Frontier physics

Biophysics

Biophysics uses physical laws and quantitative models to investigate structure, dynamics, and function in living systems across scales.

The physics–biology interface is more than applying ready-made formulas: living systems are open, thermally noisy, multicomponent, and often far from equilibrium. Choosing state variables, scales, and measurements is part of the solution, as is identifying where a simplified model fails.

D = k_B T\,\mu,\qquad \langle\Delta r^2(t)\rangle=2dDt

Definition: Brownian diffusion and thermal scale

For a spherical Brownian particle of radius a in a fluid of viscosity η, D=kBT/(6πηa) under continuum and equilibrium assumptions (Stokes–Einstein). The mean-square displacement in d dimensions is 2dDt. Crowded, active cellular environments can violate this model.

Track a Brownian particle and an ensemble of paths while varying temperature and particle size. Random paths are finite samples, not predictions for a specific cell.

From a single particle to ensembles

A single molecule reveals a noisy path; ensemble quantities such as ⟨Δr²⟩ provide a stable estimate of D. In biological media, tracking many molecules, accounting for temporal and spatial resolution, and comparing displacement distributions can distinguish ordinary diffusion from anomalous diffusion or active transport.

Example: Estimate diffusion

In 2D, a tracer has ⟨Δr²⟩=4 μm² after 1 s. Estimate D assuming ordinary Brownian motion.

Solution

For d=2, ⟨Δr²⟩=4Dt, giving D=1 μm²/s. Real data require localization-error correction and a check of linearity in time.

At thermal equilibrium, Brownian motion can be modeled by the Langevin equation $\gamma\dot{x}=F+\xi(t)$ , where the random force correlations are linked to friction by fluctuation–dissipation. Living cells consume ATP and operate out of equilibrium, so microtubules, molecular motors, and membranes can produce directed flows rather than passive diffusion. To distinguish mechanisms, researchers compare displacement distributions and temporal correlations, and measure responses to changed loads or inhibitors. Models must preserve the relevant scale: a tracer, its molecular target, and cellular structures need not move alike. Causal interpretation therefore combines tracking with interventions.

Analysis must distinguish random thermal fluctuations from active fluctuations driven by energy consumption. At equilibrium, fluctuation–response relations connect noise amplitude to linear response; violations in living matter can indicate activity, but measurement artifacts and heterogeneity must first be excluded. Tracking at multiple lags and controlling temperature and load help test the interpretation.

Quick check

For 2D Brownian diffusion, what is ⟨Δr²⟩?

What assumptions underlie the Stokes–Einstein relation?

References

Rob Phillips et al. (2012). Physical Biology of the Cell