Elastic membrane model

Following on from this work two types of mathematical model were developed that do not rely on measuring the contact area. These models are the "liquid-drop" model (Yoneda, 1973) and the elastic membrane model (Cheng, 1987a Feng and Yang, 1973 Lardner and Pujara, 1980). 

An analytical elastic membrane model was developed by Feng and Yang (1973) to model the compression of an inflated, non-linear elastic, spherical membrane between two parallel surfaces where the internal contents of the cell were taken to be a gas. This model was extended by Lardner and Pujara (1980) to represent the interior of the cell as an incompressible liquid. This latter assumption obviously makes the model more representative of biological cells. Importantly, this model also does not assume that the cell wall tensions are isotropic. The model is based on a choice of cell wall material constitutive relationships (e.g., linear-elastic, Mooney-Rivlin) and governing equations, which link the constitutive equations to the geometry of the cell during compression. 

The elastic membrane model assumes that the cell is a thin-walled sphere filled with incompressible fluid. Because the wall is thin, it may be treated as a mechanical membrane. It can be presumed that the wall cannot support out-of-plane shear stresses or bending moments. This situation is described as plane stress, as the only non-zero stresses are in the plane of the cell wall. Furthermore, the stresses can be expressed as... 

It is assumed that the cell is symmetrical across the equatorial plane and axi-symmetrical around the axis of compression, the tj axis. This symmetry allows the compression of the cell to be fully represented by a 2D curve in the positive r and p axis. To understand the elastic membrane model, the geometry of the spherical cell under compression can be represented by Figure 10. 

The elastic membrane model, formulated in terms of elastic moduli and u r), provides a significantly reduced description of insertion phenomena. More detailed analysis should account for the orientation and displacement of the lipid molecules as well as some of their internal degrees of freedom. A step in this direction has been made, for instance, in Ref. 95. At short-length scales and near nonuniformities, lipid molecules cannot attain the normal orientations typical of their mean behavior on a macroscopic scale, which must inevitably affect their elastic properties. More detailed statistical mechanical analysis and simulational studies might provide useful insight into such behavior. 

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

Initially the effect of applied voltage on membrane capacitance was attributed to the uniform electrostriction, in the manner of the elastic capacitor model [1,103], The effect of undulations was first considered by Leikin [78], In Ref. 89 the combined effect of undulations and uniform compression is studied, including the possible influence of nonlocality. The differential capacitance C is presented as... 

The set of constitutive parameters contains the (drained) elastic volumetric compliance C and two poroelastic constants the Biot stress coefficient b, and the unconstrained storage coefficient Sa = d(/dp a which can be expressed as So- = bB 1C ([13]), where B is the Skempton pore pressure coefficient. The other three parameters, a, f3, and 7 quantify the physico-chemical interactions. Both a and (3 are constrained to vary from 0 when there is no chemical interaction to 1 when the salt ions are trapped in the pore network (this limiting case is referred to as the perfect ion exclusion membrane model ). The coefficient 7 can simply be approximated by 7 x0/n, where n is the porosity of the shale. 

Meleard, P., Gerbeaud, C., Pott, T., FernandezPuente, L., Bivas, L, Mitov, M.D., Dufourcq, J., and Bothorel, P. (1997) Bending elasticities of model membranes influences of temperature and sterol content. Biophysical Journal,... 

Peeling [Dembo et al., 1988] Clamped elastic membrane Bond stress and chemical rate constants are related to bond strain Bonds are linear springs fixed in the plane of the membrane Chemical reaction of bond formation and breakage is reversible Diffusion of adhesion molecules is negligible Critical tension to overcome the tendency of the membrane to spread over the surface can be calculated Predictions of model depend whether the bonds are catch-bonds or slip-bonds If adhesion is mediated by catch-bonds, then no matter how much tension is applied, it is impossible to separate membrane and surface... 

The concept of two distinct but interlinked mechanical processes, expanded here as the coupling of hydrophobic and elastic consilient mechanisms, entered the public domain in the publication of Urry and Parker. Experimental results on elastic-contractile model proteins forged the concept, and the work of Urry and Parker extended the concept to contraction in biology. Unexpected in our examination of the relevance of this perspective to biology was to find the first clear demonstration of the concept in biology in a protein-based machine of the electron transport chain as a transmembrane protein of the inner mitochondrial membrane. Unimaginable was the occurrence of the coupled forces precisely at the nexus at which electron transfer couples to proton pumping. 

To determine the valne of the adhesive fracture energy, it is necessary to decide the mode of deformation of the pressurized layer. In the case of a relatively thin blister, the mode of deformation is considered to be mainly that of tensile deformation of the blister, and the blister is then modelled as an elastic membrane. Alternatively, in the case of a relatively thick blister, the pressurized layer is considered to deform mainly by bending, and this is modelled as an elastic circular plate with a built-in edge constraint. A further contribution to the stored elastic energy, which is available to assist growth of a debond, arises from an internal stress inherent in the test specimen.Snch stresses may be inlrodnced during... 

In the case where the membrane is deformed, the deformation profiles can be compared to a variety of theories [16,17,27, 33, 245-247]. Both in coarse-grained [30,234] and atomistic [248] simulations, it was reported that membrane thickness profiles as a function of the distance to the protein are not strictly monotonic, but exhibit a weakly oscillatory behavior. This feature is not compatible with membrane models that predict an exponential decay [16,17,27], but it is nicely captured by the coupled elastic monolayer models discussed earlier [22, 28, 30]. Coarsegrained simulations of the Lenz model showed that the coupled monolayer models describe the profile data at a quantitative level, with almost no fit parameters except the boundary conditions [30, 244]. 

The qualitative aspects of behavior of a pressurized bulge in a him are rendered most transparent when considered at the level of the membrane idealization. For example, for the most rudimentary membrane model depicted in Figure 5.32, it follows that the elastic strain energy for any given... 

In another model designed by Thompson et al. [35], pressure is created by cyclical air inflow and the amount of pressure introduced in the system is controlled by a check valve. The air comes into contact with the culture medium in the so-called driving shaft. Tubing with medium is connected to the unit where the constructs are placed this unit is called the manifold. Constructs can be accommodated in series or parallel, with a capacity of six scaffolds. Real-time flow and pressure can be monitored using an in-Kne flow meter and pressure transducer [35]. Hoerstrup et al. [43] designed a similar system in which the pulsatile flow is generated by the inflation/deflation of an elastic membrane. 

Evidently, this model entails simplifications. First, we only compute the properties of a 2D object. We study hemicylinders rather than hemispheres. A second somewhat subtle difference between the model and a real liquid surface has to do with local or global expansion and contraction. If an elastic membrane if forced to contract at some spot and to expand by the same amount somewhere else, the membrane responds with a restoring force. A liquid surface would not do this. Liquid surfaces only respond to a change in total area. This is a caveat to be kept in mind. 

A variety of thermodynamic models qualitatively capture the central equilibrium and dynamic features of microemulsion behavior (41 5). These range from phenomenological models, which treat the oil-water interface as fluctuating membranes to lattice models which describes discrete surfactants interacting with oil and water. Membrane models range from simple cubic lattice descriptions to fluctuating film models that include curvature-dependent bending elasticity to prescribe lower limits on domain size. 

In particular, elastic network models reproduce the local shape of the folding free energy landscape of a protein by an artificial effective potential that does not contain any information on the physical means by which such shape is obtained. Moreover, they intrinsically bias the global minimum to a selected conformation, and thus they are not capable to capture the protein conformational changes and exploration of multiple structural minima that may be required for exploitation of the protein function (for example, in motor proteins, transporters etc.). Finally, these models do not contain any information on the external potential of the proteins therefore, they cannot be in principle used to explore in detail interactions of proteins with the environment [i.e., with membranes, other proteins or ligands). 

This model for a hydrogel as depicted in Fig. 1 considers three phases the bath, the gel and the siuface of the gel (that is modelled as an elastic membrane). 

Fig. 1 Schematic of the gel model, the gel phase is surrounded by an elastic membrane and together they are immersed in a liquid bath...

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

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