Article pubs.acs.org/JPCB
The Dynamical Transition of Lipid Multilamellar Bilayers as a Matter of Cooperativity Judith Peters,*,†,‡ Jérémie Marion,‡,§ Francesca Natali,‡,∥ Efim Kats,‡,⊥ and Dominique J. Bicout‡,# †
Université Grenoble Alpes, LiPhy, 140 rue de la physique, 38402 Saint Martin d’Hères, France Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, 38042 Grenoble cedex 9, France § Université Grenoble Alpes, IBS, 71 avenue des Martyrs, CS 10090, 38044 Grenoble, France ∥ CNR-IOM, OGG, 71 avenue des Martyrs, CS 20156, 38042 Grenoble cedex 9, France ⊥ Landau Institute for Theoretical Physics, RAS, 142432, Chernogolovka, Moscow region, Russia # Biomathématiques et épidémiologie, EPSP − TIMC-IMAG, UMR CNRS 5525, Université Grenoble Alpes, VetAgro Sup Lyon, 69280 Marcy l’Etoile, France ‡
ABSTRACT: The present study is the application of a two-state model formerly developed by Bicout and Zaccai [Bicout, D. J.; Zaccai, G. Biophys. J. 2001, 80 (3), 1115−1123] to describe the dynamical transition exhibited in the atomic mean square displacements of biological samples in terms of dynamic and thermodynamic parameters. Data were obtained by elastic incoherent neutron scattering on 1,2-dimyristoyl-sn-glycero-3-phosphocholine lipid membranes in various hydration states and on one partially perdeuterated lipid membrane. Fitting the data with the model allowed investigating which parts of lipid molecules were mainly involved in the dynamical transition, heads, tails, or both. Clear differences were found between the fully protonated and partially deuterated membranes. These findings shed light on the question of what is the degree of dynamical cooperativity of the atoms during the transition. Whereas the level of hydration does not significantly affect it, as the dry, the intermediate dry, and fully hydrated membranes all undergo a rather broad transition, the transition of the lipid tails is much sharper and sets in at much lower temperature than that of the heads. Therefore, the dynamical cooperativity appears high among the particles in the tails. Moreover, the transition of the lipid tails has to be completed first before the one of the head groups starts.
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INTRODUCTION Biomolecules are composed of thousands of atoms apparently arranged in a nonordered way. The secondary structures are mostly bound by attractive interactions like hydrogen bonds or van der Waals interactions, thus rather weak bonds, whereas the tertiary structures are dominated by strong interactions like covalent interactions. The atoms are moreover not arranged periodically as in a crystal; thus, motional amplitudes in a given direction cannot be easily calculated. In contrast, each atom is bound to at least one neighbor, sometimes to more, via interactions that make the connections more or less flexible in the various space directions. Using the very useful harmonic oscillator model in a first approximation, these bindings can be represented as springs which maintain the atoms together and allow limited vibrations or rotations, whereas others are forbidden as, for instance, certain rotations around the main chain axis in the amino acids. It is a huge task to investigate all possible motions in such systems, but experiments such as incoherent neutron scattering give access to averaged values for the atomic displacements,2 which can be compared to molecular dynamics simulations.3 © 2017 American Chemical Society
Incoherent neutron scattering consists of measuring interferences between neutrons scattered by one and the same nucleus at different times; thus, it reflects averaged single particle motions in time and space.4 In contrast, coherent scattering requires an order in space and time so that many particles participate due to constructive interference. However, as the single particles are interconnected, the motion of one of them will influence the state of the neighboring particles. There exist examples of successful normal mode calculations to account for such effects;5 nevertheless, it remains difficult to conclude about how cooperative the processes are, e.g., how many particles experience the same dynamics in the potential of mean force. M. Bée has shown by molecular dynamics simulations on hexamethylethane6 that different kinds of motions can be separated, as they occur typically on clearly separated time scales and corresponding amplitudes are of different orders of Received: May 28, 2017 Revised: June 22, 2017 Published: June 26, 2017 6860
DOI: 10.1021/acs.jpcb.7b05167 J. Phys. Chem. B 2017, 121, 6860−6868
Article
The Journal of Physical Chemistry B
with various energy resolutions do lead to a different outcome as the slopes of the atomic mean square displacements (MSDs) as a function of temperature are instrument dependent.21 To further motivate our approach, one should notice that the equilibrium state, metastable states, kinetic barriers, and time evolution of the system are determined by respectively the global minimum, local minima, maxima, and time dependent trajectories of the very complicated and multidimensional free energy surface. Besides, particles perform their motions in a nonrigid environment (the cages are not frozen). Analysis can be made tractable if we assume that, for each thermodynamic state, the distribution of atomic configurations is sharply bimodal. Thus, we leave with the two-state model.1 Then, in the analogy with the first-order phase transitions, the actual thermodynamic transition temperature is always above the spinodal point. At the transition point, the free energies of the two coexisting phases are equal. Although the metastable state appears at the spinodal point, only above this point the “ordered phase” becomes significantly occupied. This is the essence of the heterogeneous nucleation phenomena, leading to the dynamic phase transition, we are interested in within this work. The total nucleation time is determined by the height of the barrier per nucleus. Generally, we deal with multistep processes (nucleation, growth, relaxation). In our work, we focus on measurable effects and take a pragmatic empiric approach to rationalize our experimental results in the spirit of a simple model estimate mentality. Despite this simplification, our model can match our measurements with quantitative accuracy (see below). The DT of MLB samples was investigated here in more detail within the framework of the model developed in 2001 by Bicout and Zaccai1 that takes into account both the variations in free energy (enthalpy and entropy) associated with the transition and allows probing the overall level of the dynamical cooperativity between the particles undergoing harmonic versus nonharmonic motions; the detailed distribution of the dynamical cooperativity over biomolecules is still a matter of investigation. In the context of MLB samples, the use of the technique of partial deuterating lipids allows one to selectively highlight parts of the molecule to study and address the issue of whether the dynamical transition mainly involves either heads or tails or the lipid molecule as a whole.
magnitude. Indeed, internal molecular vibrations, reorientations, and translational diffusions happen on time scales observable by incoherent neutron scattering, namely, in the range between 1 ps and 1 ns. Combining several neutron spectrometers with different energy−time resolutions permits therefore to get access to such classes of motions, when they exist in a given sample. For instance, long-range displacements can only be observed in samples in solution or at long time scales.7 The nature of movements depends also on external conditions such as temperature or pressure.8,9 It was established in the late 1980s that the very reduced molecular motions at low cryogenic temperature (12 338 (3) 22000 (160) 65.1 (6) 264 (3) 1.6 (8) 0.068 (4) 0.039 (4)
173 (5) 11300 (300) 65 (2) 135 (5) 1.7 (2) 0.62 (1) 0.57 (1)
The error in parentheses applies to the last digit of the fit results.
• MLB: obtained by rehydrating the dry MLB from pure D2O at 40 °C to achieve a high hydration level (more than 12 water molecules per lipid) • MLB heads: the fully hydrated MLB sample with partially deuterated lipids, in which the hydrogen atoms of the alkyl chains were exchanged against deuterium to highlight the head groups (DMPC-d54) • MLB tails: the same sample as the MLB, but we fitted a proportion with the parameters found for the heads and let free a second proportion corresponding to the tails An illustration of samples at two different hydrations is shown in Figure 1, and the characteristics of all samples are summarized in Table 1 in the Results section along with associated results of analyses. One wafer contained a total amount of ∼35 mg of lipids. These MLB samples were then placed in slab-shaped aluminum sample holders. Sample cells were sealed using indium wire, and the weight of the sample was monitored before and after the experiment, with no change observed, indicating a stable level of hydration. Elastic Incoherent Neutron Scattering. Elastic incoherent neutron scattering (EINS) measurements as a function of temperature were performed on the thermal (λ = 2.23 Å) highenergy resolution backscattering spectrometer IN1323 (ILL, Grenoble, France), characterized by a very large momentum transfer range (0.2 Å−1 < Q < 4.9 Å−1) with a good and nearly Q-independent energy resolution (8 μeV fwhm). IN13, therefore, allows accessing the space and time windows of 1− 30 Å and 0.1 ns, respectively. EINS scans were recorded from 20 to 340 K with a continuous temperature ramp of 0.6 K/min between 20 and 100 K, 0.45 K/min between 100 and 280, and 0.15 K/min between 280 and 340 K. To keep the signal-tonoise ratio roughly constant, a slower temperature ramp has been adopted at higher temperature to overcome the loss in elastic intensity arising from the enhanced dynamics. Data reduction was performed using the Large Array Manipulation
Program (LAMP) available at the ILL.24 Normalizing the raw data to the neutron flux, subtracting the background given by the spectrum of the empty cell, and finally normalizing to the sample measured at 20 K corrected the data. Incoherent neutron scattering is largely dominated by hydrogen atoms, because their incoherent scattering cross section is at least 40 times higher than that for any other type of atom present in biological samples.25 Moreover, about 50% of the atoms in biological systems are hydrogen atoms. The MSD thus reflects the averaged atomic motion of hydrogen atoms, which are mostly homogeneously distributed in biological macromolecules and inform over local atomic vibrations and translations around static positions. Modeling of the Mean Squared Displacements. On the basis of van Hove’s formalism26 calculated for the pair correlation function and taking into account the time resolution τ of the instrument, it can be shown that EINS data is related to the normalized elastic intensity I through el Sinc (Q ) ≈ I (Q , τ ) N
=
{− 13 Q ⟨r ⟩[1 − C (τ)]}
∑ xα exp α
2
2
α
α
(1)
where xα is the fraction of particles experiencing the same dynamics and therefore takes into account dynamical polydispersity, Q is the module of the momentum transferred between the incident and scattered neutron, ⟨rα2⟩ is the equilibrium MSD, and Cα(τ) accounts for the stationary position relaxation function in the time frame of the experimental resolution time τ (for more details, see ref 1). In the absence of dynamical polydispersity, xα = 1, all atoms are dynamically equivalent and undergo the same transition. Within the Gaussian approximation,27,28 and as it was shown in the same reference, the total MSD ⟨R2(T)⟩, which takes into 6862
DOI: 10.1021/acs.jpcb.7b05167 J. Phys. Chem. B 2017, 121, 6860−6868
Article
The Journal of Physical Chemistry B account fluctuations of all particles in the system within τ, is given by ⟨R2(T )⟩ = −3
N
el d ln Sinc (Q )
dQ 2
dynamical transition onset, Bicout and Zaccai1 empirically defined the transition temperature T0 at which 10% of large fluctuations involving free energy barrier crossing events is reached, i.e., ϕ(T0) = 0.1, thus corresponding to the onset of the dynamical transition.1 The above approach for defining T0 is quite similar to using the Lindemann criterion29 for melting of crystalline solids stating that melting occurs when the thermal motion of the atoms of the crystal reaches a critical mean square displacement of about 1/10 of the interatomic distance. To address the issue of the dynamical cooperativity between the particles undergoing harmonic versus nonharmonic motions, we consider the inverse participation ratio (IPR) that was historically introduced for classifying properties (localization) of atomic vibrations in disordered lattices.30 The IPR provides an indication on the fraction of the total number of atoms effectively participating in a given normal mode of motions. In this context, we are dealing with two-state (small and large) conformational cages separated by a free energy barrier involving two different kinds or modes of motions: intracage (harmonic) motions in the small or large cage and nonharmonic intercage motions due to free energy barrier crossings between the two cages. In this case, the IPR at temperature T reads as
= Q =0
∑ xα⟨rα 2⟩[1 − Cα(τ)] α=1
(2)
This equation indicates that experimentally visible particles are those with Cα(τ) < 1. Then, when dealing with data from the same instrument, Cα(τ) is a constant and the term [1 − Cα(τ)] can therefore be included in an effective MSD, ⟨rα2⟩. In addition, for the sake of simplicity and to ease the presentation, the explicit polydispersity can be averaged out. To simplify the approach by following the ideas exposed in the Introduction, only two types of conformational fluctuations will be considered here: small local motions, including vibrations around equilibrium positions of the atoms and methyl group rotations, and movements on a larger scale accounting, for instance, for motions induced by interactions causing the particle to agitate inside a cage defined by neighbors or produced by fluctuations of the center of mass of the whole protein. Thus, we only consider vibrational and translational motions of the atoms in each conformational substate. The vibrational motions are harmonic in both small and large conformational cages, but with different frequencies accounting for their respective sizes, and the translational motions are described in a double-well structure with nonharmonic motions between the small and large conformational cages. For this model of atom motions, Bicout and Zaccai1 showed that the MSD in eq 2, neglecting quantum effects due to zeropoint fluctuations, can be written as ⟨R2(T )⟩ = [1 − ϕ(T )]
kBT kT + ϕ(T ) B k1 k2
IPR = 1/[ϕ2 + (1 − ϕ)2 ] = 1/[1 − 2ϕ(1 − ϕ)]
The IPR (with 1 ≤ IPR ≤ 2) as a function of temperature indicates the number of different cages contributing in the motions that the atoms in the system are collectively experiencing. Indeed, it is easy to verify that IPR = 1 in the low (i.e., ϕ = 0) and high (i.e., ϕ = 1) temperature limits, indicating that the atomic (intracage) motions are localized in the small and large conformational cages, respectively, whereas, in the intermediate temperature range, IPR > 1 and reaches the maximum value 2 at T = Td where atomic (intercage) motions are delocalized between the two cages. Fitting of the Mean Squared Displacements. Data of the MSD as a function of temperature were fitted to determine dynamical (force constants) and thermo-dynamical (enthalpy and entropy) parameters for each sample considered. The fitting procedure was undertaken in the following way:16 first, to get rid of the MSD value at 20 K (corresponding mainly to zero-point energy) and reduce at the same time the number of free fitting parameters, the MSD at low temperature was fitted with a linear curve, kB (T − 20 K)/k1, up to 120 K to obtain k1. Next, all of the MSD data were fitted using eq 3, with k1 previously determined, to obtain k2, ΔH, and ΔS (and to determine Td and T0). As defined in ref 1, the apparent force constant k3 of the system at any temperature was calculated at a reference temperature of Tr = 290 K as1
(3)
where kB is the Boltzmann constant, T is the temperature, and k1 and k2 are effective force constants according to16 and defined as1 −1 ⎧ = k vs−1 + k ts−1 ⎪ k1 ⎨ ⎪ −1 −1 −1 ⎩ k 2 = k vl + k tl
(4)
where the indices stand for v as vibrational motions, t as translational motions, and s and l as small and large, respectively. They are the resulting force constants (from both the vibration and translation) to particle motions in small and large conformational cages. The probability of finding an atom in the small conformational cage at low temperature corresponding to the small translational motions is given by (1 − ϕ), where ϕ is defined by ϕ(T ) = 1/[1 + exp(β ΔG)];
(6)
⎫ [1 − ϕ(Tr)] ⎧ ϕ(Tr) ΔH 1 ⎨1 − = ϕ(Tr)⎬ + k3 k1 kBTr k2 ⎩ ⎭
ΔG = ΔH − T ΔS (5)
β−1 = kBT is the thermal energy, and ΔG, ΔH, and ΔS are the differences between the free energy, enthalpy, and entropy for the system in the large and small cages, respectively. In eq 3, [1 − ϕ(T)] represents the population at low temperature corresponding to the force constant k1 and ϕ(T) the population at high temperature corresponding to the force constant k2. And, a transition temperature Td is usually defined as the temperature at which ΔG is zero, i.e., when a compensation between enthalpy and entropy occurs, such that [1 − ϕ(Td)] = ϕ(Td) = 0.5. However, to follow the
⎧ ⎫ ΔH ⎨1 + [1 − ϕ(Tr)]⎬ kBTr ⎩ ⎭
(7)
As a check, k3 = k1 and k3 = k2 in the low (i.e., ϕ = 0) and high (i.e., ϕ = 1) temperature limits, respectively.
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RESULTS Mean Squared Displacements. The five different DMPC samples were probed on the backscattering spectrometer IN13 6863
DOI: 10.1021/acs.jpcb.7b05167 J. Phys. Chem. B 2017, 121, 6860−6868
Article
The Journal of Physical Chemistry B
Figure 2. Left: Summed intensities of fully hydrated MLB, dry MLB, MLB in an intermediate hydration state and partially deuterated MLB (MLB heads). Right: MSD, binned over 13 K, extracted according to eq 2 from the Q-range (0.29 Å−1 < Q < 2.2 Å−1). Lines connecting the data points are guides to the eyes.
Figure 3. Best fits (solid lines through the data) of the MSD of all MLB samples according to eq 5. The dashed lines correspond to the low temperature linear fits [kB (T − 20 K)/k1], extrapolated to higher temperatures for visibility. The extracted fit parameters are summarized in Table 1.
detail by M. Rheinstädter et al.32 In the low temperature region, the motions are first harmonic and then methyl group rotations set in around 120 K, independently from the hydration state.11 In contrast, the onset of the dynamical transition at temperature T0 depends on the hydration state of the sample. The fully hydrated MLB heads have the most marked and sharp transition at ∼260 K, whereas their dynamics are below the others at low temperature. The transition of the fully hydrated and protonated membrane MLB is much less pronounced and appears much broader in temperature. Such a tendency resembles the slightly lower phase transition temperature found for the MLB heads33 compared to that of MLB of DMPC.34 For the less hydrated samples, the dynamical transition appears again very broad so that almost no differences become visible on inspection by the eye between the protonated MLBs at different hydration states. The MSD at high temperature indicates a high flexibility of the lipids, which was already documented in other publications.17,35 However,
described above. It gives access to information on averaged motions of hydrogen nuclei within the lipids and therefore to the global dynamics. All samples were measured over the temperature range 20−340 K, but the data in the range T > 290 K, where phase transitions set in, were excluded because they cannot be described simultaneously in the framework of the present model. Figure 2 shows the elastic intensities summed over all accessible Q-values and MSD extracted according to the first part of eq 2 and binned over 13 K ranges, neglecting the polydispersity for the sake of simplicity and setting Cα(τ) = 0, as only one instrument was used. It can be shown that the summed intensities are inversely proportional to the square root of the MSD,31 but the summation results in smaller error bars, which makes trends in the data more visible. The Q-range corresponding to the Gaussian approximation 27,28 was identified to be 0.29−2.2 Å−1. The MSDs are associated with the sample flexibility and stand for a measure of the fluctuations, as explained in more 6864
DOI: 10.1021/acs.jpcb.7b05167 J. Phys. Chem. B 2017, 121, 6860−6868
Article
The Journal of Physical Chemistry B
Figure 4. Force constants k1 (small cage) and k3 (large cage) for the five samples, ordered in increasing hydration from left to right (see Table 1).
Figure 5. Entropy (left) and enthalpy (right) difference between the large and small conformational cages for the five samples, ordered in increasing hydration from left to right (see Table 1).
produces force constants k1 similar to the ones of the other samples but k3 values which are lower and higher than those of the other samples, respectively. Such results indicate that tails and head groups behave in opposite ways but result in an average value close to the force constant of the hydrated MLB membrane. Such findings are also supported by the results for the differences in entropy and enthalpy for the five samples, reported in Figure 5. The excess enthalpy is below 10 kJ/mol and the excess entropy between 22 and 24 J/mol/K for the three samples formed by entire lipids, whereas both excess entropy and enthalpy are much higher for the lipid head groups and the excess entropy is also enhanced for the tails. It seems that the lipid head groups need a much higher amount of energy to overcome the energy barriers, maybe due to their higher confinement. However, once the threshold in energy is reached, their MSD rises quickly (see Figure 3). The carbon chains do not need as much energy but present at the same time a higher degree of entropy due to their associated degrees of freedom. Figure 6 shows the inverse participation ratios indicating how atomic motions are distributed between small and large conformational cages. The horizontal dashed line indicates the onset of dynamical transition with IPR = 1.21, indicating that most of the atomic motions are still localized in the small cages and just a few start to delocalize to the large cage. Indeed, at very low, nonphysiological temperature (
