Helfrich parameters

The link from lipid properties to mechanical properties of the bilayers is now feasible within the SCF approach. The next step is to understand the phase behaviour of the lipid systems. It is likely that large-scale (3D) SCF-type calculations are needed to prove the conjectures in the field that particular values of the Helfrich parameters are needed for processes like vesicle fusion, etc. In this context, it may also be extremely interesting to see what happens with the mechanical parameters when the system is molecularly complex (i.e. when the system contains many different types of molecules). Then there will be some hope that novel and deep insights may be obtained into the very basic questions behind nature s choice for the enormous molecular complexity in membrane systems. 

It turns out that cjeff is a function of the flexibility of the bilayers and their mutual interaction (Van der Linden and Droge 1993). These two parameters are the mesosopic determinants. In a special case of AOT bilayers in brine, one can deduce from theory as well as from experiment (Nallet et al. 1989 Nallet 1991 Helfrich 1978), making use of the theoretical expression in Van der Linden and Droge 1993), that... 

Having established that bilayer flexibility and bilayer interaction are the mesoscopic determinants, the next question is whether these determinants can be coupled to molecular parameters. In fact, this has been done to quite some extent. In general, bilayer flexibility can be shown (both experimentally as well as theoretically by simulation methods) to be directly related to bilayer thickness, lateral interaction between heads and tails of the surfactants, type of head group (ethoxylate, sugar, etc.), type of tail (saturated, unsaturated) and specific molecular mixes (e.g. SDS with or without pen-tanol). The bilayer interaction is known to be related to characteristics such as classical electrostatics. Van der Waals, Helfrich undulation forces (stemming from shape fluctuations), steric hindrance, number, density of bilayers, ionic strength, and type of salt. Two examples will be dicussed. 

The traditional parameter for monitoring CO2 flux through the organic cycle in open community studies is oxygen (Sargent and Austin, 1954 Odum and Odum, 1955 Kohn and Helfrich, 1957 Gordon and Kelly, 1962 Kinsey, 1972). 

Case C homeotropic alignment, Ca < 0 0. In this combination of the material parameters, the linear stability analysis of the basic state does not predict a direct transition to EC since the resulting expression for U q) in Eq. (8) is negative for all q 0 (except for e in the immediate vicinity of zero, see below). The reason is that the two terms in the denominator act differently compared to the case B ca > 0, aa < 0) described in the previous subsection. The Carr-Helfrich torque is now stabilizing while the dielectric torque (oc is destabilizing. At q = qp = 0, this term dominates and describes, at the threshold Ups (see Eig. 7a), the continuous bifurcation to the Freedericksz distorted state of homogeneous (along the x direction) bend (see Fig. 8a). 

The Helfrich Hamiltonian, Eq. (1), does not include a surface tension contribution. Free membrane patches can relax and adjust their area such that they are stress-free. In many situations, however, membranes do experience mechanical stress. For example, an osmotic pressure difference between the inside and the outside of a lipid vesicle generates stress in the vesicle membrane. Stress also occurs in supported bilayer systems, or in model membranes patched to a frame. In contrast to other quantities discussed earlier (bending stiffness etc.), and also in contrast to the surface tension of demixed fluid phases, membrane stress is not a material parameter. Rather, it is akin to a (mechanical or thermodynamic) control parameter, which can be imposed through boundary conditions. 

The first tension-like quantity in planar membranes is the lateral mechanical stress in the membrane, as discussed above. If the stress is imposed by a boundary condition, such as, for instance, a craistraint on the lateral (projected) area of the membrane, it is an internal property of the membrane system that depends, among other parameters, on the area compressibility [36] and the curvature elasticity [154—161]. Alternatively, mechanical stress can be imposed externally. In that case, the projected area fluctuates, and the appropriate thermodynamic potential can be introduced into the Helfrich Hamiltonian, Eq. (1), in a straightforward manner ... 

Finally, the third tension-like parameter in membranes was introduced by Deuling and Helfrich as early as 1976 [163], and it couples to the total area of the membrane ... 

A distinguishing feature of this model is the absence of a positive feedback between the magnitude of the space charge and the orientation of the director, which is required for the Carr-Helfrich mechanism. These physical parameters are independent of each other here. 

Since the mid-1970s, much work has been done to determine the role of these mechanical properties on the overall morphological behavior of cell membranes. Canham [1], Helfrich [2] and Evans [3] have identified the different characteristics responsible for the spontaneous shape of a membrane or its resistance to deformation. Using phenomenological parameters, they gave descriptions of the membrane deformation valid on a scale where membranes can be considered as a continuum material (i.e. on a scale larger than the membrane thickness). Following this idea... 

The Helfrich theory was extended by Dubois-Violette, de Gennes, and Parodi [17] to cover the AC field. The geometrical conditions are still the same. In the distorted state, the molecules are deflected by a small angle 0 in the xz plane. The most important parameter from the point of view of charge accumulation is not exactly 0, but rather the curvature v = 30/3x of the molecular pattern y/ and the charge density, q, are used as fundamental variables. 

A nice example of double layer effects occurs with the lamellar phases discussed in Section II. We already mentioned the beautiful experiments of Safinya et al.20 where the Helfrich undulation force was clearly demonstrated, using electrostatic interactions between the layers as a control parameter. Let us try to understand how the electrostatic interlayer forces have impact upon the undulation interaction Recall (Eqn. III.3) that the counterion distribution in the neighborhood of a single charged surface falls off as x 2 for x A. Since the counterions may be approximately considered as an ideal gas,the double layer contribution to the disjoining pressure between two lamellae separated by a distance, h, is roughly... 

The parameter A is a length scale and also appears below when the Helfrich-Hurault transition is investigated. We can use the separation of variables technique and search for solutions of the form... 

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