Local curvature properties

An absolute shape characterization is obtained if a molecular contour surface is compared to some standard surface, such as a plane, or a sphere, or an ellipsoid, or any other clo.sed surface selected as standard. For example, if the contour surface is compared to a plane, then the plane can be moved along the contour as a tangent plane, and the local curvature properties of the molecular surface can be compared to the plane. This leads to a subdivision of the molecular contour surface into locally convex, locally concave, and locally saddle-type shape domains. These shape domains are absolute in the above sense, since they are compared to a selected standard, to the plane. A similar technique can be applied when using a different standard. By a topological analysis and characterization of these absolute shape domains, an absolute shape characterization of the molecular surface is obtained. 

As an example of absolute shape criteria, the local curvature properties of a MIDCO can be used for defining absolute shape domains on it [156], and for a subsequent global shape characterization. In Figure 5.1 a MIDCO G(a) is shown as an illustration of some of the concepts discussed. The simplest method [155] is based on comparisons to a reference of a tangent plane what leads to the identification of locally convex, concave, and saddle-type domains, as mentioned previously, although much finer characterizations are also possible [156,199]. 

An alternative approach to domain subdivisions of the molecular surface is based on local curvature properties. It is applicable only to differentiable molecular surfaces such as contour surfaces, e.g. the electron isodensity contour surface G(m), m being the threshold value defining the contour surface. 

The local curvature properties of the surface G(m) in each point r of the surface are given by the eigenv ues of the local Hessian matrix. Moreover, for a defined reference curvature b, the number p,(r, b) is defined as the number of local canonical curvatures (Hessian matrix eigenvalues) that are less than b. Usually b is chosen equal to zero and therefore the number p(r, 0) can take values 0,1, or 2 indicating that at the point r the molecular surface is locally concave, saddle-type, or convex, respectively. The three disjoint subsets Ao, Ai, and A2 are the collections of the surface points at which the molecular surface is locally concave, saddle-type, or convex, respectively the maximum connected components of these subsets Ao, Aj, and A2 are the surface domains denoted by Do,, Diand D2, where the index k refers to an ordering of these domains, usually according to decreasing surface size. 

Figure M-2. A subdivision of a molecular contour surface G(m), based on local curvature properties. The contour surface is subdivided into locally concave (Do, ), saddle-type (Di, ), and locally convex (D2, ) domains with respect to the local tangent plane in each point r. The second index k refers to an ordering of these domains according to decreasing surface area.

As an illustration, here we shall outline only one of the simplest of the nonvisual, topological methods for shape characterization, applicable for smooth (differentiable) molecular surfaces. This method is based on the classification of the points of a molecular surface into convex, concave, and saddle-type domains using local curvature properties, and on the representation of the mutual arrangements of these domains by a matrix or by an equivalent graph. One of the advantages of the method is the fact that the generation... 

The local curvature properties can be compared to a reference curvature parameter For each point r of the molecular surface G(a) a number p,... 

Order 0 minimization methods do not take the slope or the curvature properties of the energy surface into account. As a result, such methods are crude and can be used only with very simple energy surfaces, i.e., surfaces with a small number of local minima and monotonic behavior away from the minima. These methods are rarely used for macro-molecular systems. 

The computation of the curvatures from the bulk field differential geometry has proven to be rather imprecise. The errors produced by the use of the approximate formulas (100)-(104) are especially big if the spatial derivatives of the field sharp peaks at the phase interface. This is a common situation in the late-stage kinetics of the phase separating/ordering process, when the order parameter is saturated and the domains are separated by thin walls. Here, to calculate the curvatures, we propose a much more accurate method. It is based on the observation that the local curvatures are quantities that can be inferred solely from the shape of the interface, without appealing to the properties of the bulk field [Pg.212]

An increasingly important tool to determine the strain-induced anisotropy is MOKE (magneto-optical Kerr effect). In section 2 we mentioned already the calculations by Freeman et al. (1999). Experimentally, e.g. Ali and Watts (1999) (see also references therein) apply a bending device to induce strains in a controlled way, and determine the (local) curvature and the strains by optical interferometry or by direct measurement (stylus). The properties of the substrate are incorporated in a finite-element modelling calculation, thus allowing an absolute determination of the film properties. Compare also Stobiecki et al. (2000), who studied the strain induced anisotropy in FeB/Cu/FeB trilayers, using Kerr magnetometry (MOKE). 

AFM can also be used to probe local mechanical properties of thin films of food biopolymers, which are difficult to measure using traditional rheological methods. Several mechanical models have been developed to analyze the Young s modulus of food systems. One of the simplest models, the Hertz model, assumes that only the elastic deformation exists in a surface with spherical contacts, and the adhesion force can be neglected (Hugel and Seitz 2001). Equation (8.2) describes the relationship between the loading force, F and the penetration depth, d, where a is the radius of contact area, R the curvature of the tip radius, Vi and the Poisson s ratios of the two contact materials that have Young s modulus, Ei and E2. ... 

Convexity and curvature properties. In the above discussion and examples we have already used the concepts of convexity and locally convex domains in an intuitive manner. Whereas our goal is to provide a topological shape characterization for molecules, we shall often use geometrical tools at intermediate steps toward a topological description. These steps often involve the concepts of convexity, curvature, and a characterization of critical points of functions. 

From studies of lipid-water mixtures and isolated membranes the general functional features of the bilayer are known barrier properties, lateral diffusion, acyl chain disorder and protein association. To vmderstand the mechanisms behind a wide spectrum of membrane functions, a detailed picture at the level of local curvature is needed. Examples are fusion processes, cooperativity in receptor/ligand binding or transport through the bilayer of the proteins that are constantly synthesised for export from the endoplasmic reticulum. Some preliminary discussions of the possibilities of curved, rather than flat, membremes follow. 

The first generation methods that were proposed involved interface reconstruction and approximation of the singular interfacial term from the 2D interface properties. These models thus rely on a proper numerical procedure to locate the interface within the mesh based on the volume fraction field. The local curvature may then be calculated in each surface grid cell. 

In the most common applications of shape groups, the local shape properties are specified in terms of shape domains for example, in terms of the locally convex, concave, or saddle-type regions of MIDCOs, relative to some curvature reference parameter b. 

STEP 1. For each contour value a within a range of values for 3D property P(r), the IPCOs G(a) are partitioned into local curvature domains relative to each value b of a range of reference curvatures. 

In nanotechnology surfaces become increasingly more important because a greater fraction of the atoms are at the surfaces. In the smallest nanoparticles, all the atoms may be at the surface of the particle. This property is quite closely related to a completely different group of materials, the foams. One difference between these two situations is the local curvature of the surface. 

A particularly interesting case is that of systems in which the film has a zero interfacial tension and zero spontaneous curvature (i.e., it is spontaneously flat), together with a great flexibility, so that local curvature may nevertheless occur. P.G. de Gennes was able to predict the existence of bicontinuous phases which provide a good representation of the mysterious zone (III). Note that these sponge phases possess curvatures in opposite directions, unlike droplets in a dispersion. These points and the properties of these phases are discussed further in Chap. 5. 

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