Self-consistent field theory of protein adsorption in a non-Gaussian polyelectrolyte brush

To describe adsorption of globular protein molecules in a polyelectrolyte brush we use the strong-stretching approximation of the Edwards self-consistent field equation, combined with corrections for a non-Gaussian brush. To describe chemical potentials in this mixture of (globular) species of widely varying sizes (ions, brush polyelectrolyte segments, globular protein molecules), we use the Boublik-Mansoori-Carnahan-Starling-Leland equation of state derived for polydisperse mixtures of spherical particles. The polyelectrolyte chain is described in this approach as a string of beads with the beads of a size related to the chain diameter. We use the one-dimensional Poisson equation to describe the electrostatic field and include the ionizable character of both the brush polyions and the protein molecules. This model explains the experimental observation of high amounts of protein adsorption in a polyacid brush for $p\mathrm{H}$ values above the isoelectric point of the protein as being due to charge reversal of the protein molecules upon entry in the brush. We find a distinct minimum in protein concentration near the edge of the brush. With increasing $p\mathrm{H}$ this barrier to protein transfer becomes larger, but much less so when we increase the ionic strength, a difference that might relate to an experimentally observed difference in the protein release rate in these two cases. A free energy analysis shows that the release of small ions from the brush and the increase of brush ionization are the two driving forces for protein adsorption in a like-charged brush.

Figure 1

Density profile of non-Gaussian polyacrylic acid brush using local charge neutrality (“EN”) or the full Poisson equation (SS40, $p\mathrm{H}=10$, $c_{\infty }=10\mathrm{mM}$).

Figure 2

Spherical polyelectrolyte brush. Brush density in vol. % as function of distance from the grafting core particle, $x$ ($p\mathrm{H}=10$, $n_{\infty }=10\mathrm{mM}$). The dashed line is box model of Ref. 5, and solid lines are based on the strong-stretching approximation. “SS2” is based on a purely parabolic profile for the stretching contribution to the chemical potential; SS4, SS10, and SS40 denote higher-order corrections.

Figure 3

Electrochemical potential of the amphoteric protein molecule as a function of electrostatic potential $y$ and $p\mathrm{H}$ (solid lines). For a particle of a fixed negative charge, $\mu$ monotonically increases with decreasing $y$ (dashed line).

Figure 4

Influence of (hydrated) ion size (in Å) on protein and brush density profiles ($c_{\infty }=10\mathrm{mM}$, $p\mathrm{H}$ 5.4, $p\mathrm{I}$ 5.1, $\chi =0$). $x$ is the distance from the core particle, and $\varphi$ is the volume fraction of protein and brush segments.

Figure 5

Protein adsorption in a spherical polyanion brush as function of ionic strength and $p\mathrm{H}$. Conditions of Fig. 4.

Figure 6

Adsorbed amount of protein in a spherical brush particle, as function of $p\mathrm{H}$, ionic strength, attraction parameter $\chi$, and ion size ($0\mathrm{\AA }$, except for the curve “$+\text{ion}$ volume”).