there 
 is a need for computer methods that can calculate the aqueous solvation 
 free energies of solute molecules accurately and efficiently .
explicit - solvent models provide a physically 
 accurate and atomistically detailed model of solvent , but they can 
 be computationally expensive .
boltzmann 
 ( pb ) , generalized born ( gb ) , and 
 weighted surface area ( wsa ) approaches , are computationally efficient because they treat 
 water as a continuum .
however , they are sometimes inaccurate , smearing 
 out the particulate nature of water molecules , and they may 
 have limited transferability to situations for which they have not 
 been optimized .
solvation modeling has 
 often been improved by combining the advantages of implicit- and explicit - solvent 
 models .
one such approach is the semi - explicit assembly ( sea ) water model . in the sea approach ,
water s solvation behavior is precomputed 
 in explicit - solvent simulations of water around model spheres that 
 are then combined together as building blocks to represent any given 
 solute structure at run time .
the solvation free energy gsolv for a target solute molecule is calculated 
 from a sum over solvating waters .
sea has been shown to predict efficiently 
 the air - to - water transfer free energies ( ghyd ) of small - molecule solutes in both prospective and 
 retrospective studies with reasonable accuracy .
even 
 so , the sea model does not give accurate solvation free energies of 
 ions or solutes having high charge density .
here , we describe 
 field - sea , a considerable improvement over sea , which gives accurate 
 free energies of transfer of ions and charged solutes , at no additional 
 computational cost and with no degradation of the predictions for 
 nonpolar and uncharged polar solutes . in overview , we studied full 
 md simulations of ionic and neutral solutes in tip3p water , described 
 below , and found that the results could be captured by using a solvation 
 free - energy field surface ( in field - sea ) , instead of using precomputed 
 waters ( as in sea ) .
moreover , 
 we found that a weakly charged solute atom that is adjacent to a strongly 
 charged solute atom retains some of the restrictions of its solvating 
 water molecules that its neighboring charge has ; see figure 1 .
field - sea captures this effect using an adaptive 
 boundary method . a solvent  water molecule around a solute molecule . 

on the left , each solute atom ( weakly charged ) has a predefined radius , 
 irrespective of its neighboring solute atoms , leading to the locus 
 of water centers shown by the black dashed curve . on the right ,
one 
 solute atom is strongly charged , leading to two consequences : its 
 own solvating water molecule is pulled in tightly , and neighboring 
 solute atoms have tighter water interactions too .
such effects may not be captured in simplified solvation 
 models that treat atoms as having fixed radii , independent of neighboring 
 atoms .
below , we describe the field - sea approach to computing solvation 
 free energies . the original sea model is described elsewhere , and our related explicit - solvent and linearized poisson 
boltzmann 
 equation modeling ( lpbe ) are described 
 in the supporting information ( si ) . in both the original sea and the new 
 field - sea described here ,
first , in a precomputation step , various 
 model spheres are solvated in an explicit - solvent model , such as tip3p . 

this provides a database of component free energies that are used 
 in the second step . in the second step , at the run time ,
any given 
 solute molecule of interest is assembled from an appropriate concatenation 
 of these spheres .
the solute s solvation free energy is computed 
 by summing the component sphere free energies . on the one hand , this 
 approach provides the speed advantages of simple additivity - based 
 models as we only need to perform the first step once for a given 
 solvent model . on the other hand , this procedure is more accurate 
 than additivity - based approaches because ( 1 ) sea sums are regional , 
 not local , and ( 2 ) the database encodes the microscopic response observed 
 in explicit - solvent simulations .
the new field - sea instead captures 
 the solvation free energy using a continuous solvation field . to do this ,
the charging free energies for a series of lennard - jones 
 ( lj ) spheres solvated in tip3p water were calculated with thermodynamics 
 integration ( ti ) in the set of precomputations ( see explicit solvent 
 free energy calculations in the si ) .
we 
 construct a free - energy contour ( see si figure s1 ) as a combination of the electric field at the first solvation 
 shell boundary ( e = q / rw , where e is the signed 
 magnitude of the electric field , q is the sphere s 
 charge , and rw is the distance from the 
 sphere center to the first peak of the water oxygen s radial 
 distribution function , rdf , around the given sphere ) , the curvature c = 1/rw at this boundary , and 
 the charging free energy of the spheres . within each charge step ( q ) , all of the data of free energy , electric field , and 
 curvature were fitted to a formula , which can be taken as an expansion 
 of the born model1 equation 1 will 
 be used to calculate the free energy associated with any surface spot 
 on an arbitrary solute  solvent boundary ( esub ) .
this free energy could also be calculated from interpolation 
 between data points on the free - energy contour .
in addition 
 to a charging free - energy contour , we also need a contour for estimating 
 the explicit - solvent - accessible surface of a given solute molecule . 
 to generate this
, we calculated the rdf of water oxygen around each 
 charged sphere and picked the first peak in the rdf , denoted as the 
 boundary - sphere distance rw .
all of the rw values from these spheres and their charge 
 and lj parameters ( lj and lj ) 
 were used to build an rw contour . to generate a solvent - accessible 
 surface for a given solute
, an rw for 
 each atom is first calculated from interpolation with its partial 
 charge , lj , and lj on the rw contour .
a lee  richards surface is then constructed from the nonoverlapping 
 sections of these rw spheres centered 
 on their associated atom sites .
the fixed rw boundary is dependent upon only the individual parameters 
 of the surface atoms .
for solute molecules with multiple partial charge 
 sites ( especially molecular ions , where strong electric fields are 
 involved ) , a more physical representation of the explicit - solvent - accessible 
 surface would be one that responds to the whole solute s electric 
 field .
in other words , the surface  atom distance , rw , is determined not only by the surface atom s 
 partial charge , but also by neighboring atoms charges .
taking 
 into account such collective electrostatic effects , we adjust the 
 fixed rw boundary in an adaptive manner , 
 described by the following three - step procedure : ( 1)we cull 
 all surface segments within 1.4  of any solute atom . as we are 
 starting with a lee  richards solvent - accessible surface , the 
 minimum rw possible in the surface construction 
 is half of a water molecule diameter ( corresponding to a solute atom 
 with a 0  radius ) .
this partial - culling process simply removes 
 potential numerical instabilities from surface sites overlapping solute 
 atom centers while providing some adjustable starting surface sites 
 that penetrate within the fixed rw boundary .
we calculate 
 the electric field at a given dot a about atom a2where n is the number 
 of solute atoms , ria represents a vector from atom i to surface 
 dot a , and ria is its length ; signa = 1 when raai=1n  ( qi / ria3)ria  0 , and signa = 1 when raai=1n  ( qi / ria3)ria < 0 , and raa is the vector from atom a to dot a. from the electric field , we 
 calculate the corresponding charge by3where raa is the length of vector raa . we interpolate with this new charge and 
 atom a s lj parameters to get a new rw , rw , new .
in addition , 
 we assume that rw can not expand ( if rw , new > rw , original , we let rw , new = rw , original ) . this nonexpansion assumption is supported by solute 
solvent 
 boundary plots of model diatomic solutes ( see si figure s2 ) . finally , the rw , new s of all atom s potential surface dots are averaged 
 to yield an rw for this atom , used in 
 the later culling process . ( 3 ) we cull the buried dots adaptively . 

because each dot  atom distance ( e.g. , daa ) is adjusted independently in the above 
 procedure , the shell of potential surface dots about an atom will 
 no longer be spherical .
thus , culling buried dots is no longer a simple 
 process of eliminating dots within a neighboring atom s rw , and an adaptive culling process is needed . 
 to adaptively cull 
buried  surface dots , when we check 
 whether a given dot a is buried by a neighboring 
 atom b , we first determine a corresponding charge 
 at dot a from the electric field at this dot .
this 
 is used along with atom b s lj parameters 
 to get a new rw , rw , badp , following the above rw adjustment procedure .
atom 
 distance , dab , with rw , badp , to determine if the 
 dot is buried in atom b. if dab is less than rw , badp , it is removed from the set of potential surface 
 dots .
we call the new dot surface resulting from the above culling 
 procedure the adaptive boundary .
our term field - sea refers both to 
 the field and to the adaptive boundary .
the adaptive nature of the 
 boundary only pertains to multiatom solutes ; the adaptive and fixed rw boundaries are identical for single - atom solutes , 
 like monatomic ions . while our adaptation procedure could , in principle , 
 be applied iteratively , we found no further improvement after a single 
 calculation .
the charging 
 free energy can be calculated for any given field - sea surface via4where ndot is the total number of surface - exposed dots , mi is the total number of dots 
 ( exposed + occluded ) for corresponding atom i ( each 
 surface dot , j , of atom i , only 
 corresponds to 1/mi of 
 this atom sphere s total surface area or total solvation free 
 energy ) , and esub , j is 
 the subenergy associated with exposed surface dot j , calculated from the free - energy contour . here ,
esub , j is calculated from eq 1 using the signed magnitude of the electric field , ej , and the curvature at dot j , 1/rij .
ej is defined as negative when ejrij < 0 , where ej is the electric 
 field ( vector ) at dot j , rij is the vector from atom i to its surface dot j , and rij is this vector s length .
now , 
 we take the total solvation free energy to be the sum of polar and 
 nonpolar components5 assuming the polar component of the free energy of transfer 
 ( gpol ) can be uniquely described 
 by both the surface electric field and geometry of the solute , this 
 molecular gpol can be accurately 
 calculated from the simple summation of the surface dots energies 
 process described above .
we consider the gsolv of monatomic ions as an initial test . for molecular 
 solutes ,
to account for the role of solute conformation in the solvation 
 free energy , we average gsolv results 
 from calculations on 50 conformations ( though 10 conformations are 
 usually enough ; see the si ) . these solute 
 conformations were generated from 5 ns tip3p water md simulations 
 with a 100 ps snapshot interval .
we developed field - sea because of the errors that we 
 observed in simpler methods in solvating ions ; see figure 2 .
the explicit - solvent curve shows the hydration 
 asymmetry discussed by others , where positively charged solutes are less favored than negatively 
 charged solutes of similar size .
gsolv as the function of a model lj sphere (  
 = 0.22 nm ,  = 0.06538 kj / mol ) charge for tip3p , lpbe , and field - sea . 
 for comparison at infinite - dilution conditions ,
an ewald correction 
 is applied to the tip3p and field - sea results . here , we tested field - sea on the solvation free energies 
 of monatomic ions using ion parameters developed by aqvist or joung and cheatham ( table s2 , si ) .
these ions have different 
 sizes and charges ; therefore , they have different charge densities . 

table 1 and figure s3 ( si ) show the results of field - sea calculations , compared with 
 lpbe and tip3p .
lpbe results using the lj surface do not capture quantitatively 
 the gsolv of ions with high charge 
 densities .
in contrast , field - sea reproduces the tip3p gsolv values regardless of the specific lj parameters 
 and charges .
these results indicate that field - sea is accurately reproducing 
 explicit - solvent free energies of solvation of monatomic ions , provided 
 that their charge and lj parameters are near or encompassed within 
 the range of the rw and free - energy contours .
a common approximation in simplified 
 solvation models is to suppose that atoms have fixed 
atom 
 types  , where any particular solute atom has a given value 
 of charge and radius , independent of its atomic neighbors . however , 
 our tip3p explicit - solvent simulations described below show that this 
 approximation is a source of error . how the solvation surface is constructed 

our explicit - solvent simulations show that the solvation shell is 
 not just a simple union of the spherical surfaces of all of the atoms 
 making up the solute molecule .
imagine a diatomic solute with partially 
 charged atom a covalently connected to partially 
 charged atom b. our tip3p simulations show that when 
 atom a attracts water molecules , it also shrinks 
 the solvation shell of waters around atom b. in this 
 way , the solvation surface around the diatomic molecule a  b can be more complex than the simple union 
 of independent spherical solvation shells around atoms a and b. to address such situations , we developed 
 an adaptive method that gives a more explicit - like solvation boundary . to test our adaptive boundary method , we made up 22 fictitious 
 diatomic solutes ( see si table s4 ) , computed 
 their solvation free energies , and compared to their solvation in 
 tip3p .
we constructed these solutes by taking pairs of ordinary simulation 
 atom types , placing them at fixed covalent bond distance apart , and 
 giving each atom a charge and radius that we could vary systematically 
 over the series . because of the fictitious charges that we give them , 
 these are not molecules observed in nature
however , they are physically 
 plausible molecules that provide us with a systematic series for learning 
 about how explicit - solvent models handle solvated charges .
figure 3 shows the computed free energies of these diatomic 
 solutes from field - sea when using the fixed rw and adaptive boundaries in comparison to explicit - solvent 
 tip3p simulations .
lpbe and original sea results are shown in the si instead of here for the sake of simplicity . 
 in summary
, we find that when the tip3p gsolv is weak , for example , when the charge density of solute 
 atoms is low , the fixed rw boundary works 
 fairly well .
when the gsolv grows 
 to 100 kcal / mol and larger ( more negative ) , it becomes increasingly 
 important to use an adaptive boundary .
these strongly solvated model 
 molecules often have large , unbalanced partial charges on the solute 
 atoms , and these situations lead to exaggerated distortions of the 
 explicit solvation shell .
field - sea gsolv for model diatomic solutes ( triangles : fixed rw ; circles : adaptive boundary ) compared to tip3p simulations . 

figure 4 shows how a neighboring 
 solute atom can affect the solvation shell about another atom .
the 
 fixed rw and adaptive boundaries are identical 
 in figure 3a as both carbon atoms are weakly 
 charged and there is only minor collective electrostatic perturbation 
 on the boundary . however ,
when the solutes are more highly charged 
 and large charging energies are involved , as in figure 3b , the weakly charged carbon atom s solvation shell 
 will be distorted by its neighboring highly charged oxygen atom . in 
 this case ,
water molecules penetrate more deeply into the carbon s 
 solvent - accessible surface to better solvate the oxygen charge ( the 
 right part of the white curve in the dark blue region ) .
also , water 
 molecules pack more tightly around the carbon atom and reduce its 
 apparent rw ( the left part of the white 
 curve in the blue and light blue regions ) .
both of these effects are 
 captured well by adaptive field - sea boundaries , leading to field - sea 
 charging energies that are in excellent agreement with explicit simulation 
 results .
oxygen density around ( a ) weakly charged 
 and ( b ) strongly charged diatomic solutes .
the blue contours indicate 
 water density greater than the bulk value , with the darker blue regions 
 indicating the enhanced water probability density .
the black line 
 shows the nonadaptive fixed rw boundary . 
 for the weakly charged diatomic , the adaptive and nonadaptive boundaries 
 coincide .
the adaptive and nonadaptive boundaries differ . in order to establish that the accuracy of field - sea is not degraded 
 relative to sea on nonionic solutes , we applied field - sea with both 
 fixed rw and adaptive boundaries to a 
 standard set of 504 neutral small molecules .
this set contains an alchemically diverse 
 range of functional groups , for which solvation free energies are 
 available from experiments and tip3p simulations .
figure 5 shows 504 neutral solutes solvation free 
 energy from tip3p simulation , from the lpbe ( white diamonds ) and field - sea 
 ( white circles ) ( see si figure s4 for nonadaptive 
 fixed rw field - sea results ) .
field - sea 
 shows a rmse ( root - mean - square error ) of 1kt with a negligible mse ( mean - signed error ) , comparable to the original 
 sea and considerably better than the 
 lpbe , which bears a systematically negative error .
while the fixed rw boundary field - sea results are comparable 
 to the lpbe ( table 1 ) , it overestimates the 
 free energy when weakly charged c atoms are neighbored by highly charged 
 o or n atoms , as in the cases of alcohols , amines , ethers , and esters 
 ( si table s5 ) .
these errors arise because 
 the fixed rw boundary can not capture water 
 molecule penetration into the c atom s solvent - accessible surface 
 ( see figure 4b ) , situations the adaptive boundary 
 handles directly . therefore , while these are simply neutral solutes , 
 collective solute interactions still play a clear role in their overall 
 hydration .
figure s5 ( si ) compares 
 the total solvation free energy from field - sea with experimental results . 

the mse / rmse of field - sea to experimental solvation free energy ( 0.67/1.45 
 kcal / mol , table s6 ( si ) ) is comparable 
 to that of the much more computationally expensive tip3p water model 
 ( 0.66/1.22 kcal / mol ) .
these results indicate 
 that field - sea can accurately compute solvation free energies over 
 the full range from charge - dense ions to neutral polar and nonpolar 
 molecular solutes . here , we test field - sea on an expanded set of molecular 
 ions and biomolecules ( e.g. , acetate , butylammonium , etc . ; see si table s7 for the complete list and results ) .
these are ions that are larger and more complex than the simple 
 atomic and diatomic ions described above and should better test the 
 need for adaptive boundary considerations .
figure 5 compares solvation energies from lpbe ( orange diamond ) and 
 field - sea ( orange circle ; see si figure 
 s6 for field - sea results with a fixed rw boundary ) with tip3p results for 35 molecular ions .
the errors for 
 both lpbe and field - sea with a fixed rw boundary are around 10 kcal / mol ( table 1 ) . 

while this might be regarded as acceptable in light of the nearly 
 100 kcal / mol span of free energies covered in the tip3p simulations , 
 using an adaptive boundary with field - sea has half of this rmse .
field - sea 
 also performs well in multivalent molecular ion solvation ( si figure s7 ) and is as accurate as explicit - solvent 
 calculations compared to experimental values for ionic solute solvation 
 ( table s6 and figure s8 , si ) .
md simulations 
 of gsolv for 504 neutral solutes 
 ( white ) , 35 molecular ions ( orange ) , and 22 capped amino acid dipeptides 
 ( cyan ) in tip3p water , compared to ( a ) lpbe and ( b ) field - sea .
we also tested our methods on 22 capped amino 
 acids ( n - acetyl - x - methylamide , x = glu , arg , leu , 
 etc . ;
si table s8 ) , which are widely used 
 biological models in both theoretical studies and experiments .
these are useful precursor 
 structures for the foundation of hydrophobicity scales , used in estimating 
 the solvation of larger biomolecular structures . here
, we investigate the solvation 
 free energies of a full series of capped amino acids with field - sea . 
 as the size of solute increases , computing the total solvation
show , lpbe 
 ( cyan diamond ) and field - sea ( cyan circle ) both yield solvation free 
 energies that agree well with tip3p calculations .
again , the adaptive 
 boundary helps field - sea considerably , cutting the rmse to half of 
 that seen from the lpbe or fixed rw cases . 

these results indicate that the accuracy of field - sea does not degrade 
 as the solute size further increases .
the solvation boundaries 
 in field - sea are made from a set of surface dots .
above , we used 80 dots / atom , 
 the same as was used in previous sea studies .
what is the minimum number of grid dots that we need to properly 
 represent the first - shell boundary ? to investigate this , we performed 
 an analysis of accuracy versus relative computational time as a function 
 of the granularity of the boundary .
this analysis indicates that the 
 rmse for field - sea results on the 504 neutral molecule set is essentially 
 uncompromised even down to a granularity of 5 dots / atom , without increasing 
 the rmse above 1 kcal / mol ( figure s9 and s10 ( si ) , the granularity does not affect the accuracy for charged 
 solutes either ) . at 5 dots / atom ,
field - sea is roughly 5-fold faster 
 than dipolar sea at 80 dots / atom , the minimum surface granularity 
 recommended for this method .
these results 
 indicate that while field - sea is sensitive to the physicality of the 
 solvation boundary , it is less sensitive to the granularity of its 
 depiction .
we have 
 described field - sea , a method for computing solvation free energies 
 of solutes in water .
it improves upon an earlier method called sea 
 ( semi - explicit assembly ) .
sea captured the physics of solvation by 
 presimulating toy spheres in explicit water , collecting a database 
 of structural and energetic properties of those waters and then assembling 
 at run time the solvation physics as sums over appropriate toy spheres 
 to properly represent a given solute . in field - sea , this procedure 
 differs in using a solvation free - energy field , rather than explicit 
 waters .
furthermore , field - sea uses an adaptive boundary , allowing 
 solvating waters to approach a solute atom to different degrees depending 
 on neighboring atoms .
relative to sea , field - sea captures the solvation 
 free energies of ions and charged solutes accurately , is faster to 
 compute , and has no degradation of performance on nonpolar and polar 
 solutes .
both sea and field - sea offer advantages over implicit - solvent 
 modeling in that they entail no adjustable solute atom radii parameters . 

sea and field - sea are built upon a corresponding force field and explicit - solvation 
 model . here
one of the 
 key observations that arises from our md simulations of charged solutes 
 in tip3p water , which is captured by field - sea , is that atoms that 
 are adjacent to charged atoms in solutes acquire partial characteristics 
 of those charged atoms .
for example , a weakly charged atom s 
 solvation shell is shrunk by its neighboring big charges .
an implication 
 of this for implicit - solvent modeling is that atomic radii should 
 not be treated as fixed for solvation in water ; an atom s radius 
 in implicit - solvent modeling can depend on the nature of its neighboring 
 atom .
 previous 
 work describes a computational solvation model called semi - explicit 
 assembly ( sea ) . 
 the sea water model computes the free energies of 
 solvation of nonpolar and polar solutes in water with good efficiency 
 and accuracy . 
 however , sea gives systematic errors in the solvation 
 free energies of ions and charged solutes . 
 here , we describe field - sea , 
 an improved treatment that gives accurate solvation free energies 
 of charged solutes , including monatomic and polyatomic ions and model 
 dipeptides , as well as nonpolar and polar molecules . field - sea is 
 computationally inexpensive for a given solute because explicit - solvent 
 model simulations are relegated to a precomputation step and because 
 it represents solvating waters in terms of a solute s free - energy 
 field . 
 in essence , field - sea approximates the physics of explicit - model 
 simulations within a computationally efficient framework . 
 a key finding 
 is that an atom s solvation shell inherits characteristics 
 of a neighboring atom , especially strongly charged neighbors . 
 field - sea 
 may be useful where there is a need for solvation free - energy computations 
 that are faster than explicit - solvent simulations and more accurate 
 than traditional implicit - solvent simulations for a wide range of 
 solutes . 
Introduction
Theory and Methods
Results and Discussion
Conclusions