Statistical Mechanics Approach to Lock-Key Supramolecular Chemistry Interactions

In the supramolecular chemistry field, intuitive concepts such as molecular complementarity and molecular recognition are used to explain the mechanism of lock-key associations. However, these concepts lack a precise definition, and consequently this mechanism is not well defined and understood. Here we address the physical basis of this mechanism, based on formal statistical mechanics, through Monte Carlo simulation and compare our results with recent experimental data for charged or uncharged lock-key colloids. We find that, given the size range of the molecules involved in these associations, the entropy contribution, driven by the solvent, rules the interaction, over that of the enthalpy. A universal behavior for the uncharged lock-key association is found. Based on our results, we propose a supramolecular chemistry definition.

Figure 1

System snapshot. Macromolecules at the distance of lower free energy for the perfect lock-key match, ${\sigma }_{\mathrm{c}}={\sigma }_{\mathrm{k}}$, for a key particle eight times larger than the solvent, ${\sigma }_{\mathrm{k}}=8{\sigma }_{\mathrm{s}}$, an electrolyte concentration of 0.15 M, and for like- and highly charged lock and key particles, $z_l=z_k=32$. Red (dark) and cyan (light) particles represent the counter- and coions, respectively. Counterions are seen inside the lock partially screening the direct electrostatic interaction. Solvent particles are shown as dots to gain clarity, although they have a size of 3 Å. Even for this highly charged system, the assembled configuration is the one with the lower free energy due to the entropy gained by the solvent (there is a large increase of the solvent accessible volume when the lock-key pair is bonded).

Figure 2

Energy contributions to the lock-key net energy. Different contributions to the net energy for a system having ${\sigma }_{\mathrm{k}}/{\sigma }_{\mathrm{s}}=4$ and for $z_l=z_k=8$. The inset zooms in on the same data. The attractive (negative) contact contribution from solvent molecules and the repulsive (positive) electrostatic contribution are clearly dominant. Counterions also produce a relatively small contact and repulsive contribution while screening the direct electrostatic force. Coions play no practical role. Note that a relatively large lock-key charge ($z_l=z_k=8$) is needed to counterbalance the collective attractive force.

Figure 3

Master curve for uncharged lock-key pairs and different macroparticles-solvent size ratios. Normalized energy against normalized distance for different ${\sigma }_{\mathrm{k}}/{\sigma }_{\mathrm{s}}$ ratios, $\phi =0.2$, and for an uncharged lock-key pair, $z_l=z_k=0$. Black circles, blue squares, cyan diamonds, red up triangles, and green down triangles correspond to ${\sigma }_{\mathrm{k}}/{\sigma }_{\mathrm{s}}=1$, 2, 4, 6, and 8, respectively. All curves are normalized with their energy maximum and corresponding distance. These values are given in the inset as a function of ${\sigma }_{\mathrm{s}}/{\sigma }_{\mathrm{k}}$. Both magnitude and range increase with increasing macromolecules-solvent size ratio. Thus, increasing the relative size between the macromolecules and solvent favors attraction. However, the range increases at a lower rate than the relative size (see $x_{\max }/{\sigma }_{\mathrm{k}}$ cyan line), and thus, the attraction range, in units of key diameters, decreases.

Figure 4

Energy behavior for like-charged macromolecules. Net energy against distance for (a) $z_l=z_k=2$ and (b) a constant charge density of $0.2833\mathrm{C}/{\mathrm{m}}^2$. Both plots are obtained for a constant occupied volume fraction and having the ${\sigma }_{\mathrm{k}}/{\sigma }_{\mathrm{s}}$ ratio as a parameter. Black circles, blue squares, cyan diamonds, red up triangles, and green down triangles correspond to ${\sigma }_{\mathrm{k}}/{\sigma }_{\mathrm{s}}=1$, 2, 4, 6, and 8, respectively. The insets zoom in on the corresponding data. In (a), except for the ${\sigma }_{\mathrm{k}}={\sigma }_{\mathrm{s}}$ case, all curves show attraction at contact. In this case attraction increases on increasing the relative key-solvent size, ${\sigma }_{\mathrm{k}}/{\sigma }_{\mathrm{s}}$. In (b), the charge at both macromolecules increases with the square of their size. Consequently, at contact, the electrostatic repulsion increases faster than the collective attractive contribution from solvent molecules for increasing the lock-key size. This produces a shift of the distance of minimum free energy towards larger lock-key distances. Despite the very large charge density set for these computations, bonding is always possible.

Figure 5

Controlling the fraction of assembled lock-key pairs by shape, charge, and occupied volume fraction. The fraction of assembled lock-key pairs as a function of the occupied volume fraction as obtained from Eq. (1) and considering an initial lock and key concentration of 0.001 M. All data correspond to ${\sigma }_{\mathrm{k}}/{\sigma }_{\mathrm{s}}=4$. Black circles, blue squares, cyan diamonds, and red triangles correspond to neutral lock and key pairs with ${\sigma }_{\mathrm{k}}/{\sigma }_{\mathrm{c}}=1.0$, 0.7, 1.3, and $\infty$ (a lock without a cavity), respectively. Green down triangles correspond to charged lock and key particles with $z_l=z_k=8$. Likely charged lock-key particles may influence bonding more than imperfect matches, though a sufficiently large solvent concentration would overcome the effect anyway. Simple spheres, on the other hand, do not bond at any reasonable volume fraction.