## Theoretical Prediction

Fig. 1. A schematic of a typical protein adsorption scenario, where adsorption occurs from a flowing solution over a flat, charged surface. Near the surface, simple shear flow occurs resulting in a velocity profile v = azX, where a is the shear rate, z is the distance from the surface, and X is a unit vector parallel to the surface. A convective-diffusion boundary layer develops 8(x, t), of thickness 8, above (below) which convection^s more (less) rapid than diffusion. An applied electric field is expressed as E = E&, where E is the electric field vector, E is the magnitude and & is a unit vector normal to the surface on the protein). The flow field must be known in advance to solve Eq. 1. So long as the concentration is quite low, one may neglect the influence of the migrating proteins on the fluid and solve for the flow field using the standard Navier-Stokes equation. A typical configuration involves flow past a flat, charged surface, as depicted in Fig. 1.

The two terms on the right side of Eq. 1 represent, respectively, the contributions from thermal diffusion and electric-field-induced migration. It is interesting to consider the limiting cases where one is much larger than the other. Neglecting the second term gives pure convective diffusion. In this case, one may approximate the flux (J) to the surface by assuming steady shear flow over a flat surface acting as a perfect adsorptive sink, J(x, t) = Dcb/8(x, t), where cb is the bulk protein concentration, and is the convective diffusion boundary layer, a function of the diffusivity (D), the distance along the surface in the direction of the flow (x), the shear rate (a), and the time (t) (Brusatori et al. 2003; Calonder and Van Tassel 2001). The function /(t) is the inverse ofr(f) = [1 - (1 -/3)2/3]/2/2 for t < 1/2 and f (t) = 1 for t > 1/2. Pure convective diffusion is therefore characterized by an initially zero flux that increases to a steady-state value at dimensionless time, t = 1/2.

Inclusion of the second term on the right of Eq. 1 requires knowledge of the electric field as a function of position. A starting point toward its approximation is the Poisson-Boltzmann equation describing the distribution of electric potential in an ionic solution:

In Eq. 3, W is the electric potential, e is the elementary charge, £ is the dielectric constant, zi is the valance of the ith ionic species, c0 is the bulk concentration of the ith species, k is the Boltzmann constant, and T is the absolute temperature. An important assumption leading to Eq. 3 is the complete neglect of spatial correlations among the ions in solution. Clearly, the validity of this assumption is suspect for higher ionic concentrations. When the electric potential energy is weaker than the thermal energy, the exponential in Eq. 3 may be linearized. The result is the linear Poisson-Boltzmann equation, V2W = k2W, where k is the inverse of the Debye length, k=

e Y1 zjc0

The linearized version of Eq. 3 may be solved over the region extending away from a flat surface to yield

£k where W0 is the surface potential, z is the distance from the surface, and — is the surface charge density (the second equality follows from integrating the charge density as given by the Poisson equation). The electric field is then given by

where X is the unit vector normal to the flat surface.

Neglecting the first term on the right side of Eq. 1 is equivalent to ignoring diffusive motion. Assuming fully developed shear flow, a no-slip, perfect sink boundary condition, and an electric field given by Eq. 6, the z component velocity and the z position of a protein are given by:

Vz = 0®= q—e-Kz = dz ^ z(t) = Z0 + K-1ln (1 + q—Kie-KZ0

The velocity in the x direction is vx(t) = az(t), so the distance traveled in the x direction (x = 0 corresponds to the leading edge of the surface, as shown in Fig. 1) is:

£Z «KoW^ q—Kt e-Kzo t -e-Kzo ) - t q—K ) \ £Z

The concentration profile is thus given by:

c(x, z, t) = cbH^az0t + aK-1 ^t + — eKZ^l^1 + ^^e-KZ^ - t - x

where z0 is the implicit solution to Eq. 7, in terms of z(f) and t, and H is the Heaviside function (i.e., H(A) = 1 for A > 0 and H(A) = 0 for A < 0). The flux to the surface is thus:

Pure electrophoretic migration is therefore characterized by an initial period of zero flux, during a time needed for the argument of the Heaviside function in Eq. 9 to vanish, followed by a steady-state flux. Both of these flux predictions hold only in the limit where surface adsorption is rapid compared to transport to the surface (i. e., where the surface may be considered to be a perfect sink). This is usually the case during the initial stages of surface filling. Subsequently, the kinetics are limited by surface effects. The surface-limited rate of adsorption may be expressed as:

i where r is the density of adsorbed protein (mass per area), ka is the adsorption rate constant, \$ is the one-body cavity function, and ka,i and r are the desorption rate constant and the density of protein in the ith structural state, respectively (these states may denote various conformations, orientations, or states of aggregation; Tie et al. 2003). The cavity function is defined as \$ = {e~u/kT }r,T ,where u is the potential energy of a single molecule interacting with the surface and with all of the previously adsorbed molecules (u depends on position and orientation), k is the Boltzmann constant, T is the absolute temperature, and the brackets represent an averaged quantity, over all representations of the adsorbed layer at density r and temperature T, according to their appropriate weights, and over all orientations and positions of the single "reference" molecule. All of the quantities on the right of Eq. 11 (except cb) maybe altered by application of an electric field.

To make quantitative predictions, the potential energy of interactions between protein molecules and the charged surface must be calculated. The electrostatic contribution to this energy may be determined using the Derjaguin-Landau-Verwey-Overbeek (DLVO; Asthagiri and Lenhoff 1997; Oberholzer and Lenhoff 1999; Ravichandran and Talbot 2000; Roth and Lenhoff 1993, 1995) or density functional (Carignano and Szleifer 2002; Fang and Szleifer 2003) approaches.