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Mean Square Displacements from Elastic Incoherent Neutron Scattering Evaluated by Spectrometers Working with Different Energy Resolution on Dry and Hydrated (H2O and D2O) Lysozyme Salvatore Magazu`,*,† Federica Migliardo,†,‡ and Antonio Benedetto† Department of Physics, UniVersity of Messina, C.da Papardo Sperone n° 31, P.O. Box 55, Vill. S. Agata 98166 Messina, Italy, and Structure and Dynamics Laboratory of Molecular Materials, Unesco-L’Oreal UniVersity of Lille I, UMR CNRS 8024-59655 VilleneuVe d’Ascq CEDEX, France ReceiVed: March 17, 2010; ReVised Manuscript ReceiVed: June 6, 2010
The main aim of the present paper is the evaluation of the effects of the instrumental energy resolution on the mean square displacement (MSD) obtained by elastic incoherent neutron scattering (EINS). In particular, this study is performed in the time domain, through the time-Fourier transform of the elastically scattered neutron intensity, and is mainly focused on the connection between the system MSD and the measured MSD. It is shown how in the case of EINS, the instrumental energy resolution gives rise to the time integration of the time-dependent system MSD function weighted in time by the resolution function. The formulated approach is applied to the data collected on dry and hydrated (H2O and D2O with h ) 0.4) lysozyme samples by two spectrometers working with a different instrumental resolution (the IN10 and IN13 spectrometers of the Institute Laue-Langevin). As a result, the procedure furnishes an excellent agreement for the system MSD evaluated in the low temperature range up to T ) 40 K. I. Introduction It is well-known that neutron scattering allows the characterization of the structural and dynamical properties of a wide class of material systems, such as polymers, glasses, proteins, and so forth. These properties can be described by the timedependent spatial correlation function G(r,t) introduced by Van Hove,1 whose space-time Fourier transform corresponds to the scattering function S(Q,ω). When the system scattering cross section is mainly incoherent, the relevant contribution to the time-dependent spatial correlation function is given by the selfdistribution function Gs(r,t), which furnishes the probability to find a given particle at a distance r from a given position after a time t.1-3 The experimentally obtained neutron scattering data are also connected with the employed spectrometer instrumental features. This implies that the system observables, for example, distribution functions and mean square displacement (MSD), are influenced by instrumental effects. It is well-known that the energy window and the spanned transferred wave vector values determine the time and space ranges of the observable motions. As far as the spanned transferred wave vector range is concerned, the authors have recently formulated a procedure, the self-distribution function (SDF) procedure, for the MSD evaluation from the elastic incoherent neutron scattering (EINS) data, which is essentially based on the determination of the spatial self-distribution function.4-9 The SDF procedure allows the evaluation of both the total and the partial MSDs through the total and the partial SDFs. In particular, it has been shown that the MSD is not the simple sum of the different partial displacement contributions, but it is the weighed sum of the partial MSDs in which the * To whom correspondence should be addressed. E-mail:smagazu@ unime.it; phone: +39 0906765025; fax: +39 090395004. † University of Messina. ‡ Unesco-L’Oreal University of Lille I.
weights are obtained by a fitting procedure of measured EINS intensity data.4-9 The results of such a procedure have been compared with other approaches, relative to a spatial analysis, reported in the literature.10-12 In this frame, starting from the fact that in the ω-space the experimentally accessible quantity, the measured scattering law SR(Q,ω,∆ω), is the convolution of the scattering law with the instrumental resolution function, it is important to clarify how the measured intensity depends on the instrumental resolution.4-9 The main aim of the present paper is the evaluation of the effects of the instrumental energy resolution on the measured MSD obtained by EINS data. In this regard, several contributions are reported in the literature.13-16 In particular, Gabel and Bellissent-Funel13 have realized a dynamical analysis on C-PhycoCyanin (CPC) in the presence of trehalose starting from the EINS intensity profiles collected with different energy resolutions. These authors have taken into account the convolution of a model scattering law with the instrumental resolution functions of the IN13 and IN16 spectrometers of Institute Laue Langevin (ILL, France); this approach has allowed them to extract both the effective diffusion coefficients and the relaxation times as well as to describe the dynamic behavior of CPC hydration water over a large temperature range. In their approach, the analysis of the effects of the instrumental resolution is performed on elastic scattering data and shows the presence of a quasi elastic contribution in the elastic measured spectra related to non-Gaussian hydration water motions at temperatures higher than 235 K. Moreover, Kneller and Calandrini14 have estimated the influence of the finite instrumental resolution on the elastic intensity for a protein system starting from the assumption that, as far as the internal protein dynamics is concerned, the single particle motions can be described by fractional OrnsteinUhlenbeck processes. This study has allowed the evaluation of the missing part of the quasi elastic intensity profile, which is
10.1021/jp102436y 2010 American Chemical Society Published on Web 06/24/2010
Mean Square Displacements not accessible because of the finite instrumental resolution. The authors also furnished an estimation of the attenuation factor for the observed atomic position fluctuations both assuming a Gaussian and a triangular resolution function; as a result, when the quasi elastic neutron scattering (QENS) half width at halfmaximum increases to a relatively high value, in the lowfrequency region, the measured spectrum differs more and more in respect to the ideal one (see Figures 4 and 5 of ref 14). Finally, Becker and Smith15 have investigated the effects of the energy resolution and of dynamical heterogeneities on EINS spectra for some molecular systems. They considered the convolution of a given QENS scattering law with the instrumental resolution function (a rectangular function). The MSD has then been calculated by evaluating the second derivative of the measured scattering function; this latter consists of two contributions: the first one is connected both to the vibrational and to the elastic incoherent structure factor contributions, while the second one is connected with resolution effects (see Figures 1 and 2 of ref 15). It is also well-known that, in comparison with EINS, incoherent QENS often requires a relatively great amount of material; this limits the technique excluding the investigation of a relevant number of interesting systems, such as those of interest in the biophysical field.17-19 Furthermore, when dealing with incoherent QENS spectra in ω-space, one of the main drawbacks is constituted by a relatively high number of fitting parameters. On the other hand, EINS, through the so-called “elastic-window method” introduced by Alefeld and Kollmar,20 is one of the most effective approaches for evaluating atomic MSD in hydrogenous systems, and it is often preferred to the QENS technique in general because of both the higher neutron flux and the fact that the elastic contribution is often a factor 100 ÷ 1000 higher than the quasi-elastic one at low energy transfer. Doster and co-workers21-23 have proposed an innovative and effective way to get dynamical information by extracting the elastic component of the quasi-elastic scattering spectrum or, otherwise, by performing EINS measurements with different instrumental energy resolutions. In the present paper, we shall take into account this latter case. The proposed approach has been applied to EINS data collected, on dry and hydrated (H2O and D2O) lysozyme samples, by two spectrometers working with different instrumental resolutions; these are the IN13 and IN10 spectrometers of the Institute Laue-Langevin. Dry and hydrated lysozyme systems have been largely studied by several authors in different environments;24-28 in the present work, a comparison of the obtained results with those existing in the literature is performed.
J. Phys. Chem. B, Vol. 114, No. 28, 2010 9269 corresponding to the energy resolution are seen as elastic scatterers, whereas a decrease of the elastic intensity is observed for scattering particles which move faster. This implies that a scattering particle which moves in a time scale between the resolution time of IN13 and IN10 contributes as an elastic process in the IN13 spectra and as a nonelastic process in the IN10 spectra. Raw data were corrected for cell scattering and detector response. Partially deuterated lysozymes in dry, in D2O, and in H2O environments at a hydration value of h ) 0.4 (h ) water/protein weight fraction) have been employed. The considered hydration value has been chosen because the activity of proteins depends crucially on the presence of at least a minimum amount of solvent water.29,30 It is believed that 0.3 g of water per gram of protein is sufficient to cover most of the protein surface with one single layer of water molecules and to fully activate the protein functionality. In particular, for lysozyme, this hydration level was chosen to have a monolayer of water covering the protein surface.31 Data were collected by the IN13 spectrometer in the temperature range of 20 ÷ 310 K and by the IN10 spectrometer in the 20 ÷ 320 K temperature range. III. Theoretical Approach It is well-known that the scattering law and the intermediate scattering function are connected by a direct and inverse time Fourier transform:32,33
S(Q, ω) )
(1)
∫-∞∞ S(Q, ω)eiωtdω
(2)
√2π 1
I(Q, t) )
∫-∞∞ I(Q, t)e-iωtdt
1
√2π
The experimentally accessible quantity in the ω-space, because of the finite energy instrumental resolution ∆ω, is the convolution of the scattering law S(Q,ω) with the instrumental resolution function R(ω; ∆ω), that is, the measured scattering function SR(Q,ω;∆ω)
SR(Q, ω;∆ω) ) S(Q, ω) X R(ω;∆ω) )
∫-∞+∞ S(Q, ω - ω′)R(ω′;∆ω)dω′
(3)
that, taking into account eq 1, yields II. Experimental Section Experimental data were collected at the Institute Laue Langevin (Grenoble, France) by the IN13 and IN10 spectrometers. These spectrometers are characterized by a relatively high energy of the incident neutrons (16 meV) and allow to span a quite wide range of momentum transfer with two different energy resolutions. More specifically, for the IN13 spectrometer, the incident wavelength was 2.23 Å, the Q-range was 0.28 ÷ 4.27 Å-1, and the elastic energy resolution (fwhm) was 8 µeV, which corresponds to an elastic time resolution of 500 ps; for the IN10 spectrometer, the incident wavelength was 6.27 Å, the Q-range was 0.30 ÷ 2.00 Å-1, and the elastic energy resolution (fwhm) was 1 µeV, which corresponds to an elastic time resolution of 4000 ps. Thus, scattering particles which move in a time scale much slower than the characteristic time
SR(Q, ω;∆ω) )
[
1
√2π
]
∫-∞∞ I(Q, t)e-iωtdt
X R(ω;∆ω) )
∫-∞+∞ √ 1 ∫-∞∞ I(Q, t)e-i(ω-ω)tdtR(ω′;∆ω)dω′ ) 2π
[
]
∫-∞ I(Q, t)e-iωtdt √ 1 ∫-∞+∞ eiωtR(ω′;∆ω)dω′ ∞
2π
)
∫-∞∞ I(Q, t)R(t)e-iωtdt
(4)
In the following, the subindex R indicates that the relative function refers to the measured quantity, whereas the absence of this index indicates that the relative function is connected only to the sample.
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IV. Self-Distribution Function Procedure The SDF procedure is a recipe for the MSD evaluation from EINS experiments and has been presented in previous works.4-9 The SDF procedure is essentially based on the determination of the self-distribution function and on its use in the evaluation of the average statistical values of the physical quantity of interest 〈A〉 in agreement with the statistical mechanics definition
〈A〉 )
∫-∞∞ A(r)Gself(r)dr
(5)
The same result for the MSD in 1D space can be obtained considering directly the scalar expression of eq 7, that is, the scalar expression of the average statistical values of the displacement, r2
〈r2〉 )
(6)
(14)
In this case, the normalization condition changes and gives the following result:
∫-∞∞ Gself(r)dr ) 1
in which the spatial self-distribution function, as a probability density, may be normalized to unit
∫-∞∞ Gself(r)dr ) 1
∫-∞∞ r2Gself(r)dr
Gself(r) )
(15)
A
n exp(-r2 /4an) ∑ AnGself n (r) ) ∑ 2(πa )1/2 n
n
n
(16) In the specific case of the MSD evaluation, the dynamic observable A corresponds to the second power of the displacement, r2
〈r2〉 )
∫-∞∞ r2Gself(r)dr
〈r2〉 )
∫-∞∞ r2[4πr2Gself(r)]dr
(8)
A
n exp(-r2 /4an) ∑ AnGself n (r) ) ∑ 3/2 16(πa ) n
n
n
(10) in which ∑nAn ) 1; in addition, the MSD results
〈r2〉 ) 6
∑ Anan ) ∑ An〈r2〉n n
(11)
〈r2〉 ) 1/3〈r2〉
(12)
This implies that eq 11 in 1D yields
∑ Anan ) ∑ An〈r2〉n n
(17)
n
This result is the same as that of eq 13, which was obtained from eqs 11 and 12. To adapt the previous definition of the MSD to the case of the EINS experiments, it is necessary to consider that the instrumental energy resolution influences the physical observables as discussed in section III; therefore, eq 14 can be rewritten as follows:
〈r2〉R )
∫-∞∞ r2GRself(r)dr
(18)
Starting from this relation in the following, we shall analyze the instrumental energy resolution effects on the MSD in the frame of an EINS experiment. V. Instrumental Resolution Effects on MSD Evaluation from EINS Experiments The aim of the present work is the evaluation of the effects of the instrumental resolution on the MSD obtained from EINS experiments which allow the determination of the measured scattering law evaluated at ω ) 0, that is, SR(Q,ω ) 0,∆ω). In the following, we will start from the connection between the MSD and the self-distribution function reported in eq 18
n
in which are present the partial MSDs. This formula highlights that the MSD corresponds to a weighted sum of the different displacement contributions present in the system. 〈r2〉 represents the MSD in 3D space; if 〈r2〉 represents the MSD in 1D space, for isotropic systems, we have that
〈r2〉 ) 2
∑ Anan ) ∑ An〈r2〉n n
(9)
Considering that the self-distribution function can be written as a sum of Gaussian functions, as reported in refs 4-9, the normalization condition gives the following result:
Gself(r) )
〈r2〉 ) 2
(7)
In the case in which the system can be considered isotropic, the volume integral becomes dependent only on the scalar r. In such a case, the normalization condition and the MSD become
∫-∞∞ 4πr2Gself(r)dr ) 1
in which ∑nAn ) 1. The MSD results
n
(13)
∫-∞∞ FTr{SR(Q, ω ) 0)}r2dr ) 〈r2〉R
(19)
where FTr represents the spatial Fourier transform operator. For the evaluation of the MSD, let us consider eq 4 evaluated at ω ) 0; then, eq 19 becomes
∫-∞∞ FTr{SR(Q, ω ) 0)}r2dr ) ∫-∞∞ FTr{ ∫-∞∞ I(Q, t)R(t)dt}r2dr) ∫-∞∞ { ∫-∞∞ Gself(r, t)R(t)dt}r2dr ) ∫-∞∞ R(t){ ∫-∞∞ Gself(r, t)r2dr}dt ) ∫-∞∞ 〈r2〉(t)R(t)dt
〈r2〉R )
(20)
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Figure 1. Normalized time behavior of the system MSD at a fixed τ value, (t;τ) (in black); resolution function for different τRES values, R(t;τIN13) (in red) and R(t;τIN10) (in blue); measured MSD R evaluated by eq 21.
In the present paper, we shall consider two types of MSD: (1) the “system MSD” (i.e., 〈r2〉(t)) and (2) the “measured MSD” (i.e., 〈r2〉R). The first one is a function of time whereas the second one is the MSD value as obtained by EINS; this latter is a pure number and not a function of time. Moreover, it depends on the employed instrumental resolution. From eq 20, the measured MSD corresponds to the integration in the time domain of the product of the system MSD times the resolution function
〈r2〉R )
∫-∞∞ 〈r2〉(t)R(t)dt
(21)
Figure 1 shows the effect on the measured MSD because of employing the resolution function; as it can be seen, at the same system MSD, because of the different employed energy resolutions, different measured MSDs correspond. VI. Results and Discussion By performing EINS experiments which give a direct connection with the scattering law evaluated at ω ) 0, SR(Q,ω ) 0,∆ω), the relation between the measured MSD, 〈r2〉R, and the system MSD, 〈r2〉(t), is given by eq 21, that is, 〈r2〉R ) ∞ 〈r2〉(t)R(t)dt. ∫-∞ One can make the following remarks: (1) The comparison between the MSDs measured for two different systems with the same instrumental resolution yields
〈r2〉1,R - 〈r2〉2,R )
∫-∞∞ [〈r2〉1(t) - 〈r2〉2(t)]R(t)dt (22)
Therefore, by collecting data with the same instrument working at the same resolution, the difference between the measured MSD on two different systems does not correspond to the difference between their MSDs. (2) The comparison between the MSDs measured for the same system evaluated at different instrumental resolutions yields
〈r2〉R1 - 〈r2〉R2 )
∫-∞∞ 〈r2〉(t)[R1(t) - R2(t)]dt
(23)
Figure 2. Comparison between the measured MSDs temperature behavior obtained from data collected by the IN13 and IN10 spectrometers on dry and hydrated (H2O and D2O) lysozyme samples. The system vibrational MSDs have been evaluated by eq 25 and are reported in Table 1.
TABLE 1: Vibrational MSDs and Their Roots, Representative of Mean Displacements (MDs), for Dry and Hydrated (H2O and D2O) Lysozyme Samplesa
sample
IN10 MSD (Å2)
IN13 MSD (Å2)
IN10 MD (Å)
IN13 MD (Å)
agreement
lysozyme dry lysozyme/D2O lysozyme/H2O
0.000288 0.000262 0.000405
0.000227 0.000307 0.000447
0.017 0.016 0.020
0.015 0.017 0.021
88% 94% 95%
a Evaluated by eq 25 from EINS data collected by two ILL spectrometers working with different instrumental resolutions. In particular, the IN13 spectrometer has an energy resolution value of 8 µeV corresponding to an elastic time resolution of 500 ps; the IN10 spectrometer has an energy resolution value of 1 µeV corresponding to an elastic time resolution of 4000 ps.
In Figure 2a-c, the comparison between the measured MSDs obtained for the same systems, that is, dry and hydrated (H2O and D2O with h ) 0.4) lysozyme, respectively, by the IN13 spectrometer working at the energy resolution value of 8 µeV and by the IN10 spectrometer working at the energy resolution value of 1 µeV is shown. The MSD values have been obtained by employing the following common Q-range: 0.5 ÷ 2.0 Å-1. In agreement with
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Figure 3. Effect on the measured MSD because of the different energy resolution of the two spectrometers, IN13 (in red) and IN10 (in blue), when only vibrational motions occur; in particular, as it can be seen, at the same system MSD, different measured MSDs correspond.
eq 21, the MSD evaluated by IN10, in all the temperature ranges, is higher in respect to that evaluated by IN13. This is because that even at the lowest temperature values, where only vibrational contributions are expected to contribute, the measured MSD is the integral of the product between the resolution function and the system MSD. (3) In particular, when only vibrational motions occur, the system MSD can be considered almost constant 〈r2〉(t)f〈r2〉(V) (such a condition is approximately satisfied in the low temperature range up to T ) 40 K); in such a case, the measured MSD from eq 21 results
〈r2〉R )
∫-∞∞ 〈r2〉(t)R(t)dt ) ∫-∞∞ 〈r2〉(V)R(t)dt 〈r2〉R ) 〈r2〉(V)
∫-∞∞ R(t)dt
(24)
Then, starting from eq 24, it is possible to determine the system MSD at the lowest temperature values
〈r2〉(V) )
〈r2〉R
∫-∞∞ R(t)dt
(25)
Figure 3 shows the effect on the measured MSD because of the employment of a different energy resolution, when only vibrational motions occur; as it can be seen, at the same vibrational system MSD, different measured MSDs correspond. What is remarkable is that by applying this procedure to the data collected on the same systems by the two spectrometers IN13 and IN10, working at a different energy resolution, we obtain at the lowest temperature values the same system MSD value (see Figure 2a-c and Table 1). This furnishes an account for the validity of the procedure. VII. The Dynamical Transition Temperature Characterizing the dynamics of biological macromolecules is a key step for understanding their function since the knowledge of their structure alone is not always sufficient. One of the dynamical phenomena of great interest is the so-called “dynamic transition” which has been observed in hydrated
Figure 4. (a) Comparison between the shifted temperature behavior of the measured MSD of dry and D2O hydrated lysozyme obtained from data collected by the IN10 spectrometer. (b) Comparison between the shifted temperature behavior of the measured MSD of dry and D2O hydrated lysozyme obtained from data collected by the IN13 spectrometer. The two hydrated sample MSDs show the dynamical transition at a temperature of about (a) T ) 220 K and (b) T ) 240 K.
proteins as a sudden rise of the measured MSD at a temperature value around T ) 220-250 K,24,25,34-37 which adds itself to other MSD kinks in dry and hydrated proteins.24,25,34-36 In the previous sections, we have shown how the measured MSD depends on both the system MSD and the instrumental resolution. In particular, in the case of EINS experiments, eq 21 has been obtained. As a consequence, this implies that a kink in the temperature behavior of the measured MSD might be apparent. This interpretation has been suggested and experimentally confirmed by other authors.24,37 In particular, in ref 24, the authors showed that the observed sharp rise in the measured MSD is the result of the fact that the protein structural relaxation time reaches the limit of the experimental frequency window; in other words, at a given temperature, the system relaxation time becomes shorter than that corresponding to the slowest time observable with a given energy resolution. In this frame, the dynamical transition temperature obtained by the IN13 spectrometer for the lysozyme hydrated samples should be higher than the value obtained by the IN10 spectrometer for the same samples; in fact, the resolution time of IN10 (τRES ) 4000 ps) is higher than the resolution time of IN13 (τRES ) 500 ps). This circumstance is confirmed by our data: the dynamical transition temperature results at TD ) 220 K for IN10 data and at TD ) 240 K for the IN13 data (see Figure 4a, b). Furthermore, these values are in excellent agreement with the data reported in Figure 3 of ref 24 in which the authors reported the temperature dependence of the relaxation times in hydrated and dry lysozyme obtained from both QENS and dielectric spectroscopy experimental data. More specifically, at a temperature value of T ) 220 K, a relaxation time of about τ ) 4000 ps results in agreement with our IN10 finding; moreover, at the temperature value of T ) 240 K, a relaxation time of about τ ) 500 ps results in agreement with our IN13 finding.
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Figure 5. Conceptual map for the SDF procedure.4-9 In the triangles, the connections among the system functions, i.e., the scattering law S(Q,ω), the intermediate scattering function, I(Q,t), and the distribution function, G(r,t), via Fourier transforms together with the connections among the measured functions through the resolution function R(t,∆ω) are shown. Starting from these relations, the resolution effects evaluated in the frequency domain and the resolution effect evaluated in the time domain are reported. In this framework, through the SDF procedure,4-9 the connections between the system MSD and the measured MSD are shown. The measured MSD evaluation makes reference to the EINS case which is based on the use of the measured scattering function evaluated at ω ) 0, SR(Q,ω ) 0,∆ω). When only vibrational motions occur, e.g., at very low temperatures, it is possible to determine the vibrational contribution to the system MSD.
VIII. Conclusions Neutron scattering allows the characterization of the structural and dynamical properties of material systems. From a formal point of view, this information is expressed by the timedependent spatial correlation functions G(r,t) whose space-time Fourier transform corresponds to the scattering function S(Q,ω). The measured neutron response functions are connected, besides to the structural and dynamical properties of the investigated sample, also with the instrumental features in particular with the instrumental energy resolution, the energy window, and the spanned transferred wave vector range (see Figure 5). The main aim of the present paper is the evaluation
of the effects of the instrumental energy resolution on the measured MSD obtained by incoherent neutron scattering data; in particular, only the elastic incoherent neutron scattering (EINS) case has been considered. In Figure 5, a conceptual map is reported. It is shown that the resolution function gives rise to the time integration of the time-dependent system MSD function. Two types of MSD were considered: (1) the system MSD and (2) the measured MSD. The first one is a function of time while the second one is the MSD value obtained by EINS experiment; this latter, which is not a function of time, depends on the employed instrumental resolution. In the EINS case, the relation between the measured MSD and the system MSD is
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∞ given by eq 21, that is, 〈r2〉R ) ∫-∞ 〈r2〉(t)R(t)dt. When only vibrational motions occur and the system MSD can be considered constant, that is, 〈r2〉(t)f〈r2〉(V), it is possible to determine the system vibrational MSD contribution through eq 25, that ∞ R(t)dt. The system MSDs in the lowis, 〈r2〉(V) ) 〈r2〉R/∫-∞ temperature range were evaluated both from IN13 and IN10 spectrometers and gave an excellent agreement with each other (see Table 1). In particular, we obtain the same vibrational MSD value with an average agreement of 93%, see Table 1, which corresponds to a displacement mean amplitude of vibrational motion equal to 0.016 Å for dry lysozyme, 0.0165 Å for D2O hydrated lysozyme, and 0.0205 Å for H2O hydrated lysozyme. These findings confirm the validity of the proposed approach. To determine the system MSD in other temperature ranges, a specific physical model able to describe the time behavior of the system MSD has to be formulated. Finally, the effects of the instrumental energy resolution on the value of the dynamical transition temperature were considered. The dynamical transition temperature results were TD ) 220 K from IN10 data and TD ) 240 K from the IN13 data. These values are in excellent agreement with the data reported in Figure 3 of ref 24, in which the authors reported the temperature dependence of the relaxation times in hydrated and dry lysozyme. More specifically, the temperature value of T ) 220 K corresponds to a relaxation time of about τ ) 4000 ps in agreement with our IN10 result (the instrumental resolution time of IN10 is 4000 ps); moreover, the temperature vale of T ) 240 K corresponds to a relaxation time of about τ ) 500 ps in agreement with our IN13 result (the instrumental resolution time of IN13 is 500 ps). These findings show that the observed sharp rise in the measured MSD of the lysozyme-hydrated samples is the result of the fact that the protein’s structural relaxation reaches the limit of the experimental frequency window.
Acknowledgment. The authors wish to acknowledge the Institut Laue-Langevin for the beam time on IN13 and IN10 spectrometers. A special acknowledgement is addressed to Dr. C. Mondelli and to Dr. M. Gonzalez for the precious discussions and the guiding support during all the experiments. References and Notes (1) Van Hove, L. Phys. ReV. 1954, 95, 249. (2) Rahman, A.; Singwi, S.; Sjolander, A. Phys. ReV. 1962, 126, 985. (3) Magazu`, S. J. Mol. Struct. 2000, 523, 47. (4) Magazu`, S.; Maisano, G.; Migliardo, F.; Benedetto, A. Phys. ReV. E 2008, 77, 061802.
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