mechanical stress

a fundamental quantity for understanding how a cell senses, generates, and responds to forces.

mechanical stresses are NOT

point forces

mechanical stresses ARE

forces distributed over surfaces or within volumes

examples of mechanical stresses in a cell

— An adherent cell exerts forces on its substrate in order to migrate. — The nucleus undergoes stresses when passing through narrow pores. — The mechanical environment influences cell differentiation (mechanotransduction). — The cytoskeleton continuously adapts to balance internal and external forces.

traction force microscopy

measurement of forces exerted by the cell on the substrate.

mechanical stresses of nuclear rupture during migration

compression and shear of the nucleus

mechanical stresses of actin network under tension

transmission of forces along the cytoskeleton

mechanical stresses of confined or rigid environments

adaptation of cellular behavior to mechanical stress

continuous medium

a body assumed to be made of matter distributed homogeneously and without discontinuities, even at very small scales.

hypothesis of continuous medium

It is assumed that at every point in space, physical quantities (such as force, density, or stress) can be defined and vary continuously in space.

continuous medium can be used to describe ______ _______ with use of differential calculus tools

— Spatial variations of stresses or velocities within a cell. — Local deformations induced by internal forces. — Matter fluxes, as in actin gels.

how is a cell described as a continuous medium?

the overall behavior (mechanical, diffusive, or flow-related) is described continuously

example of a cell as a continuous medium

the cell cortex can be modeled as a thin continuous layer with an active distributed tension

limits of continuous approach

— At the molecular scale (nm), this assumption is no longer valid : a discrete (microscopic) description is required. — In highly heterogeneous regions (e.g., filament networks), a continuous average may not accurately represent the physics.

stress

the force exerted by one part of a material on another, across an imaginary surface. it is defined as force per unit area.

internal mechanical interactions are described by what in a continuous medium?

by forces distributed across imaginary internal surfaces

variables of stress

depends on the orientation of the surface may vary from one point to another within the material

biological example of stress - actin cortex of a cell

each portion exerts internal forces on its neighbors. these forces can be compressive (pressure) or shear (contractility)

stress vector

denoted as T⃗ (⃗n), where ∆F⃗ is the force exerted across a small surface ∆S. measured in N/m2 (pascal, Pa)

stress decomposition

T⃗(⃗n) = Tn⃗n + T⃗t

normal stress

Tn = T⃗ · ⃗n (tension if Tn > 0, compression if Tn < 0)

tangential stress

shear T⃗

biological example of tangential stress

lamellipodia of migrating cells generating this stress on their substrate. measured via traction force microscopy

stress tensor in 3D

on a 3D cartesian basis, a tensor is writting in matrix form σij each component: force per unit area in direction >> i which is exerted on a surface whose normal is oriented along >> j

normal stress components

σxx, σyy, σzz

tangential stress (shear) components

σxy, σyz

biological example of shear stress

the actin cytoskeleton of an epithelial cell experiences shear stresses during tissue stretching >> leading to fiber reorganization and polarization of the network along the main stress directions

stress tensors are symmetric, true or false?

true

principal stresses

σi a basis found in which the tensor is diagonal

tensor σ groups all ___ ?

stress components at a point

the stress applied on a surface oriented by n is given by?

T⃗(⃗n) = σ · ⃗n

uniaxial stress

tension/compression in a single direction ex: a cell suspended between two micropillars

plane/biaxial stress

ex: a flat cell migrating on a 2D substrate

hydrostatic stress

isotropic compression σ = −p · I p = internal pressure, all axes experience the same compression ex: osmotic pressure in a vesicle or spherical cell

pure shear

involves only tangential components ex: cytoplasmic fluid under shear or a cell in viscous flow