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Dynamics and the Free-Energy Landscape of Proteins, Explored with the Mö ssbauer Effect and Quasi-Elastic Neutron Scattering Hans Frauenfelder,† Robert D. Young,‡ and Paul W. Fenimore*,† †

T6, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States Center for Theoretical Biological Physics, Arizona State University, P.O. Box 871504, Tempe, Arizona 85287-1504, United States

‡

ABSTRACT: The Mössbauer effect and quasi-elastic neutron scattering (QENS) from hydrated proteins yield sharp elastic lines that are accompanied by broad wings. Conventionally, the elastic line and the broad wings are treated as separate phenomena. We show that there is no separation; the entire spectrum consists of Lorentzians with the natural line width. In protein crystals, the shifts of the individual lines from the elastic center above about 150 K are caused by beta fluctuations in the hydration shell. Vibrations cause shifts in the entire temperature range but are best seen below about 150 K. We construct a microscopic model for the dynamics that is based on a random walk of the proteins in their free-energy landscape. The model yields approximate values for the steps in the energy landscape. Remarkably, the quantum electrodynamic concept of gamma rays is needed to justify the model.

I. MÖ SSBAUER EFFECT AND NEUTRON SCATTERING 1

parts of the same inhomogeneous spectrum consisting of Lorentzians with the natural line width but, in the case of the quasielastic component, are heterogeneously shifted from the central line. This is in constrast to Figure 1, showing quasielastic processes as broadened but unshifted. In the present paper, we introduce a model for the quasi-elastic process in terms of the free-energy landscape. We discuss mainly Mössbauer experiments, but the conclusions are valid also for neutron scattering. In the Mössbauer effect, a radionuclide, usually 57Fe, is the source of gamma radiation. 57Fe emits a gamma ray with energy E0 = 14.4 keV and a mean life τMö = 141 ns corresponding to a rate coefficient kMö = 1/τMö = 7.1 × 107 s−1. Usually, the 57Fe nucleus recoils and the emitted gamma rays do not carry the full transition energy E0. However, if 57Fe is embedded in a solid, a fraction f(T) of the emitted gamma rays carry the full energy E0 and have the natural line width Γ = 4.7 neV. The Mössbauer spectrum is determined by gamma rays from a 57Fe source moving with a velocity v being transmitted through a stationary absorber containing the proteins. In the thin-absorber limit, the transmission, Tr(ΔEexp), is related to the scattering amplitude S(ΔEexp) by Tr(ΔEexp) = 1 − const. S(ΔEexp), where ΔEexp = E0v/c. Mössbauer spectra are evaluated by plotting Tr(ΔEexp) versus the experimental Doppler energy shift or versus the corresponding velocity v in mm/s, where 1 mm/s corresponds to 48 neV. One main measurement result is the elastic fraction f(T). Unfortunately, it is common in the literature to not

2

The Mössbauer effect, incoherent neutron scattering, and nuclear resonant spectroscopy3 are crucial tools to study the structure and dynamics of complex systems, in particular biomolecules. While the techniques differ experimentally, the underlying physics is identical. Photons or neutrons are scattered by the target, and three scattering processes occur: elastic, quasielastic, and inelastic. In elastic scattering, the kinetic energy of the incident neutron or photon is conserved in the center-of-mass frame. In inelastic scattering, the particle loses energy and the inelastic peak is clearly separated from the elastic peak. Quasielastic scattering is less well-defined. Wikipedia, for instance, states “...a limiting case of inelastic scattering, characterized by energy transfers being small....” Figure 1 shows a textbook picture of these processes. However, Figure 1 is misleading. The elastic and quasielastic lines are

Special Issue: Peter G. Wolynes Festschrift Figure 1. Schematic figure of scattering amplitude showing the components from elastic, quasielastic, and inelastic processes according to the usual definition of quasi-elastic processes. © 2013 American Chemical Society

Received: April 17, 2013 Revised: August 16, 2013 Published: August 20, 2013 13301

dx.doi.org/10.1021/jp403832n | J. Phys. Chem. B 2013, 117, 13301−13307

The Journal of Physical Chemistry B

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publish the experimentally determined f(T) but plot ⟨x2(T)⟩, the mean-square displacement (msd) given by the Lamb− Mössbauer relation f (T ) = exp[−q2⟨x 2(T )⟩]

(1)

Here q is the wave vector of the gamma rays or neutrons. In neutron scattering, the incident and scattered neutrons are selected by spectrometers with narrow energy windows. In the 57 Fe Mössbauer effect, the momentum transfer is fixed as q = 7.29 Å−1; in neutron scattering, f(T) depends on q and is called the elastic incoherent structure factor or EISF. Figure 2 shows typical examples of the temperature dependence of ⟨x2(T)⟩ as measured with the Mössbauer

Figure 3. (a) A Mössbauer transmission spectrum. Metmyoglobin crystal, 295 K, h = 0.4. The solid line is a one-parameter fit, assuming that the broadening is caused by external fluctuations. Data from Parak and Achterhold.43 (b) A QENS spectrum of perdeuterated metmyoglobin polycrystals, 297 K, q = 1.7 A−1.49

Figure 2. (a) Mean-square displacement for MbO2 in aqueous solution (squares) from Mössbauer,4 metMb polycrystals (circles).5 (b) Mean-square displacement for Mb powder (circles), h = 0.4, from QENS.6

effect4,5 and with neutron scattering.6 They show an approximately linear increase of ⟨x2(T)⟩ up to Tc. At Tc, there is a pronounced change in the slope and the msd increases rapidly above Tc. The break in the slope of the msd has been called the protein dynamical transition (PDT).6 In dehydrated proteins, the PDT is absent and the approximately linear increase of the msd continues up to 300 K. The elastic process gives only a part of the information. The quasi-elastic and inelastic processes provide deeper insight. Of particular significance are the quasi-elastic “fat wings” that are shown in Figure 3. These wings appear first at about 180 K but are dominant at ambient temperature. They appear on the Stokes and anti-Stokes side and extend above 1 μeV in the Mössbauer effect and above 1 meV in neutron scattering. In neutron scattering and in the phonon-assisted Mössbauer effect, inelastic peaks show up beyond the quasielastic region, as displayed in Figure 4.7 The quasi-elastic and inelastic spectra are quantitatively different. The quasielastic spectrum is smooth, broad, and symmetric in ΔE. The inelastic spectra are narrower, and the amplitude of the anti-Stokes line depends

Figure 4. Phonon spectra of metmyoglobin at different temperatures, taken using the phonon-assisted Mössbauer effect. The data at two temperatures show the expected behavior. At low temperature, the anti-Stokes line is absent. At high temperature, it is present.7

on the temperature but is always smaller than that of the Stokes line. 13302



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processes in protein crystals, where the α fluctuations are absent.

II. FLUCTUATIONS THAT GOVERN PROTEIN DYNAMICS Our interpretation of protein dynamics is based on fluctuations.8−10 A primary consequence of fluctuations is the existence of conformational substates (CS). Proteins are in thermal equilibrium with the surroundings, but the energy of an individual protein fluctuates constantly. The theory of fluctuations11 gives for the variance of the energy E of a protein ⟨(ΔE(T ))2 ⟩ = ⟨(E − U (T ))2 ⟩ = kBT 2Cv

III. CHALLENGES AND RESPONSES Neutron scattering and Mössbauer studies have venerable histories, but questions remain and some proposed solutions are not universally accepted. We describe here some of the questions and our solutions. 1. The Lamb−Mö ssbauer Relation. Equation 1 is ubiquitous, but the commonly used form eq 1 is only valid for motions in harmonic potentials.23 Proteins are not harmonic; they have multivalley potential surfaces where eq 1 does not apply. The conclusion: Data should be presented in terms of the measured f(T), not the derived ⟨x2⟩. The Debye− Waller factor in X-ray crystallography is a measure of atomic ⟨x2⟩ because it describes snapshots and is not affected by transitions between CS.24 2. The Protein Dynamic Transition (PDT). The PDT has generated a flood of papers, but no explanation is universally accepted.25,26 We show here that the PDT is not a real transition in the protein but that the change in slope is caused by the onset of the βh fluctuations in the hydration sheIl, which simulate the transition. 9,10 With the spectra of the β h fluctuations shown in Figure 5, the PDT can be understood. The apparent break in the msd starts when, on raising the temperature, the high-frequency wing of the βh fluctuation rate reaches the value kMö = 1/τMö = 7 × 106 s−1. The fluctuations then shift some of the sharp lines to higher and lower energies, decreasing the amplitude of the sharp line, thereby giving the illusion of a transition. The argument can be made more precise by stipulating that proteins encountering fluctuations with rate coefficients kβ below the Mössbauer rate kMö in Figure 5 are not affected by the fluctuations on Mössbauer time scale and are “recoilless”.27 Thus, f(T) is given by

(2)

Here U(T) ≡ ⟨E⟩ is the average protein energy, Cv the heat capacity at constant volume, and kB the Boltzmann constant. For a macroscopic system, the energy fluctuations are very small, but for a protein embedded in a heat bath, they are very large. With the value Cv ≈ Cp = 24 kJ/(mol K) for dehydrated Mb,12 we get ⟨E⟩ ≈ 1.35 eV ≈ 130 kJ/mol at 300 K. Hydrated proteins have even larger heat capacities.13 A myoglobin crystal hydrated to about 50% has a heat capacity of Cp ≈ 190 kJ/(mol K), resulting in ⟨E⟩ ≈ 10 eV. Thus, eq 2 implies that proteins can assume a broad range of energies and also a correspondingly broad range of conformations.14,15 Transiently, all CS can be reached. In thermal equilibrium at 300 K, however, only CS with energies of the order of kBT = 25 meV or less are appreciably occupied. The existence of multiple conformations was anticipated by farsighted scientists such as Schrödinger and Kaj Linderstrom-Lang, discovered experimentally,16 and confirmed by simulations.17−19 Protein conformational motions are transitions between CS. Three types of fluctuations, α, βh, and vibrations, are known to cause and control these transitions. The α fluctuations originate in the bulk solvent and influence the shape of the protein.20 Their rate coefficients kα(T) are inversely proportional to the solvent viscosity, and they are unobservable in solids. The βh fluctuations are mainly produced by the protein’s hydration shell; they depend on the hydration and are absent in dehydrated proteins. The spectra of the βh fluctuations for a range of temperatures are given in Figure 5. The vibrations are

f (T ) = 1 −

∫ ρβ (log k , T ) d(log k)

(3)

The recoilless fraction f(T) and the msd can be calculated by using the density of fluctuations, ρβ(log k, T), from Figure 5. The PDT is thus explained without fit parameters. 3. Quasi-Elastic Effects. The increase in ⟨x2⟩ above Tc implies that the elastic fraction f(T) decreases; fewer Mössbauer gamma rays are observed in the elastic window. Where did the missing gamma rays go? Experiments provide the answer. Above Tc, broad wings appear, as shown in Figure 3. Conventionally, the sharp Mössbauer line and the broad band are treated separately, as drawn in Figure 1.28−31 In contrast, we describe the entire spectrum at all temperatures with only one fit parameter.32 We use the theory of Singwi and Sjölander33 and the following assumptions: (a) In a protein ensemble, dephasing is exponential in time, given by the rate coefficient χkβ(T).34 (b) Emission and absorption are recoilless and have the natural line width. This fact has, for instance, been proven by the famous gravity experiment of Pound and Rebka.35 (c) The change from a sharp line to a broad spectrum is caused by the βh fluctuations. These three assumptions together explain the shape of the spectrum at all temperatures. The only fit parameter is χ, which is dimensionless and close to 1. A fit to a Mössbauer spectrum is shown in Figure 3a. 4. The Spectrum Is Dynamic. The separation of the spectrum into a sharp line and a broad band is misleading. The entire spectrum is composed of a very large number of Lorentzians with twice the natural line width but shifted away

Figure 5. Normalized dielectric relaxation spectra for myoglobin, h = 0.4.11

more complicated. Proteins have thousands of vibrational degrees of freedom.21 They cannot be taken into account individually; they are described by the vibrational density of states D(E).7 To study the coupling between the external fluctuations and the internal motions, they are examined separately and then compared.10 For the present study, the external fluctuations were measured with dielectric relaxation spectroscopy.22 Internal fluctuations were observed with the Mössbauer effect and with neutron scattering. In the following, we discuss 13303



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from E0 = 0 (Figure 6). The spectrum is dynamic. The line at E0 is not produced by a static subset of proteins. Every protein

Figure 6. A broadened Mössbauer spectrum consisting of a very large number of individual Lorentzians with twice the natural line width.

Figure 7. Schematic of the free-energy landscape (FEL). The energy of conformational substates is plotted versus a conformational coordinate. Each point stands for a few thousand coordinates that describe the positions of all atoms of the protein.

wanders continuously through the conformational space and occasionally visits E0.

IV. A MICROSCOPIC MODEL BASED ON THE FREE-ENERGY LANDSCAPE The Model. As stated above, we have explained the full Mössbauer spectrum by using the theory of Singwi and Sjölander and one dimensionless fit parameter. This feat is possible because in a quantum system coupled to a macroscopic environment the dephasing is exponential in time. The price paid for using an ensemble is the lack of a microscopic model that explains how individual proteins behave. Here we introduce a microscopic model, based on a generally accepted fact: Mössbauer gamma rays emitted by a stationary source always have the energy E0 and the natural line width Γ. If the absorber resonance does not occur at the energy E0 of the elastic line but at E0 + ΔE, the absorbing system must be in a state with the energy ΔE with respect to the elastic line. Since the spectra in Figure 4 are symmetric, the system must be capable of adding and subtracting the energy ΔE. What is the source of the energy ΔE? The obvious candidate is the protein and its surroundings. The protein fluctuates continuously from CS to CS, always exchanging energy with the surroundings and thereby increasing or decreasing its energy. The fluctuations can be caused by the βh fluctuations in the hydration shell, by the α fluctuations in the environment, and by vibrations. Here we deal with the βh fluctuations and the vibrations, but the arguments also are valid for the α fluctuations. The challenge is to connect the motions in the FEL to the shifts of the spectral lines. Figure 7 gives an extremely simplified picture of the FEL at two different temperatures. A CS is a point in the FEL, defined by the few-thousand spatial coordinates of all atoms of the protein, including those in the hydration shell. Each CS is additionally characterized by its energy E. Figure 7 is a one-dimensional cross section through the FEL, showing the energy E of selected CS versus a conformational coordinate “cc”. The βh fluctuations force random jumps from CS to CS, changing the energy of the protein by, on average, the energy δE. The protein makes n(T)

= kβ(T) τMö steps during the lifetime τMö. In the unlikely case that all jumps have the same sign, the protein reaches the maximum energy ΔEmax = n(T)δE. Because the protein makes a random walk in the FEL, we assume that the average energy reached during the time τMö is given by

ΔE ∼ δE n (T )

(4)

The experimental data in Figure 3, together with the relaxation spectra in Figure 6, permit a rough determination of δE and ΔEmax. Figure 3a shows that the fluctuations can shift the Mössbauer line by as much as a few μeV at 295 K. Figure 6 shows that kβ(T) at 295 K reaches values up to 1012 s−1 and thus the protein can make up to a few times 104 steps, reaching values of ΔEmax of a few μeV. Equation 4 then gives for the average step δE ∼ 10 neV. Similar estimates for the QENS data in Figure 3b give ΔE ∼ 1−10 meV and δE ∼ 10−100 μeV. The arguments in section II show that the energy of a protein can extend to a few eV. The fluctuations induced by the βh fluctuations thus explore only a very small part of the FEL. The effect of the fluctuations on the Mössbauer spectra depends critically on the temperature T. In Figure 7a, T is so low that only the lowest substate is occupied. Fluctuations lead only to CS with higher energies and anti-Stokes lines cannot occur. In Figure 7b, T is so high that all levels are equally populated. The random walk can start from any CS, and the protein can lose or gain energy in each step, producing Stokes and anti-Stokes lines. These arguments explain the quasi-elastic data in Figure 3. Above about 160 K, the thermal energies are much larger than ΔEmax of the βh fluctuations. Thus, all relevant CS are occupied and the anti-Stokes lines are present. The model also applies to the inelastic scattering in Figure 4. In the case where the protein has a well-defined excited state, the scattering is a two-level event with an excitation energy E*. The ratio of the populations of the excited to the ground state is given by exp{−E*/kBT}. At low T, where E*/kBT ≫ 1, the higher state is not occupied and the neutron can only excite the 13304



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upper level; there is no anti-Stokes line. (We neglect the existence of spontaneous transitions.) However, if kBT ∼ E*, the upper state is also occupied and the neutron can force it into the ground state, gaining the energy E* and producing an anti-Stokes line. Figure 4 shows an example of such an inelastic process, with E* = 30 meV, somewhat larger than kBT = 26 meV at ambient temperature. At 295 K, the anti-Stokes line is apparent. At 73 K, however, the excited state is only minimally populated and no anti-Stokes line stands out. The arguments given here depend on a crucial feature of the interaction of photons with matter. The protein is continuously joggled from CS to CS by the external fluctuations with the rate kβ(T). During the nuclear lifetime τMö, it makes n(T) = kβ(T) τMö steps. If the observation time τobs is much shorter than τβ(T) = 1/kβ(T), the protein looks stationary and no dynamic information is obtained. Structural changes can be observed if τobs ≥ τβ(T). What is the time τobs for gamma quanta? Does the Mössbauer gamma ray act like a particle with very short τobs or like a wave described by an extended wave function?36,37 The answer is given by quantum electrodynamics (QED).38 The gamma ray can be described by a “photon wave function”; even a single Mössbauer gamma ray can observe the result of the steps in the FEL during the lifetime τMö. The result of the observation is surprising. The protein makes a random walk in the FEL during the time τMö! Dynamics. The FEL is the stage in which proteins play, but what are the rules of the play? An individual protein molecule in a sample is buffeted from all sides. A complete treatment of the resulting Mössbauer spectrum is not yet in sight, but some insight is in reach. Some guidance can be found in early Mössbauer experiments. Ruby and Bolef were the first to acoustically modulate the gamma rays from a 57Fe source39 and measure the spectrum of the target. The experiment showed the expected effect: sidebands appeared. The mechanism is similar to the effect of vibrations in lattices1 and has been treated many times.40−42 The modulation can be understood. Assume that the external fluctuations move the nucleus sinusoidally with frequency Ω and amplitude x0. The spectrum then shows upshifted and downshifted sidebands separated by energies nhΩ, with weights given by |Jn(qx0)|2, where Jn(qx0) is a Bessel function and x0 a spatial amplitude. In proteins, there are an extremely large number of fluctuations and motions, each adding to the spectrum so that it looks like a continuum. In the Ruby−Bolef type experiment, the sidebands are produced by a piezoelectric crystal modulating the source. In the experiments discussed here, the incoming gamma ray is monochromatic, but the proteins in the absorber are modulated by the internal and external fluctuations.

Figure 8. (a) Temperature dependence of the 57Fe msd in metmyoglobin. (b) The density D(E) of vibrational states at 57Fe as a function of the energy E and the corresponding temperature.8 (c) Comparison of the sharp Lorentzians as measured45 and calculated using the D(E) in part b.

from the Debye law. To describe the temperature dependence of the recoilless fraction f(T), we use the model that explains the PDT.10,27 We assume that vibrations with energies above ℏkMö = 4.7 neV, corresponding to a phonon temperature TMö < 1 K, shift the Mössbauer lines. Vibrations can extend to very high energies, but above any temperature T, vibrations with E > kBT are essentially absent. Thus, the fraction 1 − f(T) of proteins that have been shifted away from the central line is given by

V. VIBRATIONS From below 20 K to Tc, the msd increases approximately linearly with increasing temperature, as shown in Figure 8a.4−6,43 Thus, the elastic fraction f(T) decreases. We propose that this effect is caused by the same mechanism that acts above 150 K, namely, shifting of the sharp Mössbauer lines into sidebands, as shown in Figure 6. Since the external fluctuations are too slow below 150 K, the shifts must be caused by other fluctuations. Vibrations are the plausible choice. The spectra of protein vibrations are described by D(E), the vibrational density of states. As expected, the spectrum is complex,7,44,45 but D(E) as shown in Figure 8b suffices for understanding the effect of the vibrations. Below about 50 K, the D(E) follows the famous Debye E2 law. Above 50 K, the shape deviates markedly

∫0

1 − ftf (T ) = const.

E(T )

D(E) dE

(5)

The subscript “tf” stands for thermal fluctuations. The lower limit on the integral is essentially zero, and the upper limit is given by E(T) = kBT. Figure 8c displays the calculated 1 − f tf(T) extracted from Figure 8b and the measured 1 − f(T) extracted from Figure 8a. Normalized at 180 K, they show the same temperature dependence. Thus, the vibrations are responsible for the decrease of the recoilless fraction with temperatures observed up to 150 K. However, vibrations do not stop at about 150 K. Figure 8c therefore shows the extension of 1 − f(T) up to 300 K. The agreement between the experimental and calculated data is remarkable. The approach 13305


The Journal of Physical Chemistry B used here cannot predict the shape of the spectrum. We have used a more sophisticated approach, based on the theory of Singwi and Sjölander,33 to explain the Mössbauer data above 150 K.32 We will extend this approach to vibrations in a separate publication. The fact that thermal fluctuations shift the Mössbauer line implies that these fluctuations cannot be purely harmonic. Thus the harmonic potential of vibrations is corrugated.

δE

ΔE 10 neV 1 μeV 1−10 meV

The energies in Table 1 are very small. Supporting evidence for such levels comes from the superb spectral hole-burning experiments by Thorn Leeson and co-workers.46,47 They studied the spectral diffusion of the heme group from 100 mK to 23 K. They found a hierarchical organization of tiers with barriers ranging from 2 to 100 meV. The fact that energies from neV to meV are observed leads to the question: Is the FEL organized in tiers as we suggested,48 or is there a continuum of conformational substates with random spacings extending from neV to tens of meV? (4) The protein is agitated simultaneously by the external fluctuations and vibrations. The large contribution of the vibrations at ambient temperatures suggests that they must be taken into account in the discussion of the protein dynamics. (5) The microscopic model presented in section IV explains quasi-elastic processes in proteins. They are caused by the random walk in the free-energy landscape in which the protein gains or loses small amounts of energy. Inelastic scattering, in contrast, is essentially a one-step process in which the incident neutron excites the protein into a vibrational level or de-excites this level at temperatures where it is measurably occupied.

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REFERENCES

(1) Chen, Y.-L.; Yang, D. P. Mössbauer Effect in Lattice Dynamics; Wiley-VCH: Weinheim, Germany, 2007. (2) Bee, M. Localized and Long-Range Diffusion in Condensed Matter: State of the Art of QENS Studies and Future Prospects. Chem. Phys. 2003, 292, 121−141. (3) Sturhahn, W. Nuclear Resonant Spectroscopy. J. Phys.: Condens. Matter 2004, 16, S497. (4) Keller, H.; Debrunner, P. G. Evidence for Conformational and Diffusional Mean Square Displacements in Frozen Aqueous Solution of Oxymyoglobin. Phys. Rev. Lett. 1980, 45, 68−71. (5) Parak, F.; Frolov, E. N.; Mössbauer, R. L.; Goldanskii, V. I. Dynamics of Metmyoglobin Crystals Investigated by Nuclear Gamma Resonance Absorption. J. Mol. Biol. 1981, 145, 825−833. (6) Doster, W.; Cusack, S.; Petry, W. Dynamical Transition of Myoglobin Revealed by Inelastic Neutron Scattering. Nature 1989, 337, 754−756. (7) Achterhold, K.; Keppler, C.; Ostermann, A.; van Bürck, U.; Sturhahn, W.; Alp, E. E.; Parak, F. G. Vibrational Dynamics of Myoglobin Determined by the Phonon-Assisted Mössbauer Effect. Phys. Rev. E 2002, 65, 051916. (8) Fenimore, P. W.; Frauenfelder, H.; McMahon, B. H.; Parak, F. G. Slaving: Solvent Fluctuations Dominate Protein Dynamics and Functions. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 16047−16051. (9) Fenimore, P. W.; Frauenfelder, H.; McMahon, B. H.; Young, R. D. Bulk-Solvent and Hydration-Shell Fluctuations, Similar to {alpha}and {beta}-Fluctuations in Glasses, Control Protein Motions and Functions. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 14408−14413. (10) Frauenfelder, H.; Chen, G.; Berendzen, J.; Fenimore, P. W.; Jansson, H.; Mc. Mahon, B. H.; Stroe, I. R.; Swenson, J.; Young, R. D. A Unified Model of Protein Dynamics. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 5129−5134. (11) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976. (12) Bakk, A. Heat Capacities of Solid State Proteins: Implications for Protein Stability in Solution. Physica A 2002, 313, 540−548. (13) Miyazaki, Y.; Matsuo, T.; Suga, H. Glass Transition of Myoglobin Crystal. Chem. Phys. 1993, 213, 303−308. (14) Cooper, A. Thermodynamic Fluctuations in Protein Molecules. Proc. Natl. Acad. Sci. U.S.A. 1976, 73, 2740−2741. (15) Cooper, A. Protein Fluctuations and the Thermodynamic Uncertainty Principle. Prog. Biophys. Mol. Biol. 1984, 44, 181. (16) Austin, R. H.; Beeson, K. W.; Eisenstein, L.; Frauenfelder, H.; Gunsalus, I. C. Dynamics of Ligand Binding to Myoglobin. Biochemistry 1975, 14, 5355. (17) Elber, R.; Karplus, M. Multiple Conformational States of Proteins: a Molecular Dynamics Analysis of Myoglobin. Science 1987, 235, 317−321. (18) Garcia, A. E.; Blumenfeld, R.; Hummer, G.; Krumhansl, J. A. Multi-Basin Dynamics of a Protein in a Crystal Environment. Physica D 1997, 107, 225−239. (19) Kitao, A.; Hayward, S.; Go, N. Energy Landscape of a Native Protein: Jumping-among-Minima Model. Proteins 1998, 33, 496−517. (20) Lubchenko, V.; Wolynes, P. G.; Frauenfelder, H. Mosaic Energy Landscapes of Liquids and the Control of Protein Conformational

Table 1. Order of Magnitude of Spacings in the FEL 1 neV 10 neV 100 μeV

ACKNOWLEDGMENTS

The work was supported by Department of Energy Contract DE-ACS206NA25396. Illuminating discussions with Ben McMahon helped us to understand the role of vibrations. Without help from Marlan Scully and Dmitri Voronine, who explained the true nature of light to us, we would still wonder if our model is based on solid ground. H.F. dedicates the work to Professor Peter G. Wolynes on the occasion of his 60th birthday, with thanks for many years of interaction and with the optimistic hope that the collaboration will continue for more years.

VI. CONCLUSIONS The present paper leads to a number of conclusions: (1) The separation of Mössbauer or QENS spectra into an elastic line and a broad band is misleading. The entire spectrum consists of Lorentzians with twice the natural line width but shifted from the center as drawn in Figure 6. (2) The spectral shifts can occur because proteins possess a large number of different conformational substates with different energies, described by the free-energy landscape. The shifts are caused by fluctuations that move the protein from substate to substate. In protein crystals, the βh fluctuations in the hydration shell dominate. During the lifetime τMö, a protein makes n(T) = kβ(T) τMö steps of average size δE in the high-dimensional free-energy landscape and winds up in one of the vast number of substates with an energy E0 + ΔE. (3) An order-of-magnitude evaluation of the step-size δE and the maximum energy ΔE yields the result summarized in Table 1.

vibrations Mössbauer QENS

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1 505 665 7744. Notes

The authors declare no competing financial interest. 13306



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Dynamics by Glass-Forming Solvents. J. Phys. Chem. B 2005, 109, 7488−7499. (21) Barth, A.; Zscherp, C. What Vibrations Tell about Proteins. Q. Rev. Biophys. 2002, 35, 369−430. (22) Kremer, F.; Schönhals, A. Broadband Dielectric Spectroscopy; Springer: Berlin, 2003. (23) Garcia, A. E.; Krumhansl, J. A.; Frauenfelder, H. Variations on a Theme by Debye and Waller: From Simple Crystals to Proteins. Proteins 1997, 29, 153−160. (24) Frauenfelder, H.; Petsko, G. A.; Tsernoglou, D. TemperatureDependent X-ray Diffraction as a Probe of Protein Structural Dynamics. Nature 1979, 280, 558−563. (25) Doster, W.; Busch, S.; Gaspar, A. M.; Appavou, M.-S.; Wuttke, J.; Scheer, H. Dynamical Transition of Protein-Hydration Water. Phys. Rev. Lett. 2010, 104, 098101. (26) Doster, W. The Two-Step Scenario of the Protein Dynamical Transition. J. Non-Cryst. Solids 2011, 357, 622−628. (27) Chen, G.; Fenimore, P. W.; Frauenfelder, H.; Mezei, F.; Swenson, J.; Young, R. D. Protein Fluctuations Explored by Inelastic Neutron Scattering and Dielectric Relaxation Spectroscopy. Philos. Mag. 2008, 88, 3877−3883. (28) Knapp, E. W.; Fischer, S. F.; Parak, F. Protein Dynamics from Mossbauer Spectra. The Temperature Dependence. J. Phys. Chem. 1982, 86, 5042−5047. (29) Parak, F.; Knapp, E. W.; Kucheida, D. Protein Dynamics: Mössbauer Spectroscopy on Deoxymyoglobin Crystals. J. Mol. Biol. 1982, 161, 177−194. (30) Nowik, I.; Bauminger, E. R.; Cohen, S. G.; Ofer, S. Spectral Shapes of Mössbauer Absorption and Incoherent Neutron Scattering from Harmonically Bound Nuclei in Brownian Motion: Applications to Macromolecular Systems. Phys. Rev. A 1985, 31, 2291−2299. (31) Mössbauer, R. L. Gamma-Resonance and X-ray Investigations of Slow Motions in Macromolecular Systems. Hyperfine Interact. 1987, 33, 199−222. (32) Young, R. D.; Frauenfelder, H.; Fenimore, P. W. Mössbauer Effect in Proteins. Phys. Rev. Lett. 2011, 107, 158102. (33) Singwi, K. S.; Sjölander, A. Resonance Absorption of Nuclear Gamma Rays and the Dynamics of Atomic Motions. Phys. Rev. 1960, 120, 1093−1102. (34) Zurek, W. H. Decoherence, Einselection, and the Quantum Origins of the Classical. Rev. Mod. Phys. 2003, 75, 715−775. (35) Pound, R. V.; Rebka, G. A. Apparent Weight of Photons. Phys. Rev. Lett. 1960, 4, 337−341. (36) Lamb, W. E., Jr. Anti-Photon. Appl. Phys. B: Lasers Opt. 1995, 60, 77−84. (37) Glauber, R. J. Nobel Lecture: One Hundred Years of Light Quanta. Rev. Mod. Phys. 2005, 78, 1267−1278. (38) Scully, M. O.; Zubairy, M. S. Quantum Optics; Cambridge University Press: Cambridge, U.K., 1997. (39) Ruby, S. L.; Bolef, D. I. Acoustically Modulated γ Rays from Fe57. Phys. Rev. Lett. 1960, 5, 5−7. (40) Shapiro, F. L. The Mössbauer Effect. Sov. Phys. Usp. 1961, 4, 881−887. (41) Abragam, A. L’Effet Mössbauer; Gordon and Breach: New York, 1964. (42) Pfeiffer, L.; Heiman, N. D.; Walker, J. C. Mössbauer Sidebands by rf Excitation of Magnetic Materials. Phys. Rev. B 1972, 6, 74−89. (43) Parak, F.; Achterhold, K. Protein Dynamics on Different Timescales. J. Phys. Chem. Solids 2005, 66, 2257−2262. (44) Adams, K. L.; Tsoi, S.; Yan, J.; Durbin, S. M.; Ramdas, A. K.; Cramer, W. A.; Sturhahn, W.; Alp, E. E.; Schulz, C. Fe Vibrational Spectroscopy of Myoglobin and Cytochrome f. J. Phys. Chem. B 2006, 110, 530−536. (45) Zeng, W.; Barabanschikov, A.; Wang, N.; Lu, Y.; Zhao, J.; Sturhahn, W.; Alp, E. E.; Sage, J. T. Vibrational Dynamics of Oxygenated Heme Proteins. Chem. Commun. 2012, 48, 6340−6342. (46) Thorn Leeson, D.; Wiersma, D. A.; Fritsch, K.; Friedrich, J. The Energy Landscape of Myoglobin: an Optical Study. J. Phys. Chem. B 1997, 101, 6331−6341.

(47) Ponkratov, V. V.; Friedrich, J.; Vanderkooi, J. M.; Burin, M. L; Berlin, Y. A. Physics of Proteins at Low Temperature. J. Low Temp. Phys. 2004, 137, 289−317. (48) Ansari, A.; Berendzen, J.; Bowne, S. F.; Frauenfelder, H.; Iben, I. E. T.; Sauke, T. B.; Shyamsunder, E.; Young, R. D. Protein States and Proteinquakes. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 5000−5004. (49) Achterhold, K.; Ostermann, A.; Moulin, M.; Haertlein, M.; Unruh, T.; Parak, F. G. Dynamical Properties of the Hydration Shell of Fully Deuterated Myoglobin. Phys. Rev. E 2011, 84, 041930-1−6.

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