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Statistical Mechanics Approach to Lock-Key Supramolecular Chemistry Interactions Gerardo Odriozola and Marcelo Lozada-Cassou* Programa de Ingenierı´a Molecular, Instituto Mexicano del Petro´leo, Eje Central La´zaro Ca´rdenas 152, 07730 Me´xico D.F., Mexico (Received 9 November 2012; published 4 March 2013) 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. DOI: 10.1103/PhysRevLett.110.105701
PACS numbers: 64.75.Yz, 05.65.+b, 81.16.Fg, 82.70.Dd
In the bottom-up approach for developing new materials, molecular self-assembly plays a fundamental role [1–4]. While there are a number of concepts, such as noncovalent chemistry, molecular complementarity, and molecular recognition, proposed to explain nanoparticle, colloidal, or supramolecular assemblies, the self-assembly mechanism is not yet well understood [5–10], nor does there seem to be a clear definition of supramolecular chemistry (SMC) [11,12]. Recently, Sacanna et al. [13] synthesized colloidal particles with geometric complementarity, i.e., using Fischer’s lock-and-key principle [14], and showed that the reversible, self-organization of their colloidal particles reduced to a simple geometrical design, regardless of composition and surface chemistry. In the past, we proposed a noncharged lock-key model which closely resembles the Sacanna experimental colloidal particles [15,16], with predictions in general accordance with their results. Here, we extend our Monte Carlo (MC) calculations to charged lock-key particles to include enthalpy effects, successfully compare them with experimental results, show a universality for uncharged lock-key interactions, and explain the lock-key self-assembly mechanism, based on general statistical mechanics principles, which leads to a supramolecular chemistry definition. In Fischer’s lock-key theory [14], the lock and key particles have geometrical complementarity. This is a key concept in molecular self-assembly, which, in turn, is the basis of SMC. SMC can be defined as the chemistry beyond the molecule, where the spatial organization of the entities is ruled by noncovalent and frequently reversible interactions [8,11,12]. In addition to geometrical complementarity, the SMC definition involves key concepts, such as molecular recognition and molecular complementarity (not necessarily geometrical) [8,11,12]. The SMC definition covers a vast area of chemistry, which goes from small solvent molecules clustering to colloidal self-assembly, passing through enzyme-substrate complex 0031-9007=13=110(10)=105701(5)
formation. Hence, colloidal self-assembly could, in a way, be considered as a branch of SMC, or the other way around, since both seem to be ruled by a similar ‘‘association’’ mechanism. On the other hand, molecular complementarity can be associated with virtually all chemical reactions or molecular self-assemblies, and the term molecular recognition borrows recognition from the field of psychology and/or machine learning, where it is well defined, but it does not properly fit into the molecular field. These concepts are not well understood in SMC [5–9,12,14], or in the nanoparticle assembly arena [10]. In short, SMC lacks a precise definition [12]. In our calculations we consider noncharged, as well as charged macromolecules, immersed in an electrolyte solution, to consider systems without and with an enthalpy contribution to their association. We will address ‘‘molecular recognition,’’ and show that this mechanistic approach can be unequivocally understood in terms of entropy gain, or equivalently, in terms of the amount of ignorance, as defined in the statistical mechanics theory [17], or uncertainty or missing information (MI), as defined in the information theory [18,19]. Our model consists of a lock-key pair, ions, and solvent particles. The lock is a hard sphere of diameter 1 , with a point charge of valence zl at its center, and a spherical cavity, of diameter 1 , whose center is located at a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ 1=2 2l 2c from the lock center. For all computations we set l ¼ 1:7c . The key and solvent particles are respectively. hard spheres of diameters k and s ¼ 3 A, The key can also have a point charge at its center of valence zk ¼ zl . Finally, a 1:1 restricted primitive model electro a concentration of 0.15 M lyte is set with e ¼ 4:25 A, (additional cations or anions are added to make the whole system electroneutral), and dielectric constant ¼ 78:5. ‘B z i z j The electrostatic interaction is given by UE ðrij Þ ¼ r , ij with ¼ 1=kB T, kB the Boltzmann constant, T ¼ 298 K
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the absolute temperature, zi and zj the valences of sites i 2 the Bjerrum length, and rij the and j, ‘B ¼ e ¼ 7:14 A distance between sites i and j. We consider both, likecharged and uncharged lock-key pairs. In the unlikecharged case, both the enthalpy and entropy contributions promote lock-key association, so we do not include this calculation here. For the uncharged case zl ¼ zk ¼ 0, and no electrolyte is added to the system: hence, no enthalpic contribution is present. As in previous work [15,16], standard NVT simulations are carried out for these systems by fixing the positions of the lock-key pair in such a way that the centers of the lock, its spherical cavity, and the key are aligned. Periodical boundary conditions are set for the three orthogonal directions and the Ewald summation formalism is employed to deal with Coulomb interactions. A snapshot of an equilibrated configuration corresponding to the system with c ¼ k , k ¼ 8s , zl ¼ zk ¼ 32, an electrolyte concentration of 0.15 M, and a total occupied volume fraction of ’ ¼ 0:2 is shown in Fig. 1. In this particular run the macroparticles are set approximately at the distance of minimum free energy. For charged systems, the electrostatic contribution to the force P acting on macroparticle i is obtained by Fel ¼ h j rUE ðrij Þi, where j runs over all sites except i. The contact contribution, which is always present, is obtained by integrating the solvent plus ions contact density, ðs ¼ cteÞ, Fc ¼ R s kB Tðs ¼ cteÞnds, where s refers to the macroparticle surface and n is a unit normal vector. After obtaining the forces for several macroparticles separation distances x
(x ¼ 0 meaning contact between the cavity and key surfaces), the energy Ris calculated by integration, i.e., by computing EðxÞ ¼ x1 Fx dx. Here Fx is the mean force between a lock and a key, at a distance x between them, and EðxÞ is its configurational energy or potential of mean force. In soft-matter many-body physics g½3 LKi ðr1 ; r2 ; r3 Þ ð2Þ ðr ; r ; r Þ=g ðr ; r Þ] is the unsymmetrical [ gð3Þ LKi 1 2 3 LK 1 2 three-particle distribution function, of species lock, key and i (solvent or ion particles); i.e., it is the non-normalized conditional probability of finding an i species particle in the position r3 , given that the lock and the key particles are at positions r1 and r2 , respectively. gð3Þ LKi ðr1 ; r2 ; r3 Þ is the threeparticle distribution function, and gð2Þ LK ðr1 ; r2 ; xÞ is the twoparticle distribution function of lock and key particles [20]. In our simulations we obtain the conditional probability for the different values of x, at infinite dilution of lock and key particles. By integration of the solvent particles in g½3 LKi ðr1 ; r2 ; r3 Þ, for all possible values of x, a maximum probability is obtained for a minimum value EðxÞ. In general, EðxÞ ¼ ln½gð2Þ LK ðr1 ; r2 ; xÞ, where the lock and key particles are at positions r1 and r2 , respectively, and such that they are a distance x apart. For a finite concentration of locks and key particles, gð2Þ LK ðr1 ; r2 ; xÞ would be the twoparticle distribution function of lock and key particles; i.e., it would give the ratio of lock-key pairs. However, since in our case we are assuming a zero concentration of lock and key particles, EðxÞ is simply an energy. Figure 2 provides an example for the output of the model and method. It shows the different contributions to the net energy, i.e., the energy necessary to bring the surface of the 5
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FIG. 1 (color online). System snapshot. Macromolecules at the distance of lower free energy for the perfect lock-key match, c ¼ k , for a key particle eight times larger than the solvent, k ¼ 8s , an electrolyte concentration of 0.15 M, and for likeand highly charged lock and key particles, zl ¼ zk ¼ 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 ˚ . Even for this highly charged system, the assembled of 3 A 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).
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FIG. 2 (color online). Energy contributions to the lock-key net energy. Different contributions to the net energy for a system having k =s ¼ 4 and for zl ¼ zk ¼ 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 (zl ¼ zk ¼ 8) is needed to counterbalance the collective attractive force.
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key particle from infinity to a distance x of the cavity surface, for k =s ¼ 4 and zl ¼ zk ¼ 8. For the given example, the net energy shows a relative maximum and a minimum, as a function of x. Thus, in this case, reversible This is despite the relabonding may occur at x ’ 2 A. tively large charge we set at the lock and key particles. Realistic systems, such as enzyme-substrate systems, usually show k =s ratios well above 4. In particular, the lock-key colloids experimentally studied by Saccana et al. have k =s ’ 14 (taking s as the depletant diameter) [13]. Hence, here we study the energy dependence of k =s , although for smaller k =s ratios, for ’ ¼ 0:2, and uncharged lock-key particles. The energy curves, as a function of x, were found to collapse, defining a master curve when normalized with the value, Emax , and location, xmax , of their maxima, showing the universality of the uncharged lock-key interaction energy, in terms of the relative key-solvent sizes. This master curve is shown in Fig. 3. The master curve is strongly negative at contact, 38E=Emax , and develops several maxima and minima of decreasing amplitude for increasing distance. The values of Emax and xmax are given in the inset as a function of s =k . Both, Emax and xmax increase for increasing relative size of the macroparticles with respect to the solvent. In other words, the magnitude and range of the solvent contact contribution to the net energy increase with the macroparticles’ size. Thus, for larger k =s ratios the attractive energy, given in absolute units, is stronger, and of longer
range, indicating a larger entropic contribution. This lockkey bonding is independent of the shape of the cavity (or cavities) in the lock particle, as long as there is a good match with the key (or keys) shape [21], since this match guaranties a maximum entropy. Thus, although in our calculations we have restricted ourselves to a particular lock-key geometry, our results suggest a universal behavior of the lock-key interaction, also for other lock-key geometries, although the corresponding master curves will not necessarily be the same; i.e., different universality classes will be found. In order to evaluate the enthalpic role in the lock-key assembly, we performed the following computer experiments. First (case a), for a constant occupied volume fraction and a constant charge in both particles, zl ¼ zk ¼ 2, we varied k =s . Second (case b), we performed the same experiment but replaced the constant charge by a fixed surface charge density on both macroparticles. At the key we set 0:2833 C=m2 and kept zl ¼ zk , so that zk and zl increased with the square of the macromolecules’ size. The results are shown in Fig. 4. In case a, with the exception of the black circles curve where the solvent and key are the same size, k =s ¼ 1, all curves show an energy minimum located at contact. These minima increase their magnitude with an increase in the k =s ratio. Thus, in this (a)
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FIG. 3 (color online). Master curve for uncharged lock-key pairs and different macroparticles-solvent size ratios. Normalized energy against normalized distance for different k =s ratios, ’ ¼ 0:2, and for an uncharged lock-key pair, zl ¼ zk ¼ 0. Black circles, blue squares, cyan diamonds, red up triangles, and green down triangles correspond to k =s ¼ 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 s =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 xmax =k cyan line), and thus, the attraction range, in units of key diameters, decreases.
FIG. 4 (color online). Energy behavior for like-charged macromolecules. Net energy against distance for (a) zl ¼ zk ¼ 2 and (b) a constant charge density of 0:2833 C=m2 . Both plots are obtained for a constant occupied volume fraction and having the k =s ratio as a parameter. Black circles, blue squares, cyan diamonds, red up triangles, and green down triangles correspond to k =s ¼ 1, 2, 4, 6, and 8, respectively. The insets zoom in on the corresponding data. In (a), except for the k ¼ s case, all curves show attraction at contact. In this case attraction increases on increasing the relative key-solvent size, k =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.
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case, the contact solvent contribution increases its dominant role with increasing k =s . The picture changes when considering a constant surface charge density, case b; i.e., increasing k =s leads to the exact opposite behavior. That is, at the macroparticles contact position, a low k =s favors attraction and repulsion is enhanced with increasing k =s . Nonetheless, the minima do not vanish but shift their location to larger distances. The minima depth, on the other hand, reaches a minimum at k =s ¼ 6, increasing for larger k =s ratios. Consequently, for colloidal particles, where a quasihomogeneous surface charge density is frequently found, the minima locations are expected at larger distances than contact, as is usual for reversible colloidal bonding, at the secondary minimum [22]. By considering an ideal behavior for lock, key, and bonded lock-key entities (diluted gases), and accounting for a configurational entropy associated with fluctuations in the relative position within the lock cavity volume vc , the following expression is obtained [13]: ½L0 ½LK ¼ expf½Emin þ lnðvc ½L0 Þ=g; ½L½K
(1)
where ½LK, ½L, and ½K stand for the concentrations of bonded lock and key, free lock, and free key particles, respectively, Emin is the bonding energy, and ½L0 ¼ ½LK þ ½L is the total lock concentration. This expression neglects the rotational degrees of freedom associated to the bonded lock and key dimer, and assumes the lock-key bonding as the only possible association. Figure 5 is built by setting Emin as the overall value of the minimum energy of our data and considering ½L0 ¼ 104 M. Notice that Eq. (1) implies finite lock and key concentrations. Hence, in accordance with the low concentrations of lock, key, and lock-key particles assumed above, which implies a noncorrelation with lock and key particles outside our simulation box, and since in our simulations we obtain the conditional probabilities of the lock-key pairs, our results are clearly valid for these finite concentration conditions. This figure reproduces the trends reported by Sacanna et al. [13]. That is (a) increasing the volume fraction always enhances attraction, (b) the perfect lock-key match maximizes attraction when the other variables are fixed, (c) a charge stabilized lock and key colloid can bind for a sufficiently high occupied volume fraction, and (d) no binding occurs without the presence of a cavity. Clearly, if higher ½L0 values were considered, our curves would shift to the left, since there would be an overall higher volume fraction. As pictured in Fig. 5, without a solvent there is no assembly. Thus, solvent molecules do not only keep suspended and hydrated colloidal particles and solutes, but they also play an important role for the assembly of macromolecules which goes beyond the known enthalpy contributions. That is, this role is certainly played by several types of interactions, such as van der Waals forces, hydrogen bonding, and dipole-dipole interactions, among others,
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FIG. 5 (color online). 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 k =s ¼ 4. Black circles, blue squares, cyan diamonds, and red triangles correspond to neutral lock and key pairs with k =c ¼ 1:0, 0.7, 1.3, and 1 (a lock without a cavity), respectively. Green down triangles correspond to charged lock and key particles with zl ¼ zk ¼ 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.
but, more importantly, by their contact (moment transference) contribution with an entropic origin to the net force. When assembly occurs, there is an increase of the accessible volume for the solvent molecules leading to a drastic increment of solvent entropy. This solvent entropy gain overcompensates by far the entropy loss associated to the lock and key bonding, since the number of solvent molecules is generally several orders of magnitude larger than the number of lock and key entities. This fact is based on first principles and so is general, independently of the lock, key, and solvent sizes and shapes, as long as there is a lockkey geometrical match. Consequently, these general principles, shown to be valid in simple mathematical and real colloidal models, should also apply in the SMC field. While the use of terms such as ‘‘molecular recognition and complementarity’’ in the SMC field are intuitively appealing, since there is a clear geometric complementarity between the lock and key particles, they are probably misleading since the ‘‘recognition process concept’’ usually involves memory, thus information, and some degree of intelligence (artificial or not) which molecules, certainly, do not posses. In terms of information, complementarity means that two or more parts complement each other to exhaust the possible information about the objects. In our case the lock, key, and solvent particles are the parts that complement each other. Thus, when lock and key molecules ‘‘geometrically complement and recognize each other’’ a
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maximization of the MI occurs. Moreover, they cannot ‘‘complement and recognize’’ in the absence of a solvent. For a given energy, the total number of microstates, in space that are accessible to the lock-key-solvent system, gives for the entropy, S ¼ klnðÞ. Because of the assumption of equal a priori probabilities of the statistical mechanics theory, in the frame of the information theory [18,19], measures the amount of ignorance, uncertainty, or MI in the system. At finite lock and key concentrations, when a fraction of keys enters into the locks, the number of microstates increases, thus maximizing the entropy or MI. Thence, the difference in MI between the final (keys inside locks) and initial (unmatched locks and keys) states is the lost information. In synthesis, locks and keys do not recognize or have information about each other; it is the system which increases the MI in the final state, or, equivalently, the amount of uncertainty on its microstate, when going from unmatched to matched positions. For finite concentrations of locks and keys, the states of maximal probability, i.e., the most probable (thermodynamical equilibrium) configurations, are those where a fraction of locks and keys is matched, according to an optimization of an energy-entropy balance. In summary, SMC is based on an enthalpy-entropy balance, where covalent interactions are not present, and where entropy gain rules the interaction, due to the molecular reorganization. We stress that our analysis does not depend on the key or lock cavity shapes, nor, for that matter, does it depend either on the number of cavities, as long as there is a match between them. Hence, we propose a SMC definition, after one of the generally accepted definitions [8]: the SMC is the chemistry beyond the molecule, where the spatial organization of the entities is ruled by entropy, and noncovalent, frequently reversible, interactions. M. L.-C. thanks Professor Jean-Marie Lehn for useful correspondence. The support of CONACYT (Grant No. 169125) is gratefully acknowledge.
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*To whom all correspondence should be addressed.
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