J Infrared Milli Terahz Waves (2014) 35:147–157 DOI 10.1007/s10762-013-0003-6

Low-frequency Spectra of a Phospholipid Bilayer Studied by Terahertz Time-domain Spectroscopy Tomoyo Andachi & Naoki Yamamoto & Atsuo Tamura & Keisuke Tominaga

Received: 13 May 2013 / Accepted: 8 July 2013 / Published online: 9 August 2013 # Springer Science+Business Media New York 2013

Abstract We have investigated the low-frequency spectra of a phospholipid bilayer composed of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) by terahertz time-domain spectroscopy (THz-TDS). We focused on the temperature and hydration dependence of the low-frequency spectra of a gel-phase sample. The spectra of the dehydrated and hydrated samples showed shoulder bands at 45 and 30 cm-1, respectively. In contrast to the dehydrated sample, in the hydrated sample spectra the slope of the temperature change of the absorption coefficient increased sharply around 240 K. This result suggests that water molecules affect the change in the low-frequency dynamics. We obtained the absorption coefficient difference spectra for different hydration levels to clarify the mechanism of the spectral change. Keywords DMPC . Hydration water . Temperature depedence

1 Introduction Cell membranes are mainly formed from self-assembled phospholipid bilayers. The membranes separate the contents of the cell from the environment, and regulate a wide range of biochemical functions in conjunction with membrane proteins, including the transport of ions and nutrients through the membrane. These functions are performed in a thermal environment surrounded with water; therefore, it is important to determine the effect of temperature and hydration on the membrane dynamics in order to understand biological functions [1].

Electronic supplementary material The online version of this article (doi:10.1007/s10762-013-0003-6) contains supplementary material, which is available to authorized users. T. Andachi : A. Tamura : K. Tominaga (*) Graduate School of Science, Kobe University, 1-1 Rokkodai-cho, Nada, Kobe 657-8501, Japan e-mail: [email protected]

N. Yamamoto : K. Tominaga Molecular Photoscience Research Center, Kobe University, 1-1 Rokkodai-cho, Nada, Kobe 657-8501, Japan

148

J Infrared Milli Terahz Waves (2014) 35:147–157

The effects of temperature and hydration on the dynamics of the lipid bilayers have been investigated with experimental techniques including neutron scattering [2, 3], dielectric spectroscopy [4, 5], and molecular dynamics simulation studies [6]. Recently, terahertz spectroscopy (1 THz ~ 33 cm-1) has drawn much attention as a powerful tool for investigating the low-frequency dynamics of biopolymers, such as proteins and DNA, and their interaction with water [7–10]. The low-frequency spectra in the THz region contain information about the dynamics on sub-picosecond to picosecond timescales. Theoretical calculations have shown that the structure of lipid bilayers also continuously fluctuates at room temperature due to thermal activation. These fluctuations are important for the biological functions of lipid bilayers and are often characterized by spectral components in the THz frequency region (0.1-10 THz). Molecular motions, such as the delocalized vibrational motion of lipid molecules, and the dynamics of hydrogen-bond forming and breaking between the head groups of lipids and water molecules, are expected to produce characteristic spectral features in this frequency region,. THz-TDS has been used to monitor the hydration effects on the low-frequency dynamics of lipid membranes. Tielrooij et al. used THz-TDS to study the dielectric relaxation dynamics of water in a model membrane consisting of 1,2-dioleoyl-sn-glycero-3-phosphocholine lipids [11]. The hydration level was expressed by the number of water molecules per lipid molecule, R = [H2O]/[lipid]. They varied the value of R from 0.3 (dehydrated sample) to 8.4 and observed three distinct types of water molecule: bulk water, water with a slow reorientation time, and partially hydrogen-bonded water with a very fast reorientation time. Hishida and Tanaka investigated the long-range hydration effects of lipid bilayer membranes formed from the model phospholipid, 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) [12–14], by changing the value of R from 19 to 190 [15]. They confirmed a long-range hydration effect on up to 4-5 layers of water. Park and coworkers studied the change in the dielectric relaxation of water during the phase transition of a DMPC lipid bilayer with R of 80 [16]. The dielectric relaxation time of water decreased initially, and then increased after the gel-to-fluid phase transition. In the present study, we used THz-TDS to investigate the temperature and hydration dependences of the low-frequency dynamics of lipid bilayers composed of DMPC. Lipid bilayers form various phase structures, including a crystalline phase, a gel phase, and a liquid-crystalline phase, depending on environmental conditions, such as the temperature and the amount of hydration water. We used lipid bilayers in the gel phase, which is formed at room temperature. The THz spectra of DMPC were measured from 123 to 293 K with a low hydration level where the value of R was 3.5 (dehydrated sample) to 7.9, in order to investigate the interplay between the water and lipid molecules and its thermal activation.

2 Material and Methods DMPC was purchased from Avanti Polar Lipids, Inc. and used without further purification. Other chemicals were purchased from Wako Pure Chemical Industries, Ltd. Gelphase samples were prepared using the following procedure [17]. In brief, DMPC was dissolved in chloroform, and then the solvent was slowly evaporated with dry air. The solvent was completely removed by placing the sample under vacuum for 12 h. Finally, a thin lipid film was formed. The phases of the sample were determined with wideangle X-ray diffraction measurements at 293 K by X-ray diffractometry (SmartLab, Rigaku).

J Infrared Milli Terahz Waves (2014) 35:147–157

149

For THz measurements, the lipid film was ground in a mortar and then compressed under a pressure of 5 MPa for 10 s to obtain pellets with a diameter of 5 mm and a thickness of 0.6-1.0 mm. The dehydrated samples were prepared by placing the sample pellet under vacuum at room temperature for more than 12 h to remove excess hydration water. For the hydrated samples, the level of hydration was controlled by incubating the sample in a container which was humidified with Milli-Q water for various incubation times. We measured the weight of the sample before and after the THz-TDS measurements and averaged them to obtain the R value. The temperature was varied from 123 to 293 K at 10° intervals by using a cryostat system. Details of the THz-TDS apparatus have been reported previously [18]. The main change we made to the system was replacing the excitation laser with a mode-locked Ti:sapphire laser (Integral Pro, Femtolasers), centered at 800 nm with a pulse duration of ~10 fs. and a repetition rate of 80 MHz. This pumped a pair of photoconductive switches that were used as a THz emitter and detector. A computer-controlled delay was used to detect the temporal waveform of the THz wave. The whole system was placed in a chamber under a flow of dry nitrogen gas. The spectral region for the measurements was 10-60 cm-1. Time-domain detection techniques can be used to observe changes in both the amplitude and phase of the THz wave as a function of the optical delay time. Therefore, the spectra of both the absorption coefficient (α) and refractive index (n) can be obtained simultaneously by Fourier transformation of the reference and sample time-domain signals. The absorption coefficient, αðe ν Þ , is expressed as !   α e ν ¼

  Z I e ν ¼

e

πe ν 1−e−βhcν

    I e ν ; ν V 3ε0 hcn e

∞ −∞

dte−i2πcνt 〈Μð0Þ⋅Μðt Þ〉;

MðtÞ ¼

e

N X

μi ðtÞ:

ð1Þ

ð2Þ

ð3Þ

i¼1

Here, I ðe ν Þ is the line shape function defined as the Fourier transform of the time-correlation function (TCF) of the total dipole moment, 〈M(0) ⋅ M(t)〉, and e ν is the wavenumber. The line shape function is represented as a normalized quantity with respect to one molecule. μi (t) is the dipole moment of each molecule and is the sum of the permanent and induced dipole moments, ε0 is the dielectric constant in vacuum, h is the Planck constant, c is the speed of light, V is the volume of the sample, and β = 1/kBT, where kB and T are the Boltzmann constant and temperature, respectively. Therefore, the absorption coefficient is associated with the TCF Fourier transform of the total dipole moment of the system.

3 Results and Discussion Figure 1 shows the X-ray diffraction patterns of the dehydrated (R = 3.5) and hydrated samples (R = 5.9). The diffraction patterns indicate the packing state of the hydrocarbon

150

J Infrared Milli Terahz Waves (2014) 35:147–157

chains of the lipid molecules. The abscissa axis is the distance, d=λ/2 sinθ, where 2θ is the scattering angle and λ is the X-ray wavelength, λ = 1.5418 Å. In the dehydrated sample, the diffraction pattern contains a major peak at 0.410 nm and a shoulder around 0.430 nm. Comparing this diffraction pattern with that of a previous study confirms that a gel-phase lipid bilayer is formed [19]. Upon hydration, the diffraction pattern changes, and three peaks are observed at 0.430, 0.404, and 0.381 nm. This indicates that the change in the diffraction pattern is caused by the transformation of the normal gel phase to a more ordered gel phase [19]; it is not caused by a phase transition in the lipid bilayer. The phase transition temperature of DMPC is reported to be 23 °C in solution [12]. Under the present experimental conditions, the hydration level is relatively low. Thus, it can be concluded that the gel-phase structure is maintained under the hydration conditions in this study. Figure 2 shows the temperature-dependent THz absorption spectra at different hydration levels (R = 3.5 (dehydrated sample), 6.8, and 7.9) for the gel-phase structure. The spectra of the refractive index and the real and imaginary parts of the complex dielectric constant are reported in the Supporting Information. The absorption coefficient increases monotonically with the wavenumber. A small shoulder is visible around 45 cm-1 for the dehydrated sample and around 30 cm-1 for the hydrated sample. The peak shoulders are discussed later. The absorption coefficient increases with the temperature. Upon hydration, the effect of temperature on the spectra becomes stronger. To clarify the differences between the dehydrated and hydrated samples, the absorption coefficient at 15, 20, and 30 cm-1 is plotted against the temperature in Figure 3. The rates of increase of the absorption coefficient are almost identical for the dehydrated sample. In contrast, during hydration the rate of the increase changes at around 240 K. This indicates that the lowfrequency dynamics in the hydrated lipid bilayers change dramatically at this temperature. Because the transition point is observed only in the hydrated samples, we propose that the water molecules play a role in the changes in the low-frequency spectra. To determine the effect of water on the spectral change, we calculate the difference spectra in the following way. The absorbance, Abs, is expressed as a product of the optical path length, l, the molar concentration, c, and the molar extinction coefficient, ε; Absðe ν Þ ¼ lcεðe ν Þ . In Figure 2, the absorption coefficient, αðe ν Þ ¼ Absðe ν Þ=l ¼ cεðe ν Þ , is plotted on the ordinate axis. We assume that the absorption coefficient is a sum of the contributions of the lipid bilayer and hydration water, n    o   ν ; R; T þ chw ðR; T Þεhw e ν ; R; T : ð4Þ Abs e ν ; R; T ¼ l cl ðR; T Þεl e

R = 3.5 R = 5.9

Intensity(arbitrary)

293 K

0.46

0.44

0.42

0.40

0.38

d (spacing) / nm Fig. 1 X-Ray diffraction patterns of the gel-phase sample at 293 K

0.36

J Infrared Milli Terahz Waves (2014) 35:147–157

151

Absorption Coefficient / cm

-1

(a) 50 123 K 153 K 183 K 213 K 243 K 273 K 293 K

40 30 20

10 0

R = 3.5 0

10

20

30

40

Wavenumber / cm

50

60

-1

Absorption Coefficient / cm

-1

(b) 50 123 K 153 K 183 K 213 K 243 K 273 K 293 K

40 30 20

10 0

R = 6.8 0

10

20

30

40

Wavenumber / cm

50

60

-1

Absorption Coefficient / cm

-1

(c) 50 123 K 153 K 183 K 213 K 243 K 273 K 293 K

40 30 20

10

R = 7.9

0

0

10

20

30

40

Wavenumber / cm

50

60

-1

Fig. 2 Temperature dependence of the absorption spectra of the DMPC lipid bilayer in the gel phase. The level of hydration is (a) R = 3.5, (b) R = 6.8, and (c) R = 7.9

Here subscript l and hw denote the lipid and the hydration water, respectively. The molar concentrations of the lipid and hydration water are estimated from the weight of the sample and the volume of the sample pellet. It should be noted that the dehydrated samples contain a small amount of hydration water. We assume that the molar extinction coefficient of the ν ; R; T Þ ¼ εhw ðe ν; T Þ . hydration water does not depend on the level of hydration, εhw ðe Figure 4 shows the difference spectra of the absorption coefficient for the dehydrated (R=R0) and hydrated samples (R) normalized with the molar concentration of the lipid and

152

J Infrared Milli Terahz Waves (2014) 35:147–157

Absorption Coefficient / cm

-1

(a) 8

R = 3.5 R = 6.8 R = 7.9

7 6 5 4

3

15 cm

2

160

200

240

-1

280

Temperature / K

Absorption Coefficient / cm

-1

(b) 12

R = 3.5 R = 6.8 R = 7.9

10 8 6 4

20 cm 160

200

240

-1

280

Temperature / K

Absorption Coefficient / cm

-1

(c) 22

R = 3.5 R = 6.8 R = 7.9

20

18 16 14 12 10

30 cm 160

200

240

-1

280

Temperature / K

Fig. 3 Temperature and hydration dependence of the DMPC lipid bilayer in the gel phase on the absorption coefficients at 15, 20, and 30 cm-1

the value of R, which are defined by   9 8  α νe; R0 ; T = 1 < α νe; R; T Δε νe; R; R0 ; T ¼ − cl ðR0 ; T Þ ; : R−R0 : cl ðR; T Þ   o   1 n  εl νe; R; T −εl νe; R0 ; T þ εhw νe; T ¼ R−R0 



ð5Þ

J Infrared Milli Terahz Waves (2014) 35:147–157

153

(a) water (293 K) ice 123 K 153 K 183 K 213 K 243 K 273 K 293 K

2

ΔR = 3.3

-1

Δε / cm M

-1

3

1

0

0

10

20

30

40

Wavenumber / cm

50

60

-1

(b) water (293 K) ice 123 K 153 K 183 K 213 K 243 K 273 K 293 K

2

ΔR = 4.4

-1

Δε / cm M

-1

3

1

0

0

10

20

30

40

Wavenumber / cm

50

60

-1

Fig. 4 Temperature dependence of the difference spectra, Δεðe ν ; R; R0 ; T Þ , of the DMPC lipid bilayer in the gel phase, where R0 = 3.5. (a) R = 6.8 (ΔR = 3.3) and (b) R = 7.9 (ΔR = 4.4). The bulk water spectrum at 293 K and ice spectrum at 263 K are also displayed

If the molar extinction coefficient of the lipid is not affected by the hydration, ν ; R0 Þ ¼ εl ðe ν ; RÞ , the difference spectra is equal to the hydration water spectrum, εl ðe ν ; T Þ . However, if the low-frequency spectrum of the lipid is affected Δεðe ν ; R; R0 ; T Þ ¼ εhw ðe by hydration, the difference is reflected in the difference spectrum. Figure 4 also shows the spectra of the frequency-dependent molar extinction coefficient of liquid water at 293 K and ice at 263 K for comparison. In the difference spectra, a positive peak is observed around 30 cm-1 and a negative peak around 45 cm-1. The peaks correspond to the shoulders in the absorption spectra of the hydrated and dehydrated samples. This is consistent with the X-ray diffraction measurements, which show that the lipid packing pattern changes upon hydration. The intensity of

154

J Infrared Milli Terahz Waves (2014) 35:147–157

the 30 cm-1 band in the difference spectrum is greater for ΔR=R−R0=6.8−3.5=3.3 (Fig. 4(a)) than for ΔR=R−R0=7.9−3.5=4.4 (Fig. 4(b)) below 240 K. This is probably caused by the factor of 1/(R − R0) in Eq. (5). However, this interpretation is not appropriate above 240 K because of the presence of spectral components observed only when ΔR=4.4. There are two possible explanations for this spectral difference: that the band at 45 cm-1 in the dehydrated sample disappears and a new band appears at 30 cm-1 after hydration; or the 45 cm-1-band gradually shifts toward the lower-frequency side after hydration. Upon hydration, the gel-phase structure becomes more ordered, and the phase transition does not occur; therefore, the latter explanation is more plausible. However, further analysis of the hydration dependence of the spectrum is required to clarify this point.

(a) ΔR = 3.3

water (293 K) ice 153 K 183 K 213 K 243 K 273 K 293 K

2

-1

ΔΔε / cm M

-1

3

1

0

0

10

20

30

40

50

Wavenumber / cm

60

-1

(b) ΔR = 4.4

water (293 K) ice 153 K 183 K 213 K 243 K 273 K 293 K

2

-1

ΔΔε / cm M

-1

3

1

0

0

10

20

30

40

Wavenumber / cm

50

60

-1

Fig. 5 Temperature dependence of the differences in the difference spectra, ΔΔεðe ν ; R; R0 ; T 1 ; T 2 Þ , of the DMPC lipid bilayer in the gel phase, where R0 = 3.5 and T1 = 123 K. (a) R = 6.8 (ΔR = 3.3) and (b) R = 7.9 (ΔR = 4.4). The bulk water spectrum at 293 K and ice spectrum at 263 K are also displayed

J Infrared Milli Terahz Waves (2014) 35:147–157

155

Previous THz-TDS studies of molecular crystals have shown that the THz spectra of small molecules, such as anthracene, contain only pure intermolecular or intramolecular vibrational bands, and that the intra- and intermolecular modes are not coupled [20]. However, for relatively large molecules the vibrational modes of the intra- and intermolecular modes are strongly coupled in the THz frequency region. Therefore, the observed absorption bands also include the low-frequency vibrational mode, which results from the mixing of the intra- or intermolecular modes. Interestingly, the observed band is shifted to a lower frequency by hydration. If the mode were a pure intermolecular mode the band would be blue shifted by hydration, because the more ordered state would increase the oscillator constant. Figure 5 shows the difference spectra of Δεðe ν ; R; R0 ; T Þ between different temperatures T1 and T2;       ΔΔε e ν ; R; R0 ; T 1 ; T 2 ¼ Δε e ν ; R; R0 ; T 2 −Δε e ν ; R; R0 ; T 1 ð6Þ We fixed the lower temperature T1 as 123 K and varied T2. The spectra of liquid water and ice are also shown for comparison. Liquid water has an intense band in the THz spectra caused by the collective rotational relaxation and structural fluctuations of the hydrogenbond network structure [21–23]. In contrast, the spectrum for ice does not contain a band below 40 cm-1, because there are no mobile water molecules present. Instead, a weak transverse acoustic phonon band is visible around 60 cm-1 [24, 25]. Figure 6 shows the temperature dependence of ΔΔεðe ν ; R; R0 ; T 1 ; T 2 Þ at fixed wavenumbers of 15, 20, and 40 cm-1. Figures 5 and 6 lead us to several conclusions. At 40 cm-1, where the lipid bilayer and ice have spectral components, ΔΔεðe ν ; R; R0 ; T 1 ; T 2 Þ changes as a function of temperature above 160 K. This probably arises from the thermal activation of the vibrational modes of the lipid bilayer or ice. In contrast, at lower frequencies, such as 15 and 20 cm-1, ΔΔεðe ν ; R; R0 ; T 1 ; T 2 Þ does not show any temperature dependence up to 230 K. This shows that the hydration water molecules are frozen and do not move freely in this temperature region. From 230 to 260 K, the spectral intensity below 30 cm-1 increases with temperature. Figure 6 shows the two experimental results for ΔR = 3.3 and 4.4 almost overlap with each other in this temperature region, indicating that this spectral change is caused by the dynamics of the hydration water, not the lipid bilayer. Above 260 K, the results for ΔR = 4.4 show a greater dependence on the temperature. This may be caused by the breakdown of ν ; R; T Þ is independent of the level of the assumption that the hydration water spectrum εhw ðe hydration. In principle, the magnitude of the changes in the dynamics of the hydration water depends on the level of hydration. Above a certain temperature, these changes may be more pronounced because of thermal activation.

4 Summary We have investigated the temperature and hydration dependence of the low-frequency spectrum of a DMPC lipid bilayer by THz-TDS. The absorption band was observed at around 45 cm-1 in the dehydrated sample. Upon hydration, a band was observed at around 30 cm-1. X-ray diffraction measurements confirmed that the packing patterns of the hydrocarbon chains changed upon hydration. Furthermore, the low-frequency dynamics of the lipid and the hydration water molecules were affected by the change in the temperature and

156

J Infrared Milli Terahz Waves (2014) 35:147–157

(a) ΔR = 3.3 ΔR = 4.4

0.4

-1

ΔΔε / cm M

-1

0.6

0.2 0.0

15 cm 150

200

-1

250

Temperature / K

(b) ΔR = 3.3 ΔR = 4.4

-1

ΔΔε / cm M

-1

0.6 0.4 0.2 0.0

20 cm 150

200

-1

250

Temperature / K

(c) ΔR = 3.3 ΔR = 4.4

-1

ΔΔε / cm M

-1

1.0

0.5

0.0

40 cm 150

200

-1

250

Temperature / K Fig. 6 Temperature and hydration dependence of the differences in the difference spectra, ΔΔεðe ν ; R; R0 ; T 1 ; T 2 Þ , of the DMPC lipid bilayer in the gel phase at 15, 20, and 40 cm-1

the level of hydration. The absorption coefficients at 15, 20, and 30 cm-1 plotted against temperature indicated that the increasing rates of the absorption coefficients were almost identical for the dehydrated sample. In contrast, upon hydration the rates of increase changed at around 240 K. This suggests that the low-frequency dynamics in this system changed dramatically at this temperature. Plotting the differences in the difference spectra at 15, 20, and 40 cm-1 against temperature showed that this transition mainly resulted from the dynamics of the hydration water. Acknowledgements This work is partially supported by Industry-Academia Collaborative R&D from the Japan Science and Technology Agency (JST).

J Infrared Milli Terahz Waves (2014) 35:147–157

157

References 1. Structure and Dynamics of Membranes. (Elsevier, Amsterdam, 1995). 2. J. Swenson, F. Kargl, P. Berntsen and C. Svanberg, J Chem Phys 129 (4), 045101 (2008). 3. M. Trapp, T. Gutberlet, F. Juranyi, T. Unruh, B. Deme, M. Tehei and J. Peters, J Chem Phys 133 (16), 164505 (2010). 4. C. Svanberg, P. Berntsen, A. Johansson, T. Hedlund, E. Axen and J. Swenson, J Chem Phys 130 (3), 035101 (2009). 5. P. Berntsen, C. Svanberg and J. Swenson, J Phys Chem B 115 (8), 1825-1832 (2011). 6. R. M. Venable, Y. H. Zhang, B. J. Hardy and R. W. Pastor, Science 262 (5131), 223-226 (1993). 7. A. G. Markelz, A. Roitberg and E. J. Heilweil, Chem Phys Lett 320 (1-2), 42-48 (2000). 8. S. Kawaguchi, O. Kambara, M. Shibata, H. Kandori and K. Tominaga, Phys Chem Chem Phys 12 (35), 10255-10262 (2010). 9. N. Yamamoto, O. Kambara, K. Yamamoto, A. Tamura, S. Saito and K. Tominaga, Soft Matter 8 (6), 19972006 (2012). 10. R. J. Falconer and A. G. Markelz, J Infrared Millim Te 33 (10), 973-988 (2012). 11. K. J. Tielrooij, D. Paparo, L. Piatkowski, H. J. Bakker and M. Bonn, Biophys J 97 (9), 2484-2492 (2009). 12. R. Koynova and M. Caffrey, Bba-Rev Biomembranes 1376 (1), 91-145 (1998). 13. F. de Meyer and B. Smit, P Natl Acad Sci USA 106 (10), 3654-3658 (2009). 14. T. P. W. Mcmullen, R. N. A. H. Lewis and R. N. Mcelhaney, Biochemistry-Us 32 (2), 516-522 (1993). 15. M. Hishida and K. Tanaka, Phys Rev Lett 106 (15), 158102 (2011). 16. D. H. Choi, H. Son, S. Jung, J. Park, W. Y. Park, O. S. Kwon and G. S. Park, J Chem Phys 137 (17), 175101 (2012). 17. L. Terminassiansaraga, E. Okamura, J. Umemura and T. Takenaka, Biochim Biophys Acta 946 (2), 417423 (1988). 18. P. Dutta and K. Tominaga, J Phys Chem A 113 (29), 8235-8242 (2009). 19. B. Tenchov, R. Koynova and G. Rapp, Biophys J 80 (4), 1873-1890 (2001). 20. O. Kambara, K. Tominaga, J. Nishizawa, T. Sasaki, H. W. Wang and M. Hayashi, Chem Phys Lett 498 (13), 86-89 (2010). 21. I. Ohmine, J Phys Chem-Us 99 (18), 6767-6776 (1995). 22. M. Cho, G. R. Fleming, S. Saito, I. Ohmine and R. M. Stratt, J Chem Phys 100 (9), 6672-6683 (1994). 23. I. Ohmine and S. Saito, Accounts Chem Res 32 (9), 741-749 (1999). 24. J. E. Bertie and E. Whalley, J Chem Phys 46 (4), 1271-1284 (1967). 25. N. H. Fletcher, The Chemical Physics of Ice (Cambridge University Press, Cambridge, 1970).