Nature Physics  Letter
The dynamics of single protein molecules is nonequilibrium and selfsimilar over thirteen decades in time
 Xiaohu Hu^{1, 2}^{, }
 Liang Hong^{3}^{, }
 Micholas Dean Smith^{1}^{, }
 Thomas Neusius^{4}^{, }
 Xiaolin Cheng^{1}^{, }
 Jeremy C. Smith^{1, 5}^{, }
 Journal name:
 Nature Physics
 Volume:
 12,
 Pages:
 171–174
 Year published:
 DOI:
 doi:10.1038/nphys3553
 Received
 Accepted
 Published online
Internal motions of proteins are essential to their function. The time dependence of protein structural fluctuations is highly complex, manifesting subdiffusive, nonexponential behaviour with effective relaxation times existing over many decades in time, from ps up to ~10^{2} s (refs 1,2,3,4). Here, using molecular dynamics simulations, we show that, on timescales from 10^{−12} to 10^{−5} s, motions in single proteins are selfsimilar, nonequilibrium and exhibit ageing. The characteristic relaxation time for a distance fluctuation, such as interdomain motion, is observationtimedependent, increasing in a simple, powerlaw fashion, arising from the fractal nature of the topology and geometry of the energy landscape explored. Diffusion over the energy landscape follows a nonergodic continuous time random walk. Comparison with singlemolecule experiments suggests that the nonequilibrium selfsimilar dynamical behaviour persists up to timescales approaching the in vivo lifespan of individual protein molecules.
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At a glance
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Although fluctuations of distances between atoms in folded proteins are necessarily spatially bounded (confined), it is conceivable that, as the timescale of observation is increased, a protein may incorporate into these fluctuations slower pathways over its energy landscape. The question then arises as to whether there is at all a finite characteristic time associated with any given structural change, or, instead, that the timescale on which a structural fluctuation is observed determines the apparent characteristic relaxation time for the motion that will be obtained. To examine this question we have performed molecular dynamics (MD) simulations to characterize the internal dynamics of three globular proteins of markedly different size and structure: one with a single structural domain (KRas), one with two structural domains (phosphoglycerate kinase (PGK)) and one with four structural domains (the Escherichia coli aminopeptidase N (ePepN)). MD simulations of different lengths (observation times) were performed. We examine in detail the motion between the two domains of PGK (Fig. 1), which is of direct functional importance^{5}. The timeaveraged meansquare displacement (TAMSD), (Supplementary Equations 1 and 2), where Δ is the lag time and t the observation time (length of the trajectory), calculated from the time series of the distance R(t) between the centres of mass of the two domains, is presented together with the corresponding normalized displacement autocorrelation function (ACF) C(Δ; t) (Supplementary Equations 3 and 5), in Fig. 2a, b. The TAMSD does not reach a plateau over the timescale examined. Furthermore, TAMSDs calculated over different observation times t are shifted relative to each other, with the slope becoming increasingly smaller with increasing t, a signature of ageing and observationtimedependent dynamics^{6}.
C(Δ; t) shifts towards longer lag times with increasing t, again consistent with ageing. Furthermore, an intriguing commonality is found in the dynamics examined on different timescales: Fig. 2c shows that τ_{c}, the characteristic time of the interdomain motion, increases in a powerlaw fashion with the observation time, t, as τ_{c}(t) ∝ t^{θ}, with θ ≈ 0.9, showing no sign of convergence. Remarkably, data from singlemolecule experiments on the distance fluctuations between sidechain pairs^{3, 4} fall close to the same powerlaw relationship (Fig. 2c), although these were obtained on t ≥ 300 s observation timescales, more than seven orders of magnitude longer than the MD, and on other proteins. Together, Fig. 2a–c reveals strong nonstationarity (ageing) of the interdomain dynamics and suggests a powerlaw dependence extending from 10^{−12} to 10^{2} s.
The residues probed in the experimental singlemolecule studies are only ~0.3–0.4 nm apart from each other^{3, 4}, in contrast to the interdomain distance explored above, the average of which is ~3.8 nm. Therefore, we investigated the distance fluctuations between the side chains of 32 selected PGK residue pairs in a variety of structural environments (see Supplementary Information for details). Although there are substantial variations in the results between individual residue pairs, the average behaviour (Fig. 2c) shows the same quantitative time dependence as described above for the interdomain centreofmass motion, with characteristic relaxation times continuously increasing with t and following practically the same power law. Thus, the nonequilibrium scaling behaviour holds both for global (for example, interdomain) protein motion and a substantial fraction of local motions, and in some cases even for distance fluctuations between adjacent residues on the same αhelix, an example of which is shown in Supplementary Fig. 8. Also, to determine whether these results are specific to PGK we also performed MD simulations of two other proteins under similar conditions: a much larger, fourdomain enzyme (ePepN) and a much smaller, singledomain GTPase, human KRas. Both these proteins exhibit the same observationtimedependent internal dynamics as shown above for PGK (see Fig. 2c and Supplementary Figs 3–5).
The power spectrum of the PGK interdomain distance fluctuation, S(f), where f is the frequency, is shown in Fig. 2d. S(f) obtained on different observation timescales can be concatenated onto a single profile. For f ≳ 0.1 THz, and over nearly five frequency decades in the MD, S(f) scales approximately as f^{−1}. Moreover, the power spectrum calculated from the singlemolecule experimental data^{4} on the ms to 10^{2} s timescales also follows the same 1/f behaviour (inset in Fig. 2d). Hence, the 1/f dependence also extends from ps up to ~10^{2} s timescales. This ‘1/fnoise’, often also referred to as ‘flicker noise’ or ‘pink noise’, indicates selfsimilarity of the corresponding dynamics on different timescales^{7}. The lack of a characteristic frequency associated with a 1/f spectrum is consistent with the observation of the timedependent relaxation time—that is, motions with everlower frequencies are sampled as the observation length is increased.
The above observations concern the dynamics of folded proteins in their broad, global, free energy minima. As the average structure of most globular proteins is well defined, at sufficiently long times the system will eventually feel a restoring force centred at the mean position—that is, a confining potential. However, within this global broad free energy well there are many small, local minima separated by barriers with different heights^{8, 9}. The dynamics of folded proteins can thus be considered as a fictitious particle diffusing on a rugged energy landscape possessing many wells of various depths^{10}. To understand the features of this landscape that give rise to the selfsimilar dynamics, we mapped the individual MD trajectories onto networks based on the transitions between different metastable conformational states^{11, 12} (see Supplementary Information for details). Clusters of similar structures form the vertices (nodes) of the network. Whenever, during the simulation, the protein transits between two clusters an edge is added between the two vertices, thus forming a conformational cluster transition network (CCTN). Similar networks have been used several times previously to describe complex dynamics, for example, refs 11,12. Graphical illustrations of obtained networks are shown in Fig. 3. The degree distributions of the networks, defined as the probability P(d) of finding a vertex connected to d direct neighbours, fully overlap on different timescales within the statistical errors, indicating topologically selfsimilar, fractal networks (Supplementary Fig. 11a). (The fractality here is in the geometry and topology of the CCTN, in contrast to the fractality of the protein structure itself, which has been related to the vibrational density of states^{13, 14}.)
A natural simplified physical description of the dynamics of the fictitious particle diffusing over the CCTN is a ‘continuous time random walk’ (CTRW; ref. 15), in which, at each step, the random walker draws from a distribution of jumping distances and waiting times. Nonergodic subdiffusive motion arises if the average waiting time diverges, as, for example, in a powerlaw waiting time distribution. Diverging waiting times can occur in diffusion on very rugged potential surfaces possessing many deep ‘traps’ in which the system is stuck for extended periods of time^{16}. Superposition of thermal noise, representing motion within the trap, leads to the noisy CTRW description introduced in ref. 17.
The decay behaviour of the tails of the ACFs at long Δ is best described by an incomplete beta function (see Supplementary Equation 7), which is the analytical ACF for a nonergodic subdiffusive CTRW in an external potential, derived from the fractional Fokker–Planck formalism^{18}. The full range ACFs obtained on all observation timescales and for all three proteins are well fitted by a noisy CTRW model (see Supplementary Equation 9 and Fig. 2b and Supplementary Figs 3–5). We note that all ACFs obtained here deviate significantly from powerlaw behaviour for large Δ. In contrast, the ACFs from related ergodic subdiffusive models, such as fractional Brownian motion or processes governed by the fractional Langevin equation, are described by a Mittag–Leffler function^{15, 19}, which decays as a power law for large lag times Δ, and is independent of t. Furthermore, projecting the motion of interest (for example, the interdomain motion) onto the CCTN filters out the noise component, leaving the subdiffusive CTRW dynamics (see Supplementary Section 7 and Supplementary Fig. 12).
If observed over a sufficiently long timescale a single, folded protein must eventually reach dynamical equilibrium. At that point the system will be ergodic (the time average of all quantities depending on the dynamics will converge to the ensembleaveraged values), the MSDs will plateau and the subdiffusive CTRW description will break down. The singlemolecule spectroscopic experiments indicate this timescale must be longer than ~10^{2} s (refs 3,4). Hence, a single protein will not reach equilibrium over most timescales on which functional processes, such as ligand binding, allostery and catalysis, usually occur (that is, μs upwards). Combination of the present results with the singlemolecule experimental data indicates that selfsimilar dynamics persists from ps through to the timescales of relevant biological processes in the cellular environment. Indeed, the median halflife of cellular proteins in yeast cells (Saccharomyces cerevisiae) has been estimated at ~45 min (ref. 20), barely an order of magnitude longer than the observation time lengths of the singlemolecule experiments. Therefore, the dynamics of an individual protein may exhibit nonequilibrium, selfsimilar dynamical behaviour throughout its typical biological lifespan.
On functional timescales, although the spatial dependence of fluctuations may evolve only extremely slowly with time, the time dependence of the associated motion remains well out of equilibrium, leading to the absence of a finite average in the powerlaw relationship revealed here between the characteristic time and observation time. This complexity means that canonical assumptions relating dynamics to function break down. Thermal equilibrium is assumed in classical theories for chemical reactions catalysed by proteins, such as the Michaelis–Menten formalism or transition state theory. However, on functional timescales any two protein molecules will sample different regions of conformational space and the associated timeaveraged dynamical properties, including catalytic rate constants, will be different. Conformational dynamics modulates enzyme catalytic activity and in some cases can determine the overall ratelimiting step^{21}. The present picture is consistent with singlemolecule experiments revealing that enzymes possess dynamic disorder^{22}, undergoing internal motions on timescales longer than those on which they function, such that the catalytic rate constant of an individual enzyme can strongly fluctuate over time^{23, 24}. The internal motions lead to fluctuations in the height of the effective reaction barrier, occurring on timescales similar to or longer than the reaction, leading to dispersed kinetics and deviation from classical Michaelis–Menten behaviour^{2, 24, 25}. In this case, the reaction rate, which would otherwise be Arrhenius, is convolved with the temporal fluctuation of the barrier height, resulting in a longtailed distribution of the rate^{24, 25, 26}. The present work indicates that the corresponding nonequilibrium dynamics is a continuous time random walk on a selfsimilar, fractal conformational cluster transition network, a general phenomenon associated with the complexity of globular protein structure. The prevalence of nonequilibrium fractal dynamics in single protein molecules on the timescales of macromolecular function may fundamentally change our appreciation of the relationship between protein dynamics and functional activity.
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Acknowledgements
Anton computer time was provided by the National Center for Multiscale Modeling of Biological Systems (MMBioS) through Grant P41GM103712S1 from the National Institutes of Health (NIH) and the Pittsburgh Supercomputing Center (PSC). The Anton machine at PSC was generously made available by D.E. Shaw Research. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the US Department of Energy under Contract No. DEAC0500OR22725 and resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under Contract No. DEAC0205CH11231. L.H. acknowledges the support from NSF China 11504231. We thank I. M. Sokolov, A. P. Sokolov and F. Noé for fruitful discussions and T. Splettstößer (http://www.scistyle.com) for rendering the 3D protein structure shown in Fig. 1.
Author information
Affiliations

Center for Molecular Biophysics, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
 Xiaohu Hu,
 Micholas Dean Smith,
 Xiaolin Cheng &
 Jeremy C. Smith

Graduate School of Genome Science and Technology, University of Tennessee, Knoxville, Tennessee 37996, USA
 Xiaohu Hu

Institute of Natural Sciences & Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
 Liang Hong

Wiesbaden Business School, RheinMain University of Applied Sciences, Bleichstr. 44, D65183 Wiesbaden, Germany
 Thomas Neusius

Department of Biochemistry and Cellular and Molecular Biology, University of Tennessee, Knoxville, Tennessee 37996, USA
 Jeremy C. Smith
Contributions
X.H. performed and conceived the research, analysed the results and wrote the manuscript. L.H. analysed the results and wrote the manuscript. M.D.S. performed the research. T.N. analysed the results and wrote the manuscript. X.C. analysed the results and wrote the manuscript. J.C.S. conceived the research, analysed the results and wrote the manuscript.
Competing financial interests
The authors declare no competing financial interests.
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Xiaohu Hu
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