Sedimentation frictional coefficient

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

The frictional coefficient varies with concentration, but at infinite dilution it reduces to the coefficient (/o) for an isolated polymer molecule moving through the surrounding fluid unperturbed by movements of other polymer molecules (see Chap. XIV). At finite concentrations, however, the motion of the solvent in the vicinity of a given polymer molecule is affected by others nearby binary encounters (as well as ones of higher order) between polymer molecules contribute also to the observed frictional effects. The influence of these interactions will persist to very low concentrations owing to the relatively large effective volume of a polymer molecule, to which attention has been directed repeatedly in this chapter. Since the sedimentation con-stant depends inversely on the frictional coefficient, s must also depend bn concentration. 

The mass of the sedimenting particle could be deduced from its rate of sedimentation at high dilution in a given field, i.e., from its sedimentation constant, if the frictional coefficient / could be determined independently. Hates of diffusion may be utilized to secure this necessary supplementary information, since the diffusion constant D depends also on the frictional coefficient. Thus ... 

Equations (29), (30), and (10) might be applied to the elucidation of the frictional coefficient in a manner paralleling the procedure applied to the intrinsic viscosity. One should then determine/o (from sedimentation or from diffusion measurements extrapolated to infinite dilution) in a -solvent in order to find the value of Kf, and so forth. Instead of following this procedure, one may compare observed frictional coefficients with intrinsic viscosities, advantage being taken of the relationships already established for the viscosity. Eliminating from Eqs. (18) and (23) we obtain ... 

Ni is Avogadro s number, / the particle friction coefficient, u the velocity and v the partial specific volume of the solute [79,80]. Inertial forces are negligible and the balancing of the above forces yields the sedimentation coefficient ... 

D=mass diffumsion coefficient Z)T=fiiermal diffusion coefficient /=friction coefficient G=(oh (centrifugal acceleration) / =Boltzniann constant meff=particle effective mass r=radius of centrifuge basket s=sedimentation coefficient T = absolute temperature =geometric volume of die channel w=channel thickness y=diermal expansion coefficient p=electrophoretic... 

As far as we are aware, only a few experimental results are available for the translational friction coefficient of polypeptides in the helix-coil transition region, and our discussion about it cannot but be very incomplete. Figure 33, taken from the work of Okita et al. (13) on the system PHPG-aqueous methanol, shows the dependence of the reduced sedimentation coefficient [s0] on the helical fraction. Here [s0] is defined as s0ri0/( 1 — i>g0), with and Q0 being the... 

A molecule in a centrifuge is acted upon not only by the applied centrifugal force but also by an opposing buoyant force that depends upon the difference in density of the sedimenting particles and the solvent and by a frictional drag, which is proportional to a frictional coefficient/. Setting the sum of these forces to zero for the hydrodynamic steady state yields Eq. 3-13, which defines the sedimentation constant s. 

To minimize the effects of concentration dependence, the sedimentation coefficient is usually determined for a particular system at several concentrations and then extrapolated to infinite dilution (S°). If, on the other hand, the diffusion coefficient is also measured at several concentrations and extrapolated to infinite dilution (D°), then values of S° and D° may be related through the frictional coefficient to determine the molecular weight of the solute molecules... 

In an SV experiment, the sedimentation and the diffusion forces that determine the net rate of movement of the solvent—solution boundary, stem from two intrinsic properties of the solute molecules, their sedimentation coefficient (s) and their diffusion coefficient (D). Whereas D depends predominantly on the shape of the solute particles, s depends both on its shape and on its mass. The diffusion coefficient, D, is defined as the ratio of the flux of molecules (Jx, moving in the direction x under diffusive forces) to the concentration gradient of the molecules (dc/dx). The dependence of D on the molecular shape stems from its relation to the frictional coefficient f ... 

Ultracentrifuge data can also be interpreted to give information about the effective shape of the dissolved polymer molecules. The actual frictional coefficient (f) can be calculated from the sedimentation constant, and this compared with the theoretical figure (f0) calculated from Stokes law (assuming a spherical molecule). The frictional ratio (f/fo) has been called the asymmetry coefficient. It expresses the deviation of the molecule in solution from a spherical shape, and can be interpreted in terms of the axial ratio of idealized particles. 

Sedimentation is a process in which gravitational or centrifugal acceleration is used to force particles through a fluid. The displacement process requires that the particles have a density different from that of the fluid. Differential displacement, leading to separation, arises because particles have different densities, different masses, and/or different friction coefficients, which may partially originate in different particle shape. 

Because the transport of large molecules is inherently sluggish (due to a large friction coefficient/, Section 4.13), the speed at which macromolecules are separated by electrophoresis and sedimentation becomes an important consideration in optimization. Since N measures the efficiency of separation, the ratio Nl time measures the time-based rate of generation of efficiency [1]. We clearly wish to maximize the Nit ratio. 

The frictional coefficient of a molecule (/ in equation 7) depends on its size, shape, and the viscosity of the medium through which it is sedimenting. For a spherical molecule or particle of radius r , the... 

For the sake of discussions in a later section we will summarize here the theoretical equations of two other important hydrodynamic properties, the translational and rotary frictional coefficients. The former, designated as / can be determined from either sedimentation (Svedberg and Pedersen, 1940) or diffusion (Einstein, 1905, 1906b Smoluchowski, 1906) measurements ... 

Here M is the molecular weight and V the partial specific volume of the solute, N the Avogadro number, k the Boltzmann constant, and T the absolute temperature s and D are the sedimentation and translational diffusion coefficients (after extrapolation to infinite dilution). The translational frictional coefficients from both measurements are regarded as identical, i.e., f, = fd. The rotary frictional coefficient, designated as f, can be determined from either flow birefringence or non-Newtonian viscosity measurements. 

Shape, too, influences the sedimentation velocity because it affects the viscous drag. The frictional coefficient/of a compact particle is smaller than that of an extended particle of the same mass. Hence, elongated particles sediment more slowly than do spherical ones of the same mass. 

Calculation of the frictional coefficient, shape factor and axial ratio for the RAGl fragment (ZDD) from the sedimentation coefficient, s°q, has been previously described in detail [5]. 

Insight into the overall shape of the ZDD dimer in solution was obtained by interpretation of sedimentation velocity and small-angle X-ray (SAXS) scattering experiments [5]. An experimental frictional coefficient, was calculated from the sedimentation... 

The diffusion constant and the sedimentation constant s are related to the frictional coefficient / of the entire chain as follows ... 

If the molecular weight of a protein is known from measurements of osmotic pressure, of sedimentation and diffusion, of sedimentation equilibrium, of light scattering, or by any other inethod, and if its partial specific volume is also known, then the frictional ratio fjf may be determined either from its sedimentation constant or its diffusion constant. The frictional coefficient / is that characteristic of a spherical unhydrated molecule of the same molecular weight and partial specific volume as the protein under consideration. If r is the radius of tliis hypothetical sphere and rj the viscosity of the medium, then... 

On the other hand, the frictional coefficient / of the actual molecule is given as a function of molecular weight and sedimentation constant s by the relation... 

Q.26.9 How is sedimentation analysis used to separate macromolecules What are some of the properties of macromolecules that might determine their frictional coefficients ... 

A.26.9 Sedimentation analysis separates macromolecules by applying a force to them while they re in solution. Their different frictional coefficients lead to different rates of sedimentation. Mass, size, and shape. A long floppy protein will sediment slower than a similarly massed compact protein. 

Considerable discrepancies exist between the theory and practice of centrifugation. Complex variables not accounted for in Equations 18.10 and 18.11, such as concentration of the suspension, nature of the medium, and characteristics of the centrifuge, will affect the sedimentation properties of a mixed population of particles. Moreover, the frictional coefficient, f in the case of an asymmetrical molecule (e.g. a protein such as myocin) can be several times the frictional coefficient (i.e. f0) of a sphere. This results in particles sedimenting at a slower rate. Equation 18.10 can, therefore, be modified to give Equation 18.12 ... 

During sedimentation under gravity, particle velocity is given by the ratio of force, Fg, to friction coefficient, B. For spherical particles B is given by the Stokes equation, B=6m r, and hence the sedimentation flux and corresponding phenomenological coefficient, an, are given by... 

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