Conformational Entropy of the RNA Phosphate Backbone and Its Contribution to the Folding Free Energy

While major contributors to the free energy of RNA tertiary structures such as basepairing, base-stacking, and charge and counterion interactions have been studied extensively, little is known about the intrinsic free energy of the backbone. To assess the magnitude of the entropic strains along the phosphate backbone and their impact on the folding free energy, we have formulated a mathematical treatment for computing the volume of the main-chain torsion-angle conformation space between every pair of nucleobases along any sequence to compute the corresponding backbone entropy. To validate this method, we have compared the computed conformational entropies against a statistical free energy analysis of structures in the crystallographic database from several-thousand backbone conformations between nearest-neighbor nucleobases as well as against extensive computer simulations. Using this calculation, we analyzed the backbone entropy of several ribozymes and riboswitches and found that their entropic strains are highly localized along their sequences. The total entropic penalty due to steric congestions in the backbone for the native fold can be as high as 2.4 cal/K/mol per nucleotide for these medium and large RNAs, producing a contribution to the overall free energy of up to 0.72 kcal/mol per nucleotide. For these RNAs, we found that low-entropy high-strain residues are predominantly located at loops with tight turns and at tertiary interaction platforms with unusual structural motifs.     

Topological Constraints and Their Conformational Entropic Penalties on RNA Folds

Functional RNAs can fold into intricate structures using a number of different secondary and tertiary structural motifs. Many factors contribute to the overall free energy of the target fold. This study aims at quantifying the entropic costs coming from the loss of conformational freedom when the sugar-phosphate backbone is subjected to constraints imposed by secondary and tertiary contacts. Motivated by insights from topology theory, we design a diagrammatic scheme to represent different types of RNA structures so that constraints associated with a folded structure may be segregated into mutually independent subsets, enabling the total conformational entropy loss to be easily calculated as a sum of independent terms. We used high-throughput Monte Carlo simulations to simulate large ensembles of single-stranded RNA sequences in solution to validate the assumptions behind our diagrammatic scheme, examining the entropic costs for hairpin initiation and formation of many multiway junctions. Our diagrammatic scheme aids in the factorization of secondary/tertiary constraints into distinct topological classes and facilitates the discovery of interrelationships among multiple constraints on RNA folds. This perspective, which to our knowledge is novel, leads to useful insights into the inner workings of some functional RNA sequences, demonstrating how they might operate by transforming their structures among different topological classes.     

Mechanism of the difference in the binding affinity of E. coli tRNAGln to glutaminyl-tRNA synthetase caused by noninterface nucleotides in variable loop

Aminoacyl-tRNA synthetases (ARSs) distinguish their cognate tRNAs from many other kinds of tRNAs, despite the very similar tertiary structures of tRNAs. Many researchers have supported the view that this recognition is achieved by intermolecular interactions between tRNA and ARS. However, one of the aptamers of Escherichia coli glutamine specific tRNA, var-AGGU, has a higher affinity to ARS than the wild-type, although the sequence difference only lies in the variable loop located on the opposite side of the binding interface with ARS. To understand the reason for the difference in affinity, we did molecular dynamics simulations on tRNAs and their complexes with ARS. We calculated the enthalpic and entropic contributions to the binding free energy with the molecular mechanics-Poisson-Boltzmann/surface area method and found that the entropic difference plays an important role in the difference in binding free energies. During the molecular dynamics simulations, dynamic rearrangements of hydrogen bonds occurred in the tertiary core region of the wild-type tRNA, whereas they were not observed in the free var-AGGU simulation. Since the internal mobility was suppressed upon complex formation with ARS, the entropy loss in the wild-type was larger than that of the aptamer. We therefore concluded that the sequence difference in the variable loop caused the difference in the internal mobility of the tertiary core region tRNAs and led to the difference in the affinity to ARS through the entropy term.     

Quantifying the Entropy of Binding for Water Molecules in Protein Cavities by Computing Correlations

Protein structural analysis demonstrates that water molecules are commonly found in the internal cavities of proteins. Analysis of experimental data on the entropies of inorganic crystals suggests that the entropic cost of transferring such a water molecule to a protein cavity will not typically be greater than 7.0 cal/mol/K per water molecule, corresponding to a contribution of approximately +2.0 kcal/mol to the free energy. In this study, we employ the statistical mechanical method of inhomogeneous fluid solvation theory to quantify the enthalpic and entropic contributions of individual water molecules in 19 protein cavities across five different proteins. We utilize information theory to develop a rigorous estimate of the total two-particle entropy, yielding a complete framework to calculate hydration free energies. We show that predictions from inhomogeneous fluid solvation theory are in excellent agreement with predictions from free energy perturbation (FEP) and that these predictions are consistent with experimental estimates. However, the results suggest that water molecules in protein cavities containing charged residues may be subject to entropy changes that contribute more than +2.0 kcal/mol to the free energy. In all cases, these unfavorable entropy changes are predicted to be dominated by highly favorable enthalpy changes. These findings are relevant to the study of bridging water molecules at protein-protein interfaces as well as in complexes with cognate ligands and small-molecule inhibitors.     

Quantitative analysis of the isolated GAAA tetraloop/receptor interaction in solution: a site-directed spin labeling study

The GNRA (N: any nucleotide; R: purine) tetraloop/receptor interaction is believed to be one of the most frequently occurring tertiary interaction motifs in RNAs, but an isolated tetraloop/receptor complex has not been identified in solution. In the present work, site-directed spin labeling is applied to detect tetraloop/receptor complex formation and estimate the free energy of interaction. For this purpose, the GAAA tetraloop/receptor interaction was chosen as a model system. A method was developed to place nitroxide labels at specific backbone locations in an RNA hairpin containing the GAAA tetraloop. Formation of the tetraloop/receptor complex was monitored through changes in the rotational correlation time of the tetraloop and the attached nitroxide. Results show that a hairpin containing the GAAA tetraloop forms a complex with an RNA containing the 11-nucleotide GAAA tetraloop receptor motif with an apparent Kd that is strongly dependent on Mg2+. At 125 mM MgCl2, Kd = 0.40 +/- 0.05 mM. The corresponding standard free energy of complex formation is -4.6 kcal/mol, representing the energetics of the tetraloop/receptor interaction in the absence of other tertiary constraints. The experimental strategy presented here should have broad utility in quantifying weak interactions that would otherwise be undetectable, for both nucleic acids and nucleic acid-protein complexes.     

Consistency in structural energetics of protein folding and peptide recognition.

We report a new free energy decomposition that includes structure-derived atomic contact energies for the desolvation component, and show that it applies equally well to the analysis of single-domain protein folding and to the binding of flexible peptides to proteins. Specifically, we selected the 17 single-domain proteins for which the three-dimensional structures and thermodynamic unfolding free energies are available. By calculating all terms except the backbone conformational entropy change and comparing the result to the experimentally measured free energy, we estimated that the mean entropy gain by the backbone chain upon unfolding (delta Sbb) is 5.3 cal/K per mole of residue, and that the average backbone entropy for glycine is 6.7 cal/K. Both numbers are in close agreement with recent estimates made by entirely different methods, suggesting a promising degree of consistency between data obtained from disparate sources. In addition, a quantitative analysis of the folding free energy indicates that the unfavorable backbone entropy for each of the proteins is balanced predominantly by favorable backbone interactions. Finally, because the binding of flexible peptides to receptors is physically similar to folding, the free energy function should, in principle, be equally applicable to flexible docking. By combining atomic contact energies, electrostatics, and sequence-dependent backbone entropy, we calculated a priori the free energy changes associated with the binding of four different peptides to HLA-A2, 1 MHC molecule and found agreement with experiment to within 10% without parameter adjustment.     

On the stability of nucleic acid structures in solution: enthalpy-entropy compensations, internal rotations and reversibility.

The thermodynamics of self-association (stacking) of free bases and nucleotides, intramolecular stacking in dinucleotides, nearest-neighbour base pair stacking interactions in duplex DNA and RNA, and the formation of hairpin loops illustrate enthalpy/entropy compensations. Large stacking exothermicities are associated with large negative entropy changes that ensure that delta G is small, permitting readily reversible associations in solution. We rationalise enthalpy/entropy compensations with reference to residual motions and torsional vibrations which make a larger entropic contribution to binding when - delta H approximately kT (thermal energy at room temperature), than when - delta H >> kT. We present a factorisation of experimental free energies for helix formation in terms of approximate contributions from the restriction of rotations, hydrophobic interactions, electrostatic interactions due to base stacking, and contributions from hydrogen bonding, and estimate the adverse free energy cost per rotor (mainly entropy) of ordering the phosphate backbone as between 1.9 and 5.4 kJ mol-1 [averaged over 12 rotors per base pair for A-U on A-U stacking (lower limit), and G-C on C-G stacking (upper limit)]. The largest cost is associated with the most exothermic stacking interactions, while the range of values is consistent with earlier conclusions from data on the fusion of hydrocarbon chains (lower value), and with entropy changes in covalent isomerisations of small molecules involving severe restrictions (upper value).     

The backbone conformational entropy of protein folding: experimental measures from atomic force microscopy

The energy dissipated during the atomic force microscopy-based mechanical unfolding and extension of proteins is typically an order of magnitude greater than their folding free energy. The vast majority of the "excess" energy dissipated is thought to arise due to backbone conformational entropy losses as the solvated, random-coil unfolded state is stretched into an extended, low-entropy conformation. We have investigated this hypothesis in light of recent measurements of the energy dissipated during the mechanical unfolding of "polyproteins" comprised of multiple, homogeneous domains. Given the assumption that backbone conformational entropy losses account for the vast majority of the energy dissipated (an assumption supported by numerous lines of experimental evidence), we estimate that approximately 19(+/-2)J/(mol K residue) of entropy is lost during the extension of three mechanically stable beta-sheet polyproteins. If, as suggested by measured peak-to-peak extension distances, pulling proceeds to near completion, this estimate corresponds to the absolute backbone conformational entropy of the unfolded state. As such, it is exceedingly close to previous theoretical and semi-empirical estimates that place this value at approximately 20J/(mol K residue). The estimated backbone conformational entropy lost during the extension of two helical polyproteins, which, in contrast to the mechanically stable beta-sheet polyproteins, rupture at very low applied forces, is three- to sixfold less. Either previous estimates of the backbone conformational entropy are significantly in error, or the reduced mechanical strength of the helical proteins leads to the rupture of a subsequent domain before full extension (and thus complete entropy loss) is achieved.     

Structural RNA has lower folding energy than random RNA of the same dinucleotide frequency

We present results of computer experiments that indicate that several RNAs for which the native state (minimum free energy secondary structure) is functionally important (type III hammerhead ribozymes, signal recognition particle RNAs, U2 small nucleolar spliceosomal RNAs, certain riboswitches, etc.) all have lower folding energy than random RNAs of the same length and dinucleotide frequency. Additionally, we find that whole mRNA as well as 5'-UTR, 3'-UTR, and cds regions of mRNA have folding energies comparable to that of random RNA, although there may be a statistically insignificant trace signal in 3'-UTR and cds regions. Various authors have used nucleotide (approximate) pattern matching and the computation of minimum free energy as filters to detect potential RNAs in ESTs and genomes. We introduce a new concept of the asymptotic Z-score and describe a fast, whole-genome scanning algorithm to compute asymptotic minimum free energy Z-scores of moving-window contents. Asymptotic Z-score computations offer another filter, to be used along with nucleotide pattern matching and minimum free energy computations, to detect potential functional RNAs in ESTs and genomic regions.     

Temperature dependence of the backbone dynamics of ribonuclease A in the ground state and bound to the inhibitor 5'-phosphothymidine (3'-5')pyrophosphate adenosine 3'-phosphate

The interaction of the dinucleotide inhibitor 5'-phosphothymidine(3',5')pyrophosphate adenosine 3'-phosphate (pTppAp) with bovine pancreatic ribonuclease A (RNase A) was characterized by calorimetry and solution NMR spectroscopy. Calorimetric data show that binding of pTppAp to RNase A is exothermic (DeltaH = -60.1 +/- 4.1 kJ/mol) with a dissociation constant of 16 nM at 298 K. At this temperature, the binding results in an entropy loss (TDeltaS = -16.8 +/- 7.3 kJ/mol) that is more favorable than that with the product analogue, 2'-CMP (TDeltaS = -31.3 +/- 0.9 kJ/mol). Temperature-dependent calorimetric experiments give a DeltaC(p) for ligand binding of -230 +/- 100 J/mol K. Binding of pTppAp results in noticeable effects on the backbone amide chemical shifts and dynamics. Amide backbone (15)N NMR spin-relaxation studies were performed on both apo RNase A and RNase A/pTppAp as a function of temperature. At each temperature, the model-free-determined order parameters, S(2), were significantly higher for RNase A/pTppAp than for the apo enzyme indicating a decrease in the conformational entropy of the protein upon ligand binding. Furthermore, the magnitude of this difference varies along the amino acid sequence specifically locating the entropic changes. The temperature dependence of S(2) at each residue enabled assessment of the local heat capacity changes (DeltaC(p)) from ligand binding. In an overall, average sense, DeltaC(p) for the protein backbone, determined from the NMR dynamics measurements, did not differ between apo RNase A and RNase A/pTppAp indicating that backbone dynamics contribute little to DeltaC(p) for protein-ligand interactions in this system. However, residue-by-residue comparison of the temperature-dependent change in entropy (DeltaS(B)) between free and bound forms reveals nonzero contributions to DeltaC(p) at individual sites. The balance of positive and negative changes reveals a redistribution of energetics upon binding. Furthermore, experiment and semiempirical estimates suggest that a large negative DeltaC(p) should accompany binding of pTppAp, and we conclude that this contribution must arise from factors other than amide backbone dynamics.     

Empirical calculation of the relative free energies of peptide binding to the molecular chaperone DnaK

We describe a methodology to calculate the relative free energies of protein-peptide complex formation. The interaction energy was decomposed into nonpolar, electrostatic and entropic contributions. A free energy-surface area relationship served to calculate the nonpolar free energy term. The electrostatic free energy was calculated with the finite difference Poisson-Boltzmann method and the entropic contribution was estimated from the loss in the conformational entropy of the peptide side chains. We applied this methodology to a series of DnaK*peptide complexes. On the basis of the single known crystal structure of the peptide-binding domain of DnaK with a bound heptapeptide, we modeled ten other DnaK*heptapeptide complexes with experimentally measured K(d) values from 0.06 microM to 11 microM, using molecular dynamics to refine the structures of the complexes. Molecular dynamic trajectories, after equilibration, were used for calculating the energies with greater accuracy. The calculated relative binding free energies were compared with the experimentally determined free energies. Linear scaling of the calculated terms was applied to fit them to the experimental values. The calculated binding free energies were between -7.1 kcal/mol and - 9.4 kcal/mol with a correlation coefficient of 0.86. The calculated nonpolar contributions are mainly due to the central hydrophobic binding pocket of DnaK for three amino acid residues. Negative electrostatic fields generated by the protein increase the binding affinity for basic residues flanking the hydrophobic core of the peptide ligand. Analysis of the individual energy contributions indicated that the nonpolar contributions are predominant compared to the other energy terms even for peptides with low affinity and that inclusion of the change in conformational entropy of the peptide side chains does not improve the discriminative power of the calculation. The method seems to be useful for predicting relative binding energies of peptide ligands of DnaK and might be applicable to other protein-peptide systems, particularly if only the structure of one protein-ligand complex is available.     

Accounting for conformational entropy in predicting binding free energies of protein-protein interactions

Protein-protein interactions are governed by the change in free energy upon binding, ΔG = ΔH - TΔS. These interactions are often marginally stable, so one must examine the balance between the change in enthalpy, ΔH, and the change in entropy, ΔS, when investigating known complexes, characterizing the effects of mutations, or designing optimized variants. To perform a large-scale study into the contribution of conformational entropy to binding free energy, we developed a technique called GOBLIN (Graphical mOdel for BiomoLecular INteractions) that performs physics-based free energy calculations for protein-protein complexes under both side-chain and backbone flexibility. Goblin uses a probabilistic graphical model that exploits conditional independencies in the Boltzmann distribution and employs variational inference techniques that approximate the free energy of binding in only a few minutes. We examined the role of conformational entropy on a benchmark set of more than 700 mutants in eight large, well-studied complexes. Our findings suggest that conformational entropy is important in protein-protein interactions--the root mean square error (RMSE) between calculated and experimentally measured ΔΔGs decreases by 12% when explicit entropic contributions were incorporated. GOBLIN models all atoms of the protein complex and detects changes to the binding entropy along the interface as well as positions distal to the binding interface. Our results also suggest that a variational approach to entropy calculations may be quantitatively more accurate than the knowledge-based approaches used by the well-known programs FOLDX and Rosetta--GOBLIN's RMSEs are 10 and 36% lower than these programs, respectively.     

Association of Helical β-Peptides and their Aggregation Behavior from the Potential of Mean Force in Explicit Solvent

Helical β-peptides have been shown to fold into well-defined structures. In aqueous solution, some β-peptides self-assemble into nanoscale fibers, aggregates, and liquid crystalline phases. Molecular simulations, at an atomistic level, are used to examine, in a systematic manner, the interactions between distinct β-peptide molecules. The relationship between side-chain chemistry (and position along the backbone) and, in particular, aggregation behaviors, is assessed by calculating the potential of mean force or dimerization free energy of two peptides in explicit water. The free energy profiles as a function of separation for helical, amphiphilic β-peptides are consistent with experimental observations, and help explain the origins of aggregate or fiber formation in solution. Close examination of the energetic and entropic contributions to the free energy reveals that, depending on the position of certain side groups along the molecule, the tendency of two peptides to aggregate can be driven by entropy or by energy, respectively. In contrast to findings from previous works that employed a coarse representation of the solvent, it is shown that water-peptide interactions play key roles in the association behavior of β-peptides.     

An improved experimental system for determining small folding entropy changes resulting from proline to alanine substitutions

Changes in protein stability can be achieved by making substitutions that increase or decrease the available conformations of the unfolded protein without altering the conformational freedom of the folded protein. Matthews and coworkers (1987) proposed that proline to alanine (P --> A) substitution would achieve this type of entropic destabilization. By comparing the Ramachandran area associated with alanine and proline residues, Matthews et al. estimated the unfolding entropy change resulting from P --> A substitution to be 4.8 cal mol(-1) K(-1). Although such an entropy difference would produce a substantial free energy change, accurately resolving such free energy changes into entropic and enthalpic components has been difficult. Here, we attempt to quantify the unfolding entropy change produced by P --> A substitution by amplifying the effect through multiple substitutions, and by decreasing the uncertainty in determining the unfolding entropy. Variants of a repeat protein, the Drosophila Notch ankyrin domain, were constructed with a varying number of P --> A substitutions at structurally conserved positions. Unfolding entropy values of the variants were determined from free energy measurements taken over a common temperature range using chemical denaturation. Our findings confirm the prediction that increasing the number of proline residues present in similar local environments increases the unfolding entropy. The average value of this increase in unfolding entropy is 7.7 +/- 4.2 cal mol(-1) K(-1), which is within error of the value estimated by Matthews et al. (1987).     

Stability of a model beta-sheet in water.

We have used molecular dynamics simulations to determine the stability in water of a model beta-sheet formed by two alanine dipeptide molecules with two intermolecular hydrogen bonds in the closely spaced antiparallel arrangement. In this paper we describe our computations of the binding free energy of the model sheet and a portion of the free energy surface as a function of a reaction co-ordinate for sheet formation. We used the free energy surface to identify stable conformations along the reaction co-ordinate. To determine whether or not the model sheet with two hydrogen bonds is more stable than a single amide hydrogen bond in water, we compared the results of the present calculations to results from our earlier study of linear hydrogen bond formation between two formamide molecules (the formamide "dimer"). The free energy surfaces for the sheet and formamide dimer each have two minima corresponding to locally stable hydrogen-bonded and solvent-separated configurations. The binding free energies of the model sheet and the formamide dimer are -5.5 and -0.34 kcal/mol, respectively. Thus, the model sheet with two hydrogen bonds is quite stable while the simple amide hydrogen bond is only marginally stable. To understand the relative stabilities of the model sheet and formamide dimer in terms of solute-solute and solute-water interactions, we decomposed the free energy differences between hydrogen-bonded and solvent-separated conformations into energetic and entropic contributions. The changes in the peptide-peptide energy and the entropy are roughly twice as large for the sheet as they are for the formamide dimer. The magnitude of the peptide-water energy difference for the sheet is less than twice (by about 3.5 kcal/mol) that for the formamide dimer, and this accounts for the stability of the sheet. The presence of the side-chains and/or blocking groups apparently prevents the amide groups in the sheet from being solvated as favorably in the separated arrangement as in the formamide dimer, where the amide groups are completely exposed to the solvent.     

Cooperativity of papain-substrate interaction energies in the S2 to S2' subsites

Enzyme-substrate contacts in the hydrolysis of ester substrates by the cysteine protease papain were investigated by systematically altering backbone hydrogen-bonding and side-chain hydrophobic contacts in the substrate and determining each substrate's kinetic constants. The observed specificity energies [defined as delta delta G obs = -RT ln [(kcat/KM)first/(kcat/KM)second)]] of the substrate backbone hydrogen bonds were -2.7 kcal/mol for the P2 NH and -2.6 kcal/mol for the P1 NH when compared against substrates containing esters at those sites. The observed binding energies were -4.0 kcal/mol for the P2 Phe side chain, -1.0 kcal/mol for the P1' C=O, and -2.3 kcal/mol for the P2' NH. The latter three values probably all significantly underestimate the incremental binding energies. The P2 NH, P2 Phe side-chain, and P1 NH contacts display a strong interdependence, or cooperativity, of interaction energies that is characteristic of enzyme-substrate interactions. This interdependence arises largely from the entropic cost of forming the enzyme-substrate transition state. As favorable contacts are added successively to a substrate, the entropic penalty associated with each decreases and the free energy expressed approaches the incremental interaction energy. This is the first report of a graded cooperative effect. Elucidation of favorable enzyme-substrate contacts remote from the catalytic site will assist in the design of highly specific cysteine protease inhibitors.     

Entropy calculations on a reversibly folding peptide: changes in solute free energy cannot explain folding behavior

The configurational entropy of a beta-heptapeptide in solution at four different temperatures is calculated. The contributions of the backbone and of the side-chain atoms to the total peptide entropy are analyzed separately and the effective contribution to the entropy arising from correlations between these terms determined. The correlation between the backbone and side-chain atoms amounts to about 17% and is rather insensitive to the temperature. The correlation of motion within the backbone and within side-chains is much larger and decreases with temperature. As the peptide reversibly folds at higher temperatures, its change in entropy and enthalpy upon folding is analyzed. The change in entropy and enthalpy upon folding of the peptide alone cannot account for the observed change in free energy on folding of the peptide in solution. Enthalpic and entropic contributions of the solvent thus also play a key role. Proteins 2001;43:45-56.     

Evidence that microRNA precursors, unlike other non-coding RNAs, have lower folding free energies than random sequences

MOTIVATION: Most non-coding RNAs are characterized by a specific secondary and tertiary structure that determines their function. Here, we investigate the folding energy of the secondary structure of non-coding RNA sequences, such as microRNA precursors, transfer RNAs and ribosomal RNAs in several eukaryotic taxa. Statistical biases are assessed by a randomization test, in which the predicted minimum free energy of folding is compared with values obtained for structures inferred from randomly shuffling the original sequences. RESULTS: In contrast with transfer RNAs and ribosomal RNAs, the majority of the microRNA sequences clearly exhibit a folding free energy that is considerably lower than that for shuffled sequences, indicating a high tendency in the sequence towards a stable secondary structure. A possible usage of this statistical test in the framework of the detection of genuine miRNA sequences is discussed.