Biomolecular solvation: from molecular to continuum modelsBiomolecular solvation: from molecular to...

Biomolecular solvation: from molecular to continuum models

Jason Wagoner, Feng Dong, Nathan Baker

Dept. of Biochemistry and Molecular BiophysicsMolecular Biophysics Graduate Program

Center for Computational BiologyWashington University in St. Louis

http://agave.wustl.edu



Biomolecular solvation: from molecular to continuum models

Jason Wagoner, Feng Dong, Nathan Baker

Dept. of Biochemistry and Molecular BiophysicsMolecular Biophysics Graduate Program

Center for Computational BiologyWashington University in St. Louis


Please tell your undergrads!

http://bp.wustl.edu/



http://bp.wustl.edu

http://bp.wustl.edu

OverviewHow does solvent behave near biomolecules of varying charge density?

How well do Poisson models describe solute-solvent interactions involving charge?

Depends on the desired accuracy and solute typeAre there implicit nonpolar solvation models that can describe the remaining solute-solvent interactions?

Yes, a WCA-like method gives good force accuracy for a variety of solutes

Much more to do!Relevance:

Understanding biomolecular solvation and electrostaticsDeveloping better methods for simulation and modeling

2

Biomolecular solvation and electrostatics

Biomolecular electrostatics: proteins

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Biomolecular electrostatics: other molecules

dsDNAApprox. linear formClose phosphate spacing2 e- per 3.4 Å

RNAStructural diversityDense phosphate spacing

SugarsLipids

5

Dermatan sulfate picture from Alberts et al

How solvent interacts with biomolecules

Water propertiesDipolar solvent (1.8 D)Hydrogen bond donor and acceptorPolarizable

Functional behavior:Bulk polarizationSite binding or specific solvationPreferential hydrationAcid/base chemistry...

Spine of hydration in DNA minor groove

(Kollman, et al.)

Carbonic anhydrase reaction mechanism

(Stryer, et al.)

How ions interact with biomoleculesNon-specific screening effects

Depends only on ionic strength (not species)Results of damped electrostatic potentialDescribed by Debye-Hückel or Poisson-Boltzmann theories for low ionic strengths

Functional behavior:Binding affinitiesRatesStructure...

7

Electrostatic potential of AChE at 0 mM and 150 mM NaCl. Rate and binding affinity decrease with [NaCl] has been

attributed to screening effects… although species-dependent influences have been observed. Radic Z, et al.

1997. J Biol Chem 272 (37): 23265-77.

How ions interact with biomoleculesSite-specific binding

Ion-specific

Site geometry, electrostatics, coordination, etc. enables favorable binding

Functional behavior: co-factors, allosteric activation, folding, etc.

8

Site of sodium-specific binding in thrombin. Sodium binding

converts thrombin to a procoagulant form by

allosterically enhancing the rate and changing substrate specificity. Pineda AO, et al. 2004. J Biol Chem 279 (30):

31842-53.

Draper DE, et al. 2005. Annu. Rev. Biophys. Biomol. Struct. 34:

221-43.

Rep + ATP kinetics influenced by specific interactions of divalent anions

with ATP binding site. Moore KJM, Lohman TM. 1994. Biochemistry 33

(48): 14565-78.

How ions interact with biomoleculesHofmeister effects (preferential hydration), ca 1888

How much salt is required to precipitate a protein? It depends on the salt...

Partitioning of ions between water and nonspecific sites on biomolecule

Dependent on ion type (solvation energy, etc.)

Dominate at high salt concentrations

Functional behavior: protein stability, membrane structure and surface potentials, protein-protein interactions

9

citrate2- > SO42- > PO42- > F- > Cl- > Br- > I- > NO3- > ClO4-

N(Me)4+ > NH4+ > Cs+ > Rb+ > K+ > Na+ > H+ > Ca2+ > Mg2+ > Al3+

weakly solvated cations

weakly solvated anions

strongly solvated cations

strongly solvated anions

most stabilizing most destabilizing

Adapted from http://www.lsbu.ac.uk/water/hofmeist.html

http://www.lsbu.ac.uk/water/hofmeist.html

http://www.lsbu.ac.uk/water/hofmeist.html

Models for biomolecular solvation

Modeling biomolecule-solvent interactions

11

Modeling biomolecule-solvent interactionsSolvent modelsQuantumExplicit

PolarizableFixed charge

Integral equationRISM3D methodsDFT

PrimitiveLangevin dipolesPoisson

PhenomenologicalGeneralized Born, et alModified Coulomb’s law

Ion modelsQuantumExplicit

PolarizableFixed charge

Integral equationRISM3D methodsDFT

Field-theoreticExtended modelsPoisson-Boltzmann

PhenomenologicalGeneralized Born, et alModified Debye-Hückel

Incr

easi

ng d

etai

l, co

st

11

Explicit solvent simulationsSample the configuration space of the system: ions, atomically-detailed water, soluteSample with respect to a particular ensemble: NpT, NVT, NVE, etc.Molecular dynamics or Monte CarloAdvantages:

High levels of detailAdditional degrees of freedom readily includedAll interactions are explicit

DisadvantagesSlow and uncertain convergenceBoundary effectsPoor scalingSome effects still not considered in may force fields... 12

Implicit solvent modelsSolute typically only accounts for 5-10% of atoms in explicit solvent simulation......so treat solvent effects implicitly:

Solvent as polarization densityIons as “mobile” charge density

Serious assumptions:Linear and local solvent response“Mean field” ion behaviorUncertain treatment of “apolar” effects

13

Solvation free energies (and mean forces)“Potentials of mean force” (PMF) and solvation free energies

Function of conformationIntegration over explicit degrees of freedom yields free energyGlobal information

Mean forcesDerivatives of PMFs for atom positionsIntegration yields PMFsLocal information

14

Polar solvation (implicit)Charging free energies

Solvent: dielectric effects through Poisson equationIons: mean-field screening effects through Poisson-Boltzmann equation

15

Poisson-Boltzmann equation

16

G[!] =14"

!

!

"#f (x)!(x)! $(x)

2["!(x)]2 +

#

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!

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Wagoner JA, Baker NA. Proc Natl Acad Sci USA, 103, 8331-6, 2006.17

Nonpolar solvation: implementationIt’s not just surface area!


Nonpolar solvation: implementationIt’s not just surface area!WCA formalism:

Cavity creationSmall length scalesLarge length scales


Nonpolar solvation: implementationIt’s not just surface area!WCA formalism:

Cavity creationSmall length scalesLarge length scales

Dispersive interactionsIntegral of potential over solvent-accessible volume

Putting it all back together

18

Adapted from: Levy RM, Zhang LY, Gallicchio E, Felts AK. 2003. J Am Chem Soc 125 (31): 9523-9530.

Solving the Poisson-Boltzmann equation

Solving the PB equation

20

Solving the PB equationParallel adaptive finite element methods

Bank and Holst, SIAM Review, 2003

A posteriori residual-based error estimators

PB-specific customization

FEtk-based solution (http://www.fetk.org/)

20

http://www.fetk.org

http://www.fetk.org

http://www.fetk.org

http://www.fetk.org

Solving the PB equationParallel adaptive finite element methods

Bank and Holst, SIAM Review, 2003

A posteriori residual-based error estimators

PB-specific customization

FEtk-based solution (http://www.fetk.org/)

Parallel focusing methodsBaker et al, Proc Natl Acad Sci, 2001

Loosely related to Bank-Holst method

PMG-based solution (http://www.fetk.org/)

20

http://www.fetk.org

http://www.fetk.org

http://www.fetk.org

http://www.fetk.org

http://www.fetk.org

http://www.fetk.org

http://www.fetk.org

http://www.fetk.org

Implicit solvent toolsAPBS (http://apbs.sf.net/)

PB electrostatics calculationsFreely availableFast finite element (FEtk) and multigrid (PMG) solvers from Holst group (http://fetk.org) Works with most popular visualization software (VMD, PMV, PyMOL)Links with CHARMM, AMBER, TINKER*

PDB2PQR (http://pdb2pqr.sf.net/)

21

Baker NA, et al Proc Natl Acad Sci USA, 98, 10037, 2001; *Schnieders MJ, et al. J Chem Phys, 126, 124114, 2007.

http://apbs.sf.net

http://apbs.sf.net

http://fetk.org/

http://fetk.org/

http://cam.ucsd.edu/~mholst/codes/pmg/index.html

http://cam.ucsd.edu/~mholst/codes/pmg/index.html

http://fetk.org

http://fetk.org

http://pdb2pqr.sf.net


PDB2PQRPDB2PQR (http://pdb2pqr.sf.net/)

Collaborative project: Jens Nielsen, Jan Jensen, and Gerhard Klebe groupsPrepares PDB files for other calculationsAssigns titration states (PROPKA) and optimizes hydrogen positions“Repairs” missing heavy atomsAssigns parametersWeb-based and command-lineFreely available (GPL or BSD) and extensible

22

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Dolinsky TJ, et al. Nucleic Acids Res, 35, W522-5, 2007; Dolinsky TJ, et al. Nucleic Acids Res, 32, W665-7, 2007.



Balancing accuracy and efficiency in solvation

models

OverviewHow does solvent behave near biomolecules of varying charge density?

Are there implicit polar solvation models that can describe solute-solvent interactions involving charge?Are there implicit nonpolar solvation models that can describe the remaining solute-solvent interactions?

Relevance:Understanding biomolecular solvation and electrostaticsDeveloping better methods for simulation and modeling

24

Testing molecular extremesHow well do these models work?Assess solvation forcesVarious systems:

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Hexanes (low charge density, small size) -- in progress!

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Hexanes


Hexanes (low charge density, small size) -- in progress! IFABP (medium charge density, large size)

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IFABPHexanes


Hexanes (low charge density, small size) -- in progress! IFABP (medium charge density, large size)IRE (high charge density)

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IFABP IREHexanes


Hexanes (low charge density, small size) -- in progress! IFABP (medium charge density, large size)IRE (high charge density)

Future studies:Free energiesHybrid models

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IFABP IREHexanes

Methods (or a “reality” check)Comparing explicit solvent simulations to implicit solvent modelsSolvation forces (and energies)Compare polar and nonpolar forces:

Run long simulation of flexible charged biomolecule in explicit solvent

Cluster conformations

For each rigid conformational cluster representative:

Run simulation of charged biomolecule

Run simulation of uncharged biomolecule

26

F(np)i (x) = !

!!U (np)

uv

!xi

"

uncharged

F(p)i (x) = !

!!U (p)

uv

!xi+

!U (np)uv

!xi

"

charged

+

!!U (np)

uv

!xi

"

uncharged

Wagoner JA, Baker NA, J Comput Chem, 25, 1623-9, 2004; Wagoner JA, Baker NA. Proc Natl Acad Sci USA, 103, 8331-6, 2006. Adapted from Roux B, Simonson T. Biophys Chem, 78, 1-20, 1999.

A medium charge density test caseIntestinal fatty acid binding protein (IFABP)Very open structureMedium-to-low charge density

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IFABP

A medium charge density test caseIntestinal fatty acid binding protein (IFABP)Very open structureMedium-to-low charge densityShould be relatively “easy” for implicit model description:

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IFABP


Weak solvent polarization

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IFABP


Weak solvent polarizationLimited ion importance*

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IFABP

Force-fitting procedure1. Choose optimal solvent probe radius σs

2. Fit surface tension ϒ to area dependence and pressure p to volume

3. Evaluate mean-squared error χ2

4. Evaluate confidence intervals

28

IFABP solvationNMR structures (PDB ID 1AEL)Molecular dynamics

NpT: 300 K, 1 atm

AMBER99 parameters, TIP3P water

~150 mM NaCl,+1 e protein charge

1 ns trajectory per conformation (water converged)

PB calculations:

Spline-based surfaces

APBS (http://apbs.sf.net/)

0/150 mM ionic strength

29

Wagoner JA, Baker NA, J Comput Chem, 25, 1623-9, 2004.

http://apbs.sf.net

http://apbs.sf.net

IFABP polar solvation

Comparison of explicit and implicit solvent forces

Unoptimized parameters (Bondi) show reasonable correlation but poor overall agreement (average error ≈ 6 kcal mol-1 Å-1)

Optimization of radii based on force and energy shows strong correlation and good agreement (average error ≈ 1 kcal mol-1 Å-1)

Optimized radii tested for a variety of (small protein) systems

30

Wagoner JA, Baker NA, J Comput Chem, 25, 1623-9, 2004; Swanson JMJ, et al, J Chem Theor Comput, 3, 170-83, 2007.

IFABP nonpolar solvation


Best-fit implicit solvent parameters: γ=0±1 cal mol-1 Å-2, p=55±2 cal mol-1 Å-3, σ=1.25 (1.16-1.39) Å

SASA is unable to describe the total nonpolar solvation force

Repulsive nonpolar solvation force dominated by SAV pressure

31

Wagoner JA, Baker NA. Proc Natl Acad Sci USA, 103, 8331-6, 2006.

R = 0.94R = 0.72

IFABP nonpolar solvation


Best-fit implicit solvent parameters: γ=0±1 cal mol-1 Å-2, p=55±2 cal mol-1 Å-3, σ=1.25 (1.16-1.39) Å

SASA is unable to describe the total nonpolar solvation force

Repulsive nonpolar solvation force dominated by SAV pressure

31

Wagoner JA, Baker NA. Proc Natl Acad Sci USA, 103, 8331-6, 2006.

R = 0.94R = 0.72

Need better surfaces!

Summary of IFABP results

32

Summary of IFABP resultsPolar solvation:

Reasonable agreement in forces between explicit and implicit forces with AMBER radiiImprovement of agreement upon optimizationTransferability to other protein models

32



Nonpolar solvation:Excellent agreement in forcesImportance of dispersion and volume for describing nonpolar solvationArea-tension contributions apparently unimportant

32



Nonpolar solvation:Excellent agreement in forcesImportance of dispersion and volume for describing nonpolar solvationArea-tension contributions apparently unimportant

PB + nonpolar provide a good coarse-grained description of IFABP solvation

32

A low charge density test caseVery low charge density

Solvation dominated by nonpolar contributions

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Hexanes



Motivated by:Suggestion from Ron Levy

33Gallicchio E, Kubo MM, Levy RM. J Phys Chem B, 104, 6271-85, 2000.




Simplicity (can calculate energies)

Strong conformational dependence

33

hydrogens from hexane (ggg) all of the previously calculatedhydration thermodynamic properties of cyclohexane werereproduced within statistical uncertainties.The hydration free energies of the hexane rotamers are

linearly related to the SASA. However, the linear scalingparameter ! of the hydration free energy of the hexane rotamerswith respect to the SASA is very different from the scalingparameter corresponding to the alkane series from methane tohexane (ttt). A linear fit shows that for the normal alkane seriesfrom methane to hexane (ttt) ! = 9 cal/(mol Å2), while for thehexane rotamers we obtain ! = 70 cal/(mol Å2), a value almost1 order of magnitude greater. Interestingly, the value of !corresponding to the alkane cavities studied in this work (! =73 cal/(mol Å2)) is very similar to the one corresponding to thehexane rotamers.These observations suggest that the variation in hydration free

energies with conformational changes are dominated by thework of cavity formation. This is confirmed by examining thehydration thermodynamics of the hexane rotamer cavities (Table6). The rotamer cavity hydration free energies scale linearlywith the SASA; a linear scaling parameter of ! = 74 cal/(molÅ2) is obtained. Consequently, the cavity and total free energychanges in going from the ttt rotamer to the ggg rotamer arevery similar, approximately -2.7 kcal/mol in either case. Theentropic and enthalpic signatures are also very similar in the

two cases. The ttt to ggg transition is favored equally by entropyand enthalpy for either the full or cavity rotamers.The thermodynamics of the hexane rotamers can be rational-

ized by observing (see Table 5) that, even though there is asubstantial SASA change, the solute-solvent interaction ofhexane hardly changes in going from the ttt rotamer to the gggrotamer. In these conformational transformations the solute-solvent interaction is almost totally uncorrelated with the SASA.The good correlation between atomic composition and

solute-solvent interaction energy and the corresponding poorcorrelation between the SASA and the solute-solvent interac-tion energy appears to be a general feature of hydrophobichydration. Ashbaugh et al.34 studied the conformational depen-dence of the hydration free energies of the linear alkanes. Theyobserved that the hydration free energies of each alkane indifferent conformations correlate very well with the molecularsurface area and that, moreover, the proportionality constant issimilar for all of the alkanes and roughly equal to theproportionality constant measured for the corresponding alkanecavities. The observations of Ashbaugh et al. imply that thesolute-solvent attractive interaction energy is only weaklydependent on molecular surface area and that, instead, it mainlydepends on the atomic composition. A weak correlation betweenSASA and solute-solvent interaction energy was also observedby Pitarch et al.,91 who observed that the interaction energybetween the methane dimer and water is nearly independent ofthe methane-methane separation distance; they concluded thatthe cavity contribution is the main driving force for thehydrophobic interaction between the methane molecules.3.5. Theoretical Estimates of the Work of Cavity Forma-

tion. In this paper the work of cavity formation of the alkanesin water is calculated by the FEP method. The method, althoughaccurate, is computationally very intensive. It is thereforeinteresting from both a theoretical and practical point of viewto compare the free energies of cavity formation calculated herewith estimates from theoretical models such as scaled particletheory (SPT)45 and information theory (IT).46

The reversible work associated with creating a void volumeof a particular shape in a solvent is related to the probability p0of finding a natural occurrence of such a cavity in a sample ofthe pure solvent.44 Both SPT and IT give estimates of p0 andthus, ultimately, of the free energy of solvation of the cavity.The SPT estimate of the free energy of hydration of the linear

alkanes from 1 to 10 carbon atoms have been reported by Irisaet al.92 They set to 1.99 Å the radius of all the CHn groups (n) 2, 3, or 4) which are considered spherical. The excludedvolume of water is also considered spherical with radius 1.375

TABLE 5: Calculated Hydration Free Energies (!G), Enthalpies (!H), Entropies (-T!S), Solute-Solvent Energy Changes(!Uuv), and Solvent Reorganization Energies (!Uvv) of the Hexane Rotamersa

molecule !G !H -T!S !Uuv !Uvvhexane (ttt) 3.781 ( 0.064 -5.682 ( 0.299 9.462 ( 0.306 -12.384 ( 0.040 6.703 ( 0.296hexane (tgt) 3.386 ( 0.085 -4.708 ( 0.403 8.094 ( 0.412 -12.192 ( 0.063 7.484 ( 0.398hexane (tgg) 2.600 ( 0.098 -5.175 ( 0.480 7.775 ( 0.490 -12.214 ( 0.078 7.039 ( 0.474hexane (ggg) 1.099 ( 0.117 -5.879 ( 0.578 6.979 ( 0.590 -12.295 ( 0.095 6.416 ( 0.570

a Values reported in kcal/mol at 298 K.

TABLE 6: Calculated Hydration Free Energies (!G), Enthalpies (!H), Entropies (-T!S), Solute-Solvent Energy Changes(!Uuv) and Solvent Reorganization Energies (!Uvv) of the Hexane Rotamer Cavitiesa

molecule !G !H -T!S !Uuv !Uvvhexane (ttt) 18.037 ( 0.069 10.487 ( 0.310 7.550 ( 0.318 1.850 ( 0.048 8.636 ( 0.307hexane (tgt) 17.811 ( 0.077 9.907 ( 0.362 7.904 ( 0.370 1.826 ( 0.054 8.081 ( 0.358hexane (tgg) 16.970 ( 0.083 9.207 ( 0.402 7.763 ( 0.410 1.728 ( 0.058 7.479 ( 0.398hexane (ggg) 15.317 ( 0.092 8.166 ( 0.453 7.151 ( 0.462 1.604 ( 0.065 6.562 ( 0.448


Figure 8. Calculated hydration free energies of the n-hexane rotamers(ttt, tgt, tgg, and ggg) compared with the alkane hydration free energiesfrom Figure 3. Calculated free energies are shown taking as a referencethe experimental hydration free energy of methane. See Figure 3 forthe symbol keys.

Alkane Hydration Free Energies J. Phys. Chem. B, Vol. 104, No. 26, 2000 6281

Gallicchio E, Kubo MM, Levy RM. J Phys Chem B, 104, 6271-85, 2000.




Simplicity (can calculate energies)

Strong conformational dependence

Apparent scaling with surface area

33



































Gallicchio E, Kubo MM, Levy RM. J Phys Chem B, 104, 6271-85, 2000.

Hexanes

34

Hexane conformers

OPLS force field gives good solvation energies

Molecular dynamics simulationAMBER9NVT: 300 K (equilibrated at 1 atm with NpT)OPLS parameters, TIP3P water1.5 ns trajectory per conformation (observables converged)

Calculation of:Solvation forcesSolvation energies (in progress!)

ttt

tg-t

g+g-g+

tg+g-

tg-g-g-g-g-

-0.5 0 0.5

Explicit solvent forces (kcal mol-1

A-1

)

-0.5

0

0.5

Impli

cit

solv

ent

forc

es (

kca

l m

ol-1

A-1

)

Optimal hexane resultsLarge difference between SASA-only errors and other modelsOptimal solvent radii are very small!Optimal ≠ realistic......but might suggest a problem with surface definitions or model

35

Table 1: Optimal model parameters for best-fit solvent radii and associated quality of fit metricsfor di!erent implicit solvent nonpolar force models. 99% confidence intervals for the parametersare given in brackets. See text for details.

Parameters SASA only SAV only SASA + SAV!s, A 0.02 0.58 0.69

[0.0 - 2.2] [0.31- 0.87] [0.42 - 1.10]", cal mol!1 A!2 53.63 – 5.63

[37.73 - 69.53] [2.65 - 8.54]p, cal mol!1 A!3 - 82.35 66.33

[74.72 - 89.97] [55.23 - 77.82]r 0.7385 0.9685 0.9713R 0.8555 0.9820 0.9854#2 0.0284 0.00381 0.00308(kcal2 mol!2 A!2)

Table 2: Optimal model parameters for best-fit solvent radii and associated quality of fit metricsfor di!erent implicit solvent repulsive nonpolar force models. 99% confidence intervals for theparameters are given in brackets. See text for details.

Parameters SASA only SAV only SASA + SAV!s, A 0.02 0.58 0.69

[0.0 - 2.2] [0.29 - 0.97] [0.41 - 1.17]", cal mol!1 A!2 58.95 – 6.52

[43.19 - 74.72] [3.58 - 9.45]p, cal mol!1 A!3 - 84.88 66.42

[76.90 - 92.86] [55.23 - 77.72]r 0.7959 0.9693 0.9777R 0.8921 0.9845 0.9888#2 0.0278 0.00419 0.00306(kcal2 mol!2 A!2)

tested three variants of the nonpolar implicit solvent model; each variant includes the attractivedispersion forces (Eq. 8) as well as versions of the repulsive force (Eq. 5) which include only the"A(x) term (“SASA only”), only the pV (x) term (“SAV only”), or both terms (“SASA + SAV”).

Table 1 shows the results of simultaneously optimizing ", p, and !s (the solvent radius) tominimize #2 as described previously4. The quality of fit for these parameters was measured by theregression coe"cient r, Pearson correlation coe"cient R, and the mean-squared error #2. Table 3presents the data for nonpolar forces obtained using optimal " and p parameters for the popular 1.4A solvent probe radius, as well as optimal parameters determined by a previous study4. Confidenceintervals for these parameters were assigned using a standard F-test [cite Numerical Recipes] at the99% confidence level. Figure 1 shows the range of the #2 statistic for all values of solvent proberadius in order to demonstrate sensitivity with respect to this parameter.

In order to better understand the contributions of the SASA and SAV terms to the repulsiveforces, we performed a similar optimization over just repulsive forces, with the mean-square errordefined as

#2rep =

118N

6!

c=1

N!

i=1

"""f (rep)i,c ! F(rep)

i,c

"""2

(10)

Table 2 displays the results of this set of optimizations.

4

Range of statistically-indistinguishable values99% confidence interval allows probe radii as large as 1.13 Å

Increase to “realistic” radii involves little more error

Consistent with results for IRE and IFABP...

...however, area contribution is more significant than with macromolecules

36

≈1.13 Å

Range of statistically-indistinguishable values99% confidence interval allows probe radii as large as 1.13 Å

Increase to “realistic” radii involves little more error

Consistent with results for IRE and IFABP...

...however, area contribution is more significant than with macromolecules

36

Table 2: Optimal model parameters for specific solvent radii and associated quality of fit metrics for di!erent implicit solvent nonpolarforce models. 99% confidence intervals for the parameters are given in brackets. See text for details.

Parameters SASA only SAV only SASA + SAVProbe Radii, !s, A 1.13 1.4 1.13 1.4 1.13 1.4", cal mol!1 A!2 24.03 21.07 – – 6.62 6.70

[16.53 - 31.53] [14.63 - 27.51] [4.07 - 9.09] [4.26 - 9.16]p, cal mol!1 A!3 – – 57.87 49.65 46.69 38.54

[50.94 - 64.80] [42.70 - 56.59] [38.09 - 55.68] [30.12 - 46.86]r 0.6722 0.6738 0.9523 0.9371 0.9646 0.9556R 0.8143 0.8159 0.9678 0.9556 0.9810 0.9755#2 ! 103 0.03576 0.03549 0.00676 0.00928 0.00400 0.00515(kcal2 mol!2 A!2)

6

Range of statistically-indistinguishable values99% confidence interval allows probe radii as large as 1.13 ÅConsistent with previous results for IRE and IFABP......however, area contribution is more significant than with macromolecules

31

Table 2: Optimal model parameters for specific solvent radii and associated quality of fit metrics for di!erent implicit solvent nonpolarforce models. 99% confidence intervals for the parameters are given in brackets. See text for details.

Parameters SASA only SAV only SASA + SAVProbe Radii, !s, A 1.13 1.4 1.13 1.4 1.13 1.4", cal mol!1 A!2 24.03 21.07 – – 6.62 6.70

[16.53 - 31.53] [14.63 - 27.51] [4.07 - 9.09] [4.26 - 9.16]p, cal mol!1 A!3 – – 57.87 49.65 46.69 38.54

[50.94 - 64.80] [42.70 - 56.59] [38.09 - 55.68] [30.12 - 46.86]r 0.6722 0.6738 0.9523 0.9371 0.9646 0.9556R 0.8143 0.8159 0.9678 0.9556 0.9810 0.9755#2 ! 103 0.03576 0.03549 0.00676 0.00928 0.00400 0.00515(kcal2 mol!2 A!2)

6

-0.5 0 0.5

Explicit solvent force (kcal mol-1

A-1

)

-0.5

0

0.5

Imp

lici

t so

lven

t fo

rce

(kca

l m

ol-1

A-1

) !s = 1.13 Å

Model transferabilityHow well does this model work for:

Results from nonpolar calculations on other systems?Typical water probe radii?

Reasonable errors with previous parametersMore emphasis on SASA with typical water probe radii

37

Table 3: Optimal model parameters for specific solvent radii and associated quality of fit metricsfor di!erent implicit solvent nonpolar force models. 99% confidence intervals for the parametersare given in brackets. See text for details.

Parameters SASA only SAV only SASA + SAVProbe Radii, !s, A 1.4 1.4 1.25 1.4", cal mol!1 A!2 21.07 – 0 6.70

[14.63 - 27.51] [4.26 - 9.16]p, cal mol!1 A!3 – 49.65 55 38.54

[42.70 - 56.59] [30.12 - 46.86]r 0.6738 0.9371 0.9661 0.9556R 0.8159 0.9556 0.9628 0.9755#2 0.03549 0.00928 0.00791 0.00515(kcal2 mol!2 A!2)

We note that the parameterization process is particularly sensitive to the set of solute radii.All results are presented using solute radii of ri = !i

2 . If the force optimization is performed usingsolute radii of ri =

!!i!s

2 , the observed solvation forces are significantly less accurate (data notshown). Can this discussion be extended to explain why the optimal radii are so small?

These results indicate the sensitive nature of surface parameterization, and to justify the set ofradii used in this work we additionally performed a force optimization using independent solventprobe radii for Carbon and Hydrogen atoms. This represents a full parameterization of the Hexanesurface with results independent of solute radii (data not shown). The optimal forces from thisanalysis were not significantly more accurate than those presented and, for this work, do not justifya parameterization scheme that includes more than one solvent probe radius. However, complexsolutes that contain more than two atom types may exhibit more significant sensitivity to thismethod of parameterization, and future analysis should be directed at resolving such issues.

Need to discuss in light of IFABP and related results. Is SASA+SAV significantly better than SASAonly for the realistic radii?

4 Conclusion

5 Acknowledgments

JAW summer support and NAB grant support, Rohit, etc. Where should Emilio Gallicchio go? Author?

References

[1] C. Chothia, “Hydrophobic bonding and accessible surface area in proteins,” Nature, vol. 248,no. 5446, pp. 338–339, 1974.

[2] K. A. Sharp, A. Nicholls, R. F. Fine, and B. Honig, “Reconciling the magnitude of the micro-scopic and macroscopic hydrophobic e!ects,” Science, vol. 252, no. 5002, pp. 106–109, 1991.

[3] J. D. Weeks, D. Chandler, and H. C. Andersen, “Role of repulsive forces in determining theequilibrium structure of simple liquids,” Journal of Chemical Physics, vol. 54, no. 12, pp. 5237–47, 1971.

6




37



[14.63 - 27.51] [4.26 - 9.16]p, cal mol!1 A!3 – 49.65 55 38.54

[42.70 - 56.59] [30.12 - 46.86]r 0.6738 0.9371 0.9661 0.9556R 0.8159 0.9556 0.9628 0.9755#2 0.03549 0.00928 0.00791 0.00515(kcal2 mol!2 A!2)



!!i!s




4 Conclusion

5 Acknowledgments


References




6

-0.5 0 0.5


A-1

)

-0.5

0

0.5

Imp

lici

t so

lven

t fo

rce

(kca

l m

ol-1

A-1

)

σs = 1.25 Å, ϒ = 0 cal mol-1 Å-2, p = 55 cal mol-1 Å-3




37



[14.63 - 27.51] [4.26 - 9.16]p, cal mol!1 A!3 – 49.65 55 38.54

[42.70 - 56.59] [30.12 - 46.86]r 0.6738 0.9371 0.9661 0.9556R 0.8159 0.9556 0.9628 0.9755#2 0.03549 0.00928 0.00791 0.00515(kcal2 mol!2 A!2)



!!i!s




4 Conclusion

5 Acknowledgments


References




6

-0.5 0 0.5


A-1

)

-0.5

0

0.5

Impli

cit

solv

ent

forc

e (k

cal

mol-1

A-1

)

σs = 1.40 Å, ϒ and p optimized

Solvation energiesForce-optimized parameters give reasonable solvation energies

38

Levy et al explicitForce-optimized implicit

Conformer

Solvation energy (kcal/mol)

Solvation energiesForce-optimized parameters give reasonable solvation energiesWork in progress:

Free energy decomposition (cavity vs. dispersion)Conformational PMFOther optimization schemes

38

Levy et al explicitForce-optimized implicit

Conformer

Solvation energy (kcal/mol)

Summary of hexane results

39

Summary of hexane resultsWCA-like model predicts nonpolar solvation forces with good accuracy

39

Summary of hexane resultsWCA-like model predicts nonpolar solvation forces with good accuracyAmbiguity on optimal reference repulsive model:

Best results obtained with both area-tension and volume-pressure componentsVolume-pressure model is nearly as good as combined modelArea-tension model alone is not predictive

39



Good transferability from other parameters

39



Good transferability from other parametersAdditional solvation energy analyses needed

39

Implicit solvent models at “molecular extremes”

After throwing away all molecular details of solvent...

40


After throwing away all molecular details of solvent...Nonpolar model

Good agreement for a variety of moleculesTransferability of parametersWCA-like and volume terms appear to be essential ingredients

40




Polar modelReasonable agreement for IFABPIon behavior still to be explored

40





Implicit solvent models offer a reasonable coarse-grained description of solvation -- even for challenging systems

40





Implicit solvent models offer a reasonable coarse-grained description of solvation -- even for challenging systemsForces aren’t everything!

40





Implicit solvent models offer a reasonable coarse-grained description of solvation -- even for challenging systemsForces aren’t everything! Work still remains for parameterization and improvement of models

40

Baker groupCurrent

Mike Bradley (Mol Biophys grad student)Dave Gohara (Programmer) Stephen Gradwohl (BME undergrad)Yong Huang (Programmer)Sechin Jain (BME undergrad)Peter Jones (Programmer)Sunjoo Lee (Mol Biophys grad student)

Michal Lijowski (Staff scientist)Marcelo Marucho (Senior scientist)Brett Olsen (Mol Cell Biol grad student)Rachel Rice (Mol Biophys grad student)Samir Unni (BME undergrad)

Selected alumniTodd Dolinsky (HP)Feng Dong (Merck)Tom Richner (U Wisc)Jason Wagoner (Stanford)

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Other acknowledgments

FundingNIH (NIGMS, NCI, NIHGRI, NCRR)DoDNSFTeragridCDISloan Foundation

CollaboratorsDave Cerutti (Vanderbilt)Emilio Gallicchio (Rutgers)Kathleen Hall (Wash U)

Mike Holst (UCSD)Jan Jensen (U Copenhagen)Ron Levy (Rutgers)Hui Lu (UIC)Andy McCammon (UCSD)Rohit Pappu (Wash U)Jay Ponder (Wash U)Jens Nielsen (UC Dublin)Gerard Wong (UIUC)

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Biomolecular solvation: from molecular to continuum modelsBiomolecular solvation: from molecular to...

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