Shape dynamics

In resting neutrophils it is estimated that there are about 11-23 microtubules per cell, with a diameter of approximately 25 nm and a wall width of 5 nm. They are long, tubular structures made by the helical formation of tubulin molecules, which are either a- or /3-subunits, each with a relative molecular mass of 55 kDa (Fig. 4.9). Each subunit is present in equimolar amounts in a tubulin molecule, and these subunits exist as dimers of one a- and one /3-subunit. Because microtubules are polar, growth of the fibre is biased towards one end, termed the plus end. A number of microtubule-associated proteins (MAPs) affect the dynamic shape of the microtubule, and in the resting neutrophil about 35-40% of the tubulin pool is assembled, whilst the remainder can be assembled very rapidly after cell stimulation. 

The apphcation of the laser diffraction technique is sometimes questioned becanse it measures geometric instead of aerod5mamic particle diameters. However, the aerod5mamic diameter can be calcnlated when the dynamic shape factor and density are known. Moreover, the dynamic shape factor (x) of micronized particles will often be only shghtly higher than 1 and so is the tme particle density (1.0 < pp < 1.4 g cm ). As a conseqnence, the aerodynamic diameter differs only slightly from the eqnivalent volume diameter (see Eq. 3.1). 

Hansen PL, Wagner JB, Helveg S, Rostrup-Nielsen JR, Clausen BS, Topsoe H. Atom-resolved imaging of dynamic shape changes in supported copper nanocrystals. Science. 2002 295 2053. 

To use Stokes law with chains or fibers, several approaches are available. Traditionally a correction factor k, known as the dynamic shape factor, is defined such that... 

Fuchs [30] introduced a dynamic shape factor, y/p, to relate volume and Stokes diameters ... 

The dynamic shape factor /ris defined as the ratio of the resistance to motion of a given particle divided by the resistance of a spherical particle of the same volume. 

Galai Dynamic Shape Analyzer DSA-10 is a complete shape characterization system for particles in motion. All particles are classified by maximum and minimum diameters, area and perimeter, aspect ratio, shape factor and more. A video microscope camera synchronized with a strobe light takes still pictures continuously of particles in dynamic flow, generating shape information on tens of thousands of particles, in the 1 to 6,000 pm range in minutes. 

Galai CIS-1000 is an on-line particle size analyzer. A bypass from the process line feeds the sample into the sensor unit where it is sized and either drained off or fed back to the line. Full compatibility with the laboratory instrument is maintained since it uses the identical combination of laser-based time-of-transit particle sizing using the 1001 sensor and dynamic shape analysis using the 1002 sensor. The size range covered is from 2 pm to 3600 pm with measurement of size, area, volume, shape, concentration and estimated surface area with a cycling time of 300 s. 

Dynamic Shape Properties Conformational Freedom and Electronic Excitation... 

For sake of simplicity in describing the essential ideas within this and the following sections, first we shall consider a simplified model of a formal, static nuclear arrangement for each molecule. This constraint on the model will be released when dynamic shape analysis is discussed. 

We can formulate the above ideas more precisely by considering the dynamic shape properties of molecules within a nuclear configuration space M. 

Consider two different subsets of the same space D, or subsets of two dynamic shape spaces D and D of two different stoichiometric families of molecules. One may compare those domains of the two subsets that belong to the same shape group H 2. Since within these domains the nuclear configuration is not fully specified, that is, there exists some configurational freedom within these domains, the above approach provides a description of the dynamic similarity of molecular shapes. We shall return to the problems of dynamic shape similarity in Chapter 6. 

Each (a,b)-map can be regarded as a subset of the dynamic shape space D. Such a subset contains all points of D where the internal coordinates corresponding to the nuclear arrangement are fixed. 

If the entire range of curvature parameter b is considered, then a list of the finite number of distinct shape matrices and those curvature values bj where a change of the shape matrix occurs, gives a detailed, numerical shape characterization of the MIDCO surface G(a). In the most general case of variations in the two parameters a and b, as well as in the nuclear configuration K, one can study the dynamic shape space invariance domains, the (a,b)-maps, and various projections of the invariance domains of shape matrices, following the principles [158] applied for the shape group invariance domains of the dynamic shape space D. 

On the simplest level, one can consider a semiclassical model of limited motions of various parts of the molecule relative to one another. Within such approximation, the dynamic shape variations due to internal motions, for example, those due to vibrations, can be modeled by an infinite family of geometrical arrangements. Within this approach, we consider a family of shapes occurring for these arrangements and study the common, invariant topological features. 

As it has been pointed out in Section 5.2, it is natural to formulate dynamic shape analysis aproaches in terms of the dynamic shape space D described earlier [158]. The reader may recall that the dynamic shape space D is a composition of the nuclear configuration space M, and the space of the parameters involved in the shape representation, for example, the two-dimensional parameter space defined by the possible values of the density threshold a, and the reference curvature parameter b of a given MIDCO surface. 

We shall distinguish two types of methods for dynamic shape analysis. The methods of the first type are used to determine which nuclear arrangements are associated with a given topological shape. The methods of the second type determine the available topological shapes compatible with some external conditions, for example, with an energy bound. 

Within the simplest formulation of a dynamic shape analysis method of the first type, the invariance of topological descriptors within domains of the dynamic shape... 

The theory of fuzzy sets [382-385] has numerous contemporary applications in various fields of engineering [386-388], in the description of quantum mechanical uncertainty [389-393], in the study of molecular identity preserving deformations [106,251], in new approaches to the description of approximate symmetry [252,394,395], as well as in both static and dynamic shape characterization and dynamic shape similarity analysis of molecules [55,396]. 

P.G. Mezey, "Dynamic Shape Analysis of Biomolecules Using Topological Shape Codes", in The Role of Computational Models and Theories in Biotechnology. J. Bertran (Ed.), pp. 83-104, Kluwer Academic Publishers, Dordrecht, 1992. 

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