Cavity-ligand binding in a simple two-dimensional water model
By means of Monte Carlo computer simulations in the isothermal-isobaric ensemble, we investigated the interaction of a hydrophobic ligand with the hydrophobic surfaces of various curvatures (planar, convex and concave). A simple two-dimensional model of water, hydrophobic ligand and surface was used...
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irk-123456789-1557762019-06-18T01:29:23Z Cavity-ligand binding in a simple two-dimensional water model Mazovec, G. Lukšič, M. Hribar-Lee, B. By means of Monte Carlo computer simulations in the isothermal-isobaric ensemble, we investigated the interaction of a hydrophobic ligand with the hydrophobic surfaces of various curvatures (planar, convex and concave). A simple two-dimensional model of water, hydrophobic ligand and surface was used. Hydration/dehidration phenomena concerning water molecules confined close to the molecular surface were investigated. A notable dewetting of the hydrophobic surfaces was observed together with the reorientation of the water molecules close to the surface. The hydrogen bonding network was formed to accommodate cavities next to the surfaces as well as beyond the first hydration shell. The effects were most strongly pronounced in the case of concave surfaces having large curvature. This simplified model can be further used to evaluate the thermodynamic fingerprint of the docking of hydrophobic ligands. Використовуючи комп’ютерне моделювання в iзотермiчно-iзобаричному ансамблi, дослiджено взаємодiю гiдрофобного лiганда з гiдрофобними поверхнями рiзної кривизни (плоскi, випуклi та вгнутi). Використано просту двовимiрну модель води, гiдрофобний лiганд та гiдрофобну поверхню. Дослiджено явища гiдратацiї/дегiдратацiї молекул води близьких до молекулярної поверхнi. Спостерiгалось суттєве незмочування гiдрофобних поверхонь поряд з переорiєнтацiєю молекул води поблизу поверхнi. Спостережено формування мережi водневих зв’язкiв так що порожнини розташовуються бiля поверхнi, а також поза межами першої гiдратної оболонки. Згаданi ефекти найбiльш вираженi у випадку ввiгнутих поверхонь з великою кривизною. Цю спрощену модель можна використовувати щоб оцiнити термодинамiчнi аспекти докування гiдрофобних лiгандiв. 2016 Article Cavity-ligand binding in a simple two-dimensional water model / G. Mazovec, M. Lukšič, B. Hribar-Lee // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13004: 1–6 . — Бібліогр.: 20 назв. — англ. 1607-324X DOI:10.5488/CMP.19.13004 arXiv:1603.02163 PACS: 02.70.Uu, 05.10.Ln, 61.20.Gy, 61.20.Ja, 61.30.Hn http://dspace.nbuv.gov.ua/handle/123456789/155776 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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By means of Monte Carlo computer simulations in the isothermal-isobaric ensemble, we investigated the interaction of a hydrophobic ligand with the hydrophobic surfaces of various curvatures (planar, convex and concave). A simple two-dimensional model of water, hydrophobic ligand and surface was used. Hydration/dehidration phenomena concerning water molecules confined close to the molecular surface were investigated. A notable dewetting of the hydrophobic surfaces was observed together with the reorientation of the water molecules close to the surface. The hydrogen bonding network was formed to accommodate cavities next to the surfaces as well as beyond the first hydration shell. The effects were most strongly pronounced in the case of concave surfaces having large curvature. This simplified model can be further used to evaluate the thermodynamic fingerprint of the docking of hydrophobic ligands. |
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Mazovec, G. Lukšič, M. Hribar-Lee, B. |
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Mazovec, G. Lukšič, M. Hribar-Lee, B. Cavity-ligand binding in a simple two-dimensional water model Condensed Matter Physics |
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Mazovec, G. Lukšič, M. Hribar-Lee, B. |
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Mazovec, G. |
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Cavity-ligand binding in a simple two-dimensional water model |
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Cavity-ligand binding in a simple two-dimensional water model |
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Cavity-ligand binding in a simple two-dimensional water model |
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Cavity-ligand binding in a simple two-dimensional water model |
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Cavity-ligand binding in a simple two-dimensional water model |
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cavity-ligand binding in a simple two-dimensional water model |
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Інститут фізики конденсованих систем НАН України |
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Cavity-ligand binding in a simple two-dimensional water model / G. Mazovec, M. Lukšič, B. Hribar-Lee // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13004: 1–6
. — Бібліогр.: 20 назв. — англ. |
series |
Condensed Matter Physics |
work_keys_str_mv |
AT mazovecg cavityligandbindinginasimpletwodimensionalwatermodel AT luksicm cavityligandbindinginasimpletwodimensionalwatermodel AT hribarleeb cavityligandbindinginasimpletwodimensionalwatermodel |
first_indexed |
2025-07-14T08:00:52Z |
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2025-07-14T08:00:52Z |
_version_ |
1837608524719325184 |
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Condensed Matter Physics, 2016, Vol. 19, No 1, 13004: 1–6
DOI: 10.5488/CMP.19.13004
http://www.icmp.lviv.ua/journal
Cavity-ligand binding in a simple two-dimensional
water model∗
G. Mazovec, M. Lukšič, B. Hribar-Lee
University of Ljubljana, Faculty of Chemistry and Chemical Technology,
Večna pot 113, SI-1000 Ljubljana, Slovenia
Received November 14, 2015, in final form December 22, 2015
By means of Monte Carlo computer simulations in the isothermal-isobaric ensemble, we investigated the in-
teraction of a hydrophobic ligand with the hydrophobic surfaces of various curvatures (planar, convex and
concave). A simple two-dimensional model of water, hydrophobic ligand and surface was used. Hydration/de-
hidration phenomena concerning water molecules confined close to the molecular surface were investigated.
A notable dewetting of the hydrophobic surfaces was observed together with the reorientation of the water
molecules close to the surface. The hydrogen bonding network was formed to accommodate cavities next to
the surfaces as well as beyond the first hydration shell. The effects were most strongly pronounced in the case
of concave surfaces having large curvature. This simplified model can be further used to evaluate the thermo-
dynamic fingerprint of the docking of hydrophobic ligands.
Key words: cavity-ligand binding, water confinement, surface hydration, potential of mean force, Monte Carlo
computer simulation, two-dimensional water model
PACS: 02.70.Uu, 05.10.Ln, 61.20.Gy, 61.20.Ja, 61.30.Hn
1. Introduction
Due to the important influence of the confinement on the structural, thermodynamical, and phase
equilibrium properties of simple fluids, these systems have been extensively studied recently [1–3]. The
topic is also of great interest in the studies of biological systems in which many unanswered questions
are concerned with hydration and dehydration of molecules of arbitrary shapes where water as a solvent
finds itself confined close to the molecular surface.
One of the important issues that have begun to emerge is the effect of the shape, curvature, and
roughness of a surface on its interaction with a ligand [4–10]. In order to understand the phenomena
such as binding of the ligand to the receptor, or drugs to proteins, it is crucial to know the potential
of the mean force between two biomolecules in question. Virtually all the binding sites in biology have
a concave shape which imposes very particular geometrical constraints to the solvated water [11] and,
therefore, the findings referring to the potential of the mean force between two spherical surfaces cannot
be generalized to these problems.
To summarize the recent molecular dynamics studies on model systems of purely hydrophobic cav-
ities [10, 12–14], it has been shown that water appears to be an active component in cavity-ligand asso-
ciation. An important impact of changes in water structure during the binding process has been noticed
[10, 14]. While the cavity hydration can be altered by changing its radius, in weakly hydrated cavity re-
gions the reorganization of water molecules and suppression of the solvent fluctuations leads to enthalpy
driven association [15]. A different interpretation of the computer simulation results have been given by
Graziano [17]. His analysis shows that the Gibbs free energy gain upon association of a ligand in a concave
hydrophobic cavity is mainly due to the decrease in the solvent-excluded volume, that translates in a gain
of configurational-translational entropy of water molecules. This entropic driving force is masked by a
∗
Dedicated to the 65th birthday of Prof. Dr. Stefan Sokołowski.
© G. Mazovec, M. Lukšič, B. Hribar-Lee, 2016 13004-1
http://dx.doi.org/10.5488/CMP.19.13004
http://www.icmp.lviv.ua/journal
G. Mazovec, M. Lukšič, B. Hribar-Lee
large enthalpy gain associated with the reorganization of water-water hydrogen bonds upon association
of the two nonpolar objects [17].
In this work we have focused on the investigation of the potential of the mean force between a hy-
drophobic ligand and a planar, concave, and convex purely hydrophobic surface with different curva-
tures. Our main interest was in the interpretation of the potential of the mean force through the water
microstructural changes due to the confinement. To better visualize these effects we have chosen a simple
two-dimensional Mercedes-Benz (MB) water model [16].
The paper is organized as follows: After this short introduction, the model and method are described.
Next, the results are presented and discussed, and the conclusions are given in the end.
2. Model and method
The Mercedes-Benz model [16] was used to describe water molecules. In this model, the water mole-
cule is represented as a two-dimensional Lennard-Jones (LJ) disk with three equally separated hydrogen
bonding arms, and interacts with another water molecule through the potentialUww:
Uww(Xi ,X j ) =ULJ(ri j )+UHB(Xi ,X j ). (2.1)
Xi denotes the vector with coordinates and orientation of the i -th particle, and ri j is the distance between
the centres of molecules i and j . Lennard-Jones 12-6 potential is:
ULJ(ri j ) = 4εLJ
[(
σLJ
ri j
)12
−
(
σLJ
ri j
)6]
, (2.2)
where εLJ represents the depth of the potential well, and σLJ is the Lennard-Jones diameter.
The hydrogen bond strength is a Gaussian function of the intermolecular distance and the angle be-
tween two hydrogen bonding arms:
UHB(Xi ,X j ) = εHBG(ri j − rHB)
3∑
k,l=1
G(îk · ûi j −1)G(ĵl · ûi j +1), (2.3)
whereG(x) is an unnormalized Gaussian function:
G(x) = exp
(−x2/2σ2) . (2.4)
The unit vector îk represents the k-th arm of the i -th molecule (k = 1, 2, 3). Unit vector ûi j is the direction
vector of the line joining the centres of the i -th and j -th molecule. Parameters εHB = −1 and rHB = 1
determine the energy and the length of the optimal hydrogen bond. Values for the model parameters are
as follows: σLJ = 0.7rHB = 0.7, εLJ = 0.1 |εHB| = 0.1, and σ= 0.085.
The hydrophobic ligand was represented as a Lennard-Jones disk of the same size (σLJ = 0.7) as the
water molecule.
To model a hydrophobic surface of a given curvature, a molecular wall was formed from Lennard-
Jones discs (see figure 1). One surface was planar, while the others were bent. The interaction of the test
water or hydrophobic ligand with the surfaces having positive curvatures (concave), and with negative
curvatures (convex) was tested. The two radii used for concave surfaces were R∗ = 4.55 and 1.40, while
for the convex surfaces the radii were 2.80 and 5.95. The hydrophobic particle and the water molecules
interactedwith the particles forming the hydrophobic surface through the Lennard-Jones potential [equa-
tion (2.2)]. The LJ parameters (σLJ and εLJ) were the same for all particles in question.
Reduced units were used throughout the paper: r∗ = r /rHB, T ∗ = kBT /|εHB|, p∗ = pr 2
HB
/|εHB|.
Isobaric-isothermal Monte Carlo computer simulations [18] were performed for systems with 120 MB
water molecules and a single hydrophobic ligand (test particle). The hydrophobic surface was placed in
the centre of the simulation box. Reduced temperature, T ∗
, and pressure, p∗
, of the systems were 0.20
and 0.19, respectively. Initial configuration of water molecules was chosen randomly and then equili-
brated in 107
cycles long simulation. Statistics were then collected over 108
cycles long production run
13004-2
Cavity-ligand binding in a simple two-dimensional water model
A B C
Figure 1. Schematic representation of the surfaces used in simulations. All surfaces were made of three
layers of hydrophobic particles with Lennard-Jones (LJ) diameter σLJ = 0.7. Planar surface (A) was com-
posed of 3 ·7 = 21 LJ discs, while curved surfaces (B and C) were made of 7+9+11 = 27 LJ discs. Radii of
the inner and outer layer of the surface B were R∗ = 4.55 and 5.95, respectively, while for the surface C
the inner and the outer radii were 1.40 and 2.80, respectively. The surfaces were placed in the middle of
the simulation box.
which started from equilibrated configuration. In one cycle, either rotation or displacement (probabilities
for rotation and displacement being equal) of a water molecule was attempted. Maximal displacement
and rotation were adjusted throughout the simulation, so that approximately one half of Monte Carlo
moves were accepted. After every 120th cycle, an attempt to change the volume of the simulation box
was made, where the maximal change was adjusted in the same way as maximal displacement/rotation.
The pair correlation and angular distribution functions of water next to the surface were calculated
using the histogram method, while the potential of the mean force between the hydrophobic ligand and
the surface was calculated using the Widom’s insertion technique [19].
3. Results and discussion
Here, we present the results of the Monte Carlo simulations of interaction of the hydrophobic ligand
with the hydrophobic surface in water. All the results are given for T ∗ = 0.2 and p∗ = 0.19. Figure 2 (a)
shows the potentials of the mean force (PMFs) between the hydrophobic particle and the central particle
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
0 1 2 3 4 5
a
0 1 2 3 4 5
b
Figure 2. Potentials of the mean force (PMFs) between a hydrophobic ligand (test particle) and the central
particle of the hydrophobic surface. Centre-to-centre distance between test particles is given by r∗ (in the
direction of the x-axis; see the insets). Results for the convex shaped surfaces are shown in panel (a), and
for the concave case in panel (b). The reduced PMF,W ∗(r ) = W (r )/kBT , for the case of linear surface is
shown in both panels (circles). Two curvatures of the surface were tested— panel (a): 5.95 (squares) and
2.80 (triangles), panel (b): 4.55 (squares) and 1.40 (triangles). T∗ = 0.20 and p∗ = 0.19.
13004-3
G. Mazovec, M. Lukšič, B. Hribar-Lee
Figure 3. Representative simulation snapshots for various surfaces. White circles represent MB water
molecules with three hydrogen bonding arms (Mercedes-Benz logo). The grey circles are hydrophobic
particles forming the surfaces. T∗ = 0.20 and p∗ = 0.19.
of the surface for convex surfaces, while the PMFs for concave surfaces are shown in figure 2 (b). The
PMF with planar surface is shown for comparison in both panels. An important difference from the PMF
between two hydrophobic discs inwater (figure 1 of reference [6]) is observed. There is a larger difference
in the values of the peaks belonging to the contact minimum (CM) and solvent separated minimum (SSM)
compared to the homogeneous system (two hydrophobic discs); CM state is here much more stable than
the SSM state. This suggests that the hydrophobic particle would prefer to stay close to the surface. Not
much difference between the PMFs for different surface curvatures is observed in the convex cases, while
in the case of strongly concave surface, the contact state is additionally stabilized and the position of the
SSM is shifted further away from the surface.
To interpret these results in view of water microstructure, we analysed the hydration of the sur-
faces more in detail. Characteristic simulation snapshots are shown in figure 3. A dehydration of the
hydrophobic surface is noticed for all surfaces studied. The water molecules are pushed away and cavi-
ties are formed at the surface contact as well as beyond the first layer of water molecules. This is further
confirmed by the water-surface radial distribution functions shown in figure 4. All g (r ) show a layered
structure of water molecules away from the surface. There is no significant qualitative difference in the
g (r ) between the planar surface case and convex bent surfaces. The value of the contact peak slightly
decreases with an increasing curvature [see panel (a) of figure 4].
On the other hand, a qualitatively different behaviour is observed in the case of strongly concave
surface. Here, water is completely pushed away from the cavity, forming the first hydration layer ap-
proximately two hydrogen bonds away from the surface [figure 4 (b)]. This is in agreement with the pre-
0 1 2 3 4 5
b
0.0
0.4
0.8
1.2
1.6
0 1 2 3 4 5
a
Figure 4. Radial distribution functions between a test water molecule and the central particle of the hy-
drophobic surface. Distance r∗ has the same meaning as in figure 2. Surface parameters, notations and
conditions (T∗, p∗
) are the same as in figure 2.
13004-4
Cavity-ligand binding in a simple two-dimensional water model
1.2
1.4
1.6
1.8
2.0
2.2
2.4
0 10 20 30 40 50 60
/ °
a
0 10 20 30 40 50 60
/ °
b
Figure 5. Angular distribution of the first hydration shell water with respect to the hydrophobic surface.
Central particle of the surface closest to the test water was chosen to gather statistics. The angle φ is
defined 0° when one of the water’s hydrogen bonding arms points towards the centre of the surface’s
particle [see the sketch in panel (a)]. Results for the convex shaped surfaces are shown in panel (a), and
for the concave case in panel (b). For comparison, the unnormalized distribution, Nϕ, for the case of
linear surface is shown in both panels (circles). Surface parameters, notations and conditions (T∗, p∗
)
are the same as in figure 2.
vious studies by Baron et al. and Graziano [10, 12, 13, 15, 17]. Due to the incapability of forming hydrogen
bonds, a water molecule in such a state is entropically stabilized.
Further, the orientation of the surface water molecules belonging to the first hydration shell was
examined. Figure 5 (a) shows the results for the convex surfaces, and panel (b) for the concave surfaces.
The case of planar wall is shown in both panels for comparison. As already noticed by Southall et al.
[20], a model water molecule close to the planar surface can no longer form all three hydrogen bonds
with neighbouring water molecules. To compensate for this loss, it “wastes” one of the hydrogen bonds
by pointing it towards the surface. This is seen from the angular distribution where the most probable
orientation of the watermolecule at the planar surface is the one pointing the hydrogen bond towards the
surface (ϕ= 0°). The two hydrogen bonds pointing away from the surface participate in the formation of
cavities facilitating a favourable SSM in the PMFs.
As the surface adopts the convex shape, the angular preference of the water to the surface is much
less expressed. For the case in panel (a) of figure 5, we can conclude that angular preferences of water
have a negligible contribution to the shape of the PMFs.
The situation gets reversed in the case of strongly concave surface [triangles in figure 5 (b)]. The
preferential angle of water molecule in this case is the one where the hydrogen bonding arm points at an
angle 60° with respect to the centre-centre coordinate [see the insert in panel (a)], suggesting the strongly
obstructed hydrogen bond formation due to the lack of space in the cavity. This way a water molecule
was stabilized by an increase of the rotational entropy.
4. Conclusions
The results for the potential of the mean force between hydrophobic solute and hydrophobic surface
of various shapes in the model water solutions were presented. The results were analysed in view of hy-
dration water structure and orientation. All the hydrophobic surfaces studied showed dewetting which
was most pronounced for the bent concave surface. The angular orientation of the water molecules fa-
cilitated the formation of cavities which stabilized the non-covalent direct and water separated binding
of the hydrophobic molecule to the surface. In the future work, temperature dependence of the phenom-
ena will be investigated and the results will be used for the enthalpy-entropy decomposition of the free
energy of ligand-surface interaction.
13004-5
G. Mazovec, M. Lukšič, B. Hribar-Lee
Acknowledgements
This work was supported by the Slovenian Research Agency through grant P1-0103-0201 and Slovene-
Korean bilateral grant BI-KR/13-14-003. M.L. and B.H.-L. acknowledge the support of the NIH Grant
2R01GM063592-14.
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Зв’язки порожнина-лiганд у простiй двовимiрнiй
моделi води
Г.Мазовець,М. Лукшич, Б. Грiбар-Лi
Унiверситет Любляни, факультет хiмiї i хiмiчної технологiї, Любляна, Словенiя
Використовуючи комп’ютерне моделювання в iзотермiчно-iзобаричному ансамблi, дослiджено взаємо-
дiю гiдрофобного лiганда з гiдрофобними поверхнями рiзної кривизни (плоскi, випуклi та вгнутi). Вико-
ристано просту двовимiрну модель води, гiдрофобний лiганд та гiдрофобну поверхню. Дослiджено явища
гiдратацiї/дегiдратацiї молекул води близьких до молекулярної поверхнi. Спостерiгалось суттєве незмочу-
вання гiдрофобних поверхонь поряд з переорiєнтацiєю молекул води поблизу поверхнi. Спостережено
формування мережi водневих зв’язкiв так що порожнини розташовуються бiля поверхнi, а також поза
межами першої гiдратної оболонки. Згаданi ефекти найбiльш вираженi у випадку ввiгнутих поверхонь з
великою кривизною. Цю спрощену модель можна використовувати щоб оцiнити термодинамiчнi аспе-
кти докування гiдрофобних лiгандiв.
Ключовi слова: зв’язки порожнина-лiганд, утримання води, поверхнева гiдратацiя, потенцiал середньої
сили, комп’ютерне моделювання Монте Карло, двовимiрна модель води
13004-6
http://dx.doi.org/10.1002/9781118949702.ch5
http://dx.doi.org/10.1063/1.1818677
http://dx.doi.org/10.1021/jp0736847
http://dx.doi.org/10.1021/la036036x
http://dx.doi.org/10.1021/jp994203p
http://dx.doi.org/10.1016/S0301-4622(02)00167-9
http://dx.doi.org/10.1021/j100009a052
http://dx.doi.org/10.1063/1.440540
http://dx.doi.org/10.1021/jp055043m
http://dx.doi.org/10.1021/ja1050082
http://dx.doi.org/10.1126/science.2011744
http://dx.doi.org/10.1021/ct300121r
http://dx.doi.org/10.1021/jp309373q
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http://dx.doi.org/10.1016/j.cplett.2012.03.020
http://dx.doi.org/10.1021/jp992860b
Introduction
Model and method
Results and discussion
Conclusions
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