Functional Dynamics of Substrate Recognition in TEM Beta-Lactamase.

The beta-lactamase enzyme provides effective resistance to beta-lactam antibiotics due to substrate recognition controlled by point mutations. Recently, extended-spectrum and inhibitor-resistant mutants have become a global health problem. Here, the functional dynamics that control substrate recognition in TEM beta-lactamase are investigated using all-atom molecular dynamics simulations. Comparisons are made between wild-type TEM-1 and TEM-2 and the extended-spectrum mutants TEM-10 and TEM-52, both in apo form and in complex with four different antibiotics (ampicillin, amoxicillin, cefotaxime and ceftazidime). Dynamic allostery is predicted based on a quasi-harmonic normal mode analysis using a perturbation scan. An allosteric mechanism known to inhibit enzymatic function in TEM beta-lactamase is identified, along with other allosteric binding targets. Mechanisms for substrate recognition are elucidated using multivariate comparative analysis of molecular dynamics trajectories to identify changes in dynamics resulting from point mutations and ligand binding, and the conserved dynamics, which are functionally important, are extracted as well. The results suggest that the H10-H11 loop (residues 214-221) is a secondary anchor for larger extended spectrum ligands, while the H9-H10 loop (residues 194-202) is distal from the active site and stabilizes the protein against structural changes. These secondary non-catalytically-active loops offer attractive targets for novel noncompetitive inhibitors of TEM beta-lactamase.

JEDi: java essential dynamics inspector - a molecular trajectory analysis toolkit.

BACKGROUND: Principal component analysis (PCA) is commonly applied to the atomic trajectories of biopolymers to extract essential dynamics that describe biologically relevant motions. Although application of PCA is straightforward, specialized software to facilitate workflows and analysis of molecular dynamics simulation data to fully harness the power of PCA is lacking. The Java Essential Dynamics inspector (JEDi) software is a major upgrade from the previous JED software. RESULTS: Employing multi-threading, JEDi features a user-friendly interface to control rapid workflows for interrogating conformational motions of biopolymers at various spatial resolutions and within subregions, including multiple chain proteins. JEDi has options for Cartesian-based coordinates (cPCA) and internal distance pair coordinates (dpPCA) to construct covariance (Q), correlation (R), and partial correlation (P) matrices. Shrinkage and outlier thresholding are implemented for the accurate estimation of covariance. The effect of rare events is quantified using outlier and inlier filters. Applying sparsity thresholds in statistical models identifies latent correlated motions. Within a hierarchical approach, small-scale atomic motion is first calculated with a separate local cPCA calculation per residue to obtain eigenresidues. Then PCA on the eigenresidues yields rapid and accurate description of large-scale motions. Local cPCA on all residue pairs creates a map of all residue-residue dynamical couplings. Additionally, kernel PCA is implemented. JEDi output gives high quality PNG images by default, with options for text files that include aligned coordinates, several metrics that quantify mobility, PCA modes with their eigenvalues, and displacement vector projections onto the top principal modes. JEDi provides PyMol scripts together with PDB files to visualize individual cPCA modes and the essential dynamics occurring within user-selected time scales. Subspace comparisons performed on the most relevant eigenvectors using several statistical metrics quantify similarity/overlap of high dimensional vector spaces. Free energy landscapes are available for both cPCA and dpPCA. CONCLUSION: JEDi is a convenient toolkit that applies best practices in multivariate statistics for comparative studies on the essential dynamics of similar biopolymers. JEDi helps identify functional mechanisms through many integrated tools and visual aids for inspecting and quantifying similarity/differences in mobility and dynamic correlations.

Victory Tax: A Holistic Income Tax System.

How can an income tax system be designed to exploit human nature and a free market to create a poverty free society, while balancing budgets without disproportional tax burdens? Such a tax system, with universal character, is deduced from the following guiding principles: (1) a single tax rate applies to all income types and levels; (2) the tax rate adjusts to satisfy budget projections; (3) government transfer only supplements the income of households with self-generated income below the poverty line; (4) deductions for basic living expenses, itemized investments and capital losses are allowed; (5) deductions cannot be applied to government transfer. A general framework emerges with three parameters that determine a minimum allowed tax deduction, a maximum allowed itemized deduction, and a maximum deduction defined by income percentage. An income distribution that mimics the United States, and a series of log-normal distributions are considered to quantitatively compare detailed characteristics of this tax system to progressive and flat tax systems. To minimize government dependency while maximizing after-tax income, the effective tax rate (ETR) as a function of income percentile takes the shape of the letter, V, inspiring the name victory tax, where the middle class has the lowest ETR.

Rigidity and flexibility characteristics of DD[E/D]-transposases Mos1 and Sleeping Beauty.

DD[E/D]-transposases catalyze the multistep reaction of cut-and-paste DNA transposition. Structurally, several DD[E/D]-transposases have been characterized, revealing a multi-domain structure with the catalytic domain possessing the RNase H-like structural motif that brings three catalytic residues (D, D, and E or D) into close proximity for the catalysis. However, the dynamic behavior of DD[E/D]-transposases during transposition remains poorly understood. Here, we analyze the rigidity and flexibility characteristics of two representative DD[E/D]-transposases Mos1 and Sleeping Beauty (SB) using the minimal distance constraint model (mDCM). We find that the catalytic domain of both transposases is globally rigid, with the notable exception of the clamp loop being flexible in the DNA-unbound form. Within this globally rigid structure, the central ß-sheet of the RNase H-like motif is much less rigid in comparison to its surrounding α-helices, forming a cage-like structure. The comparison of the original SB transposase to its hyperactive version SB100X reveals the region where the change in flexibility/rigidity correlates with increased activity. This region is found to be within the RNase H-like structural motif and comprise the loop leading from beta-strand B3 to helix H1, helices H1 and H2, which are located on the same side of the central beta-sheet, and the loop between helix H3 and beta-strand B5. We further identify the RKEN214-217DAVQ mutations of the set of hyperactive mutations within the catalytic domain of SB transposase to be the driving factor that induces change in residue-pair rigidity correlations within SB transposase. Given that a signature RNase H-like structural motif is found in DD[E/D]-transposases and, more broadly, in a large superfamily of polynucleotidyl transferases, our results are relevant to these proteins as well.

JED: a Java Essential Dynamics Program for comparative analysis of protein trajectories.

BACKGROUND: Essential Dynamics (ED) is a common application of principal component analysis (PCA) to extract biologically relevant motions from atomic trajectories of proteins. Covariance and correlation based PCA are two common approaches to determine PCA modes (eigenvectors) and their eigenvalues. Protein dynamics can be characterized in terms of Cartesian coordinates or internal distance pairs. In understanding protein dynamics, a comparison of trajectories taken from a set of proteins for similarity assessment provides insight into conserved mechanisms. Comprehensive software is needed to facilitate comparative-analysis with user-friendly features that are rooted in best practices from multivariate statistics. RESULTS: We developed a Java based Essential Dynamics toolkit called JED to compare the ED from multiple protein trajectories. Trajectories from different simulations and different proteins can be pooled for comparative studies. JED implements Cartesian-based coordinates (cPCA) and internal distance pair coordinates (dpPCA) as options to construct covariance (Q) or correlation (R) matrices. Statistical methods are implemented for treating outliers, benchmarking sampling adequacy, characterizing the precision of Q and R, and reporting partial correlations. JED output results as text files that include transformed coordinates for aligned structures, several metrics that quantify protein mobility, PCA modes with their eigenvalues, and displacement vector (DV) projections onto the top principal modes. Pymol scripts together with PDB files allow movies of individual Q- and R-cPCA modes to be visualized, and the essential dynamics occurring within user-selected time scales. Subspaces defined by the top eigenvectors are compared using several statistical metrics to quantify similarity/overlap of high dimensional vector spaces. Free energy landscapes can be generated for both cPCA and dpPCA. CONCLUSIONS: JED offers a convenient toolkit that encourages best practices in applying multivariate statistics methods to perform comparative studies of essential dynamics over multiple proteins. For each protein, Cartesian coordinates or internal distance pairs can be employed over the entire structure or user-selected parts to quantify similarity/differences in mobility and correlations in dynamics to develop insight into protein structure/function relationships.

Decomposing Dynamical Couplings in Mutated scFv Antibody Fragments into Stabilizing and Destabilizing Effects.

Conformational fluctuations within scFv antibodies are characterized by a novel perturbation-response decomposition of molecular dynamics trajectories. Both perturbation and response profiles are stratified into stabilizing and destabilizing conditions. The linker between the VH and VL domains exhibits the dominant dynamical response by being coupled to nearly the entire protein, responding to both stabilizing and destabilizing perturbations. Perturbations within complementarity-determining regions (CDR) induce rich behavior in dynamic response. Among many effects, stabilizing any CDR loop in the VH domain triggers a destabilizing response in all CDR loops in the VL domain and vice versa. Destabilizing residues within the VL domain are likely to stabilize all CDR loops in the VH domain, and, when these residues are not buried, the CDR loops in the VL domain are also likely to be stabilized. These effects, described by shifts in normal mode characteristics, initiate a propensity for dynamic allostery with possible functional implications in bispecific antibodies.

Statistical Measures to Quantify Similarity between Molecular Dynamics Simulation Trajectories.

Molecular dynamics simulation is commonly employed to explore protein dynamics. Despite the disparate timescales between functional mechanisms and molecular dynamics (MD) trajectories, functional differences are often inferred from differences in conformational ensembles between two proteins in structure-function studies that investigate the effect of mutations. A common measure to quantify differences in dynamics is the root mean square fluctuation (RMSF) about the average position of residues defined by Cα-atoms. Using six MD trajectories describing three native/mutant pairs of beta-lactamase, we make comparisons with additional measures that include Jensen-Shannon, modifications of Kullback-Leibler divergence, and local p-values from 1-sample Kolmogorov-Smirnov tests. These additional measures require knowing a probability density function, which we estimate by using a nonparametric maximum entropy method that quantifies rare events well. The same measures are applied to distance fluctuations between Cα-atom pairs. Results from several implementations for quantitative comparison of a pair of MD trajectories are made based on fluctuations for on-residue and residue-residue local dynamics. We conclude that there is almost always a statistically significant difference between pairs of 100 ns all-atom simulations on moderate-sized proteins as evident from extraordinarily low p-values.

Mutations in Antibody Fragments Modulate Allosteric Response Via Hydrogen-Bond Network Fluctuations.

A mechanical perturbation method that locally restricts conformational entropy along the protein backbone is used to identify putative allosteric sites in a series of antibody fragments. The method is based on a distance constraint model that integrates mechanical and thermodynamic viewpoints of protein structure wherein mechanical clamps that mimic substrate or cosolute binding are introduced. Across a set of six single chain-Fv fragments of the anti-lymphotoxin-ß receptor antibody, statistically significant responses are obtained by averaging over 10 representative structures sampled from a molecular dynamics simulation. As expected, the introduced clamps locally rigidify the protein, but long-ranged increases in both rigidity and flexibility are also frequently observed. Expanding our analysis to every molecular dynamics frame demonstrates that the allosteric responses are modulated by fluctuations within the hydrogen-bond network where the native ensemble is comprised of conformations that both are, and are not, affected by the perturbation in question. Population shifts induced by the mutations alter the allosteric response by adjusting which hydrogen-bond networks are the most probable. These effects are compared using response maps that track changes across each single chain-Fv fragment, thus providing valuable insight into how sensitive allosteric mechanisms are to mutations.

Rigidity Emerges during Antibody Evolution in Three Distinct Antibody Systems: Evidence from QSFR Analysis of Fab Fragments.

The effects of somatic mutations that transform polyspecific germline (GL) antibodies to affinity mature (AM) antibodies with monospecificity are compared among three GL-AM Fab pairs. In particular, changes in conformational flexibility are assessed using a Distance Constraint Model (DCM). We have previously established that the DCM can be robustly applied across a series of antibody fragments (VL to Fab), and subsequently, the DCM was combined with molecular dynamics (MD) simulations to similarly characterize five thermostabilizing scFv mutants. The DCM is an ensemble based statistical mechanical approach that accounts for enthalpy/entropy compensation due to network rigidity, which has been quite successful in elucidating conformational flexibility and Quantitative Stability/Flexibility Relationships (QSFR) in proteins. Applied to three disparate antibody systems changes in QSFR quantities indicate that the VH domain is typically rigidified, whereas the VL domain and CDR L2 loop become more flexible during affinity maturation. The increase in CDR H3 loop rigidity is consistent with other studies in the literature. The redistribution of conformational flexibility is largely controlled by nonspecific changes in the H-bond network, although certain Arg to Asp salt bridges create highly localized rigidity increases. Taken together, these results reveal an intricate flexibility/rigidity response that accompanies affinity maturation.

Dynamics and thermodynamic properties of CXCL7 chemokine.

Chemokines form a family of signaling proteins mainly responsible for directing the traffic of leukocytes, where their biological activity can be modulated by their oligomerization state. We characterize the dynamics and thermodynamic stability of monomer and homodimer structures of CXCL7, one of the most abundant platelet chemokines, using experimental methods that include circular dichroism (CD) and nuclear magnetic resonance (NMR) spectroscopy, and computational methods that include the anisotropic network model (ANM), molecular dynamics (MD) simulations and the distance constraint model (DCM). A consistent picture emerges for the effects of dimerization and Cys5-Cys31 and Cys7-Cys47 disulfide bonds formation. The presence of disulfide bonds is not critical for maintaining structural stability in the monomer or dimer, but the monomer is destabilized more than the dimer upon removal of disulfide bonds. Disulfide bonds play a key role in shaping the characteristics of native state dynamics. The combined analysis shows that upon dimerization flexibly correlated motions are induced between the 30s and 50s loop within each monomer and across the dimer interface. Interestingly, the greatest gain in flexibility upon dimerization occurs when both disulfide bonds are present, and the homodimer is least stable relative to its two monomers. These results suggest that the highly conserved disulfide bonds in chemokines facilitate a structural mechanism that is tuned to optimally distinguish functional characteristics between monomer and dimer.

Variations within class-A ß-lactamase physiochemical properties reflect evolutionary and environmental patterns, but not antibiotic specificity.

The bacterial enzyme ß-lactamase hydrolyzes the ß-lactam ring of penicillin and chemically related antibiotics, rendering them ineffective. Due to rampant antibiotic overuse, the enzyme is evolving new resistance activities at an alarming rate. Related, the enzyme's global physiochemical properties exhibit various amounts of conservation and variability across the family. To that end, we characterize the extent of property conservation within twelve different class-A ß-lactamases, and conclusively establish that the systematic variations therein parallel their evolutionary history. Large and systematic differences within electrostatic potential maps and pairwise residue-to-residue couplings are observed across the protein, which robustly reflect phylogenetic outgroups. Other properties are more conserved (such as residue pKa values, electrostatic networks, and backbone flexibility), yet they also have systematic variations that parallel the phylogeny in a statistically significant way. Similarly, the above properties also parallel the environmental condition of the bacteria they are from in a statistically significant way. However, it is interesting and surprising that the only one of the global properties (protein charge) parallels the functional specificity patterns; meaning antibiotic resistance activities are not significantly constraining the global physiochemical properties. Rather, extended spectrum activities can emerge from the background of nearly any set of electrostatic and dynamic properties.

Changes in Lysozyme Flexibility upon Mutation Are Frequent, Large and Long-Ranged.

We investigate changes in human c-type lysozyme flexibility upon mutation via a Distance Constraint Model, which gives a statistical mechanical treatment of network rigidity. Specifically, two dynamical metrics are tracked. Changes in flexibility index quantify differences within backbone flexibility, whereas changes in the cooperativity correlation quantify differences within pairwise mechanical couplings. Regardless of metric, the same general conclusions are drawn. That is, small structural perturbations introduced by single point mutations have a frequent and pronounced affect on lysozyme flexibility that can extend over long distances. Specifically, an appreciable change occurs in backbone flexibility for 48% of the residues, and a change in cooperativity occurs in 42% of residue pairs. The average distance from mutation to a site with a change in flexibility is 17-20 Å. Interestingly, the frequency and scale of the changes within single point mutant structures are generally larger than those observed in the hen egg white lysozyme (HEWL) ortholog, which shares 61% sequence identity with human lysozyme. For example, point mutations often lead to substantial flexibility increases within the ß-subdomain, which is consistent with experimental results indicating that it is the nucleation site for amyloid formation. However, ß-subdomain flexibility within the human and HEWL orthologs is more similar despite the lowered sequence identity. These results suggest compensating mutations in HEWL reestablish desired properties.

Ab initio gene prediction for protein-coding regions.

Motivation: Ab initio gene prediction in nonmodel organisms is a difficult task. While many ab initio methods have been developed, their average accuracy over long segments of a genome, and especially when assessed over a wide range of species, generally yields results with sensitivity and specificity levels in the low 60% range. A common weakness of most methods is the tendency to learn patterns that are species-specific to varying degrees. The need exists for methods to extract genetic features that can distinguish coding and noncoding regions that are not sensitive to specific organism characteristics. Results: A new method based on a neural network (NN) that uses a collection of sensors to create input features is presented. It is shown that accurate predictions are achieved even when trained on organisms that are significantly different phylogenetically than test organisms. A consensus prediction algorithm for a CoDing Sequence (CDS) is subsequently applied to the first nucleotide level of NN predictions that boosts accuracy through a data-driven procedure that optimizes a CDS/non-CDS threshold. An aggregate accuracy benchmark at the nucleotide level shows that this new approach performs better than existing ab initio methods, while requiring significantly less training data. Availability and implementation: https://github.com/BioMolecularPhysicsGroup-UNCC/MachineLearning.

Switch II mutants reveal coupling between the nucleotide- and actin-binding regions in myosin V.

Conserved active-site elements in myosins and other P-loop NTPases play critical roles in nucleotide binding and hydrolysis; however, the mechanisms of allosteric communication among these mechanoenzymes remain unresolved. In this work we introduced the E442A mutation, which abrogates a salt-bridge between switch I and switch II, and the G440A mutation, which abolishes a main-chain hydrogen bond associated with the interaction of switch II with the γ phosphate of ATP, into myosin V. We used fluorescence resonance energy transfer between mant-labeled nucleotides or IAEDANS-labeled actin and FlAsH-labeled myosin V to examine the conformation of the nucleotide- and actin-binding regions, respectively. We demonstrate that in the absence of actin, both the G440A and E442A mutants bind ATP with similar affinity and result in only minor alterations in the conformation of the nucleotide-binding pocket (NBP). In the presence of ADP and actin, both switch II mutants disrupt the formation of a closed NBP actomyosin.ADP state. The G440A mutant also prevents ATP-induced opening of the actin-binding cleft. Our results indicate that the switch II region is critical for stabilizing the closed NBP conformation in the presence of actin, and is essential for communication between the active site and actin-binding region.

Protein Function Analysis through Machine Learning.

Machine learning (ML) has been an important arsenal in computational biology used to elucidate protein function for decades. With the recent burgeoning of novel ML methods and applications, new ML approaches have been incorporated into many areas of computational biology dealing with protein function. We examine how ML has been integrated into a wide range of computational models to improve prediction accuracy and gain a better understanding of protein function. The applications discussed are protein structure prediction, protein engineering using sequence modifications to achieve stability and druggability characteristics, molecular docking in terms of protein-ligand binding, including allosteric effects, protein-protein interactions and protein-centric drug discovery. To quantify the mechanisms underlying protein function, a holistic approach that takes structure, flexibility, stability, and dynamics into account is required, as these aspects become inseparable through their interdependence. Another key component of protein function is conformational dynamics, which often manifest as protein kinetics. Computational methods that use ML to generate representative conformational ensembles and quantify differences in conformational ensembles important for function are included in this review. Future opportunities are highlighted for each of these topics.

Nonadditivity in conformational entropy upon molecular rigidification reveals a universal mechanism affecting folding cooperativity.

Previously, we employed a Maxwell counting distance constraint model (McDCM) to describe α-helix formation in polypeptides. Unlike classical helix-coil transition theories, the folding mechanism derives from nonadditivity in conformational entropy caused by rigidification of molecular structure as intramolecular cross-linking interactions form along the backbone. For example, when a hydrogen bond forms within a flexible region, both energy and conformational entropy decrease. However, no conformational entropy is lost when the region is already rigid because atomic motions are not constrained further. Unlike classical zipper models, the same mechanism also describes a coil-to-ß-hairpin transition. Special topological features of the helix and hairpin structures allow the McDCM to be solved exactly. Taking full advantage of the fact that Maxwell constraint counting is a mean field approximation applied to the distribution of cross-linking interactions, we present an exact transfer matrix method that does not require any special topological feature. Upon application of the model to proteins, cooperativity within the folding transition is yet again appropriately described. Notwithstanding other contributing factors such as the hydrophobic effect, this simple model identifies a universal mechanism for cooperativity within polypeptide and protein-folding transitions, and it elucidates scaling laws describing hydrogen-bond patterns observed in secondary structure. In particular, the native state should have roughly twice as many constraints as there are degrees of freedom in the coil state to ensure high fidelity in two-state folding cooperativity, which is empirically observed.

Characterization of the pre-force-generation state in the actomyosin cross-bridge cycle.

Myosin is an actin-based motor protein that generates force by cycling between actin-attached (strong binding: ADP or rigor) and actin-detached (weak binding: ATP or ADP.P(i)) states during its ATPase cycle. However, it remains unclear what specific conformational changes in the actin binding site take place on binding to actin, and how these structural changes lead to product release and the production of force and motion. We studied the dynamics of the actin binding region of myosin V by using fluorescence resonance energy transfer (FRET) to monitor conformational changes in the upper-50-kDa domain of the actin binding cleft in the weak and strong actin binding states. Steady-state and lifetime data monitoring the FRET signal suggest that the cleft is in a more open conformation in the weak actin binding states. Transient kinetic experiments suggest that a rapid conformational change occurs, which is consistent with cleft closure before actin-activated phosphate release. Our results have identified a pre-force-generation actomyosin ADP.P(i) state, and suggest force generation may occur from a state not yet seen by crystallography in which the actin binding cleft and the nucleotide binding pocket are closed. Computational modeling uncovers dramatic changes in the rigidity of the upper-50-kDa domain in different nucleotide states, which suggests that the intrinsic flexibility of this domain allows myosin motors to accomplish simultaneous tight nucleotide binding (closed nucleotide binding pocket) and high-affinity actin binding (closed actin binding cleft).

Molecular function recognition by supervised projection pursuit machine learning.

Identifying mechanisms that control molecular function is a significant challenge in pharmaceutical science and molecular engineering. Here, we present a novel projection pursuit recurrent neural network to identify functional mechanisms in the context of iterative supervised machine learning for discovery-based design optimization. Molecular function recognition is achieved by pairing experiments that categorize systems with digital twin molecular dynamics simulations to generate working hypotheses. Feature extraction decomposes emergent properties of a system into a complete set of basis vectors. Feature selection requires signal-to-noise, statistical significance, and clustering quality to concurrently surpass acceptance levels. Formulated as a multivariate description of differences and similarities between systems, the data-driven working hypothesis is refined by analyzing new systems prioritized by a discovery-likelihood. Utility and generality are demonstrated on several benchmarks, including the elucidation of antibiotic resistance in TEM-52 beta-lactamase. The software is freely available, enabling turnkey analysis of massive data streams found in computational biology and material science.

Allosteric response is both conserved and variable across three CheY orthologs.

A computational method to identify residues likely to initiate allosteric signals has been developed. The method is based on differences within stability and flexibility profiles between wild-type and perturbed structures as computed by a distance constraint model. Application of the approach to three bacterial chemotaxis protein Y (CheY) orthologs provides a comparison of allosteric response across protein family divergence. Interestingly, we observe a rich mixture of both conservation and variability within the identified allosteric sites. While similarity within the overall response parallels the evolutionary relationships, >50% of the best scoring putative sites are only identified in a single ortholog. These results suggest that detailed descriptions of intraprotein communication are substantially more variable than structure and function, yet do maintain some evolutionary relationships. Finally, structural clusters of large response identify four allosteric hotspots, including the ß4/α4 loop known to be critical to relaying the CheY phosphorylation signal.

Helix/coil nucleation: a local response to global demands.

A complete description of protein structure and function must include a proper treatment of mechanisms that lead to cooperativity. The helix/coil transition serves as a simple example of a cooperative folding process, commonly described by a nucleation-propagation mechanism. The prevalent view is that coil structure must first form a short segment of helix in a localized region despite paying a free energy cost (nucleation). Afterward, helical structure propagates outward from the nucleation site. Both processes entail enthalpy-entropy compensation that derives from the loss in conformational entropy on helix formation with concomitant gain in favorable interactions. Nucleation-propagation models inherently assume that cooperativity arises from a sequential series of local events. An alternative distance constraint model asserts there is a direct link between available degrees of freedom and cooperativity through the nonadditivity in conformational entropy. That is, helix nucleation is a concerted manifestation of rigidity propagating through atomic structure. The link between network rigidity and nonadditivity of conformational entropy is shown in this study by solving the distance constraint model using a simple global constraint counting approximation. Cooperativity arises from competition between excess and deficiency in available degrees of freedom in the coil and helix states respectively.