Role of fluctuations in membrane models: thermal versus non-thermal

We study the comparative importance of thermal to non-thermal fluctuations for membrane-based models in the linear regime. Our results, both in 1+1 and 2+1 dimensions, suggest that non-thermal fluctuations dominate thermal ones only when the relaxation time $\tau$ is large. For moderate to small values of $\tau$, the dynamics is defined by a competition between these two forces. The results are expected to act as a quantitative benchmark for biological modelling in systems involving cytoskeletal and other non-thermal fluctuations.

💡 Research Summary

The paper investigates the relative importance of thermal (equilibrium) and non‑thermal (active) fluctuations in membrane models within the linear regime, focusing on both 1+1‑dimensional strings and 2+1‑dimensional membranes. Starting from a Helfrich‑type free energy that includes surface tension (T) and bending rigidity (B), the overdamped dynamics is written as γ∂ₜZ = −B∇⁴Z + T∇²Z + η_th + η_nth. The thermal noise η_th obeys the fluctuation‑dissipation theorem and is taken as white Gaussian noise with strength D₀, while the non‑thermal noise η_nth is modeled as an exponentially correlated stochastic force with correlation ⟨η_nth(x,t)η_nth(x′,t′)⟩ = F₀ e^{−|t−t′|/τ}δ(x−x′). The amplitude F₀ scales as 1/τ to satisfy normalization, and τ is the characteristic relaxation time of the active process (e.g., cytoskeletal activity).

By Fourier transforming the equations, the authors obtain the mode‑dependent decay rate α(k)=B k⁴+T k²/γ. For the non‑thermal case the two‑point correlation function splits into two contributions: a τ‑dependent term proportional to τ⁻¹ exp(−|t−t′|/τ) and a term proportional to α(k) exp(−α(k)|t−t′|). The thermal contribution is the familiar D₀/(γ²) exp(−α(k)|t−t′|)/α(k). In the limit τ→0 the non‑thermal term reduces to the thermal form, whereas for τ→∞ the τ‑dependent term remains finite and dominates over the thermal part.

The same analysis is carried out for the 2+1‑dimensional membrane, where angular integration introduces a Bessel function J₀(kX). In the X→0 limit (same spatial point) the Bessel factor approaches unity, and the structure of the correlation functions mirrors the 1+1‑dimensional case. Thus the qualitative conclusion is unchanged: large τ leads to dominance of active fluctuations, while small τ yields a competition between the two noise sources.

To quantify the competition the authors define a ratio R(τ)=⟨Z_nth²⟩/(D₀γ⁻²J), where J is an integral over k‑space of the thermal propagator. Using realistic parameter values (B≈11.8 k_BT, T≈5650 k_BT μm⁻², γ≈4.7×10⁶ k_BT s μm⁻⁴, D₀≈2 k_BT M), they compute R(τ) numerically. The results show that for τ of the order of a few seconds (the range observed in experiments on cell membranes and red blood cells) R exceeds unity, indicating that active, non‑thermal fluctuations dominate the membrane dynamics. For τ in the millisecond or sub‑millisecond range, the two contributions are comparable, leading to a mixed regime where neither can be neglected.

The paper concludes that the relaxation time τ is the key parameter governing whether thermal noise or active noise sets the scale of membrane fluctuations. This provides a quantitative benchmark for biophysical modeling: when τ is large, active processes such as cytoskeletal remodeling must be explicitly included; when τ is small, a purely thermal description may suffice. The authors note that future work will address nonlinear extensions and spatially correlated active noise, which could further refine the theoretical framework.