Monte Carlo study of the frame, fluctuation and internal tensions of fluctuating membranes with fixed area

Effects of Normal and Lateral Electric Fields on Membrane Mechanical Properties

As a core component of biological and synthetic membranes, lipid bilayers are key to compartmentalizing chemical processes. Bilayer morphology and mechanical properties are heavily influenced by electric fields, such as those caused by biological ion concentration gradients. We present atomistic simulations exploring the effects of electric fields applied normally and laterally to lipid bilayers. We find that normal fields decrease membrane tension, while lateral fields increase it. Free energy perturbation calculations indicate the importance of dipole-dipole interactions to these tension changes, especially for lateral fields. We additionally show that membrane area compressibilities can be related to their cohesive energies, allowing us to estimate changes in membrane bending rigidity under applied fields. We find that normal and lateral fields decrease and increase bending rigidity, respectively. These results point to the use of directed electric fields to locally control membrane stiffness, thereby modulating associated cellular processes.

The Plasma Membrane and Mechanoregulation in Cells

Cells inhabit a mechanical microenvironment that they continuously sense and adapt to. The plasma membrane (PM), serving as the boundary of the cell, plays a pivotal role in this process of adaptation. In this Review, we begin by examining well-studied processes where mechanoregulation proves significant. Specifically, we highlight examples from the immune system and stem cells, besides discussing processes involving fibroblasts and other cell types. Subsequently, we discuss the common molecular players that facilitate the sensing of the mechanical signal and transform it into a chemical response covering integrins YAP/TAZ and Piezo. We then review how this understanding of molecular elements is leveraged in drug discovery and tissue engineering alongside a discussion of the methodologies used to measure mechanical properties. Focusing on the processes of endocytosis, we discuss how cells may respond to altered membrane mechanics using endo- and exocytosis. Through the process of depleting/adding the membrane area, these could also impact membrane mechanics. We compare pathways from studies illustrating the involvement of endocytosis in mechanoregulation, including clathrin-mediated endocytosis (CME) and the CLIC/GEEC (CG) pathway as central examples. Lastly, we review studies on cell-cell fusion during myogenesis, the mechanical integrity of muscle fibers, and the reported and anticipated roles of various molecular players and processes like endocytosis, thereby emphasizing the significance of mechanoregulation at the PM.

Piezo1 mechanosensing regulates integrin-dependent chemotactic migration in human T cells

T cells are crucial for efficient antigen-specific immune responses and thus their migration within the body, to inflamed tissues from circulating blood or to secondary lymphoid organs, plays a very critical role. T cell extravasation in inflamed tissues depends on chemotactic cues and interaction between endothelial adhesion molecules and cellular integrins. A migrating T cell is expected to sense diverse external and membrane-intrinsic mechano-physical cues, but molecular mechanisms of such mechanosensing in cell migration are not established. We explored if the professional mechanosensor Piezo1 plays any role during integrin-dependent chemotaxis of human T cells. We found that deficiency of Piezo1 in human T cells interfered with integrin-dependent cellular motility on ICAM-1-coated surface. Piezo1 recruitment at the leading edge of moving T cells is dependent on and follows focal adhesion formation at the leading edge and local increase in membrane tension upon chemokine receptor activation. Piezo1 recruitment and activation, followed by calcium influx and calpain activation, in turn, are crucial for the integrin LFA1 (CD11a/CD18) recruitment at the leading edge of the chemotactic human T cells. Thus, we find that Piezo1 activation in response to local mechanical cues constitutes a membrane-intrinsic component of the 'outside-in' signaling in human T cells, migrating in response to chemokines, that mediates integrin recruitment to the leading edge.

Mesoscale simulation of biomembranes with FreeDTS

We present FreeDTS software for performing computational research on biomembranes at the mesoscale. In this software, a membrane is represented by a dynamically triangulated surface equipped with vertex-based inclusions to integrate the effects of integral and peripheral membrane proteins. Several algorithms are included in the software to simulate complex membranes at different conditions such as framed membranes with constant tension, vesicles and high-genus membranes with various fixed volumes or constant pressure differences and applying external forces to membrane regions. Furthermore, the software allows the user to turn off the shape evolution of the membrane and focus solely on the organization of proteins. As a result, we can take realistic membrane shapes obtained from, for example, cryo-electron tomography and backmap them into a finer simulation model. In addition to many biomembrane applications, this software brings us a step closer to simulating realistic biomembranes with molecular resolution. Here we provide several interesting showcases of the power of the software but leave a wide range of potential applications for interested users.

Mechano-regulation by clathrin pit-formation and passive cholesterol-dependent tubules during de-adhesion

Adherent cells ensure membrane homeostasis during de-adhesion by various mechanisms, including endocytosis. Although mechano-chemical feedbacks involved in this process have been studied, the step-by-step build-up and resolution of the mechanical changes by endocytosis are poorly understood. To investigate this, we studied the de-adhesion of HeLa cells using a combination of interference reflection microscopy, optical trapping and fluorescence experiments. We found that de-adhesion enhanced membrane height fluctuations of the basal membrane in the presence of an intact cortex. A reduction in the tether force was also noted at the apical side. However, membrane fluctuations reveal phases of an initial drop in effective tension followed by saturation. The area fractions of early (Rab5-labelled) and recycling (Rab4-labelled) endosomes, as well as transferrin-labelled pits close to the basal plasma membrane, also transiently increased. On blocking dynamin-dependent scission of endocytic pits, the regulation of fluctuations was not blocked, but knocking down AP2-dependent pit formation stopped the tension recovery. Interestingly, the regulation could not be suppressed by ATP or cholesterol depletion individually but was arrested by depleting both. The data strongly supports Clathrin and AP2-dependent pit-formation to be central to the reduction in fluctuations confirmed by super-resolution microscopy. Furthermore, we propose that cholesterol-dependent pits spontaneously regulate tension under ATP-depleted conditions.

Early tension regulation coupled to surface myomerger is necessary for the primary fusion of C2C12 myoblasts

Here, we study the time-dependent regulation of fluctuation–tension during myogenesis and the role of the fusogen, myomerger. We measure nanometric height fluctuations of the basal membrane of C2C12 cells after triggering differentiation. Fusion of cells increases fluctuation–tension but prefers a transient lowering of tension (at ∼2–24 h). Cells fail to fuse if early tension is continuously enhanced by methyl-β-cyclodextrin (MβCD). Perturbing tension regulation also reduces fusion. During this pre-fusion window, cells that finally differentiate usually display lower tension than other non-fusing cells, validating early tension states to be linked to fate decision. Early tension reduction is accompanied by low but gradually increasing level of the surface myomerger. Locally too, regions with higher myomerger intensity display lower tension. However, this negative correlation is lost in the early phase by MβCD-based cholesterol depletion or later as differentiation progresses. We find that with tension and surface-myomerger’s enrichment under these conditions, myomerger clusters become pronouncedly diffused. We, therefore, propose that low tension aided by clustered surface-myomerger at the early phase is crucial for fusion and can be disrupted by cholesterol-reducing molecules, implying the potential to affect muscle health.

Frame tension governs the thermal fluctuations of a fluid membrane: new evidence

Two different tensions can be defined for a fluid membrane: the internal tension, γ, conjugated to the real membrane area in the Hamiltonian, and the frame tension, τ, conjugated to the projected (or frame) area. According to the standard statistical description of a membrane, the fluctuation spectrum is governed by γ. However, using rotational invariance arguments, several studies argued that the fluctuation spectrum must be governed by the frame tension τ instead. These studies disagree on the origin of the result obtained with the standard description yet: either a miscounting of configurations, quantified with the integration measure, or the use of a quadratic approximation of the Helfrich Hamiltonian. Analyzing the simplest case of a one-dimensional membrane, for which the arc length offers a natural parametrization, we give a new proof that the fluctuations are driven by τ, and show that the origin of the issue with the standard description is a miscounting of membrane configurations. The origin itself of this miscounting depends on the thermodynamic ensemble in which calculations are made.

Effect of heterogeneous substrate adhesivity of follower cells on speed and tension profile of leader cells in primary keratocyte collective cell migration

In single keratocyte motility, membrane tension is reported to be high at cell-fronts and believed to establish front coherence. To understand role of membrane mechanics in collective cell migration, we study membrane height fluctuations in cell sheets from fish scales using interference reflection microscopy (IRM). We report the monolayer to have cells lacking substrate adhesion and show that such ‘non-sticky’ cells can form bridges between leader cells and far-away follower cells. Do such interactions alter motility and membrane mechanics in such leaders? We find non-significant, but reduced speed for leaders with ‘non-sticky’ followers in comparison to other leaders. Cells show high phenotypic variability in their membrane fluctuation tension profiles. On average, this tension is found to be lower at cell fronts than the mid-section. However, leaders with non-sticky followers are more prone to display higher tension at their front and have a negative correlation between cell speed and front-mid tension difference. Thus, we conclude that intracellular tension gradients are heterogeneous in cell sheets and substrate adhesivity of followers can control the coupling of the gradient to cell speed.

Summary: Understanding how leading cells in keratocyte cell sheets are affected when their followers ‘ride’ on them and how this alters their basal membrane's height fluctuations and fluctuation tension.

Undulation of a moving fluid membrane pushed by filament growth

Biomembranes experience out-of-equilibrium conditions in living cells. Their undulation spectra are different from those in thermal equilibrium. Here, we report on the undulation of a fluid membrane pushed by the stepwise growth of filaments as in the leading edge of migrating cells, using three-dimensional Monte Carlo simulations. The undulations are largely modified from equilibrium behavior. When the tension is constrained, the low-wave-number modes are suppressed or enhanced at small or large growth step sizes, respectively, for high membrane surface tensions. In contrast, they are always suppressed for the tensionless membrane, wherein the wave-number range of the suppression depends on the step size. When the membrane area is constrained, in addition to these features, a specific mode is excited for zero and low surface tensions. The reduction of the undulation first induces membrane buckling at the lowest wave-number, and subsequently, other modes are excited, leading to a steady state.

Virtual bending method to calculate bending rigidity, saddle-splay modulus, and spontaneous curvature of thin fluid membranes

A method to calculate the bending rigidity κ, saddle-splay modulus κ[over ¯], and spontaneous curvature C_{0} of a fluid membrane is proposed. Virtual work for the bending deformations into cylindrical and spherical shapes is calculated for a flat membrane. This method does not require a force decomposition, unlike the existing stress-profile method. The first derivative of the deformation gives κC_{0} and is a discrete form of the first moment of the stress profile. The second derivatives give κ and κ[over ¯] and include the variance terms of the first derivatives, which are not accounted for in the stress-profile method. This method is examined for a solvent-free meshless membrane model and a dissipative-particle-dynamics two-bead amphiphilic molecular model. It is concluded that κ and κ[over ¯] of a thin membrane can be accurately calculated, whereas for a thick membrane or one with an explicit solvent, a further extension to include the volume-fluctuation effects is required for an accurate estimation. The amplitude of the volume-fluctuation effects can be evaluated using the parameter dependence in the present method.

Fluctuations and conformational stability of a membrane patch with curvature inducing inclusions

Membranes with curvature inducing inclusions display a range of cooperative phenomena, which can be linked to biomembrane function, e.g. membrane tubulation, vesiculation, softening and spontaneous tension. We investigate how these phenomena are related for a fluctuating, framed membrane through analysis of a descretized membrane model by Monte Carlo simulation techniques. The membrane model is based on a dynamically triangulated surface equipped with non-interacting, up-down symmetry breaking inclusions where only terms coupled linearly to mean-curvature are maintained. We show that the lateral configurational entropy plays a key role for the mechanical properties of the semi-flexible membrane, e.g. a pronounced softening at intermediate inclusion coverages of the membrane and generation of membrane tension. Tensionless framed membranes will remain quasi-flat up to some threshold coverage, where a shape instability occurs with formation of pearling or tubular membranes, which below full coverage is associated with segregation of inclusions between the curved and flat membrane geometries. For inclusions with preference for highly curved membranes the instability appears at dilute inclusion coverages and is accompanied by strong configurational fluctuations.

Simple differences in the protein-membrane attachment mechanism have functional consequences for surface mechanics

This paper describes two methods for propagating coupled membrane and embedded particle dynamics with ensembles that are valid to second order in the deformation of the membrane. Proteins and functional lipids associate with cellular membranes, and their attachments influence membrane physical and dynamical properties. Therefore, it is necessary to accurately model the coupled dynamics of the membrane and any associated material of interest. We have developed two methods for coupling membrane and particle dynamics that differ in the binding mechanism of the particle to the surface. The "on-surface" mechanism should be used for particles that slide along the membrane; this description leads to an effective reduction in the membrane surface tension. The "in-surface" mechanism treats the particles as tightly bound to the lipidic binding sites; the method avoids double counting lateral entropy of implicitly modeled lipids. We emphasize the differences between these two mechanisms, when it is appropriate to use them, and how the methods differ from previously used dynamic methods.

Membrane stretching elasticity and thermal shape fluctuations of nearly spherical lipid vesicles

One of the most widely used methods for determination of the bending elasticity modulus of model lipid membranes is the analysis of the shape fluctuations of nearly spherical lipid vesicles. The theoretical basis of this analysis is given by Milner and Safran [Phys. Rev. A 36, 4371 (1987)0556-279110.1103/PhysRevA.36.4371]. In their theory the stretching effects are not considered. In the present study we generalized their approach including the stretching effects deduced after application of the statistical mechanics to vesicles.

Structure of lateral heterogeneities in a coarse-grained model for multicomponent membranes

We study the lateral domain structure in a coarse-grained molecular model for multicomponent lipid bilayers by semi-grandcanonical Monte Carlo simulations. The membranes are filled with liquid ordered (lo) domains surrounded by a liquid disordered (ld) matrix. Depending on the membrane composition and temperature, we identify different morphological regimes: one regime (I) where the lo domains are small and relatively compact, and two regimes (II, II') where they are larger and often interconnected. In the latter two regimes, the ld matrix forms a network of disordered trenches separating the lo domains, with a relatively high content of interdigitated line defects. Since such defects are also a structural element of the modulated ripple phase in one component membranes, we argue that the regimes II, II' may be amorphous equivalents of the ripple phase in multicomponent membranes. We also analyze the local structure and provide evidence that the domains in regime I are stabilized by a monolayer curvature mechanism postulated in earlier work [S. Meinhardt et al., PNAS, 2013, 110, 4476].

Biophysics of membrane curvature remodeling at molecular and mesoscopic lengthscales

At the micron scale, where cell organelles display an amazing complexity in their shape and organization, the physical properties of a biological membrane can be better-understood using continuum models subject to thermal (stochastic) undulations. Yet, the chief orchestrators of these complex and intriguing shapes are a specialized class of membrane associating often peripheral proteins called curvature remodeling proteins (CRPs) that operate at the molecular level through specific protein-lipid interactions. We review multiscale methodologies to model these systems at the molecular as well as at the mesoscopic and cellular scales, and also present a free energy perspective of membrane remodeling through the organization and assembly of CRPs. We discuss the morphological space of nearly planar to highly curved membranes, methods to include thermal fluctuations, and review studies that model such proteins as curvature fields to describe the emergent curved morphologies. We also discuss several mesoscale models applied to a variety of cellular processes, where the phenomenological parameters (such as curvature field strength) are often mapped to models of real systems based on molecular simulations. Much insight can be gained from the calculation of free energies of membranes states with protein fields, which enable accurate mapping of the state and parameter values at which the membrane undergoes morphological transformations such as vesiculation or tubulation. By tuning the strength, anisotropy, and spatial organization of the curvature-field, one can generate a rich array of membrane morphologies that are highly relevant to shapes of several cellular organelles. We review applications of these models to budding of vesicles commonly seen in cellular signaling and trafficking processes such as clathrin mediated endocytosis, sorting by the ESCRT protein complexes, and cellular exocytosis regulated by the exocyst complex. We discuss future prospects where such models can be combined with other models for cytoskeletal assembly, and discuss their role in understanding the effects of cell membrane tension and the mechanics of the extracellular microenvironment on cellular processes.

The tension of framed membranes from computer simulations

We have analyzed the behavior of a randomly triangulated, self-avoiding surface model of a flexible, fluid membrane subject to a circular boundary by Wang-Landau Monte Carlo computer simulation techniques. The dependence of the canonical free energy and frame tension on the frame area is obtained for flexible membranes. It is shown that for low bending rigidities the framed membrane is only stable above a threshold tension, suggesting a discontinuous transition from the collapsed (branched polymer) state to a finite tension extended state. In a tension range above this threshold tension the membranes display power-law characteristics for the equation of state, while higher tension levels includes both an extended linear (elastic) as well as a highly non-linear stretching regime. For semi-flexible membranes a transition from extended to buckled conformations takes place at negative frame tensions. Our analysis indicates that at zero frame tension the crumpling transition of fluid membranes show characteristics of both critical behavior and a discontinuous transition at low bending rigidities.

Excess area dependent scaling behavior of nano-sized membrane tethers

Thermal fluctuations in cell membranes manifest as an excess area ([Formula: see text]) which governs a multitude of physical process at the sub-micron scale. We present a theoretical framework, based on an in silico tether pulling method, which may be used to reliably estimate [Formula: see text] in live cells. We perform our simulations in two different thermodynamic ensembles: (i) the constant projected area and (ii) the constant frame tension ensembles and show the equivalence of our results in the two. The tether forces estimated from our simulations compare well with our experimental measurements for tethers extracted from ruptured GUVs and HeLa cells. We demonstrate the significance and validity of our method by showing that all our calculations performed in the initial tether formation regime (i.e. when the length of the tether is comparable to its radius) along with experiments of tether extraction in 15 different cell types collapse onto two unified scaling relationships mapping tether force, tether radius, bending stiffness κ, and membrane tension σ. We show that [Formula: see text] is an important determinant of the radius of the extracted tether, which is equal to the characteristic length [Formula: see text] for [Formula: see text], and is equal to [Formula: see text] for [Formula: see text]. We also find that the estimated excess area follows a linear scaling behavior that only depends on the true value of [Formula: see text] for the membrane, based on which we propose a self-consistent technique to estimate the range of excess membrane areas in a cell.

Fluctuation tension and shape transition of vesicles: renormalisation calculations and Monte Carlo simulations

It has been known for long that the fluctuation surface tension of membranes r, computed from the height fluctuation spectrum, is not equal to the bare surface tension σ, which is introduced in the theory either as a Lagrange multiplier to conserve the total membrane area or as an external constraint. In this work we relate these two surface tensions both analytically and numerically. They are also compared to the Laplace tension γ, and the mechanical frame tension τ. Using the Helfrich model and one-loop renormalisation calculations, we obtain, in addition to the effective bending modulus κ_eff, a new expression for the effective surface tension σ_eff = σ - εk_BT/(2a_p) where k_BT is the thermal energy, a_p the projected cut-off area, and ε = 3 or 1 according to the allowed configurations that keep either the projected area or the total area constant. Moreover we show that the crumpling transition for an infinite planar membrane occurs for σ_eff = 0, and also that it coincides with vanishing Laplace and frame tensions. Using extensive Monte Carlo (MC) simulations, triangulated membranes of vesicles made of N = 100-2500 vertices are simulated within the Helfrich theory. As compared to alternative numerical models, no local constraint is applied and the shape is only controlled by the constant volume, the spontaneous curvature and σ. It is shown that the numerical fluctuation surface tension r is equal to σ_eff both with radial MC moves (ε = 3) and with corrected MC moves locally normal to the fluctuating membrane (ε = 1). For finite vesicles of typical size R, two different regimes are defined: a tension regime for [small sigma, Greek, circumflex]_eff = σ_effR²/κ_eff > 0 and a bending one for -1 < [small sigma, Greek, circumflex]_eff < 0. A shape transition from a quasi-spherical shape imposed by the large surface energy, to more deformed shapes only controlled by the bending energy, is observed numerically at [small sigma, Greek, circumflex]_eff ≃ 0. We propose that the buckling transition, observed for planar supported membranes in the literature, occurs for [small sigma, Greek, circumflex]_eff ≃ -1, the associated negative frame tension playing the role of a compressive force. Hence, a precise control of the value of σ_eff in simulations cannot but enhance our understanding of shape transitions of vesicles and cells.

Relaxation rate of the shape fluctuation of a fluid membrane immersed in a near-critical binary fluid mixture

We consider the two-time correlation of the shape fluctuation of a fluid membrane immersed in a near-critical binary fluid mixture. Usually one component of the mixture is preferably attracted by the membrane. Adsorption layers, where the preferred component is more concentrated, are generated on both sides of the membrane significantly because of the near-criticality. The resultant gradient of the local mass-density difference between the two components generates additional stress, including the osmotic pressure, to influence the membrane motion. Assuming the mixture to be in the homogeneous phase near, but not too close to, the demixing critical point, we use the Gaussian free-energy functional to calculate the relaxation rate for a wavelength much longer than the correlation length of the mixture. Our calculation supposes weak preferential attraction and weak dependence of the mixture viscosity on the mass-density difference, and is performed within the linear approximation with respect to the undulation amplitude. It is shown for small wave number that the additional stress makes the relaxation more rapid independently of whether the preferred component is more viscous or not and that the relaxation rate can be regarded as proportional to the wave number even for a tensionless membrane. This linear dependence comes from the balance between the frictional force due to the mixture viscosity and the restoring force of the adsorption layer.