Polyelectrolyte Brush Interacting with Ampholytic Nanoparticles as Coarse-Grained Models of Globular Proteins: Poisson–Boltzmann Theory

Abstract

The self-consistent field Poisson–Boltzmann framework is applied to analyze equilibrium partitioning of ampholytic nanoparticles (NPs) between buffer solution and polyelectrolyte (PE) polyanionic brush. We demonstrate that depending on pH and salt concentration in the buffer solution, interactions between ionizable (acidic and basic) groups on the NP surface and electrostatic field created by PE brush may either lead to the spontaneous uptake of NPs or create an electrostatic potential barrier, preventing the penetration of NPs inside PE brush. The capability of PE brush to absorb or repel NPs is determined by the shape of the insertion free energy that is calculated as a function of NP distance from the grafting surface. It is demonstrated that, at a pH value below or slightly above the isoelectric point (IEP), the electrostatic free energy of the particle is negative inside the brush and absorption is thermodynamically favorable. In the latter case, the insertion free energy exhibits a local maximum (potential barrier) at the entrance to the brush. An increase in pH leads to the shallowing of the free energy minimum inside the brush and a concomitant increase in the free energy maximum, which may result in kinetic hindering of NP uptake. Upon further increase in pH the insertion free energy becomes positive, making NP absorption thermodynamically unfavorable. An increase in salt concentration diminishes the depth of the free energy minimum inside the brush and eventually leads to its disappearance. Hence, in accordance with existing experimental data our theory predicts that an increase in salt concentration suppresses absorption of NPs (protein globules) by PE brush in the vicinity of IEP. The interplay between electrostatic driving force for NP absorption and osmotic repelling force (proportional to NP volume) indicates that for large NPs with relatively small number of ionizable groups osmotic repulsion overcomes electrostatic attraction preventing thereby absorption of NPs by PE brush.

Polyelectrolyte Brushes with Protein-Like Nanocolloids

“…Insertion of the ampholytic nanocolloidal particle from the bulk of the solution into the PE brush leads to the change in the ionization degree of both basic and acidic monomer groups. The respective free energy change can be presented 68,69 as…”

“…Insertion of the ampholytic nanocolloidal particle from the bulk of the solution into the PE brush leads to the change in the ionization degree of both basic and acidic monomer groups. The respective free energy change can be presented , as normalΔ F i o n ( z ) / N Σ k B T = f + ln ( 1 − α + false( z false) 1 − α b + ) goodbreak+ ( 1 − f + ) .25em ln ( 1 − α − false( z false) 1 − α b − ) where α + ( z ), α b + and α – ( z ), α b – are the respective degrees of ionization of basic and acidic groups on the surface of the NPs placed at distance z from the grafting surface and in the bulk of the solution (at z = ∞), respectively. We remark that eq is a particular case of the more general equation derived in ref for the case when the particle surface is decorated by multiple types of cationic and anionic groups.…”

“…The position z * can be found from the equation ψ ( z * ) = prefix− δ normalp H b log ( e ) which implies that pH( z *) = pH IEP . By taking the derivative of the free energy Δ F ion ( z ), eq , with respect to z, it is straightforward to demonstrate that Δ F ion ( z ) exhibits a maximum in the point of the charge inversion, z = z * true( ∂ normalΔ F i o n ( z ) ∂ z true) z = z * = 0 ; .25em true( ∂ 2 normalΔ F i o n ( z ) ∂ z 2 true) z = z * < 0 ; goodbreak0em2em⁣ normalΔ F i o n ( z * ) > 0 (see ref for detailed discussion).…”

“…Insertion of the ampholytic nanocolloidal particle from the bulk of the solution into the PE brush leads to the change in the ionization degree of both basic and acidic monomer groups. The respective free energy change can be presented , as where α + ( z ), α b + and α – ( z ), α b – are the respective degrees of ionization of basic and acidic groups on the surface of the NPs placed at distance z from the grafting surface and in the bulk of the solution (at z = ∞), respectively. We remark that eq is a particular case of the more general equation derived in ref for the case when the particle surface is decorated by multiple types of cationic and anionic groups.…”

“…13 It is anticipated that in general cases, both mechanisms driving spontaneous protein absorption by the PE brush both below and above the IEP may be involved. 64 In our previous studies, 68,69 we have systematically studied the effects of the buffer pH b and salt concentration on the uptake of nanoparticles (NPs) by negatively charged PE brushes using a simplified model of NPs comprising equal numbers of ionizable cationic and anionic groups on its surface with equal acidic ionization constants, K + = K − . The position-dependent differential free energy of the NP insertion into the brush and net particle charge were calculated within the nonlinear Poisson−Boltzmann approximation.…”