces. In addition, we find that for each cell there is a range of stiffness where a homeostatic configuration of the cell can be achieved, outside of which the cell either relaxes its cytoskeletal forces and detaches from the very weak substrate, or generates an increasingly strong pulling force through stress fibers with a positive feedback loop on very stiff substrates. In this way, the theory offers the underlying mechanism for the myofibroblast conversion in wound healing and smooth muscle cell dysfunction in cardiac disease

Mechanosensitivity of the 2nd Kind: TGF-β Mechanism of Cell Sensing the Substrate Stiffness.

Cells can sense forces applied to them, but also the stiffness of their environment. These are two different phenomena, and here we investigate the mechanosensitivity of the 2nd kind: how the cell can measure an elastic modulus at a single point of adhesion-and how the cell can receive and interpret the chemical signal released from the sensor. Our model uses the example of large latent complex of TGF-β as a sensor. Stochastic theory gives the rate of breaking of latent complex, which initiates the signaling feedback loop after the active TGF-β release and leads to a change of cell phenotype driven by the α-smooth muscle actin. We investigate the dynamic and steady-state behaviors of the model, comparing them with experiments. In particular, we analyse the timescale of approach to the steady state, the stability of the non-linear dynamical system, and how the steady-state concentrations of the key markers vary depending on the elasticity of the substrate. We discover a crossover region for values of substrate elasticity closely corresponding to that of the fibroblast to myofibroblast transition. We suggest that the cell could actively vary the parameters of its dynamic feedback loop to 'choose' the position of the transition region and the range of substrate elasticity that it can detect. In this way, the theory offers the unifying mechanism for a variety of phenomena, such as the myofibroblast conversion in fibrosis of wounds and lungs and smooth muscle cell dysfunction in cardiac disease.This work was supported by the EPSRC Critical Mass grant for Theoretical Condensed Matter, the Sims Scholarship, and the Cambridge Trusts, and the University of Sydney.This is the final version of the article. It first appeared from PLOS via http://dx.doi.org/10.1371/journal.pone.013995

A diagram of a contracted myofibroblast and less contracted fibroblast, reprinted by permission from Macmillan Publishers Ltd: Nature [21], copyright 2002.

<p>The large increse in <i>α</i>-SMA expression is evident (as is widely reported increase in active TGF-<i>β</i> on stiff substrates)—but our conclusion about <i>λ</i> (linked to the other two markers) suggests that there are only large focal adhesions, and no/few individual LLC on the cell surface on stiff substrates.</p

Constrained parameters.

<p>Constrained parameters.</p

Plot of km(f˜,κ˜) (in real units of [s⁻¹]) for three different values of scaled substrate stiffness κ˜.

<p>The red cross represents the value of the dimensionless scaled force <math><mrow><msub><mi>f</mi><mo>˜</mo>eq</msub></mrow></math> in cell equilibrium, corresponding to the steady state value of <i>α</i>. In the lowest plot (for stiff substrate), the red cross corresponds to a value of <i>k</i>_<i>m</i> = 0.01 s⁻¹.</p

Experimentally determined values.

<p>Experimentally determined values.</p

A diagram showing the mechanosensor system, from [16].

<p>The large latent complex of TGF-<i>β</i> binds to ECM proteins attached to a substrate, which may be deformable. The other end of LLC is attached to the cytoskeleton via an integrin complex that transmits the pulling force. If released, the free active TGF-<i>β</i> can bind to a set of receptors on the cell surface to initiate the signalling loop discussed in this paper.</p

A graph showing variation of concentration of steady-state values of <i>α</i>, <i>β</i> and <i>λ</i> (in units of particle per cell) against thescaled substrate stiffness κ˜ (expressed in decimal logarithmic units).

<p>Initial conditions were random. The dashed lines show the approximate location of the transition region and the arrow marks the position of myofibroblast transition.</p

Graphs showing the evolution of <i>α</i>, <i>β</i>, and <i>λ</i> on logarithmic scales for.

<p>(a) low value of substrate stiffness <math><mrow><mi>κ</mi><mo>˜</mo><mo>=</mo><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></mrow></math>, and (b) high substrate stiffness <math><mrow><mi>κ</mi><mo>˜</mo><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn></mrow></math>, from different initial conditions. The final steady state reached for (a): <i>λ</i> = 412, <i>β</i> < 1 and <i>α</i> = 1.6, and for (b): <i>λ</i> < 1, <i>β</i> = 108 and <i>α</i> = 2718.</p