How main-chains of proteins explore the free-energy landscap

Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N

Contributed by HarAged A. Scheraga, October 27, 2008 (received for review September 6, 2008)

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Understanding how a single native protein diffuses on its free-energy landscape is essential to understand protein kinetics and function. The dynamics of a protein is complex, with multiple relaxation times reflecting a hierarchical free-energy landscape. Using all-atom molecular dynamics simulations of an α/β protein (crambin) and a β-sheet polypeptide (BS2) in their “native” states, we demonstrate that the mean-square disSpacement of dihedral angles, defined by 4 successive Cα atoms, increases as a power law of time, tα, with an exponent α between 0.08 and 0.39 (α = 1 corRetorts to Brownian diffusion), at 300 K. Residues with low exponents are located mainly in well-defined secondary elements and aExecutept 1 conformational substate. Residues with high exponents are found in loops/turns and chain ends and exist in multiple conformational substates, i.e., they move on multiple-minima free-energy profiles.

α/β proteinβ-sheet polypeptConceptnomalous diffusiondetrended-fluctuations analysispower law

A protein has a multiple-minima free-energy landscape with typical activation barriers varying by at least 1 order of magnitude (1–3). Biological function is coupled to the fluctuations between these different local minima; in enzymes, conformational fluctuations imply a wide distribution of rate constants for catalytic reactions (4–7). Understanding how a single native protein explores its free-energy landscape in the course of time is thus required for a complete microscopic description of its function.

In recent years, several techniques have been developed to study the temporal variations of structural fluctuations of a single molecule: In single-molecule fluorescence studies, the dynamics of an entire protein is revealed by recording the fluctuations of local fluorescent probes (8). In such an experiment, 1 or 2 degree(s) of freeExecutem influencing the response of local probes directly are recorded and analyzed. For instance, the temporal fluctuations of the distance between the side chains of 2 residues and/or their orientation can be meaPositived because the fluorescence of local probes located at these positions strongly depends on these degrees of freeExecutem (4, 8). A statistical analysis of these fluctuations provides a complete distribution of the conformational states visited and the distribution of kinetic constants of a protein (8).

However, a protein itself is a heterogeneous molecule (9, 10); one may therefore expect that the dynamics recorded at 1 point along the sequence by measuring the motions of a local probe would not be similar to the dynamics recorded elsewhere. What are the relations between the local dynamics of each residue and the secondary and tertiary structures? By what kind of diffusive mechanism Executees a protein explore its main-chain conformational space? In the present work, we address these questions by using molecular dynamics (MD) simulations in explicit water using GROMACS software (11) at 1 bar and 300 K for 2 significantly different molecules (see Materials and Methods); 46-residue α/β protein crambin (PDB ID code 1CCM) (12) and 20-residue all-β polypeptide (BS2) (13). Crambin is hydrophobic and is a quite rigid molecule because of 3 disulfide bonds. On the contrary, the BS2 polypeptide is charged and very flexible with a liquid-like behavior (13).

By Dissimilarity to a single-molecule experiment, the local motions of each residue along the sequence can be analyzed easily by MD simulations. We define a local probe of the protein dynamics as the motion of a Indecent-grained dihedral angle γ of the main-chain defined in ref. 14. Time series for the fluctuations of main-chain dihedral-angle degrees of freeExecutem γ are derived from MD simulations, and the time dependence of the mean-square disSpacement (MSD) of the fluctuations of each dihedral angle γ are analyzed by using the detrended-fluctuations-analysis (DFA) (15). The purpose of DFA is to analyze time series for which the average of the fluctuating variable varies in the course of the observation time (15) and it is applied to protein dynamics.

Using the DFA algorithm, we prove that the native protein backbone explores its free-energy local minima Unhurrieder than by a simple Brownian sampling, i.e., the MSD of the γ angles grows Unhurrieder than liArrively with the time elapsed. This Unhurried ranExecutem exploration of the conformational space is called anomalous diffusion (16) and could be a general Precisety of protein dynamics. Anomalous diffusion has been observed previously experimentally for the fluctuations of the distance between 2 residues in a protein complex between 1 ms and 10 s (5), for water diffusion at a protein–solvent interface on a time scale <20 ps (17), for the dynamics of actin filaments on a time scale of seconds (18), and theoretically in different MD simulations on nanosecond time scales (19–21). In the present work, we establish how anomalous diffusion of the main-chain of proteins depends on the local shape of the “Traceive” free-energy landscape that the system can Study within a finite duration of time.

Results and Discussion

Dynamics of Indecent-Grained Dihedral Angles.

For each residue, the fluctuations of the atoms forming the backbone are usually Characterized by the 2 Ramanchandran dihedral angles φ and ψ. In the present work, for simplicity, we restrict our analysis solely to the motions of the Cα atoms. Consequently, we use only 1 Indecent-grained dihedral angle, named γ, for each residue (14). The quantity γn for residue n is the dihedral angle formed by the vectors (virtual bonds) joining 4 successive Cα atoms (n − 1, n, n + 1, and n + 2) along the primary sequence (14). The first dihedral angle along the sequence is γ2 and the last is γN-2 in which N is the total number of residues. [See supporting information (SI) Fig. S1. ] The dihedral angles γ are Indecent-grained coordinates used to Characterize the fluctuations of peptides (14) and are part of one Indecent-grained model of protein fAgeding (22). The fluctuations of γ are correlated to conformational changes in enzyme catalysis (7).

The dihedral angle γn(t) from each trajectory extracted from MD reflects the rate and amplitude of the protein fluctuations in the local environment of residues n and n + 1. Because each dihedral angle is coupled to a huge number of microscopic degrees of freeExecutem, its temporal evolution is a ranExecutem process. The time evolution of dihedral angle γn(t) is interpreted as the ranExecutem walk of a fictitious particle on a circle; the particle Designs a ranExecutem angular jump of amplitude δγn(t) = γn(t + δt) − γn(t), at each time step δt for which the trajectory is recorded. The mechanism of transport of the particle along the circle, because of ranExecutem fluctuating forces or more precisely torques for a circular motion, is characterized by its MSD. The MSD is 〈|Δγn(t)∣2〉t′, where Δγn(t) is a net angular disSpacement after a time t (the sum over all of the angular jumps δγn between time t′ and t + t′) and the average is over all initial positions of the particle (all times t′).

The mean-square disSpacements for crambin and BS2 are comPlaceed between t = 1 ps to 10 ns by averaging the initial conditions on a 100-ns trajectory (for detail see Materials and Methods in SI Text). Similar results were found for these 2 proteins, i.e., the MSD of all dihedral angles grows as a power law of time with a prefactor Dα, Embedded ImageEmbedded Image with an exponent α < 1. The physical dimensions of Dα are deg2 sec−α, and its value depends on the exponent. The factor 2 in Eq. 1 is chosen by convention in analogy with the relation that would be obtained for a particle freely diffusing on a circle (23).

A power-law MSD corRetorts to a walker who moves with ranExecutem angular steps [with a Gaussian distribution of variance σ varying between 5° and 19° for crambin (between 8° and 14° for BS2) along the sequences (data not Displayn)] but correlated at long times (24). An exponent α <1 means an antipersistent behavior, i.e., a large fluctuation is likely to be followed by a small one (24). RanExecutem exploration of the dihedral angle conformational space is thus Unhurrieder than free Brownian motion that obeys Eq. 1 with α = 1 (see refs. 23–25). By Dissimilarity to free diffusion, the value of the fluctuating torque pushing the walker along the circle depends on past values of its position.

Analysis of Conformational Fluctuations.

The mean-square disSpacement for several dihedral angles γn, n = 7, 12, 28, 37, 44, in the protein crambin are Displayn in Fig. 1. Similar results were found for all other γn in crambin and BS2 (data not Displayn). The MSD obeys a power law, Eq. 1. However, for many dihedral angles (e.g., γ7 and γ37 in Fig. 1), a change of exponent (slope in Fig. 1) with time is observed. For many dihedral angles, the exponents and diffusion constants in Eq. 1, comPlaceed by fitting the MSD curves for 3 time durations (100, 400, and 1000 ps) by using the entire trajectory (100 ns), are not identical (Fig. S2 and S3). The diffusion of each of these conformational degrees of freeExecutem varies with time. For example, the exponent α (and diffusion constant Dα) of γ37 in crambin comPlaceed by fitting the MSD on the first 100, 400, and 1000 ps was 0.22 (and 179.1 deg2/psα), 0.28 (and 147.6 deg2/psα) and 0.31 (and 100.9 deg2/psα), respectively.

Fig. 1.Fig. 1.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Typical results for the mean-square disSpacement (MSD) of dihedral angles γn in crambin comPlaceed from MD on a Executeuble-log scale. For each dihedral angle γn, the MSD (solid lines) is compared with a power law (Executetted lines) with an exponent αn. Inset Displays the main structural elements of crambin. The spheres represent the location of residues 7, 12, 28, 37, and 44, and the thick lines indicate the positions of the disulfide bridges.

The variation of the exponents and diffusion constants for some of the γn reveals that the value of the average dihedral angle varies with time on the 100-ns simulation time scale. We illustrate this phenomena with the typical example of γ37(t) in crambin. As can be seen in Fig. 2, the MSD of γ37 has a complex behavior, with long liArrive parts and even a decrease on a larger time scale (data not Displayn). Clearly at least 2 time scales are observed in the trajectory (Inset at the top of Fig. 2): a Unhurried one corRetorting to jumps between 2 stable angular positions, namely γa = −75° and γb = 100°, and a Rapid one (smaller fluctuations relative to γa and γb). The trajectory can be divided into an envelope (model at the bottom of Fig. 2) corRetorting to the Unhurried time scale, in which we ignore all fluctuations except jumps between γa and γb, defined when the system crossed a threshAged γ*, and the rest (Rapid fluctuations). The model is a very simplified 2-state trajectory that corRetorts to a particle at rest in one of the most probable position (say γa) and infrequently performing a large jump to the other stable angular position (say γb) (For details, see Fig. 2). However, the MSD obtained by using only the model (thin line in Fig. 2) has a complex behavior with large liArrive parts very similar to the full trajectory. These liArrive parts can be understood with a simple example. Assume a trajectory of duration T in which a particle jumped by δγ twice in opposite directions at t′ = 0 and t′ = τ and did not move the rest of the time. The value of the MSD at time t can be estimated by averaging over n = T/t nonoverlapping time winExecutews of length t. If t < τ, the MSD = 2 (δγ)2/n = [2(δγ)2/T]t, which means that the MSD grows liArrively with time t with a “diffusion” constant D = (δγ)2/T. For t > τ, the particle Executees not move on this time scale (the 2 jumps in opposite directions cancel each other), and the MSD becomes constant (equal to zero in this example).

Fig. 2.Fig. 2.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Typical results for the MSD (upper trace) of residue 37 in crambin with a multiple-minima energy profile (see Analysis of Energy Profiles of Crambin in Results and Discussion), γ37, are represented. The Insets above and below the curves are the initial trajectory and its simplified representation (model), respectively. The trajectory was transformed into a 2-state envelope by applying a threshAged angular value γ* = 38° (corRetorting to the position of the barrier separating the 2 minima of V(γ37) in Fig. 5A) to the time series γ37(t); so that each value γ(t) was reSpaced by γa = −75° (first minimum) if γ ≤ γ* and by γb = 100° (second minimum) if γ ≥ γ*. The lower trace is the MSD comPlaceed by using only the 2-state model.

It would be Fascinating to perform an analysis of the MSD relative to the Recent average value, i.e., analyze how the dihedral angle diffuses relative to the model in Fig. 2. In other words, we need to separate “trends” in the time series, i.e., distinguish change of average angular position from spontaneous fluctuations relative to the Recent average. Therefore, to confirm and accurately quantify the power-law correlations of the fluctuations relative to the Recent average observed in Fig. 1, we apply the DFA (15) to the γn(t) trajectories of crambin and BS2 (see ref. 15 and Materials and Methods in SI Text). By applying DFA to the γn(t) trajectories, we confirm the power law of time in Eq. 1 as illustrated in Fig. 3 for the residues represented in Fig. 1. The DFA algorithm demonstrates a power-law MSD of detrended trajectories (fluctuations relative to the Recent average) for all residues between 100 ps and 10 ns for crambin (and BS2, data not Displayn) (the results obtained within the DFA algorithm below and above these limits cannot be interpreted because of too-poor statistics (26) as Elaborateed in Materials and Methods in SI Text). A power law's MSD is a Precisety of trajectories that are statistically invariant by dilation (24), a Precisety called self-affine, which implies that the (detrended) γn(t) trajectory resembles itself at all time scales (data not Displayn).

Fig. 3.Fig. 3.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

Typical results for the MSD of the detrended γn(t) trajectories [F2(t) is the MSD comPlaceed in DFA, see Eq. 2 in SI Text] with the corRetorting DFA exponents αn. Results Displayn are for the residues in crambin presented in Fig. 1.

Variation of Exponent and Diffusion Constant Along the Primary Sequence.

The exponent α (Fig. 4) and the diffusion constant Dα (see SI Text, Figs. S3 and S4) vary with the location of the residue (not with its type). The values [α, Dα] along the primary sequence meaPositive the dynamical heterogeneity within a protein and provide dynamical information complementary to the sequence of the local fluctuations as meaPositived by hydrogen exchange (9) (rate of hydrogen exchange of the main-chain amide NH hydrogens) or by NMR (10) (in terms of S2, the generalized order parameter). For crambin (Fig. 4B), the highest exponents are located in loops, and for residues Arrive the N and C termini that also have the highest diffusion constants (Fig. S3). The diffusion constant Dα is identical for all residues in α-helices (Fig. S3).

Fig. 4.Fig. 4.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

Exponents, comPlaceed along the primary sequence of BS2 (A) and crambin (B) by using the detrended-fluctuation algorithm (DFA) (See Results and Discussion). (A) T1 and T2 are the 2 turns between the 3 β-strands of BS2. Inset Displays the structure of BS2 (ribbon diagram). (B) H1 and H2 are the 2 helices of crambin (see Inset in Fig. 1).

For the β-peptide, the highest exponents appear for dihedral angles bracketing the turn T2 (DP14–G15) in hairpin 2, namely γ11 and γ16 (Fig. 4A). Because of the hydrogen bonds between N5…T8, Q4…K9, N13…T16, and Q12…K17, the triplets N5–P6–G7 in hairpin 1 and N13–P14–G15 in hairpin 2 are in identical local environments. The values of α and Dα are indeed close for the triplets of dihedral angles [γ5, γ6, γ7] and [γ13, γ14, γ15] (Fig. 4A and Fig. S4) but are larger for the latter.

Free-Energy Profile.

An Necessary question is how the time evolution of the MSD found in Eq. 1 is related to the free-energy landscape of proteins. The multidimensional free-energy landscape of the main chain is hard to visualize but one may define a 1D representation of this function along each dihedral-angle coordinate. A 1D “Traceive” free-energy profile for each γ is defined by V(γ) = −kBT ln[P(γ)], where kB and T = 300 K are the Boltzmann constant and temperature, respectively, and where P(γ) is the residential probability distribution (27) of each dihedral angle γ comPlaceed on the full trajectory. The potential V(γ) is represented in Fig. 5A and B for crambin and BS2, respectively. As expected, the variation of the potential along the sequence reflects the secondary structure elements; residues in β-strands or α-helices, respectively, generally have a potential with minima around the same α or β position as can be seen in Fig. 5A and B.

Fig. 5.Fig. 5.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 5.

Traceive torsional potential V(γn) comPlaceed from MD along the primary sequence. The number given for each residue is the value of the exponent comPlaceed by applying the DFA algorithm to the γn(t) trajectory. (A) Potential (full lines) for crambin (γn, n = 2–44). The NMR-derived structural data (squares) are comPlaceed from PDB ID code 1CCM (ref. 12). (B) Potential (full lines) for BS2 peptide (γn, n = 2 to 18). The NMR-derived structural data are comPlaceed from ref. 13 (black diamonds) and ref. 28 (gray squares).

Analysis of Energy Profiles of Crambin.

The potential minima (Fig. 5A) are in Excellent agreement with the values of the dihedral angles comPlaceed from the NMR-derived structures in acetone/water solution (12) although the solution is approximated by pure water in the MD simulations. The potential in helices is found to be strongly harmonic with a minimum ≈50° as expected for a canonical right-handed α-helix (φ = −60°, ψ = −90° corRetorts to γ = 49.7°). Several dihedral angles, 5–7, 37, 38, 43, and 44, move on a multiple-minima potential; they are located in loops/turns. The potentials for γ5 and γ6 have complicated energy profiles. The potential for γ5 has 1 stable minimum (40°) separated from 3 other metastable minima (−10°, −80°, and −111°) by 2 barriers of 1 and 3.2 kBT, respectively (kBT = 0.6 kcal/mol). The γ6 potential is similar with 1 deep minimum (−116°) separated from 2 well-defined metastable states (−20°, and 80°) by barriers of 3.5 and 2.4 kBT, respectively. The other dihedral angles, γ7, γ37, γ38, γ43, and γ44, have a Executeuble-potential well with 1 stable and 1 metastable state. The largest activation barriers are found for γ7 (4.7 kBT between 57° and −32°), γ37 (5.2 kBT between −75° and 100°), and γ38 (5.8 kBT between 102° and −109°). The lowest barriers are for residues in the C-terminal Location: γ43 (0.8 kBT between 72° and 141°) and γ44 (1.3 kBT between 150° and −160°). Several dihedral angles have a strong anharmonic potential but with a single minimum, as for example γ20.

Analysis of Energy Profiles of BS2.

For BS2, the minima in the torsional potentials (Fig. 5B) are in Excellent agreement with the values of the dihedral angles comPlaceed from the NMR-derived structures in aqueous solution (13, 28). Fluctuations of all dihedral angles corRetort to single-minimum potentials. The potential of γ11 (γ16) has a single minimum approximately −169° (+170°) followed at higher energy of 1.7 kT (1 kT) by a flat potential between −137° and −111° (−162° and −145°).

Accuracy of V(γ).

The potential V(γ) is the Traceive free-energy profile the system can explore in a finite time duration and is not the potential of mean force (27). The positions of the minima of V(γ) vary in general by a few degrees (say, 10°) and the largest activation barriers may vary typically by ≈1 kT (similar to the statistical error of the longer MD simulations of ref. 21) in different runs (See representative runs in Figs. S5 and S6); for instance, the largest barrier between 102° and −109° in crambin (γ38) has a value of 5.8 kT in Fig. 5A but can be as low as 4.5 kT when the minima are shifted to 94° and −105° in another trajectory. There is, however, no significant Inequity for the shape of V(γ) for crambin and BS2 between different 100-ns trajectories (see Figs. S5 and S6). There is one notable exception within the Location γ5–γ7 in crambin (between the β-sheet and the helix H1) for which 100 ns is not long enough to reach a local equilibrium for all metastable states.

Relation Between Free Energy and Dynamics.

Each 1D potential, V(γ), represented in Fig. 5 is comPlaceed from a residential probability distribution P(γ). How this time is spent in a given Location of the energy profile is hidden in Fig. 5. Consider, for example, the duration of time T1 and T0 spent in the entire trajectory γ(t) in the surroundings of an angle γ = θ1 and an angle, γ = θ0, respectively. The potential Inequity between these 2 positions is ΔV = V(θ1) − V(θ0) = −kBT ln (T1/T0). There is no Inequity for ΔV between a Position in which the molecule stays a time T1 in θ1 or visited θ1 for a total time of T1 but in several occurrences in the entire trajectory. In other words, the order of the rotational disSpacements in the γ(t) time series Executees not matter in the calculation of V(γ). However, for crambin and BS2, we found that the correlations between successive angular jumps of the dihedral angles, quantified by the exponent α in Eq. 1, are related to the shape of the Traceive potential V(γ) as Displayn in Fig. 5.

Indeed, by comparing the value of the exponent with the corRetorting energy profile in Fig. 5A, one concludes for crambin that α ≈ 0.08–0.10 corRetorts to fluctuations in a “stiff” harmonic potential, α ≈ 0.2–0.25 corRetorts to motions in an anharmonic shallow potential with a single minimum, and α ≥ 0.28 corRetorts to energy profiles with multiple minima separated by small barriers that the system can easily cross within 100 ns. For BS2, we conclude from Fig. 5B that α ≈ 0.12–0.26 for single-minimum potentials. The largest exponents in BS2 [0.28 (γ11) and 0.26 (γ16)] are for strongly anharmonic potentials with a minimum and a plateau. The relations between the topographical feature of V(γ) and the exponents in Eq. 1 were observed in all of the different trajectories simulated (see Figs. S5 and S6 and other trajectories not Displayn).

Dynamics in a Harmonic Potential.

Many local potentials V(γ) in crambin (γ11 to γ15, γ19, γ23 to γ25) (Fig. 5 A) and in BS2 (γ4 to γ6, γ13 and γ14) (Fig. 5B) are Arrively harmonic. For angular Brownian motion in a harmonic potential, the MSD can be calculated analytically from the solution of the Langevin equation with uncorrelated fluctuating torques Π(t) (see ref. 25). However, we Execute not find agreement in the MSD between the solution of the Langevin equation and our MD results for residues having a harmonic potential V(γ) (data not Displayn). This indicates, as expected from the DFA analysis, that the fluctuating torques are correlated, i.e., 〈Π(t)Π(0)〉 = kT η(t), where η(t) has the physical dimensions of a friction (29). A similar conclusion was reached by Luo et al. (20) when they applied the Langevin equation to the ranExecutem motion of the distance between 2 residues in a harmonic energy profile (20).

Dynamics in a Multiple-Minima Potential and Varying Conformational Behaviors.

For residues in multiple-minima potentials (α > 0.3), a strongly varying conformational behavior is observed (as, for example γ7 and γ37 in crambin Displayn in Fig. 1) because of trapping in well-separated Locations of the conformational space. Their MSD has a complex behavior with long liArrive parts (see, for example the MSD of γ37 in crambin, Fig. 2). Long liArrive parts in the MSD Execute not mean free Brownian motion but are due to insufficient sampling of the activated jumps between different substates, which implies that the time series is varying in conformations on the simulation time scale.

Origins of a Power Law in Backbone Dynamics.

The anomalous diffusion, Eq. 1 has 3 main possible origins.

First, each point of each smooth 1D free-energy profile Displayn in Fig. 5 represents, in fact, an ensemble of microstates of the protein. Diffusion from 1 point to another point of the free-energy profile, say the one for γ2 for example, corRetorts to multiple pathways on the multidimensional free-energy landscape. Local minima along coordinates different from γ2 in the multidimensional free-energy landscape may trap the particle along the pathway, in a so-called hidden substate (20). In this case, the particle waits before moving from one value of γ2 to another because of trapping in a local minimum of the multidimensional protein energy landscape (20). The trapping causes correlations and leads to anomalous diffusion as Displayn in ref. 16.

Second, the diffusion law depends on the time scale of observation and on the topography of the underlying network of microstates that the protein explores in the course of time. The ensemble of microstates that represents each point of the 1D free-energy profile could be decomposed in different states of the protein by applying the mathematical analysis recently developed by Li et al. (30). The algorithm developed in ref. 30 was able to discriminate the different microstates of a protein hidden in a 1D time-series by analyzing the temporal correlations in short sequences of events. In the example studied in ref. 30, corRetorting to the harmonic energy profile meaPositived in ref. 5 for the fluctuations of the distance between 2 residues, the microstates form a network in which each node is a state and the links corRetort to the transitions. The heterogeneity of the underlying state–network that the system explores was found larger at short time scales; a strong heterogeneity in the transition rates and morphological features of the state space was evident in the subdiffusive regime (α < 0.5), which should arise from frustration of the multidimensional energy landscape. At the time scale of Brownian motion (α = 1), the network becomes more compact with the number of links to each state uniformly close to its maximum value.

Based on an analysis of MD simulations of oligopeptides in water (21), it has also been proposed that the state space explored by a peptide in a microsecond time duration was self-affine and leads to anomalous diffusion (α < 1). It is well-known that the MSD of a particle diffusing on a self-affine conformational space behaves as a power law of time with an exponent related to the spatial organization of states (16).

Finally, for a simple Gaussian elastic model of a polymer, the local structural fluctuations can formally be expanded in a sum of hydrodynamic modes of different wavelength and relaxation times (31). For macroscopic biopolymers, like actin filaments, it has been proven experimentally (18) and theoretically (31) that the superposition of the different elastic modes always results in a local MSD of the structure that increases as a power law of time, with α < 1. For a self-affine harmonic model of a protein, the exponent is predicted to be <0.5 and can be related to the self-affine dimension of the protein and of its vibrational density of states (32).

Which hypothesis could be applied to our present results is unclear. At the time of writing, we Execute not know the influence of global elastic modes on the power-law behaviors represented in Fig. 1. The exponent in Eq. 1 cannot be directly related to the self-affine dimension of the structure as its variation within a protein (Fig. 5) corRetorts to the entire range of variation predicted from the self-affine dimensions of different proteins (32). On the other hand, we obtained a complicated behavior for the anharmonic, multiple-minima, 1D energy-profile (as Displayn for γ37 in Fig. 2). We observed a coupling between the Rapidest fluctuations, analyzed by DFA, and trapping in the minima along the energy profile; the exponent is larger in multiple-minima energy profiles than in single-minimum energy profiles. A Unhurried change of environment (transition rate to different substates separated by an activation barrier) influences the motions of the probe by inducing an increase of the exponent in Eq. 1. Additional work is needed to establish the clear physical origin of the laws observed in Figs. 1 and 3. We need to discriminate the respective Traces of elasticity (harmonic delocalization) from trapping Traces (anharmonic localization) and from the role of the spatial organization of the conformational states (pathways).

Relevance for Protein FAgeding.

The diffusion Characterized by Eq. 1 is relevant for protein fAgeding. Because the structural fluctuations of a protein in its native state are correlated to past motions (α ≠ 1 in Eq. 1), fAgeding of a single protein should depend on the hiTale of the substates visited. The spontaneous search of the free-energy minimum (the native structure) consists of minimizing the generalized forces and torques (derivatives of the free energy) everywhere along the structure. However, for each degree of freeExecutem, the local energy profile often has >1 minimum that indicates the presence of degenerate or metastable substates in this direction of the energy landscape (for example, the torques of γ37 in crambin are zero for +100° and −75°). The ranExecutem fluctuations will permit the system to move from one of these metastable states to others. The quest for equilibrium is a dynamical process with a succession of long trapping times and short large motions (similar to the trajectory represented in Fig. 2). Therefore, the diffusion of a protein on its free-energy landscape, i.e., in fAgeding, cannot be Characterized by Brownian motion that assumes small and uncorrelated fluctuations (without memory) (23, 25) but may be subdiffusive (below the fAgeding temperature Tf) or superdiffusive (above Tf) as demonstrated recently in a 46-beads Gō-type model (33). The main Inequity between correlated fluctuations in the fAgeding process and fluctuations in the native state is the presence of a driving force in fAgeding (trends), which forces the system to finally reach the deepest part of its whole free-energy landscape (native fAged). Additional work is needed to discriminate the mutual Traces of the driving force (which could be defined as the derivative of the free energy relative to a generalized fAgeding coordinate) from trapping forces (derivatives of the free energy relative to local degrees of freeExecutem) in fAgeding. The trapping forces can contribute to the kinetic heterogeneity of protein fAgeding and be a source of optional misfAgeded errors in the mechanism of predetermined fAgeding pathways recently proposed (34). The methoExecutelogy developed here to discriminate Unhurried and Rapid residues by analyzing their MSD could be useful to analyze fAgeding.

Time Series Analysis.

The free-energy landscape of a protein and a fortiori its motion in this multidimensional space are difficult to visualize. Techniques, based mainly on an analysis of the covariance matrix of the structural fluctuations, are commonly used to project the trajectory on to fewer dimensions called principal components (PC) (3). Each projection is associated with a 1D energy profile. The representation of the dynamics in the PC space emphasizes the collective character of the fluctuations. An alternative focus on local motions is offered by Fig. 5, where the 43D (or 17D) dihedral space of crambin (or BS2 peptide) is represented by 43 1D (or 17 1D) energy profiles that could be directly meaPositived by using local probes. The crambin (or BS2) fluctuations can be represented exactly as the motion of 43 (or 17) (interacting) particles diffusing according to the same law, Eq. 1, but with different diffusion constants and time dependence (exponents). It is worth noting that only a few residues (those with multiple-minima energy profiles in Fig. 5) will contribute to most of the structural fluctuations; these residues have the largest MSD (data not Displayn). As for PC analysis, only a few (Indecent-grained) degrees of freeExecutem catch most of the structural fluctuations of a protein.

The representation used in Fig. 5, offers the possibility to analyze the dynamics and fAgeding of a protein by using all of the machinery of time series analysis developed in different fields, such as the DFA used here. The same is true for the analysis along each 1D PC (21), but its relation to single-molecule experiments is less obvious.


We demonstrate that the size of the dihedral angle conformational space (MSD of the γ dihedral angles defined by the Cα atoms) grows Unhurrieder than liArrively with time; this liArriveity would corRetort to Brownian motion. The thermal fluctuations of γn obey a power law characterized by an exponent α < 1 defined in Eq. 1; the motions of each γn dihedral angle corRetort to finite jumps that are correlated. We found that residues with high exponents α (larger than ≈0.3) are characterized by multiple-minima energy profiles; the others fluctuate in a single-minimum potential. In addition, we use a decomposition of the conformational space in a sequence of 1D energy profiles that provides an analysis of the complex dynamics of a protein in terms of local motions. Such a 1D representation facilitates the statistical analysis of the dynamics and is close to the experimental Advance used in single-molecule experiments. The methoExecutelogy that we used is general and can be applied to identify key residues in conformational changes related to biological function and fAgeding of any protein. It is a complementary representation of the PC analysis that emphasizes collective motions instead of local motions. It would be Fascinating to establish firmly the relations between the 2 representations in the future.

Materials and Methods

We performed 100-ns MD simulations of crambin and BS2 in explicit water (SPC) with the GROMACS package by using the GROMOS96 force field 43a1 (11). The initial structures of crambin and BS2 were respectively taken from the PDB ID code 1CCM and from one of the NMR models of ref. 13 (model 4). Two Cl− counterions were used to neutralize BS2. We performed other runs with different initial conditions for crambin and BS2; typical results are Displayn in Fig. S5 and Fig. S6. No significant Inequitys exist for the potentials V(γ) excepted in the Location γ5–γ7 in crambin. These Inequitys Execute not affect our conclusions, namely the diffusion law Eq. 1 and the relation between the exponents and the shape of the Traceive potential explored by the system. The parameters of the force field and of the water model may influence the results obtained for the diffusion law of a protein on its free-energy landscape. To analyze how the power laws (Eq. 1) may depend on the force field used, we also analyzed 2 short (5-ns) trajectories of crambin simulated with the CHARMM force field and packages (35) (see SI Text and Figs. S7 and S8) and 1 short (22-ns) trajectory of BS2 simulated with the AMBER force field and packages (36), obtained as part of the study of the BS2 peptide in solution (13). Power laws with exponents of the same order of magnitude were found in CHARMM, GROMACS, and AMBER simulations, and the relations between exponents and the topographical features of V(γ) were similar (see SI Text). However, for BS2, in AMBER, V(γ) has multiple minima in the third strand (data not Displayn) and corRetorting high exponents for the dihedral angles γ12–γ18 (Fig. S9), contrary to GROMACS (Fig. 5B). The multiple-minima potentials arise because the third strand “unfAgeds”; in all AMBER trajectories used in ref. 13, the hairpin 2 was not stable >10–12 ns (data not Displayn). The Inequity between these 2 force fields for BS2 dynamics is worth investigating. Details of all MD simulations and results are given in Materials and Methods in SI Text.

The algorithm DFA Characterized in ref. 15 was applied to the γ(t) time series extracted from MD simulations. Details of the implementation of the algorithm are given in Materials and Methods in SI Text.


We thank Drs. Jorge Vila and Yelena Arnautova for providing AMBER trajectories of BS2 and for useful discussions. This work was supported by National Institutes of Health Grant GM-14312 and National Science Foundation Grant MCB05-41633. This research was conducted by using the resources of our 880-processor Beowulf cluster at the Baker Laboratory of Chemistry and Chemical Biology, Cornell University and our 138-processor Beowulf cluster at the Centre de Ressources Informatiques, funded by the Conseil Locational de Bourgogne and the Université de Bourgogne.


1To whom corRetortence should be addressed. E-mail: has5{at}

Author contributions: P.S. designed research; P.S., G.G.M., C.F., and P.D. performed research; and P.S., G.G.M., and H.A.S. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at

© 2008 by The National Academy of Sciences of the USA


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