Cellular Mechanics

Actin Filaments (Microfilaments)

F-actin is a semiflexible polymer with persistence length $l_p \approx 17\,\mu$m and diameter 7 nm. Its bending modulus is:

Actin forms the cell cortex (~100-500 nm thick layer beneath the membrane), stress fibers in adherent cells, and the leading edge lamellipodium during migration. The elastic modulus of a crosslinked actin network depends on filament concentration$c$ as $G \sim c^{5/2}$ (in the semiflexible regime).

Microtubules

Microtubules are hollow cylinders of tubulin dimers with diameter 25 nm and persistence length $l_p \approx 5.2$ mm — making them essentially rigid on cellular length scales. Their bending stiffness is:

Microtubules bear compressive loads in cells (unlike actin and intermediate filaments, which primarily bear tension). They buckle at forces of order$F_c = \pi^2 \kappa/L^2 \sim 1$ pN for a 10 $\mu$m filament.

Intermediate Filaments

Intermediate filaments (vimentin, keratin, lamin) have persistence length$l_p \approx 0.3\text{-}1\,\mu$m and are much more extensible than actin or microtubules, sustaining strains of 250-350% before failure. They provide mechanical resilience to cells and tissues, acting as a safety net against large deformations.

Why Cells Are Viscoelastic

Cells are neither purely elastic (solid) nor purely viscous (fluid). They exhibit viscoelastic behavior: their mechanical response depends on the timescale of deformation. The key features are:

• • Creep: Under constant stress, strain increases over time
• • Stress relaxation: Under constant strain, stress decreases over time
• • Hysteresis: Loading and unloading curves differ
• • Frequency dependence: Stiffness increases with deformation rate

The linear viscoelastic response is fully characterized by either the relaxation modulus $G(t)$ or the creep compliance $J(t)$:

Complex Modulus

For oscillatory deformation $\varepsilon(t) = \varepsilon_0 e^{i\omega t}$, the stress response is $\sigma(t) = G^*(\omega)\varepsilon(t)$ where:

$G'(\omega)$ is the storage modulus(elastic energy stored per cycle) and $G''(\omega)$ is the loss modulus (energy dissipated per cycle). The loss tangent $\tan\delta = G''/G'$ quantifies the relative importance of dissipation. For cells, typically $\tan\delta \sim 0.2\text{-}0.5$.

Model Construction

The Kelvin-Voigt model places a spring (modulus $E$) and dashpot (viscosity $\eta$) in parallel. Since both elements share the same strain:

The total stress is the sum of spring and dashpot stresses:

Creep Response Derivation

Apply constant stress $\sigma_0$ at $t = 0$. The ODE becomes:

This is a first-order linear ODE. With initial condition $\varepsilon(0) = 0$(no instantaneous deformation because the dashpot cannot respond instantly):

The creep compliance is:

The material creeps toward a finite equilibrium strain $\varepsilon_\infty = \sigma_0/E$, which is solid-like behavior. The retardation time $\tau = \eta/E$ characterizes how quickly equilibrium is reached.

Model Construction

The Maxwell model places a spring and dashpot in series. Both elements carry the same stress, and the total strain is additive:

Differentiating and using $\dot{\varepsilon}_{\text{spring}} = \dot{\sigma}/E$ and$\dot{\varepsilon}_{\text{dashpot}} = \sigma/\eta$:

Stress Relaxation Derivation

Apply constant strain $\varepsilon_0$ at $t = 0$ (so $\dot{\varepsilon} = 0$for $t > 0$). The ODE becomes:

With initial condition $\sigma(0) = E\varepsilon_0$ (instantaneous elastic response):

The relaxation modulus is:

The stress decays to zero, which is fluid-like behavior. Under constant stress (creep), the Maxwell model gives $J(t) = 1/E + t/\eta$ — an unbounded linear increase, characteristic of viscous flow.

Complex Moduli

For oscillatory deformation:

The Maxwell model is liquid-like at low frequency ($G' \to 0$ as $\omega \to 0$) and solid-like at high frequency. The Kelvin-Voigt model is always solid-like ($G' = E$ constant) with increasing dissipation at high frequency.

Standard Linear Solid (Zener Model)

Neither the Maxwell nor Kelvin-Voigt model alone captures real cell behavior. The standard linear solid combines a Maxwell element in parallel with a spring, giving both creep and relaxation:

where $E_0$ is the instantaneous modulus and $E_\infty$ is the equilibrium (relaxed) modulus. The creep compliance is:

where $\tau' = \tau E_0/E_\infty$ is the retardation time (different from the relaxation time). This model gives a solid-like equilibrium ($J(\infty) = 1/E_\infty$) with an instantaneous elastic response ($J(0) = 1/E_0$).

Power-Law Rheology of Cells

Many cell types exhibit power-law creep rather than exponential relaxation:

with $\beta \approx 0.1\text{-}0.5$ for most cell types. This implies a continuous distribution of relaxation times, consistent with the soft glassy rheology (SGR) model. The exponent $\beta$ relates to the "noise temperature"$x = 1 + \beta$:

• • $x \to 1$ ($\beta \to 0$): glass-like, very slow relaxation
• • $x = 1.5$ ($\beta = 0.5$): approaching fluid-like behavior

Active vs Passive Materials

Living cells are active materials: they continuously consume energy (ATP) to generate forces and drive motion. This fundamentally changes their mechanical behavior compared to passive (equilibrium) materials:

• • Violation of FDT: The fluctuation-dissipation theorem breaks down; fluctuations can exceed thermal predictions
• • Active stress: Molecular motors generate contractile stresses ($\sigma_{\text{active}} \sim 100\text{-}1000$ Pa in cells)
• • Self-organization: Spontaneous pattern formation, cortical flows, cell division

The active stress tensor in the actin cortex can be written as:

where $\zeta$ is the activity coefficient, $\Delta\mu$ is the chemical potential difference driving motor activity, and $\mathbf{n}$ is the local filament orientation.

Active Brownian Particles

The simplest active matter model is the active Brownian particle (ABP):

The mean-square displacement transitions from ballistic to diffusive:

where $\tau_p = 1/D_R$ is the persistence time. At long times, the effective diffusion coefficient is$D_{\text{eff}} = D_T + v_0^2/(2D_R)$ in 2D, which can be orders of magnitude larger than thermal diffusion alone.

The Persistent Random Walk Model

Cell migration on 2D surfaces follows a persistent random walk: cells move in a roughly straight line for a persistence time $\tau_p$ (typically 10-60 minutes) before randomly reorienting.

The velocity autocorrelation decays exponentially:

The MSD has two limits:

• • Short times ($t \ll \tau_p$): $\langle r^2 \rangle \approx v_0^2 t^2$ (ballistic)
• • Long times ($t \gg \tau_p$): $\langle r^2 \rangle \approx 4D_{\text{eff}} t$ (diffusive)

Force Generation During Migration

Cell migration involves a cyclic process:

• • Protrusion: Actin polymerization at the leading edge generates force $F \sim k_BT/(l \cdot \delta) \sim 1\text{-}10$ pN per filament
• • Adhesion: New focal adhesions form at the front
• • Contraction: Myosin II generates tension in stress fibers (~nN forces)
• • Retraction: Rear adhesions release, cell body translocates

The polymerization force at the leading edge can be estimated from the Brownian ratchet model:

where $\delta \approx 2.7$ nm is the actin monomer size and$[G]$ is the free monomer concentration. Typical values give$F_{\text{poly}} \approx 1\text{-}5$ pN per filament, with ~100 filaments per $\mu$m of leading edge, generating ~100 pN/$\mu$m of force.