Order of steps in the cytochrome C folding pathway: evidence for a sequential stabilization mechanism

Previous work used hydrogen exchange (HX) experiments in kinetic and equilibrium modes to study the reversible unfolding and refolding of cytochrome c (Cyt c) under native conditions. Accumulated results now show that Cyt c is composed of five individually cooperative folding units, called foldons, which unfold and refold as concerted units in a stepwise pathway sequence. The first three steps of the folding pathway are linear and sequential. The ordering of the last two steps has been unclear because the fast HX of the amino acid residues in these foldons has made measurement difficult. New HX experiments done under slower exchange conditions show that the final two foldons do not unfold and refold in an obligatory sequence. They unfold separately and neither unfolding obligately contains the other, as indicated by their similar unfolding surface exposure and the specific effects of destabilizing and stabilizing mutations, pH change, and oxidation state. These results taken together support a sequential stabilization mechanism in which folding occurs in the native context with prior native-like structure serving to template the stepwise formation of subsequent native-like foldon units. Where the native structure of Cyt c requires sequential folding, in the first three steps, this is found. Where structural determination is ambiguous, in the final two steps, alternative parallel folding is found.     

Folding units govern the cytochrome c alkaline transition

The alkaline transition of cytochrome c is a model for protein structural switching in which the normal heme ligand is replaced by another group. Stopped flow data following a jump to high pH detect two slow kinetic phases, suggesting two rate-limiting structure changes. Results described here indicate that these events are controlled by the same structural unfolding reactions that account for the first two steps in the reversible unfolding pathway of cytochrome c. These and other results show that the cooperative folding-unfolding behavior of protein foldons can account for a variety of functional activities in addition to determining folding pathways.     

Branching in the sequential folding pathway of cytochrome c

Previous results indicate that the folding pathways of cytochrome c and other proteins progressively build the target native protein in a predetermined stepwise manner by the sequential formation and association of native-like foldon units. The present work used native state hydrogen exchange methods to investigate a structural anomaly in cytochrome c results that suggested the concerted folding of two segments that have little structural relationship in the native protein. The results show that the two segments, an 18-residue omega loop and a 10-residue helix, are able to unfold and refold independently, which allows a branch point in the folding pathway. The pathway that emerges assembles native-like foldon units in a linear sequential manner when prior native-like structure can template a single subsequent foldon, and optional pathway branching is seen when prior structure is able to support the folding of two different foldons.     

The foldon substructure of staphylococcal nuclease

To search for submolecular foldon units, the spontaneous reversible unfolding and refolding of staphylococcal nuclease under native conditions was studied by a kinetic native-state hydrogen exchange (HX) method. As for other proteins, it appears that staphylococcal nuclease is designed as an assembly of well-integrated foldon units that may define steps in its folding pathway and may regulate some other functional properties. The HX results identify 34 amide hydrogens that exchange with solvent hydrogens under native conditions by way of large transient unfolding reactions. The HX data for each hydrogen measure the equilibrium stability (ΔG_HX) and the kinetic unfolding and refolding rates (k_op and k_cl) of the unfolding reaction that exposes it to exchange. These parameters separate the 34 identified residues into three distinct HX groupings. Two correspond to clearly defined structural units in the native protein, termed the blue and red foldons. The remaining HX grouping contains residues, not well separated by their HX parameters alone, that represent two other distinct structural units in the native protein, termed the green and yellow foldons. Among these four sets, a last unfolding foldon (blue) unfolds with a rate constant of 6 × 10^− 6 s^− 1 and free energy equal to the protein's global stability (10.0 kcal/mol). It represents part of the β-barrel, including mutually H-bonding residues in the β4 and β5 strands, a part of the β3 strand that H-bonds to β5, and residues at the N-terminus of the α2 helix that is capped by β5. A second foldon (green), which unfolds and refolds more rapidly and at slightly lower free energy, includes residues that define the rest of the native α2 helix and its C-terminal cap. A third foldon (yellow) defines the mutually H-bonded β1-β2-β3 meander, completing the native β-barrel, plus an adjacent part of the α1 helix. A final foldon (red) includes residues on remaining segments that are distant in sequence but nearly adjacent in the native protein. Although the structure of the partially unfolded forms closely mimics the native organization, four residues indicate the presence of some nonnative misfolding interactions. Because the unfolding parameters of many other residues are not determined, it seems likely that the concerted foldon units are more extensive than is shown by the 34 residues actually observed.     

Protein folding and misfolding: mechanism and principles

Two fundamentally different views of how proteins fold are now being debated. Do proteins fold through multiple unpredictable routes directed only by the energetically downhill nature of the folding landscape or do they fold through specific intermediates in a defined pathway that systematically puts predetermined pieces of the target native protein into place? It has now become possible to determine the structure of protein folding intermediates, evaluate their equilibrium and kinetic parameters, and establish their pathway relationships. Results obtained for many proteins have serendipitously revealed a new dimension of protein structure. Cooperative structural units of the native protein, called foldons, unfold and refold repeatedly even under native conditions. Much evidence obtained by hydrogen exchange and other methods now indicates that cooperative foldon units and not individual amino acids account for the unit steps in protein folding pathways. The formation of foldons and their ordered pathway assembly systematically puts native-like foldon building blocks into place, guided by a sequential stabilization mechanism in which prior native-like structure templates the formation of incoming foldons with complementary structure. Thus the same propensities and interactions that specify the final native state, encoded in the amino-acid sequence of every protein, determine the pathway for getting there. Experimental observations that have been interpreted differently, in terms of multiple independent pathways, appear to be due to chance misfolding errors that cause different population fractions to block at different pathway points, populate different pathway intermediates, and fold at different rates. This paper summarizes the experimental basis for these three determining principles and their consequences. Cooperative native-like foldon units and the sequential stabilization process together generate predetermined stepwise pathways. Optional misfolding errors are responsible for 3-state and heterogeneous kinetic folding.     

Comparison between denaturant- and temperature-induced unfolding pathways of protein: a lattice Monte Carlo simulation

Denaturant-induced unfolding of protein is simulated by using a Monte Carlo simulation with a lattice model for protein and denaturant. Following the binding theory for denaturant-induced unfolding, the denaturant molecules are modeled to interact with protein by nearest-neighbor interactions. By analyzing the conformational states on the unfolding pathway of protein, the denaturant-induced unfolding pathway is compared with the temperature-induced unfolding pathway under the same condition; that is, the free energies of unfolding under two different pathways are equal. The two unfoldings show markedly different conformational distributions in unfolded states. From the calculation of the free energy of protein as a function of the number fraction (Q0) of native contacts relative to the total number of contacts, it is found that the free energy of the largely unfolded state corresponding to low Q0 (0.1 < Q0 < 0.5) under temperature-induced unfolding is lower than that under denaturant-induced unfolding, whereas the free energy of the unfolded state close to the native state (Q0 > 0.5) is lower in denaturant-induced unfolding than in temperature-induced unfolding. A comparison of two unfolding pathways reveals that the denaturant-induced unfolding shows a wider conformational distribution than the temperature-induced unfolding, while the temperature-induced unfolding shows a more compact unfolded state than the denaturant-induced unfolding especially in the low Q0 region (0.1 < Q0 < 0.5).     

Elucidating quantitative stability/flexibility relationships within thioredoxin and its fragments using a distance constraint model

Numerous quantitative stability/flexibility relationships, within Escherichia coli thioredoxin (Trx) and its fragments are determined using a minimal distance constraint model (DCM). A one-dimensional free energy landscape as a function of global flexibility reveals Trx to fold in a low-barrier two-state process, with a voluminous transition state. Near the folding transition temperature, the native free energy basin is markedly skewed to allow partial unfolded forms. Under native conditions the skewed shape is lost, and the protein forms a compact structure with some flexibility. Predictions on ten Trx fragments are generally consistent with experimental observations that they are disordered, and that complementary fragments reconstitute. A hierarchical unfolding pathway is uncovered using an exhaustive computational procedure of breaking interfacial cross-linking hydrogen bonds that span over a series of fragment dissociations. The unfolding pathway leads to a stable core structure (residues 22-90), predicted to act as a kinetic trap. Direct connection between degree of rigidity within molecular structure and non-additivity of free energy is demonstrated using a thermodynamic cycle involving fragments and their hierarchical unfolding pathway. Additionally, the model provides insight about molecular cooperativity within Trx in its native state, and about intermediate states populating the folding/unfolding pathways. Native state cooperativity correlation plots highlight several flexibly correlated regions, giving insight into the catalytic mechanism that facilitates access to the active site disulfide bond. Residual native cooperativity correlations are present in the core substructure, suggesting that Trx can function when it is partly unfolded. This natively disordered kinetic trap, interpreted as a molten globule, has a wide temperature range of metastability, and it is identified as the "slow intermediate state" observed in kinetic experiments. These computational results are found to be in overall agreement with a large array of experimental data.     

Using an amino acid fluorescence resonance energy transfer pair to probe protein unfolding: application to the villin headpiece subdomain and the LysM domain

Previously, we have shown that p-cyanophenylalanine (Phe CN) and tryptophan (Trp) constitute an efficient fluorescence resonance energy transfer (FRET) pair that has several advantages over commonly used dye pairs. Here, we aim to examine the general applicability of this FRET pair in protein folding-unfolding studies by applying it to the urea-induced unfolding transitions of two small proteins, the villin headpiece subdomain (HP35) and the lysin motif (LysM) domain. Depending on whether Phe CN is exposed to solvent, we are able to extract either qualitative information about the folding pathway, as demonstrated by HP35, which has been suggested to unfold in a stepwise manner, or quantitative thermodynamic and structural information, as demonstrated by LysM, which has been shown to be an ideal two-state folder. Our results show that the unfolding transition of HP35 reported by FRET occurs at a denaturant concentration lower than that measured by circular dichroism (CD) and that the loop linking helix 2 and helix 3 remains compact in the denatured state, which are consistent with the notion that HP35 unfolds in discrete steps and that its unfolded state contains residual structures. On the other hand, our FRET results on the LysM domain allow us to develop a model for extracting structural and thermodynamic parameters about its unfolding, and we find that our results are in agreement with those obtained by other methods. Given the fact that Phe CN is a non-natural amino acid and, thus, amenable to incorporation into peptides and proteins via existing peptide synthesis and protein expression methods, we believe that the FRET method demonstrated here is widely applicable to protein conformational studies, especially to the study of relatively small proteins.     

Temperature-dependent downhill unfolding of ubiquitin. II. Modeling the free energy surface

To provide evidence for the interpretation of temperature‐dependent unfolding kinetics and the downhill unfolding scenario presented in the accompanying experimental article (Part I), the free energy surface of ubiquitin unfolding is calculated using statistical mechanical models of the Muñoz‐Eaton (ME) form. The models allow only two states for each amino acid residue, folded or unfolded, and permutations of these states generate an ensemble of microstates. One‐dimensional free energy curves are calculated using the number of folded residues as a reaction coordinate. The proposed sequential unfolding of ubiquitin's β‐sheet is tested by mapping the free energy onto two reaction coordinates inspired by the experiment as follows: the number of folded residues in ubiquitin's stable β‐strands I and II and those of the less stable strands III–V. Although the original ME model successfully captures folding features of zipper‐like one‐dimensional folders, it misses important tertiary interactions between residues that are far from each other in primary sequence. To take tertiary contacts into account, partially folded microstates based on a spherical growth model are included in the calculation and compared with the original model. By calculating the folding probability of each residue for a given point on the free energy surface, the unfolding pathway of ubiquitin is visualized. At low temperature, thermal unfolding occurs along a sequential unfolding pathway as follows: disruption of the β‐strands III–V followed by unfolding of the strands I and II. At high temperature, multiple unfolding routes are formed. The heterogeneity of the transition state explains the global nonexponential unfolding observed in the T‐jump experiment at high temperature. The calculation also reports a high stability for the α‐helix of ubiquitin. Proteins 2008. © 2008 Wiley‐Liss, Inc.     

Unfolding of the loggerhead sea turtle (Caretta caretta) myoglobin: A (1)H-NMR and electronic absorbance study

The effect of urea concentration on the backbone solution structure of the cyanide derivative of ferric Caretta caretta myoglobin (at pH 5.4) is reported. By addition of urea, sequential and long-range nuclear Overhauser effects (NOEs) are gradually lost. By using the residual NOE constraints to build the molecular model, a picture of the unfolding pathway was obtained. When the urea concentration is raised to 2.2 M, helices A and B appear largely disordered; helices C, D, and F loose structural constraints at 3.0 M urea. At urea concentration >6 M, the protein appears to be fully unfolded, including the GH hairpin and helix E stabilizing the prosthetic group. Reversible and cooperative denaturation isotherms obtained by following NOE peaks are considerably different from those obtained by monitoring electronic absorption changes. The reversible and cooperative urea-dependent folding-unfolding process of C. caretta myoglobin follows the minimum three-state mechanism N long left and right arrow X long left and right arrow D, where X represents a disordered globin structure (occurring at approximately 4 M urea) that still binds the heme.     

Proteolysis as a measure of the free energy difference between cytochrome c and its derivatives.

Limited cleavage of oxidized and reduced horse heart cytochrome c (Cyt c) and the azide complex of Cyt c by proteinase K at room temperature yields a single cut within the central loop (36-60 in the sequence). Using an assay that allows spectroscopic evaluation of the fraction of intact protein as a function of time, together with a simple kinetic model for proteolysis, fluctuation opening of the loop can be related to the free energy of the corresponding protein. This allows us to estimate quantitatively the free energy difference between the oxidized form of Cyt c and other states using proteolysis as a probe. The results we obtain indicate that oxidized Cyt c is 2.0 kcal mol(-1) less stable than the reduced form, and 0.07 kcal mol(-1) is more stable than the Cyt c: azide complex at 25 degrees C. These values agree in magnitude with results from hydrogen exchange and unfolding studies, suggesting that the stability of a protein can be directly related to its structural dynamics.     

Discrete Kinetic Models from Funneled Energy Landscape Simulations

A general method for facilitating the interpretation of computer simulations of protein folding with minimally frustrated energy landscapes is detailed and applied to a designed ankyrin repeat protein (4ANK). In the method, groups of residues are assigned to foldons and these foldons are used to map the conformational space of the protein onto a set of discrete macrobasins. The free energies of the individual macrobasins are then calculated, informing practical kinetic analysis. Two simple assumptions about the universality of the rate for downhill transitions between macrobasins and the natural local connectivity between macrobasins lead to a scheme for predicting overall folding and unfolding rates, generating chevron plots under varying thermodynamic conditions, and inferring dominant kinetic folding pathways. To illustrate the approach, free energies of macrobasins were calculated from biased simulations of a non-additive structure-based model using two structurally motivated foldon definitions at the full and half ankyrin repeat resolutions. The calculated chevrons have features consistent with those measured in stopped flow chemical denaturation experiments. The dominant inferred folding pathway has an “inside-out”, nucleation-propagation like character.     

Two structural subdomains of barstar detected by rapid mixing NMR measurement of amide hydrogen exchange

Equilibrium amide hydrogen exchange studies of barstar have been carried out at pH 6.7, 32° SDC using one- and two-dimensional nuclear magnetic resonance. An unusually large fraction of the backbone amide hydrogens of barstar exchange too fast to be measured, and the exchange rates of only fifteen slow-exchanging amide sites including indole amides of two tryptophans could be measured in the presence of 0 to 1.8 M guanidine hydrochloride (GdnHCl). Measurement of exchange occurring in tens of seconds in the unfolding transition region was possible by the use of a fast stopped-flow mixing method. The observed exchange rates have been simulated in the EX2 limit according to a two-process model that incorporates two exchange-competent states: a transiently unfolded state (U*) in which many amide hydrogens are completely accessible to solvent-exchange, and a near-native locally unfolded state (N*), in which only one or a few amide hydrogens are completely accessible to solvent-exchange. The two-process model appears to account for the observed exchange behavior over the entire range of GdnHCl concentrations studied. For several measurable slow-exchanging amide hydrogens, the free energies of production of exchange-competent states from the exchange-incompetent native state are significantly higher than the free-energy of production of the equilibrium unfolded state from the native state, when the latter is determined from circular dichroism- or fluorescence-monitored equilibrium unfolding curves. The result implies that U*, which forms transiently in the strongly native-like conditions used for the hydrogen exchange studies, is higher in energy than the equilibrium-unfolded state. The higher energy of this transiently unfolded exchange-competent state can be attributed to either proline isomerization or to the presence of residual structure. On the basis of the free energies of production of exchange-competent states, the measured amide sites of barstar appear to define two structural subdomains—a three-helix unit and a two-β-strand unit in the core of the protein. Proteins 30:295–308, 1998. © 1998 Wiley-Liss, Inc.     

Mapping the energy landscape of repeat proteins using NMR-detected hydrogen exchange

Repeat proteins contain tandem arrays of a simple structural motif. In contrast to globular proteins, repeat proteins are stabilized only by interactions between residues that are relatively close together in the sequence, with no ”long-range” interactions. Our work focuses on the tetratricopeptide repeat (TPR), a 34 amino acid helix-turn-helix motif found in tandem arrays in many natural proteins. Earlier, we reported the design and characterization of a series of consensus TPR (CTPR) proteins, which are built as arrays of multiple tandem copies of a 34 amino acid consensus sequence. Here, we present the results of extensive hydrogen exchange (HX) studies of the folding-unfolding behavior of two CTPR proteins (CTPR2 and CTPR3). We used HX to detect and characterize partially folded species that are populated at low frequency in the nominally folded state. We show that for both proteins the equilibrium folding-unfolding transition is non-two-state, but sequential, with the outermost helices showing a significantly higher probability than inner helices of being unfolded. We show that the experimentally observed unfolding behavior is consistent with the predictions of a simple Ising model, in which individual helices are treated as ”spin-equivalents”. The results that we present have general implications for our understanding of the thermodynamic properties of repeat proteins.     

Free energy landscape and folding mechanism of a beta-hairpin in explicit water: a replica exchange molecular dynamics study

The free energy landscape and the folding mechanism of the C-terminal beta-hairpin of protein G is studied by extensive replica exchange molecular dynamics simulations (40 replicas and 340 ns total simulation time), using the GROMOS96 force field and the SPC explicit water solvent. The study reveals that the system preferentially adopts a beta-hairpin structure at biologically important temperatures, and that the helix content is low at all temperatures studied. Representing the free energy landscape as a function of several types of reaction coordinates, four local minima corresponding to the folded, partially folded, molten globule, and unfolded states are identified. The findings suggest that the folding of the beta-hairpin occurs as the sequence: collapse of hydrophobic core --> formation of H-bond --> formation of the turn. Identifying the folded and molten globule states as the main conformations, the free energy landscape of the beta-hairpin is consistent with a two-state behavior with a broad transition state. The temperature dependence of the folding-unfolding transition is investigated in some detail. The enthalpy and entropy jumps at the folding transition temperature are found to be about three times lower than the experimental estimates, indicating that the folding-unfolding transition in silico is less cooperative than its in vitro counterpart.     

Conformational states of HRPA1 induced by thermal unfolding: effect of low molecular weight solutes

Fluorescence, CD, and activity measurements were used to characterize the different conformational states of horseradish peroxidase A1 induced by thermal unfolding. Picosecond time-resolved fluorescence studies showed a three-exponential decay dominated by a picosecond lifetime component resulting from energy transfer from tryptophan to heme. Upon thermal unfolding a decrease in the preexponential factor of the picosecond lifetime and an increase in the quantum yield were observed approaching the characteristics observed for apoHRPA1. The fraction of heme-quenched fluorophore decreased to 0.4 after unfolding as shown by acrylamide quenching. A new unfolding pathway for HRPA1 was proposed and the effect of the low molecular weight solutes trehalose, sorbitol, and melezitose on this pathway was analyzed. Native HRPA1 unfolds with an intermediate between the native and the unfolded conformation. The unfolded conformation can refold to the native state or to a native-like conformation with no calcium ions upon cooling or can give an irreversible denatured state. The refolded conformation with no calcium ions was clearly identified in a second thermal scan in the presence of EDTA and shows secondary and tertiary structures, heme reincorporation in the cavity, and at least 59% of activity. Melezitose stabilized the refolded Ca2+-depleted protein and induced a more complex mechanism for heme disruption. The effect of sorbitol and trehalose were mainly characterized by an increase in the temperature of unfolding.     

THE ROLES OF PLANT HORMONES IN STYLE AND STIGMA GROWTH IN GAILLARDIA GRANDIFLORA (ASTERACEAE)

Style and stigma elongation and stigma unfolding, and the roles of plant hormones in these processes in Gaillardia grandiflora Van Houtte were investigated. Style and stigma elongation in vivo began just after anthesis, and style elongation was accompanied by epidermal cell elongation (greatest near the stigma) and a fresh weight increase, but not by cell division or a dry weight increase. The stigma unfolded after the style and stigma elongated. Style-stigma units excised from young disc flowers of this composite were measured as they responded to plant growth regulators applied singly, as well as in sequential and simultaneous combinations, in vitro. Style elongation was promoted by auxin, was inhibited by gibberellins and ethylene, and was unaffected by other growth regulators. Stigma elongation followed a similar pattern of response. Endogenous auxin levels and ethylene production showed parallel variation and endogenous gibberellin levels showed inverse variation with style and stigma elongation. Stigma unfolding was more sensitive to auxin applications and was promoted by applied ethylene. Ethylene production showed parallel variation and endogenous auxin levels showed inverse variation with stigma unfolding. AVG and Co²⁺ applications decreased auxin-induced style elongation and fusicoccin promoted all of the growth responses of style-stigma units in vitro. A gibberellin-auxin-ethylene-acid growth interaction mode of control is proposed for these three growth processes.     

Nonadditive Interactions in Protein Folding: The Zipper Model of Cytochrome <Emphasis Type="Italic">c</Emphasis>

Hydrogen exchange experiments (Krishna et al. in J. Mol. Biol. 359:1410, 2006) reveal that folding–unfolding of cytochrome c occurs along a defined pathway in a sequential, stepwise manner. The simplified zipper-like model involving nonadditive coupling is proposed to describe the classical “on pathway” folding–unfolding behavior of cytochrome c. Using free energy factors extracted from HX experiments, the model can predict and explain cytochrome c behavior in spectroscopy studies looking at folding equilibria and kinetics. The implications of the proposed model are discussed for such problems as classical pathway vs. energy landscape conceptions, structure and function of a native fold, and interplay of secondary and tertiary interactions.     

Equilibrium unfolding studies of monellin: the double-chain variant appears to be more stable than the single-chain variant

To improve our understanding of the contributions of different stabilizing interactions to protein stability, including that of residual structure in the unfolded state, the small sweet protein monellin has been studied in both its two variant forms, the naturally occurring double-chain variant (dcMN) and the artificially created single-chain variant (scMN). Equilibrium guanidine hydrochloride-induced unfolding studies at pH 7 show that the standard free energy of unfolding, ΔG°(U), of dcMN to unfolded chains A and B and its dependence on guanidine hydrochloride (GdnHCl) concentration are both independent of protein concentration, while the midpoint of unfolding has an exponential dependence on protein concentration. Hence, the unfolding of dcMN like that of scMN can be described as two-state unfolding. The free energy of dissociation, ΔG°(d), of the two free chains, A and B, from dcMN, as measured by equilibrium binding studies, is significantly lower than ΔG°(U), apparently because of the presence of residual structure in free chain B. The value of ΔG°(U), at the standard concentration of 1 M, is found to be ～5.5 kcal mol(-1) higher for dcMN than for scMN in the range from pH 4 to 9, over which unfolding appears to be two-state. Hence, dcMN appears to be more stable than scMN. It seems that unfolded scMN is stabilized by residual structure that is absent in unfolded dcMN and/or that native scMN is destabilized by strain that is relieved in native dcMN. The value of ΔG°(U) for both protein variants decreases with an increase in pH from 4 to 9, apparently because of the thermodynamic coupling of unfolding to the protonation of a buried carboxylate side chain whose pK(a) shifts from 4.5 in the unfolded state to 9 in the native state. Finally, it is shown that although the thermodynamic stabilities of dcMN and scMN are very different, their kinetic stabilities with respect to unfolding in GdnHCl are very similar.     

Force-induced unfolding of human telomeric G-quadruplex: A steered molecular dynamics simulation study

We study the unfolding of a parallel G-quadruplex from human telomeric DNA by mechanical stretching using steered molecular dynamics (MD) simulation. We find that the force curves and unfolding processes strongly depend on the pulling sites. With pulling sites located on the sugar-phosphate backbone, the force-extension curve shows a single peak and the unfolding proceeds sequentially. Pulling sites located on the terminal nucleobases lead to a force-extension curve with two peaks and the unfolding is more cooperative. Simulations of the refolding of partially unfolded quadruplexes show very different behavior for the two different pulling modalities. In particular, starting from an unfolded state prepared by nucleobase pulling leads to a long-lived intermediate state whose existence is also corroborated by the free energy profile computed with the Jarzynski equation. Based on this observation, we propose a novel folding pathway for parallel G-quadruplexes with the human telomere sequence.