Acid-Base Equilibrium and Dielectric Environment Regulate Charge in Supramolecular Nanofibers

Abstract

Peptide amphiphiles are a class of molecules that can self-assemble into a variety of supramolecular structures, including high-aspect-ratio nanofibers. It is challenging to model and predict the charges in these supramolecular nanofibers because the ionization state of the peptides are not fixed but liable to change due to the acid-base equilibrium that is coupled to the structural organization of the peptide amphiphile molecules. Here, we have developed a theoretical model to describe and predict the amount of charge found on self-assembled peptide amphiphiles as a function of pH and ion concentration. In particular, we computed the amount of charge of peptide amphiphiles nanofibers with the sequence C ₁₆ - V ₂ A ₂ E ₂. In our theoretical formulation, we consider charge regulation of the carboxylic acid groups, which involves the acid-base chemical equilibrium of the glutamic acid residues and the possibility of ion condensation. The charge regulation is coupled with the local dielectric environment by allowing for a varying dielectric constant that also includes a position-dependent electrostatic solvation energy for the charged species. We find that the charges on the glutamic acid residues of the peptide amphiphile nanofiber are much lower than the same functional group in aqueous solution. There is a strong coupling between the charging via the acid-base equilibrium and the local dielectric environment. Our model predicts a much lower degree of deprotonation for a position-dependent relative dielectric constant compared to a constant dielectric background. Furthermore, the shape and size of the electrostatic potential as well as the counterion distribution are quantitatively and qualitatively different. These results indicate that an accurate model of peptide amphiphile self-assembly must take into account charge regulation of acidic groups through acid-base equilibria and ion condensation, as well as coupling to the local dielectric environment.

Keywords: charge regulation; dielectric constant; ion condensation; peptide amphiphiles; theory.

FIGURE 1

Atomistic simulation snapshot and density profiles. (A) chemical structure of C ₁₆ − V ₂ A ₂ E ₂. (B) Snapshot of the last simulation configuration. C ₁₆ − V ₂ A ₂ E ₂ nanofiber is highlighted at the center, with glutamic acid residue colored in orange and hydrogen atoms omitted for display. Na ⁺ ions are represented by blue beads, and water by black dots. Blue solid lines denote the simulation box. Aggregation number is 17.3 ± 0.1 PA/nm. (C) Volume fraction of PA-fiber and number density of Glu units as a function of radial cylindrical coordinate.

FIGURE 2

The average degree of dissociation as function of reservoir pH for increasing (A) RbCl and (B) NaCl salt concentrations. The line labeled ‘ideal’ represent the ideal solution behavior.

FIGURE 3

The average degree of protonation, deprotonation and ion-condensation of the glutamic acid residue as function of reservoir pH for a concentration of [RbCl] = 50 mM.

FIGURE 4

The radial cylindrical number density distribution of the total, and the protonated, deprotonated and Rb⁺ condensed Glu acid residues of the PA-nanofiber. For a reservoir pH of 7.4 and a RbCl concentration of [RbCl] = 50 mM. The inset shows the relative dielectric constant.

FIGURE 5

The average degree of protonation, as a function of reservoir pH for various carboxylate-Rb dissociation free energies. Form top to bottom the dissociation free energy, or equivalent the binding free energy decreases. The reservoir has a RbCl concentration of [RbCl] = 50 mM.

FIGURE 6

The radial cylindrical number density distribution of free Rb⁺ ions (solid lines) and bound Rb⁺ ions (dashed lines). The different lines correspond to different carboxylate-Rb dissociation free energies. The reservoir has a pH of 7.4 and a RbCl concentration of [RbCl] = 50 mM.

FIGURE 7

The average degree of charge as function of reservoir pH for fixed dielectric constant, varying dielectric constant and varying dielectric constant plus electrostatic self-energy. The reservoir has a RbCl concentration of [RbCl] = 50 mM.

FIGURE 8

The electrostatic potential (A) and total charge number density (B) as function of radial coordinate for fixed dielectric constant (dashed lines) and varying dielectric constant plus electrostatic solvation energy (solid lines). The inset shows the total Rb⁺ ion concentration including free and bound ions. The reservoir has a pH value of 7.4 and a RbCl concentration of [RbCl] = 50 mM.