Statistical mechanics of thermal fluctuations of nearly spherical membranes: the influence of bending and stretching elasticities
Nicholay S. Tonchev
备注：
This is a review based on a lecture at the Conference in memory of Vyatcheslav Borisovich Priezzhev held in the Bogoliubov Laboratory of Theoretical Physics (JINR-Dubna, 10 September 2019), 26 pages
Theoretical studies of nearly spherical vesicles and microemulsion droplets, that present typical examples for thermally-excited systems that are subject to constraints, are reviewed. We consider the shape fluctuations of such systems constrained by fixed area $A$ and fixed volume $V$, whose geometry is presented in terms of scalar spherical harmonics. These constraints can be incorporated in the theory in different ways. After an introductory review of the two approaches: with an exactly fixed by delta-function membrane area $A$ [Seifert, Z. Phys. B, 97, 299, (1995)] or approximatively by means of a Lagrange multiplier $σ$ conjugated to $A$ [Milner and Safran, Phys. Rev. A, 36, 4371 (1987)], we discuss the determined role of the stretching effects, that has been announced in the framework of a model containing stretching energy term, expressed via the membrane vesicle tension [Bivas and Tonchev, Phys.Rev.E, 100, 022416 (2019)]. Since the fluctuation spectrum for the used Hamiltonian is not exactly solvable an approximating method based on the Bogoliubov inequalities for the free energy has been developed. The area constraint in the last approach appears as a self-consistent equation for the membrane tension. In the general case this equation is intractable analytically. However, much insight into the physics behind can be obtained either imposing some restrictions on the values of the model parameters, or studying limiting cases, in which the self-consistent equation is solved. Implications for the equivalence of ensembles have been discussed as well.
The dynamics of microcapsules in steady shear flow was studied using a theoretical approach based on three variables: The Taylor deformation parameter $α_{\rm D}$, the inclination angle $θ$, and the phase angle $ϕ$ of the membrane rotation. It is found that the dynamic phase diagram shows a remarkable change with an increase in the ratio of the membrane shear and bending elasticities. A fluid vesicle (no shear elasticity) exhibits three dynamic modes: (i) Tank-treading (TT) at low viscosity $η_{\rm {in}}$ of internal fluid ($α_{\rm D}$ and $θ$ relaxes to constant values), (ii) Tumbling (TB) at…
Transition from bending-dominated to stretching-dominated elastic response in semi-flexible fibrous networks plays an important role in the mechanical behavior of cells and tissues. It is induced by changes in network connectivity and relies on construction of new cross-links. We propose a simple continuum model of this transition with macroscopic strain playing the role of order parameter. An unusual feature of this Landau-type theory is that it is based on a single-well potential. We predict that bending-to-stretching transition proceeds through propagation of the localized fronts separating…
Stretching and bending of (really) thick elastic plates
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）
We study the spontaneous out-of-plane bending of a planar untwisted ribbon composed of nematic polymer networks activated by a change in temperature. Our theory accounts for both stretching and bending energies, which compete to establish equilibrium. We show that when equilibrium is attained these energy components obey a \emph{complementarity} relation: one is maximum where the other is minimum. Moreover, we identify a \emph{bleaching} regime: for sufficiently large values of an activation parameter (which measures the mismatch between the degrees of order in polymer organization in the refe…
We study the effect of intersegment interactions on the effective bending and stretching moduli of a semiflexible polymer chain with a finite stretching modulus. For an interaction potential of a screened Debye-H\" uckel type renormalization of the stretching modulus is derived on the same level of approximation as the celebrated Odijk-Skolnick-Fixman result for the bending modulus. The presence of mesoscopic intersegment interaction potentials couples the bending and stretching moduli in a manner different from that predicted by the macroscopic elasticity theory. We advocate a fundamental cha…
The balance between stretching and bending deformations characterizes shape transitions of thin elastic sheets. While stretching dominates the mechanical response in tension, bending dominates in compression after an abrupt buckling transition. Recently, experimental results in suspended living epithelial monolayers have shown that, due to the asymmetry in surface stresses generated by molecular motors across the thickness $e$ of the epithelium, the free edges of such tissues spontaneously curl out-of-plane, stretching the sheet in-plane as a result. This suggests that a competition between be…
We develop further a statistical model coupling denaturation and chain conformations in DNA (Palmeri J, Manghi M and Destainville N 2007 Phys. Rev. Lett. 99 088103). Our Discrete Helical Wormlike Chain model takes explicitly into account the three elastic degrees of freedom, namely stretching, bending and torsion of the polymer. By integrating out these external variables, the conformational entropy contributes to bubble nucleation (opening of base-pairs), which sheds light on the DNA melting mechanism. Because the values of monomer length, bending and torsional moduli differ significantly in …
We show that gels formed by arrested spinodal decomposition of protein solutions exhibit elastic properties in two distinct frequency domains, both elastic moduli exhibiting a remarkably strong dependence on volume fraction. Considering the large difference between the protein size and the characteristic length of the network we model the gels as porous media and show that the high and low frequency elastic moduli can be respectively attributed to stretching and bending modes. The unexpected decoupling of the two modes in the frequency domain is attributed to the length scale involved: while s…
We study the shapes of human red blood cells using continuum mechanics. In particular, we model the crenated, echinocytic shapes and show how they may arise from a competition between the bending energy of the plasma membrane and the stretching/shear elastic energies of the membrane skeleton. In contrast to earlier work, we calculate spicule shapes exactly by solving the equations of continuum mechanics subject to appropriate boundary conditions. A simple scaling analysis of this competition reveals an elastic length which sets the length scale for the spicules and is, thus, related to the num…
Nematic polymeric networks are (heat and light) activable materials, which combine the features of rubber and nematic liquid crystals. When only the stretching energy of a thin sheet of nematic polymeric network is minimized, the intrinsic (Guassian) curvature of the shape it takes upon (thermal or optical) actuation is determined. This, unfortunately, produces a multitude of possible shapes, for which we need a selection criterion, which may only be provided by a correcting bending energy depending on the extrinsic curvatures of the deformed shape. The literature has so far offered approximat…
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. In their theory the stretching effects are not considered. In the present study we generalized their approach including the stretching effects deduced after an application of statistical mechanics of vesicles.
Some important biomolecules (for instance, bacterial FtsZ and eukaryotic DNA) are known to posses spontaneous (intrinsic) curvature. Using a simple extension of the wormlike chain model, we study the response of a weakly bending filament in two dimensions to a pulling force applied at its ends (a configuration common in classical in-vitro experiments and relevant to several in-vivo cell cases). The spontaneous curvature of such a chain or filament can in general be arc-length dependent and we study a case of sinusoidal variation, from which an arbitrary case can be reconstructed via Fourier tr…
Shells are ubiquitous thin structures that can undergo large nonlinear elastic deformations while exhibiting combined modes of bending and stretching, and have profound modern applications. In this paper, we have proposed a new Isogeometric formulation, based on classical Koiter nonlinear shell theory, to study instability problems like wrinkling and buckling in thin shells. The use of NURBS-basis provides rotation-free, conforming, higher-order spatial continuity, such that curvatures and membrane strains can be computed directly from the interpolation of the position vectors of the control p…
We present a multi-resolution methodology for modelling F-actin filaments. It provides detailed microscopic information at the level of individual monomers at a lower computational cost by replacing the monomer-based model in parts of the simulated filament by a rod-based macroscopic model. In the monomer-based description, G-actin is represented by ellipsoids bound at the surface in a double helical configuration to form F-actin. The rod-based model is coarser, in which F-actin is described using a Cosserat model, as seen in the preceding paper [arXiv:2112.01480]. The multi-resolution methodo…
We study the adhesion of an elastic sheet on a rigid spherical substrate. Gauss'Theorema Egregium shows that this operation necessarily generates metric distortions (i.e. stretching) as well as bending. As a result, a large variety of contact patterns ranging from simple disks to complex branched shapes are observed as a function of both geometrical and material properties. We describe these different morphologies as a function of two non-dimensional parameters comparing respectively bending and stretching energies to adhesion. A complete configuration diagram is finally proposed.
Networks of elastic beams can deform either by stretching or bending of their members. The primary mode of deformation (bending or stretching) crucially depends on the specific details of the network architecture. In order to shed light on the relationship between microscopic geometry and macroscopic mechanics, we characterize the structural features of networks which deform uniformly, through the stretching of the beams only. We provide a convenient set of geometrical criteria to identify such networks, and derive the values of their effective elastic moduli. The analysis of these criteria el…
We present a perturbation theory to find the response of an anisotropic DNA to the external tension. It is shown that the anisotropy has a nonzero but small contribution to the force-extension curve of the DNA. Thus an anisotropic DNA behaves like an isotropic one with an effective bending constant equal to the harmonic average of its soft and hard bending constants.
Mechanochemical simulations of actomyosin networks are traditionally based on one-dimensional models of actin filaments having zero width. Here, and in the follow up paper, approaches are presented for more efficient modelling which incorporates stretching, bending, shearing and twisting of actin filaments. Our modelling of a semi-flexible filament with a small but finite width is based on the Cosserat theory of elastic rods, which allows for six degrees of freedom at every point on the filament's backbone. In the variational models presented in this paper, a small and discrete set of paramete…
We present an effective medium theory that includes bending as well as stretching forces, and we use it to calculate mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability $p$. We find a rigidity threshold $p_b$ which has the same value for all positive bending rigidity and a crossover characterizing bending-, stretching-, and bend-stretch coupled elastic regimes controlled by…
We study the elasticity of random stiff fiber networks. The elastic response of the fibers is characterized by a central force stretching stiffness as well as a bending stiffness that acts transverse to the fiber contour. Previous studies have shown that this model displays an anomalous elastic regime where the stretching mode is fully frozen out and the elastic energy is completely dominated by the bending mode. We demonstrate by simulations and scaling arguments that, in contrast to the bending dominated \emph{elastic energy}, the equally important \emph{elastic forces} are to a large extent…
The flexibility of short DNA fragments is studied by a Hamiltonian model which treats the inter-strand and intra-strand forces at the level of the base pair. The elastic response of a set of homogeneous helices to externally applied forces is obtained by computing the average bending angles between adjacent base pairs along the molecule axis. The ensemble averages are performed over a room temperature equilibrium distribution of base pair separations and bending fluctuations. The analysis of the end-to-end distances and persistence lengths shows that even short sequences with less than $100$ b…
The study of elastic membranes carrying topological defects has a longstanding history, going back at least to the 1950s. When allowed to buckle in three-dimensional space, membranes with defects can totally relieve their in-plane strain, remaining with a bending energy, whose rigidity modulus is small compared to the stretching modulus. In this paper, we study membranes with a single edge-dislocation. We prove that the minimum bending energy associated with strain-free configurations diverges logarithmically with the size of the system.
We investigate the effect on biomembrane mechanical properties due to the presence an external potential for a non-conductive non-compressible membrane surrounded by different electrolytes. By solving the Debye-Huckel and Laplace equations for the electrostatic potential and using the relevant stress-tensor we find: in (1.) the small screening length limit, where the Debye screening length is smaller than the distance between the electrodes, the screening certifies that all electrostatic interactions are short-range and the major effect of the applied potential is to decrease the membrane tens…
Single molecule experiments on B-DNA stretching have revealed one or two structural transitions, when increasing the external force. They are characterized by a sudden increase of DNA contour length and a decrease of the bending rigidity. It has been proposed that the first transition, at forces of 60--80 pN, is a transition from B to S-DNA, viewed as a stretched duplex DNA, while the second one, at stronger forces, is a strand peeling resulting in single stranded DNAs (ssDNA), similar to thermal denaturation. But due to experimental conditions these two transitions can overlap, for instance f…
We present a theory for the elasticity of cross-linked stiff polymer networks. Stiff polymers, unlike their flexible counterparts, are highly anisotropic elastic objects. Similar to mechanical beams stiff polymers easily deform in bending, while they are much stiffer with respect to tensile forces (``stretching''). Unlike in previous approaches, where network elasticity is derived from the stretching mode, our theory properly accounts for the soft bending response. A self-consistent effective medium approach is used to calculate the macroscopic elastic moduli starting from a microscopic charac…
We perform Monte Carlo simulations to study the elastic properties of the helix-coil worm-like chain model of alpha-helical polypeptides. In this model the secondary structure enters as a scalar (Ising like) variable that controls the local chain bending modulus. We compute numerically the bending and stretching compliances of this molecule as well as the nonlinear interaction between stretching and torque over a wide range of model parameters. The numerical results agree well with previous mean-field and perturbative calculations where they are expected to do so. The Monte Carlo simulations a…
We model the spontaneous assembly of a capsid (a virus's closed outer shell) from many copies of identical units, using entirely irreversible steps and only information local to the growing edge. Our model is formulated in terms of (i) an elastic Hamiltonian with stretching and bending stiffness and a spontaneous curvature, and (ii) a set of rate constants for addition of new units or bonds. An ensemble of highly irregular capsids is generated, unlike the well-known icosahedrally symmetric viruses, but (we argue) plausible as a way to model the irregular capsids of retroviruses such as HIV. We…
We employ computer simulations and thermodynamic integration to analyse the effects of bending rigidity and slit confinement on the free energy cost of tying knots, $ΔF_{\rm knotting}$, on polymer chains under tension. A tension-dependent, non-zero optimal stiffness $κ_{\rm min}$ exists, for which $ΔF_{\rm knotting}$ is minimal. For a polymer chain with several stiffness domains, each containing a large amount of monomers, the domain with stiffness $κ_{\rm min}$ will be preferred by the knot. A {\it local} analysis of the bending in the interior of the knot reveals that local stretching of cha…
We use computer simulations and a simple free energy model to study the response of a bilayer membrane to the application of a negative (compressive) mechanical tension. Such a tension destabilizes the long wavelength undulation modes of giant vesicles, but it can be sustained when small membranes and vesicles are considered. Our negative tension simulation results reveal two regimes - (i) a weak negative tension regime characterized by stretching-dominated elasticity, and (ii) a strong negative tension regime featuring bending-dominated elastic behavior. This resembles the findings of the cla…
Using a fully analytical theory, we compute the leading order corrections to the translational, rotational and translation-rotation coupling mobilities of an arbitrary axisymmetric particle immersed in a Newtonian fluid moving near an elastic cell membrane that exhibits resistance towards stretching and bending. The frequency-dependent mobility corrections are expressed as general relations involving separately the particle's shape-dependent bulk mobility and the shape-independent parameters such as the membrane-particle distance, the particle orientation and the characteristic frequencies ass…
