Chemistry Central Journal Open Access
Research article
Average and extreme multi-atom Van der Waals interactions: Strong coupling of multi-atom Van der Waals interactions with covalent bonding Alexei V Finkelstein Address: Institute of Protein Research, Russian Academy of Sciences,142290, Pushchino, Moscow Region, Russia Email: Alexei V Finkelstein -
[email protected]
Published: 30 July 2007 Chemistry Central Journal 2007, 1:21
doi:10.1186/1752-153X-1-21
Received: 30 March 2007 Accepted: 30 July 2007
This article is available from: http://journal.chemistrycentral.com/content/1/1/21 © 2007 Finkelstein et al This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract Background: The prediction of ligand binding or protein structure requires very accurate force field potentials – even small errors in force field potentials can make a 'wrong' structure (from the billions possible) more stable than the single, 'correct' one. However, despite huge efforts to optimize them, currently-used all-atom force fields are still not able, in a vast majority of cases, even to keep a protein molecule in its native conformation in the course of molecular dynamics simulations or to bring an approximate, homology-based model of protein structure closer to its native conformation. Results: A strict analysis shows that a specific coupling of multi-atom Van der Waals interactions with covalent bonding can, in extreme cases, increase (or decrease) the interaction energy by about 20–40% at certain angles between the direction of interaction and the covalent bond. It is also shown that on average multi-body effects decrease the total Van der Waals energy in proportion to the square root of the electronic component of dielectric permittivity corresponding to dipoledipole interactions at small distances, where Van der Waals interactions take place. Conclusion: The study shows that currently-ignored multi-atom Van der Waals interactions can, in certain instances, lead to significant energy effects, comparable to those caused by the replacement of atoms (for instance, C by N) in conventional pairwise Van der Waals interactions.
Background Van der Waals (VdW) forces, which are very important for the structure and interactions of biological molecules, are usually treated as a simple sum of pairwise inter-atomic interactions even in dense systems like proteins [1-5]. However, multi-atom VdW interactions are usually ignored. This seems to follow the Axilrod-Teller theory [6] which predicts a drastic (stronger than for pairwise interactions) decrease of three-atom interactions with distance; and indeed, detailed computations of single-atom liquids [7] and solids [8,9] show that MB (multi-body) effects
amount to only ~5% of the total energy. However, this work shows that multi-atom VdW interactions can become quite large in the presence of covalent bonds. This finding, which equally concerns atomic interactions in biological molecules and solvents, implies a necessity to revise the all-atom force fields currently used.
Results and Discussion Theory Following earlier studies [6,7,10-12], each atom i(i = 1, 2, ..., n) is considered as a three-dimensional (3D) harmonic Page 1 of 9 (page number not for citation purposes)
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quantum oscillator, where an electron with mass mi oscillates harmonically with a frequency ωi, if unaffected by other atoms. Atom i is fixed in the 3D point ri; it has an instantaneous dipole moment exi, where e is the elementary charge and xi the 3D vector of instantaneous displacement (whose equilibrium value is zero). Thus, a classic Hamiltonian for the system of coupled oscillators is 2 θij m ⎛ dx ⎞ m ω2 xi x j ; H = ∑ i ⎜ i ⎟ + ∑ i i x 2i + ∑ ∑ e 2 2 | rij |3 i i j >i i 2 ⎝ dt ⎠
(1) here θij/|rij|3 = (δ - 3nij 丢 nij)/|rij|3 is the usual dipoledipole interaction tensor (where δ is the 3D unit matrix, nij 丢 nij the tensor product of vectors nij = rij/|rij|, and rij = ri - rj; i ≠ j). As is usually done (see Refs. [6,10-12]), relatively week quadruple-dipole and so on interactions of oscillators are ignored in addition to possible inharmoniousness of the oscillators. Energy of multi-body VdW interactions
Using the substitution y i = x i mi [12], a conventional quantum mechanical Hamiltonian operator for coupled oscillators is obtained from [1]: 2
= Hˆ = − 2
∂2
∑ ∂y 2 + ∑ i
i
i
θij ω i2 2 e2 yi + ∑ ∑ yy ; 3 i j 2 i j >i mi m j | rij |
(2) here = is the reduced Planck constant. By the introduction of a 3n-dimensional vector Y = [y1, ..., yn] composed of n 3D vectors yi, and a 3n × 3n matrix B = ||bij|| composed
of
n
×
bij [bij = ω i2δ if i = j, bij = (e 2
n
3D
energy is W =
3n
=
∑2' s =1
s
=
= ⎡ 1/ 2 ⎤ in the presence of Sp B ⎦ 2 ⎣
dipole-dipole interactions, while the energy of the same but non-interacting oscillators is W0 =
= n ∑ 3ωi . Thus, 2 i =1
the energy of VdW dispersion forces is
⎞ (4) ⎟. ⎟ ⎠ In general, one can compute all 3n eigenvalues (Ωi)2 of 3n × 3n matrix B(and thus eigenvalues Ωi of B1/2 and their sum Sp [B1/2]) in a time proportional to (3n)3. Computationally, this solves a problem of exact calculation of VdW dispersion forces for any system of polarizable dipoles. ΔW = W − W0 =
= ⎛ ⎡ 1/ 2 ⎤ n − 3ω ⎜ Sp B ⎦ ∑ i 2 ⎜⎝ ⎣ i =1
However, to get a physical understanding of the main terms contributing to these forces, one has to consider the main terms of Sp[B1/2]. Matrix B can be presented as ⎡ ω12δ 0 ⎢ ⎢ 0 ω 22δ B = B0 + ΔB = ⎢ ... ⎢ ... ⎢ 0 ⎣ 0
0 ⎤ ⎡ 0 ⎥ ⎢ ... 0 ⎥ ⎢ b21 ⎥+ ... ... ⎥ ⎢ ... ⎥ ⎢ ... ω n2δ ⎦ ⎣ bn1 ...
b12 0 ... bn2
... b1n ⎤ ... b2n ⎥⎥ , ... ... ⎥ ⎥ ... 0 ⎦
(5)
blocks
mi m j )(θij / | rij |3 ) if i ≠ j]
, we obtain a simple Hamiltonian
1 =2 d2 Hˆ = − + YBY 2 dY 2 2
determined matrix B1/2 (while in the case of too large dipole-dipole interactions the system becomes unstable, and at least one Ω2 becomes negative). The frequencies Ω > 0 of the stable system of oscillators determine the ground state energy of this system [12]: the system's
where matrix B0 corresponds to uncoupled oscillators and ΔB to weak coupling of the oscillators. Now let us consider an auxiliary matrix Bλ = B0 + λΔB (where λ is a small 1/ 2
(3)
for oscillation in a 3n dimensional potential well determined by the 3n × 3n matrix B. The 3n eigenvalues (Ω1)2,
multiplier), and present its square root, Bλ
, as a series
B1λ/ 2 = B10/ 2 + λ Z1 + λ 2 Z 2 + ... , where B10/ 2 is a diagonal 1/ 2
matrix with 3D blocks ( B0
1/ 2
)ij = ωiδ if i = j, and ( B0
..., (Ω3n)2 of matrix B are squared frequencies of 3n inde-
= 0 if i≠j, and matrices Z1, Z2,... have to be calculated.
pendent one-dimensional oscillations along its eigenvectors. If dipole-dipole interactions are absent (i.e., all θij ≡
Since
0, where 0 is the zero 3D matrix), Ω1 = Ω2 = Ω3 = ω1, ...,
Ω3n-2 = Ω3n-1 = Ω3n = Ωn. If dipole-dipole interactions are
)ij
B1λ/ 2B1λ/ 2 = B0 + λ[B10/ 2 Z1 + Z1B10/ 2 ] + λ 2[B10/ 2 Z 2 + Z1Z1 + Z 2B10/ 2 ] + ... 1/ 2 1/ 2
and Bλ Bλ
= B0 + λΔB ,
small, Ω1 > 0, ..., Ω3n > 0 are eigenvalues of the positively
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is a sum of conventional London's energies ΔWik [10-13] of pairwise interactions: since Sp[θik θki] = 6,
B10/ 2 Z1 + Z1B10/ 2 = ΔB B10/ 2 Z 2 + Z 2B10/ 2 = − Z1Z1
ΔWik = −
" B10/ 2 Z m + Z mB10/ 2 = −(Z1Z m−1 + ... + Z m−1Z1 ); and so on. (6) This system can be solved recursively: equation MZ + ZM = Y (where matrices consist of equal-size blocks (M)ij,
3=(ω i + ω k ) 2 ⋅ ( γ ik ) , 2
(10)
where
γ ik =
(Z)ij, (Y)ij, i, j = 1, ..., n) unambiguously determines Z for
ω iω k ⎛ α iα k ⋅⎜ ω i + ω k ⎜⎝ | rik |3
⎞ ⎟ ⎟ ⎠
(11)
any Y when M = ||ωiδijδ|| is a positively determined diag-
is a convenient dimensionless parameter for the interac-
onal
tion of oscillators i and k, and αi = e2/(mi ω i2 ) is the elec-
matrix:
(MZ + ZM)ij = ∑ ⎡⎣ ω iδ ip δ ⋅ (Z)pj + (Z)ip ⋅ ω pδ pj δ ⎤⎦ = [ω i + ω j ](Z)ij
tronic polarizability of atom i.
, and from [ωi + ωj](Z)ij = (Y)ij = (Y)ij we have (Z)ij = (Y)ij/
The term
p
(ωi + ωj). 1/(ωi
Denoting
ωj)
+
bij / (ωi + ω j ) = { e2
as
μij
and
} ( θij / | rij |3 )
[ mi m j (ω i + ω j )] ⋅
as hij, we have
(
⎧⎪ = ω i + ω k + ω p ⎫⎪ = Sp [ Z 3 ] = ∑ ∑ ∑ ⎨ ⋅ ⋅ Sp[hik hkp h pi ] ⎬ 2 ω iω kω p ⎪⎭ i
