A wave-mechanical model of incoherent quasielastic scattering in complex systems

Significance

Quasielastic incoherent neutron scattering (QENS) is a key tool for the exploration of complex systems, such as liquids, polymers, glasses, and biomolecules. A considerable number of neutron facilities exist and more are being planned. Understanding QENS is important, both for comprehending and applying the science and making efficient use of the facilities. We claim that the present explanation of QENS is incomplete. We propose a wave-mechanical model, consistent with neutron diffraction. It is based on the free-energy landscape and treats the neutrons as de Broglie wave packets. The model is supported by experiments and has predictive power. It provides significant insight into the dynamics of proteins and may lead to a better understanding of biological processes.

Abstract

Quasielastic incoherent neutron scattering (QENS) is an important tool for the exploration of the dynamics of complex systems such as biomolecules, liquids, and glasses. The dynamics is reflected in the energy spectra of the scattered neutrons. Conventionally these spectra are decomposed into a narrow elastic line and a broad quasielastic band. The band is interpreted as being caused by Doppler broadening due to spatial motion of the target molecules. We propose a quantum-mechanical model in which there is no separate elastic line. The quasielastic band is composed of sharp lines with twice the natural line width, shifted from the center by a random walk of the protein in the free-energy landscape of the target molecule. The walk is driven by vibrations and by external fluctuations. We first explore the model with the Mössbauer effect. In the subsequent application to QENS we treat the incoming neutron as a de Broglie wave packet. While the wave packet passes the protons in the protein and the hydration shell it exchanges energy with the protein during the passage time of about 100 ns. The energy exchange broadens the ensemble spectrum. Because the exchange involves the free-energy landscape of the protein, the QENS not only provides insight into the protein dynamics, but it may also illuminate the free-energy landscape of the protein–solvent system.

Quasielastic effects are a key to understanding the dynamics of complex systems, from water to proteins (1, 2). A novice trying to understand quasielastic incoherent neutron scattering (QENS) is easily mystified. “Quasielastic” is usually taken to mean broadening of the elastic line due to spatial diffusion of the scattering particle. This definition is vague. We introduce a model that permits an unambiguous definition. It describes the QENS of proteins as involving a random walk in the free-energy landscape (FEL), driven by external fluctuations and by thermal vibrations. During the walk, the neutrons exchange energy with the protein, thus broadening the energy spectrum.

In QENS the energy spectrum I(ΔE) of the scattered neutrons is measured as a function of the energy transfer ΔE relative to the energy of the elastic line at ΔE = 0. At present the QENS spectra are separated into a narrow elastic peak and a broad quasielastic band shown schematically in Fig. 1A. The band is taken to consist of broad Lorentzians with width Γ_h centered at ΔE = 0 as sketched in Fig. 1B. The broadening is attributed to spatial motion of the target atoms, for instance by continuous diffusion, by jumps from one lattice site to another, or by conformational changes in proteins. The motions lead to different width Γ_h for different proteins. We call this model SMM, for “spatial motion model,” and discuss it in more detail later. We have introduced a radically different model, ELM, for “energy landscape model” (3). In the ELM, there is no separate elastic line pinned to the center. The entire spectrum is composed of a very large number of spectral lines with twice the natural line width as shown in Fig. 1C. Such a spectrum is called “inhomogeneous” (4, 5). The lines are shifted from the center by transitions among the conformational substates of the FEL. Different proteins experience different energy shift. The shift energies are taken from the spectrum of low-energy soft modes of the system. We explain the ELM in more detail below. The two models are complementary because every transition in the energy landscape involves a change in the protein conformation and vice versa. The ideal model for the QENS would treat both aspects together. Such a model does not yet exist and we are left exploring which model explains the experimental data more convincingly, does not contradict experimental evidence, and uses fewer fit parameters. The present work treats protein; other systems such as water may lead to different conclusions.

Fig. 1.

(A) Conventionally the elastic line and the quasielastic band in neutron scattering are treated as separate phenomena. (B) The broad band is usually assumed to be composed of Lorentzians of different widths and amplitudes, centered at ΔE = 0 (black curves). The sum is shown in red. (C) The proposed model (ELM) is composed of a very large number of narrow, shifted Lorentzians and has no separate elastic line. B and C adapted from ref. 4.

Mössbauer Effect

Neutron scattering is not the best technique to study the concepts of protein dynamics because its instrumental energy resolution is poor. This fact is evident in Fig. 2A, where the resolution function R(ΔE) hides the central part of the QENS spectrum (6). The spectra of the SMM and of the ELM have similar wings, but differ unmistakably near the center. The SMM claims to see a sharp elastic line and an underlying broad band; the ELM spectrum is smooth and without a separate sharp line. To distinguish the two models, the energy resolution must have about the same width as the elastic line. The QENS violates this condition. Fortunately there is a stand-in for the QENS with a superb energy resolution, namely the Mössbauer effect (7, 8), which also displays quasielastic effects. We therefore treat the Mössbauer effect first and then apply what we learned to the QENS. In the Mössbauer experiments a radionuclide, usually ⁵⁷Fe, is the source of the gamma radiation. The nuclide ⁵⁷Fe emits a γ-ray with energy E_Mö = 14.412497 keV and a mean life τ_Mö = 141 ns corresponding to a rate coefficient k_Mö = 1/τ_Mö = 7.1 × 10⁷ s⁻¹ and a natural line width Γ_Mö = 4.66 neV. Usually, the ⁵⁷Fe nucleus recoils; the emitted gamma ray loses the recoil energy and shifts out of resonance with the 14.4-keV transition. However, if the ⁵⁷Fe atom is embedded in a solid, some of the atoms do not recoil so that the emitted gamma rays carry the full energy E_Mö and have the natural line width Γ_Mö. The Mössbauer spectrum is measured by the transmission of γ-rays from a ⁵⁷Fe source moving with a velocity v through a stationary sample containing ⁵⁷Fe embedded for instance in a protein and kept at the temperature T. A fraction f(T) of the incoming recoilless Mössbauer photons elastically excites the 14.4-keV level. In the thin-absorber limit the transmission Tr(ΔE) is related to the scattering amplitude S(ΔE) by Tr (ΔE) = 1 − const. S(ΔE),where ΔE = E_Mö v/c. Mössbauer spectra are evaluated by plotting Tr(ΔE) versus ΔE or versus the corresponding source velocity v in mm/s, where 1 mm/s corresponds to 48.8 neV. Fig. 2 B and C displays Mössbauer spectra (9, 10). At 80 K, the spectrum can be fit with a single Lorentzian with about twice the natural line width Γ_Mö. At 295 K the spectrum is broad and can be fit either with a sharp line and a broad band (SMM) or with a broad spectrum consisting of a very large number of Mössbauer lines without a central narrow line (ELM).

Fig. 2.

(A) Energy spectrum of perdeuterated metmyoglobin measured with QENS (red circles). The resolution function R(E) is scaled to maximum at zero energy and assumed to be approximately Gaussian (blue lines). The spectrum involves 72% H atoms from hydration water and 28% from the protein. Adapted from Achterhold et al. (6). (B) Mössbauer spectrum for carbonmonoxy–myoglobin at low temperature. Adapted from ref. 9. (C) The spectrum measured using the Mössbauer effect for hydrated metmyoglobin at 295 K. Adapted from ref. 10. Hydration is 0.4 for A and C. Note the different energy scales in A and B.

ELM

The Mössbauer photons emitted by a stationary ⁵⁷Fe source always have the energy E_Mö = 14.4 keV, the lifetime τ_Mö = 141 ns, and the natural line width Γ_Mö = 4.66 neV. The resonance levels in the ⁵⁷Fe absorber have the same energy and line width. In each observed event, a Mössbauer photon is resonantly absorbed by a ⁵⁷Fe atom in the target. At low temperatures the spectrum shows a narrow line with a width of about 2Γ_Mö as in Fig. 2B. If the target ⁵⁷Fe atom is in a protein and observed at ambient temperature, the spectrum shows broad wings as in Fig. 2C. In the ELM we do not introduce a separate sharp central line, but interpret the observed spectrum as being smooth, composed of lines with width 2Γ_Mö (Fig. 1C). Each ⁵⁷Fe atom has a different resonance energy owing to the protein being in a different conformational substate. An incoming quantum with energy E_Mö can only be absorbed by a transition with the same energy E_Mö. If the absorption spectrum does not show the line at E_Mö, but at E_Mö + ΔE, the target must have provided the energy ΔE during the lifetime τ_Mö. We propose that the energy fluctuations in the protein–solvent system are responsible for the energy shifts. A protein can assume a large number of different conformations with energies up to a few eV (11–14). The ⁵⁷Fe atom is coupled to the protein–solvent system and its FEL. At very low temperatures, transitions between substates are too slow to be observed. A protein in a given substate remains in that substate, and the Mössbauer spectrum consists of a single narrow line. At high temperatures, however, a protein fluctuates rapidly among substates. The Mössbauer photon is a wave packet (15) that exchanges energy with the ⁵⁷Fe atom during the passage time given by the Mössbauer lifetime τ_Mö. During this time the protein makes a random walk in the energy landscape as shown in Fig. 3 (3). When the Mössbauer quantum is registered the spectrometer records the absorption line at E_Mö + ΔE. ΔE does not depend on the energy of the initial substate and can be positive or negative. The result is a broad band. If ΔE << k_BT, the band is symmetric with the center set at ΔE = 0 as in Figs. 1 and 2. The energy for the random walk is provided by the heat bath in which the protein lives (16). The transitions in the FEL are driven by three types of fluctuations known from the physics of solids, glasses, and supercooled liquids (17). They are the α-fluctuations in the bulk solvent (13, 18), the β_h-fluctuations in the hydration shell (19–21), and vibrations (22). The α-processes are structural fluctuations in the solvent; they modulate the shape of the protein and can thereby induce transitions among the substates. Their rate coefficient k_α(T) is inversely proportional to the solvent viscosity; the α-fluctuations are unobservable in solids. The β_h-fluctuations are dielectric fluctuations in the hydration shell. Their rate coefficient k_β(T) depends on the degree of hydration and they are absent in dehydrated proteins (23, 24). Here we use experimental data from systems where the α-fluctuations are absent. We have treated the effect of thermal vibration previously (3). Thus, we restrict the treatment on the effect of the β_h-fluctuations.

Fig. 3.

Random walk of a protein in the energy landscape. In the Mössbauer effect, the incoming photon hits a protein in a specific substate. During the lifetime τ_Mö, the protein makes a random walk in the energy landscape, gaining or losing the energy ΔE. The jumps in the FEL are caused by the β_h-fluctuations in the hydration shell. The time for one jump of magnitude ±δE is τ_β. The total energy shift is approximately given by ΔE ∼ ± δE(τ_Mö/τ_β)1/2. The model is assumed to apply also to QENS.

The exploration of the ELM starts with the elastic fraction f(T), the primary result of most experiments. Unfortunately many papers do not report f(T), but invert the Lamb–Mössbauer relation

$$f\left(T\right)=\text{exp}\left(-q^2\langle x^2\left(T\right)\rangle \right)$$ [1]

and publish the mean-squared displacement (msd), <x²(T)>.

Here q is the momentum transfer. This relation is only valid in the Gaussian approximation, which can be wrong in complex systems (25). This leaves us in a quandary. We can either use Eq. 1 to extract f(T) or we can use the msd despite its limited validity. We select the second route and plot in Fig. 4 the msd from three Mössbauer experiments (26–28) and three QENS experiments (29, 30). The figure shows four striking features: (i) The curves are all similar despite the fact that they involve very different targets, techniques, samples, and times. (ii) The msd increases nearly linearly from about 10 K to a temperature T_D ∼ 180 K. The slope is similar for the Mössbauer experiments and the QENS. T_D is approximately the same for QENS and the Mössbauer effect. (iii) At T_D, the slope of the msd in hydrated proteins increases dramatically. This effect is called “protein dynamical transition,” or PDT (29). (iv) In dehydrated proteins, the PDT is absent and the nearly linear T dependence of the msd continues to at least 300 K. We now compare the two models in their ability to explicate these features. The SMM can explain feature (ii) as being caused by vibrations (31), but has little to say for the rest. The ELM explains all features: (i) The similarity implies underlying general mechanisms. The principal features of the ELM, namely the existence of the FEL and the control through fluctuations, are similar in all systems in Fig. 4. (ii) The approximately linear increase with temperature of the msd below about 180 K is explained in both models as being caused by the thermal vibrations (3). (iii) The ELM quantitatively explains the PDT: The change in slope is due to the kinetic onset of the β_h-fluctuations in the hydration shell (3, 18, 19, 32). Fig. 3 implies that sizable shifts can only be observed if the β_h-fluctuations are faster than the characteristic Mössbauer rate or if

$${\tau }_{\beta }<{\tau }_{Mö}.$$ [2]

This simple relation is significant because it pinpoints the temperature T_D where the protein dynamics changes from vibration-dominated to external-fluctuation controlled. Below T_D thermal vibrations dominate and proteins are essentially nonfunctional. Above T_D the external fluctuations are crucial; they shift the lines from the center thereby decreasing f(T), increasing the msd, and producing the broad spectrum. (iv) In the absence of the external fluctuations no dynamical transition occurs.

Fig. 4.

msd <x²> for three samples each of Mössbauer effect (open symbols) and QENS (closed symbols). Mössbauer effect: open squares, oxymyoglobin in aqueous solution (27); green diamonds, Fe²⁺ in glycerol–water solution (28); blue circles, hydrated deoxymyoglobin crystals (26). QENS: closed squares, GST powder with 0.5 hydration by D₂O from ref. 30; green diamonds, crambin in 50% water–ethanol with full deuteration (30); blue circles, hydrated metmyoglobin powder at 0.4 hydration (29).

The β_h-spectrum with one free parameter predicts the shape of the Mössbauer spectrum (33) and the temperature dependence of the heat flow measured by differential scanning calorimetry (34). Additional support for the claims that the Mössbauer spectra are inhomogeneous, composed of narrow lines, and that external fluctuations produce these narrow sidebands comes from experiments where the fluctuations are produced by piezoelectric crystals (35, 36).

QENS Explained

The key experimental data in the Mössbauer effect and in the QENS are similar: The elastic intensities have about the same temperature dependence, the PDT appears at about the same temperature T_D, and the spectra have quasielastic wings above T_D. This similarity suggests that the concepts found in the Mössbauer experiments can also explain the QENS data. An additional clue comes from experiments with very cold neutrons reflected from a glass surface. Sidebands with widths of about 3 neV and energy splittings of 3–9 neV are observed when the surface is vibrated with a piezoelectric transducer (37). Conventionally, low-energy neutron scattering treats the incoming neutrons as plane waves and the scattered neutrons as spherical waves (38). The insight gained from the Mössbauer effect suggests, however, that the neutron should be described by a wave packet. Strong support for this assertion comes from interference and diffraction experiments with thermal neutrons (39–43). Thus, we postulate that in the QENS the neutron is a wave packet with an effective transit time τ_n and transit (coherence) length δ_n. This packet exchanges energy with the target proton during the time τ_n and emerges with an energy shift ΔE. What characterizes this wave packet? The wave packet of the Mössbauer quantum is essentially a 1D structure described by the nuclear lifetime τ_Mö. The wave packet of a neutron, however, is a 3D structure, produced and shaped by the experimental setup. The coherence lengths are different in the longitudinal, transverse, and vertical directions (42). Such a 3D packet hits a protein of typical diameter 5 nm. The resulting interaction is complex (44, 45). For the interpretation of the scattering data, the values of τ_n and δ_n are crucial. Interference experiments, where the length is determined by varying the transit time in two different branches of an interferometer, yield a coherence length of about 10 nm. In contrast, diffraction experiments produce wave packets with a length of about 100 μm (40). Assuming a neutron velocity of 1 km/s, the corresponding passage times τ_n are about 10 ps for the double-slit interference and 100 ns for the diffraction experiment. Which value of τ_n explains the experimental observations? The coherence time is determined by collimators in the neutron transport system, typically leading to a value on the order of 100 ns (46). This choice is supported by the data in Fig. 4. Eq. 2 implies that T_D is given by the condition τ_β ∼ τ_Mö. Fig. 4 shows that T_D is similar for the QENS and the Mössbauer effect. Assuming that the hydration shell fluctuations are also similar, the near-coincidence implies that the passage time τ_n for neutrons is about the same as for the Mössbauer photons, about 100 ns, in agreement with the value stated above. We adopt this value to obtain the line width of the QENS lines. With τ_n ∼ 100 ns and the Heisenberg uncertainty relation, the line width is Γ_n ∼ 10 neV, essentially the same as for the Mössbauer photon. This value is supported by the elegant experiment of Felber et al. (37), which shows individual lines with widths of a few neV. The ELM thus predicts that the broad energy spectrum in neutron scattering is inhomogeneous, composed of individual narrow lines as sketched in Fig. 1C. A word of caution is, however, appropriate: Scattered neutrons lose energy because the target recoils. If the recoil energy loss were large compared with typical energy shifts in Figs. 1C and 3 the concept of an inhomogeneous spectrum would lose validity. Fortunately, the recoil energy loss is small under the usual conditions of small energies and small momentum transfers so that the concept of an inhomogeneous spectrum as in Fig. 1C is valid.

Summary and Conclusions

We propose an alternative model for the quasielastic effects in the Mössbauer effect and neutron scattering on proteins. The key observations to be explained are shown in Figs. 2 and 4. At temperatures well below 200 K, the spectrum shows the elastic line with about twice the natural line width (Fig. 2B). Above about 180 K the elastic line sprouts wings and the msd increases rapidly (Fig. 4). For the past half-century the quasielastic effects in neutron scattering and in the Mössbauer effect have been treated by assuming that the sharp line and the broad band represent two separate processes as sketched in Fig. 1A. The broad band is taken to be caused by spatial motions such as diffusion. We call this model the SMM. This model has problems. A fit to Fig. 2C shows no evidence for an elastic line. The bandwidth of the broad band in the SMM exceeds the natural line width by orders of magnitude, but no mechanism for such an enormous broadening has been proposed. The SMM has no mechanism that predicts the PDT, the temperature T_D, and the effect of hydration. The ELM, does not have these problems. It treats the entire spectrum as being composed of components with essentially twice the natural line width as in Fig. 1C. It predicts the PDT and the time dependence of the msd from below 50 K to above 300 K for hydrated and dehydrated myoglobin. The center of the spectrum is not permanently occupied by a specific protein; each protein in the sample can occupy it at some time as suggested by Fig. 3. Thus, we believe that the ELM, the model based on diffusion in the FEL, explains the experimental data from the QENS and the Mössbauer effect better than the current model. Because the passage time τ_n can be changed experimentally, there is some hope of gaining some insight into the FEL.