Intrinsic curvature

The classification of critical points in one dimension is based on the curvature or second derivative of the function evaluated at the critical point. The concept of local curvature can be extended to more than one dimension by considering partial second derivatives. d2f/dqidqj, where qt and qj are x or y in two dimensions, or x, y, or z in three dimensions. These partial curvatures are dependent on the choice of the local axis system. There is a mathematical procedure called matrix diagonalization that enables us to extract local intrinsic curvatures independent of the axis system (Popelier 1999). These local intrinsic curvatures are called eigenvalues. In three dimensions we have three eigenvalues, conventionally ranked as A < A2 < A3. Each eigenvalue corresponds to an eigenvector, which yields the direction in which the curvature is measured. 

A similar kind of triangular unit was proposed by Mao and coworkers [70] (see Fig. 15a). Here only one species of triangle was produced, with sticky ends chosen so that each vertex would join with either one or two other triangles as in Fig. 15b, leading to the formation of two-dimensional lattices (Fig. 15c, d). This approach and that previously described [69] do not yield extended structures, possibly because of intrinsic curvature and/or excessive flexibility. To overcome these limitations, stiffer structures based on DX-type junctions were considered. 

Figure 6.14 X-ray structure of glycoluril-based host 6.36 showing the intrinsic curvature.17...

Glycoluril units have the capacity to form multiple hydrogen bonds by utilising the NH and C=0 groups that they contain. Glycoluril units also have an intrinsic curvature to them, helping to form shapes that are mutually compatible (Fig. 53). 

Gruner SM. Intrinsic curvature hypothesis for hiomemhrane lipid-composition-A role for nonbilayer lipids. Proc. Natl. Acad. Sci. U. S. A. 1985 82 3665-3669. 

The reason that this structure forms is as follows units of octahedral W3C crystallise on one side of an interface, that separates solid W3C from Fe. The Fe atoms are drawn into the interstices of the W3C elements, with a driving force dependent on the Gaussian curvature of the interface. This is one way of looking at intrinsic curvature in solids. As will become apparent, there are other ways that link the interactions between atoms, molecules and larger aggregates to local curvature. 

The concept of intrinsic curvature is particularly useful when dealing with the intricate and beautiful structures formed by zeolites. Zeolites are commonly used as technical materials. They exhibit many special properties, due to their extraordinary ability to selectively absorb a large range of molecules. The forces that act are weak, physical not chemical, and we characterise them by invoking the idea of intrinsic curvature. 

Figure 2.12 Plot of the area per T-atom vertex SI) versus the average ring size, n, for a variety of zeolites, silica clathrasils and dense silicates. (All zeolites have a silicon aluminium ratio exceeding three, so that the approximate stoichiometry of all these frameworks is Si02). Zeolite and clathrasil frameworks are labelled by the code adopted by the International Zeolite Association [18]). The shaded domain indicates the window of geometrically accessible values of as a function of the ring size. Despite the allowed geometric variability, the value of D is close to 12.2A2 for all these "silicates", regardless of the ring size and consequent intrinsic curvature.

In all the cases analysed here, those frameworks realised in the laboratory (or in nature) do lie near to the dotted curves deduced from reasonable bond lengths and angles. Remarkably though, the variation in bulk density follows the locus of the curves assuming constant area. The intersection of the two curves determines the preferred intrinsic curvature, characterised by the ring size. 

How do these unconventional ideas link with the standard view of a solid as a close packed array of atoms Evidently most of the frameworks discussed above cannot be so characterised. The two-dimensional hyperbolic picture does break down for very dense structures. Thus the densest four-coordinated silicate, coesite, violates this universality (see Fig. 2.12). (Its ring size is less than that of trid5m[ ite, cristobalite, keatite or quartz, in spite of its higher density.) This polymorph is too dense for a two-dimensional description to be useful and the Aree-dimensional description takes over. The notion of intrinsic curvature is less rigid for silicates than for the other frameworks, because the Si-O-Si angle usually differs from 180 . 

In 1980 it was pointed out that the prolamellar body is a perfect example of a Cp structure [4]. (Later, more detailed, analyses have revealed that it may also be a Cd structure cf. section 7.2.) Following work on the structure of cubic phases, it was also realised that two-dimensional analogues are possible. This in turn suggested that a phase transition involving changes in the intrinsic curvature of membranes might be possible [29, 30]. Such a mechanism has far reaching implications. Clear evidence for such transitions between bilayer conformations has been reported [9]. This membrane bilayer model will be described below. 

Epand RM, Fuller N, Rand RP. Role of the position of unsaturation on the phase behavior and intrinsic curvature of phosphatidylethanolamines. Biophys J 1996 71 1806-1810. 

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