Membrane undulations driven by force fluctuations of active proteins

Red blood cell flickering activity locally controlled by holographic optical tweezers

Red blood cells possess a singular mechanobiology, enabling efficient navigation through capillaries smaller than their own size. Their plasma membrane exhibits non-equilibrium shape fluctuation, often reported as enhanced flickering activity. Such active membrane motion is propelled by motor proteins that mediate interactions between the spectrin skeleton and the lipid bilayer. However, modulating the flickering in living red blood cells without permanently altering their mechanical properties represents a significant challenge. In this study, we developed holographic optical tweezers to generate a force field distributed along the equatorial membrane contour of individual red blood cells. In free-standing red blood cells, we observed heterogeneous flickering activity, attributed to localized membrane kickers. By employing holographic optical forces, these active kickers can be selectively halted under minimal invasion. Our findings shed light on the dynamics of membrane flickering and established a manipulation tool that could open new avenues for investigating mechanotransduction processes in living cells.

Odd elasticity and topological waves in active surfaces

Odd elasticity describes active elastic systems whose stress-strain relationship is not compatible with a potential energy. As the requirement of energy conservation is lifted from linear elasticity, new antisymmetric (odd) components appear in the elastic tensor. In this work we study the odd elasticity and non-Hermitian wave dynamics of active surfaces, specifically plates of moderate thickness. These odd moduli can endow the vibrational modes of the plate with a nonzero topological invariant known as the first Chern number. Within continuum elastic theory, we show that the Chern number is related to the presence of unidirectional shearing waves that are hosted at the plate's boundary. We show that the existence of these chiral edge waves hinges on a distinctive two-step mechanism. Unlike electronic Chern insulators where the magnetic field at the same time gaps the spectrum and imparts chirality, here the finite thickness of the sample gaps the shear modes, and the odd elasticity makes them chiral.

Fluctuations and Shape Dependence of Microphase Separation in Systems with Long-Range Interactions

The combination of phase separation and long-ranged, effective, Coulomb interactions results in microphase separation. We predict the sizes and shapes of such microdomains and uniquely their dependence on the macroscopic sample shape which also affects the effective interfacial tension of fluctuations of the lamellar phase. These are applied to equilibrium salt solutions and block copolymers. Nonequilibrium phase separation in the presence of chemical reactions (e.g., cellular condensates) is mapped to the Coulomb theory to which our predictions apply. In some cases, the effective interfacial tension can be ultralow.

Clustering of lipids driven by integrin

Integrin is an important transmembrane receptor protein which remodels the actin network and anchors the cell membrane towards the extracellular matrix via mechanochemical pathways. The clustering of specific lipids and lipid-anchored proteins, which is essential for a certain type of endocytosis process, is facilitated at integrin-mediated active regions. To study this, we propose a minimal exactly solvable model which includes the interplay of stochastic shuttling between integrin on and off states with the intrinsic dynamics of the membrane. We propose a two-step mechanism in which the integrin induces an aster-like arrangement in the actin network, followed by clustering of lipids in that region. We obtain an analytic expression for the deformation and local membrane velocity, and thereby the evolution of clustering mediated by a single integrin. The deformation evolves nonmonotonically and its dependence on the stochastic shuttling timescales and membrane properties is elucidated. Our estimates of the area of the deformed region and the number of lipids in it indicate strong clustering.

Mesoscale, long-time mixing of chromosomes and its connection to polymer dynamics

Chromosomes are arranged in distinct territories within the nucleus of animal cells. Recent experiments have shown that these territories overlap at their edges, suggesting partial mixing during interphase. Experiments that knock-down of condensin II proteins during interphase indicate increased chromosome mixing, which demonstrates control of the mixing. In this study, we use a generic polymer simulation to quantify the dynamics of chromosome mixing over time. We introduce the chromosome mixing index, which quantifies the mixing of distinct chromosomes in the nucleus. We find that the chromosome mixing index in a small confinement volume (as a model of the nucleus), increases as a power-law of the time, with the scaling exponent varying non-monotonically with self-interaction and volume fraction. By comparing the chromosome mixing index with both monomer subdiffusion due to (non-topological) intermingling of chromosomes as well as even slower reptation, we show that for relatively large volume fractions, the scaling exponent of the chromosome mixing index is related to Rouse dynamics for relatively weak chromosome attractions and to reptation for strong attractions. In addition, we extend our model to more realistically account for the situation of the Drosophila chromosome by including the heterogeneity of the polymers and their lengths to account for microphase separation of euchromatin and heterochromatin and their interactions with the nuclear lamina. We find that the interaction with the lamina further impedes chromosome mixing.

Capsule self-oscillating gels showing cell-like nonthermal membrane/shape fluctuations

A primary interest in cell membrane and shape fluctuations is establishing experimental models reflecting only nonthermal active contributions. Here we report a millimeter-scaled capsule self-oscillating gel model mirroring the active contribution effect on cell fluctuations. In the capsule self-oscillating gels, the propagating chemical signals during a Belousov-Zhabotinsky (BZ) reaction induce simultaneous local deformations in the various regions, showing cell-like shape fluctuations. The capsule self-oscillating gels do not fluctuate without the BZ reaction, implying that only the active chemical parameter induces the gel fluctuations. The period and amplitude depend on the gel layer thickness and the concentration of the chemical substrate for the BZ reaction. Our results allow for a solid experimental platform showing actively driven cell-like fluctuations, which can potentially contribute to investigating the active parameter effect on cell fluctuations.

Mean first-passage time of an active fluctuating membrane with stochastic resetting

We study the mean first-passage time of a one-dimensional active fluctuating membrane that is stochastically returned to the same flat initial condition at a finite rate. We start with a Fokker-Planck equation to describe the evolution of the membrane coupled with an Ornstein-Uhlenbeck type of active noise. Using the method of characteristics, we solve the equation and obtain the joint distribution of the membrane height and active noise. In order to obtain the mean first-passage time (MFPT), we further obtain a relation between the MFPT and a propagator that includes stochastic resetting. The derived relation is then used to calculate it analytically. Our studies show that the MFPT increases with a larger resetting rate and decreases with a smaller rate, i.e., there is an optimal resetting rate. We compare the results in terms of MFPT of the membrane with active and thermal noises for different membrane properties. The optimal resetting rate is much smaller with active noise compared to thermal. When the resetting rate is much lower than the optimal rate, we demonstrate how the MFPT scales with resetting rates, distance to the target, and the properties of the membranes.

Physical-Chemical Regulation of Membrane Receptors Dynamics in Viral Invasion and Immune Defense

Mechanical cues dynamically regulate membrane receptors functions to trigger various physiological and pathological processes from viral invasion to immune defense. These cues mainly include various types of dynamic mechanical forces and the spatial confinement of plasma membrane. However, the molecular mechanisms of how they couple with biochemical cues in regulating membrane receptors functions still remain mysterious. Here, we review recent advances in methodologies of single-molecule biomechanical techniques and in novel biomechanical regulatory mechanisms of critical ligand recognition of viral and immune receptors including SARS-CoV-2 spike protein, T cell receptor (TCR) and other co-stimulatory immune receptors. Furthermore, we provide our perspectives of the general principle of how force-dependent kinetics determine the dynamic functions of membrane receptors and of biomechanical-mechanism-driven SARS-CoV-2 neutralizing antibody design and TCR engineering for T-cell-based therapies.

Stochastic resetting and first arrival subjected to Gaussian noise and Poisson white noise

We study the dynamics of an overdamped Brownian particle subjected to Poissonian stochastic resetting in a nonthermal bath, characterized by a Poisson white noise and a Gaussian noise. Applying the renewal theory we find an exact analytical expression for the spatial distribution at the steady state. Unlike the single exponential distribution as observed in the case of a purely thermal bath, the distribution is double exponential. Relaxation of the transient spatial distributions to the stationary one, for the limiting cases of Poissonian rate, is investigated carefully. In addition, we study the first-arrival properties of the system in the presence of a delta-function sink with strength κ, where κ=0 and κ=∞ correspond to fully nonreactive and fully reactive sinks, respectively. We explore the effect of two competitive mechanisms: the diffusive spread in the presence of two noises and the increase in probability density around the initial position due to stochastic resetting. We show that there exists an optimal resetting rate, which minimizes the mean first-arrival time (MFAT) to the sink for a given value of the sink strength. We also explore the effect of the strength of the Poissonian noise on MFAT, in addition to sink strength. Our formalism generalizes the diffusion-limited reaction under resetting in a nonequilibrium bath and provides an efficient search strategy for a reactant to find a target site, relevant in a range of biophysical processes.

Long-range correlations in pinned athermal networks

We derive exact results for displacement fields that develop as a response to external pinning forces in two-dimensional athermal networks. For a triangular lattice arrangement of particles interacting through soft potentials, we develop a Green's function formalism which we use to derive exact results for displacement fields produced by localized external forces. We show that in the continuum limit the displacement fields decay as 1/r at large distances r away from a force dipole. Finally, we extend our formulation to study correlations in the displacement fields produced by the external pinning forces. We show that uncorrelated pinned forces at each vertex give rise to long-range correlations in displacements in athermal systems, with a nontrivial system size dependence. We verify our predictions with numerical simulations of athermal networks in two dimensions.

Inertial effects on the Brownian gyrator

The recent interest into the Brownian gyrator has been confined chiefly to the analysis of Brownian dynamics both in theory and experiment despite the applicability of general cases with definite mass. Considering mass explicitly in the solution of the Fokker-Planck equation and Langevin dynamics simulations, we investigate how inertia can change the dynamics and energetics of the Brownian gyrator. In the Langevin model, the inertia reduces the nonequilibrium effects by diminishing the declination of the probability density function and the mean of a specific angular momentum, j_{θ}, as a measure of rotation. Another unique feature of the Langevin description is that rotation is maximized at a particular anisotropy while the stability of the rotation is minimized at a particular anisotropy or mass. Our results suggest that the Langevin dynamics description of the Brownian gyrator is intrinsically different from that with Brownian dynamics. In addition, j_{θ} is proven to be essential and convenient for estimating stochastic energetics such as heat currents and entropy production even in the underdamped regime.

Dynamic Nuclear Structure Emerges from Chromatin Cross-Links and Motors

The cell nucleus houses the chromosomes, which are linked to a soft shell of lamin protein filaments. Experiments indicate that correlated chromosome dynamics and nuclear shape fluctuations arise from motor activity. To identify the physical mechanisms, we develop a model of an active, cross-linked Rouse chain bound to a polymeric shell. System-sized correlated motions occur but require both motor activity and cross-links. Contractile motors, in particular, enhance chromosome dynamics by driving anomalous density fluctuations. Nuclear shape fluctuations depend on motor strength, cross-linking, and chromosome-lamina binding. Therefore, complex chromosome dynamics and nuclear shape emerge from a minimal, active chromosome-lamina system.

More than just a barrier: using physical models to couple membrane shape to cell function

The correct execution of many cellular processes, such as division and motility, requires the cell to adopt a specific shape. Physically, these shapes are determined by the interplay of the plasma membrane and internal cellular driving factors. While the plasma membrane defines the boundary of the cell, processes inside the cell can result in the generation of forces that deform the membrane. These processes include protein binding, the assembly of protein superstructures, and the growth and contraction of cytoskeletal networks. Due to the complexity of the cell, relating observed membrane deformations back to internal processes is a challenging problem. Here, we review cell shape changes in endocytosis, cell adhesion, cell migration and cell division and discuss how by modeling membrane deformations we can investigate the inner working principles of the cell.

On the Role of Curved Membrane Nanodomains, and Passive and Active Skeleton Forces in the Determination of Cell Shape and Membrane Budding

Biological membranes are composed of isotropic and anisotropic curved nanodomains. Anisotropic membrane components, such as Bin/Amphiphysin/Rvs (BAR) superfamily protein domains, could trigger/facilitate the growth of membrane tubular protrusions, while isotropic curved nanodomains may induce undulated (necklace-like) membrane protrusions. We review the role of isotropic and anisotropic membrane nanodomains in stability of tubular and undulated membrane structures generated or stabilized by cyto- or membrane-skeleton. We also describe the theory of spontaneous self-assembly of isotropic curved membrane nanodomains and derive the critical concentration above which the spontaneous necklace-like membrane protrusion growth is favorable. We show that the actin cytoskeleton growth inside the vesicle or cell can change its equilibrium shape, induce higher degree of segregation of membrane nanodomains or even alter the average orientation angle of anisotropic nanodomains such as BAR domains. These effects may indicate whether the actin cytoskeleton role is only to stabilize membrane protrusions or to generate them by stretching the vesicle membrane. Furthermore, we demonstrate that by taking into account the in-plane orientational ordering of anisotropic membrane nanodomains, direct interactions between them and the extrinsic (deviatoric) curvature elasticity, it is possible to explain the experimentally observed stability of oblate (discocyte) shapes of red blood cells in a broad interval of cell reduced volume. Finally, we present results of numerical calculations and Monte-Carlo simulations which indicate that the active forces of membrane skeleton and cytoskeleton applied to plasma membrane may considerably influence cell shape and membrane budding.

Dynamics of passive and active membrane tubes

Utilising Onsager's variational formulation, we derive dynamical equations for the relaxation of a fluid membrane tube in the limit of small deformation, allowing for a contrast of solvent viscosity across the membrane and variations in surface tension due to membrane incompressibility. We compute the relaxation rates, recovering known results in the case of purely axis-symmetric perturbations and making new predictions for higher order (azimuthal) m-modes. We analyse the long and short wavelength limits of these modes by making use of various asymptotic arguments. We incorporate stochastic terms to our dynamical equations suitable to describe both passive thermal forces and non-equilibrium active forces. We derive expressions for the fluctuation amplitudes, an effective temperature associated with active fluctuations, and the power spectral density for both the thermal and active fluctuations. We discuss an experimental assay that might enable measurement of these fluctuations to infer the properties of the active noise. Finally we discuss our results in the context of active membranes more generally and give an overview of some open questions in the field.

Large-scale simulation of biomembranes incorporating realistic kinetics into coarse-grained models

Biomembranes are two-dimensional assemblies of phospholipids that are only a few nanometres thick, but form micrometre-sized structures vital to cellular function. Explicit molecular modelling of biologically relevant membrane systems is computationally expensive due to the large number of solvent particles and slow membrane kinetics. Coarse-grained solvent-free membrane models offer efficient sampling but sacrifice realistic kinetics, thereby limiting the ability to predict pathways and mechanisms of membrane processes. Here, we present a framework for integrating coarse-grained membrane models with continuum-based hydrodynamics. This framework facilitates efficient simulation of large biomembrane systems with large timesteps, while achieving realistic equilibrium and non-equilibrium kinetics. It helps to bridge between the nanometer/nanosecond spatiotemporal resolutions of coarse-grained models and biologically relevant time- and lengthscales. As a demonstration, we investigate fluctuations of red blood cells, with varying cytoplasmic viscosities, in 150-milliseconds-long trajectories, and compare kinetic properties against single-cell experimental observations.

Active Contact Forces Drive Nonequilibrium Fluctuations in Membrane Vesicles

We analyze the nonequilibrium shape fluctuations of giant unilamellar vesicles encapsulating motile bacteria. Owing to bacteria-membrane collisions, we experimentally observe a significant increase in the magnitude of membrane fluctuations at low wave numbers, compared to the well-known thermal fluctuation spectrum. We interrogate these results by numerically simulating membrane height fluctuations via a modified Langevin equation, which includes bacteria-membrane contact forces. Taking advantage of the lengthscale and timescale separation of these contact forces and thermal noise, we further corroborate our results with an approximate theoretical solution to the dynamical membrane equations. Our theory and simulations demonstrate excellent agreement with nonequilibrium fluctuations observed in experiments. Moreover, our theory reveals that the fluctuation-dissipation theorem is not broken by the bacteria; rather, membrane fluctuations can be decomposed into thermal and active components.

How a lipid bilayer membrane responds to an oscillating nanoparticle: Promoted membrane undulation and directional wave propagation

Mechanical forces acting on a plasma membrane are of essential importance to cellular functioning via inducing delicate change of the membrane shape with the underlying mechanism yet to be elucidated. Here, we introduce an oscillating nanoparticle (NP) interaction with a lipid bilayer membrane, using the coarse-grained simulation to investigate the dynamic membrane response to constrained mechanical stimulation, which is ubiquitous in biology. Our results demonstrate that, the membrane responds to an oscillating NP by generating nanoscale undulation waves, which immediately propagate through the membrane. In dynamics, propagation of the generated membrane undulation waves always starts from flattening of the region where the NP locates, thus producing a lateral force to propel the waves away from the point of stimulation. The speed of membrane undulation wave propagation is proportional to that of NP oscillation and accelerated by increasing the integral membrane surface tension, suggesting that both the membrane bending and stretching contribute to the energy driving the unique response of membrane undulation wave propagation.

Statistical Mechanics of an Elastically Pinned Membrane: Equilibrium Dynamics and Power Spectrum

In biological settings, membranes typically interact locally with other membranes: the extracellular matrix in the exterior or internal cellular structures such as the cytoskeleton, locally pinning the membrane. Characterizing the dynamical properties of such interactions presents a difficult task. Significant progress has been achieved through simulations and experiments, yet analytical progress in modeling pinned membranes has been impeded by the complexity of governing equations. Here, we circumvent these difficulties by calculating analytically the time-dependent Green's function of the operator governing the dynamics of an elastically pinned membrane in a hydrodynamic surrounding and subject to external forces. This enables us to calculate the equilibrium power spectral density for an overdamped membrane pinned by an elastic, permanently attached spring subject to thermal excitations. By considering the effects of the finite experimental resolution on the measured spectra, we show that the elasticity of the pinning can be extracted from the experimentally measured spectrum. Membrane fluctuations can thus be used as a tool to probe mechanical properties of the underlying structures. Such a tool may be particularly relevant in the context of cell mechanics, in which the elasticity of the membrane's attachment to the cytoskeleton could be measured.

Theoretical study of vesicle shapes driven by coupling curved proteins and active cytoskeletal forces

Eukaryote cells have a flexible shape, which dynamically changes according to the function performed by the cell. One mechanism for deforming the cell membrane into the desired shape is through the expression of curved membrane proteins. Furthermore, these curved membrane proteins are often associated with the recruitment of the cytoskeleton, which then applies active forces that deform the membrane. This coupling between curvature and activity was previously explored theoretically in the linear limit of small deformations, and low dimensionality. Here we explore the unrestricted shapes of vesicles that contain active curved membrane proteins, in three-dimensions, using Monte-Carlo numerical simulations. The activity of the proteins is in the form of protrusive forces that push the membrane outwards, as may arise from the cytoskeleton of the cell due to actin or microtubule polymerization occurring near the membrane. For proteins that have an isotropic convex shape, the additional protrusive force enhances their tendency to aggregate and form membrane protrusions (buds). In addition, we find another transition from deformed spheres with necklace type aggregates, to flat pancake-shaped vesicles, where the curved proteins line the outer rim. This second transition is driven by the active forces, coupled to the spontaneous curvature, and the resulting configurations may shed light on the formation of sheet-like protrusions and lamellipodia of adhered and motile cells.

Guided by curvature: shaping cells by coupling curved membrane proteins and cytoskeletal forces

Eukaryote cells have flexible membranes that allow them to have a variety of dynamical shapes. The shapes of the cells serve important biological functions, both for cells within an intact tissue, and during embryogenesis and cellular motility. How cells control their shapes and the structures that they form on their surface has been a subject of intensive biological research, exposing the building blocks that cells use to deform their membranes. These processes have also drawn the interest of theoretical physicists, aiming to develop models based on physics, chemistry and nonlinear dynamics. Such models explore quantitatively different possible mechanisms that the cells can employ to initiate the spontaneous formation of shapes and patterns on their membranes. We review here theoretical work where one such class of mechanisms was investigated: the coupling between curved membrane proteins, and the cytoskeletal forces that they recruit. Theory indicates that this coupling gives rise to a rich variety of membrane shapes and dynamics, while experiments indicate that this mechanism appears to drive many cellular shape changes.This article is part of the theme issue 'Self-organization in cell biology'.

Signatures of motor susceptibility to forces in the dynamics of a tracer particle in an active gel

We study a model for the motion of a tracer particle inside an active gel, exposing the properties of the van Hove distribution of the particle displacements. Active events of a typical force magnitude can give rise to non-Gaussian distributions having exponential tails or side peaks. The side peaks are predicted to appear when the local bulk elasticity of the gel is large enough and few active sources are dominant. We explain the regimes of the different distributions and study the structure of the side peaks for active sources that are susceptible to the elastic stress that they cause inside the gel. We show how the van Hove distribution is altered by both the duty cycle of the active sources and their susceptibility, and suggest it as a sensitive probe to analyze microrheology data in active systems with restoring elastic forces.

Living Matter: Mesoscopic Active Materials

An introduction to the physical properties of living active matter at the mesoscopic scale (tens of nanometers to micrometers) and their unique features compared with "dead," nonactive matter is presented. This field of research is increasingly denoted as "biological physics" where physics includes chemical physics, soft matter physics, hydrodynamics, mechanics, and the related engineering sciences. The focus is on the emergent properties of these systems and their collective behavior, which results in active self-organization and how they relate to cellular-level biological function. These include locomotion (cell motility and migration) forces that give rise to cell division, the growth and form of cellular assemblies in development, the beating of heart cells, and the effects of mechanical perturbations such as shear flow (in the bloodstream) or adhesion to other cells or tissues. An introduction to the fundamental concepts and theory with selected experimental examples related to the authors' own research is presented, including red-blood-cell membrane fluctuations, motion of the nucleus within an egg cell, self-contracting acto-myosin gels, and structure and beating of heart cells (cardiomyocytes), including how they can be driven by an oscillating, mechanical probe.

Broken detailed balance and non-equilibrium dynamics in living systems: a review

Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.

Lipid self-assembly and lectin-induced reorganization of the plasma membrane

The plasma membrane represents an outstanding example of self-organization in biology. It plays a vital role in protecting the integrity of the cell interior and regulates meticulously the import and export of diverse substances. Its major building blocks are proteins and lipids, which self-assemble to a fluid lipid bilayer driven mainly by hydrophobic forces. Even if the plasma membrane appears-globally speaking-homogeneous at physiological temperatures, the existence of specialized nano- to micrometre-sized domains of raft-type character within cellular and synthetic membrane systems has been reported. It is hypothesized that these domains are the origin of a plethora of cellular processes, such as signalling or vesicular trafficking. This review intends to highlight the driving forces of lipid self-assembly into a bilayer membrane and the formation of small, transient domains within the plasma membrane. The mechanisms of self-assembly depend on several factors, such as the lipid composition of the membrane and the geometry of lipids. Moreover, the dynamics and organization of glycosphingolipids into nanometre-sized clusters will be discussed, also in the context of multivalent lectins, which cluster several glycosphingolipid receptor molecules and thus create an asymmetric stress between the two membrane leaflets, leading to tubular plasma membrane invaginations.This article is part of the theme issue 'Self-organization in cell biology'.

Tracking fast cellular membrane dynamics with sub-nm accuracy in the normal direction

Cellular membranes are important biomaterials with highly dynamic structures. Membrane dynamics plays an important role in numerous cellular processes, but precise tracking it is challenging due to the lack of tools with a highly sensitive and fast detection capability. Here we demonstrate a broad bandwidth optical imaging technique to measure cellular membrane displacements in the normal direction at sub-nm level detection limits and 20 μs temporal resolution (1 Hz-50 kHz). This capability allows us to study the intrinsic cellular membrane dynamics over a broad temporal and spatial spectrum. We measured the nanometer-scale stochastic fluctuations of the plasma membrane of HEK-293 cells, and found them to be highly dependent on the cytoskeletal structure of the cells. By analyzing the fluctuations, we further determine the mechanical properties of the cellular membranes. We anticipate that the method will contribute to the understanding of the basic cellular processes, and applications, such as mechanical phenotyping of cells at the single-cell level.

Cytoplasmic Flow and Mixing Due to Deformation of Motile Cells

The cytoplasm of a living cell is a dynamic environment through which intracellular components must move and mix. In motile, rapidly deforming cells such as human neutrophils, bulk cytoplasmic flow couples cell deformation to the transport and dispersion of cytoplasmic particles. Using particle-tracking measurements in live neutrophil-like cells, we demonstrate that fluid flow associated with the cell deformation contributes to the motion of small acidic organelles, dominating over diffusion on timescales above a few seconds. We then use a general physical model of particle dispersion in a deforming fluid domain to show that transport of organelle-sized particles between the cell periphery and the bulk can be enhanced by dynamic deformation comparable to that observed in neutrophils. Our results implicate an important mechanism contributing to organelle transport in these motile cells: cytoplasmic flow driven by cell shape deformation.

Mapping Cell Membrane Fluctuations Reveals Their Active Regulation and Transient Heterogeneities

Shape fluctuations of the plasma membrane occur in all cells, are incessant, and are proposed to affect membrane functioning. Although studies show how membrane fluctuations are affected by cellular activity in adherent cells, their spatial regulation and the corresponding change in membrane mechanics remain unclear. In this article, we study how ATP-driven activities and actomyosin cytoskeleton impact basal membrane fluctuations in adherent cells. Using interference imaging, we map height fluctuations within single cells and compare the temporal spectra with existing theoretical models to gain insights about the underlying membrane mechanics. We find that ATP-dependent activities enhance the nanoscale z fluctuations but stretch out the membrane laterally. Although actin polymerization or myosin-II activity individually enhances fluctuations, the cortex in unperturbed cells stretches out the membrane and dampens fluctuations. Fitting with models suggest this dampening to be due to confinement by the cortex. However, reduced fluctuations on mitosis or on ATP-depletion/stabilization of cortex correlate with increased tension. Both maps of fluctuations and local temporal autocorrelation functions reveal ATP-dependent transient short-range (<2 μm) heterogeneities. Together, our results show how various ATP-driven processes differently affect membrane mechanics and hence fluctuations, while creating distinct local environments whose functional role needs future investigation.

Nonequilibrium fluctuations of lipid membranes by the rotating motor protein F1F0-ATP synthase

The shape of biological membranes is constantly remodeled and maintained out of equilibrium by active proteins. The functional capacity of membrane deformation is mainly determined by the mechanical interplay between protein activity and bending elasticity. In our experiments, we find that ATP synthase, a rotating membrane protein that synthesizes the biochemical energy in cells through proton-pumping activity across the membrane, promotes localized nonequilibrium membrane fluctuations when reconstituted in giant lipid vesicles. The large membrane deformations emerge from the pumping action of rotating proteins clustered at specific emplacements in the membrane. Our results pave the way to new experimental realizations to explore the collective effects of rotating ATP synthases and their possible biological implications for biomembrane organization and protein functionality.

ATP synthase is a rotating membrane protein that synthesizes ATP through proton-pumping activity across the membrane. To unveil the mechanical impact of this molecular active pump on the bending properties of its lipid environment, we have functionally reconstituted the ATP synthase in giant unilamellar vesicles and tracked the membrane fluctuations by means of flickering spectroscopy. We find that ATP synthase rotates at a frequency of about 20 Hz, promoting large nonequilibrium deformations at discrete hot spots in lipid vesicles and thus inducing an overall membrane softening. The enhanced nonequilibrium fluctuations are compatible with an accumulation of active proteins at highly curved membrane sites through a curvature−protein coupling mechanism that supports the emergence of collective effects of rotating ATP synthases in lipid membranes.

On the origin of shape fluctuations of the cell nucleus

The nuclear envelope (NE) presents a physical boundary between the cytoplasm and the nucleoplasm, sandwiched in between two highly active systems inside the cell: cytoskeleton and chromatin. NE defines the shape and size of the cell nucleus, which increases during the cell cycle, accommodating for chromosome decondensation followed by genome duplication. In this work, we study nuclear shape fluctuations at short time scales of seconds in human cells. Using spinning disk confocal microscopy, we observe fast fluctuations of the NE, visualized by fluorescently labeled lamin A, and of the chromatin globule surface (CGS) underneath the NE, visualized by fluorescently labeled histone H2B. Our findings reveal that fluctuation amplitudes of both CGS and NE monotonously decrease during the cell cycle, serving as a reliable cell cycle stage indicator. Remarkably, we find that, while CGS and NE typically fluctuate in phase, they do exhibit localized regions of out-of-phase motion, which lead to separation of NE and CGS. To explore the mechanism behind these shape fluctuations, we use biochemical perturbations. We find the shape fluctuations of CGS and NE to be both thermally and actively driven, the latter caused by forces from chromatin and cytoskeleton. Such undulations might affect gene regulation as well as contribute to the anomalously high rates of nuclear transport by, e.g., stirring of molecules next to NE, or increasing flux of molecules through the nuclear pores.

Mechanics of active surfaces

We derive a fully covariant theory of the mechanics of active surfaces. This theory provides a framework for the study of active biological or chemical processes at surfaces, such as the cell cortex, the mechanics of epithelial tissues, or reconstituted active systems on surfaces. We introduce forces and torques acting on a surface, and derive the associated force balance conditions. We show that surfaces with in-plane rotational symmetry can have broken up-down, chiral, or planar-chiral symmetry. We discuss the rate of entropy production in the surface and write linear constitutive relations that satisfy the Onsager relations. We show that the bending modulus, the spontaneous curvature, and the surface tension of a passive surface are renormalized by active terms. Finally, we identify active terms which are not found in a passive theory and discuss examples of shape instabilities that are related to active processes in the surface.

Fluctuation spectra and force generation in nonequilibrium systems

Many biological systems are appropriately viewed as passive inclusions immersed in an active bath: from proteins on active membranes to microscopic swimmers confined by boundaries. The nonequilibrium forces exerted by the active bath on the inclusions or boundaries often regulate function, and such forces may also be exploited in artificial active materials. Nonetheless, the general phenomenology of these active forces remains elusive. We show that the fluctuation spectrum of the active medium, the partitioning of energy as a function of wavenumber, controls the phenomenology of force generation. We find that, for a narrow, unimodal spectrum, the force exerted by a nonequilibrium system on two embedded walls depends on the width and the position of the peak in the fluctuation spectrum, and oscillates between repulsion and attraction as a function of wall separation. We examine two apparently disparate examples: the Maritime Casimir effect and recent simulations of active Brownian particles. A key implication of our work is that important nonequilibrium interactions are encoded within the fluctuation spectrum. In this sense, the noise becomes the signal.

Dynamics of a membrane interacting with an active wall

Active motions of a biological membrane can be induced by nonthermal fluctuations that occur in the outer environment of the membrane. We discuss the dynamics of a membrane interacting hydrodynamically with an active wall that exerts random velocities on the ambient fluid. Solving the hydrodynamic equations of a bound membrane, we first derive a dynamic equation for the membrane fluctuation amplitude in the presence of different types of walls. Membrane two-point correlation functions are calculated for three different cases: (i) a static wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale. We focus on the mean squared displacement (MSD) of a tagged membrane describing the Brownian motion of a membrane segment. For the static wall case, there are two asymptotic regimes of MSD (∼t^{2/3} and ∼t^{1/3}) when the hydrodynamic decay rate changes monotonically. In the case of an active wall, the MSD grows linearly in time (∼t) in the early stage, which is unusual for a membrane segment. This linear-growth region of the MSD is further extended when the active wall has a finite intrinsic time scale.

Direct Cytoskeleton Forces Cause Membrane Softening in Red Blood Cells

Erythrocytes are flexible cells specialized in the systemic transport of oxygen in vertebrates. This physiological function is connected to their outstanding ability to deform in passing through narrow capillaries. In recent years, there has been an influx of experimental evidence of enhanced cell-shape fluctuations related to metabolically driven activity of the erythroid membrane skeleton. However, no direct observation of the active cytoskeleton forces has yet been reported to our knowledge. Here, we show experimental evidence of the presence of temporally correlated forces superposed over the thermal fluctuations of the erythrocyte membrane. These forces are ATP-dependent and drive enhanced flickering motions in human erythrocytes. Theoretical analyses provide support for a direct force exerted on the membrane by the cytoskeleton nodes as pulses of well-defined average duration. In addition, such metabolically regulated active forces cause global membrane softening, a mechanical attribute related to the functional erythroid deformability.

Dynamics of two-component membranes surrounded by viscoelastic media

We discuss the dynamics of two-component fluid membranes which are surrounded by viscoelastic media. We assume that membrane-embedded proteins can diffuse laterally and induce a local membrane curvature. The mean squared displacement of a tagged membrane segment is obtained as a generalized Einstein relation. When the elasticity of the surrounding media obeys a power-law behavior in frequency, an anomalous diffusion of the membrane segment is predicted. We also consider the situation where the proteins generate active non-equilibrium forces. The generalized Einstein relation is further modified by an effective temperature that depends on the force dipole energy. The obtained generalized Einstein relations are useful for membrane microrheology experiments.

Direct measurement of the cortical tension during the growth of membrane blebs

Mechanics is at the heart of many cellular processes and its importance has received considerable attention during the last two decades. In particular, the tension of cell membranes, and more specifically of the cell cortex, is a key parameter that determines the mechanical behavior of the cell periphery. However, the measurement of tension remains challenging due to its dynamic nature. Here we show that a noninvasive interferometric technique can reveal time-resolved effective tension measurements by a high-accuracy determination of edge fluctuations in expanding cell blebs of filamin-deficient melanoma cells. The introduced technique shows that the bleb tension is ~10-100 pN/μm and increases during bleb growth. Our results directly confirm that the subsequent stop of bleb growth is induced by an increase of measured tension, possibly mediated by the repolymerized actin cytoskeleton.

Minimal model of stochastic athermal systems: origin of non-Gaussian noise

For a wide class of stochastic athermal systems, we derive Langevin-like equations driven by non-Gaussian noise, starting from master equations and developing a new asymptotic expansion. We found an explicit condition whereby the non-Gaussian properties of the athermal noise become dominant for tracer particles associated with both thermal and athermal environments. Furthermore, we derive an inverse formula to infer microscopic properties of the athermal bath from the statistics of the tracer particle. We apply our formulation to a granular motor under viscous friction and analytically obtain the angular velocity distribution function. Our theory demonstrates that the non-Gaussian Langevin equation is the minimal model of athermal systems.

Nonequilibrium collective dynamics in photoexcited lipid multilayers by time resolved diffuse x-ray scattering

We study the nonequilibrium shape fluctuations in fluorescence labeled phospholipid multibilayers composed of the model lipid DOPC and the well-known lipid dye Texas red, driven out of equilibrium by short laser pulses. The temporal evolution of the lipid bilayer undulations after excitation was recorded by time resolved x-ray diffraction. Already at moderate peak intensities (Pp≤10(5)  W/cm2), pulsed laser illumination leads to significant changes of the undulation modes in a well-defined lateral wavelength band. The observed phenomena evolve on nano- to microsecond time scales after optical excitation, and can be described in terms of a modulation instability in the lipid multilamellar stack.

Coarse-grain simulations of active molecular machines in lipid bilayers

A coarse-grain method for simulations of the dynamics of active protein inclusions in lipid bilayers is described. It combines the previously proposed hybrid simulations of bilayers [M.-J. Huang, R. Kapral, A. S. Mikhailov, and H.-Y. Chen, J. Chem. Phys. 137, 055101 (2012)], based on molecular dynamics for the lipids and multi-particle collision dynamics for the solvent, with an elastic-network description of active proteins. The method is implemented for a model molecular machine which performs active conformational motions induced by ligand binding and its release after reaction. The situation characteristic for peripheral membrane proteins is considered. Statistical investigations of the effects of single active or passive inclusions on the shape of the membrane are carried out. The results show that the peripheral machine produces asymmetric perturbations of the thickness of two leaflets of the membrane. It also produces a local saddle in the midplane height of the bilayer. Analysis of the power spectrum of the fluctuations of the membrane midplane shows that the conformational motion of the machine perturbs these membrane fluctuations. The hydrodynamic lipid flows induced by cyclic conformational changes in the machine are analyzed. It is shown that such flows are long-ranged and should provide an additional important mechanism for interactions between active inclusions in biological membranes.

Marcus canonical integral for non-Gaussian processes and its computation: pathwise simulation and tau-leaping algorithm

The stochastic integral ensuring the Newton-Leibnitz chain rule is essential in stochastic energetics. Marcus canonical integral has this property and can be understood as the Wong-Zakai type smoothing limit when the driving process is non-Gaussian. However, this important concept seems not well-known for physicists. In this paper, we discuss Marcus integral for non-Gaussian processes and its computation in the context of stochastic energetics. We give a comprehensive introduction to Marcus integral and compare three equivalent definitions in the literature. We introduce the exact pathwise simulation algorithm and give the error analysis. We show how to compute the thermodynamic quantities based on the pathwise simulation algorithm. We highlight the information hidden in the Marcus mapping, which plays the key role in determining thermodynamic quantities. We further propose the tau-leaping algorithm, which advance the process with deterministic time steps when tau-leaping condition is satisfied. The numerical experiments and its efficiency analysis show that it is very promising.

Stochastic energetics for non-Gaussian processes

By introducing a new stochastic integral, we investigate the energetics of classical stochastic systems driven by non-Gaussian white noises. In particular, we introduce a decomposition of the total energy difference into the work and the heat for each trajectory, and derive a formula to calculate the heat from experimental data on the dynamics. We apply our formulation and results to a Langevin system driven by a Poisson noise.

Effective tension and fluctuations in active membranes

We calculate the fluctuation spectrum of the shape of a lipid vesicle or cell exposed to a nonthermal source of noise. In particular, we take constraints on the membrane area and the volume of fluid that it encapsulates into account when obtaining expressions for the dependency of the membrane tension on the noise. We then investigate three possible origins of the nonthermal noise taken from the literature: A direct force, which models an external medium pushing on the membrane, a curvature force, which models a fluctuating spontaneous curvature, and a permeation force coming from an active transport of fluid through the membrane. For the direct force and curvature force cases, we compare our results to existing experiments on active membranes.

Temperature and scattering contrast dependencies of thickness fluctuations in surfactant membranes

Temperature and scattering contrast dependencies of thickness fluctuations have been investigated using neutron spin echo spectroscopy in a swollen lamellar phase composed of nonionic surfactant, water, and oil. In the present study, two contrast conditions are examined; one is the bulk contrast, which probes two surfactant monolayers with an oil layer as a membrane, and the other is the film contrast, which emphasizes an individual surfactant monolayer. The thickness fluctuations enhance dynamics from the bending fluctuations, and are observed in a similar manner in both contrast conditions. Thickness fluctuations can be investigated regardless of the scattering contrast, though film contrasts are better to be employed in terms of the data quality. The thickness fluctuation amplitude is constant over the measured temperature range, including in the vicinity of the phase boundary between the lamellar and micellar phases at low temperature and the boundary between the lamellar and bicontinuous phases at high temperature. The damping frequency of the thickness fluctuations is well scaled using viscosity within the membranes at low temperature, which indicates the thickness fluctuations are predominantly controlled by the viscosity within the membrane. On the other hand, in the vicinity of the phase boundary at high temperature, thickness fluctuations become faster without changing the mode amplitude.

Continuum simulations of biomembrane dynamics and the importance of hydrodynamic effects

Traditional particle-based simulation strategies are impractical for the study of lipid bilayers and biological membranes over the longest length and time scales (microns, seconds and longer) relevant to cellular biology. Continuum-based models developed within the frameworks of elasticity theory, fluid dynamics and statistical mechanics provide a framework for studying membrane biophysics over a range of mesoscopic to macroscopic length and time regimes, but the application of such ideas to simulation studies has occurred only relatively recently. We review some of our efforts in this direction with emphasis on the dynamics in model membrane systems. Several examples are presented that highlight the prominent role of hydrodynamics in membrane dynamics and we argue that careful consideration of fluid dynamics is key to understanding membrane biophysics at the cellular scale.

Effective temperature of red-blood-cell membrane fluctuations

Biologically driven nonequilibrium fluctuations are often characterized by their non-Gaussianity or by an "effective temperature", which is frequency dependent and higher than the ambient temperature. We address these two measures theoretically by examining a randomly kicked particle, with a variable number of kicking motors, and show how these two indicators of nonequilibrium behavior can contradict. Our results are compared with new experiments on shape fluctuations of red-blood cell membranes, and demonstrate how the physical nature of the motors in this system can be revealed using these global measures of nonequilibrium.

Effective 3D viscoelasticity of red blood cells measured by diffraction phase microscopy

We present optical measurements of nanoscale red blood cell fluctuations obtained by highly sensitive quantitative phase imaging. These spatio-temporal fluctuations are modeled in terms of the bulk viscoelastic response of the cell. Relating the displacement distribution to the storage and loss moduli of the bulk has the advantage of incorporating all geometric and cortical effects into a single effective medium behavior. The results on normal cells indicate that the viscous modulus is much larger than the elastic one throughout the entire frequency range covered by the measurement, indicating fluid behavior.

Elastic moderation of intrinsically applied tension in lipid membranes

Tension in lipid membranes is often controlled externally, by pulling on the boundary of the membrane or changing osmotic pressure across a curved membrane. But modifications of the tension can also be induced in an internal fashion, for instance as a byproduct of changing a membrane's electric potential or, as observed experimentally, by activity of membrane proteins. Here we develop a theory that demonstrates how such internal contributions to the tension are moderated through elastic stretching of the membrane when the membrane is initially in a low-tension floppy state.

Forming transmembrane channels using end-functionalized nanotubes

Using dissipative particle dynamics (DPD) simulations, we examine the interaction between amphiphilic nanotubes and lipid bilayer membranes. The nanotubes are represented by a hydrophobic shaft that is end-functionalized with hydrophilic groups. Nanotubes that are capped by a monolayer of hydrophilic beads or also encompass hydrophilic "hairs" on just one end of the shaft are found to spontaneously penetrate and assume a transmembrane position; the process, however, depends critically on the membrane tension. On the other hand, nanotubes that include hydrophilic hairs at both ends of the hydrophobic shaft are not observed to spontaneously self-organize into the bilayer. When the membrane is stretched to form a pore, the nanotubes with two hairy ends adsorb on the edge of the pore and become localized in the membrane, thus forming a transmembrane channel. The findings from these studies provide guidelines for creating biomimetic nanotube channels that are capable of selectively transporting molecules through the membrane in response to changes in the local environment.