Cooperativity in Noncovalent Interactions - Chemical Reviews (ACS

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Cooperativity in Noncovalent Interactions A. Subha Mahadevi and G. Narahari Sastry* Centre for Molecular Modelling, CSIR-Indian Institute of Chemical Technology, Tarnaka, Hyderabad, India 500607 11. 3. Catalysis 11. 4. Material Science 12. Prospects and Outlook Author Information Corresponding Author Notes Biographies Acknowledgments References

1. INTRODUCTION After conquering the atomic structure about a century ago, chemists have been largely interested in understanding the concept of chemical bond and the formation of a molecule from its constituent atoms.1 Electrons, nucleus, as well as the nuclear particles have their individual characteristics and give rise to different types of elements, which make up our periodic table. Molecules, which are formed from these atoms, possess their characteristically exquisite properties depending not only on the type of atoms but also on the way in which they are connected. These connections, known as chemical bonds, are at the heart of chemistry and provide the rational power to engage in a beautiful excursion toward the design of molecules through making and breaking bonds. Therefore, in the contemporary age whenever a chemist, or for that matter any natural scientist, refers to a pure compound, such as benzene, water, or even more complicated structures such as taxol, fullerenes, etc., they refer to their molecular structures. Clearly, molecules are individual building blocks, and the way in which they organize themselves have a great bearing on their properties. We all know that the structure of most matter is determined by the way in which molecules associate with themselves, giving rise to different states of matter, and it is particularly interesting that in the condensed phases, the properties of molecules behave very differently depending on the temperature, pressure, volume, the presence of other molecules, interfaces with surfaces, environment, and under varying electric or magnetic fields. In all cases, while the molecule as an entity is intact, the way in which noncovalent forces operate with molecules of the same type or with other molecules or surfaces is the key in determining these variations. Understanding the nature of noncovalent interactions is thus extremely important to see what causes these variations in the properties.2−9 The most dramatic changes in the properties of the molecules occur in condensed phases, in liquid and solid, as the way in which noncovalent interactions operate. Further, the way in which two molecules interact with each other are largely governed by a combination of dispersion2,8 and electrostatic

CONTENTS 1. Introduction 2. Origin of Cooperativity 3. Hydrogen Bonding and Cooperativity 3. 1. Hydrogen-Bonded Clusters 3. 2. Hydrogen Bonding in Peptides and Carbohydrates 3. 3. Hydrogen Bonding-Interplay with Other Noncovalent Interactions 4. Quantitative Definitions for Cooperativity 4. 1. Many Body Analysis 4. 2. Double and Triple Mutant Analysis 4. 3. Measures for Estimation of Cooperativity in Noncovalent Interactions 4. 3. 1. Interaction Energy and Energy Decomposition Analysis 4. 3. 2. Spectoscopic Parameters 4. 3. 3. Structural Analysis 5. How Does a Pair of Noncovalent Interactions Mutually Influence Each Other? 5.1. Cation-Induced Cooperativity 5.2. Anion-Induced Cooperativity 6. Strong Manifestation of Weak Interactions 7. Interplay of π−π Interactions 7.1. With Other Noncovalent Interactions 7.2. In Molecular Clusters 8. Modulation of Cooperativity Due to Solvation 9. Cooperativity in Biochemistry 9.1. Allostery and Binding Cooperativity 9.2. Multimeric Proteins, Ion Channels, and Enzymes 9.3. Drug Receptor Interactions 9.4. Protein Folding 9.5. Transcription 9.6. DNA/RNA 10. Anticooperativity 11. Manifestation of Cooperativity 11. 1. Self-Assembly 11. 2. Chelate Cooperativity

© 2016 American Chemical Society

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Received: June 30, 2014 Published: February 3, 2016 2775

DOI: 10.1021/cr500344e Chem. Rev. 2016, 116, 2775−2825

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Noncovalent interactions were first recognized by J. D. van der Waals in the later part of 19th century helping in reformulation of the equation of state for real gases.2 Unlike covalent interaction which leads to the formation of a classical molecule, noncovalent interaction leads to the formation of molecular clusters. Some noncovalent interactions are also known to act at long distances of several angstroms unlike covalent bonds.3−7 For example, the electrostatic, induction, and dispersion interactions are longrange interactions, while the exchange-repulsion and chargetransfer interactions are classified as short-range interactions. The short-range interactions have their origin in the overlap of molecular orbitals and do not act at long distances. Although these interactions are traditionally considered to be weak, their strength covers a substantial range, starting from a few kJ/mol to several hundreds of kJ/mol depending on the type of interaction. Earlier reports by Kollman have indicated how using the study of noncovalent complexes indeed involves a fruitful interplay between theory and experiment.3 Conventionally, from a chemical point of view, the nature of a given chemical bond, for example a C−C bond, remains largely unaffected by its surroundings.1 This is the basis for developing molecular mechanics force fields, hybridization, and as such giving an estimation of bond energies of a typical bond pair as shown below.

interactions. So much so that hydrogen bonding is one of the most popular noncovalent interactions and in a colloquial sense is almost synonymous with noncovalent interactions.10−13 On the basis of the structure of interacting molecules, cation−π interaction,14−17 π−π stacking,18 CH···π interaction,18 anion−π interaction,19−23 halogen bond,24 lone pair interaction25 and so on and so forth have emerged (Figure 1). Indeed, the origin of

E (r N ) =

∑ bonds

Figure 1. Representation of the different kinds of noncovalent interactions.

+

ki (li − li ,0)2 + 2

∑ torsions

attraction in these interactions is electrostatic, induction, and dispersion interactions. While most of these interactions are in general weaker than covalent interactions, the strength of some of the bonds such as hydrogen bond, cation−π, and anion−π can have a varied range and in some particular instances might be even slightly stronger than a weak covalent bond.

∑ angles

ki (θi − θi ,0)2 2

Vn [1 + cos(nω − γ )] 2

(1)

Signature of a typical covalent bond is that its bond length, bond strength, as well as the spectroscopic signatures are usually characteristic, and any modulations in their strength by neighboring covalent bonds tend to be a small percentage of its absolute value. Thus, in general, variations in the covalent

Figure 2. Manifestation of cooperativity in highly varied fields. 2776

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assemblies, catalysis, and biological function is taken up. An extensive review of the available literature on cooperativity also reveals how cooperative behavior has aided substantially toward understanding the conformational transitions and allosteric interactions in proteins. Cooperativity is thus a concept of vital importance in chemistry, biology, material science, and allied areas.26−36 The synergistic interplay between different kinds of noncovalent interactions is crucial to maintain the structure of important biomacromolecules like DNA and proteins and in retaining the fidelity of information processing needed for normal life. In the field of chemistry too, cooperativity of noncovalent interactions has an essential role in templatedirected synthesis, the transmission of stereochemical information, and in determining the structure and properties of materials.39−43

bonds are not very dramatic. However, noncovalent interactions can vary remarkably. Let us consider the C−C single bond in ethane and propane which are comparable in their length and strength which remains largely so in the congeneric series of alkanes. However, let us take a dimer of water connected through a hydrogen bond and compare it with a trimer, tetramer, and a larger polymer. It has been unambiguously established that the average hydrogen bond strength dramatically increases, with the concomitant shortening of hydrogen bond length, becoming almost double when it reaches a decamer.13 Thus, the way in which the noncovalent interactions interact with themselves is certainly not additive. Hydrogen bond strength ranges from a very weak interaction to somewhat comparable with a weak covalent bond. Similarly, the strengths of cation−π and π−π interactions can be altered by two to three times, higher or lower depending on the environment. One more important aspect in any given process of aggregation is the large number of noncovalent interactions that operate simultaneously. Thus, noncovalent interactions are a topic of outstanding importance, in all areas of chemistry, material science, biology, and medicine. The additive scheme used in eq 1 for covalent bonds encounters utter inadequacies when applied to systems governed by noncovalent interactions. The underlying reason for this failure is well-known nonadditivity. This nonadditivity arises due to the cooperativity or anticooperativity of noncovalent interactions. In the current review, we try to describe the concept of cooperativity among noncovalent interactions.26−32 When a pair of noncovalent interactions strengthen each other, it is called cooperative while when they weaken each other they are defined as operating in an anticooperative manner. Cooperativity implies that the sum of at least two interactions is larger than the simple addition of the individual interactions. The terms cooperative and anticooperative have different origins and meanings in different fields28,33−36 and are extensively used in biology (Figure 2). Although the qualitative concept of cooperativity is quite clear, quantitative experimental measures barring some spectroscopic approaches are rather scarce. Computational quantum chemical methods provide the most relevant means of probing and quantifying cooperativity. It is interesting to examine the strength and limitations of quantum chemical methods and the dependence of the cooperativity measure on method, basis set employed, basis set superposition error, etc. Knowledge of how noncovalent interactions manifest themselves in small molecular clusters and the quantification of their cooperativity and anticooperativity thus appears to be essential for comprehension of supramolecular assembly.37,38 In the current review, we start with a discussion on the origin of cooperativity. Historically, cooperativity has been studied extensively in hydrogen-bonded systems and is thus discussed as a prelude to the section on its quantification and measurement. From a fundamental point of view, it is interesting to examine how a pair of noncovalent interactions mutually influences each other. In addition to hydrogen bonding, noncovalent interactions such as cation−π, anion−π, and other weak interactions have a strong manifestation in controlling supramolecular structure and function. Among the noncovalent interactions, cation−π may be argued to be stronger, while π−π and CH−π interactions are weaker compared to hydrogen bonding. We divide the review into sections on cooperativity in hydrogenbonded clusters, cation−π induced cooperativity, anion−π induced cooperativity, and cooperativity among weak interactions. Finally, the impact of cooperativity in understanding various phenomena such as formation of supramolecular

2. ORIGIN OF COOPERATIVITY An explanation of the origin of cooperativity is helpful for understanding various cooperative effects. The cooperativity reported in this review can be classified into the following types based on origin: (a) many body interaction, (b) secondary interaction, (c) chelate effects, and (d) cooperativity and anticooperativity induced by conformation change. The many body interaction is a major source of the cooperativity in the clusters of small molecules and is the origin of hydrogen bond cooperativity. It not only considers the sum of pairwise interactions but also includes three-body terms, fourbody terms, and so on. The contribution of different intermolecular forces to many body interaction is very well summarized in the book, The Theory of Intermolecular Forces by A. J. Stone.44 The main contributions to forces between molecules may be classified into long-range and short-range. Three kinds of long-range effects are mentioned below. The electrostatic effect arises from the classical interaction between the static charge distributions of the two molecules. They are strictly pairwise additive and may be either attractive or repulsive in nature. Dispersion refers to the attractive term of the vdW equation. This attractive part of vdW interactions is an electron correlation (quantum mechanical many-body) effect known as London dispersion.9 The contribution of dispersion to the many body interaction is small. Induction effects arise from the distortion of a particular molecule in the electric field of all its neighbors and are always attractive. As the fields of several neighboring molecules may reinforce each other or cancel out, induction is strongly nonadditive.44 Because the origin of the induction interaction is the induced polarization by electric field, the many body interaction is particularly significant in cases where the interacting molecule or ion has a strong electric field such as the Mg2+ ion. Another vital aspect of the many body effects is the modification of intermolecular forces between molecules in the presence of a solvent. The secondary interaction is another important source of cooperativity.45−51 The secondary electrostatic interactions rationalize the relative stabilities of multiple hydrogen-bonded complexes. In multiple hydrogen-bonded complexes, the arrangement of the hydrogen bond donor (D) and acceptor (A) decides the total stability of the complex. Using molecular mechanics calculations on nucleic acid base pairs, Jorgensen et al. elucidated that secondary interactions between neighboring heteroatoms can contribute additional stabilization, provided that hydrogen bond donors with their positive partial charges are present in one molecule and acceptors in the other molecule (DD, AA).45−47 Repulsion between diagonally placed acceptor 2777

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quite a remarkable statement, and spectroscopic advances and computational methods have enabled one to quantitatively estimate the extent of cooperativity after almost 3−4 decades after the observation. Dannenberg and co-workers have undertaken a series of studies on amide clusters and ascertained the cooperativity in these hydrogen-bonded clusters.65−67 Besides, a plethora of experimental and computational studies which evaluate cooperativity in hydrogen-bonded clusters have also been reported over the last two decades.68−151 The areas which have witnessed the liberal use of the word cooperativity in the context of hydrogen bonding include delineation of peptide and carbohydrate structure152−176 and also the concept of resonance-assisted hydrogen bonding.177−179

and donor functions will destabilize the primary hydrogen bond when both molecules contain alternating sequences (e.g., AD, DA, etc.). Jorgensen et al. also reported the importance of secondary electrostatic interactions due to polar functional groups located closely to hydrogen bonding sites.46 Studies by Uchimaru et al. revealed that although the main source of the stabilization for hydrogen-bonding association was the electrostatic contribution, the secondary electrostatic and polarization interactions due to polar functional groups located closely to the hydrogen bonding sites also significantly alter the magnitude of hydrogen-bonding stabilization.48 Uchimaru and co-workers also reported that the hydrogen bond donor−acceptor orientation had an important role in cooperativity.49 Their case study in three iso-complexes of C8H9N5O2 showed that when a hydrogen bond is formed at the neighboring hydrogen bond site, the intrinsic hydrogen bond forming abilities of hydrogen bonding sites in the DA type molecules are increased, while those in the AA and DD type molecules are decreased. Another important source of cooperativity is the chelate effect.28,52−54 The loss of the entropy of translation which is associated with the cluster formation increases Gibbs free energy. If molecules can bind by two interactions, the formation of the second binding does not accompany the loss of the entropy of translation. This is the origin of chelate effect. Williams and coworkers52,53 have extensively discussed the general relevance of enthalpy/entropy compensations to binding interactions of biological importance in particular with respect to the binding of agonists versus antagonists to a common receptor site. The chelate effect is one of the important sources of the stability of βsheet structure, in which many hydrogen bonds exist. The major source of cooperativity in biological systems is one which is induced by conformational change. Typical examples of cooperativity in biology are the binding of oxygen to hemoglobin, wherein binding at each of its four binding sites increases the oxygen affinity of the other sites, and in the folding of proteins and nucleic acids that are characterized by sharp melting transitions.26 Indeed the origin of this kind of allosteric cooperativity is quite different from the origin of cooperativity found in clusters of small molecules bound by intermolecular forces. The hydrophobic interaction55 has its origin in the change of the structure of liquid water associated with the hydrophobic molecules. A study by Chandler56 suggests that water molecules can still form a hydrogen bond network around a small solute molecule, although it is constrained by the presence of the solute. The solvent entropy is reduced while the enthalpy remains unaffected. The hydrophobic force has long been considered as the major driving force of protein folding.57 When a protein folds, the hydrophobic side chains are embedded within the protein, leaving the hydrophilic side chains at the surface.

3. 1. Hydrogen-Bonded Clusters

Water clusters have remained as the most favored model systems while investigating the intermolecular forces that act among them, especially with regard to nonadditive interaction energies.11,68−95 Beginning in the late 1960s Morokuma and Pedersen68 and Kollman and Allen69 were among the first to carry out ab initio Hartree−Fock calculations for a pair of interacting water molecules. In 1970, Moskowitz and co-workers employed SCF calculations to investigate the potential of interaction for dimers and trimers of water molecules.70 The study clearly revealed how three-molecule nonadditivities are large in magnitude and vary in sign depending on the hydrogen bond pattern involved. Huyskens reported a study on factors governing the influence of a first hydrogen bond on the formation of a second one by the same molecule or ion.11 Several experimental studies have addressed the structure and spectra of small water clusters, ice, as well as larger threedimensional water clusters.43,71−74,93 A few experimental studies which were employed to study hydrogen bond cooperativity are mentioned below. A study of self-association and hydrogen bonding of 3,4,5-trichlorophenol with water using matrixisolation FT-IR spectroscopy provided experimental proof for the dominating role of the induction energy term in the three body effect.77 Saykally and co-workers reported the quantification of hydrogen bond cooperativity in the water by measuring the far-infrared vibration−rotation tunnelling spectrum of the perdeuterated water tetramer.43 Pate and co-workers have employed broadband rotational spectroscopy on water clusters produced in a pulsed molecular jet expansion to determine the oxygen atom geometry in three isomers of the nonamer and two isomers of the decamer.75 The cooperativity effects revealed by the hydrogen bond O−O distance variations in this study were shown to be consistent with a simple model for hydrogen bonding in water that takes into account the cooperative and anticooperative bonding effects of nearby water molecules. Schmidt and co-workers reported an alternative interpretation of the structure of the IR vibrational mode [υ(OH) band] of pure water.76 The reinterpretation was based on the influence of the cooperative hydrogen bonding arising from a network of hydrogen bonds in the liquid. The insights obtained by foregoing experimental studies have generated a lot of interest among theoretical and computational chemists. Ab initio studies for the ground states of the linear water dimer with Cs symmetry and cyclic water tetramer with S4 symmetry were undertaken by Lesyng and co-workers.79 Employing a cooperativity parameter based on the two-body, non-neighbor interaction energy, plus three- and four-body contributions, including one-body deformation terms in relation to the total interaction energy of the water tetramer, they demonstrated an

3. HYDROGEN BONDING AND COOPERATIVITY Hydrogen bonding has been a subject of extensive study, and several key reviews are available in literature revealing its wideranging importance.58−64 In most cases, hydrogen-bonded clusters have been employed to model cooperativity. Historically, the first mention (to our knowledge) of cooperativity can be seen in the paper of H. S. Frank and W. Wen in Discussions of Faraday Society.10 The authors made a dramatic observation that “the formation of hydrogen bonds in water is predominantly a cooperative phenomenon, so that, in most cases, when one bond forms, several (perhaps “many”) will form, and when one bond breaks, then, typically, a whole cluster will “dissolve” ”. This is 2778

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Figure 3. Pictorial representation of four different kinds of arrangements of water clusters (W1D, W2D, W2DH, and W3D clusters) employed to evaluate cooperativity as a function of arrangement of individual molecules in a cluster of particular size. Edited and reprinted from ref 84. Copyright 2010 American Chemical Society.

energy ratios ([H2O10:2], [H2O20:10]) revealed that the strength of complexation energy increases by 33, 38, 35, and 91%, respectively, as cluster size increases from 2−10 in four different kinds of arrangements analyzed. As the cluster size increases from 10−20; however, the increase in strength of hydrogen bonding is in the range of 4−6% only. Hence the nonadditivity of hydrogen bond strength or the associated cooperativity was much more evident as cluster size increased from 3−10 rather than from 10− 20, where the augmentation of hydrogen bond strength is only marginal for all the four arrangements. Thus, the presence of a definite amount of cooperativity in hydrogen-bonded water clusters where the addition of a subsequent monomer to a hydrogen-bonded cluster augments the strength of existing interactions was inferred. The mode of arrangement, in particular, how these interactions were arranged was shown to play a crucial role on the strength of interaction. Application of combined infrared spectroscopy and DFT approaches by Ohno et al. demonstrated how the formation of one hydrogen bond in a hydrogen-bonded water chain cooperatively enhances or diminishes the strength of another hydrogen bond.85 An observation of how cooperativity of the hydrogen bonding of water molecules affects the corresponding OH stretching bands was made. Quantum calculations performed on a series of water clusters in order to mimic water molecules found in restricted environment unlike bulk water show that cooperative effects must be taken into account in the treatment of hydrogen bonds and water clusters in such bounded systems.86 Ruckenstein and co-workers focused on the role of many-body interactions on the structure of ordinary ice and liquid water revealing that 62−63% of hydrogen bonds must be broken to disintegrate a large hexagonal ice piece used to simulate water molecules in the form of a cube into small clusters.87 Stokely et al. performed a study on the lowtemperature phase behavior of liquid water by combining mean field calculations and Monte Carlo simulations.88 While emphasizing key physical quantities that determine scenarios which describe water, they report that it is the amount of cooperativity in relation to the strength of the directional component of the hydrogen bond that establishes which scenario prevails. A recent study explores the impact of hydrogen bond cooperativity in eight low lying water hexamers.89 Stabilizing

energy gain of 29% based on cooperativity in the S4 water tetramer with the MP3/6-31G** approximation. Suhai convincingly elucidated that hydrogen bonding in ice is a highly cooperative phenomenon where the optimization of the structure of the infinite water polymer at the MP2 level yielded a relative enhancement of 47% over the corresponding dimer value.80 Further, the study revealed that the cohesive energy of ice results from a delicate balance between different repulsive and attractive terms in third and fourth order, which exhibit different long-range behaviors. Xantheas studied the significance of all higher-order components for interaction energy, in particular, the three-body term among the nonadditive terms for water clusters ranging in size from clusters trimer through pentamer.81 Hydrogen-bonding networks wherein donor−acceptor arrangements existed between all water molecules were associated with the largest nonadditivities among other networks present in lowlying minima of small water clusters. Yáñez and co-workers performed an ab initio study on water trimers and identified a global minimum which corresponded to an asymmetric cyclic structure.82 This structure presented significant cooperative effects compared to a Cs dimer as reflected in several parameters such as a stiffer intermolecular potential, shorter O−O distances, longer donor O−H bond lengths, larger energies per hydrogen bond, and greater shifts of the donor O−H bond stretching frequencies than the Cs dimer. Studies on hydrogen bonding in phenol, water, and phenol-water clusters by Subramanian and coworkers have demonstrated the enhanced stability on account of hydrogen bonding, although for clusters with a similar hydrogenbonding pattern, intermolecular interaction in phenol clusters was slightly stronger than in water clusters.83 The impact of varying arrangements of water molecules in different kinds of clusters as a function both of size of cluster (n = 2−20) and method of computation used have been studied by our group using extensive density functional theory (DFT) and ab initio calculations (Figure 3).84 These model systems served to explain associated cooperativity in hydrogen bonding. The ratio of complexation energy per hydrogen bond from decamer to dimer [H2O10:2] and eicosamer to decamer [H2O20:10] was calculated to quantify the strength of hydrogen bond with increase in cluster size and was used as an indicator of cooperativity that is seen in the water clusters. The complexation 2779

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subjected to free optimization without any emphasis on the kind of order in the structure that is attained on optimization. The interaction energy per monomer increased from dimer to 15mer by 90% in the case of circular arrangement, by 76% in the case of linear arrangement, and by 34% in the case of a third standard arrangement, respectively. The cooperativity in hydrogen bonding for acetamide clusters was evaluated by considering the variation in geometrical parameters including bond distances and electron density values at the bond critical point in the different model systems. Similarly, DFT calculations were performed on four arrangements of formamide clusters [HCONH2]n, (n = 1−10) linear, circular, helical, and stacked forms.109 These studies reveal maximum cooperativity, based on interaction energy per monomer, in the stacked arrangement followed by the circular, helical, and linear arrangements, respectively. In all arrangements, an increasing trend in cooperativity of hydrogen bonding was observed as a function of increasing cluster size. Parra and co-workers reported DFT studies on intermolecular bifurcated hydrogen-bonding interactions in diformamide chains110 and carbonic acid clusters,111 showing the existence of significant cooperative effects in a linear network of three-center bifurcated hydrogen bonds as well as two-dimensional ringlike networks. There are two types of bifurcated three-center hydrogen bond interactions they have explored: (a) one that involves a hydrogen atom and two acceptor atoms denoted as A1HA2 and (b) one that involves one acceptor atom and two hydrogen atoms denoted as H1AH2. Positive cooperativity observed in these studies helps to rationalize the common occurrence of three-center H bonds in the crystals structures of many molecular systems. Cooperativity in intramolecular bifurcated hydrogen-bonding interactions has also been reported by them.112 Ludwig et al. analyzed the temperature dependence of NMR chemical shifts and quadrupole coupling constants in neat N-methylacetamide using experimental and ab initio quantum cluster equilibrium (QCE) theory.113 Strong cooperative effects were found in the linear Nmethylacetamide molecular clusters (n = 3−5) as reflected in the chemical shifts and quadrupole coupling values for each species. Theoretical calculations on smaller (HCN)n clusters where n = 1−4 at the RHF and MP2 level and for larger clusters where n = 5−7 at the RHF level of theory have shown that large cooperative effects of hydrogen bonding are reflected in increasing hydrogen bond energies, decreasing intermolecular separations, increasing average dipole moments and in C−H stretching frequencies shifts and the corresponding intensities as a consequence of increasing cluster size.114 Theoretical computations on systems consisting of up to four hydrazine molecules illustrate a significant contribution of cooperative phenomena to the interaction energy, amounting to as much as 12% of the overall interaction energy.115 DFT calculations to examine the effect of hydrogen bond cooperativity on the magnitude of the NMR chemical shifts and spin−spin coupling constants in a C4hsymmetric guanine-quartet and in structures consisting of six cyanamide monomers have been reported.116 The study showed that the magnitude of the NMR properties along the hydrogen bond network, for example the |1JNH| coupling and 1H and 15N chemical shifts of the hydrogen-bonding amino N−H group and the |h2JNN| trans-hydrogen bond coupling, increased for structures containing a larger number of monomers. Wang and co-workers have proposed a rapid method to predict the hydrogen bond cooperativity in long N-methylacetamide chains containing up to 200 monomers.117,118 It is based on parameters obtained from fittings to the hydrogen-bonding energies in N-

cooperativity observed in linear hydrogen-bonded water systems diminished as clusters move from nearly planar to threedimensional structures. Water molecules which donated both hydrogens to form double-donating interactions had increased stabilization, whereas waters which accepted two hydrogen bonds experienced a decrease in stabilization due to cooperative and anticooperative effects. Studies on hydrogen-bonded clusters are not limited to water clusters alone. 116−148 Dannenberg and co-workers have contributed substantially to quantifying hydrogen bond cooperativity by performing quantum chemical calculations on several molecular clusters.65−67,101−107 Starting with ab initio and semiempirical studies on 1,3 diones,65 dimers, and trimers of acetic acid66 and acetylacetone67 to evaluate the strength of hydrogen bonding, they further investigated how cooperative interactions dictate the hydrogen-bonding structure. Employing DFT calculations on chain and ribbonlike arrangement of molecules, they demonstrated a high degree of cooperativity in hydrogen-bonded chains of urea and formamide molecules.101−103 Linear chains of hydrogen-bonded formamide molecules containing from 3 to 15 monomeric units revealed a cooperative effect, wherein the strongest hydrogen bonds, typically those nearest the center of the formamide chain, have a bond strength which approaches 200% that of the dimer.104 Further, vibrational frequencies of the coupled N−H, CO stretches, and C−N stretch/CNH bend revealed clear shifts that reflect the cooperative stabilization of the hydrogen bonds.105 We have performed a structural and energetic comparison of linear, circular and standard arrangements of (acetamide)n clusters (n = 1−15) at the B3LYP/D95** level of theory to reveal significant cooperativity of hydrogen-bonding and sizedependent structural preferences (Figure 4).108 The standard arrangement in this study is defined as one where the maximum number of hydrogen-bonding interactions are found while being

Figure 4. Two factors considered while studying cooperativity in acetamide clusters (ref 108), size of clusters and difference in types of arrangements for clusters of the same size. BSSE corrected interaction energy per hydrogen bond (in kcal/mol) calculated employing the B3LYP/D95** level of theory with dispersion correction is indicated in the figure for selected clusters. 2780

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Figure 5. Representation of (a) different water, formamide, and acetamide clusters employed to understand modulation of hydrogen bond cooperativity in the presence of H+, Na+, Mg2+, OH−, and Cl− ions using DFT calculations (ref 131). (b) Plot showing BSSE uncorrected sequential binding energy values obtained for one particular arrangement of water molecules named as W1D with five different ions and increasing cluster size.

H···π) in a cyclic complex involving one acetylene and two water molecules has been studied.128 The total interaction energy of the three hydrogen bonds was shown to increase as the number of methyl group substitutions in the complex increases. MP2 and DFT studies on linear (urea)n=3−10 clusters revealed how cooperativity effects significantly enhance the N−H···O hydrogen bond from its strength of −7.90 kcal/mol in a dimer to −11.39 kcal/mol in a decamer.129 Gadre and co-workers recently reported the cooperative contribution toward hydrogen bonding in para-substituted calix[n]arenes (CX[n]) (n = 4, 5) and their thio analogues.130 Cooperativity was found to be nearly 5 times larger in the parent structure than that to its thio analogue. More recently, work done in our group evaluated the impact that the presence of ions, such as Mg2+, Na+, H+, Cl−, and OH−, has on hydrogen-bonded clusters of increasing size (water, formamide, and acetamide [n = 1−10]) in the context of their associated cooperativity using DFT calculations (Figure 5).131 The presence of an ion provides contrasting insight into the evaluated sequential binding energies of hydrogen-bonded clusters of different sizes. Dramatically higher sequential binding energies were seen in the presence of ions initially for smallersized molecular clusters compared to parent clusters. With increasing cluster size, the difference in the sequential binding energies between ionic clusters and parent clusters becomes reduced. However, Mg2+-bearing clusters continued to exhibit substantially higher sequential binding energies and cooperativity, even with increasing cluster size. Monovalent ions had a reduced impact on hydrogen bond cooperativity from hexamer onward, and this may be indicative of the short-range over which their influence prevails. The study helped reinforce the extent of impact that divalent cations have on hydrogen-bonded chains as against monovalent ions. Zabardasti et al. reported presence of anticooperativity in hydrogen-bonded clusters in two separate studies.132,133 Ab initio calculations on dihydrogen-bonded clusters of BeH42− with 1−4 molecules of NH3 revealed how cooperative effect decreased with the increasing size of the clusters.132 In another study employing ab initio and DFT calculations on hydrogenbonded clusters of water cyanuric acid, the presence of both cooperative and anticooperative effects were seen depending on the geometry of the structures.133 Albrecht et al. considered the

methylacetamide chains containing 2 to 7 monomeric units generated using ab initio calculations. A DFT study has revealed significant contribution of cooperativity in the interactions of hydrazoic acid clusters consisting of up to four monomers.119 Scheiner’s group probed into the underlying nature of the CH···O interaction using the ab initio methods.120−124 They thoroughly explored the cooperativity aspect of hydrogen bonds, wherein a chain of n hydrogen-bonded molecules is held together more strongly than would be expected based on the energetics of the single hydrogen bond within a dimer.122 As mentioned earlier in the section on origin of cooperativity, hydrogen bond cooperativity is typically attributed in large measure to the polarization induced in each subunit by the presence of its hydrogen-bonding partner. Scheiner and co-workers focused on evaluating the ability of one hydrogen bond in a chain to affect others by comparing the CH···O bonds in (H2CO)n and (HFCO)n to the OH···O bonds in (H2O)n.120,121 Although the degree of cooperativity is generally proportional to the strength of the hydrogen bond, the CH···O bonds in (HFCO)n were shown to display a disproportionately high degree of cooperativity. It was estimated that the mean hydrogen bond energy in an infinite chain of H2CO molecules was 25% greater than the same quantity in a dimer, while the long water chain exhibits a 66% enhancement over (H2O)2. Inspite of containing substantially weaker individual hydrogen bonds than those in water chains, the study revealed that (HFCO)n manifests an energetic cooperativity that is nearly as large as that of the OH··· O congeners. Cooperativity between the O−H···O and C−H··· O hydrogen bonds have been investigated by quantum chemical calculations.125 The interaction energies of the O−H···O and C− H···O hydrogen bonds were increased by 53% and 58% respectively, demonstrating the presence of large cooperativity. Grabowski and co-workers reported a study on H2CO···(HF)n (n = 1, ..., 9) complexes using the MP2 method and showed how cooperativity effect significantly enhances F−H···O hydrogen bond.126 Chen and co-workers employed energy decomposition approaches to quadruple and double hydrogen-bonded model systems to emphasize how cooperativity is the most important factor determining stability of the complex particularly in quadruple hydrogen bond dimers.127 The cooperativity between three types of hydrogen bonds (O−H···O, C−H···O, and O− 2781

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Figure 6. Selected crystal structures showing coexistence of different noncovalent interactions along with corresponding references where these structures are reported. (a) Hydrogen bonding, π−π, and halogen interactions as seen in ref 182, (b) hydrogen bonding and π−π interactions noted in refs 375, 372, and 373, (c) anion−π and π−π interactions seen in ref 302, (d) cation−π and anion−π interactions from ref 19, and (e) π−π interactions and halogen bonds observed in refs 381 and 382.

series. Using theoretical studies, Wu et al. clearly established the existence of significant cooperativity in the formation of 310 and α-helices, whereas no cooperativity was found in the formation of β-strands and 27-ribbons for a series of polyglycine models containing up to 14 amino acid residues.163 The absence of significant cooperativity in terms of enthalpy contribution in βsheets was further shown by employing a model system consisting of a dimer of a tripeptide to obtain repeating units for the parallel and antiparallel β-sheets.164 Baker and co-workers dwelt on the origin and relative importance of the contributions to helical cooperativity.165 They also reported the role of cooperative hydrogen bonding in amyloid formation.166 Dannenberg and co-workers performed ONIOM (DFT/ AM1) calculations on capped parallel-β-sheets of acetylVQIVYK-NHCH 3 , acetyl-Q-NHCH 3, and acetyl-alanineNHCH3 as model systems to study the importance of hydrogen bonding between glutamine side chains to the formation of

changes in atomic energy in the clusters versus the isolated monomer for small clusters of methanol, water, and formaldehyde. A variety of stabilities were observed within these hydrogen-bonded clusters, including indications of cooperative and anticooperative interactions.134 3. 2. Hydrogen Bonding in Peptides and Carbohydrates

Hydrogen bonds are the most important interactions between amino acid residues in peptides and proteins as they provide the stability for secondary structures like helices or sheets. Several theoretical studies using α helix and β sheet model systems have been performed to understand their inherent cooperativity.153−173 Kemp et al. used the experimental demonstration of an enthalpic component to the cooperativity of α-helical peptides as evidence for hydrogen bond cooperation in these structures.162 This was done considering the helicity measured at different temperatures in water for a solubilized polyalanine 2782

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amyloid VQIVYK parallel β-sheets.167,168 The Q sheets in these studies exhibit cooperativity that is reminiscent of chains of formamides104,105 and 4-pyridones,107 where the interactions become stronger as the chain or sheet grows. Using both classical electrostatic methods and DFT methods, the energetics of fibril formation for the first three layers was shown to be cooperative. Ireta et al. emphasized the importance of cooperativity in the stability of α-helices.169 Two limiting cases were considered, one of an isolated hydrogen bond and the other of an infinite α helical chain for DFT studies. Cooperativity within an infinite network of hydrogen bonds was clearly shown to strengthen each individual bond by more than a factor of 2. Jensen and co-workers reported a computational methodology for backbone amide proton chemical shift (δH).170 Interestingly, cooperative hydrogen-bonding effects were found to have a significant impact on δH values by affecting the primary hydrogen bond geometry and polarizing the electron density around the amide proton. Magistrato and co-workers employed DFT and hybrid DFT/MM simulations both in vacuum and in aqueous solution on model polyglutamine β-sheet structures characteristically observed in Huntington’s disease.153 Cooperativity of glutamine side chains was shown to affect both directions, perpendicular and parallel to the backbone. The unusual behavior observed in these β-sheets provided significant extrastabilization of polyglutamine aggregation. Schreiber and coworkers evaluated direct and cooperative contributions while studying the strength of buried hydrogen bonds and salt bridges.171 Employing a modifed multiple mutant cycle protocol to selected interactions between TEM-1-β-lactamase and its protein inhibitor BLIP, they demonstrated how formation of network of interactions helps establish cooperative effects which can convert even unfavorable interactions to favorable ones, more so for salt bridges and to a lesser extent for hydrogen bonds. Dashnau and co-workers used molecular dynamics (MD) simulations combined with water−water hydrogen bond angle analysis, calculation of solvent accessible surface area, and approximate free energy of solvation to show how intramolecular hydrogen bond cooperativity was closely associated with changes in water structure surrounding the aldohexopyranose stereoisomers.174 Gadre and co-workers report a direct estimation of individual intramolecular O−H···O interaction energies in sugar molecules using the molecular tailoring approach (MTA).175 These studies reveal a contribution to the hydrogen bond energy from the cooperativity to be typically between 0.1 and 0.6 kcal/ mol when hydrogen bonds were a part of a relatively weak equatorial−equatorial hydrogen bond network and higher between 0.5 and 1.1 kcal/mol when hydrogen bonds participated in an axial−axial hydrogen bond network.

reported.183 On the basis of a reductionist approach for the construction of a micelle, Fernández and co-workers have shown how highly directional hydrogen bonds build a frame on top of which the dispersive forces give the aggregate its final shape.184 Lesarri and co-workers have determined the structures for phenol dimer and trimer through the use of chirped pulse Fourier transform microwave spectroscopy in the 2−8 GHz band.185 The dimeric structure was found to represent a case, where an interplay between dispersion and hydrogen bonding played an essential role in fixing the complexation geometry. A few studies also explore cooperativity of the dihydrogen bond with other noncovalent interactions.150,151,186,187 Ren and co-workers describe a cooperativity effect between the dihydrogen bonding and H−M···π interactions (M = Li, Na, and K) in ternary complexes of FH···HM···C2H2/C2H4/C6H6 using the DFT and MP2 level calculations.186 Analyses of the charge of the hydrogen atoms in H···H moiety, atoms in molecule (AIM) analysis, and electron density shifts methods were adopted, and the cooperativity effect of the dihydrogen bond on the H−M···π interaction was shown to be more pronounced than that of the M···π bond on the H···H interaction. The coexistence of both dihydrogen bonding and metal−σ interaction was explored by Grabowski and co-workers in the case of three aggregates H2···LiH···H2, H2···NaH···H2, and H2···HBeH···H2 from among a series of complexes formed by hydrogen with metal hydrides.187 They however report that the modulus of the cooperativity energy is not greater than 0.05 kcal/ mol, suggesting that no significant cooperative effect is observed in these instances. In recent years, several studies on the interplay between hydrogen bonding and the halogen bond have been reported.188−195 Li and co-workers have performed ab initio calculations on H3N···XY···HF triads (X, Y = F, Cl, and Br), each having a halogen bond and a hydrogen bond.191 An analysis of molecular geometries, binding energies, and infrared spectra of monomers, dyads, and triads to examine cooperative effects indicate significant cooperativity between the halogen and hydrogen bonds in these complexes. Further effect of a halogen bond on a hydrogen bond was more pronounced than that of a hydrogen bond on a halogen bond. In another study, by means of cooperativity of halogen bond with hydrogen bond and substitution effect, the change of halogen bond from the chlorine-shared one to an ion-pair one was successfully realized.192 The substitution of alkali metal greatly strengthened the halogen bond, while the cooperativity not only made the halogen bond have a large change but also led to a large change in the strength of the hydrogen bond. Meng and co-workers noted the positive cooperativity between the HOX···OH/SH halogen bond and the Y−H···(H)OX hydrogen bond in OH/SH··· HOX···HY (X = Cl and Br; Y = F, Cl, and Br) complexes by means of MP2 level calculations.193 Further, the interplay between hydrogen bonding and lithium bonding in the HLiNCH-NCH complex was studied with ab initio calculations. An increase in binding energies by about 19% and 61% for the lithium and hydrogen bonds, respectively, in the trimer were noted.196 Yáñ ez and co-workers recently addressed cooperativity between hydrogen bonds and beryllium bonds in (H2O)n BeX2 (n = 1−3, X = H, F) complexes.197 The changes in the atomic energy components were correlated with the changes in the strength of the interactions, thereby accounting for cooperative or anticooperative effects. Russo and co-workers studied the interplay between hydrogen bonding and both anion−π and

3. 3. Hydrogen Bonding-Interplay with Other Noncovalent Interactions

Hydrogen bond operates in combination with numerous other noncovalent interactions (Figure 6). A brief discussion on some of the studies exploring their coexistence and their mutual impact is given in the following section.180−214 In this context, Geerlings and co-workers have investigated the interplay between aromatic stacking and hydrogen bonding in nucleobases by employing high-level quantum chemical calculations.180 Hydrogen-bonding capacity of the N3 and O2 atoms of cytosine was shown to increase linearly with the electrostatic repulsion between the stacked rings. Superstructures of diketopyrrolopyrrole donors and perylenediimide acceptors formed by a combination of hydrogen bonding and π···π stacking have also been recently 2783

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approach to study condensed phases as it offers a method to arrive at the energy of an N-molecule system that can be expressed as a sum of lower-order interactions which may be obtained with reasonably high accuracy. Xantheas reported the importance of the nonadditive terms in particular the three-body terms as being responsible for determining the relative stabilities of various trimer through pentamer isomers of water clusters.81 Almeida and co-workers demonstrate through many body decomposition analysis, the importance of a proper representation of the cooperative effects in hydrogen fluoride by way of inclusion of at least the three-body terms.145 Recent advances in this context include the development of a 3-body:many-body integrated fragmentation method for weakly bound clusters, specifically finding application in the case of water clusters (H2O)n, n = 3−10, 16, and 17.217 This technique captures all 1, 2, and 3-body interactions with a high-level electronic structure method, whereas a less demanding method is used to obtain 4body and higher-order interactions. Another variant in this approach particularly with respect to evaluation of the interaction energies of large clusters (H2O)n=6,16,24 from many-body expansion involves reconstruction of the cluster from small subclusters with a much lower computational cost.218 This is done by applying progressively lower-level methods for subsequent terms in the many-body expansion. Rapid convergence of the many-body expansion is the key to such a stratified approximation many-body approach. While studying complex microscopic nonadditivity effects in hydrophobic interactions and their impact on protein folding, Chan and coworkers evaluated potentials of mean force of three-body hydrophobic association of three methanes in water.219 Recently, Matsumoto has reported a study on the four-body cooperativity observed while surveying the effective attraction force between hydrated methane molecules.220

lone-pair−π interactions for the global stability of halidedichlorotetraoxacalix[2]-arene[2]triazine using first principle computations, highlighting that their combination leads to a cooperativity effect.198 Emphasizing how hydrogen bonds have indeed become a crucial functional and structural element in modern inorganic chemistry, a combination of transition metal ions and hydrogen-bonding interactions have been studied.199 Cooperativity between weak hydrogen bonds was revealed in molecular clusters by examination of the structure, internal dynamics, and origin of the weak intermolecular forces between sevoflurane (1,1,1,3,3,3-hexafluoro-2-(fluoromethoxy)propane) and a benzene molecule, using multi-isotopic broadband rotational spectra.200 Thus, a large number of systematic investigations mentioned in the section above have established the modulation of hydrogen bond strength through cooperativity.

4. QUANTITATIVE DEFINITIONS FOR COOPERATIVITY The quantification of cooperativity differs based on the approach and the nature of the system employed.215−253 4. 1. Many Body Analysis

Several studies attempt to get a quantitative account of the cooperative effects by decomposing the interaction energy of a system of n bodies and by inclusion of the many body terms in the analysis.145,215−227 Axilrod and Teller215 and Muto216 were apparently the first to provide an explicit treatment of the nonadditive interaction energies in 1943. Hankins et al. investigated the potential of interaction for pairs and triplets of water molecules and made a representation for the individual terms in the many body expansion for these small water clusters.70 A quantitative measure of the cooperative effect is made by decomposing the interaction energy of a system of n bodies. Typically, the energy of the cluster in consideration can be broken down into 1, 2,..., n body contributions via the many body decomposition, where the 1-body term provides the monomer distortion or relaxation energy. The many body expansion of interaction energy for a system with N monomers in it is given below N

E(1, .., N ) =

4. 2. Double and Triple Mutant Analysis

Thermodynamic double mutant cycles and triple mutant boxes are widely employed for the experimental quantification of noncovalent interactions and cooperative effects in proteins.26,228−234 Double mutant cycles were originally devised to investigate the interactions in proteins.228,229 Chemical double mutant cycles help to measure the magnitude of a particular functional group interaction in both weakly and strongly bound complexes.231 Any difference between the functional group interaction energies in two systems was shown to provide a measure of the magnitude of the enthalpic chelate effect in the complexes.233 Two double mutant cycles can be formally combined to produce a triple mutant box, providing a general method for quantification of cooperative effects. Hunter and coworkers have performed extensive studies using double and triple mutant cycle experiments to evaluate cooperativity in noncovalent interactions.26,231−233 Quantification of intermolecular functional group interactions in hydrogen-bonded zipper complexes in chloroform231 and in two different doubly hydrogen-bonded motifs in carbon tetrachloride, chloroform, 1,1,2,2-tetrachloroethane, and cyclohexane form part of their studies.233 Schreiber and co-workers have provided an experimental approach to evaluate the net binding free energy of buried hydrogen bonds and salt bridges in selected interactions between TEM-1-β-lactamase and its protein inhibitor, BLIP.171 Results from this study demonstrate the importance of forming networks of buried salt bridges and suggest that the cooperative networking effect results from the favorable contribution of the

N

∑V

1B

(i ) +

i

∑ V 2B(i , j) i