Description

Computation of binding free energy with molecular dynamics and grand canonical Monte Carlo simulations

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

Computation of binding free energy with molecular dynamicsand grand canonical Monte Carlo simulations
Yuqing Deng
1
and Benoît Roux
1,2,a
1
Biosciences Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, Illinois 60439, USA
2
Department of Biochemistry and Molecular Biology, Gordon Center for Integrative Science,University of Chicago, 929 57th Street, Chicago, Illinois 60637, USA
Received 13 December 2007; accepted 17 January 2008; published online 18 March 2008;publisher error corrected 26 March 2008
The binding of a ligand to a receptor is often associated with the displacement of a number of boundwater molecules. When the binding site is exposed to the bulk region, this process may be sampledadequately by standard unbiased molecular dynamics trajectories. However, when the binding siteis deeply buried and the exchange of water molecules with the bulk region may be difﬁcult tosample, the convergence and accuracy in free energy perturbation
FEP
calculations can beseverely compromised. These problems are further compounded when a reduced system includingonly the region surrounding the binding site is simulated. To address these issues, we couplemolecular dynamics
MD
with grand canonical Monte Carlo
GCMC
simulations to allow thenumber of water to ﬂuctuate during an alchemical FEP calculation. The atoms in a spherical innerregion around the binding pocket are treated explicitly while the inﬂuence of the outer region isapproximated using the generalized solvent boundary potential
GSBP
. At each step duringthermodynamic integration, the number of water in the inner region is equilibrated with GCMC andenergy data generated with MD is collected. Free energy calculations on camphor binding to adeeply buried pocket in cytochrome P450cam, which causes about seven water molecules to beexpelled, are used to test the method. It concluded that solvation free energy calculations with theGCMC/MD method can greatly improve the accuracy of the computed binding free energycompared to simulations with ﬁxed number of water. ©
2008 American Institute of Physics
.
DOI: 10.1063/1.2842080
I. INTRODUCTION
A number of recent studies have shown very encourag-ing results in calculations of standard
absolute
binding freeenergies using atomistic molecular dynamics
MD
simula-tions
see Ref. 1 for a review
.
2–10
The progress is explained,in part, by several developments both formal and technical.Over the last years, many of the critical issues concerning thetreatment of the standard state and the usage of restraint po-tentials to improve convergence have been resolved,
2,7,8,11–14
and strategies for the treatment of multiple ligand orienta-tions and protein conformations have been elaborated.
6,8,9
Inorder to decrease the computational cost of free energy com-putations, methods have been designed to simulate reducedmodel systems in which the inﬂuence of the surrounding isincorporated implicitly via an effective “boundarypotential.”
15–20
This progress signals that reliable free energy computa-tions are increasingly becoming a reality. However, achiev-ing a proper and complete equilibration of the solvent mol-ecules during free energy computations can still represent apractical and important problem. While representative sol-vent conﬁgurations are generated spontaneously by unbiasedMD trajectories when a binding site is exposed to the bulk phase, difﬁculties in sampling solvent conﬁgurations can be-come particularly acute when a binding site is deeply buriedand inaccessible. In this case, the exchange of water mol-ecules with the bulk region may be very slow and the con-vergence and accuracy in free energy perturbation
FEP
cal-culations based on unbiased molecular dynamics trajectoriescan be severely compromised. Difﬁculties in properly sam-pling the hydration states of a binding site are further com-pounded when a reduced system is simulated with an effec-tive boundary potential.A typical example is provided by the active site of cyto-chrome P450, a protein that oxidizes endogenous and xeno-biotic substrates.
21
Previous simulation studies have shownthat some conformational changes, including side chain rota-meric states, are required to open up a channel allowing ex-change between the cavity and the bulk phase;
21,22
about sixwater molecules are expelled from the buried cavity uponsubstrate binding. Such a change in hydration state must becaptured in free energy calculations to yield accurate andmeaningful results.Advance simulation methodologies to en-hance sampling, such as replica exchange moleculardynamics
23
might be able to alleviate some of those prob-lems. For example, recent applications
24,25
of Hamiltonianreplica exchange
26
have shown the ability to accelerate pro-tein folding by introducing hydrophobic replicas. However,constructing a novel replica exchange algorithm designed toenhance the sampling of water occupancy inside conﬁnedbinding pockets is not straightforward.To address this problem, Helms and Wade designed analchemical transformation to include speciﬁc water mol-ecules in a calculation of the binding free energy of camphor
a
Electronic mail: roux@chicago.edu.
THE JOURNAL OF CHEMICAL PHYSICS
128
, 115103
2008
0021-9606/2008/128
11
/115103/8/$23.00 © 2008 American Institute of Physics
128
, 115103-1
to cytochrome P450.
27
Accordingly, the water moleculesknown to be displaced were
“
switched on
”
as the ligandcamphor was
“
switched off
”
during the alchemical transfor-mation. While this procedure takes into account the waterdisplacement in the binding process, it implicitly assumesthat one knows in advance the number of water moleculesinvolved in the process. Systematic methods to account forthe hydration level of internal cavities have been designed,though they require the speci
ﬁ
c evaluation of all probabilityof water occupancy.
3,12,27
–
31
Rigorously speaking, the num-ber of water molecules in a reduced system should be inopen thermodynamic equilibrium with the bulk phase. Thus,proper con
ﬁ
gurational averages are sampled only if the num-ber of water molecules is allowed to
ﬂ
uctuate according toan effective grand canonical ensemble.
32
In this context, freeenergy computations with reduced systems should optimallybe combined with a grand canonical Monte Carlo
32
–
36
GCMC
to yield proper statistical averages. As the hydra-tion state of a binding site is important,
31,37
–
39
a generallyapplicable simulation strategy able to naturally generate theproper hydration states would be highly desirable.In this paper, we develop and test a free energy simula-tion framework that combines GCMC with MD generatedwith a reduced system. The method used in the present studyis called the generalized solvent boundary potential
GSBP
.
20
GSBP is a generalization of the spherical solventboundary potential
SSBP
, which was designed to simulatea solute in bulk water.
18
The reduction in system size af-forded by GSBP permits inexpensive free energy computa-tions quickly and with reasonable accuracy.
6,7,40
The presentwork is an extension of the developments in Ref. 32. It is ourhope that the combination of GCMC and GSBP will providea robust and ef
ﬁ
cient framework for the computations of binding free energies.The rest of the paper is organized as follows. In thefollowing section, we present the fundamental theory for freeenergy calculation with GCMC/MD. We then proceed to testcalculations of solvation free energy and camphor/ cytochrome P450 binding free energy. Last, we conclude thepaper with a summary of the main results.
II. THEORYA. GCMC and free energy calculation
Let us consider a solute inside a large solvent reservoirof
N
solvent molecules. Without loss of generality, we as-sume that the center of mass of the solute is
ﬁ
xed at thesrcin and that the internal degrees of freedom of the soluteare represented by
X
. The degrees of freedom of a solventmolecule, assumed here to be rigid for the sake of simplicity,are represented by
x
, which includes the center-of-masstranslation and rigid body rotation. The total potential energyof the full system is
U
X
,
x
1
,...,
x
N
;
, where
is a ther-modynamic switching parameter such that the solute is non-interacting
decoupled
when
=0 and fully interacting when
=1. By de
ﬁ
nition, the solvation free energy
G
of thesolute is
e
−
G
=
d
X
d
x
1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
d
X
d
x
1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
=0
.
1
Let us deﬁne a spherical inner region, centered on the solutein the solvent. The space outside the inner region is referredto as the outer region. For any instantaneous conﬁguration of the system, it is possible to monitor the total number of sol-vent molecules inside the inner region. It is given by thediscrete function
n
r
1
,
r
2
,...,
r
N
, deﬁned as
n
r
1
,
r
2
, ... ,
r
N
=
i
=1
N
H
r
i
,
2
where
r
i
is the position of the center of the
i
th solvent mol-ecule and
H
is a step-function equal to 1 with the moleculeinside the inner region. The probability
P
n
of having ex-actly
n
solvent molecules inside the inner region is calculatedfrom the average
P
n
=
nn
=
d
X
d
x
1
¯
d
x
N
n
,
n
e
−
U
X
,
x
1
,...,
x
N
;
d
X
d
x
1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
,
3
where
nn
is a Kroenecker discrete delta function,
nn
=
1 if
n
=
n
r
1
,
r
2
, ... ,
r
N
0 otherwise.
4
By construction, the probabilities
P
n
are normalized, i.e.,
n
P
n
=1, via the completeness of the Kroenecker delta.Inserting the Kroenecker function in the expression for
G
and summing over all possible number of solventmolecules yields,
e
−
G
=
n
d
X
d
x
1
¯
d
x
N
n
,
n
e
−
U
X
,
x
1
,...,
x
N
;
d
X
d
x
1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
=0
.
5
It is useful to introduce the case
n
=0 and
=0 as a referencestate,
e
−
G
=
n
d
X
d
x
1
¯
d
x
N
n
,
n
e
−
U
X
,
x
1
,...,
x
N
;
d
X
d
x
1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
=0
d
X
d
x
1
¯
d
x
N
0,
n
e
−
U
X
,
x
1
,...,
x
N
;
=0
d
X
d
x
1
¯
d
x
N
0,
n
e
−
U
X
,
x
1
,...,
x
N
;
=0
=
P
0
= 0
n
d
X
d
x
1
¯
d
x
N
n
,
n
e
−
U
X
,
x
1
,...,
x
N
;
d
X
d
x
1
¯
d
x
N
0,
n
e
−
U
X
,
x
1
,...,
x
N
;
=0
,
6
where
P
0
=0
is the probability of ﬁnding the inner regionempty of any solvent molecules with a decoupled solute,
P
0
= 0
=
d
X
d
x
1
¯
d
x
N
0,
n
e
−
U
X
,
x
1
,...,
x
N
;
=0
d
X
d
x
1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
=0
.
7
Since
P
0
=0
is the probability of spontaneous occurrencefor a conﬁguration with zero solvent molecule in the innerregion, it is associated with the reversible work
G
hs
=
−
k
B
T
ln
P
0
=0
for inserting a hard-sphere correspond-ing to radius of the inner region into the solvent. To compute
115103-2 Y. Deng and B. Roux J. Chem. Phys.
128
, 115103
2008
the con
ﬁ
gurational integral with exactly
n
solvent moleculesin the inner region, one can arbitrarily pick the
ﬁ
rst
n
mol-ecules and enforce that the remaining solvent molecules arerestricted to the outer region. Since all the solvent moleculesare identical, there are
N
!
/
N
−
n
!
n
!
N
n
/
n
! equivalentchoices, which gives the expression,
e
−
G
=
P
0
= 0
n
N
!
n
!
N
−
n
!
d
X
in
d
x
1
¯
d
x
n
out
d
x
n
+1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
d
X
out
d
x
1
¯
d
x
n
out
d
x
n
+1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
=0
=
P
0
= 0
n
N
n
n
!
d
X
in
d
x
1
¯
d
x
n
out
d
x
n
+1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
d
X
out
d
x
1
¯
d
x
n
out
d
x
n
+1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
=0
=
P
0
= 0
n
N
n
n
!
d
X
in
d
x
1
¯
d
x
n
e
−
W
X
,
x
1
,...,
x
n
;
d
X
out
d
x
1
¯
d
x
n
e
−
W
X
,
x
1
,...,
x
n
;
=0
,
8
where
W
is the potential of mean force
PMF
for the solute and the
n
tagged solvent molecules, de
ﬁ
ned as
e
−
W
X
,
x
1
,...,
x
n
;
out
d
x
n
+1
¯
d
x
N
e
−
U
X
,
x
1
,...,
x
N
;
.
9
It is useful to express the total potential energy
U
in terms of contributions from the inner and outer regions,
U
X
,
x
1
, ... ,
x
N
;
=
U
ii
X
,
x
1
, ... ,
x
n
;
+
U
io
X
,
x
1
, ... ,
x
N
;
+
U
oo
x
n
+1
, ... ,
x
N
,
10
where
U
ii
denotes the potential energy of the molecules within the inner region,
U
io
denotes the interaction energy between themolecules in the inner region and the ones in the outer region, and
U
oo
denotes the potential energy in the outer region only.From this decomposition, the PMF may be expressed as
20
e
−
W
X
,
x
1
,...,
x
n
;
=
e
−
U
ii
X
,
x
1
,...,
x
n
;
out
d
x
n
+1
¯
d
x
N
e
−
U
io
X
,
x
1
,...,
x
N
;
+
U
oo
x
n
+1
,...,
x
N
out
d
x
n
+1
¯
d
x
N
e
−
U
oo
x
n
+1
,...,
x
N
=
e
−
U
ii
X
,
x
1
,...,
x
n
;
+
W
X
,
x
1
,...,
x
n
;
.
11
By de
ﬁ
nition,
W
is equivalent to the solvation free energyof the solute and
n
solvent molecules in a
ﬁ
xed con
ﬁ
gurationembedded in its associated spherical exclusion region. It isequivalent to the reaction
ﬁ
eld solvent potential that is gen-erated by GSBP.
20
Furthermore, one may note thatlim
x
1
,...,
x
n
→
W
X
,
x
1
, ... ,
x
n
;
= 0
=
n
s
+
u
int
X
,
12
where
s
is the excess solvation free energy of the solventmolecules in the bulk region
far away from the inner region
and
u
int
X
is the internal potential function of the solute
decoupled from the solvent
. By virtue of the translationaland rotational invariance of the bulk region and Eq.
12
, thecon
ﬁ
gurational integral in the denominator in Eq.
8
may berewritten as
d
X
out
d
x
1
¯
d
x
n
e
−
W
X
,
x
1
,...,
x
n
;
=0
=
V
n
e
−
n
s
8
2
n
d
X
e
−
u
int
X
,
13
where
V
is the volume of the bulk region and
8
2
is theintegral over the internal degrees of freedom of a solventmolecule,
N.B.,
out
d
x
i
=
V
8
2
. If the solvent moleculeswere
ﬂ
exible, the above expressions would be modi
ﬁ
ed toinclude con
ﬁ
gurational integral over internal degrees of freedom. It follows that the solvation free energy of the solute is
e
−
G
=
P
0
= 0
n
¯
n
n
!
d
X
in
d
x
1
¯
d
x
n
e
−
W
X
,
x
1
,...,
x
n
;
−
n
s
8
2
n
d
X
e
−
u
int
X
,
14
where
¯
=
N
/
V
is the average solvent density in the bulk.Equivalently, the solvation free energy of the solute may beexpressed as
e
−
G
=
P
0
= 0
n
¯
n
K
n
,
15
where
K
n
are effective equilibrium association constants,
K
n
=1
n
!
d
X
in
x
1
¯
d
x
n
e
−
W
X
,
x
1
,...,
x
n
;
−
n
s
8
2
n
d
X
e
−
u
int
X
,
16
note that
K
0
is equal to 1
. Equations
15
and
16
form thebasis of the quasichemical free energy formalism thatGrabowski
et al.
have used to discuss the solvation free en-ergy of ions.
41
One may also compute the solvation free energy using athermodynamic integration,
115103-3 Computation of binding free energy J. Chem. Phys.
128
, 115103
2008
G
=
−
k
B
T
01
d
ln
P
0
= 0
n
¯
n
n
!
d
X
in
d
x
1
¯
d
x
n
e
−
W
X
,
x
1
,...,
x
n
;
−
n
s
8
2
n
d
X
e
−
u
int
X
=
01
d
n
P
n
d
X
in
d
x
1
¯
d
x
n
W
/
e
−
W
X
,
x
1
,...,
x
n
;
d
X
in
d
x
1
¯
d
x
n
e
−
W
X
,
x
1
,...,
x
n
;
=
01
d
n
P
n
W
,
n
,
17
where the bracket
¯
,
n
implies a constrained average with
ﬁ
xed coupling constant
and
ﬁ
xed number of solvent mol-ecules in the inner region, weighted by the probabilities
P
n
of
n
solvent molecules occupying the inner region,
P
n
=
¯
n
K
n
m
¯
m
K
m
,
18
where Eqs.
4
,
11
, and
12
have been used.Occupancy of an inner region by
n
solvent moleculesdistributed according to the probability
P
n
arises naturallyin a GCMC simulation.
32
The reversible work expression canbe written in terms of a grand canonical average of
W
/
,
G
=
01
d
W
.
19
This formulation shows how the
ﬂ
uctuations in the numberof solvent molecules are naturally incorporated into the freeenergy calculation performed in a
ﬁ
nite inner region. Equa-tion
19
is the central result of this paper.A formulation of free energy calculations based on Eq.
19
is essentially equivalent to familiar thermodynamic in-tegration
TI
,
42
where the averaging of
W
/
has beengeneralized to include the average over con
ﬁ
guration andnumber of water molecules in the open ensemble. Interest-ingly, it is not possible to convert Eq.
19
into a standardFEP.
43
In the open system ensemble with
ﬂ
uctuating numberof water molecules, FEP between two values of the couplingparameters
1
and
2
requires a reequilibration of the num-ber of water molecules for each end-state
1
and
2
. Thishighlights important formal differences between the two
ﬁ
-nite representations of an in
ﬁ
nite thermodynamic systemimplemented respectively with SSBP
Ref. 18
and GSBP.
20
In GSBP, the radius of the inner region is
ﬁ
xed.
20
Accord-ingly, proper con
ﬁ
gurational averages can only be achievedif the number of water molecules is allowed to
ﬂ
uctuate ac-cording to the grand canonical ensemble.
32
In contrast, theradius of the inner region
ﬂ
uctuates dynamically in SSBPwhile the number of explicit water molecules within the in-ner region is constant.
18
For this reason, free energy calcula-tions can rigorously be performed according to both the TIand FEP protocoles with the effective PMF in SSBP.
18
Al-though one should rigorously use only TI with GSBP, inpractice, it is possible to use FEP if the perturbation proceedsby small steps for which
P
n
does not vary signi
ﬁ
cantly.
B. Free energy decomposition
To enhance the sampling ef
ﬁ
ciency, we separate thebinding free energy into the following components, de
ﬁ
nedas the progressive switching on of repulsive, dispersive, elec-trostatic
see Ref. 40 for details on the repulsive and disper-sive contributions
, and introduction/removal of restraintpotentials,
6,7,40
G
b
°
=
G
repsite
−
G
repbulk
+
G
dispsite
−
G
dispbulk
+
G
elecsite
−
G
elecbulk
+
G
rstr
°
,
20
where
G
repsite
and
G
repbulk
are contributions from repulsion inbinding site
restraints on
and bulk solvent, respectively,
G
dispsite
and
G
dispbulk
are contributions from dispersion in bind-ing site
restraints on
and bulk solvent, respectively,
G
elecsite
and
G
elecbulk
are contributions from electrostatics in bindingsite
restraints on
and bulk solvent, respectively, and
G
rstro
is the contribution from translational and rotational restraintpotential with standard volume unit in liters. Here, we do notuse restraint on the ligand conformation because the cam-phor does not have any rotable bonds. The solvation freeenergy is the sum of all bulk contributions,
G
solv
=
G
repbulk
+
G
dispbulk
+
G
elecbulk
.
21
III. COMPUTATIONAL DETAILS
The
CHARMM PARAM27
force
ﬁ
eld was used for the pro-tein and for the heme in its reduced state.
44
The solvent watermodel was TIP3P.
45
The benzene model,
46
after which the
CHARMM22
phenylalanine was parametrized,
44
was used inthe benzene solvation free energy calculations. The general-ized AMBER force
ﬁ
eld was used for camphor;
47
the ligandtopology and parameters were generated with
ANTECHAMBER1.27
48
and with AM1-BCC partial charges.
49,50
Crystal structures of the cytochrome P450/camphorcomplex in the Protein Data Bank
2CPP
were used to gen-erate the all-atom structures. Hydrogen atoms were addedusing the HBUILD
51
facility in
CHARMM
.
52
The missing sidechain coordinates of LYS205 were predicted from
SCWRL
.
53
The explicit simulation spheres were henceforth constructedfrom those all-atom structures. A 15
Å
radius sphere wasde
ﬁ
ned around the center of the binding pocket with 121generalized multiple basis functions for GSBP.
20
The systemwas hydrated with 20 cycles of MC and MD
10 000 MCmoves followed by 10 000 MD steps with 2 fs time step
.
32
The MC consists of rigid body translation, rotation, andGCMC moves with equal probability. The anchor atoms forthe restraint were selected at random on the protein and cam-phor ligand. The reference ligand center of mass and orien-tation angle values were calculated as the average of a 20 psunrestrained MD trajectory.All the free energy perturbation calculations were carried
115103-4 Y. Deng and B. Roux J. Chem. Phys.
128
, 115103
2008
out with the
CHARMM
program
52
version c34a2 modi
ﬁ
ed forthe present study. The simulations in the binding site consistof 18 windows of ligand repulsion,
ﬁ
ve windows of liganddispersion, ten windows of ligand electrostatics, and 15 win-dows of restraint removal.
6
Two trajectories were run foreach of the windows. Each of the trajectories consists of sixcycles of GCMC/MD run
8000 MC steps followed by 16 psMD
to equilibrate the system and 8 ps data collection. Afurther 16 ps MD and 8 ps data collection was done after theGCMC/MD cycles
112 ps total for each window
. For the
ﬁ
xed water number simulation, each trajectory was 20 ps of equilibration and 40 ps of data collection. No GCMC wasapplied in the removal of the restraint potentials on theligand.In the bulk GCMC/MD, FEP calculations were done in a10
Å
sphere with nine basis functions of generalized mul-tiples for GSBP. Ten cycles of GCMC/MD run
10 000 MCsteps followed by 20 ps MD
were run to equilibrate thesystem and 8 ps data collection. A further 20 ps MD and8 ps data collection was done after the GCMC/MD cycle
96 ps total for each trajectory
. The
ﬁ
xed water number cal-culation in the bulk
G
intbulk
calculation was done withSSBP.
18,40
A solute molecule and 400 water molecules wereintegrated with the 2 fs time step Langevin dynamics. Thelength of each simulation window was 40 ps equilibrationfollowed by 40 ps of sampling. The relevant coupling param-eters were the same for the bulk and site calculations. Allbonds involving hydrogen atoms were
ﬁ
xed with theSHAKE algorithm.
54
To ensure convergence, we performeda series of free energy calculations for each system, startingfrom the con
ﬁ
gurations of each window saved at the end of the last run.
IV. RESULTS AND DISCUSSIONSA. Solvation free energy
The excess chemical potential
hydration free energy
of water and benzene has been calculated
ﬁ
rst to test and vali-date the simulation strategy. The number of water is shownin Fig. 1 as the simulation progresses from a noninteractingwater to a fully interacting one. As expected for a watermolecule solute, the number of water solvent molecules
ﬂ
uc-tuates around the mean value, with no net change. The aver-ages are reported in Table I. It is observed that about twosolvent water molecules are driven out of the simulationsphere to accommodate the solute. Because of its size, ben-zene is expected to replace several water molecules when itis fully solvated. As the progression in Fig. 1 shows, thenumber of water molecules steadily decreases when the re-pulsive potential of benzene is gradually introduced
stages1
–
10
. The change is more obvious during the initial stagesof the solute insertion, when the purely repulsive cavity isswitched on. As the dispersive/attractive potential is added
stages 10
–
14
, the number of water molecules increasesslightly, consistent with a slight stabilization by the van derWaals attractive interaction of water around a nonpolarsolute.
34
Because the polarity of benzene is almost negli-gible, the addition of electrostatic potential
stages 14
–
24
does not cause any further change in the number of watermolecules.Table I reports the free energy decomposition for thehydration free energies of water and benzene. For water, thefree energy values from the decomposition match closelywith the earlier calculation computed from periodic bound-ary conditions
PBC
and SSBP.
40
The chemical potential
−
6.2 kcal
/
mol
used as input in the GCMC moves agreesquite well with the
−
6.09
kcal
/
mol chemical potentialcomputed from FEP. This is encouraging and re
ﬂ
ects on theself-consistency of the method. The statistical errors wereestimated from the last
ﬁ
ve runs of a series of 11 runs of simulations. For water, the precision of the free energy val-ues is high, in spite of
ﬂ
uctuations in the number of watermolecules in the GSBP sphere. One may note that in theGSBP sphere the average number of water molecules in thewater calculation is slightly higher than 140, the number of water in a 10
Å
sphere with bulk density of 0.0334
Å
−
3
. Thisoverestimated number might be due to
ﬁ
nite size effects; thesurface tension causing an increase in the local pressure anddensity inside the droplet. The agreement of benzene hydra-tion free energies with previous PBC and SSBPcalculations
40
is not perfect but is quite reasonable. Thenearly exact match of the current benzene solvation value tothe experiment is partly fortuitous and we would not expectthis small difference to hold for other molecules. Given thesensitivity of the repulsive free energy contribution to thedetails of the simulations,
40
such small differences are notsurprising. These simulations show that the currentGCMC/MD protocol can be used for calculating solvationfree energies with
ﬁ
nite simulation systems and GSBP.
FIG. 1. Shown in the plot is the number of water molecules during solvationcalculation. The upper panel is the solvation of water. The lower panel is thesolvation of benzene. The simulation is divided into 24 stages, starting fromnoninteracting in stage one and progressing to full-interacting solute
wateror benzene
in stage 24. In stages 1
–
10, the solute repulsion is turned ongradually. In stages 10
–
14, the solute attraction is added. At last, in stages14
–
24, the solute particle charges are turned on. The errors are standarddeviation of
ﬁ
ve complete runs of solvation free energy calculations.
115103-5 Computation of binding free energy J. Chem. Phys.
128
, 115103
2008

Search

Similar documents

Tags

Related Search

Binding Free Energy CalculationsMolecular Dynamics and Monte Carlo MethodsMolecular dynamics and dockingAb Initio Molecular Dynamics Calculations of Molecular DynamicsGibbs Free EnergyMolecular Genetics and biotechnology of fisheThe Covenants of the Prophet Muhammad with thAb initio Molecular DynamicsHeresy of the Free Spirit

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks