constitutive relationships in biology

— The cytoplasm may behave like a viscous fluid over time — The nucleus resists external forces elastically — The actin cortex or the membrane may combine elastic and viscous effects — Some tissues stiffen when stretched : nonlinear behavior.

linear elastic material (hooke's law)

An elastic material returns to its initial shape after deformation. The stress–strain relationship is linear: σ = E ε where E is Young’s modulus (stiffness). This is a reversible behavior.

linear elasticity applies when

deformations are small and proportional to the applied stresses

during a simple tensile test, the stress of a spring is defined as?

σ = F/S0 with S0 the initial cross-sectional area.

biological examples of linear elasticity

— Cytoskeleton under small deformation — Stretched cell membrane

poisson's ratio

When a solid is subjected to tension, it elongates along the direction of the force but contracts in the perpendicular directions. This transverse contraction is characterized by Poisson’s ratio ν, defined as : ∆a/a0= ∆b/b0 = −ν x ∆L/L0 = −νε11

stress-strain curve

During a tensile test, a typical curve is obtained with three phases: — Phase OA (elastic) : stress is proportional to strain, reversible. — Phase AB to C (plastic) : irreversible deformation after exceeding the elastic limit. — Point C (fracture) : material failure.

elastic deformation

— A material is elastic if it returns to its shape after unloading. — For small deformations, the response is linear : σ = Eε — Transverse contraction is characterized by the Poisson’s ratio ν. — Young’s modulus E measures stiffness (slope of the linear phase).

viscous material (newtonian fluid)

A viscous material flows when stress is applied. The stress is proportional to the strain rate : σ = η ε ̇ where η is the viscosity and ε ̇ the shear rate. This is a dissipative behavior (energy loss).

biological examples of viscous material

— Cytoplasm ; — Fluid extracellular medium ; — Nucleoplasm (over long timescales).

perfectly plastic material

A plastic material undergoes irreversible deformation beyond a threshold (yield limit). This model is rare in biology but can be useful in extreme cases.

behavior of perfectly plastic material

— Before the threshold: elastic response ; — Beyond the threshold: permanent deformation.

biological example of perfectly plastic material

— Rupture of an intercellular junction ; — Irreversible nuclear creep under prolonged stress.

elastic vs. viscous vs. plastic materials

— Elastic : σ = Eε – Reversible, rigid (e.g., cell cortex) — Viscous : σ = ηε ̇ – Fluid, dissipative (e.g., cytoplasm) — Plastic : irreversible deformation beyond a threshold (e.g., membrane rupture)

linear viscoelastic laws

using combinations of springs (elastic elements) and dashpots (viscous elements). many biological materials exhibit both elastic and viscous properties

maxwell model -> viscous + elastic (in series)

The constitutive equation of the model is : dσ/dt + σ/τ = E * dε/dt, with τ = η/E This equation links the rate of change of stress and strain. It can be solved for common cases like stress relaxation. If a constant strain ε = ε0 is suddenly applied at t = 0, the solution is : σ(t) = E ε0 e^−t/τ

characteristics of the maxwell model

— Good model for viscoelastic fluids. — Reproduces exponential decay of stress under constant strain. — Characteristic time τ = η/E controls relaxation rate.

characteristics of the kelvin-voigt model

— Creep — No stress relaxation: stress remains constant if strain is imposed. — Good model for elastic and viscous structures in parallel, such as the cell cortex or soft tissues.

biological interpretation of kelvin-voigt model

In an indentation test on a cell (AFM, micro-needle), one would observe a progressive indentation increase, reflecting the combined viscous + elastic behavior of the cortex.

creep

the gradual increase in strain under constant stress

creep time equation

τ = η/E quantifies how quickly deformation evolves toward equilibrium

tension/compression test

stretching or compressing a cell or tissue to extract Young’s modulus.

indentation

pressing a probe (AFM tip or bead) into a cell to measure local stiffness.

micropipette aspiration

suction of a membrane fragment to evaluate its tension and viscoelasticity.

passive microrheology

tracking the Brownian motion of microbeads in the cytoplasm.

active microrheology

applying forces using optical or magnetic tweezers to probe local mechanical properties.

quantities measured by mechanical characterization experiments

— Young’s modulus E : measure of the material’s elasticity (stiffness). — Viscosity η : resistance to flow or slow deformation. — Creep or relaxation time : temporal characteristics of viscoelastic responses.

biological examples that can be measured with mech. characterization experiments

— Tumor cells: often softer than healthy cells, detectable by AFM or magnetic beads. — Embryology: stiffness of embryonic tissues evolves during development, influencing morphogenesis. — Aging or differentiation: cell mechanical properties change with physiological state.

cell mechanical properties like stiffness, viscosity ___

are accessible via well-established experiments (tension, AFM, microrheology). they help distinguish cellular states (normal vs. tumor, differentiated vs undifferentiated) and shed light on biological processes