Self-assembly and bilayer-micelle transition of fatty acids studied by replica-exchange constant pH molecular dynamics

Abstract

Recent interest in the development of surfactant-based nanodelivery systems targeting tumor sites has sparked our curiosity in understanding the detailed mechanism of the self-assembly and phase transitions of pH-sensitive surfactants. Toward this goal, we applied a state-of-the-art simulation technique, continuous constant pH molecular dynamics (CpHMD) with the hybrid-solvent scheme and pH-based replica-exchange protocol, to study the de novo self-assembly of 30 and 40 lauric acids, a simple model titratable surfactant. We observed the formation of a gel-state bilayer at low and intermediate pH and a spherical micelle at high pH, with the phase transition starting at 20-30% ionization and being completed at 50%. The degree of cooperativity for the transition increases from the 30-mer to the 40-mer. The calculated apparent or bulk pKa value is 7.0 for the 30-mer and 7.5 for the 40-mer. Congruent with experiment, these data demonstrate that CpHMD is capable of accurately modeling large conformational transitions of surfactant systems while allowing the simultaneous proton titration of constituent molecules. We suggest that CpHMD simulations may become a useful tool in aiding in the design and development of pH-sensitive nanocarriers for a variety of biomedical and technological applications.

Figure 1

Top. Cumulative unprotonated fraction of the 40-mer system at all pH values as a function of simulation time. Only the last 20 ns is shown. 1 ps is equivalent to 1 exchange step. Bottom. All replicas walk through the entire pH ladder. For clarity the pH conditions visited by two randomly chosen replicas as a function of time are shown.

Figure 2

Representative snapshots of the 40-mer at pH 4 (left) and pH 9 (right) in the simulation box. Alkyl tails are shown in cyan, while the protonated and unprotonated headgroups are represented by blue and red beads, respectively. For clarity water is not shown.

Figure 3

Distribution of the angle between two surfactants in the 30- (top) and 40-mer (bottom) systems. For clarity only the integer pH conditions are shown with the exception of pH 6.5 and 7.5 displayed as dashed orange and blue curves for the 40-mer. Magenta curve represents the distribution obtained from the simulation of a spherical micelle comprising 60 laurate ions.

Figure 4

Probability distribution of the alkyl-chain length (defined as the C₁-C₁₂ distance) in the bilayer (top) and micelle (bottom). The 30- and 40-mers are displayed in red and blue, respectively. The dash black curve represents the cumulative probability (integration of the probability density). Since it is very similar for the two aggregates, only one curve is shown.

Figure 5

Blue bars indicate the fractional overlap, which is defined as (2 * L − T)/L, where L represents the average monomer length (distance between the C₁ and C₁₂ atoms in both leaflets) and T represents the average bilayer thickness (distance between the C₁ atoms of the upper and lower leaflets).

Figure 6

Top. Titration curves for the 30- and 40-mers. The fraction of laurate ions was calculated at each pH condition based on the last 20-ns data. Bottom. Calculated pK_a values for the individual lauric acids in the 30-mer (red) and 40-mer (blue) aggregate. The dash lines represent the average pK_a’s.

Figure 7

Relative free energy (kcal/mol) as a function of P₂ order parameter and pH for the 30-(top) and 40-mer (bottom).

Figure 8

Top. Distribution of the angles between two surfactants in the transition states of the 40-mer at pH 6.5. Bottom. Representative snapshot of a transition state. Colors are the same as in Figure 2.

Figure 9

Top. Radial distribution functions between carboxyl carbons in the 40-mer at indicated pH conditions. Plots are shifted vertically to facilitate visualization. Bottom. Average number of hydrogen bonds between laurate and lauric acid headgroups for the 40-mer at different pH conditions. Error bars represent standard deviation.

Figure 10

Top. Probability distribution of the angles between octanoic acids obtained from the simulation the self-assembled 20-mer. Data was collected from 15–30 ns for each pH replica. Bottom. Unprotonated fraction of octanoic acids at different pH conditions. Solid curve is the best fit of the data using the Hill equation.