Renormalized mechanics and stochastic thermodynamics of growing vesicles

TL;DR

It is shown that the renormalization of membrane mechanical properties by nonequilibrium driving gives rise to a morphological transition between a weakly driven regime, in which growing vesicles remain quasispherical, and a strongly driven regime, in which vesicles accommodate rapid membrane uptake by developing surface wrinkles.

Abstract

Uncovering the rules governing the nonequilibrium dynamics of the membranes that define biological cells is of central importance to understanding the physics of living systems. We theoretically and computationally investigate the behavior of flexible quasispherical vesicles that exchange membrane constituents, internal volume, and heat with an external reservoir. The excess chemical potential and osmotic pressure difference imposed by the reservoir act as generalized thermodynamic driving forces that modulate vesicle morphology. We show that the renormalization of membrane mechanical properties by nonequilibrium driving gives rise to a morphological transition between a weakly driven regime, in which growing vesicles remain quasispherical, and a strongly driven regime, in which vesicles accommodate rapid membrane uptake by developing surface wrinkles. Additionally, we propose a minimal vesicle growth-shape law, derived using insights from stochastic thermodynamics, that robustly describes vesicle growth dynamics even in strongly driven, far-from-equilibrium regimes.

Figures (12)

• Figure 1: Examples of membrane growth processes in living cells and schematic representation of our Monte Carlo vesicle model. Membrane growth is central to the dynamic structure and function of cells and their internal compartments, and it plays a key role in active processes such as (a) cell division, (b) phagocytosis of pathogens by macrophages, and (c) the shape dynamics of organelles such as mitochondria. (d) Schematic of our Monte Carlo simulations for $d=3$. A fluctuating vesicle is modeled as a quasispherical triangulated mesh that exchanges heat, surface particles (vertices), and volume with a reservoir characterized by temperature $T$, chemical potential $\mu$, and osmotic pressure difference $\Delta p$. For strongly nonequilibrium growth conditions ($\mu \gg \mu_\text{eq}$), we observe a morphological transition between a near-equilibrium regime in which the shapes of growing vesicles remain quasispherical and a far-from-equilibrium regime with persistent wrinkling.
• Figure 2: Phenomenology of vesicle growth dynamics for various imposed excess chemical potentials. (a) Ensembles of growth trajectories for various values of excess chemical potential $\Delta\mu = \mu - \mu_\mathrm{eq}$, with the net influx of surface particles (vertices) ($\Delta N = N(t) - N(0)$) plotted as a function of the number of elapsed Monte Carlo sweeps $t$. Light gray lines correspond to individual trajectories and blue lines correspond to the ensemble average $\langle \Delta N(t) \rangle$. (b) Representative snapshots of vesicle configurations after number of sweeps $t$, for different values of the excess chemical potential $\Delta\mu/k_\mathrm{B}T$. Images are centered on the corresponding values of $(t,\;\Delta\mu)$. The vesicle surface is colored by the normalized radius $R/R_0$, in which $R_0$ is the average radius at $t=0$. For sufficiently large nonequilibrium driving (large $\Delta\mu$), growing vesicles exhibit highly deformed morphologies, with significant out-of-plane undulations. For these simulations, the particle reservoir exchange attempt rate is $p_\mathrm{exchange} = 1$, the imposed osmotic pressure difference is $\Delta p = 0$, and the number of samples is $n_\mathrm{samples}=500$.
• Figure 3: Power spectra and renormalized mechanical properties of growing vesicles at various excess chemical potentials. (a) Mean squared amplitude $\langle |u_\ell|^2 \rangle$ of spherical harmonic modes of degree $\ell$ for three different values of excess chemical potential $\Delta\mu$. Solid curves show fits to Eq. \ref{['eq:spectrum']}, an effective elastic shell model with $\Delta\mu$-dependent renormalized tension $\gamma$ (in units of $k_\mathrm{B}T/\ell_0^2$), Young's modulus $Y$ (in units of $k_\mathrm{B}T/\ell_0^2$), and bending rigidity $\kappa$ (in units of $k_\mathrm{B} T$). Representative vesicle configurations illustrate the transition from quasispherical to highly deformed morphologies with increasing $\Delta\mu$. (b-d) Variation of the renormalized parameters with $\Delta\mu/k_\mathrm{B}T$. The near-equilibrium regime ($\Delta\mu \approx 0$) shows approximately linear behavior characterized by slopes $c_\gamma$, $c_Y$, and $c_\kappa$. Here, the particle exchange attempt rate is $p_\mathrm{exchange}=1$, the osmotic pressure difference is $\Delta p = 0$, the cutoff time is $\tau=5000$ sweeps, and the number of samples is $n_\mathrm{samples} = 500$.
• Figure 4: Renormalized material parameters predict the onset of wrinkling. Variation of the effective tension $\gamma$ with increasing excess chemical potential $\Delta\mu/k_\mathrm{B} T$. Blue points show values extracted from fitting vesicle shape fluctuation spectra to Eq. \ref{['eq:spectrum']}. The gray curve shows the predicted zero-temperature critical buckling tension $\gamma_{c,0}=-(2/R)\sqrt{\kappa Y}$, while the black curve shows the temperature-dependent critical tension $\gamma_c=\gamma_{c,0}\Psi(\mathsf{ET})$ predicted by RG calculations from Ref. kosmrlj_statistical_2017. The light blue line shows a linear fit in the near-equilibrium regime. The shaded region indicates where buckling is predicted to occur based on the RG calculations, with $\gamma < \gamma_c$. Also shown are representative final configurations corresponding to the values of $\Delta \mu$ indicated by the dashed lines. Here, $p_\mathrm{exchange}=1$, $\Delta p = 0$, and $\tau=5000$ sweeps, and the number of samples is $n_\mathrm{samples} = 500$.
• Figure 5: Relationship between applied pressure, chemical potential, and vesicle stability. (a) Variation of the effective tension $\gamma$ with excess chemical potential $\Delta\mu/k_\mathrm{B}T$ for different values of osmotic pressure difference $\Delta p$. The measured effective tensions are compared with the predicted finite-temperature critical tension $\gamma_c$. The intersections (stars) indicate predicted morphological transition points, above which $\gamma < \gamma_c$. (b) Phase diagram in $\Delta p$-$\Delta\mu$ space. The region where $\gamma > \gamma_c$ corresponds to stable quasispherical growth, while $\gamma < \gamma_c$ indicates unstable growth with persistent out-of-plane deformations. The solid line represents a linear fit. Here, the excess chemical potential is computed relative to the equilibrium chemical potential at finite pressure difference: $\Delta\mu = \mu - \mu_\mathrm{eq}|_{\Delta p}$. For these data, $p_\mathrm{exchange}=1$, $\tau=5000$ sweeps, and the number of samples is $n_\mathrm{samples} = 500$.