Hydrodynamic resistance matrix

The resistance matrix depends on co-ordinates of all particles, in nonlinear manner. The situation is illustrated in Appendix F for the case of two particles. To avoid the non-linear problem, one uses the preliminary averaging of the hydrodynamic resistance matrix (Kirkwood and Riseman 1948 Zimm... 

The degree of completeness of the analogies between Eqs. (331) and (49) is quite remarkable oll and Jfoll are each symmetric, positive-definite forms, as are their direct submatrices too. The positive-definiteness of o stems from the positivity of the rate of irreversible entropy production. In contrast to the proof of the symmetry of the hydrodynamic resistance matrix (B22), the corresponding proof of the symmetry of the diffusion matrix is trivial. The latter may be taken to be symmetric by definition since its antisymmetric part gives rise to no observable macroscopic physical consequence. 

Equation (350) furnishes a wholly independent proof of the symmetry and positive-definiteness of the diffusivity matrix, for these characteristics now follow from the comparable properties of the hydrodynamic resistance matrix. 

This equation shows that the resistance-drag force for a certain particle depends on the relative velocities of all the particles of the macromolecule and also on the relative distance between the particles. This expression determines an approximate matrix of hydrodynamic resistance... 

The exact components of the matrix of hydrodynamic resistance for a two-particle chain are shown in Appendix F. 

The matrix of the hydrodynamic resistance for two particles can exactly be determined based on results of Section 2.2. The components of the matrix are as follows... 

Multibody hydrodynamic interactions have generally been ignored in simulations (for reasons of computational cost) with the notable exception of [243,264] for a monolayer system involving a small number of particles. Satoh et al. [266,267] approximate the multibody hydrodynamic forces by assuming additivity of the velocities. This does, however, not guarantee positive definiteness of the mobility matrix (inverse of the resistance matrix), imless a short cutoff radius of the hydrodynamic interactions is used [266,267]. 

Equations (25) - (29) determine the simplest approach to the dynamics of a macromolecule, even so, it appears to be rather complex if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. Due to these effects, all the beads in the chain ought to be considered to interact with each other in a non-linear way. To tackle with the problem, this set of coupled non-linear equations is usually simplified. There exist the different simpler approaches originating in works of Kirkwood and Riseman [46], Rouse [2], Zimm [5], Cerf [4], Peterlin [6] to the dynamics of a bead-spring chain in the flow of viscous liquid. The linearization is usually achieved by using preliminary-averaged forms of the matrix of hydrodynamic resistance (hydrodynamic interaction) [5] and the matrix of the internal viscosity [4]. In the last case, to ensure the proper covariance properties when the coil is rotated as a whole, Eq. (29) must be modified and written thus... 

The non-diagonal terms of the matrixes Hai and Gai are connected with mutual influence of the particles of the chain. One can admit that, in accordance with the works by Edwards and Freed (1974) and Freed and Edwards (1974, 1975), the hydrodynamic interaction in the system between the particles of the chain becomes negligible, and one can introduce a diagonal matrix of external resistance, but, in virtue of relation (3.5), one cannot introduce non-zero diagonal matrix of internal resistance, so that the simplest forms of the matrixes are... 

The extracellular space of tissues is an aqueous gel of proteins and polysaccharides. This gel potentially provides an additional resistance to the diffusion of molecules in the extracellular space due to volume exclusion and hydrodynamic interactions. Reconstituted gels of extracellular matrix components (e.g., collagen) are often used to evaluate the magnitude of this resistanee. In general, the diffusion coefficient for a protein depends on the properties of the gel and the size of the protein (Figure 4.10). 

An effective medium approach, which uses hydraulic permeabilities to define the resistance of the fiber network to diffusion, has been used to estimate reduced diffusion coefficients in gels [77]. For a particle diffusing within a fiber matrix, the rate of particle diffusion is influenced by steric effects (due to the volume excluded by the fibers in the gel, which is inaccessible to the diffusing particle) and hydrodynamic effects (due to increased hydrodynamic drag on the diffusing particle caused by the presence of fibers). Recently, it was proposed that these two effects are multiplicative [78], so that the diffusion coefficient observed for particles in a fiber mesh can be predicted from ... 

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