Hyeon, C. ; Dima, R. I. ; Thirumalai, D. Pathways and kinetic barriers in mechanical unfolding and refolding of RNA and proteins. Structure 14, 1633-45.Abstract
Using self-organized polymer models, we predict mechanical unfolding and refolding pathways of ribozymes, and the green fluorescent protein. In agreement with experiments, there are between six and eight unfolding transitions in the Tetrahymena ribozyme. Depending on the loading rate, the number of rips in the force-ramp unfolding of the Azoarcus ribozymes is between two and four. Force-quench refolding of the P4-P6 subdomain of the Tetrahymena ribozyme occurs through a compact intermediate. Subsequent formation of tertiary contacts between helices P5b-P6a and P5a/P5c-P4 leads to the native state. The force-quench refolding pathways agree with ensemble experiments. In the dominant unfolding route, the N-terminal alpha helix of GFP unravels first, followed by disruption of the N terminus beta strand. There is a third intermediate that involves disruption of three other strands. In accord with experiments, the force-quench refolding pathway of GFP is hierarchic, with the rate-limiting step being the closure of the barrel.
Hyeon, C. ; Dima, R. I. ; Thirumalai, D. Size, shape, and flexibility of RNA structures. J Chem Phys 125, 194905.Abstract
Determination of sizes and flexibilities of RNA molecules is important in understanding the nature of packing in folded structures and in elucidating interactions between RNA and DNA or proteins. Using the coordinates of the structures of RNA in the Protein Data Bank we find that the size of the folded RNA structures, measured using the radius of gyration R(G), follows the Flory scaling law, namely, R(G)=5.5N(1/3) A, where N is the number of nucleotides. The shape of RNA molecules is characterized by the asphericity Delta and the shape S parameters that are computed using the eigenvalues of the moment of inertia tensor. From the distribution of Delta, we find that a large fraction of folded RNA structures are aspherical and the distribution of S values shows that RNA molecules are prolate (S>0). The flexibility of folded structures is characterized by the persistence length l(p). By fitting the distance distribution function P(r), that is computed using the coordinates of the folded RNA, to the wormlike chain model we extracted the persistence length l(p). We find that l(p) approximately 1.5N(0.33) A which might reflect the large separation between the free energies that stabilize secondary and tertiary structures. The dependence of l(p) on N implies that the average length of helices should increase as the size of RNA grows. We also analyze packing in the structures of ribosomes (30S, 50S, and 70S) in terms of R(G), Delta, S, and l(p). The 70S and the 50S subunits are more spherical compared to most RNA molecules. The globularity in 50S is due to the presence of an unusually large number (compared to 30S subunit) of small helices that are stitched together by bulges and loops. Comparison of the shapes of the intact 70S ribosome and the constituent particles suggests that folding of the individual molecules might occur prior to assembly.
Stan, G. ; Brooks, B. R. ; Lorimer, G. H. ; Thirumalai, D. Identifying natural substrates for chaperonins using a sequence-based approach. Protein Sci 14, 193-201.Abstract
The Escherichia coli chaperonin machinery, GroEL, assists the folding of a number of proteins. We describe a sequence-based approach to identify the natural substrate proteins (SPs) for GroEL. Our method is based on the hypothesis that natural SPs are those that contain patterns of residues similar to those found in either GroES mobile loop and/or strongly binding peptide in complex with GroEL. The method is validated by comparing the predicted results with experimentally determined natural SPs for GroEL. We have searched for such patterns in five genomes. In the E. coli genome, we identify 1422 (about one-third) sequences that are putative natural SPs. In Saccharomyces cerevisiae, 2885 (32%) of sequences can be natural substrates for Hsp60, which is the analog of GroEL. The precise number of natural SPs is shown to be a function of the number of contacts an SP makes with the apical domain (N(C)) and the number of binding sites (N(B)) in the oligomer with which it interacts. For known SPs for GroEL, we find approximately 4 < N(C) < 5 and 2
Tarus, B. ; Straub, J. E. ; Thirumalai, D. Probing the initial stage of aggregation of the Abeta(10-35)-protein: assessing the propensity for peptide dimerization. J Mol Biol 345, 1141-56.Abstract
Characterization of the early stages of peptide aggregation is of fundamental importance in elucidating the mechanism of the formation of deposits associated with amyloid disease. The initial step in the pathway of aggregation of the Abeta-protein, whose monomeric NMR structure is known, was studied through the simulation of the structure and stability of the peptide dimer in aqueous solution. A protocol based on shape complementarity was used to generate an assortment of possible dimer structures. The structures generated based on shape complementarity were evaluated using rapidly computed estimates of the desolvation and electrostatic interaction energies to identify a putative stable dimer structure. The potential of mean force associated with the dimerization of the peptides in aqueous solution was computed for both the hydrophobic and the electrostatic driven forces using umbrella sampling and classical molecular dynamics simulation at constant temperature and pressure with explicit solvent and periodic boundary conditions. The comparison of the two free energy profiles suggests that the structure of the peptide dimer is determined by the favorable desolvation of the hydrophobic residues at the interface. Molecular dynamics trajectories originating from two putative dimer structures indicate that the peptide dimer is stabilized primarily through hydrophobic interactions, while the conformations of the peptide monomers undergo substantial structural reorganization in the dimerization process. The finding that the phi-dimer may constitute the ensemble of stable Abeta(10-35) dimer has important implications for fibril formation. In particular, the expulsion of water molecules at the interface might be a key event, just as in the oligomerization of Abeta(16-22) fragments. We conjecture that events prior to the nucleation process themselves might involve crossing free energy barriers which depend on the peptide-peptide and peptide-water interactions. Consistent with existing experimental studies, the peptides within the ensemble of aggregated states show no signs of formation of secondary structure.
Barsegov, V. ; Thirumalai, D. Dynamics of unbinding of cell adhesion molecules: transition from catch to slip bonds. Proc Natl Acad Sci U S A 102, 1835-9.Abstract
The unbinding dynamics of complexes involving cell-adhesion molecules depends on the specific ligands. Atomic force microscopy measurements have shown that for the specific P-selectin-P-selectin glycoprotein ligand (sPSGL-1) the average bond lifetime t initially increases (catch bonds) at low (< or =10 pN) constant force, f, and decreases when f > 10 pN (slip bonds). In contrast, for the complex with G1 anti-P-selectin monoclonal antibody t monotonically decreases with f. To quantitatively map the energy landscape of such complexes we use a model that considers the possibility of redistribution of population from one force-free state to another force-stabilized bound state. The excellent agreement between theory and experiments allows us to extract energy landscape parameters by fitting the calculated curves to the lifetime measurements for both sPSGL-1 and G1. Surprisingly, the unbinding transition state for P-selectin-G1 complex is close (0.32 nm) to the bound state, implying that the interaction is brittle, i.e., once deformed, the complex fractures. In contrast, the unbinding transition state of the P-selectin-sPSGL-1 complex is far (approximately 1.5 nm) from the bound state, indicative of a compliant structure. Constant f energy landscape parameters are used to compute the distributions of unbinding times and unbinding forces as a function of the loading rate, rf. For a given rf, unbinding of sPSGL-1 occurs over a broader range of f with the most probable f being an order of magnitude less than for G1. The theory for cell adhesion complexes can be used to predict the outcomes of unbinding of other protein-protein complexes.
Dima, R. I. ; Hyeon, C. ; Thirumalai, D. Extracting stacking interaction parameters for RNA from the data set of native structures. J Mol Biol 347, 53-69.Abstract
A crucial step in the determination of the three-dimensional native structures of RNA is the prediction of their secondary structures, which are stable independent of the tertiary fold. Accurate prediction of the secondary structure requires context-dependent estimates of the interaction parameters. We have exploited the growing database of natively folded RNA structures in the Protein Data Bank (PDB) to obtain stacking interaction parameters using a knowledge-based approach. Remarkably, the calculated values of the resulting statistical potentials (SPs) are in excellent agreement with the parameters determined using measurements in small oligonucleotides. We validate the SPs by predicting 74% of the base-pairs in a dataset of structures using the ViennaRNA package. Interestingly, this number is similar to that obtained using the measured thermodynamic parameters. We also tested the efficacy of the SP in predicting secondary structure by using gapless threading, which we advocate as an alternative method for rapidly predicting RNA structures. For RNA molecules with less than 700 nucleotides, about 70% of the native base-pairs are correctly predicted. As a further validation of the SPs we calculated Z-scores, which measure the relative stability of the native state with respect to a manifold of higher free energy states. The computed Z-scores agree with estimates made using calorimetric measurements for a few RNA molecules. Structural analysis was used to rationalize the success and failures of SP and experimentally determined parameters. First, from the near perfect linear relationship between the number of native base-pairs and sequence length, we show that nearly 46% of nucleotides are not in stacks. Second, by analyzing the suboptimal structures that are generated in gapless threading we show that the SPs and experimentally determined parameters are most successful in predicting stacks that end in hairpins. These results show that further improvement in secondary structure prediction requires reliable estimates of interaction parameters for loops, bulges, and stacks that do not end in hairpins.
Hyeon, C. ; Thirumalai, D. Mechanical unfolding of RNA hairpins. Proc Natl Acad Sci U S A 102, 6789-94.Abstract
Mechanical unfolding trajectories, generated by applying constant force in optical-tweezer experiments, show that RNA hairpins and the P5abc subdomain of the group I intron unfold reversibly. We use coarse-grained Go-like models for RNA hairpins to explore forced unfolding over a broad range of temperatures. A number of predictions that are amenable to experimental tests are made. At the critical force, the hairpin jumps between folded and unfolded conformations without populating any discernible intermediates. The phase diagram in the force-temperature (f, T) plane shows that the hairpin unfolds by an all-or-none process. The cooperativity of the unfolding transition increases dramatically at low temperatures. Free energy of stability, obtained from time averages of mechanical unfolding trajectories, coincides with ensemble averages, which establishes ergodicity. The hopping time between the native basin of attraction (NBA) and the unfolded basin increases dramatically along the phase boundary. Thermal unfolding is stochastic, whereas mechanical unfolding occurs in "quantized steps" with great variations in the step lengths. Refolding times, upon force quench, from stretched states to the NBA are at least an order of magnitude greater than folding times by temperature quench. Upon force quench from stretched states, the NBA is reached in at least three stages. In the initial stages, the mean end-to-end distance decreases nearly continuously, and there is a sudden transition to the NBA only in the last stage. Because of the generality of the results, we propose that similar behavior should be observed in force quench refolding of proteins.
Cheung, M. S. ; Klimov, D. ; Thirumalai, D. Molecular crowding enhances native state stability and refolding rates of globular proteins. Proc Natl Acad Sci U S A 102, 4753-8.Abstract
The presence of macromolecules in cells geometrically restricts the available space for poplypeptide chains. To study the effects of macromolecular crowding on folding thermodynamics and kinetics, we used an off-lattice model of the all-beta-sheet WW domain in the presence of large spherical particles whose interaction with the polypeptide chain is purely repulsive. At all volume fractions, phi(c), of the crowding agents the stability of the native state is enhanced. Remarkably, the refolding rates, which are larger than the value at phi(c) = 0, increase nonmonotonically as phi(c) increases, reaching a maximum at phi(c)=phi(c)(*). At high values of phi(c), the depletion-induced intramolecular attraction produces compact structures with considerable structure in the denatured state. Changes in native state stability and folding kinetics at phi(c) can be quantitatively mapped onto confinement in a volume-fraction-dependent spherical pore with radius R(s) approximately (4pi/3phi(c))(1/3) R(c) (R(c) is the radius of the crowding particles) as long as phi(c)< or =phi(c)(*). We show that the extent of native state stabilization at finite phi(c) is comparable with that in a spherical pore. In both situations, rate enhancement is due to destabilization of the denatured states with respect to phi(c) = 0.
Zheng, W. ; Brooks, B. R. ; Doniach, S. ; Thirumalai, D. Network of dynamically important residues in the open/closed transition in polymerases is strongly conserved. Structure 13, 565-77.Abstract
The open/closed transition in polymerases is a crucial event in DNA replication and transcription. We hypothesize that the residues that transmit the signal for the open/closed transition are also strongly conserved. To identify the dynamically relevant residues, we use an elastic network model of polymerases and probe the residue-specific response to a local perturbation. In a variety of DNA/RNA polymerases, a network of residues spanning the fingers and palm domains is involved in the open/closed transition. The similarity in the network of residues responsible for large-scale domain movements supports the notion of a common induced-fit mechanism in the polymerase families for the formation of a closed ternary complex. Multiple sequence alignment shows that many of these residues are also strongly conserved. Residues with the largest sensitivity to local perturbations include those that are not so obviously involved in the polymerase catalysis. Our results suggest that mutations of the mechanical "hot spots" can compromise the efficiency of the enzyme.
Thiagarajan, V. ; Ramamurthy, P. ; Thirumalai, D. ; Ramakrishnan, V. T. A novel colorimetric and fluorescent chemosensor for anions involving PET and ICT pathways. Org Lett 7 657-60.Abstract
A novel colorimetric and fluorescent chemosensor ADDTU-1 bearing dual receptor sites, which shows specific optical signaling for AcO-, H2PO4-, and F- over other anions and dual response toward AcO- and F- via PET and ICT mechanisms, is described. [structure: see text]
Stan, G. ; Brooks, B. R. ; Thirumalai, D. Probing the "annealing" mechanism of GroEL minichaperone using molecular dynamics simulations. J Mol Biol 350, 817-29.Abstract
Although the intact chaperonin machinery is needed to rescue natural substrate proteins (SPs) under non-permissive conditions the "minichaperone" alone, containing only the isolated apical domain of GroEL, can assist folding of a certain class of proteins. To understand the annealing function of the minichaperone, we have carried out molecular dynamics simulations in the NPT ensemble totaling 300ns for four systems; namely, the isolated strongly binding peptide (SBP), the minichaperone, and the SBP and a weakly binding peptide (WBP) in complex with the minichaperone. The SBP, which is structureless in isolation, adopts a beta-hairpin conformation in complex with the minichaperone suggesting that favorable non-specific interactions of the SPs confined to helices H and I of the apical domains can induce local secondary structures. Comparison of the dynamical fluctuations of the apo and the liganded forms of the minichaperone shows that the stability (needed for SP capture) involves favorable hydrophobic interactions and hydrogen bond network formation between the SBP and WBP, and helices H and I. The release of the SP, which is required for the annealing action, involves water-mediated interactions of the charged residues at the ends of H and I helices. The simulation results are consistent with a transient binding release (TBR) model for the annealing action of the minichaperone. According to the TBR model, SP annealing occurs in two stages. In the first stage the SP is captured by the apical domain. This is followed by SP release (by thermal fluctuations) that places it in a different region of the energy landscape from which it can partition rapidly to the native state with probability Phi or be trapped in another misfolded state. The process of binding and release can result in enhancement of the native state yield. The TBR model suggests "that any cofactor that can repeatedly bind and release SPs can be effective in assisting protein folding." By comparing the structures of the non-chaperone alpha-casein (which has no sequence similarity with the apical domain) and the minichaperone and the hydrophobicity profiles we show that alpha-casein has a pair of helices that have similar sequence and structural profiles as H and I. Based on this comparison we identify residues that stabilize (destabilize) alpha-casein-protein complexes. This suggests that alpha-casein assists folding by the TBR mechanism.
Thirumalai, D. ; Hyeon, C. RNA and protein folding: common themes and variations. Biochemistry 44, 4957-70.Abstract
Visualizing the navigation of an ensemble of unfolded molecules through the bumpy energy landscape in search of the native state gives a pictorial view of biomolecular folding. This picture, when combined with concepts in polymer theory, provides a unified theory of RNA and protein folding. Just as for proteins, the major folding free energy barrier for RNA scales sublinearly with the number of nucleotides, which allows us to extract the elusive prefactor for RNA folding. Several folding scenarios can be anticipated by considering variations in the energy landscape that depend on sequence, native topology, and external conditions. RNA and protein folding mechanism can be described by the kinetic partitioning mechanism (KPM) according to which a fraction (Phi) of molecules reaches the native state directly, whereas the remaining fraction gets kinetically trapped in metastable conformations. For two-state folders Phi approximately 1. Molecular chaperones are recruited to assist protein folding whenever Phi is small. We show that the iterative annealing mechanism, introduced to describe chaperonin-mediated folding, can be generalized to understand protein-assisted RNA folding. The major differences between the folding of proteins and RNA arise in the early stages of folding. For RNA, folding can only begin after the polyelectrolyte problem is solved, whereas protein collapse requires burial of hydrophobic residues. Cross-fertilization of ideas between the two fields should lead to an understanding of how RNA and proteins solve their folding problems.
Chauhan, S. ; Caliskan, G. ; Briber, R. M. ; Perez-Salas, U. ; Rangan, P. ; Thirumalai, D. ; Woodson, S. A. RNA tertiary interactions mediate native collapse of a bacterial group I ribozyme. J Mol Biol 353, 1199-209.Abstract
Large RNAs collapse into compact intermediates in the presence of counterions before folding to the native state. We previously found that collapse of a bacterial group I ribozyme correlates with the formation of helices within the ribozyme core, but occurs at Mg2+ concentrations too low to support stable tertiary structure and catalytic activity. Here, using small-angle X-ray scattering, we show that Mg2+-induced collapse is a cooperative folding transition that can be fit by a two-state model. The Mg2+ dependence of collapse is similar to the Mg2+ dependence of helix assembly measured by partial ribonuclease T1 digestion and of an unfolding transition measured by UV hypochromicity. The correspondence between multiple probes of RNA structure further supports a two-state model. A mutation that disrupts tertiary contacts between the L9 tetraloop and its helical receptor destabilized the compact state by 0.8 kcal/mol, while mutations in the central triplex were less destabilizing. These results show that native tertiary interactions stabilize the compact folding intermediates under conditions in which the RNA backbone remains accessible to solvent.
Morrison, G. ; Thirumalai, D. The shape of a flexible polymer in a cylindrical pore. J Chem Phys 122, 194907.Abstract
We calculate the mean end-to-end distance R of a self-avoiding polymer encapsulated in an infinitely long cylinder with radius D. A self-consistent perturbation theory is used to calculate R as a function of D for impenetrable hard walls and soft walls. In both cases, R obeys the predicted scaling behavior in the limit of large and small D. The crossover from the three-dimensional behavior (D --> infinity) to the fully stretched one-dimensional case (D --> 0) is nonmonotonic. The minimum value of R is found at D approximately 0.46R(F), where R(F) is the Flory radius of R at D --> infinity. The results for soft walls map onto the hard wall case with a larger cylinder radius.
Dunlavy, D. M. ; O'Leary, D. P. ; Klimov, D. ; Thirumalai, D. HOPE: a homotopy optimization method for protein structure prediction. J Comput Biol 12, 1275-88.Abstract
We use a homotopy optimization method, HOPE, to minimize the potential energy associated with a protein model. The method uses the minimum energy conformation of one protein as a template to predict the lowest energy structure of a query sequence. This objective is achieved by following a path of conformations determined by a homotopy between the potential energy functions for the two proteins. Ensembles of solutions are produced by perturbing conformations along the path, increasing the likelihood of predicting correct structures. Successful results are presented for pairs of homologous proteins, where HOPE is compared to a variant of Newton's method and to simulated annealing.
Barsegov, V. ; Thirumalai, D. Probing protein-protein interactions by dynamic force correlation spectroscopy. Phys Rev Lett 95, 168302.Abstract
We develop a formalism for single molecule dynamic force spectroscopy to map the energy landscape of protein-protein complex (P(1)P(2)). The joint distribution P(tau(1),tau(2)) of unbinding lifetimes tau(1) and tau(2), measurable in a compression-tension cycle, which accounts for the internal relaxation dynamics of the proteins under tension, shows that the histogram of tau(1) is not Poissonian. The theory is applied to the forced unbinding of protein P1, modeled as a wormlike chain, from P(1)P(2). We propose a new class of experiments which can resolve the effect of internal protein dynamics on the unbinding lifetimes.
Ziv, G. ; Haran, G. ; Thirumalai, D. Ribosome exit tunnel can entropically stabilize alpha-helices. Proc Natl Acad Sci U S A 102, 18956-61.Abstract
Several experiments have suggested that newly synthesized polypeptide chains can adopt helical structures deep within the ribosome exit tunnel. We hypothesize that confinement in the roughly cylindrical tunnel can entropically stabilize alpha-helices. The hypothesis is validated by using theory and simulations of coarse-grained off-lattice models. The model helix, which is unstable in the bulk, is stabilized in a cylindrical cavity provided the diameter (D) of the cylinder exceeds a critical value D*. When D < D* both the helical content and the helix-coil transition temperature (T(f)) decrease abruptly. Surprisingly, we find that the stability of the alpha-helix depends on the number (N) of amino acid residues. Entropic stabilization, as measured by changes in T(f), increases nonlinearly as N increases. The simulation results are in quantitative agreement with a standard helix-coil theory that takes into account entropy cost of confining a polypeptide chain in a cylinder. The results of this work are in qualitative accord with most of the findings of a recent experiment in which N-dependent ribosome-induced helix stabilization of transmembrane sequences was measured by fluorescence resonance energy transfer.
Klimov, D. K. ; Thirumalai, D. Symmetric connectivity of secondary structure elements enhances the diversity of folding pathways. J Mol Biol 353, 1171-86.Abstract
The influence of native connectivity of secondary structure elements (SSE) on folding is studied using coarse-grained models of proteins with mixed alpha and beta structure and the analysis of the structural database of wild-type proteins. We found that the distribution of SSE along a sequence determines the diversity of folding pathways. If alpha and beta SSE are localized in different parts of a sequence, the diversity of folding pathways is restricted. An even (symmetric) distribution of alpha and beta SSE with respect to sequence midpoint favors multiple folding routes. Simulations are supplemented by the database analysis of the distribution of SSE in wild-type protein sequences. On an average, two-thirds of wild-type proteins with mixed alpha and beta structure have symmetric distribution of alpha and beta SSE. The propensity for symmetric distribution of SSE is especially evident for large proteins with the number of SSE > or = 10. We suggest that symmetric SSE distribution in protein sequences may arise due to nearly random allocation of alpha and beta structure along wild-type sequences. The tendency of long sequences to misfold is perhaps compensated by the enhanced pathway diversity. In addition, folding pathways are shown to progress via hierarchic assembly of SSE in accordance with their proximity along a sequence. We demonstrate that under mild denaturation conditions folding and unfolding pathways are similar. However, the reversibility of folding/unfolding pathways is shown to depend on the distribution of SSE. If alpha and beta SSE are localized in different parts of a sequence, folding and unfolding pathways are likely to coincide.
Barsegov, V. ; Thirumalai, D. Influence of surface interactions on folding and forced unbinding of semiflexible chains. J Phys Chem B 109, 21979-88.Abstract
We have investigated the folding and forced unbinding transitions of adsorbed semiflexible polymer chains using theory and simulations. These processes describe, at an elementary level, a number of biologically relevant phenomena that include adhesive interactions between proteins and tethering of receptors to cell walls. The binding interface is modeled as a solid surface, and the wormlike chain (WLC) is used for the semiflexible chain (SC). Using Langevin simulations, in the overdamped limit we examine the ordering kinetics of racquet-like and toroidal structures in the presence of an attractive interaction between the surface and the polymer chain. For a range of interactions, temperature, and the persistence length, l(p), we obtained the monomer density distribution, n(x), (x is the perpendicular distance of a tagged chain end from the surface) for all of the relevant morphologies. There is a single peak in n(x) inside the range of attractive forces, b, for chains in the extended conformations, whereas in racquet and toroidal structures there is an additional peak at x approximately b. The simulated results for n(x) are in good agreement with theory. The formation of toroids on the surface appears to be a first-order transition as evidenced by the bimodal distribution in n(x). The theoretical result underestimates the simulated n(x) for x < b and follows n(x) closely for x >/= b; the calculated density agrees exactly with n(x) in the range x < b. The chain-surface interaction is probed by subjecting the surface structures to a pulling force, f. The average extension, x( f), as a function of f exhibits a sigmoidal profile with sharp all-or-none transition at the unfolding force threshold f = f(c) which increases for more structured states. Simulated x(f) compare well with the theoretical predictions. The critical force, f(c), is a function of l(s)/l(c) for a fixed temperature, where l(c) and l(s) are the length scales that express the strength of the intramolecular and SC-surface attraction, respectively. For a fixed l(s), f(c) increases as l(p) decreases.
Caliskan, G. ; Hyeon, C. ; Perez-Salas, U. ; Briber, R. M. ; Woodson, S. A. ; Thirumalai, D. Persistence length changes dramatically as RNA folds. Phys Rev Lett 95, 268303.Abstract
We determine the persistence length l(p) for a bacterial group I ribozyme as a function of concentration of monovalent and divalent cations by fitting the distance distribution functions P(r) obtained from small angle x-ray scattering intensity data to the asymptotic form of the calculated P(WLC)(r) for a wormlike chain. The l(p) values change dramatically over a narrow range of Mg(2+) concentration from approximately 21 Angstroms in the unfolded state (U) to approximately 10 Angstroms in the compact (I(C)) and native states. Variations in l(p) with increasing Na(+) concentration are more gradual. In accord with the predictions of polyelectrolyte theory we find l(p) alpha 1/kappa(2) where kappa is the inverse Debye-screening length.