Multi-component modeling of quasielastic neutron scattering from phospholipid membranes U. Wanderlingh, G. D’Angelo, C. Branca, V. Conti Nibali, A. Trimarchi, S. Rifici, D. Finocchiaro, C. Crupi, J. Ollivier, and H. D. Middendorf Citation: The Journal of Chemical Physics 140, 174901 (2014); doi: 10.1063/1.4872167 View online: http://dx.doi.org/10.1063/1.4872167 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Cholesterol enhances surface water diffusion of phospholipid bilayers J. Chem. Phys. 141, 22D513 (2014); 10.1063/1.4897539 Shock wave interaction with a phospholipid membrane: Coarse-grained computer simulations J. Chem. Phys. 140, 054906 (2014); 10.1063/1.4862987 Vibrational spectroscopy of water in hydrated lipid multi-bilayers. II. Two-dimensional infrared and peak shift observables within different theoretical approximations J. Chem. Phys. 135, 164506 (2011); 10.1063/1.3655671 Hydration dependent studies of highly aligned multilayer lipid membranes by neutron scattering J. Chem. Phys. 133, 164505 (2010); 10.1063/1.3495973 Structural relaxations of phospholipids and water in planar membranes J. Chem. Phys. 130, 035101 (2009); 10.1063/1.3054141

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THE JOURNAL OF CHEMICAL PHYSICS 140, 174901 (2014)

Multi-component modeling of quasielastic neutron scattering from phospholipid membranes U. Wanderlingh,1,a) G. D’Angelo,1 C. Branca,1 V. Conti Nibali,2 A. Trimarchi,1 S. Rifici,1 D. Finocchiaro,1 C. Crupi,3 J. Ollivier,4 and H. D. Middendorf5,b) 1

Dipartimento di Fisica e Scienze della Terra, University of Messina, I-98166 Messina, Italy Institute for Physical Chemistry II, Ruhr-University Bochum, Bochum, Germany 3 IPCF-V.le F. Stagno D’Alcontres, n. 37, Messina 98158, Italy 4 Institut Laue-Langevin, 6 rue J. Horowitz, BP 156, F-38042 Grenoble, France 5 Clarendon Laboratory, University of Oxford, Oxford, United Kingdom 2

(Received 7 October 2013; accepted 9 April 2014; published online 5 May 2014) We investigated molecular motions in the 0.3–350 ps time range of D2 O-hydrated bilayers of 1palmitoyl-oleoyl-sn-glycero-phosphocholine and 1,2-dimyristoyl-sn-glycero-phosphocholine in the liquid phase by quasielastic neutron scattering. Model analysis of sets of spectra covering scale lengths from 4.8 to 30 Å revealed the presence of three types of motion taking place on well-separated time scales: (i) slow diffusion of the whole phospholipid molecules in a confined cylindrical region; (ii) conformational motion of the phospholipid chains; and (iii) fast uniaxial rotation of the hydrogen atoms around their carbon atoms. Based on theoretical models for the hydrogen dynamics in phospholipids, the spatial extent of these motions was analysed in detail and the results were compared with existing literature data. The complex dynamics of protons was described in terms of elemental dynamical processes involving different parts of the phospholipid chain on whose motions the hydrogen atoms ride. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4872167] I. INTRODUCTION

Lipid bilayers are among the most versatile and significant biomaterials in Nature. They are essential components of all biological membranes and in particular of cell membranes where they act not only as barriers between the inside and outside of a cell but also as matrices for receptors and pores enabling highly selective transport processes. Because of the enormous structural and dynamical complexity of real biomembranes, model systems consisting of relatively simple lipid bilayers are frequently studied. The biologically relevant functions of these model bilayers depend strongly on their basic structural and dynamical properties, so it is not surprising that a great deal of experimental and theoretical work is devoted to elucidate their intriguing physicochemical aspects.1–8 While the structure of model phospholipid bilayers under different conditions has been the subject of a large number of publications,2, 4, 9–14 less attention has been paid to their dynamical properties in an aqueous environment at temperatures at which they are in the liquid-crystalline phase.2 The high degree of complexity that characterizes lipid bilayers under these conditions is reflected in their dynamics by a greater heterogeneity of motions over a wider range of time and space. These dynamics encompass, for example, the long-wavelength undulation and bending modes of bilayers in the micro- to nanosecond range as well as density fluctuations with shorter wavelengths in the nano- to picosecond range.2, 15, 16 Several techniques have been used to study the relaxation dynamics in lipid bilayers, such as neutron a) Electronic mail: [email protected] b) Deceased.

0021-9606/2014/140(17)/174901/10/$30.00

scattering,15, 17–20 inelastic x-ray scattering,21 NMR,15, 22 dielectric spectroscopy,23 and dynamic light scattering.24 Among these, energy-resolving neutron techniques are uniquely able to access dynamic structure factors S(Q, ω) over ranges of momentum transfer ¯Q and energy transfer ¯ω that are directly relevant to molecular dynamics (MD) simulations. Quasielastic neutron scattering (QENS), in particular, is an especially appropriate method for the study of the dynamical properties of lipid bilayers since it covers the important transition region from damped low-frequency vibrations to a variety of diffusional processes up to time scales of the order of nanoseconds. Earlier neutron scattering studies25, 26 have described the dynamics of whole phospholipid molecules in terms of a rotational diffusion about the long axes along with in-plane and out of plane local diffusion. Interestingly, although the existence of two different in-plane and out of plane diffusive motions has been confirmed,27–29 a ballistic, flow-like motion30, 31 has been invoked recently to explain the overall long-range diffusion of phospholipid molecules in the membrane plane. At shorter scale lengths, on the other hand, the dynamics of the acyl chains has been described in terms of kink diffusion concepts,26 and the fast dynamics of hydrogen atoms has been modelled by two-site and three-site jump processes.32 During the past 5–10 years, with the commissioning of new spectrometers at intense pulsed neutron sources, as well as ambitious upgrades of classical instruments at reactor centers, previous flux and instrument time limitations are gradually being overcome. As a result, the scope for biodynamics experiments on membrane systems in the 270 ≤ T ≤ 330 K range is expanding substantially.33 Even without added components of functional interest (e.g., cholesterol,

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alcohols, anaesthetics, peptides), hydrated bilayer assemblies of phospholipid molecules are in effect ternary systems because of the fundamentally different headgroup and acyl chain dynamics. In practice, therefore, the challenge in QENS studies using state-of-the-art instruments is to decompose sets of high-quality spectra into distinct, statistically significant components and to attempt to relate these to the dynamics of specific membrane constituents. While it can be argued that any comprehensive characterisation of membrane dynamics will require MD simulations of the full dynamic structure factors for incoherent and coherent scattering, Sinc (Q, ω) and Scoh (Q, ω), there can be no question that analytical models will continue to play a central role in the acquisition and interpretation of data from energy-resolving neutron techniques, for three main reasons: (i) it is important to be able to constrain the large parameter space of complex MD simulations by experimental results from techniques covering the Q, ω-domain of interest; (ii) analytical models for elementary atomic and molecular processes contributing to the dynamics are highly developed and tested, giving not only insight into dominant modes of interaction but also connecting closely with well-established neutron work on biophysical hydration (at the head group-water interface) as well as on chain dynamics (via neutron polymer studies);34–36 and (iii) optimising data acquisition on very fast third-generation QENS spectrometers at MW spallation sources will require lineshape decomposition analyses during experiments in progress.

II. EXPERIMENTAL DETAILS A. Sample preparation

1,2-Dimyristoyl-sn-glycero-phosphocholine (DMPC) and 1-palmitoyl-oleoyl-sn-glycero-phosphocholine (POPC) were purchased as powder from Avanti Polaris Lipids and ROF Corporation, respectively. We prepared highly oriented, hydrated multilayer stacks following Hallock et al.,37 using as supporting substrate high quality mica, V-1 grade Muscovite (KAl2 (Si3 Al)O10 (OH, F)4 ). Mica was purchased from Ted Pella Inc. in rectangular sheets of about 25 × 75 mm2 and 220 μm thick. Each sheet was split in thinner foils, about 30 μm thick, by means of a micro tip dissection needle and by wetting the separation area. The most suitable foils were cut to obtain 25 × 35 mm2 mica sheets. The phospholipids were dissolved in a solution of 2:1 CHCl3 /CH3 OH (chloroform/methanol). After drying the phospholipid solution, it was mixed in a solution 2:1 CHCl3 /CH3 OH containing a 1:1 molar ratio of naphthalene (C10 H8 ) to lipid so that, for each mg of substrate, the lipid was dissolved again in 15 μl of this solution. This solution was only applied to one face of the mica sheets, giving a phospholipid coverage of ∼1.5 mg/cm2 . Naphthalene and any residual organic solvent were removed by means of an overnight vacuum drying. The lipid coated mica sheets were hydrated at 40 ◦ C in 96% relative humidity by exposing them to a saturated potassium sulphate D2 O solution for 12 days, after which appropriate amount of D2 O per mole of lipid was added. Each sample was then built up stacking 6 substrate plates such that the last one

J. Chem. Phys. 140, 174901 (2014) TABLE I. Total cross section (barn) for a single phospholipid (DMPC) with 28 hydration water molecules. Percentage values for constituent groups are given relative to the total cross section for a whole phospholipid.

28D2 O DMPC

σ inc 114.8 5779.27

σ coh 431.7 374.

%σ 8.2 91.8

3CH3 4CH2 + CH + 2C N + 8O + P 24CH2 2CH3

722.3 722.3 0.5 3852.5 481.6

32.5 54.7 48.2 217.6 21.6

12.3 12.6 0.8 66.1 8.2

Water Phospholipid Head methyls Head linkers Head phosphate Tail acyl chains Tail methyls

was not coated. It was equilibrated at 4 ◦ C for 12 additional days. B. Spectrometer

QENS experiments were performed using the time-offlight spectrometer IN5 at the Institut Laue-Langevin, Grenoble, France. The incident wavelength was 8 Å and the Q-range covered 0.2–1.3 Å−1 . The set up of the spectrometer was chosen to best suite the expected dynamic response of the investigated system. At high Q the covered energy range (−10 : 0.5 meV) allows to observe fast motions while at low angle, the energy resolution (half width 10 μeV) measured by a 0.5 mm vanadium plate, allows to well resolve conformational motions. Moreover, thanks to the almost triangular shape of the resolution and to the quality of the data it is also possible to detect sub resolution broadening in the wings of the elastic peak. For these instrument parameters, the time scale range of the dynamics probed almost extends from approximately nano to picosecond. All spectra were recorded for the 135◦ orientation of planar membrane stacks relative to the incident beam, at 298 K and 303 K for POPC and DMPC, respectively. Both were hydrated with 28 D2 O molecules per lipid. After monitor normalization, the spectra collected were scaled to vanadium intensity and corrected for absorption and self shielding. Corrections for multiple scattering were estimated to be negligible because the high transmission of the samples (≥93%). Finally, the corrected spectra were converted to dynamic structure factor S(Q, ¯ω) and interpolated to get sets of spectra at constant Q values. The bound cross sections for a whole DMPC molecule, its component groups, and 28 D2 O molecules, were calculated and are listed in Table I. The corresponding cross sections for POPC are identical except for slightly different acyl chain values. As can be seen the incoherent cross section for a DMPC molecule is about 95% of the total and the water contribution (mostly coherent) amounts to about 8%, thus validating the basic assumption of predominant incoherent scattering reflecting single-particle correlation of hydrogens which act as markers of the dynamics. III. DATA ANALYSIS

The dynamic structure factors S(Q, ¯ω) measured in energy-resolving neutron scattering experiments always

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FIG. 1. Measured QENS spectra of DMPC at 303 K for 4 different Q values together with the corresponding model function fits (red line, see text). The elastic peak and the broadened quasielastic components are also shown.

comprise an incoherent and a coherent contribution. In the present case, to a good approximation, the former accounts for the phospholipids dynamic, Slip (Q, ¯ω), whereas the latter carries information on the D2 O hydration dynamics, Swat (Q, ¯ω). Both are convoluted with R(¯ω), the experimental resolution function: S(Q, ¯ω) = [Slip (Q, ¯ω) + Swat (Q, ¯ω)] ⊗ R(¯ω).

A. Experimental results and theoretical model

(1)

Lipid dynamics: As can be seen from Table I, ∼92% of the total scattering intensity is due to incoherent scattering from the phospholipid molecules. Therefore, Slip (Q, ¯ω) was fitted by a phenomenological model consisting of the sum of a delta function A0 δ(¯ω), representing the elastic response, and of three Lorentzians, accounting for three different kinds of relaxational motions that cause the quasielastic broadening of the elastic peak:25, 32, 38 Slip (Q, ¯ω) = A0 δ(¯ω) +

3  Ai i=1

π

i2

i . + ¯ω2

the basis of the water model of Teixeira et al.39 in conjunction with Vineyard’s40, 41 convolution approximation. Details of the procedure used are given in the supplementary material.42

(2)

Each relaxation is characterized by a quasielastic incoherent structure factor, that corresponds to the integral Ai of the relevant Lorentzian function. The HWHMs of the latter,  i , are directly related to the characteristic times associated with the motions of the scattering nuclei involved, τ i = ¯/ i . Water dynamics: The contribution of hydration water, D2 O, which accounts for the ∼8% of the total scattered intensity, has been explicitly considered in the data fitting by means of an analytical S(Q, ¯ω). This was evaluated on

Examples of the quality of the fit obtained by Eq. (1) for DMPC at 303 K are shown in Fig. 1 at four values of the momentum transfer Q. As can be seen, the phenomenological model describes fairly well the experimental spectra in the whole Q range probed. For the three Lorentzians used to fit the lipid dynamics of DMPC and POPC, the Q-dependence of their integrated areas Ai and half widths  i are reported in Figs. 2 and 3, respectively. Rather interestingly, we observe that the values found for  i differ by orders of magnitude over the entire Q-range, thus corroborating the existence of three dominant dynamical processes taking place on different time scales. Moreover, the observed weak dependence of  i on Q indicates that the motions involved are localized. Although, in principle, useful information on the geometry of such motions can be inferred from the Elastic Incoherent Structure Factor (EISF),43 the elastic scattering contributions due to different dynamical processes in these phospholipid systems are not easily separable. Consequently the calculation of the relevant EISF functions is not straightforward. In order to adequately describe these processes, we have turned to a microscopic model for the hydrogen dynamics. Bearing in mind the composite structure of phospholipid molecules, the confinement effects resulting from the bilayer packing, and the experimental evidence for

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FIG. 4. Illustration of the decomposition of the vector position of H atoms in the phospholipid molecule.

FIG. 2. Q dependence of the integrated areas, Ai , of the Lorentzian functions for DMPC (a) and POPC (b).

three separated dynamical processes, we write S(Q, ¯ω) = (A(Q)δ(¯ω) + (1 − A(Q))) (¯ω)2a+2 a

⊗ (B(Q)δ(¯ω) + (1 −

B(Q))) (¯ω)2b+2 b

⊗ (C(Q)δ(¯ω) + (1 − C(Q))) (¯ω)2c+2 . c

(3)

Here we have assumed that the dynamical structure factor S(Q, ¯ω) can be modelled by a convolution of three types of motion. Each of these accounts for a confined dynamics and is represented as the sum of an EISF and a quasielastic term. This  expression is equivalent to consider the proton position, R(t),    as the sum of three position vectors, (Ra (t) + Rb (t) + Rc (t)), each relative to a generalized center of mass of the relevant structure, which drags the hydrogen atoms along as time progresses (see Fig. 4). Specifically, the fastest motion can be identified with the hydrogen motion around the carbon atom, the intermediate motion can be related to the dynamics of carbon atom in the acyl chain, and the slowest motion can be ascribed to the diffusive motion of the whole phospholipid. Under the condition of dynamical processes taking place on widely separated time scales, this expression well represents the phenomenological model used to fit the experimental data.42 In fact, by expanding Eq. (4) and considering that Lorentzian curves with close HWHM are not experimentally distinguishable in a fitting procedure, we obtain (see the supplementary material): S(Q, ¯ω) = ABC · δ(¯ω) + (1 − A)BC · La (¯ω) +(1 − B)C · Lb (¯ω) + (1 − C) · Lc (¯ω),

FIG. 3. Plot of the Q dependence of  i of the Lorentzian functions for DMPC (empty symbol) and POPC (solid symbol).

(4)

which is coincident with the phenomenological model used to fit the experimental data, Eq. (2), and provides a physical meaning to the parameters used in the model in terms of EISF

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for the observed dynamical processes: A0 = ABC

A1 = (1 − A)BC,

A2 = (1 − B)C

A3 = (1 − C).

(5)

At the same time Eq. (4) allows to calculate the EISFs relative to the experimentally observed dynamics from the parameters that gave best fits to the lineshapes: C(Q) = 1 − A3

B(Q) = 1 − A(Q) = 1 −

A2 , 1−A3 A1 . 1−A3 −A2

(6)

In this expression, C(Q), corresponds to the EISF of the fastest dynamics, EISFfast , B(Q), corresponds to the intermediate dynamics, EISFmed , and A(Q), to the slowest dynamics, EISF slow . It is worth noticing that for the calculation of the EISF , except for the faster one, it is necessary to include the areas of all the quasielastic components. The results obtained using Eq. (6) are shown in Sec. IV along with an attempt to develop a more detailed description of the dynamical processes observed.

FIG. 5. Q dependence of the experimental EISFfast for the POPC (blue symbol) and DMPC (red symbol) sample. Solid lines are fits to the uniaxial rotation model.

follows: EI SFf ast (Q) =

∞  (2l + 1)jl2 (QR)Sl2 (μ),

(7)

l=0

IV. DISCUSSION

As mentioned previously, the fast dynamics observed in the experimental spectra can be attributed to rapid movements of H atoms bound to C atoms, the intermediate motion to conformational changes of the acyl chains and the slowest motion to overall rotations and translations of the whole phospholipid molecule. Starting from these considerations, geometrical models that provide an appropriate description for each of these dynamics are presented and the resulting insights into phospholipid dynamics are discussed and compared for the DMPC and POPC samples.

A. Fast dynamics

The ∼2 meV Lorentzian HWHM associated to the fastest component, attributed to the motion of hydrogen with respect to the bounding carbon atom, corresponds to a characteristic time (¯/) of about 0.33 ps. As can be seen from Table I this contribution is largely due to the CH2 groups although the fraction from CH3 groups is not negligible. Methylene groups in acyl chains are known to participate to cis-trans conformational jumps whereas methyl groups perform a three-site jump rotational diffusion, but both these motions show longer characteristic times (3–8 ps).44 To describe the fast dynamics we assume that hydrogen atoms perform fast, non oscillatory, librational motions around the bounding carbon atom. This may be modelled as uniaxial rotational diffusion45 similar to analyses of QENS from liquid crystals.46, 47 According this model, hydrogen atoms perform hindered rotational diffusion, peaked around a preferential direction. The corresponding scattering law is a delta function, multiplied by the EISF , plus a series of Lorentzian functions45 with increasing widths, and depends on the angle θ between Q and the bond direction. The EISF, averaged over all the possible orientations of the bond direction with respect to Q can be written as

where R is the distance between the proton H and the C atom and Sl (μ) is an orientational order parameter dependent on the width of the angular distribution, μ. Sl (μ) folSl + Sl−1 with S0 lows the recurrence relation: Sl+1 = − 2l+1 μ 1 = 1 and S1 = coth δ − μ . In particular S1 is related to the average amplitude of the oscillations, α, by the relation α = cos −1 (S1 ). The experimental EISF for the fastest component is shown in Fig. 5 for both samples along with the fitting model described. To fit the experimental data with Eq. (7) we have taken into account the first six terms. This gave μ = 1.339 and μ = 1.548 for DMPC and POPC, respectively, corresponding to angular spreads of α = 66◦ ± 0.5◦ and α = 63◦ ± 0.5◦ . As shown, the model satisfactorily reproduces the experimental data, thus suggesting that the origin of this dynamics could be a fast, small amplitude fluctuation of dihedral angle in the acyl chain and an hindered rotation of the methyl groups. B. Lipid chains dynamics

The measured Lorentzian HWHM for the intermediate dynamics corresponds to a characteristic time (τ = ¯/) of 5–50 ps which is consistent with the characteristic times of conformational dynamics in the lipid chains and of methyl group rotational diffusion.26, 32, 48–50 On the other hand, the model adopted implies that this motion should be referred to the center of mass around which the hydrogen atoms perform the observed fast movements, that is around the bounding carbon atoms. Because of the elevate number of carbon atoms in the acyl chains we expect that this contribution is related to the segmental dynamics of chains. At the same time, since also methyl-group dynamics falls in the same energy windows the following analysis of the intermediate contribution will include both kinds of dynamics with a proper weight. To obtain information on the geometry of this motion, the EISFs obtained were fitted with the sum of two contributions, one

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accounting for the methyl and the other for the acyl chains dynamics. The relative weight for each contribution was determined from the cross section fractions for CH2 and CH3 . The methyl group dynamics has been modelled by a threesite jump rotational diffusion44 for which EISF is given as EI SFmeth =

√ 1 (1 + 2j0 (QRm 3), 3

(8)

where j0 (QR) is the spherical Bessel function of zero order and Rm = 1.03 Å is the rotation radius. The weight for this contribution is 0.205 for DMPC and 0.180 for POPC. As for the chain dynamics, this can be described in terms of the transitions of several carbon atoms between anti and gauche conformations in the acyl chain.51 NMR measurements indicate that carbon atoms in the chains experience greater disorder with increasing distance from the glycerol group.52 In earlier work on phospholipid membranes,25 the movement of any given carbon atom was modelled as restricted diffusion within a sphere, the radii of which were increasing along the chain. In our analysis we model the chain dynamics in terms of a Gaussian exploration of the spatial regions swept out during conformational motions, rather than in terms of a continuous distribution of spherical regions. Extending this description to all atoms, we picture the dynamics of the whole phospholipid as shown schematically in Fig. 7. Satisfactory results were obtained using a distribution of five and six Gaussians for DMPC and POPC, respectively, considering about three carbon atoms in each segmented volume. We have kept the number of components used in the fitting procedures as small as possible. An increase in the number of terms used does not correspond to a sensible increase of the quality of the data fit, while decreasing the number of components results in a lower quality. Our limitation in the observation of motions of the chains with greater spatial detail is determined by the limited range of Q available. According to this model the EISF can be expressed as EI SFmed =

N 1  exp (−(Qςi )2 ), N i=0

(a)

(9)

where ς i is the standard deviation. The weights for this contribution are 0.795 for DMPC and 0.820 for POPC. In Fig. 6, the experimental EISFmed for POPC and DMPC is shown as a function of Q. The full line is the fit of Eq. (9) to the data according to our model of restricted diffusion. As can be seen, the fitted curves agree very well with the experimental data. The fitting parameters ς i are listed in Table II along with the results obtained by using a distribution of five and six spheres with increasing radius r. The corresponding fitted curves are not shown since they nearly coincide with the Gaussian ones. It is interesting to note that both models yield the same spatial extension for the dynamics observed; in fact the resulting sphere radii are about twice the standard deviation. It is evident, moreover, that DMPC and POPC display similar dynamics, even though POPC chains may explore slightly larger regions. Although our sphere model gives some insights into the geometry of motions occurring in the system, it does not reveal further characteristics of these motions nor does it indicate which parts of the phospholipid participate in

(b)

FIG. 6. Q dependence of the experimental EISFmed for DMPC (a) and POPC (b). The solid and dashed lines are fits to the Gaussian distribution model and the bead model, respectively, described in the text.

the dynamics probed. To elaborate on this we have developed an explicit “bead model” for the phospholipid dynamics: we model a phospholipid molecule as a structure made up of several linked beads representing head groups, chain segments, and tail methyls (see Fig. 7). The five beads that refer to CH3 groups (three in the polar head and two at the tail end) are assumed to perform a three-site jump rotational diffusion as described by Eq. (8). The four beads that refer to CH2 groups (one for the polar head and three for segments of the lipid chains) are assumed to perform two site jumps between the two equilibrium sites with a common jump length (d) and different jump rate probabilities (τ 1 and τ 2 ). The expression for

TABLE II. Radii r, with their standard deviations ς , as deduced from the sphere model described in the text. DMPC r 0.0 0.2 0.5 2.7 3.2

POPC ς

r

ς

0.0 0.0 0.3 1.2 1.5

0.0 0.0 0.0 1.1 2.1 4.2

0.0 0.0 0.0 0.5 1.0 2.0

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J. Chem. Phys. 140, 174901 (2014) TABLE III. Values for the ratio ρ i of the jump rate probabilities τ 1 and τ 2 , for the 4 chain positions considered in Eq. (11). DMPC

POPC

0.00 0.08 0.42 1.00

0.00 0.03 0.38 1.00

ρh ρa ρb ρc

states. A similar observation of an increase in the mobility along the lipid chains has been reported recently in a QENS study on lipid vesicles.38 Although the quality of the data fitting obtained by the “bead model” is not as good as that obtained by using the Gaussians or the spheres models, we found this model quite useful to quantify the observed gradient in the mobility of the chains and for the indication that this mobility has to be strongly related with transition between anti and gauche conformation of the lipid chains. FIG. 7. Schematic picture of the chain dynamics of DMPC. Left panel: Sketch of the spherical (dashed areas) and Gaussian (shaded areas) regions. Right panel: Illustration of the bead model. See text for details.

the EISF in this case is EI SFi (Q) =

1 (1 + ρi2 + 2ρi j0 (Qd)), (1 + ρi )2

(10)

where ρ i = τ 1 /τ 2 is the ratio of the jump rates and the subscript (i = h, a, b, c) indicates the head or a segment position along the lipid chain. A value of d = 2.49 Å was used for the jump length, since this corresponds to the carbon atom transit distance between anti and gauche conformations. Keeping in mind that the motion of each segment has to be convoluted with the motion of the previous one, the overall dynamics (relative to a center of mass positioned at the head-tail contact) is then obtained as the sum of all contributions. Thus the EISF bead for this model can be expressed as EI SFbead = A Eisfmeth Eisfh + B Eisfh C (Eisfa + Eisfa Eisfb + Eisfa Eisfb Eisfc ) 3 +D Eisfa Eisfb Eisfc Eisfmeth , (11)

+

where A, B, C, and D are the fractional cross sections (see Table I) for the CH3 groups in the polar head, the CH2 groups in the polar head, the CH2 groups in the tails, and the terminal CH3 groups, respectively. Results of fitting the experimental EISF with Eq. (11) are shown in Fig. 6 and the best-fit parameters obtained are given in Table III. It is worth noting that, despite its crudeness, this model is able to reproduce fairly well the experimental data, suggesting in particular that a jump process based on a transition between anti and gauche conformations is fundamental to the chain dynamics. Moreover, these findings indicate that the polar head and the first part of the lipid chain do not move appreciably on an intermediate time scale. A somewhat restricted conformational dynamics is possible for the central part of the lipid chain whereas the last part of the chain freely explores the two conformational

C. Slow dynamics

The HWHM values for the slowest component are listed in Fig. 3. They show a plateau at low Q followed by a monotonic increase with Q, a behavior characteristic of a confined diffusion process. Moreover, the decay of the correspondent EISF indicates that the size of the region explored during this motion is comparable to the phospholipid head dimension. These findings suggest that the slow dynamics observed can be ascribed to the motion of whole phospholipid molecules, which in our model are center-of-mass motions. For this case we have also tested several models with the aim of interpreting the EISFslow obtained. A model of restricted diffusion in a cylindrical volume,53 consistent with the symmetry of the bilayer, gave the best results. According to this model, the motion is due to the coupling of two independent dynamics: a linear diffusion along the cylinder axis L, and an isotropic planar diffusion inside a circle of radius R perpendicular to the former. The resulting scattering law can be expressed as the convolution product between the relevant Sinc (Q, ¯ω) functions. Assuming that the cylinder is aligned along the bilayer normal (z-axis), and that this forms an angle with Q whose components are Qz and Qr relative to the bilayer plane, the scattering law can be written as S(Q, ¯ω) = S ⊥ (Qz , ¯ω) ⊗ S  (Qr , ¯ω),

(12)

where S ⊥ (Qz , ¯ω) and S  (Qr , ¯ω), referred to as out of plane and in-plane components, are the dynamical structure factors for diffusion along a segment and for diffusion inside a circle. The detailed derivation of the complete scattering law can be found in the original work.53 Here we present the combined expression which is the sum of an elastic term, EISFcyl , and a triple infinity of Lorentzian lines. The former is given by  EI SFcyl (Q) =

j02

Qz L 2



2J1 (Qr R) Qr R

2 .

(13)

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FIG. 8. Top: Q-dependence of the experimental EISFslow for DMPC. The solid line represents the fit to the EISFcyl model. Bottom: Q2 -dependence of , the corresponding HWHM. Also shown is the DQ2 law.

The latter are quasielastic structure factors with HWHMs expressed by   2 2 l π (xmn )2 n , (14) lm = D + L2 R2 d where xmn is the (m + 1)th root of the equation dx Jn (x) = 0. n The J are nth order cylindrical Bessel functions of first kind. According to this model the resulting HWHM for the ∗ sum of Lorentzians attains a constant value, (Q)  =  , in −1 2 the low Q region where Q ξ . Here ξ = (R + L2 /4) is defined as the maximum distance between the center of the cylinder and the surface. For Q  ξ −1 , moreover, the HWHM of the sum of Lorentzians approaches the limiting form (Q) = DQ2 , where D is the free diffusion coefficient inside the confinement volume. This is also related to the plateau value  ∗ by   L D/ξ 2 , (15) ∗ = λ 2R

where λ(x) is a function that varies from 2.5 to 5.8. By applying the above model (Eq. (13)) to the experimental EISFslow data we can deduce information about the dimensions L and R of the confinement region. Also, from the estimation of the plateau value  ∗ together with Eq. (15), we can derive the diffusion coefficient inside that region. The results are illustrated in Figs. 8 and 9 for DMPC and POPC, respectively. In both cases there is a good agreement between experimental data and theoretical predictions. The resulting

J. Chem. Phys. 140, 174901 (2014)

FIG. 9. Top: Q-dependence of the experimental EISFslow for POPC. The solid line represents the fit to the EISFcyl model. Bottom: Q2 -dependence of , the corresponding HWHM. Also shown is the DQ2 law.

parameters L, 2R,  ∗ , and D are listed in Table (IV): The values obtained for the length of the cylinder L are consistent with the protrusion of phospholipids. Furthermore, the values for the cylinder cross-section 2π R2 , i.e., 57 Å2 for DMPC and 82 Å2 for POPC, may be compared with the correspondent polar head area obtained experimentally11 as 61 Å2 and 69 Å2 , respectively. The values found for the diffusion coefficient within the confinement region agree with those found in similar systems by means of QENS techniques.54, 55 These values are approximately two order of magnitude faster than the long range lateral diffusion coefficient found by macroscopic techniques.56–59 We surmise that the slightly higher values of lateral confinement and diffusion coefficient resulting for POPC could be due to their unsaturated chain. Several recent papers about the diffusion of this kind of systems suggest that there is also a flow-like component to the motion of the lipid molecules.31, 60–63 The values we have found for the diffusion coefficients and confinement sizes TABLE IV. Values of height, diameter, HWHM at plateau level, and diffusion coefficient for POPC and DMPC, from the model for confined diffusion within a cylindrical region.

L (Å) 2R (Å)  ∗ (μeV) D (10−2 Å2 /ps )

DMPC

POPC

3.73 8.5 1.6 1.2

3.50 10.2 1.9 1.9

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suggest that, on the nanosecond time scale, the center of mass of a phospholipid, in both samples, diffuse in a region with dimensions very close to the available area per polar head. Although this findings may suggest a rattling and jump diffusion process, they also imply that in the tightly packed fluid state of the bilayer a single phospholipid should wander inside a volume which, due to steric hindrance, is already occupied by the other molecules. Such a circumstance can be explained in terms of a re-arrangement of the nearest neighbour molecules thus providing a picture compatible with flow-like diffusion.

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tion. This is of considerable practical interest in view of the increasing scope for studying complex multi-component systems opened up by the new generation of spectrometers at high-intensity pulsed neutron sources.64

ACKNOWLEDGMENTS

We acknowledge the Institut Laue-Langevin, Grenoble, for access to facilities, J. Ollivier, M. Zbiri, and H. Schober for experimental assistance, and the Clarendon Laboratory, University of Oxford, for hospitality to H.D.M. This work was supported by MIUR Research Projects.

V. CONCLUSIONS

Using a state-of-the-art multichopper spectrometer on the world brightest continuous neutron source, we have measured sets of high-quality QENS spectra from D2 O-hydrated bilayer stacks of two phospholipids, DMPC and POPC, both in the liquid crystal phase near 300 K. The dynamic structure factors derived from these spectra were shown to be well fitted by a phenomenological model consisting of the sum of a δ-function reflecting the elastic response plus three (Q, ω)-dependent Lorentzians accounting for the quasielastic response. Within the domain of the experimental parameters covered, i.e., ∼4.8–30 Å in space and from ∼0.3 to 350 ps in time, we were able to identify three dominant dynamical processes with well separated time scales. These were ascribed to: (i) a fast uniaxial rotational diffusion of hydrogens around their bonding carbons; (ii) a conformational, jump-like, dynamics of the acyl chain along with three-site rotation of methyl groups; and (iii) a confined diffusion process of whole phospholipid molecules within their first-neighbours cages, compatible with a flow-like diffusion. The basic finding of a clear-cut time scale separation allowed us to proceed with some detailed model building for each of these dynamics. We employed the fundamental tool kit of analytical expressions for neutron scattering from elementary atomic and molecular motions, as developed and widely used in QENS studies since the 1980s for describing a variety of hindered translational, librational, and rotational diffusion processes in soft-matter systems such as polymer melts, liquid crystals, and carbohydrate gels. We have thus provided a highly intuitive description of two prototypical phospholipid membrane systems differing mainly in their acyl chain dynamics, and have shown in particular how the difference in chain saturation slightly affects their dynamics. Within this framework of model analysis we have also presented, for the first time, a detailed decomposition of the EISF functions which quantify spatial aspects of the restricted motions of relevant phospholipid components. Although based on some ad hoc assumptions about the nature of the dynamical processes contributed by the principal constituents of the phospholipids investigated, and in this respect less general than MD simulations of the full dynamic structure factors, our results demonstrate the utility of analytical models which relate closely to well-established work using QENS. In the context of current efforts to promote biophysical applications of energy-resolving neutron techniques, we emphasize the potential for developing approaches of this kind as useful platforms for rapid QENS data analysis and interpreta-

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