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J. Phys. Chem. B 2010, 114, 4536–4550
Structural Properties of Hydration Shell Around Various Conformations of Simple Polypeptides Dariusz Czapiewski and Jan Zielkiewicz* Gdan´sk UniVersity of Technology, Department of Chemistry Narutowicza 11/12, 80-952 Gdan´sk, Poland ReceiVed: September 7, 2009; ReVised Manuscript ReceiVed: January 20, 2010
In this paper we investigate structural properties of water within the solvation shell around the peptide core created by a well-defined conformation of polypeptide chain. The following secondary structures are investigated: linear (straight chain), and three helices PII (polyproline-like), 310, and R. We propose using the two-particle contribution to entropy as a rational measure of the water structural ordering within the solvation layer. This contribution divides into two terms, depending on the peptide-water and water-water interactions, respectively, and in this paper both terms are investigated. The structure of “solvation” water is described by the second term, and therefore it mainly attracts our attention. Determination of this term, however, is not an easy task, requiring some controversial approximations. Therefore, we have transformed this term to the form of some rational parameter which measures the local structural ordering of water within the solvation shell. Moreover, the results of several independent investigations are reported: we adopt the harmonic approximation for an independent estimation of the water entropy within the solvation shell, and we also study structure of the water-water hydrogen bond network, mean geometry of a single hydrogen bond, the self-diffusion coefficients (both translational and rotational) of water, and the mean lifetimes of water-water and water-peptide hydrogen bonds. All the obtained results lead to the conclusion that the local structure of water within the solvation shell changes only slightly in comparison to the bulk one. If so, the measure of local water ordering proposed by us is exploited with the aim to gain the deeper insight on the structural properties of “solvation” water. It has been shown that this parameter can be factored into three terms, which measure translational, configurational, and orientational ordering, respectively. Using this factoring, the ordering map for a precise description of the water local ordering has been built. An interesting correlation is observed: the points on this map lie approximately on the straight line, while the linear conformations clearly deviate from the general tendency. Further analysis of the obtained results allows us to express the supposition that an increasing local ordering of water around given secondary structure corresponds to an increasing relative stability of this structure in aqueous solution. Analyzing the geometry of the water-water hydrogen bond network within the solvation layer, we find some systematic deviations of this geometry from the bulk water properties. We also observe that the alanine peptides (excluding the linear form) disturb the hydrogen bond network in the less range, and in another way than the various conformations of polyglycine, while the linear form of polyalanine behaves very similarly to the glycine ones. Next, investigating the dynamic properties, we also conclude that water near the peptide surface creates a pseudorigid structure, a “halo” around the peptide core. This “halo” is stabilized by slightly higher energy of the hydrogen bonds network: we have found that within this region the hydrogen bonds network is slightly less distorted, the water-water hydrogen bonds are a little more stable and their mean lifetime is clearly longer that that of bulk water. Significant differences between the alanine- and glycine-based polypeptides are also visible. It has also been found that this solvation layer interacts with the polyalanine in another way than with polyglycine. Although in the case of the glycine-based polypeptide this layer slides relatively freely over the peptide surface, for the alaninebased polypeptide this sliding is strongly hindered by the presence of the methyl groups, and this effect is additionally enhanced by a rise in the solvation layer rigidity. Thus, the survey of various dynamic properties allows us to perceive and to explain distinct differences in behavior of water within the solvation shell around both glycine and alanine peptides. Introduction It is a well-known fact that water plays a crucial role in determining the secondary (tertiary, quaternary) structure of proteins in a solution.1–3 The influence of the solvent on aminoacid preferences of adopting various secondary structures would account for the fact why prediction of secondary structure is successful in 70% cases only. Also the tendency of various * Corresponding author. E-mail:
[email protected].
amino acids to create an R-helical structure strongly depends on the solvent.4,5 An impressive example that illustrates the solvent role is the extreme stability of an unsolvated (in vacuum) R-helical form of polyalanine: it remains almost completely helical up to 725 K, whereas in an aqueous solution the same helix is only marginally stable.6 To give another persuasive example, we resort to the results obtained by Sorin et al.7 They showed, by using computer simulations and by manipulation of the Lennard-Jones parameters of water, that the character of
10.1021/jp9086199 2010 American Chemical Society Published on Web 03/16/2010
Structural Properties of Simple Polypeptides the solvation interface induces several preferential conformations of the alanine-based polypeptide. Other works also confirm that the solvent environment has a significant effect on the stability of various polyalanine secondary structures.8,9 Not only the secondary (tertiary, quaternary) protein structure is solvent dependent but also the protein conformational dynamics (which determines its vital functionality) is tightly connected to the structure and the dynamics of the surrounding water.10–14 It is even claimed that protein and solvent dynamics are so closely related that they should be perceived as a single entity.2 It is easy to understand all these facts: the structural stability of a particular peptide conformation is determined by the free energy difference between considered conformational states. The free energy can be divided into two terms: the energetic and the entropic one. To evaluate the influence of the solvent, therefore, we should estimate the solvent dependent contribution to each term. The energetic term may be readily calculated from the simulation data; calculation of the entropic term, however, is a very hard problem. Although the conformational entropy change of the peptide chain can be estimated using, for example, the covariance matrix method,15–17 the calculation of hydration entropy still remains an unresolved problem, even though its estimation seems to be very important.18,19 Recently, Lin et al.20,21 have proposed an approximate method for calculation of the water entropy using the two-phase (2PT) model. This method has been adopted to the entropy calculations of “solvation” water within minor and major grooves of DNA,22 and as far as we know, this is the only attempt of the local entropy change calculations. We keep in mind that the entropy changes concomitant with the solvation (hydration) process reflect changes in the structure of the surrounding water. These structural changes influence, undoubtedly, many physical properties of water within the solvation layer.3,10,12 Hence, not only the direct protein-water interactions but also the modification of water structure should be taken into account. In other words, this means that not only water influences the peptide structure but also the structure of the surrounding water “fits” to a particular peptide conformation. Thus, a following question arises: how does the structure (and also various physical properties) of water within the solvation shell around the peptide molecule depend on the given secondary structure formed by this biopolymer? To answer this question, we resort to the concept of n-particle correlation functions, which offer a valuable tool for this study. Preliminary results of such studies were reported previously.23 Now, we extend our investigations onto very important measurements of the solvation layer water-water structural ordering. As far as we know, this is the first attempt of such approach. Moreover, some other methods are also used with the aim of the independent confirmation of our deductions. So, in this paper we present the results of the investigations of some structural and dynamic properties of water within the solvation shell formed around four basic conformations of the polypeptide chain: linear (an extended chain), and three helices: PII (polyproline-like), 310, and R. Method Molecular Dynamics Simulations. The simulations were carried out using the Amber1024 molecular dynamics package. Polypeptide (Ala15) was built using 15 alanine residues; the Amber ff03 force field was used. Four conformations of the peptide molecule have been investigated: linear (straight chain), and three helices: PII (polyproline-like), 310, and R. The peptide molecule was centered within the box; the minimal distance of
J. Phys. Chem. B, Vol. 114, No. 13, 2010 4537 the peptide molecule from the box walls was 1.5 nm for each conformation. The peptide backbone atoms were positionally restrained (using harmonic forces, k ) 6 kcal/(mol Å2)) to prevent the conformational changes. For our investigations we use the TIP4P model of water instead of the SPC one used previously.23 There are two reasons for such a choice. First, we investigate some dynamic properties of water, while the TIP4P model better describes such properties than the SPC one. Second, it is a chance to examine if our deductions depend on the choice of water model. Besides the peptide-water systems, the system containing only 2410 TIP4P water molecules was also investigated; it has served as a reference system, to determinate the properties of pure water. The simulations of all systems were carried out using NVT conditions; the temperature (T ) 298 K) was kept constant by the weak coupling to an external bath (τT ) 0.2 ps). Every system was initially equilibrated for at least 2 ns in NPT conditions, afterward 0.2 ns in NVT conditions, and then the trajectory was written to a file for further analysis. In this paper we wish to find various properties of the water surrounding a well-defined fragment of the polypeptide secondary structure. Therefore, to avoid the “end effect”, two residues at both peptide ends were omitted, and the middle region representing the well-defined structural part of the polypeptide chain was taken into account. So, the analyzed part of the peptide chain is 11 residues long. Hydrogen Bond Analysis. We have adopted the hydrogen bond definition described in the Wernet et al.’s paper.25 According to this definition, the hydrogen bond between two water molecules is formed if the oxygen-oxygen distance fulfills the relation
ROO e -0.000044βOOH2 + 0.33 [nm]
(1)
where βOOH is the O-O-H angle (in degrees). We demanded, moreover, the total interaction energy between two “hydrogenbonded” molecules to be negative; it is roughly equivalent to demanding that the force acting between these molecules be attractive. This total interaction energy has been calculated as the sum of both Lennard-Jones and electrostatic terms between two molecules for which relation 1 is fulfilled; its absolute value, averaged over all H-bonded molecules, was assumed as the mean H-bond energy of the system. We determine both energy and geometry of the hydrogen bond as a function of the distance r from the polypeptide axis. Diffusion Coefficients and the Hydrogen Bond Lifetime. We wish to determine these quantities separately for both the first and the second solvation shell around the peptide molecule. The diffusion coefficients and the lifetimes of the hydrogen bond have been calculated from the appropriate time correlation functions. The following procedures were adopted. Let vK(t) denote a water molecule’s vector of translational (K ) T), or rotational velocity (K ) R). The diffusion coefficients: translational (DT), and rotational (DR), were calculated from the appropriate velocity autocorrelation function using the well-known Green-Kubo relation26
DK )
1 lim( 3 tf∞
∫0t CK(t) dt)
(2)
where CK(t) symbolizes the translational (or angular) velocity autocorrelation function:
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CK(t) ) lim
Czapiewski and Zielkiewicz
( T1 ∫ v (x) v (x+t) dx) ) 〈v (x) v (x+t)〉
Tf∞
T
0
K
K
K
K
{
{
(3)
[
NW 3kB - kB ln Moreover, we decompose the vector vK(t) into their components within the “internal” coordinate system of the water molecule. This “internal” coordinate frame (we assume it to be righthanded) originates at the water particle center of mass, and the coordinate axes are the same as the main axes of the inertia tensor; the axes are ordered such that Ix < Iy < Iz. Then, the “directional”, normalized autocorrelation functions, CKi(t), have been calculated for each of ith component (i ) x, y, z) of the vK(t) vector. The kinetics of a single hydrogen bond is described by the time correlation function
SA(t) ) 〈h(t0) h(t0+t)〉
(4)
where the binary function h(t) ) 1 if the investigated hydrogen bond between two water molecules exists continuously (without intermediate breaking) from t0 up to t, and h(t) ) 0 otherwise. In other words, SA(t) yields the probability that a single hydrogen bond formed at time t0 remains bonded all the time up to t. For more details about these calculations, see our previous paper.23 Error Estimation in Diffusion Coefficients and the Hydrogen Bond Lifetimes. The estimation of obtained results uncertainty is a separate problem, and it is a hard task. We solve this problem by repeating, for all systems, our calculations several (3-5) times, and on this base the error limits for all the investigated quantities have been estimated. Two-Particle Contribution to Entropy. Using the trajectory obtained from the molecular dynamics calculations, the position and orientation of each water molecule were determined; a set of these quantities reflects the internal structure of water in the vicinity of the peptide. To obtain a precise description, we need some measure of that structural ordering, and the absolute entropy of water can serve as such a measure. However, calculations of this quantity are a very hard task. We resort to the Green concept, which allows us to decompose the total system entropy into the n-particle contributions. The entropy of an N-particle system can be calculated as a sum of the series
S ) sid + s(2) + s(3) + ... + s(N) ) sid + sexc
(5)
where sid symbolizes the entropy of an ideal gas and s(n) describes the contributions to the entropy originating from an n-particle correlation (n ) 2, 3, ..., N). This expression was first derived by Green;27 later, Nettleton and Green,28 and much later also Raveche´,29 obtained a similar expansion for the entropy of open systems, and this idea has been followed by many authors.30–36 This method allows us to calculate 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 series (5) are omitted in practicesthis is the so-called two-particle approximation. Within this approximation, the expression for the entropy reduces to the form
S = sid + s(2)
[
sid ) NPsPid + NWsid W ) NP 3kB - kB ln
]}
FPσP 3 λ λ Ω TP RP
]}
FWσW λ 3λ Ω TW RW
+
(7)
where indexes P and W denote protein and water, respectively, N represents the total number of molecules, F is the number density (F ) N/V, V is the volume of system), σ is the symmetry number (for water σW ) 2, for peptide, in general, σP ) 1). The other symbols are
λT )
h
λR )
√2πmkBT
(√ ) h
2πkBT
3
·
1
√IxIyIz Ω ) 8π2
(8)
where I symbolizes a molecule’s main moments of inertia, m is its mass, and h and kB represent Planck and Boltzmann constants, respectively. The two-particle contribution to entropy for such two-component system has the form34,36
s(2) ) -
FP2
∫
(2) kB {gPP (rP1,rP2,ωP1,ωP2) 2!Ω2 (2) (2) ln[gPP (rP1,rP2,ωP1,ωP2)] - gPP (rP1,rP2,ωP1,ωP2) + 1} F PF W (2) kB {gPW (rP,rW,ωP,ωW) drP1 drP2 dωP1 dωP2 2!Ω2 (2) (2) ln[gPW (rP,rW,ωP,ωW)] - gPW (rP,rW,ωP,ωW) + 1}
∫
drP drW dωP dωW -
FW2
∫
kB {g(2) WW(rW1,rW2,ωW1,ωW1) 2!Ω2 (2) ln[g(2) WW(rW1,rW2,ωW1,ωW2)] - gWW(rW1,rW2,ωW1,ωW2) + 1} (2) (2) + sPW + s(2) drW1 drW2 dωW1 dωW2 ) sPP WW
(9)
where g(2) symbolizes the two-particle correlation function, depending on the space positions r and the orientations ω of both regarded molecules. The integrals are calculated over accessible positions and orientations of all molecular pairs. Now we wish to exploit the above relation for estimation of the water entropy around the various conformations of the polypeptide chain. Using this relation requires, however, some care. (2) (2) (2) , sPW , and sWW , will be Therefore, each of the terms, sPP considered by us separately. Subsections a and b presented below describe our argument. (a) At the beginning, we note that the first term in relation 9, (2) , describes the peptide-peptide interactions, and therefore sPP it is out of our interest. The second term in this expression, (2) , describes the protein-water interactions. It depends on sPW twelve variables in summary. If the investigated system, however, is homogeneous and isotropic (translation and rotation invariant), the number of variables reduces to six, and this term is equal to (2) sPW ) -NPkB
FW 2!Ω
(2) (2) (2) (r,ω) ln gPW (r,ω) - gPW (r,ω) + ∫ [gPW
1] dr dω
(10)
(6)
For the peptide-water binary mixture term sid can be expressed by the relation
where r and ω ) (R, β, γ) denote the position and the three Euler angles, which determine position and orientation of a water molecule relative to the peptide one. Moreover, we assume, for
Structural Properties of Simple Polypeptides
J. Phys. Chem. B, Vol. 114, No. 13, 2010 4539
simplicity, that the investigated fragment of the secondary structure formed by the polypeptide chain has a cylindrical symmetry. Under this assumption, we need only four variables: the distance r of the water molecule (as its center of mass) from the peptide axis, and the three Euler angles, ω ) (R, β, γ), describing its orientation relative to this axis. Definitions of these variables are the same as in our previous paper.23 For the description of such ordering, we determine the function (2) (2) (r,R,β,γ) ) gPW (r,ω), defined in the cylindrical coordinates gPW as follows (2) gPW (r,ω) )
Ω dN(r,ω) Ω dN(r,ω) ) FW dV dω FW 2πrLdr dω
(11)
where L symbolizes the cylinder length, dN(r,ω) denotes the number of water particles that are present within the distance range (r,r+dr) and that are orientated within the range (ω,ω+dω), and
dω ) sin R dR dβ dγ 0eR Rc, and for a completely (2) (r,ω) ) 1, and random distribution we have (by definition) gPW (2) ) 0. On the other hand, an therefore we also read sPW,orient increasing ordering of water molecules produces smaller and (2) , and for an ideal, crystalsmaller (negative) values of sPW,orient like arrangement this value goes (as it may easily be shown) to minus infinity. Moreover, by integration of Iorient(r) within the (2) (r) function: (0, r) interval we obtain the sPW,orient
(2) sPW,orient (r) ) -NPkB
FW 2!
∫0r Iorient(r) dV ) r -NPkBFWLπ ∫0 Iorient(r)r dr
(19)
The function Iorient(r) contains information about orientational (2) (r) effects present at distance r, while the function sPW,orient measures a local orientational ordering of water in the surrounding space at the distance up to r. This allows us to estimate, in a rational manner, both the scope and the magnitude of the water structural ordering around the peptide core. Note that all (2) these remarks may also be adapted, of course, to the sPW,orient contribution, which describes a translational ordering of water (2) around the peptide, and to the sPW quantity. (b) A similar role, as a measure of structural ordering, is played by the last term in eq 9. It depends on the water-water interactions, and therefore its value serves as the measure of the water structural ordering in the vicinity of the peptide surface. So, it is highly desired that its estimation be done. Calculation of this term is, however, very difficult, because in the vicinity of the peptide the two-particle correlation function, (2) , depends on all twelve variables. For a given pair of water gWW molecules, these variables describe the position and the orientation of the each one relative to the peptide core. So, it seems that (accurate) determination of the third term in integral 9 is, up to today, quite impossible in practice, and several approximations are required. Thus, our argument is as follows.
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Czapiewski and Zielkiewicz
At first, we take into account the water molecules belonging to both the first and the second solvation layer around investigated part of the polypeptide core; these are the central ones. This space region represents a drilled cylinder of outer radius Rsol around the investigated peptide core. Now, we assume this region to be homogeneous, and under this assumption the (2) transforms to contribution gWW
s(2) WW
FW ) -NWkB 2Ω2
orientational ones.39 This factoring was made in a way quite (2) . We read similar to the one described above for factoring gWW
g(2) WW(r,ω) )
∫ {g(2)WW(r,ω1,ω2) ln[g(2)WW(r,ω1,ω2)] -
2 g(2) WW(r,ω1,ω2) + 1}r dr dω1 dω2
)
(
)
VΩ2 dN(r,ω1,ω2) V dN(r) ) · N r2 dr dω dω N Ω r2 dr 1 2 1 Ω1 dN(r,ω1) Ω2 dN(r,ω1,ω2) · dN(r) dω1 dN(r,ω1) dω2
(
)(
(2) (2) g(2) r (r) · gconf(r,ω1) · gorient(r,ω1,ω2)
)
(23)
(20) where we define
where r and ω1 ) (φ, θ) represent the distance and the two angles (in a spherical coordinate system) describing the position of a surrounding water molecule relative to the central one, while ω2 ) (R, β, γ) symbolizes the three Euler angles determining orientation of a surrounding water particle relative to the central one. Thus, a set of six variables, (r, ω1, ω2), describes relative position and orientations of two water molecules. We have, of course
dω1 ) sin θ dφ dθ Ω1 )
∫ dω1 ) 4π
dω2 ) sin R dR dβ dγ Ω2 )
∫ dω2 ) 8π2
where 0
