Equilibrium Shape Equation and Geometrically Permissible Condition for Two-Component Lipid Bilayer Vesicles

Equilibrium shapes of vesicles composed of a mixture of partially miscible amphiphiles are investigated. To take into account the influences of the composition, a simple phenomenological coupling between the co mposition and the curvatures, including the mean curvature and the Gauss curvature of the membrane surface, is suggested. By minimizing the potential functional, the general shape equation is obtained and solved analytically for vesicles with simple shapes. Besides, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are put forward. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. The close relations between the predictions and existing experimental phenomena published recently are shown.     

General Mathematical Frame for Open or Closed Biomembranes (Part I): Equilibrium Theory and Geometrically Constraint Equation

This paper aims at constructing a general mathematical frame for the equilibrium theory of open or closed biomembranes. Based on the generalized potential functional, the equilibrium differential equation for open biomembrane (with free edge) or closed one (without boundary) is derived. The boundary conditions for open biomembranes are obtained. Besides, the geometrically constraint equation for the existence, formation and disintegration of open or closed biomembranes is revealed. The physical and biological meanings of the equilibrium differential equation and the geometrically constraint equation are discussed. Numerical simulation results for axisymmetric open biomembranes show the effectiveness and convenience of the present theory.     

Geometric and thermodynamic restrictions for the self-assembly of glycosphingolipid-phospholipid systems

The thermodynamic and geometrical features of possible self-assembled structures of a series of chemically related glycosphingolipids differing in the complexity of their polar headgroup, and of their mixture with phospholipids, have been predicted according to the theory of self-assembly of hydrocarbon amphiphiles of Israelachvili et al. ((1980) Q. Rev. Biophys. 13, 340-357). The type and number of carbohydrate residues in the oligosaccharide chain of the polar headgroup are of paramount importance to determine the characteristics and thermodynamic stability of the possible self-assembled structure. In single component systems, the general prediction of the theory is that smaller aggregates may form as the polar headgroup of the glycosphingolipid is more complex and as the lateral surface pressure is smaller. In noninteracting two-component glycosphingolipid-phospholipid systems, the thermodynamic stability and the overall geometry of the possible aggregate appear to be determined by the proportion and type of glycosphingolipid present. Large and abrupt changes of the possible free energy per molecule, radius of curvature, and predicted asymmetry ratio for a particular glycosphingolipid may be triggered by relatively small changes of the molecular parameters, lipid composition, lateral surface pressure or vice-versa. If intermolecular interactions are taken into account with respect to the predictions for an ideal, noninteracting system, the theory indicates that two-component bilayer vesicles of polysialoganglioside-phosphatidylcholine may be thermodynamically and geometrically more stable. On the other hand, for systems constituted by phosphatidylcholine and neutral glycosphingolipids or monosialogangliosides, the possible bilayer vesicle is predicted to be less stable than in the ideal, noninteracting case. The results emphasize the general validity of the theory as applied to glycosphingolipid-containing systems.     

Mathematical and numerical models for transfer of low-density lipoproteins through the arterial walls: a new methodology for the model set up with applications to the study of disturbed lumenal flow

In this work we introduce and discuss several mathematical models, based on partial differential equations, devised to study the coupled transport of macromolecules as low-density lipoproteins in the blood stream and in the arterial walls. These models are accurate provided that a suitable set of physical parameters characterizing the physical properties of the molecules and of the wall layers are available. Here we turn our attention on this aspect, and propose a new methodology to compute the physical parameters needed for the model set up, starting from available in vivo measurements. Then, we focus on the study of the accumulation of low-density lipoproteins in vascular districts featuring a highly disturbed flow. Our results demonstrate that mathematical models whose set up procedure benefits from an experimental feedback provide reliable information not only qualitatively, but also quantitatively. Their application to geometrically perturbed vascular districts (as for example a severe stenosis) shows that geometrical parameters such as curvature and variations of the lumenal section strongly influence the accumulation of low-density lipoproteins within the wall. For instance, in a stenotic segment with 75% area constriction, the LDL concentration at the lumenal side of the wall is about 10% higher than for the undisturbed segment.     

A curvature-mediated mechanism for localization of lipids to bacterial poles

Subcellular protein localization is a universal feature of eukaryotic cells, and the ubiquity of protein localization in prokaryotic species is now acquiring greater appreciation. Though some targeting anchors are known, the origin of polar and division-site localization remains mysterious for a large fraction of bacterial proteins. Ultimately, the molecular components responsible for such symmetry breaking must employ a high degree of self-organization. Here we propose a novel physical mechanism, based on the two-dimensional curvature of the membrane, for spontaneous lipid targeting to the poles and division site of rod-shaped bacterial cells. If one of the membrane components has a large intrinsic curvature, the geometrical constraint of the plasma membrane by the more rigid bacterial cell wall naturally leads to lipid microphase separation. We find that the resulting clusters of high-curvature lipids are large enough to spontaneously and stably localize to the two cell poles. Recent evidence of localization of the phospholipid cardiolipin to the poles of bacterial cells suggests that polar targeting of some proteins may rely on the membrane's differential lipid content. More generally, aggregates of lipids, proteins, or lipid-protein complexes may localize in response to features of cell geometry incapable of localizing individual molecules.     

Unsupervised neural learning on lie group

The present paper aims at introducing the concepts and mathematical details of unsupervised neural learning with orthonormality constrains. The neural structures considered are single non-linear layers and the learnable parameters are organized in matrices, as usual, which gives the parameters spaces the geometrical structure of the Euclidean manifold. The constraint of orthonormality for the connection-matrices further restricts the parameters spaces to differential manifolds such as the orthogonal group, the compact Stiefel manifold and its extensions. For these reasons, the instruments for characterizing and studying the behavior of learning equations for these particular networks are provided by the differential geometry of Lie groups. In particular, two sub-classes of the general Lie-group learning theories are studied in detail, dealing with first-order (gradient-based) and second-order (non-gradient-based) learning. Although the considered class of learning theories is very general, in the present paper special attention is paid to unsupervised learning paradigms.     

Optimal models for social change

By assigning coordinates to the environmental function space comprising all physical and mental stimuli, mathematical interpretations can be based on such terms as adaptability, and reactivity which relate to individuals interacting with their environment within a society. These psychometric concepts are incorporated into a framework of functional analysis, which permits the optimization of social change by maximizing the satisfaction integral through the use of variational or dynamic programming methods in conjunction with some optimal social policy. The approach provides a mathematical connection between psychology and sociology, and further demonstrates that existing forms of government are simulated by differential equations belonging to the same general class. The synthesis of new classes of functional equations describing social progress is visualized as a legitimate objective for abstract mathematical sociology.     

A Lyapunov function for piecewise-independent differential equations: stability of the ideal free distribution in two patch environments

In this article we construct Lyapunov functions for models described by piecewise-continuous and independent differential equations. Because these models are described by discontinuous differential equations, the theory of Lyapunov functions for smooth dynamical systems is not applicable. Instead, we use a geometrical approach to construct a Lyapunov function. Then we apply the general approach to analyze population dynamics describing exploitative competition of two species in a two-patch environment. We prove that for any biologically meaningful parameter combination the model has a globally stable equilibrium and we analyze this equilibrium with respect to parameters.      

Elastic curvature constants of lipid monolayers and bilayers

Bending elasticity is an important property of lipid vesicles, non-lamellar lipid phases and biological membranes. Experimental values of the mean curvature moduli, k(c), of lipid bilayers and of the monolayer leaflets of inverted hexagonal (H(II)) phases of lipids are tabulated here for easy reference. Experimental estimates of the Gaussian curvature modulus, k (c), are also included. Consideration is given to the relation between the bending moduli of bilayers and the constituent monolayer leaflets. Useful mathematical relations involving the bending moduli and spontaneous curvature are summarized.     

Mechanical collapse of confined fluid membrane vesicles

Compact cylindrical and spherical invaginations are common structural motifs found in cellular and developmental biology. To understand the basic physical mechanisms that produce and maintain such structures, we present here a simple model of vesicles in confinement, in which mechanical equilibrium configurations are computed by energy minimization, balancing the effects of curvature elasticity, contact of the membrane with itself and the confining geometry, and adhesion. For cylindrical confinement, the shape equations are solved both analytically and numerically by finite element analysis. For spherical confinement, axisymmetric configurations are obtained numerically. We find that the geometry of invaginations is controlled by a dimensionless ratio of the adhesion strength to the bending energy of an equal area spherical vesicle. Larger adhesion produces more concentrated curvatures, which are mainly localized to the “neck” region where the invagination breaks away from its confining container. Under spherical confinement, axisymmetric invaginations are approximately spherical. For extreme confinement, multiple invaginations may form, bifurcating along multiple equilibrium branches. The results of the model are useful for understanding the physical mechanisms controlling the structure of lipid membranes of cells and their organelles, and developing tissue membranes.     

Kinematics,material symmetry,and energy densities for lipid bilayers with spontaneous curvature

Continuum mechanical tools are used to describe the deformation, energy density, and material symmetry of a lipid bilayer with spontaneous curvature. In contrast to conventional approaches in which lipid bilayers are modeled by material surfaces, here we rely on a three-dimensional approach in which a lipid bilayer is modeling by a shell-like body with finite thickness. In this setting, the interface between the leaflets of a lipid bilayer is assumed to coincide with the mid-surface of the corresponding shell-like body. The three-dimensional deformation gradient is found to involve the curvature tensors of the mid-surface in the spontaneous and the deformed states, the deformation gradient of the mid-surface, and the transverse deformation. Attention is also given to the coherency of the leaflets and to the area compatibility of the closed lipid bilayers (i.e., vesicles). A hyperelastic constitutive theory for lipid bilayers in the liquid phase is developed. In combination, the requirements of frame indifference and material symmetry yield a representation for the energy density of a lipid bilayer. This representation shows that three scalar invariants suffice to describe the constitutive response of a lipid bilayer exhibiting in-plane fluidity and transverse isotropy. In addition to exploring the geometrical and physical properties of these invariants, fundamental constitutively associated kinematical quantities are emphasized. On this basis, the effect on the energy density of assuming that the lipid bilayer is incompressible is considered. Lastly, a dimension reduction argument is used to extract an areal energy density per unit area from the three-dimensional energy density. This step explains the origin of spontaneous curvature in the areal energy density. Importantly, along with a standard contribution associated with the natural curvature of the lipid bilayer, our analysis indicates that constitutive asymmetry between the leaflets of the lipid bilayer gives rise to a secondary contribution to the spontaneous curvature.     

Effects of curvature and composition on α-synuclein binding to lipid vesicles

Parkinson's disease is characterized by the presence of intracellular aggregates composed primarily of the neuronal protein α-synuclein (αS). Interactions between αS and various cellular membranes are thought to be important to its native function as well as relevant to its role in disease. We use fluorescence correlation spectroscopy to investigate binding of αS to lipid vesicles as a function of the lipid composition and membrane curvature. We determine how these parameters affect the molar partition coefficient of αS, providing a quantitative measure of the binding energy, and calculate the number of lipids required to bind a single protein. Specific anionic lipids have a large effect on the free energy of binding. Lipid chain saturation influences the binding interaction to a lesser extent, with larger partition coefficients measured for gel-phase vesicles than for fluid-phase vesicles, even in the absence of anionic lipid components. Although we observe variability in the binding of the mutant proteins, differences in the free energies of partitioning are less dramatic than with varied lipid compositions. Vesicle curvature has a strong effect on the binding affinity, with a >15-fold increase in affinity for small unilamellar vesicles over large unilamellar vesicles, suggesting that αS may be a curvature-sensing protein. Our findings provide insight into how physical properties of the membrane may modulate interactions of αS with cellular membranes.     

A parameter identification problem arising from a model of canalicular bile formation

We develop a simple mathematical model for bile formation and analyze some features of the model that suggest the design for future physiological experiments. The mathematical model results in a boundary value problem for a system of functional differential equations depending on several physical parameters. From the observability of the boundary values we can identify, both qualitatively and quantitatively, some of these physical parameters. This identification then suggests physical experiments from which one could infer some of the bile transport phenomena that are not, at present, directly observable. The mathematical parameter identification problem is solved by converting the boundary value problem to a transition time problem for a quadratic system of ordinary differential equations on the plane where we are able to employ some special properties of quadratic systems in order to obtain a solution.The author was supported by the Air Force Office of Scientific Research and the National Science Foundation under the grants AF-AFOSR-89-0078 and DMS-9022621The author was supported by National Institutes of Health under grant number R37 DK-27623     

球形拓扑下对budding和multi-budding生物膜泡形状的研究

流体相磷脂双分子层在水溶液中能自组织的形成球形拓扑的膜泡,膜泡的平衡形状是由其弯曲弹性能决定的.Budding是流体相磷脂双分子膜泡的一个显著的形状.从自发曲率(SC)模型的弯曲能量和双层(BC)模型的弯曲能量在参数替换情况下,我们看到自发曲率(SC)和双层(BC)模型具有相同的形状方程.本文从双层(BC)模型的相图出发,在双层(BC)模型下,通过建立一些与目标形状相近似的初始形状,在面积A,体积V和平均曲率的积分M的约束条件和使用约化量的情况下,经Surface Evolver软件的逐次细分并长时间演化,得到了一些budding和multi-budding型的生物膜泡形状.     

Synergetics of the Membrane Self-Assembly: A Micelle-to-Vesicle Transition

A model approach is developed to study intermediate steps and transientstructures in a course of the membrane self-assembly. The approach isbased on investigation of mixed lipid/protein-detergent systems capable ofthe temperature induced transformation from a solubilized micellar stateto closed membrane vesicles. We performed a theoretical analysis ofself-assembling molecular structures formed in binary mixtures ofdimyristoylphosphatidylcholine (DMPC) and sodium cholate (NaC). Thetheoretical model is based on the Helfrich theory of curvature elasticity,which relates geometrical shapes of the structures to their free energy inthe Ginzburg-Landau approximation. The driving force for the shapetransformation is spontaneous curvature of amphiphilic aggregates which isnonlinearly dependent on the lipid/detergent composition. An analysis ofthe free energy in the regular solution approximation shows that theformation of mixed structures of different shapes (discoidal micelles,rod-like micelles, multilayer membrane structures and vesicles) ispossible in a certain range of detergent/lipid ratios. A transition fromthe flat discoidal micelles to the rod-like cylindrical micelles isinduced by curvature instabilities resulting from acyl chain melting andinsertion of detergent molecules into the lipid phase. Nonideal mixing ofthe NaC and DMPC molecules results in formation of nonideal cylindricalaggregates with elliptical cross section. Further dissolution of NaCmolecules in DMPC may be accompanied with a change of their orientation inthe lipid phase and leads to temperature-induced curvature instabilitiesin the highly curved cylindrical geometry. As a result the rod-likemicelles fuse into less curved bilayer structures which transformeventually to the unilamellar and multilamellar membrane vesicles. Thetheoretical analysis performed shows that a sequence of shapetransformations in the DMPC/NaC mixed systems is determined by thesynergism of four major factors: detergent/lipid ratio, temperature (acylchain melting), DMPC and NaC mixing, and reorientation of NaC molecules inmixed aggregates.     

A novel leaflet-selective fluorescence labeling technique reveals differences between inner and outer leaflets at high bilayer curvature

Understanding the differences in the physical properties of the inner and outer leaflet of membranes and how the leaflets are coupled to each other requires methods that can selectively label both the outer and inner leaflets. In this report we introduce a combined chromatography/cyclodextrin method for selective labeling of the inner leaflet. Combining this method with selective labeling of the outer leaflet, we are able to show that there is a distinct difference in polar headgroup physical properties of the inner and outer leaflet headgroups in small unilamellar vesicles composed of a wide variety of phosphatidylcholines and a phosphaticylcholine/sphingomyelin mixture. It appears that the inner leaflet headgroups are more tightly packed than those of the outer leaflet. This differential packing disappears when vesicle size increases, showing that it is a consequence of membrane curvature. Differential packing is also reduced as acyl chain length is decreased. In the future, selective leaflet labeling is likely to be a powerful tool for investigating the properties of asymmetric lipid vesicles.     

Light scattering characterization of extruded lipid vesicles

By modeling extruded unilamellar lipid vesicles as thin-walled ellipsoidal shells, mathematical analysis provides simple equations which relate the mean elongation and other morphological characteristics of a vesicle population to quantities readily obtained from combined static and dynamic light scattering measurements. For SOPC vesicles extruded through a 100 nm pore-size filter into a 72.9 mm NaCl solution, the inferred elongation ratio (vesicle long axis to short axis) is approximately 3.7±0.6. When these vesicles were dialyzed into hypertonic or hypotonic solutions, this elongation ratio varied from 1 (for spherical liposomes) in strongly hypotonic solutions to greater than 6 in increasingly hypertonic solutions, beyond which abrupt morphological transformations appear. These results are quantitatively consistent with a mechanism of vesicle formation by extrusion and with the expectation that vesicle volumes change to equalize internal and external osmolarity via water flow, subject to the constraint of constant bilayer area. Our analysis also provides simplified equations to assess the effects of vesicle elongation and polydispersity on liposome parameters that are commonly required to characterize vesicle preparations for diverse applications. The implications of this study for routine light scattering characterization of extruded vesicles are discussed. Received: 31 July 1998 / Revised version: 26 October 1998 / Accepted: 5 November 1998