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J. Phys. Chem. B 2008, 112, 7810–7815
Two-Particle Entropy and Structural Ordering in Liquid Water Jan Zielkiewicz† Department of Chemistry, Gdan´sk UniVersity of Technology, Narutowicza 11/12, 80-952 Gdan´sk, Poland ReceiVed: October 27, 2007; ReVised Manuscript ReceiVed: March 31, 2008
Entropies of simple point charge (SPC) water were calculated over the temperature range 278-363 K using the two-particle correlation function approximation. Then, the total two-particle contribution to the entropy of the system was divided into three parts, which we call translational, configurational, and orientational. The configurational term describes the contribution to entropy, which originates from spatial distribution of surrounding water molecules (treated as points, represented by the center of mass) around the central one. It has been shown that this term can serve as the metric of the overall orientational ordering in liquid water. Analyzing each of these three terms as a function of intermolecular distance, r, we also find a rational definition of the hydration shell around the water molecule; the estimated radii of the first and second hydration shells are 0.35 nm and 0.58 nm, respectively. We find, moreover, that the first hydration shell around the water molecule participates roughly in 70% of the total orientational entropy of water, and this rate is roughly temperature independent. Introduction Water is the most popular solvent; it is also the most important solvent for proteins. The structure of proteins strongly depends on the solvent environment. The role of water in stabilizing protein structure has been the subject of many papers,1–3 and this problem is still relevant. Analyzing the influence of water on protein structure, we take into account not only the direct protein-water interactions but also changes of the water structure within the layer at the immediate vicinity of the protein surface. Many physical properties of water within this solvation layer differ from the bulk ones.4 The reason these differences occur is the change of the structure of water within the solvation layer. The hydrogen bonds between water molecules are distorted or even disrupted, and, as a result, we observe significant changes of structural ordering in water. These structural changes are responsible for the so-called hydrophobic effect; it is widely agreed that the hydrogen bond interactions between water molecules are a key to understanding this effect.5 So, in this work we wish to find a link between entropy and the local structure of water. Our intention is to adopt (in the near future) this linkage for investigations into the structure of water within the solvation shell around the polypeptides. Another conclusion derived from the above remarks is that it is very important to use such water models, which properly reflect the structure of liquid water in molecular dynamics simulations. The most rational and “compact” metric of water structure is absolute entropy. In our recent paper6 it was found that, among the most popular water models, the simple point charge (SPC) model best represents the experimental value of the absolute entropy of water. A similar conclusion can also be drawn from an independent estimation of entropy of water given by Henchman.7 Therefore, we chose the SPC model of water for our further investigations. Method Molecular Dynamics Simulations. Simulations were carried out at nine temperatures: 5, 10, 15, 25, 35, 45, 60, 75, and 90 † E-mail:
[email protected].
°C, and at constant volume corresponding to a solvent density of 1.000 g/cm3; periodic boundary conditions were used. The system contained 512 rigid SPC water molecules within a cubic box. The Lennard-Jones parameters for the SPC model are ε0 ) 0.65017 kJ mol-1 and σO ) 0.31656 nm; the partial charges on atoms are qO ) -0.82e and qH ) +0.41e; the bond length is rO-H ) 0.1 nm, and the angle between bonds is ΘHOH ) 109.47°. The GROMOS968 program was used for simulation. The simulation temperature was kept constant by weakly coupling to a temperature bath with a relaxation time equal to 0.1 ps. Every system was initially equilibrated for at least 1 ns at a given temperature, then the trajectory was written to the file every 4 fs for further analysis. Calculations of the Entropy of Systems. In this work we adopt the expression of entropy into a series of the n-particle correlation functions. The entropy of an N particle system can be calculated as a sum of the series
S/Nk ) sid + s(2) + s(3) + · · · + s(N) ) sid + sexc
(1)
where sid symbolizes the entropy of an ideal gas, and s(n) describes the contributions to the entropy originating from n-particle correlations (n ) 2, 3, . . ., N). This expression was first derived by Green;9 Nettleton and Green10 and, much later, Reveche´11 derived a similar expansion for the entropy of open systems, and this idea was followed by many authors.12–16 This method allows for calculation of the absolute entropy of the system; however, it is difficult–or even impossible–to determine the n-particle correlation functions for higher n (n > 2). Therefore, the higher terms in the series (1) are omitted in practice; this is so-called two-particle approximation. It is difficult to estimate the error resulting from truncation of the series. Some results obtained for the Lennard-Jones fluid indicate12 that, in this case, the two-particle term contributes about 85-95% to the total excess entropy, sexc. We return to this problem later on in this paper. Within the two-particle approximation we omit all the higher terms, and expression for entropy reduces to the form14,15
S(2)/Nk ) sid + s(2) where
10.1021/jp7103837 CCC: $40.75 2008 American Chemical Society Published on Web 06/06/2008
(2)
Two-Particle Entropy in Liquid Water
J. Phys. Chem. B, Vol. 112, No. 26, 2008 7811
( FσΩ λ λ )
sid ) 3 - ln
3 T rot
and
s2 ) -
F2 2!Ω2
∫ (g(2)ln g(2) - g(2) + 1)dr1dω1dr2dω2
In the above relation F ) N/V), ri and ωi are coordinates of the center of mass and the Euler angles (describing rotations around the center of mass), determining the positions and spatial orientations of both particles, respectively. Other symbols are Ω ) ∫ dω ) 8π2, τ is the symmetry number (for water τ ) 2), λT and λrot are defined as
λT )
h
g(2)(r, ω1) )
√2πmkT h
√2πIxkT · √2πIykT · √2πIzkT
0 e R < π,
dω1 ) sin ϑdφdϑ,
dω2 ) sin RdRdβdγ
S (t) ) S∞ -
A B + √t(ln t + C)
∫ dω1 ) ∫02π dφ∫0π sin ϑdϑ ) 4π
In the above relation gr(r) is the well-known “ordinary” radial distribution function, and gconf describes the “configuration ordering” of surrounding particles around the central water molecule. Taking into consideration the remaining variables (three Euler angles), we find the final form of factorization: (2) (2) g(2)(r, ω1, ω2) ) g(2) r (r) · gconf(ω1|r) · gorient(ω2|r, ω1) (7)
[∫ I r
0
]dV
(2) trans + Iconf + Iorient - gr (r) + 1
0 e γ < 2π
)
(4)
where distance r and angles (φ,ϑ) determine the position of the surrounding particle (as the center of mass) in the spherical coordinate system whose origin is placed in the center of mass of the central molecule, and three Euler angles (R, β, γ) determine the orientation of the surrounding molecule relative to the central one. In this work we determine both the g(2) function and the integral (4) from the molecular dynamics results. The function g(2) has been determined within 0 < r < 1.2 nm interval by building the six-dimensional histogram. We used the distance step dr ) 0.02 nm, and the angle step deg ) 10°; the histogram values were normalized according to the definition of the two-particle correlation functions.17 The absolute entropy of the system under two-particle approximation was calculated according to eqs 2–4. Total simulation time was equal to 12.8 ns at each temperature, and the intermediate results of the entropy calculations were saved every 0.4 ns for further analysis. As it was found previously,6 the following function (2)
Ω1 )
s(2)(r) FN )k 2!
0eϑ 0.58 nm) are negligible in practice. The boundaries between these regions correspond to the position of the first and second minimum on the radial distribution function (see Figure 2b). Therefore, these regions may be identified with the first and second solvation (hydration) shell around the water molecule. This confirms the well-known fact that “short-range ordering” exists in liquid water. Our results provide the basis for estimating the scope of this ordering: it reaches approximately 0.58 nm. (3) As we note above, the major part (∼70%) of the entropy contribution, s(2), originates from the first solvation shell; moreover, this rate is almost temperature independent. Therefore, it is especially interesting to analyze the local ordering of water molecules within this shell; the word “local” means that we limit ourselves to the first solvation shell. The local structure of water is mainly determined by hydrogen bonds formed between molecules. In this work, we use a very generous criterion for defining the hydrogen bond, which allows for a
wide range of angles (up to 45°), and the mean number of hydrogen bonds per water molecule can serve as a simple metric of this local ordering. Some more sophisticated metrics of the local ordering may also be used, such as a parameter for tetrahedral ordering.21 It seems, however, that the most intuitive description of the local structure is given by the distribution (2) function gconf (r,φ,ϑ), defined by eq 6 and investigated over the first solvation shell (at r < 0.35 nm). Figure 3 shows this function, calculated at two temperatures: 278 and 363 K. From this figure, the tetrahedral arrangement of water molecules is clearly visible. The spatial distribution (given by the function (2) gconf (r,φ,ϑ)r < 0.35) seems to be more illustrative than the mean number of hydrogen bonds per water molecule. Moreover, from (2) function gconf (r,φ,ϑ), we directly calculate (using the second relation in eq 9) the configurational contribution to the entropy of the system, s(2) conf, and these calculations are based on rigorous statistical-mechanical considerations. It is interesting to note (2) here that both these quantities (sconf and the mean number of hydrogen bonds per molecule) are roughly proportional (see Figure 4). The above considerations can be summarized as follows: the most rational metric of local ordering in water is (2) the configurational part, sconf , of the orientational contribution to the entropy of the system. (2) In section 2 of our discussion, we conclude that the sum sor (2) (2) ) sconf + sorient measures the overall orientatonal ordering in water. On the other hand, from our last considerations, it seems (2) that the sconf quantity is sufficient for the description of local ordering in water. As it may be intuitively expected, both metrics (2) (2) of ordering–local, sconf , and overall, sor –are not independent. (2) Now, we wish to demonstrate that sconf can serve as a proper metric of total orientational effects in water. First, Figure 5 shows that s(2) conf (r) varies (as a function of r) in the same manner (2) (2) as the sum s(2) or (r) ) sconf (r) + sorient(r). Second, Figure 6 shows that the relation (2) (2) (2) s(2) conf + sorient ) sor ) f(sconf) (2) sconf
(15)
is almost strictly linear; in other words, is proportional to (2) (2) (2) (2) the sum sor ) sconf + sorient . Therefore, we can use sconf as a
7814 J. Phys. Chem. B, Vol. 112, No. 26, 2008
Figure 4. Correlation between the mean number of hydrogen bonds per water molecule, nHB, and the configurational contribution (in (2) J/mol · K units) to the entropy of water, sconf .
Zielkiewicz numerical example. In this work, we simulate 512 water molecules; even by taking into analysis 3.2 × 106 configurations (2) (12.8 ns of total simulation time), the calculated sor values still differ from its asymptotic (calculated from eq 5) values by 1.2-1.3 J/mol · K. In other words, a full convergence is still not achieved. These calculations require approximately 2 weeks of computer time, using a single processor Itanium 2, 1.6 GHz. (2) For calculations of the sconf quantity, however, only 104-105 configurations are sufficient for achieving the full convergence; it requires only a few hours of computer time. Recently, Errington and Debenedetti22 showed that a combination of two measures of structural order in water produces an integrated metric of the overall amount of order in the liquid. Following this idea, Esposito et al.23 have proposed to use both (2) (2) translational, strans , and orientational, sor (in our notation), (2) contributions to entropy for this purpose. Replacing s(2) or by sconf, it is possible to exploit this procedure for investigating structural changes in water within the solvation shell around solute molecules (such as polypeptides). Such calculations are in progress. Conclusion
Figure 5. The configurational part of the contribution (in J/mol · K (2) units) to entropy of water, sconf , calculated as a function of radius r and at various temperatures in the range 278-363 K. The arrow in this plot indicates the direction of increasing temperature.
Using factorization of the two-particle entropy, S(2), into three terms, we find linear dependence between the total orientational (2) contribution to absolute entropy, sor (defined by eq 14), and (2) the configurational term, sconf (eq 9). From this relation, it turns (2) out that the configurational term, sconf , is sufficient for the description of both local and overall structure in liquid water. (2) This statement allows one to use sconf instead of the total (2) orientational entropy, sor ; as a result, we were able to significantly simplify the calculations of the metric of orientational effects in liquid water. Moreover, an analysis of contribu(2) (2) tions s(2) trans(r), sconf(r), and sorient(r) plotted as a function of radius r (see Figures 2a and 5) offers a rational definition of the first and second hydration shells and clearly indicates boundaries between them. We also find that the first solvation shell (2) participates in ca. 70% of the total orientational entropy, sor ; it is interesting to note that this rate seems to be temperature independent. Acknowledgment. Calculations were carried out at the Academic Computer Centre (TASK) in Gdan´sk, and using a computer at the Centre of Excellence ChemBioFarm, located at the Department of Chemistry, Gdan´sk University of Technology. References and Notes
Figure 6. Linear relation between the total orientational contribution (2) (2) to the entropy, sor , and its configurational part, sconf . The straight line on this graph was plotted using the least-squares method; the entropy values are given in J/mol · K units. (2) measure of orientational effects instead of the sor quantity, at least within the investigated temperature range 278-363 K. (2) (2) Replacing sor by sconf allows one to save computer time used in entropy calculations. This can be illustrated by the following
(1) Dagetti, V. Chem. ReV 2006, 106, 1898. (2) Prabhu, N.; Sharp, K. Chem. ReV. 2006, 106, 1616. (3) Raschke, T. M. Curr. Opin. Struct. Biol. 2006, 16, 152. Nguyen, H. G.; Marchut, A. J.; Hall, C. K. Protein Sci. 2004, 13, 2909. (4) Baghi, B. Chem. ReV. 2005, 105, 3197. (5) Pratt, L. R. Annu. ReV. Phys. Chem. 2002, 53, 409. (6) Zielkiewicz, J. J. Chem. Phys. 2005, 123, 104501; 2006, 124, 109901. (7) Henchman, R. H. J. Chem. Phys. 2007, 126, 064504. (8) van Gunsteren, W. F.; Billeter, S. R.; Eising, A. A.; Hu¨nenberger, P. H.; Kru¨ger, P.; Mark, A. E.; Scott, W. R.; Tironi, I. G. Biomolecular Simulation: The GROMOS96 Manual and User Guide; Biomos b.v.: Zu¨rich, Groningen, 1996. (9) Green, H. S. The Molecular Theory of Fluids; North-Holland: Amsterdam, 1952; Chapter III. (10) Nettleton, N. E.; Green, M. S. J. Chem. Phys. 1958, 29, 1365. (11) Reveche´, H. J. J. Chem. Phys. 1971, 55, 2242. (12) Baranyai, A.; Evans, D. J. Phys. ReV. A 1989, 40, 3817; 1990, 42, 849. (13) Wallace, D. C. J. Chem. Phys. 1987, 87, 2282; Phys. ReV. A 1989, 39, 4843.
Two-Particle Entropy in Liquid Water (14) Lazaridis, T.; Paulaitis, M. E. J. Phys. Chem. 1992, 96, 3847. Smith, D. E.; Laird, B. B.; Haymet, A. D. J. J. Phys. Chem. 1993, 97, 5788. Lazaridis, T.; Paulaitis, M. E. J. Phys. Chem. 1992, 97, 5789. (15) Lazaridis, T.; Karplus, M. J. Chem. Phys. 1996, 105, 4294. (16) Prestipino, S.; Giaquinta, P. V. J. Stat. Phys. 1999, 96, 135. (17) Mu¨nster, A. Statistical Thermodynamics; Springer Verlag: Berlin/ Heidelberg/New York; Academic Press: London/New York, 1969; Vol. 1. (18) Press, W. H.; Flannery, B. P.; Teukolsky S. A.; Vetterling, W. T. Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed.; Cambridge University: Cambridge, U.K., 1999.
J. Phys. Chem. B, Vol. 112, No. 26, 2008 7815 (19) Wernet, Ph.; Nordlund, D.; Bergmann, U.; Cavalleri, M.; Odelius, M.; Ogasawara, H.; Na¨slund, L. A.; Hirsch, T. K.; Ojama¨e, L.; Glatzel, P.; Pettersson, L. G. M.; Nilsson, A. Science 2004, 304, 995. (20) Saija, F.; Saitta, A. M.; Giaquinta, P. V. J. Chem. Phys. 2003, 119, 3587. (21) Chau, P-L.; Hardwick, A. J. Mol. Phys. 1998, 93, 511. (22) Errington, J. R.; Debenedetti, P. G. Nature 2001, 409, 318. (23) Esposito, R.; Saija, F.; Saitta, A. M.; Giaquinta, P. V. Phys. ReV. E 2006, 73, 040502.
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