The minimized dead-end elimination criterion and its application to protein redesign in a hybrid scoring and search algorithm for computing partition functions over molecular ensembles

One of the main challenges for protein redesign is the efficient evaluation of a combinatorial number of candidate structures. The modeling of protein flexibility, typically by using a rotamer library of commonly-observed low-energy side-chain conformations, further increases the complexity of the redesign problem. A dominant algorithm for protein redesign is dead-end elimination (DEE), which prunes the majority of candidate conformations by eliminating rigid rotamers that provably are not part of the global minimum energy conformation (GMEC). The identified GMEC consists of rigid rotamers (i.e., rotamers that have not been energy-minimized) and is thus referred to as the rigid-GMEC. As a postprocessing step, the conformations that survive DEE may be energy-minimized. When energy minimization is performed after pruning with DEE, the combined protein design process becomes heuristic, and is no longer provably accurate: a conformation that is pruned using rigid-rotamer energies may subsequently minimize to a lower energy than the rigid-GMEC. That is, the rigid-GMEC and the conformation with the lowest energy among all energy-minimized conformations (the minimized-GMEC) are likely to be different. While the traditional DEE algorithm succeeds in not pruning rotamers that are part of the rigid-GMEC, it makes no guarantees regarding the identification of the minimized-GMEC. In this paper we derive a novel, provable, and efficient DEE-like algorithm, called minimized-DEE (MinDEE), that guarantees that rotamers belonging to the minimized-GMEC will not be pruned, while still pruning a combinatorial number of conformations. We show that MinDEE is useful not only in identifying the minimized-GMEC, but also as a filter in an ensemble-based scoring and search algorithm for protein redesign that exploits energy-minimized conformations. We compare our results both to our previous computational predictions of protein designs and to biological activity assays of predicted protein mutants. Our provable and efficient minimized-DEE algorithm is applicable in protein redesign, protein-ligand binding prediction, and computer-aided drug design.     

Maintaining solvent accessible surface area under rotamer substitution for protein design

Although quantities derived from solvent accessible surface areas (SASA) are useful in many applications in protein design and structural biology, the computational cost of accurate SASA calculation makes SASA-based scores difficult to integrate into commonly used protein design methodologies. We demonstrate a method for maintaining accurate SASA during a Monte Carlo search of sequence and rotamer space for a fixed protein backbone. We extend the fast Le Grand and Merz algorithm (Le Grand and Merz, J Comput Chem, 14, 349), which discretizes the solvent accessible surface for each atom by placing dots on a sphere and combines Boolean masks to determine which dots are exposed. By replacing semigroup operations with group operations (from Boolean logic to counting dot coverage) we support SASA updates. Our algorithm takes time proportional to the number of atoms affected by rotamer substitution, rather than the number of atoms in the protein. For design simulations with a one hundred residue protein our approach is approximately 145 times faster than performing a Le Grand and Merz SASA calculation from scratch following each rotamer substitution. To demonstrate practical effectiveness, we optimize a SASA-based measure of protein packing in the complete redesign of a large set of proteins and protein-protein interfaces.     

Radical performance enhancements for combinatorial optimization algorithms based on the dead-end elimination theorem

Recent advances in protein design have demonstrated the effectiveness of optimization algorithms based on the dead-end elimination theorem. The algorithms solve the combinatorial problem of finding the optimal placement of side chains for a set of backbone coordinates. Although they are powerful tools, these algorithms have severe limitations when the number of side chain rotamers is large. This is due to the high-order time dependence of the aspect of the calculation that deals with rotamer doubles. We present three independent algorithmic enhancements that significantly increase the speed of the doubles computation. These methods work by using quantities that are inexpensive to compute as a basis for forecasting which expensive calculations are worthwhile. One of the methods, the comparison of extrema, is derived from analytical considerations, and the remaining two, the “magic-bullet” and the “q_rs” and “q_uv” metrics, are based on empirical observation of the distribution of energies in the system. When used together, these methods effect an overall speed improvement of as much as a factor of 47, and for the doubles aspect of the calculation, a factor of 95. Together, these enhancements extend the envelope of inverse folding to larger proteins by making formerly intractable calculations attainable in reasonable computer time. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1505–1514, 1998     

Rotamer optimization for protein design through MAP estimation and problem‐size reduction

The search for the global minimum energy conformation (GMEC) of protein side chains is an important computational challenge in protein structure prediction and design. Using rotamer models, the problem is formulated as a NP‐hard optimization problem. Dead‐end elimination (DEE) methods combined with systematic A* search (DEE/A*) has proven useful, but may not be strong enough as we attempt to solve protein design problems where a large number of similar rotamers is eligible and the network of interactions between residues is dense. In this work, we present an exact solution method, named BroMAP (branch‐and‐bound rotamer optimization using MAP estimation), for such protein design problems. The design goal of BroMAP is to be able to expand smaller search trees than conventional branch‐and‐bound methods while performing only a moderate amount of computation in each node, thereby reducing the total running time. To achieve that, BroMAP attempts reduction of the problem size within each node through DEE and elimination by lower bounds from approximate maximum‐a‐posteriori (MAP) estimation. The lower bounds are also exploited in branching and subproblem selection for fast discovery of strong upper bounds. Our computational results show that BroMAP tends to be faster than DEE/A* for large protein design cases. BroMAP also solved cases that were not solved by DEE/A* within the maximum allowed time, and did not incur significant disadvantage for cases where DEE/A* performed well. Therefore, BroMAP is particularly applicable to large protein design problems where DEE/A* struggles and can also substitute for DEE/A* in general GMEC search. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

NMR Studies of Solvent Effect on Internal Rotation in Some Non-Symmetrical 1,2-Disubstituted Ethanes

The A₂B₂ system of PMR spectra of 3-bromopropionic acid, 3-chloropropionic acid, 3-bromopropionitrile, and 3-chloropropionitrile exhibit appreciable solvent effect at room temperature. NMR spectroscopic parameters of A₂B₂ spectrum as well as physical parameters related to internal rotation, i. e. the highest energy barrier and the energy difference between rotamers, were determined for these compounds in the medium of various solvents. It was found that in the case of 3-bromopropionic acid, the trans rotamer is more stable than the gauche rotamers, and the energy difference decreases with increasing dielectric constant of solvent. While in the case of 3-bromopropionitrile, the gauche rotamers were found to be more stable than the trans rotamer and the energy difference increases with increasing dielectric constant of solvent. In the remaining two compounds, 3-chloropropionic acid and 3-chloropropionitrile, both trans and gauche rotamers are equally stable in a solvent of low dielectric constant, however in a solvent of higher dielectric constant, the gauche rotamers become more stable than the trans rotamer and the energy difference becomes more pronounced with increasing dielectric constant of solvent.     

Importance of thorough conformational analysis in modelling transition metal-mediated reactions: Case studies on pincer complexes containing phosphine groups

Advances in processing capabilities of computer clusters have allowed for the full modeling of organometallic complexes that previously would have been simplified to reduce computational cost. Increased feasibility of computational modeling offers new challenges, not only in terms of limitations of methods and theory, but attention should be paid to complexes that can exist in many conformations, as the appropriate choice of conformer may be easily overlooked. In this work a series of pincer complexes with isopropyl and cyclopentyl substituents have been chosen as examples to demonstrate the importance of conformational analysis. The complexes examined contain four isopropyl or cyclopentyl groups on phosphor atoms generating between 27 and 324 possible rotamers. The importance of conformational search in a mechanistic investigation is demonstrated with the CO₂ insertion into a nickel hydride bond of POCOP ⁱPr nickel hydride complex. Results show that the reaction energy profile can be both exergonic and endergonic depending on rotamer choice. Specifically, the POCOP^iPr Ni-formato complex product of the CO₂ insertion reaction had an energy difference between the lowest and highest energy rotamer as high as 16.8 kcal/mol. The significant energy differences between rotamers highlight the importance of thorough conformational analysis and should be taken into consideration when evaluating the energy profile of related reactions.     

Matrix-isolation infrared spectroscopy of the rotational isomers of 1,2-, 1,3-, and 1,4-benzenedicarboxaldehyde

Rotational isomers (rotamers) of the three structural isomers of benzenedicarboxaldehydes (1,2-, 1,3-, and 1,4-derivatives) have been investigated in detail using matrix-isolation infrared spectroscopy in the 600-4000 cm-1 region, combined with UV photoexcitation and density-functional theory (DFT) calculations. Two rotamers were identified for 1,2- and 1,4-benzenedicarboxaldehyde (1,2- and 1,4-BDA, respectively), while three rotamers were identified for 1,3-benzenedicarboxaldehyde (1,3-BDA) in infrared spectra upon UV-irradiation. Most of the observed infrared bands of each rotamer have been assigned. The energetic relationships among the rotamers were revealed based on the infrared data and the DFT calculations. It is shown that the intramolecular C-H...H-C interaction in the H-syn rotamer or the C-H...O=C hydrogen bonding in the anti rotamer of 1,2-BDA results in the blue-shift of the aldehyde C-H stretching band and the shortening of the aldehyde C-H bond length. Both photoinduced rotational isomerization and rearrangement were observed upon UV irradiation for 1,2-BDA. The structure of the major enol isomer formed as the result of the photochemical rearrangement of 1,2-BDA is determined.     

Kinetic control of proline amide rotamers: total synthesis of trans,trans- and cis,cis-ceratospongamide

Ceratospongamide is a cyclic peptide natural product that is biosynthesized as a mixture of two proline rotamers. Remarkably, these rotamers do not detectably interconvert at temperatures up to 100 degress C. Here we report high-yielding syntheses of each rotamer and demonstrate that the threonine-derived oxazoline plays a critical role in determining the kinetic distribution of conformational isomers.     

1-萘酚的紫外共振双光子电离光谱

利用脉冲分子束技术, 在305-322 nm范围内研究了1-萘酚(1NP)的共振双光子电离(R2PI)光谱. 1NP分子存在cis和trans两种旋转异构体, 但实验中仅观测到trans异构体的电子振动跃迁光谱, 其S1←S0跃迁的(0-0)带头出现在317.90 nm(即31456 cm-1)位置. 利用光谱选律及ab initio和密度泛函(DFT)计算, 对trans异构体在S1态的振动模进行标识, 得出主要对应于对称性为a'的平面内振动模. 计算显示, cis异构体在电子基态S0的能量较trans异构体高出439 cm-1, 而第一激发能却比trans异构体的低1216 cm-1, 与之相应的实验值分别是220和274 cm-1. 计算数值与实验结果在能量变化趋势上完全一致. 共振双光子电离谱中没有观测到cis异构体的光谱信号, 其原因可归结为分子束的有效冷却效应使得处于基态的cis异构体的布居数密度相对trans异构体极低, 导致cis光谱信号太小而未能被探测到.     

Vibrational OH-stretching overtone spectroscopy of jet-cooled resorcinol and hydroquinone rotamers

We have measured the OH-stretching fundamental and overtone spectra of resorcinol and hydroquinone in a supersonic jet using nonresonant ionization detected infrared/near-infrared spectroscopy. Anharmonic oscillator local mode calculations of the OH-stretching frequencies and intensities and Boltzmann populations of the stable rotamers have been calculated at the B3LYP/6-311++G(3df,2pd) level to help interpret the observed spectra. Resorcinol has three stable rotamers and in the recorded second and third OH-stretching overtone spectra there is evidence of two distinguishable rotamers. Hydroquinone has two stable rotamers; however, the OH-stretching oscillators of each rotamer are so similar in nature that even up to the fourth OH-stretching overtone the transitions coincide. These results place a limit on the ability of the jet-cooled overtone spectroscopy technique to distinguish between rotamers.     

Fluorescence of cis-1-amino-2-(3-indolyl)cyclohexane-1-carboxylic acid: a single tryptophan chi(1) rotamer model

A constrained derivative, cis-1-amino-2-(3-indolyl)cyclohexane-1-carboxylic acid, cis-W3, was designed to test the rotamer model of tryptophan photophysics. The conformational constraint enforces a single chi(1) conformation, analogous to the chi(1) = 60 degrees rotamer of tryptophan. The side-chain torsion angles in the X-ray structure of cis-W3 were chi(1) = 58.5 degrees and chi(2) = -88.7 degrees. Molecular mechanics calculations suggested two chi(2) rotamers for cis-W3 in solution, -100 degrees and 80 degrees, analogous to the chi(2) = +/-90 degrees rotamers of tryptophan. The fluorescence decay of the cis-W3 zwitterion was biexponential with lifetimes of 3.1 and 0.3 ns at 25 degrees C. The relative amplitudes of the lifetime components match the chi(2) rotamer populations predicted by molecular mechanics. The longer lifetime represents the major chi(2) = -100 degrees rotamer. The shorter lifetime represents the minor chi(2) = 80 degrees rotamer having the ammonium group closer to C4 of the indole ring (labeled C5 in the cis-W3 X-ray structure). Intramolecular excited-state proton transfer occurs at indole C4 in the tryptophan zwitterion (Saito, I.; Sugiyama, H.; Yamamoto, A.; Muramatsu, S.; Matsuura,T. J. Am. Chem. Soc. 1984, 106, 4286-4287). Photochemical isotope exchange experiments showed that H-D exchange occurs exclusively at C5 in the cis-W3 zwitterion, consistent with the presence of the chi(2) = 80 degrees rotamer in solution. The rates of two nonradiative processes, excited-state proton and electron transfer, were measured for individual chi(2) rotamers. The excited-state proton-transfer rate was determined from H-D exchange and fluorescence lifetime data. The excited-state electron-transfer rate was determined from the temperature dependence of the fluorescence lifetime. The major quenching process in the -100 degrees rotamer is electron transfer from the excited indole to carboxylate. Electron transfer also occurs in the 80 degrees rotamer, but the major quenching process is intramolecular proton transfer. Both quenching processes are suppressed by deprotonation of the amino group. The results for cis-W3 provide compelling evidence that the complex fluorescence decay of the tryptophan zwitterion originates in ground-state heterogeneity with the different lifetimes primarily reflecting different intramolecular excited-state proton- and electron-transfer rates in various rotamers.     

Rotamer libraries of spin labelled cysteines for protein studies

Studies of structure and dynamics of proteins using site-directed spin labelling rely on explicit modelling of spin label conformations. The large computational effort associated with such modelling with molecular dynamics (MD) simulations can be avoided by a rotamer library approach based on a coarse-grained representation of the conformational space of the spin label. We show here that libraries of about 200 rotamers, obtained by iterative projection of a long MD trajectory of the free spin label onto a set of canonical dihedral angles, provide a representation of the underlying trajectory adequate for EPR distance measurements. Rotamer analysis was performed on selected X-ray structures of spin labelled T4 lysozyme mutants to characterize the spin label rotamer ensemble on a single protein site. Furthermore, predictions based on the rotamer library approach are shown to be in nearly quantitative agreement with electron paramagnetic resonance (EPR) distance data on the Na(+)/H(+) antiporter NhaA and on the light-harvesting complex LHCII whose structures are known from independent cryo electron microscopy and X-ray studies, respectively. Suggestions for the selection of labelling sites in proteins are given, limitations of the approach discussed, and requirements for further development are outlined.     

Conformational preferences of 2-methoxy, 2-methylthio, and 2-methylselenocyclohexyl-N,N-dimethylcarbamate: a theoretical and experimental investigation

Studies on the conformational equilibria of 2-methoxy, 2-methylthio, and 2-methylselenocyclohexyl-N,N-dimethylcarbamate are reported. DNMR spectroscopy experiments at 203 K provided the percentages of each conformer in equilibrium. Theoretical calculations using the MP2, B3LYP, and B971 methods with cc-pVDZ basis set were applied to determine the differences in energy between the conformers. The analysis of the potential energy surface (PES) for each conformer showed the presence of two rotamers. NBO analysis provided an explanation of the factors (hyperconjugative and steric interactions) that drive rotamer and conformer preferences.     

Conformational behavior of cis-2-methoxy, cis-2-methylthio, and cis-2-methylselenocyclohexanol: a theoretical and experimental investigation

Studies on the conformational equilibria of 2-methoxy, 2-methylthio, and 2-methylselenocyclohexanol are reported. Dynamic NMR spectroscopy experiments at 203-210 K were performed, which provided the percentages of each conformer in equilibrium. Theoretical calculations using the B3LYP method and aug-cc-pvdz basis set were applied to determine the differences in energy between the conformers. The analysis of the potential energy surface of each conformer showed the presence of two rotamers. Natural bond orbital analysis provided an explanation of which factors are driving the rotamer and conformer preferences.     

Solvent effects on CD and MCD spectra of phenylalanine and its related compounds

Solvent effects and pH effects on CD and MCD spectra of phenylalanine and its derivatives were observed. CD spectra of N-acetyl-L -phenylalanine ethyl ester in various solvents were analyzed using the rotamer population determined by NMR . The magnitude of optical activity due to each rotamer was determined for two CD bands. The signs of Cotton effect of rotamers I and II are positive and that of rotamer II is negative. The optical activity is greater in polar than in nonpolar solvents. MCD spectra of phenylalanine reveal a remarkable pH effect. In the alkaline pH range it is of the toluene type, from which it turns to a weak spectrum in the acid pH range.     

Conformational analysis of 1-butyl-3-methylimidazolium by CCSD(T) level ab initio calculations: effects of neighboring anions

Conformational energies for the butyl group of 1-butyl-3-methylimidazolium (bmim) were calculated by high-level ab initio methods. Estimated relative energies for the TT, GT and G'T rotamers of an isolated bmim cation at the CCSD(T)/cc-pVTZ level are 0.0 -0.02 and -0.50 kcal/mol, respectively. The close contact of a Cl anion to theC(2)-H of imidazolium considerably increases the relative stability of the GT rotamer. Estimated relative energies for the three rotamers of the [bmim]Cl complex, in which the Cl anion exists close to the C(2)-H, are 0.0, -1.61 and -0.25 kcal/mol, respectively. The GT rotamer is favored by the strong attractive electrostatic interaction between the bmim cation and Cl anion. The C(2)-H group in the GT rotamer has a larger positive charge compared with those in the TT and G'T rotamers. The contact of a Br anion to the C(2)-H also stabilizes the GT rotamer. The effects of the Cl anion close to the C(4)-Hand C(5)-Hare small. The anion effects suggest that the GT rotamer is the most stable in ionic liquids. The positive charge on imidazolium ring does not largely change the conformational energies. Estimated relative energies for the three rotamers of N-butylimidazole (0.0, -0.29 and -0.75 kcal/mol, respectively) are not largely different from those for isolated bmim. Calculated MP2/cc-pVTZ level torsional potential for the C im-N im-C-C bond has a minimum when the torsional angle is close to 90 degrees. Coplanar conformation is not a stable structure. Calculated torsional barrier height between the two nonplanar minima is less than 1 kcal/mol.     

Unique Structural Properties of 2,4,6‐Tri‐tert‐butylanilide: Isomerization and Switching between Separable Amide Rotamers through the Reaction of Anilide Enolates

Herein, we report a unique structural property of 2,4,6‐tri‐tert‐butylanilide, which can be separated into its amide rotamers at room temperature. Interconversion between the rotamers of anilide enolates occurs readily at room temperature and their reaction with electrophiles gives mixtures of the rotamers in a ratio that depends on the reactivity of the corresponding electrophile. That is, the reaction of the 2,4,6‐tri‐tert‐butylacetanilide enolate with reactive electrophiles, such as allyl bromide or protic acids, gives mixtures of the anilide rotamers in which the E rotamer is the major component, whereas less‐reactive electrophiles, such as 1‐bromopropane and 2‐iodopropane, yield mixtures of the rotamers in which the Z rotamer is the major component. The rotameric ratio of the product is also strongly dependent on the reactivity of the anilide enolate. Switching between the anilide rotamers can be achieved through protonation of a less‐reactive enolate by a less‐reactive protic acid and thermal isomerization of the anilide.     

Intramolecular quenching of tryptophan fluorescence by the peptide bond in cyclic hexapeptides

Intramolecular quenching of tryptophan fluorescence by protein functional groups was studied in a series of rigid cyclic hexapeptides containing a single tryptophan. The solution structure of the canonical peptide c[D-PpYTFWF] (pY, phosphotyrosine) was determined in aqueous solution by 1D- and 2D-(1)H NMR techniques. The peptide backbone has a single predominant conformation. The tryptophan side chain has three chi(1) rotamers: a major chi(1) = -60 degrees rotamer with a population of 0.67, and two minor rotamers of equal population. The peptides have three fluorescence lifetimes of about 3.8, 1.8, and 0.3 ns with relative amplitudes that agree with the chi(1) rotamer populations determined by NMR. The major 3.8-ns lifetime component is assigned to the chi(1) = -60 degrees rotamer. The multiple fluorescence lifetimes are attributed to differences among rotamers in the rate of excited-state electron transfer to peptide bonds. Electron-transfer rates were calculated for the six preferred side chain rotamers using Marcus theory. A simple model with reasonable assumptions gives excellent agreement between observed and calculated lifetimes for the 3.8- and 1.8-ns lifetimes and assigns the 1.8-ns lifetime component to the chi(1) = 180 degrees rotamer. Substitution of phenylalanine by lysine on either side of tryptophan has no effect on fluorescence quantum yield or lifetime, indicating that intramolecular excited-state proton transfer catalyzed by the epsilon-ammonium does not occur in these peptides.     

Z and E rotamers of N‐formyl‐1‐bromo‐4‐hydroxy‐3‐methoxymorphinan‐6‐one and their interconversion as studied by 1H/13C NMR spectroscopy and quantum chemical calculations

N‐Formyl‐1‐bromo‐4‐hydroxy‐3‐methoxymorphinan‐6‐one (compound 2 ), an important intermediate in the NIH Opiate Total Synthesis, presumably exists as a mixture of two rotamers (Z and E) in both CHCl₃ and DMSO at room temperature due to the hindered rotation of its N‐C18 bond in the amide moiety. By comparing the experimental ¹H and ¹³C chemical shifts of a single rotamer and the mixture of compound 2 in CDCl₃ with the calculated chemical shifts of the geometry optimized Z and E rotamers utilizing density functional theory, the crystalline rotamer of compound 2 was characterized as having the E configuration. The energy barrier between the two rotamers was also determined with the temperature dependence of ¹H and ¹³C NMR coalescence experiments, and then compared with that from the reaction path for the interconversion of the two rotamers calculated at the level of B3LYP/6‐31G*. Detailed geometry of the ground state and the transition states of both rotamers are given and discussed. Copyright © 2012 This article is a US Government work and is in the public domain in the USA.