Random Coil Model Research Articles

Protein ensemble modeling and analysis with MMMx.

Proteins, especially of eukaryotes, often have disordered domains and may contain multiple folded domains whose relative spatial arrangement is distributed. The MMMx ensemble modeling and analysis toolbox (https://github.com/gjeschke/MMMx) can support the design of experiments to characterize the distributed structure of such proteins, starting from AlphaFold2 predictions or folded domain structures. Weak order can be analyzed with reference to a random coil model or to peptide chains that match the residue-specific Ramachandran angle distribution of the loop regions and are otherwise unrestrained. The deviation of the mean square end-to-end distance of chain sections from their average over sections of the same sequence length reveals localized compaction or expansion of the chain. The shape sampled by disordered chains is visualized by superposition in the principal axes frame of their inertia tensor. Ensembles of different sizes and with weighted conformers can be compared based on a similarity parameter that abstracts from the ensemble width.

Exploring Nearest Neighbor Interactions and Their Influence on the Gibbs Energy Landscape of Unfolded Proteins and Peptides.

The Flory isolated pair hypothesis (IPH) is one of the corner stones of the random coil model, which is generally invoked to describe the conformational dynamics of unfolded and intrinsically disordered proteins (IDPs). It stipulates, that individual residues sample the entire sterically allowed space of the Ramachandran plot without exhibiting any correlations with the conformational dynamics of its neighbors. However, multiple lines of computational, bioinformatic and experimental evidence suggest that nearest neighbors have a significant influence on the conformational sampling of amino acid residues. This implies that the conformational entropy of unfolded polypeptides and proteins is much less than one would expect based on the Ramachandran plots of individual residues. A further implication is that the Gibbs energies of residues in unfolded proteins or polypeptides are not additive. This review provides an overview of what is currently known and what has yet to be explored regarding nearest neighbor interactions in unfolded proteins.

Randomizing of Oligopeptide Conformations by Nearest Neighbor Interactions between Amino Acid Residues.

Flory’s random coil model assumes that conformational fluctuations of amino acid residues in unfolded poly(oligo)peptides and proteins are uncorrelated (isolated pair hypothesis, IPH). This implies that conformational energies, entropies and solvation free energies are all additive. Nearly 25 years ago, analyses of coil libraries cast some doubt on this notion, in that they revealed that aromatic, but also β-branched side chains, could change the 3J(HNHCα) coupling of their neighbors. Since then, multiple bioinformatical, computational and experimental studies have revealed that conformational propensities of amino acids in unfolded peptides and proteins depend on their nearest neighbors. We used recently reported and newly obtained Ramachandran plots of tetra- and pentapeptides with non-terminal homo- and heterosequences of amino acid residues to quantitatively determine nearest neighbor coupling between them with a Ising type model. Results reveal that, depending on the choice of amino acid residue pairs, nearest neighbor interactions either stabilize or destabilize pairs of polyproline II and β-strand conformations. This leads to a redistribution of population between these conformations and a reduction in conformational entropy. Interactions between residues in polyproline II and turn(helix)-forming conformations seem to be cooperative in most cases, but the respective interaction parameters are subject to large statistical errors.

An MP2/Molecular Dynamics study of the solvent effects on the conformational equilibrium of the glycine dipeptide

The solvent effects on the conformational equilibrium of the Glycine dipeptide are studied using a Quantum Mechanics/Molecular Mechanics method based on the mean field approximation. Glycine is the structurally simplest amino acid found in nature and is usually considered the amino acid that best conforms to the basic assumptions of the random coil model. It has been experimentally found, however, that in gas phase the conformational landscape of the glycine dipeptide is dominated by only two structures (C5 and C7) characterized by the formation of an intramolecular hydrogen bond (HB). In this work we show that a similar situation is found in water solution: the conformational equilibrium is also dominated by only two structures (PPII and α) that, furthermore, are not stable in gas phase and in which the intramolecular HB is replaced by an intermolecular HB. This behavior mimics that of other peptides which display less conformational freedom. We show that relative stability results from the interplay of two free energy components that operate in opposite directions: internal energy and dipeptide-water interaction energy. The influence exerted by the solvent structure and the dipeptide charge distribution on the relative stability of the different conformers is also analyzed.

Analyses of experiment based ramachandran plots of tri-, tetra- and pentapeptides reveal residue dependent interactions between neighbors

Flory's canonical random coil model is based on the assumption that conformational fluctuations of amino acid residues in unfolded poly(oligo)peptides and proteins are uncorrelated (isolated pair hypothesis, IPH). It implies that conformational energies, entropies, and solvation free energies are all additive. Nearly 25 years ago, analyses of coil libraries casted some doubt on this notion in that they revealed that aromatic residues could change the 3J(HNHCα) coupling of their neighbors. Since then multiple lines of bioinformatical, computational and lately also experimental evidence indicate that the conformational propensities of amino acids in unfolded peptides and proteins depend on their nearest neighbors and possibly even on the nature of the second neighbor.

Quantitative Analysis of Protein Unfolded State Energetics: Experimental and Computational Studies Demonstrate That Non-Native Side-Chain Interactions Stabilize Local Native Backbone Structure.

Proteins fold on relatively smooth free energy landscapes which are biased toward the native state, but even simple topologies which fold rapidly can experience roughness on their free energy landscape. The details of these interactions are difficult to elucidate experimentally. Closely related to the problem of deciphering the details of the free energy landscape is the problem of defining the interactions in the denatured state ensemble (DSE) which is populated under native conditions, that is, under conditions where the native state is stable. The DSE of many proteins deviates from random coil models, but quantifying and defining the energetics of the transiently populated interactions in this ensemble is extremely challenging. Characterization of the DSE of proteins which fold to compact structures is also relevant to studies of intrinsically disordered proteins (IDPs) since interactions in the dynamic ensemble populated by IDPs can modulate their behavior. Here we show how experimental thermodynamic and pKa measurements can be combined with computational thermodynamic integration to quantify interactions in the DSE. We show that non-native side chain interactions can stabilize native backbone structure in the DSE and demonstrate that that even rapidly folding proteins can form energetically significant non-native interactions in their DSE. As an example, we characterize a non-native salt bridge that stabilizes local native backbone structure in the DSE of a widely studied fast-folding protein, the villin headpiece helical domain. The combined computational experimental approach is applicable to other protein unfolded states and provides insight that is impossible to obtain with either method alone.

Understanding and applications of Ser/Gly linkers in protein engineering.

Peptide linkers consisting of repeats of glycine and serine residues are commonly chosen by protein engineers to introduce flexible and hydrophilic spacers between protein domains. Given the popularity of these linkers, gaining a quantitative insight in their conformational behavior is important to understand the effect on functional properties of fusion proteins, including energy transfer efficiency in luminescent sensor proteins, intramolecular domain interactions and (multivalent) binding. In this chapter, we discuss how the conformational behavior of Ser/Gly linkers can be described using random coil models, and how measuring FRET as a function of linker length can be used to obtain empirical values for the stiffness of linkers containing different Ser-to-Gly ratios. Subsequently, we show how these models and the experimentally determined linker stiffness can be used to explain and predict the functional properties of multidomain proteins, providing useful rules-of-thumb and design tools for optimal linker engineering.

Heat Effects on Coil Hydrodynamic Size Reveal the Energetics of Denatured State Conformational Bias

Conformational equilibria in the protein denatured state has key roles regulating folding, stability, and function. Yet detailed information on the denatured state has proven elusive because of the low solubility of unfolded protein and the difficulty in deciphering the character of individual conformations from the ensemble average. To better understand the structural preferences of the denatured state, we have analyzed both the sequence dependence to the mean hydrodynamic size of disordered proteins in water and the impact of heat on the coil dimensions, showing that the sequence dependence and thermodynamic energies associated with intrinsic biases for the α and PPII backbone conformations can be obtained. These results are important for two reasons. First, survey of the protein coil library reveals that the α and PPII basins of the Ramachandran map are the dominant structural features of the protein denatured state. Our results demonstrate how to measure the temperature dependence to the α and PPII populations. Second, it has long been known that the dimensions of denatured proteins in 6 M guanidinium chloride are successfully predicted by the random coil model, irrespective of numerous experiments showing they can maintain residual structures. Hence, the protein hydrodynamic size is thought to give poor structural detail. Our results show that this is not the case for unfolded proteins in water. Rather, surprisingly, the hydrodynamic size of the unfolded state is found to be an effective reporter on the extent of the biases for the α and PPII backbone conformations.

Nearest Neighbor Effects in Homopeptide Segments of Short Peptides Explored by Circular Dichroism and NMR Spectroscopy

Contrary to the view based on the classical random coil model, it is now clearly established that individual amino acid residues in unfolded peptides and proteins differ with regard to their conformational propensities. The prime example is alanine which shows a very high preference for polyproline II (pPII). Recently, our research groups conducted a conformational analysis of GxyG host-guest peptides with selected mixtures of aliphatic (A,V,L) and more polar/charged residues (S,D,K) as guest residues x and y.

Random coil chemical shifts for serine, threonine and tyrosine phosphorylation over a broad pH range

Phosphorylation is one of the main regulators of cellular signaling typically occurring in flexible parts of folded proteins and in intrinsically disordered regions. It can have distinct effects on the chemical environment as well as on the structural properties near the modification site. Secondary chemical shift analysis is the main NMR method for detection of transiently formed secondary structure in intrinsically disordered proteins (IDPs) and the reliability of the analysis depends on an appropriate choice of random coil model. Random coil chemical shifts and sequence correction factors were previously determined for an Ac-QQXQQ-NH2-peptide series with X being any of the 20 common amino acids. However, a matching dataset on the phosphorylated states has so far only been incompletely determined or determined only at a single pH value. Here we extend the database by the addition of the random coil chemical shifts of the phosphorylated states of serine, threonine and tyrosine measured over a range of pH values covering the pKas of the phosphates and at several temperatures (www.bio.ku.dk/sbinlab/randomcoil). The combined results allow for accurate random coil chemical shift determination of phosphorylated regions at any pH and temperature, minimizing systematic biases of the secondary chemical shifts. Comparison of chemical shifts using random coil sets with and without inclusion of the phosphoryl group, revealed under/over estimations of helicity of up to 33%. The expanded set of random coil values will improve the reliability in detection and quantification of transient secondary structure in phosphorylation-modified IDPs.

Anticooperative Nearest-Neighbor Interactions between Residues in Unfolded Peptides and Proteins

Growing evidence suggests that the conformational distributions of amino acid residues in unfolded peptides and proteins depend on the nature of the nearest neighbors. To explore whether the underlying interactions would lead to a breakdown of the isolated pair hypothesis of the classical random coil model, we further analyzed the conformational propensities that were recently obtained for the two guest residues (x,y) of GxyG tetrapeptides. We constructed a statistical thermodynamics model that allows for cooperative as well as for anticooperative interactions between adjacent residues adopting either a polyproline II or a β-strand conformation. Our analysis reveals that the nearest-neighbor interactions between most of the central residues in the investigated GxyG peptides are anticooperative. Interaction Gibbs energies are rather large at high temperatures (350 K), at which point many proteins undergo thermal unfolding. At room temperature, these interaction energies are less pronounced. We used the obtained interaction parameter in a Zimm-Bragg/Ising-type approach to calculate the temperature dependence of the ultraviolet circular dichroism (CD) of the MAX3 peptide, which is predominantly built by KV repeats. The agreement between simulation and experimental data was found to be satisfactory. Finally, we analyzed the temperature dependence of the CD and 3J(HNHα) parameters of the amyloid β1–9 fragment. The results of this analysis and a more qualitative consideration of the temperature dependence of denatured proteins probed by CD spectroscopy further corroborate the dominance of anticooperative nearest-neighbor interactions. Generally, our results show that unfolded peptides—and most likely also proteins—exhibit some similarity with antiferromagnetic systems.

Intrinsically disordered inhibitor of glutamine synthetase is a functional protein with random-coil-like pKa values.

The sequential action of glutamine synthetase (GS) and glutamate synthase (GOGAT) in cyanobacteria allows the incorporation of ammonium into carbon skeletons. In the cyanobacterium Synechocystis sp. PCC 6803, the activity of GS is modulated by the interaction with proteins, which include a 65-residue-long intrinsically disordered protein (IDP), the inactivating factor IF7. This interaction is regulated by the presence of charged residues in both IF7 and GS. To understand how charged amino acids can affect the binding of an IDP with its target and to provide clues on electrostatic interactions in disordered states of proteins, we measured the pKa values of all IF7 acidic groups (Glu32, Glu36, Glu38, Asp40, Asp58, and Ser65, the backbone C-terminus) at 100 mM NaCl concentration, by using NMR spectroscopy. We also obtained solution structures of IF7 through molecular dynamics simulation, validated them on the basis of previous experiments, and used them to obtain theoretical estimates of the pKa values. Titration values for the two Asp and three Glu residues of IF7 were similar to those reported for random-coil models, suggesting the lack of electrostatic interactions around these residues. Furthermore, our results suggest the presence of helical structure at the N-terminus of the protein and of conformational changes at acidic pH values. The overall experimental and in silico findings suggest that local interactions and conformational equilibria do not play a role in determining the electrostatic features of the acidic residues of IF7.

Determinants of the pKa values of ionizable residues in an intrinsically disordered protein

Intrinsically disordered proteins (IDPs) are prevalent in eukaryotes; in humans, they are often associated with diseases. The protein NUPR1 is a multifunctional IDP involved in the development and progression of pancreatic cancer; therefore, it constitutes a target for drug design. In an effort to contribute to the understanding of the conformational features of NUPR1 and to provide clues on amino acid interactions in disordered states of proteins, we measured the pKa values of all its acidic groups (aspartic and glutamic residues, and backbone C terminus) by using NMR spectroscopy at low (100 mM) and high (500 mM) NaCl concentration. At low ionic strength, the pKa values were similar to those reported for random-coil models, except for Glu18 and Asp19, suggesting electrostatic interactions around these residues. Molecular modelling and simulation indicate an additional, significant role of nearby proline residues in determining the polypeptide conformational features and water accessibility in the region around Glu18, modulating the titration properties of these amino acids. In the other acidic residues of NUPR1, the small deviations of pKa values (compared to those expected for a random-coil) are likely due to electrostatic interactions with charged adjacent residues, which should be reduced at high NaCl concentrations. In fact, at high ionic strength, the pKa values of the aspartic residues were similar to those in a random coil, but there were still small differences for those of glutamic acids, probably due to hydrogen-bond formation. The overall findings suggest that local interactions and hydrophobic effects play a major role in determining the electrostatic features of NUPR1, whereas long-range charge contributions appear to be of lesser importance.

Conformational Entropies of Unfolded Peptides: The Source of a Realistic Estimation of the Entropy of Unfolded Peptides and Proteins

All theories of protein folding agree in suggesting that the folded state is promoted by a gain in enthalpy, whereas the unfolded state is favored by entropy. The entropy of the latter and thus the entropic loss associated with protein folding is generally estimated by utilizing the rather random conformational distributions of the so-called random coil model which assumes that all amino acid residues sample the entire sterically allowed region of the Ramachandran space. However, recent spectroscopic studies as well as analyses of coil libraries have revealed residue specific and generally less pronounced conformational sampling than suggested by the random coil model. These findings imply that unfolded peptides and proteins carry less entropy than expected. We used experimentally obtained Ramachandran plots of amino acid residues in GxG and GxyG host guest systems (x,y,: selected amino acid residues) in water to calculate the Boltzmann entropy of individual residues in the absence and presence of nearest neighbor interactions. Our results suggest that entropic chains of polypeptides exhibit conformational entropies which at room temperature contribute 3-5 kJ/mol less to the Gibbs/Helmholtz free energy. This suggest that classical random coil distributions lead to a substantial overestimation of conformational entropies. Thermodynamic data inferred from the temperature dependence of 1H 3J(HNHα) coupling constants and UVCD spectra of the investigated peptides allowed an estimation of the temperature dependence of the conformational entropies and a comparison with changes of the enthalpies and entropies of solvation.

Local order in the unfolded state: conformational biases and nearest neighbor interactions.

The discovery of Intrinsically Disordered Proteins, which contain significant levels of disorder yet perform complex biologically functions, as well as unwanted aggregation, has motivated numerous experimental and theoretical studies aimed at describing residue-level conformational ensembles. Multiple lines of evidence gathered over the last 15 years strongly suggest that amino acids residues display unique and restricted conformational preferences in the unfolded state of peptides and proteins, contrary to one of the basic assumptions of the canonical random coil model. To fully understand residue level order/disorder, however, one has to gain a quantitative, experimentally based picture of conformational distributions and to determine the physical basis underlying residue-level conformational biases. Here, we review the experimental, computational and bioinformatic evidence for conformational preferences of amino acid residues in (mostly short) peptides that can be utilized as suitable model systems for unfolded states of peptides and proteins. In this context particular attention is paid to the alleged high polyproline II preference of alanine. We discuss how these conformational propensities may be modulated by peptide solvent interactions and so called nearest-neighbor interactions. The relevance of conformational propensities for the protein folding problem and the understanding of IDPs is briefly discussed.

Relationship between nonlinear pressure-induced chemical shift changes and thermodynamic parameters.

NMR chemical shift analysis is a powerful method to investigate local changes in the environment of the observed nuclear spin of a polypeptide that are induced by application of high hydrostatic pressure. Usually, in the fast exchange regime, the pressure dependence of chemical shifts is analyzed by a second order Taylor expansion providing the first- and second-order pressure coefficient B1 and B2. The coefficients then are interpreted in a qualitative manner. We show here that in a two-state model, the ratio of B2/B1 is related to thermodynamic parameters, namely the ratio of the difference of compressibility factors Δβ' and partial molar volumes ΔV. The analysis is applied to the random-coil model peptides Ac-Gly-Gly-Xxx-Ala-NH2, with Xxx being one of the 20 proteinogenic amino acids. The analysis gives an average Δβ'/ΔV ratio of 1.6 GPa(-1) provided the condition |ΔG(0)| ≪ 2RT holds for the difference of the Gibbs free energies (ΔG(0)) of the two states at the temperature (T0) and the pressure (p0). The amide proton and nitrogen B2/B1 of a given amino acid Xxx are strongly correlated, indicating that their pressure-dependent chemical shift changes are due to the same thermodynamic process. As a possible physical mechanism providing a two-state model, the hydrogen bonding of water with the corresponding amide protein was simulated for isoleucine in position Xxx. The obtained free energy could satisfy the relation |ΔG(0)| ≪ 2RT. The derived relation was applied to the β-amyloid peptide Aβ and the phosphocarrier protein HPr from S. carnosus. For the transition of state 1 to state 2' of Aβ, the derived relation of B2/B1 to Δβ'/ΔV can be confirmed experimentally. The HPr protein is characterized by substantially higher negative B2/B1 values than those found in the tetrapeptides with an average value of approximately -5.1 GPa(-1) (Δβ'/ΔV of 5.1 GPa(-1) provided |ΔG(0)| ≪ 2RT holds). Qualitatively, the B2/B1 ratio can be used to predict regions of the HPr protein involved in the interaction with enzyme I or HPr-kinase/phosphatase.

PH Dependence of Amylin Fibrillization

In type 2 diabetics, the hormone amylin misfolds into amyloid plaques implicated in the destruction of the pancreatic β-cells that make insulin and amylin. The aggregative misfolding of amylin is pH-dependent, and exposure of the hormone to acidic and basic environments could be physiologically important. Amylin has two ionizable residues between pH 3 and 9: the α-amino group and His18. Our approach to measuring the pKa values for these sites has been to look at the pH dependence of fibrillization in amylin variants that have only one of the two groups. The α-amino group at the unstructured N-terminus of amylin has a pKa near 8.0, similar to the value in random coil models. By contrast, His18, which is involved in the intermolecular β-sheet structure of the fibrils, has a pKa that is lowered to 5.0 in the fibrils compared to the random coil value of 6.5. The lowered pKa of His18 is due to the hydrophobic environment of the residue, and electrostatic repulsion between positively charged His18 residues on neighboring amylin molecules in the fibril. His18 acts as an electrostatic switch inhibiting fibrillization in its charged state. The presence of a charged side chain at position 18 also affects fibril morphology and lowers amylin cytotoxicity toward a MIN6 mouse model of pancreatic β-cells. In addition to the two expected pKa values, we detected an apparent pKa of ~4.0 for the amylin-derived peptide NAc-SNNFGAILSS-NH2, which has no titratable groups. This pKa is due to the pH-induced ionization of the dye thioflavin T. By using alternative methods to follow fibrillization such as the dye Nile Red or turbidimetry, we were able to distinguish between the titration of the dye and groups on the peptide. Large differences in reaction kinetics were observed between the different methods at acidic pH, because of charges on the ThT dye, which hinder fibril formation much like the charges on the protein.

Enthalpy-Entropy Compensation and Isoequilbria Implicate Solvation as the Driving Force for Amino Acid Conformational Propensity

The existence of residual structure in intrinsically disordered proteins (IDPs) raise questions regarding how disorder/order preferences are encoded in the peptide sequence. In addition to formation of local secondary structure, individual amino acids sample unique and restricted conformational space, contrary to the predictions of the random coil model. To understand the thermodynamic driving forces underlying these newly discovered amino acid conformational biases, we have examined the thermodynamics governing these intrinsic conformational propensities in a series of host-guest experiments on model “GxG” peptides in aqueous solution. Global two-state pPII/β thermodynamic analysis of UVCD and HNMR data revealed the existence of a nearly exact enthalpy-entropy compensation for the pPII-β equilibrium for all residues. The obtained ΔH/ΔS values exhibit a nearly perfect linear relationship reflecting compensation temperature of 295K ± 2 K. We identified iso-equilibria for two subsets of the investigated peptides, indicating that pPII and β propensities of most residues become similar at physiological temperature, in spite of rather large differences between respective enthalpic and entropic differences. Interestingly, alanine, aspartic acid, and threonine, which have largely biased intrinsic conformational propensities, are the only amino acid residues that do not share iso-equilibria. We assign the isoequilibria to a common enthalpy-entropy compensation which occurs in the hydration shell of side chains with considerable hydrophobic content. Aspartic acid and threonine deviate from this behavior owing to their capability to form hydrogen bonds with the backbone even in extended structures, whereas the solvation energy of alanine is dominated by backbone - solvent interactions.

Nearest Neighbor Interactions Attenuate Intrinsic Amino Acid Conformational Preferences: A Combined Vibrational and NMR Study

Contrary to the isolated pair hypothesis, a pivotal ingredient of the random coil model, conformational distributions of amino acid residues in unfolded proteins cannot be considered as isolated from their nearest neighbors. To enable structural predictions of unfolded proteins or peptides (IDPs) knowledge about intrinsic propensities of amino acid residues must be complemented by information on nearest-neighbor interactions. To explore this influence, we preformed a joint vibrational and 2D-NMR spectroscopy study of selected “GxyG” host-guest peptides in aqueous solution: GDyG, GSyG, GxLG, GxVG, where x/y={A,K,LV}. The choice of D and S (L and V) for the x (y) position was motivated by their documented ability to drastically change the distribution of alanine in xAy tripeptides of truncated coil libraries. The observed spectral information were analyzed in terms of superpositions of 2D-Gaussian functions in the Ramachandran space, representing sub-ensembles of pPII-, β-strand-, helical-, and turn-like conformations. Results reveal that “turn forming residues” and hydrophobic residues have very strong effects on the conformation of their nearest neighbors. A direct comparison of the conformational distribution for S and D in GSAG ,GSLG, and GDAG with corresponding distributions in GxG peptiess reveals that hydrophobic neighbors cause drastic reductions of turn propensities. As a neighbor, K eliminates the asx turn propensity of D. In addition, insertion of these 'turn forming' residues (S, and D) causes conformational redistributions of L, A, and K from pPII to more extended β strand-like conformations while at the same time increasing the overall turn content. Concomitantly, the positions of pPII and particularly of β distributions are shifted to lower ψ values. Taken together, our results indicate that Dx and Sx motifs might act as conformational randomizers in proteins, attenuating intrinsic propensities of neighboring residues.

A quantitative measure for protein conformational heterogeneity

Conformational heterogeneity is a defining characteristic of proteins. Intrinsically disordered proteins (IDPs) and denatured state ensembles are extreme manifestations of this heterogeneity. Inferences regarding globule versus coil formation can be drawn from analysis of polymeric properties such as average size, shape, and density fluctuations. Here we introduce a new parameter to quantify the degree of conformational heterogeneity within an ensemble to complement polymeric descriptors. The design of this parameter is guided by the need to distinguish between systems that couple their unfolding-folding transitions with coil-to-globule transitions and those systems that undergo coil-to-globule transitions with no evidence of acquiring a homogeneous ensemble of conformations upon collapse. The approach is as follows: Each conformation in an ensemble is converted into a conformational vector where the elements are inter-residue distances. Similarity between pairs of conformations is quantified using the projection between the corresponding conformational vectors. An ensemble of conformations yields a distribution of pairwise projections, which is converted into a distribution of pairwise conformational dissimilarities. The first moment of this dissimilarity distribution is normalized against the first moment of the distribution obtained by comparing conformations from the ensemble of interest to conformations drawn from a Flory random coil model. The latter sets an upper bound on conformational heterogeneity thus ensuring that the proposed measure for intra-ensemble heterogeneity is properly calibrated and can be used to compare ensembles for different sequences and across different temperatures. The new measure of conformational heterogeneity will be useful in quantitative studies of coupled folding and binding of IDPs and in de novo sequence design efforts that are geared toward controlling the degree of heterogeneity in unbound forms of IDPs.