We first suppose that our particles are smooth, rigid and spherical and totally buoyant, i. e. their movement is not influenced by terrestrial gravity.
A particle suspended in a fluid can be transported by convection (i. e. entrained by the motion of the fluid) or by diffusion. The key parameter is the relative motion of the fluid with respect to the adsorbent. Far from the surface, the flow is uninfluenced by the surface; at the surface, on the other hand, friction dictates that the fluid is stationary (ignoring the possibility of slip); and the velocity in between is constantly diminishing. Hence near a surface, a particle will move by diffusion, and far from the surface by convection. At a certain intermediate distance, typically of the order of tens of micrometres, the transport régime will cross over from convection to diffusion. This distance is known as the diffusion boundary distance (6). A comprehensive treatment is given by Levich (1962). The flow in certain geometries, such as the tube or plate, can be solved analytically. These geometries are therefore particularly favourable for kinetic evaluation, a factor which should be borne in mind when designing flow cuvettes. For flow in a tube, where D is the diffusivity of the protein, F the volumetric flow rate, and C a constant depending on the dimensions of the tube.
When choosing flow rates, it should be remembered that only laminar flow régimes can be analysed conveniently, i. e. up to Reynolds numbers of at most around 1000. In principle a diffusion boundary still exists in the case of turbulent flow, but the motion is much more complicated than in the case of laminar flow.
Hence, a protein-sized particle will be moving diffusively already at a distance of the order of a thousand times its own dimensions ( a few nm in diameter) from the adsorbing surface.
At a distance of the order of ten times its own dimensions from the adsorbing surface, the particle may begin to be influenced by the long range hydrophilic repulsion (see Cacace et al. (1997) for a discussion of intermolecular forces), which will considerably retard its rate of arrival at the surface. When designing the flow conditions for an experiment, it is only necessary to ensure that convective-diffusion is rapid enough to replenish the particles lost from the solution in the vicinity of the surface by attachment to it. This is of practical importance given that many biological samples of carefully purified macromolecules are available only in extremely small quantities.
Good quantitative approximations for analysing the flow régimes can be derived from the equations of Fick and Smoluchowski (see Ramsden (1998) for a more complete discussion). If the surface (at z = 0, where z is the coordinate normal to the surface) is a perfect sink for the adsorbate, then the bulk (solution) concentration cb is zero at z = 0, the concentration gradient will be approximately linear, and the rate of accumulation is
It is a good idea to compare the experimentally observed rate with this maximum upper limit (which may, however, be exceeded if there is a long range attractive force, e. g. electrostatic, between particle and surface).
The effect of any energy barrier is to retard accumulation. In the immediate vicinity of the surface, the local bulk concentration cv will be much
higher than zero (although still less than cb). It is convenient to consider that the rate of accumulation at the surface is given by the product of cv and a chemical rate coefficient ka, which is directly related to the repulsive energy barrier (Spielman and Friedlander 1974):
and the Fick-Smoluchowski régime (linear concentration gradient) applies to the zone above this vicinal region. Hence dCv
where V and S are unit volume and surface respectively. Strictly speaking the distance of the vicinal layer from the surface should be subtracted from 6 in the denominator, but since that distance is of the order of molecular dimensions, i.e. only a few nm, whereas 6 is of the order of a few or a few tens of microns, this correction can be neglected. If desorption of the material also has to be taken into account a term with a chemical desorption coefficient kd can be included:
In a great many cases accumulation of material is limited to a monolayer, or to occupying a monolayer of receptors, in which case a function 0 must be introduced, which gives the fraction of the surface still available for adsorption or binding (i. e. the probability that space is available). Our kinetic equation then becomes:
We shall discuss 0—which obviously depends on M or v—below. One important implication is that as the surface fills up, i. e. as 0 ^ 0, cv will tend to approach cb, and dM/dt will asymptotically approach zero regardless of the flow régime. This can be immediately seen by letting the left hand side of whichever of the previous three equations is appropriate go to zero, yielding an explicit expression for cv, e. g.
if ka = 1 and kd = 0, which can then be substituted into Eq. 35, in which
The pure diffusion régime
If there is no flow at all then the vicinal layer is replenished by diffusion only, i. e. there is no distance at which concentration is maintained constant by effectively an infinite reservoir as in convective diffusion. This leads to the well-known result of Smoluchowski, according to which the flux to the surface constantly diminishes as t-fi, where fi = 0.5 for standard diffusion (other exponents have been found for the diffusion of highly non-spherical, non-compact proteins such as tenascin (Ramsden 1992).
If the flow rate is slow enough for the rate of adsorption to be limited by transport alone, or, more quantitatively, if the dimensionless parameter.
becomes large relative to unity, then the initial rate of adsorption is given by Eq. 37, from which D may be obtained. This is a useful way to determine the diffusion coefficient in solution.
If a repulsive potential barrier U(z) exists between protein and surface, where z is the distance between them, not every arriving protein will adsorb, even if there is space for it to do so, and v will be diminished by a rate coefficient ka, which can be found by integrating U(z):
Sometimes the denominator of the right hand term is called the "adsorption length", 6a. The interaction potential U can be approximated by the sum of the particle-surface interaction free energies, corresponding to the Lifschitz-van der Waals (LW), electrostatic (el) and electron donor-acceptor (da) interactions:
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