A differential equation for the Generalized Born radii

This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 9783--9791 9783

Cite this: Phys. Chem.Chem.Phys.,2013,15, 9783

A differential equation for the Generalized Born radii

Federico Fogolari,*ab Alessandra Corazzaab and Gennaro Espositoab

The Generalized Born (GB) model offers a convenient way of representing electrostatics in complex

macromolecules like proteins or nucleic acids. The computation of atomic GB radii is currently performed

by different non-local approaches involving volume or surface integrals. Here we obtain a non-linear

second-order partial differential equation for the Generalized Born radius, which may be solved using

local iterative algorithms. The equation is derived under the assumption that the usual GB approximation

to the reaction field obeys Laplace’s equation. The equation admits as particular solutions the correct GB

radii for the sphere and the plane. The tests performed on a set of 55 different proteins show an overall

agreement with other reference GB models and ‘‘perfect’’ Poisson–Boltzmann based values.

1 Introduction

Electrostatics is of central importance for complex biomolecularsystems1,2 in solution. As electrostatic effects arise from the(correlated) interactions of a large number of solvent and soluteatoms, implicit treatments of solvent have been widely adoptedto treat electrostatics in solution. When the focus is on the soluteatoms, the potential of mean force, depending only on soluteatoms, takes implicitly into account solvent interactions.3 One of themost fundamental approaches involves the Poisson–Boltzmannequation4–6 although alternative and improved approacheshave been proposed.7

Although the solution of the Poisson–Boltzmann equationprovides the electrostatic potential in space, in many applica-tions Green’s function is sought rather than the potential. Anexample is the computation of pKa’s in proteins where theprotonation state of each titratable group is subjected to MonteCarlo moves and the interaction with all other titratable groupsin their ionization states is computed.8 Another area that wouldbenefit from the availability of Green’s function and its deriva-tives with respect to atomic coordinates is molecular dynamicssimulation. Forces obtained by numerical solutions of thePoisson–Boltzmann equation9 satisfy only approximately thethird principle of dynamics. The Generalized Born model hasbecome the choice of election for molecular dynamics simulationsof complex biomolecules.10–12 The many-body problem is encodedin a single atomic parameter (the Generalized Born radius) and an

analytical form of Green’s function has been proposed.13 Withaccurate computation of Born radii using the GBR6 model,14,15 thelatter form reproduces well the reference Green’s function obtainedby the repeated solutions of the Poisson–Boltzmann equation.15

Unfortunately since Born radii depend on the shape of themolecule which in turn depends on atomic coordinates, Green’sfunction is not easily differentiable and the expression for thederivatives of the potential of mean force can be obtained efficientlyonly through approximations.16–18

Atomic Generalized Born radii are currently obtained viavolume or, equivalently, surface integrals14,15,19,20 and, comparedto earlier approaches, the solvation energies corresponding tothe computed radii match very closely the solvation energiescomputed via the Poisson–Boltzmann equation, which isconsidered the reference standard. The radii that provide thePoisson–Boltzmann solvation energy when substituted in theBorn equation are referred to as ‘‘perfect’’ Born radii and offer abenchmark for Generalized Born radii computations.21

Although the surface integral expressions are based on reason-able approximations (like the Coulomb field approximation19) orexact results for spheres,14 the methods for computing GB radiiappear still largely empirical, implying often ad hoc correction terms.

Here we make a first attempt to use a more fundamentalapproach to GB radii computation by proposing an equation forthe self energy of a charge inside a complex biomolecularshape. As we detail hereafter this is far from being the ultimatesolution to the problem, but lays the ground for an alternativeway of computing Generalized Born radii. There are a fewnotable advantages in this approach: (1) it depends only onthe assumption on the form of Green’s function and it istherefore self-consistent; (2) the solution may be obtained bya local iterative algorithm, compared to (depending on theproblem) costly non-local integrals.

a Dipartimento di Scienze Mediche e Biologiche, Universita’ di Udine,

Piazzale Kolbe, 4 - 33100 Udine, Italy. E-mail: [email protected];

Fax: +39 0432 494301; Tel: +39 0432 494320b Istituto Nazionale Biostrutture e Biosistemi, Viale medaglie d’Oro,

305 - 00136 Roma, Italy

Received 18th March 2013,Accepted 11th April 2013

DOI: 10.1039/c3cp51174j

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The solutions obtained by solving the equation proposed inthis work agree reasonably with those computed by surfaceintegral and with ‘‘perfect’’ Born radii obtained by solving thePoisson–Boltzmann equation, showing that the approach isfeasible and calling for its further development.

Approximate solutions are proposed that are as accurate andfast as other state-of-the-art methods to compute GeneralizedBorn radii.

2 Methodology

We have recently suggested a differential equation for theGeneralized Born radius within a given shape and proved thatit is accurate for a sphere and for the space within two infiniteplanes.22 We discuss the approach in some detail hereafter.

2.1 Generalized Born model

In continuum electrostatic models any atomic charge qi atposition -

ri and embedded in the molecular low dielectricmedium surrounded by the solvent high dielectric medium isassociated with a reaction field U rf

i (-r ) that is due to polarization

charges at dielectric boundaries. In the following we neglect forthe sake of simplicity contributions from salts. The electricpotential within molecular boundaries is the sum of a directCoulomb potential:

UCouli ð~r Þ ¼ 1

4peine0

qi

j~r�~rij(1)

where ein and e0 are the solute relative dielectric constantand the vacuum absolute dielectric constant, respectively, andthe reaction field U rf

i (-r ) that obeys Laplace’s equation within

molecular boundaries:

Ui(-r ) = U Coul

i (-r ) + U rf

i (-r ) (2)

In Generalized Born models the reaction field due to chargeqi is approximated at the position -

rj of any other charge q j

using an analytical approximation that depends on the distance|-rj �

-ri| and on two parameters, the so-called Generalized

Born (GB) radii of the interacting charges. The GB radius ofcharge qi is the radius of a spherical particle possessing thesame charge and the same solvation energy of the charge qi

embedded in the molecule. The GB radii are computed byseveral different fast methods that aim at reproducing thesolvation energy computed using the reference Poisson–Boltzmann model.

The GB model approximation of the reaction field at theposition -

rj of charge q j is

Urf

i ~rj� �

¼ � 1

4pe0

1

ein� 1

eout

� �

� qiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~rj �~ri�� ��2 þ aiaj exp

� ~rj�~rik k2K 0aiaj

s (3)

where the two parameters ai and aj are the Generalized Bornradii of charge qi and charge qj, respectively, and parameter K0

is typically taken equal to 4.0 although a value of 8 has beensuggested in order to achieve better agreement with referencePoisson–Boltzmann calculations.23 A smaller value has beenused for improved pKa predictions.24 The hat (4) symbolindicates that this is an approximation to the exact U rf

i (-r ).

2.2 Derivation of a differential equation for GB radii

Note that once the boundary is defined the GB radius ak of chargeqk depends only on its position -rk and in general, for a givenboundary, we can define the function a(-

r ) that associates with eachposition within the boundary its Generalized Born radius. Inparticular a(-rk) = ak for all charges qk. This point is crucial to thefollowing discussion because we can consider now the charge qi asthe source charge and write the GB approximation to the reactionfield at the general position -

r within the molecular boundaries as:

Urf

i ð~rÞ ¼ �1

4p1

ein� 1

eout

� �

� qiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~r�~rik k2 þ aiað~rÞ exp

� ~r�~rik k2K 0aiað~rÞ

r (4)

where the dependence of U rfi (-

r ) on -r is both explicit and

implicit through the GB radius at position -r, as indicated bythe argument of the function a().

Since U rfi (-

r ) obeys the Laplace equation within the lowdielectric region of the solute molecule we assume that itsapproximation U rf

i (-r ) satisfies the same equation, i.e.:

r2U rfi (-

r ) = 0 (5)

By substituting eqn (4) in eqn (5) we obtain an equation fora(-r) that depends in general on the position -

ri of the sourcecharge qi. The equation provides the condition that a(-

r ) mustsatisfy for eqn (5) to hold. In order to obtain a self consistentequation for a(-r) we consider the same equation when -

r --ri

and a(-r) - a(-ri). In this limit:

Urf

i ð~rÞ �1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

~r�~rik k2 1� 1

K 0

� �þ aiað~r Þ

s

� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiaiað~r Þ

p 1� 1

2

~r�~rik k2 1� 1

K 0

� �aiað~r Þ

0BB@

1CCA

� ai�12að~r Þ�

12 � ai

�32að~r Þ�

321

2~r�~rik k2 1� 1

K 0

� �

(6)

The Laplacian of U rf assumes therefore the form of adifferential equation on a(-

r ). In particular the Laplacian ofthe first term in the above eqn (6) is:

ai�12

3

4að~r Þ�

52 ~ra�

~ra�

� 1

2að~r Þ�

32r2a

� �(7)

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The Laplacian of the second term of eqn (6) involves moreterms, but the only term surviving in the limit -

r --ri is the term

�ai�32að~r Þ�

321

2r2 ~r�~rik k2�

1� 1

K 0

� �(8)

Finally by letting -r --ri and therefore a(-r ) - ai, and multi-

plying the equation by 2a3 (a is different from zero inside theboundary) we obtain the following general non-linear partialdifferential equation (nl-PDE) for the Generalized Born radiusa(-r)

ar2a� 3

2~ra�

~ra�

þ 6K ¼ 0 (9)

where K depends on the choice of the parameter K0 through the

relation: K ¼ 1� 1

K 0. Note that this formulation indicates that

K (i.e. implicitly K0) plays the role of a scaling parameter for theBorn radii and therefore the choice of the parameter, whichaffects only pairwise interactions, also affects Born radii in thepresent approach because it affects the derivatives of thereaction field at the position of the same charge.

The accuracy of the solution obviously depends on theaccuracy of the approximation: U rf

i (-r ) E U rf

i (-r ). One may ques-

tion its applicability close to the source charge position, but itshould be considered that the approximation is also well-definedat the position of the source charge, so the issue might ratherconcern the derivatives of the above approximation close to theposition of the source charge instead of the approximation itself.It is likely that the approximation depends on the molecularshape. We considered a small protein with 774 atoms andcomputed the reaction field at all atomic positions due to a unitcharge at every single atomic position. The comparison of thesevalues with the reaction field obtained using 4 shows an excellentagreement with correlation coefficient 0.997 and fitting slope 0.96for K0 = 4 and 1.01 for K0 = 16. Thus the above approximationseems to hold at least for globular proteins.

2.3 Tests on systems with simple geometry

Eqn (9) is a non-linear second order partial differential equa-tion that is not easily tractable. Establishing the well-posednessof the problem goes beyond the scope of this communication,and it will be a subject for future investigations.

We consider in the following systems with simple geome-tries, where the general solution may be obtained, and later weresort to numerical solution for complex biomolecular shapeslike those of proteins. Note that for planes and spheres the Stillet al. formula provides the correct reaction field with theexponential replaced by 1, i.e. with K0 - N, whereas forproteins K0 = 4 is typically used. The boundary condition isthat the GB radius at the boundary be zero.

For the sphere the equation is radially symmetric andbecomes:

ad2adr2þ 2a

r

dadr� 3

2

dadr

� �2þ6K ¼ 0 (10)

It is easy to verify that the correct solution for a sphere with

radius R, aðrÞ ¼ R2 � r2

R, obtained e.g. by the image charge

method, satisfies the equation by direct substitution.For the plane eqn (9) reads:

ad2adx2� 3

2

dadx

� �2þ6K ¼ 0 (11)

where x is the distance from the boundary. Also here it is easy toverify by substitution that the correct solution a(r) = 2x satisfiesthe equation.

Both for the sphere and the plane the boundary condition isa = 0 as can be seen from the correct solution. We will assumethe same boundary condition for all subsequent analysis, asdiscussed in the next subsection.

2.4 Tests on protein systems

After checking that the equation leads to correct results for thesphere and the plane (a limiting case of the sphere), we assess theaccuracy of eqn (9), resting on the approximation U rf

i (-r ) E U rfi (-r ),

for the complex shapes of proteins.The boundary surface used in the finite difference solution

of eqn (9) is given by the solvent accessible surface defined, e.g.by Edelsbrunner and Koehl,25 as the locus of points which areaccessible to the center of a probe solvent sphere. The resultingGB radii are corrected afterwards (see below) in order to matchthe reference molecular surface dielectric boundary model.

Using the solvent accessible surface has the advantage of (i)being easily implemented within the program used to solve theequation; and (ii) avoiding narrow clefts for which the zeroboundary condition might not be appropriate.

Eqn (9) is solved numerically by discretizing the problemon a grid with spacing 0.5 Å for proteins (resulting in anaverage number of grid points equal to 95 � 95 � 95 for thechosen protein test set) and turning the equation in a finitedifferences equation which reads at each node as a seconddegree equation for ai,j,k with coefficients determined by thevalues at neighbouring nodes:

�ai;j;k26

h2þai;j;k

aiþ1;j;kþai�1;j;kþai;jþ1;kþai;j�1;kþai;j;kþ1þai;j;k�1h2

�3

2

aiþ1;j;k�ai�1;j;k� �2þ ai;jþ1;k�ai;j�1;k

� �2þ ai;j;kþ1�ai;j;k�1� �2

4h2

þ6K ¼ 0 (12)

The equation is then solved iteratively by taking the positiveroot of the above equation and assigning it to the radius atnode i,j,k. As soon as a novel estimation is available it is used inthe same iteration. When the relative change in the estimatesof the radius at one node is less than a threshold value (10�8 inthe present implementation) that node is excluded from furtheriterations.

Boundary conditions are imposed by setting all Born radiioutside the low dielectric region of the solute to zero. Thisamounts to assuming a grounded conducting boundary surfacewhich may be a good approximation for convex shapes, but may

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9786 Phys. Chem. Chem. Phys., 2013, 15, 9783--9791 This journal is c the Owner Societies 2013

provide a very poor approximation for complex molecularshapes. In particular any narrow cleft will be assigned zeroGB radius, and thus nearby atoms will be assigned small GBradii, independent of the width of the cleft. Another potentialproblem could be related to small inner cavities that couldresult in artifactually low Generalized Born radii for the atomssurrounding the cavity.

A boundary square surface element is assigned between twogrid points belonging to the outer region and to the innerregion, respectively. The square surface element is orthogonalto the line connecting the two grid elements (Fig. 1).

The parameter K in eqn (9), which is set to 1.0 in thefollowing, is expected to depend on the position of the pointand on the shape of the molecule as well because it is knownthat it is 1.0 for a sphere and there are suggestions that 0.75better describes interactions within proteins. We would thusexpect that Born radii computed according to eqn (9) should

overestimate Born radii by a factor 1:15 ¼ 1ffiffiffiffiffiffiffiffiffi0:75p

� �for proteins.

On the other hand setting to zero the Born radius at theboundary strongly overestimates solvation, thus reducing thecomputed Born radii.

After eqn (9) has been solved numerically on the grid a valuesare interpolated at atomic positions using tri-linear interpolationand compared with the corresponding values obtained by surfaceintegral or by Poisson–Boltzmann calculations.

In order to compare the results with the reference ‘‘perfectradii’’ obtained by Poisson–Boltzmann calculations that use themolecular surface as the dielectric boundary instead of thesolvent accessible surface, the GB radii are corrected using acubic function whose coefficients are best fitted to ‘‘perfectradii’’. We will refer to this procedure for solving the equationand obtaining the estimated GB radii as the PDE_fd_0.5 model.

2.5 Fast approximations to the solution of the nl-PDE forGB radii

The finite-difference solution of the equation used in thepreliminary tests is not practical because it requires a largegrid, and it is not straightforward to include the molecular

surface as a dielectric boundary. One simple way to turn themethod into a fast method is to use a large mesh (1.0 Å) and a lessstringent convergence criterion (10�3 Å). With this parameters’choice (which will be referred to as the PDE_fd_1.0 model), theexecution time is similar to other fast approximated methods.

A drawback of eqn (9) is that since the boundary conditionimposes that the GB radius be zero at the boundary, innercrevices or cavities result in artifactually low GB radii atneighbouring atomic positions. To face the problem and tomake calculations faster we sought an approximation to thesolution of the equation only at atomic positions (where theatomic radius must be estimated). For this purpose we estimateboth the Laplacian operator and the gradient based only on theavailable GB radii at neighbouring atomic or surface positions.This way of proceeding has the advantage of strongly reducingthe calculations and it results in a less abrupt change in GBradii for atomic positions close to internal crevices or cavities.

An atom is listed as neighbour of a reference atom whentheir distance is less than the sum of their van der Waals radiiplus 2.0 Å. All surface points (whose GB radius is by definitionzero) of neighbouring atoms are considered neighbours. Thealgorithm proceeds iteratively:

Step (1) assign each GB radius as the van der Waals radius;Step (2) estimate the gradient and the Laplacian of the GB

radius from the GB radius to be updated at the referenceposition and at neighbouring points:

~ra � 1

n

Xi¼1;n

a ~rið Þ � a ~r0ð Þð Þ~ri �~r0k k

~ri �~r0~ri �~r0k k (13)

Da � 6

n

Xi¼1;n

a ~rið Þ � a ~r0ð Þ~ri �~r0k k2

(14)

where n is the number of neighbouring points.When we substitute the above equations in eqn (9) we obtain

a second degree equation for a(-r0), which can be obtained andupdated straightforwardly;

Step (3) iterate Step (2) until convergence, i.e. when theabsolute difference of the update with respect to the previousvalue is less than 0.001.

The dielectric boundary used here is the molecular surfacegenerated by the program MSMS,26 at very low surface pointdensity (1 point per 10 Å2), for computational efficiency.

Due to the rather large local environment in which thederivatives are approximated the computed radii are correctedusing a cubic function whose coefficients are best-fitted to‘‘perfect radii’’. We will refer to this procedure for solving theequation and obtaining the estimated GB radii as the PDE_fda(finite difference approximation) model.

2.6 Reference Poisson–Boltzmann calculations

Poisson–Boltzmann calculations have been performed usingthe program APBS27 and the program UHBD.28 The solventprobe radius was set to 1.4 Å. The ionic strength was set to0.0 M. For all 93 240 atoms in the dataset of 55 proteins describedbelow ‘‘perfect’’ Generalized Born radii were computed from

Fig. 1 Grid representation of the boundary surface shown in two dimensions.The shaded area represents the volume within the dielectric boundary. Thethicker line represents the grid and the thickest lines are for the boundarysurface elements.

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the computed self-solvation energy. In practice all atom chargesare set to zero except that of the atom whose GB radius is beingcomputed, which is set to 1 protonic charge. The solvationenergy is computed using the grid-based method reported byLuty et al.29 which amounts to subtracting the total electrostaticgrid energy in vacuo from that in solution, using exactly thesame grid.

The program UHBD was used with a grid of 160 � 160 � 160nodes with a variable spacing ranging from 0.48 to 0.75 Å. Thespacing was chosen in order to have at least 30 Å margin fromthe bounding box for all atoms of the protein. The number ofsurface points per atom is 500. For the molecular surface thekeyword nmap was used with a solvent probe radius equalto 1.4 Å. Dielectric boundary smoothing was applied.30 Theboundary conditions were set to 0.0 at the bounding box. Theconvergences criterion was set to 1 � 10�5.

Similar parameters were used in the program APBS forcomputing solvation energies.

2.7 Reference Generalized Born models

The results obtained by the present approach are comparedwith existing methods for computing Generalized Born radii(GBR) which differ by the theoretical model they are based onand by the approximations involved in the computation. Forthe fastest methods accuracy is decreased as a tradeoff forspeed. The methods considered here are:

(1) GBR_OBC model – GBR model based on the Onufriev,Bashford and Case volume integral approximation as output byGROMACS after suitable modification of the code.

(2) GBR6_si model – GBR6 based on the Grycuk approachand calculated exactly by surface integral. This is equivalent tothe GBR6 volume integral version. 500 surface points per atomwere used to define the surface for better accuracy.

(3) GBR6_si_ld model – as GBR6_si model but using a verylow surface point density (1 point per 10 Å2).

(4) GBR6_avi – GBR6 based on the Grycuk approach butcomputed according to the approximations to the volumeintegral implemented in the program GBR6 by Tjong andZhou,15 available from URL: http://gbr6.sourceforge.net/. Thisis much faster than the surface integral method (similar toGBR_OBC) but less accurate, as far as GB radii are concerned,than the GBR6_si model. The target dielectric boundary is thevan der Waals dielectric boundary. For this reason the GB radiiwill not be compared with the ones computed using themolecular surface as the dielectric boundary. Note that withsimple suitable linear corrections it can however estimateaccurately total solvation energies for proteins.

The issue of the dielectric boundary surface to be used incontinuum electrostatic calculations has been recentlyaddressed by Pang and Zhou31 and Rocchia and coworkers.32

We adopt here the choice of the molecular surface for thereference Poisson–Boltzmann calculations because it is widelyused and it is theoretically well justified. Other choices implyingoften correction on the results are equally legitimate dependingon the application of the calculation itself.31

2.8 Protein test set

We consider a set of 55 different proteins that have beenpreviously used as a test set by Tjong and Zhou.15 This setincludes proteins with different lengths, shapes thus providinga rather general test set.

3 Results3.1 Convergence of finite difference solution

The peculiar nonlinearity of eqn (9), and in particular the square ofthe gradient of a, prevents the application of solution methods thathave been used for solving other linear and non-linear equations,notably the Poisson–Boltzmann equation.33 The solution methodused here was chosen because it is very easy to implement, thusallowing fast exploration of the features of the proposed equation.Other methods of solution could be in principle applied, like finiteelement methods.

Several tests have been performed on the influence of initialconditions on the convergence of the solution. Differentschemes were implemented where all inner grid nodes (1) wereassigned the same random value (ranging from 0 to 6 Å); (2)were assigned different random values (ranging from 0 to 6 Å);(3) were assigned a value dependent on the distance from thecenter of geometry of the system, mimicking charges embedded ina sphere. The latter choice was adopted in further tests. With ad hocreassignment of values at initial steps when the determinant ofeqn (12) was less than zero or the computed values were less thanthe nodes’ spacing, the solutions always converged at the samevalues, with an average root mean square deviation of 0.04 Å. Withthe convergence criterion adopted here, i.e. a node value is notsubjected to further change when the variation is less than 10�8 Å,and an average grid size of about 100 � 100 � 100, the timeneeded to solve the equation on a protein of 3000 atoms is ca. 30 son a laptop. When the grid mesh is set to 1.0 Å with an averagegrid size of 50 � 50 � 50 the computational time is less than 1 s.More efficient resolution methods will be considered in futuredevelopments.

3.2 Test on proteins: Generalized Born radii

As a test of the equations the Generalized Born radii have beencomputed for all the atoms of the 55 protein test set usingeqn (9) and using the solvent accessible surface as the dielectricboundary (i.e. the van der Waals surface inflated by the solventprobe radius taken here as 1.4 Å) and then applying a simplecubic correction (PDE_fd_0.5 model, see Methods section). Theresulting radii were compared with the corresponding radiicomputed by solving the Poisson–Boltzmann equation usingthe molecular surface dielectric boundary definition.

Note that each ‘‘perfect’’ radius is obtained by solvingthe Poisson–Boltzmann equation in a single calculation andtherefore the calculation is quite time consuming compared tothe single calculation of the method proposed here. Thesolution of eqn (9) after suitable correction provides radii closeto ‘‘perfect’’ radii, with a correlation coefficient of 0.979 and a rootmean square difference of 0.51 Å. The agreement of computed

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versus perfect radii is overall very good as can be appreciated inFig. 2. The good agreement is maintained when the grid meshis enlarged to 1.0 Å with a correlation coefficient of 0.963 and aroot mean square difference of 0.67 Å (Fig. 4). Deviations affectin a similar way the whole radius range. There is a number ofoutlier values which is however very small compared to the93 240 points displayed in the figure. The regions with thelarger deviations correspond to internal clefts where the veryconcept of a dielectric constant is not well defined.

3.3 Test on proteins: solvation energies

Based on the computed GB radii solvation energies have beencomputed using the equation:

DGsolv ¼Xi;j

DGsolv;ij (15)

with

DGsolv;ij ¼ �1

8pe0

1

ein� 1

eout

� �qiqjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rij2 þ aiaj exp�rij24aiaj

r (16)

The solvation energies computed in this way appear to beunderestimated with respect to the reference Poisson–Boltzmanncomputations. For this reason a simple scaling factor (1.05) isapplied in order to better reproduce reference values.

With this correction the agreement with Poisson–Boltzmanncomputed solvation energies is very good, notwithstanding thedifferent treatment of dielectric boundaries. Using the solventaccessible surface and correcting afterwards the computedradii and solvation energy thus provide an effective procedure.The correlation coefficient between the solvation energiescomputed using the present approach and by solving the PBequation is 0.994 and the average root mean square erroramounts to less than 9% of the average absolute value of thesolvation energy (Fig. 3). Note that we are comparing absolutesolvation energies, rather than binding solvation energies, andtherefore absolute errors do not cancel out in comparison.

3.4 Comparison with other methods

The methods for computation of Generalized Born radii mustbe fast to be practical. A method based on the solution of adifferential equation is obviously not fast unless a coarse meshis employed or approximations are used. For this reason weconsider here only those methods based on eqn (9) that run asfast as widely used methods. In particular we consider the finitedifference solution using a mesh of 1 Å and the fast approxi-mation described in the Methods section. With both thesemethods the average execution time on the protein test set isca. 1.0 second per protein on a laptop. In this respect eqn (9)provides an alternative practical route to the computation ofGeneralized Born radii.

Here we compare the results with other available methods,namely the reference Onufriev, Bashford and Case model34

as implemented in the software Gromacs35 and the surfaceintegral GBR6 model22 using the molecular surface generatedwith very low density (1 point every 10 Å2) in such a way that theaverage running time is around 1.0 second per protein compar-able to other methods.

We do not compare here our results with the resultsobtained with the volume integral approximation of the GBR6model as implemented in the GBR6 program15 because thetarget radii of the program are for the van der Waals dielectricboundary rather than for the molecular surface dielectricboundary, which is the reference here for Poisson–Boltzmanncalculations.

The results compare well with the most accurate GBR6_simethod and its fast, low surface point density, versionGBR6_si_ld. The main difference is that for both methodsbased on eqn (9) corrections (similar to the GBR_OBC model)based on a cubic function of the computed radii are necessary,whereas for the GBR6_si models no correction is needed.

It is apparent from Fig. 4 that the accuracy of both thePDE_fda and the PDE_fd_1.0 models is lesser but comparable

Fig. 2 Born radii computed solving eqn (9) using 0.5 Å grid mesh spacing versus‘‘perfect’’ radii computed solving the Poisson–Boltzmann equation for all 93 240atoms in the 55 protein test set (see the text for details).

Fig. 3 Electrostatic solvation energies obtained by the PDE_fd_0.5 model versusthe reference Poisson–Boltzmann ones.

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to that of the GBR6_si model. The GBR_OBC model appears toreproduce well ‘‘perfect radii’’ at short radii, but gives poorprecision at larger radii.

The usage of a cubic function for the PDE_fda model leadsoccasionally to radii smaller than the atomic radii, and for thisreason the corrected value is reset to the atomic radius, afeature apparent in Fig. 4. Some of the differences may alsobe ascribed to differences in the dielectric boundary surfacedefinition. Indeed the differences observed for the GBR6_simodel are not observed in the same plot using the solventaccessible surface as the dielectric boundary for both PB andGBR6_si calculations.

In Table 1 the correlation coefficients, residual and averagerelative error after linear fitting are reported. As can be seen the

fast methods based on eqn (9) perform as well or better thanother fast methods.

A comparison is due with the GBR6_avi method, because it hasbeen reported to reproduce accurately binding solvation energies.

Fig. 4 Comparison of ‘‘fast’’ GB radii calculation methods. In the upper row results obtained by the methods based on eqn (9). Finite difference approximation(PDE_fda) (left panel) and finite difference solution using a mesh size of 1.0 Å (PDE_fd_1.0) (right panel). In the lower part results obtained by the Onufriev, Bashford,Case (OBC) model (left panel) and the low surface point density GBR6 surface integral (GBR6_si_ld) model (right panel). See the text.

Table 1 GB model versus reference PB model

Model

GBR vs. ‘‘perfect radii’’ GB vs. PB solv. en.

CPUtime (s)

Corr.coeff.

RMSD(Å)

Av. rel.err.

Corr.coeff.

RMSD(kJ mol�1)

Av. rel.err.

PDE_fd_0.5 0.979 0.51 0.11 0.994 589 0.09 18.5PDE_fd_1.0 0.964 0.67 0.13 0.991 677 0.10 1.2PDE_fda 0.964 0.66 0.13 0.932 1883 0.15 1.0OBC 0.665 1.88 0.42 0.983 956 0.16 0.15GBR6_si_ld 0.965 0.66 0.14 0.937 1815 0.14 0.76

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9790 Phys. Chem. Chem. Phys., 2013, 15, 9783--9791 This journal is c the Owner Societies 2013

As mentioned above, the target GB radii of the model arethose for the van der Waals surface dielectric boundary.When the same radii (i.e. assuming the van der Waals surfaceas the dielectric boundary) are computed using the Poisson–Boltzmann equation there is an apparent underestimation ofthe GB radii by both the GBR6_avi method and the PDE_fd_0.5method (assuming the van der Waals surface as the dielectricboundary). The agreement with the ‘‘perfect radii’’ is poorbecause the range of GB radii computed by the GBR6_avimethod and the PDE_fd_0.5 is rather small. Both approximatemethods show a similar trend although correlation between thetwo is not very high (0.769) with a root mean square deviationof 0.3 Å. On the other hand, because the disagreement mostlyinvolves the largest radii, the agreement with the Poisson–Boltzmann computed solvation energies is very good, with acorrelation coefficient 0.995 and a root mean square deviationafter linear fitting of 664 kJ mol�1 for the GBR6_avi model anda correlation coefficient 0.996 for the PDE_fd_0.5 model with aroot mean square deviation of 618 kJ mol�1. The solvationenergies computed using the two models display very similarbehaviour with average root mean square deviation (358 kJ mol�1)lesser than the respective ones from PB solvation energies. Thecorrelation coefficient between the two models is 0.999.

Note that the comparison may not be completely fairbecause here we use a different set of charges than those usedfor optimization of the GBR6_avi model,15 but it illustrates thepoint that the two approaches lead to very similar results.

4 Conclusions

The Generalized Born model is a practical approach to thesimulation of electrostatic solvation effects in macromolecules.Born radii are mostly computed via volume or surface integralswith approximations and heuristic corrections. Here we haveshown that it is possible to write a differential equationconsistent with the functional form chosen in GeneralizedBorn models for the reaction field. The solution of this equa-tion agrees well with the results obtained from the Poisson–Boltzmann equation.

The approach has some potential advantages over standardBorn radii computation because the slow variation of the Bornradii in the inner part of the molecules could be exploited in thenumerical solution of the equation. The same equation, restingonly on the assumption on the functional form of the reactionfield, can be used for any molecular shape. The kind of non-linearity of the equation prevents the application of well-knownnumerical resolution methods, nevertheless, the solutionobtained in this work by the straightforward finite differencesolution of the equation (or by approximations thereof) showsthat the approach is computationally competitive with surfaceor volume integral computations of Generalized Born radii.

In its present form the approach is suitable for fast solvationenergy calculations, but not for force calculation because GBradii are not computed as a function of pairwise contributions.Further work to turn the solution of eqn (9) in a function of

pairwise distances will be needed to use the approach for forcecalculations.

Acknowledgements

Prof. R. Vermiglio is gratefully acknowledged for helpful dis-cussion and suggestions. Dr I. M. Lait is acknowledged forvaluable support.

References

1 M. E. Davis and J. A. McCammon, Chem. Rev., 1990, 90, 509–521.

2 B. Honig and A. Nicholls, Science, 1995, 268, 1144–1149.3 T. Hill, An introduction to statistical Mechanics, Dover Pub-

lications, New York, 1956.4 F. Fogolari, A. Brigo and H. Molinari, J. Mol. Recognit., 2002,

15, 377–392.5 M. T. Neves-Petersen and S. B. Petersen, Biotechnol. Annu.

Rev., 2003, 9, 315–395.6 N. A. Baker, Curr. Opin. Struct. Biol., 2005, 15, 137–143.7 P. Koehl, H. Orland and M. Delarue, Phys. Rev. Lett., 2009,

102, 087801.8 J. Antosiewicz, J. A. McCammon and M. K. Gilson, J. Mol.

Biol., 1994, 238, 415–436.9 M. K. Gilson, M. E. Davis, B. A. Luty and J. A. McCammon,

J. Phys. Chem., 1993, 97, 3591–3600.10 D. Bashford and D. A. Case, Annu. Rev. Phys. Chem., 2000,

51, 129–152.11 P. Koehl, Curr. Opin. Struct. Biol., 2006, 16, 142–151.12 J. Chen, C. L. Brooks and J. Khandogin, Curr. Opin. Struct.

Biol., 2008, 18, 140–148.13 W. C. Still, A. Tempczyk, R. C. Hawley and T. Hendrickson,

J. Am. Chem. Soc., 1990, 112, 6127–6129.14 T. Grycuk, J. Chem. Phys., 2003, 119, 4817–4826.15 H. Tjong and H. X. Zhou, J. Phys. Chem., 2007, 111,

3055–3061.16 G. D. Hawkins, C. J. Cramer and D. G. Truhlar, J. Phys.

Chem., 1996, 100, 19824–19839.17 A. Onufriev, D. Bashford and D. A. Case, J. Phys. Chem.,

2000, 104, 3712–3720.18 E. Gallicchio and R. M. Levy, J. Comput. Chem., 2004, 25,

479–499.19 A. Ghosh, C. S. Rapp and R. A. Friesner, J. Phys. Chem. B,

1998, 102, 10983–10990.20 J. Mongan, W. A. Svrcek-Seiler and A. Onufriev, J. Chem.

Phys., 2007, 127, 185101.21 A. Onufriev, D. A. Case and D. Bashford, J. Comput. Chem.,

2002, 23, 1297–1304.22 F. Fogolari, A. Corazza, V. Yarra, A. Jalaru and G. Esposito,

BMC Bioinf., 2012, 13(suppl. 4), S18.23 M. S. Lee, M. Feig, F. R. Salsbury and C. L. Brooks,

J. Comput. Chem., 2003, 24, 1348–1356.24 B. Jayaram, Y. Liu and D. L. Beveridge, J. Chem. Phys., 1998,

109, 1465–1471.

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15 2

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View Article Online

This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 9783--9791 9791

25 H. Edelsbrunner and P. Koehl, in Combinatorial and Com-putational Geometry, ed. J. E. Goodman, J. Pach andE. Welzl, Cambridge University Press, 2005, pp. 243–275.

26 M. Sanner, J.-C. Spehner and A. Olson, Biopolymers, 1996,38, 305–320.

27 N. A. Baker, D. Sept, S. Joseph, M. J. Holst andJ. A. McCammon, Proc. Natl. Acad. Sci. U. S. A., 2001, 98,10037–10041.

28 J. D. Madura, J. M. Briggs, R. Wade, M. E. Davis, B. A. Luty,A. Ilin, J. A. Antosiewicz, M. K. Gilson, B. Bagheri, L. R. Scott andJ. A. McCammon, Comput. Phys. Commun., 1995, 91, 57–95.

29 B. A. Luty, M. E. Davis and J. A. McCammon, J. Comput.Chem., 1992, 13, 768–771.

30 M. E. Davis and J. A. McCammon, J. Comput. Chem., 1991,12, 909–912.

31 X. Pang and H.-X. Zhou, Comput. Phys. Commun., 2013, 13,1–12.

32 S. Decherchi, J. Colmenares, C. E. Catalano, M. Spagnuolo,E. Alexov and W. Rocchia, Comput. Phys. Commun., 2013, 13,61–89.

33 F. Fogolari, A. Brigo and H. Molinari, Biophys. J., 2003, 85,159–166.

34 A. Onufriev, D. Bashford and D. A. Case, Proteins: Struct.,Funct., Genet., 2004, 55, 383–394.

35 H. J. C. Berendsen, D. van der Spoel and R. van Drunen,Comput. Phys. Commun., 1995, 91, 43–56.

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