Volume 13 Supplement 4
Bluues: a program for the analysis of the electrostatic properties of proteins based on generalized Born radii
© Fogolari et al.; licensee BioMed Central Ltd. 2012
Published: 28 March 2012
The Poisson-Boltzmann (PB) equation and its linear approximation have been widely used to describe biomolecular electrostatics. Generalized Born (GB) models offer a convenient computational approximation for the more fundamental approach based on the Poisson-Boltzmann equation, and allows estimation of pairwise contributions to electrostatic effects in the molecular context.
We have implemented in a single program most common analyses of the electrostatic properties of proteins. The program first computes generalized Born radii, via a surface integral and then it uses generalized Born radii (using a finite radius test particle) to perform electrostic analyses. In particular the ouput of the program entails, depending on user's requirement:
1) the generalized Born radius of each atom;
2) the electrostatic solvation free energy;
3) the electrostatic forces on each atom (currently in a dvelopmental stage);
4) the pH-dependent properties (total charge and pH-dependent free energy of folding in the pH range -2 to 18;
5) the pKa of all ionizable groups;
6) the electrostatic potential at the surface of the molecule;
7) the electrostatic potential in a volume surrounding the molecule;
Although at the expense of limited flexibility the program provides most common analyses with requirement of a single input file in PQR format. The results obtained are comparable to those obtained using state-of-the-art Poisson-Boltzmann solvers. A Linux executable with example input and output files is provided as supplementary material.
Generalized Born models
Electrostatic effects arising due to the interaction of solute charges among themselevs and with solvent and ion charges, are of utmost importance for biomolecular structure and function. Continuum methods based on the Poisson-Boltzmann equation have been widely used for calculating electrostatic effects [1–5]. In the last decades much interest has been devoted to developing fast approximations to the solution of the Poisson-Boltzmann (PB) equation.
Onufriev and coworkers have developed an analytical approximation to the exact potential inside and outside the low dielectric region of a sphere , that performs surprisingly well also for the complex shape of proteins [7–10] and is therefore more general than the generalized Born (GB) models.
When only self- and interaction energies and forces are sought, the most widely used approach is based on generalized Born (GB) models. Recent reviews summarize the approach and highlight most interesting recent results [3, 11–14].
Central to these models is the estimation of polarization charge contributions to: i) the self-energy of each charge (embedded in the solute); ii) the interaction energy of each pair of charges.
This expression was found to be more accurate than the exact expression for two charges embedded in a conducting sphere, although the coefficient of 8.0 instead of 4.0 in the exponential has been suggested  and other similar forms have been proposed [17, 18].
Computation of Born radii
Born radii could be in principle computed by solving a system of linear equations for the polarization charges at the boundaries of the solute volume [19–25]. Under the approximation that the ionic solution provides complete screening, amounting to the assumption that the surface behaves like a grounded conductor, polarization charges at the surface are such that the integral of their electric field at any outer surface point is exactly the opposite of the source charge field. This approximation amounts to setting ε out = ∞. In a different context the same approximation has been proposed many years ago as the conducting surface model (COSMO) [26, 27] and successfully used since then. Under this condition it is possible to obtain simple formulae for the generalized Born radius for a low dielectric region delimited by a sphere or a plane. The reaction field satisfies Laplace equation inside the surface and the solution may be obtained solving the equation using suitable basis functions with Dirichlet boundary conditions at discretized points on the surface .
where is the unit vector normal to the surface at point and pointing outwards.
The above expression which will be referred hereafter as GBR6 following Grycuk  and Tjong and Zhou [31, 32], has been analysed in detail by Mongan et al. . Equation 7 was found to perform extremely well also for "very" non spherical shapes and in the context of biomolecular models. Occasional large differences with respect to Poisson-Boltzmann calculations were found for inner cavities and local concavities at the surface, i.e. in conditions where a continuum model is anyway questionable. That study concluded that with a correction for a small systematic error, the GBR6 model is a sufficiently accurate continuum electrostatic model .
Tjong and Zhou [31, 32] used an analytical implementation for the estimation of Born radii based on volume integrals (corresponding to the above formula (7) that uses a surface integral instead) and showed its superior accuracy compared to other existing methods for a set of 55 very different proteins. In their approach it is made clear that standard estimations of Born radii compute in fact only geometric properties, and they provide empirical formulae for the correct Born energy depending on the inner and outer dielectric constants, ionic strength, total charge and number of atoms.
In the absence of a theory which could be cast in a fast computational framework, fitting approaches have been successfully followed and tested on large sets of proteins showing excellent agreement with Poisson-Boltzmann calculation results. Romanov et al.  proposed that a linear combination of integrals I3 to I6, in the present notation, and a constant term could fit the self-polarization energy and thus be used to compute generalized Born radii (see also the discussion by Mongan et al. ).
Applications of the generalized Born model
Based on the correct estimation of Born radii, the solvation contribution to the interaction between any two charges may be computed by using equation (2). Derivation of equation (2) with respect to atomic positions gives the electrostatic solvation forces acting on atoms. The implicit dependence of Born radii on atomic positions makes computation of forces far from trivial [35–40] for the generalized Born model as well as for the parent Poisson-Boltzmann model where the boundaries depend on atomic positions . The possibility of computing energies and forces faster with respect to the reference Poisson-Boltzmann equation has made the Generalized Born model the choice of election for implicit solvent molecular dynamics simulations. Also, the computation of pairwise solvation energies allows for fast computation of pKa of multiple titrating groups as we discuss in the Methods section.
Applications of generalized Born model (and other implicit solvent methods) have been reviewed elsewhere [3, 11–14]. At variance with the reference Poisson-Boltzmann model, the computation of electrostatic potential in space is outside the scope of the Generalized Born model where only interactions are considered thorugh equation 2. Here we use a finite radius test charge in order to define a potential within the frame of the generalized Born model.
Aim of this work
the generalized Born radius of each atom;
the solvation electrostatic free energy;
the electrostatic forces on each atom (the theory of electrostatic forces based on surface integrals is developed here and implementation is still at a developmental stage);
the pH-dependent properties (total charge and pH-dependent free energy of folding in the pH range -2 to 18);
the pKa of all ionizable groups;
the electrostatic potential at the surface of the molecule;
the electrostatic potential in a volume surrounding the molecule.
Results and discussion
Generalized Born radii
Electrostatic solvation free energy
Electrostatic solvation forces
Since solvation energies depend, according to equation 2, on atomic positions explicitly and implicitly through Born radii, the computation of solvation electrostatic forces is quite complex, and strongly dependent on the interface model chosen [38–40] (see also for a general discussion ).
In order to estimate how important are effects due to the dependence of Born radii on atomic positions, solvation electrostatic forces have been computed for each atom of the 55 proteins as the derivative of the solvation energy under the approximation that Born radii are constant. Albeit approximate this way of computing electrostatic forces preserve by definition zero total electrostatic force as expected for isotropic media and as found by the correct expression for the force . Due to the different way of computing forces we did not attempt comparing ionic boundary, dielectric boundary and charge times electrostatic field components of the electrostatic force, but we rather compared the total solvation force. The results are not very accurate with the average root mean square deviation, with respect to the forces computed using APBS using the same parameters, equal to 2.7 kJ/(Åmol) compared to an average square root value of the force of 5.0 kJ/(Åmol). The correlation coefficient is 0.84.
pH-dependent properties (total charge, pH-dependent free energy of folding, and pKa of ionizable groups
The test set provided by Gunner and coworkers has been used to test the prediction of pKa's in proteins . We compared the results obtained with the results obtained by running the program Propka2.0 [44–47]. The latter program is very fast and provides predictions whose accuracy is comparable to that of more computationally intensive programs. We chose this program as a reference because it is widely used and it is seemingly the fastest available with a very good accuracy. A wide range of programs and methods have been compared recently  and the reader is referred to that work for more extensive comparisons of existing softwares.
Predicted vs. experimental pKa shifts
Propka v. 2.0
On the other hand some scaling of contributions is done and therefore large deviations from experimental results are not found. In this respect, Propka that uses ten adjustable parameters and other parameters chosen for best performance  apparently does not prevent very large shifts to be predicted. As a consequence the global RMSD from experimental data is large and could thus easily be reduced. The execution time of Propka is in the range of seconds, while our program runs in minutes. No optimization or approximation has been implemented as yet.
Surface electrostatic potential
The evaluation of the potential in regions outside the molecule is beyond the scope of the generalized Born approach which focuses instead on self and interaction energies. This is at variance with approaches that approximate the potential computed using the Poisson-Boltzmann equation. In particular Onufriev and coworkers [7, 8] have found an approximation (not simply amounting to a truncation) based on Kirkwood's series expansion of the solution for a sphere . Their analytical model performs surprisingly well for a large set of proteins and enables fast calculation of the potential at the surface and in the volume surrounding the molecule.
Electrostatic potential in a volume surrounding the molecule
A program for the analysis of the electrostatic properties of proteins based on the surface integral computation of generalized Born radii has been presented. Further work will be devoted to improve the efficiency of calculations in particular for what concerns electrostatic solvation forces. The program, together with examples is given as supplementary material (Additional file 1).
Reference APBS and UHBD calculations and parameters
Reference electrostatic calculations were performed with the programs APBS  and UHBD [54, 55]. Except where noted, standard parameters were used: inner dielectric constant 1.0, outer dielectric constant 78.54, temperature 298.15 K, ionic strength 0.150 M. In APBS the mesh of the grid was 0.25 Å which resulted in very large grid size (2573 points). In UHBD, used for surface potential calculations, a focusing procedure was used with the final mesh size of 0.8 Å.
The boundary surface was defined in the two programs in order to match as close as possible the definition used in our program, i.e. the molecular surface for atoms with radii inflated by the radius of the solvent. The surface point density was 10 Å-2.
Except where noted the same temperature, dielectric constant and ionic strength were used in our program.
Solvent accessible surface
Solvent accessible surface points and surface normal vectors have been generated by considering the van der Waals sphere of each atom inflated by the radius of the solvent (e.g., for an atom of radius 1.9Å and a solvent radius of 1.4Å the inflated radius was 3.3Å).
n pt was 200 corresponding for an atom with radius 1.9Å to a density of 4.4 pts Å-2.
Atoms are mapped on a grid and the position of each surface point is compared only with the position of the list of atoms associated with neighboring grid points. This procedure was found to be efficient and robust without any failure.
Each surface point within the inflated van der Waals volume of a different atom is considered buried. All surface integrals have been performed as the corresponding finite sums over exposed (non buried) surface points.
Generalized Born radii
where I n is defined by equation 8, or by fitting the reference solvation self energy by linear combinations of the solvation energies corresponding to different α n 's (n ranging here from 3 to 6) as done by Romanov et al. . Fitting was necessary because the published coefficients did not provide good results, as a result of the different procedures used to compute surfaces and solvation free energies.
An equation for the generalized Born radius
where the constant K depends on the coefficient used in the exponential and is 0.75 for the commonly adopted Still formula.
For the conducting grounded surface approximation the boundary condition is that the Born radius approaches 0 as the boundary surface is approached. It is easy to check that this equation is satisfied for the sphere and the plane where K is equal 1.
where J0(x) is the Bessel function of order zero. We may consider only order 0 functions due to cylindrical symmetry. The coefficients are obtained by imposing that at the boundary the potential (U react + U source ) is equal to zero and using the orthogonality relation: . The parameters k are chosen such that J0(kR) = 0.
The solution of equation (10) is obtained by bisection, guessing first a value at the midpoint of the axis of symmetry and integrating the equation using an adaptive Runge-Kutta fourth order method  and requiring that the value of the Born radius at the boundary be 0.
Electrostatic solvation energies in ionic solutions
where the functions ϕi,jare the Green's functions at points i and j but do not include the self potential of each charge in the inner dielectric medium and δi,jis Kronecker's delta.
Electrostatic solvation forces
For a unit charge placed at 7.5 Å from the center of a globular protein of radius 10 Å, assuming an inner dielectric constant 4.0, the resulting force is 14 kJ/(mol Å). An opposite force of the same magnitude is applied on the center of the sphere defining the boundaries. This simple example shows that forces arising from variations of Born radii are not negligible.
The derivative of is not straightforward as the domain of integration depends on atomic coordinates.
Note that in the above equations the operator operates only on r S coordinates, i.e. at the point where the integrand function is evaluated.
where the apices i and j in I5 means that the integral involves the vector from the surface points to the coordinates of atoms i and j respectively. The computation of forces via surface integral according to the above equations is not practical, unless a cutoff on the distance between pairs of atoms and surface points is used. Indeed for each of the pairs of atoms j and i, where N is the number of atoms, surface integrals must be computed (see equation (21)). This is the result of the brute force application of our surface integral definition of Born radius. Work is under way in our laboratory to make this computation efficient.
pH-dependent properties (total charge and pH-dependent free energy of folding in the pH range -2 to 18)
The method described by Antosiewicz et al.  is implemented here with some modifications. PDB entries corresponding to proteins for which data are available in the pKa database made available by Gunner and coworkers  were prepared with pdb2pqr .
The method implemented here is a single conformer method, at variance with other methods like the QM/MM/TI method by Simonson et al. , MCCE  and the constant pH molecular dynamics method by McCammon and coworkers . These methods are expected to be more accurate, at the expense of large computational time.
Approaches similar to that implemented here have been used by Onufriev and coworkers [64, 65] where a significant speedup for larger proteins, with respect to our implementation, is due to clustering of interacting sites.
Other approaches which use a continuum method have been proposed recently which achieve very good agreement with experimental values, by using a pentapetide reference model for the titratable group, and by optimizing the hydrogen positions .
Improvements similar to those mentioned in the above paragraphs will be implemented in the future in our program.
We summarize here the theory that is discussed at length by Antosiewicz et al. .
Energy of ionization
where z is the charge of the group upon ionization, R is the gas constant, T is the temperature (298.15 K in the present calculations).
It is assumed that pKa of protein ionizable groups in an unfolded protein are the same as those of model compound. We assume here that the the pKa are the following: 4.5 for Glu, 3.8 for Asp, 6.5 for His, 12.5 for Arg, 10.5 for Lys, 9.0 for Tyr, 8.0 for Cys, 8.0 for the N-terminal ammine and 3.2 for the C-terminal carboxyl.
where each term describes a contribution to the free energy of ionization: the first term is the free energy of ionization in the model compound, the second term is the difference in ionization self-energy, the third term describes the difference in interaction of ionization charges with partial charges in the unionized protein and model compound and the fourth term describes the interaction among ionization charges. Following Antosiewicz et al. the inner dielectric constant is set to 20.0. A wide range of inner dielectric constants has been used for pKa calculations. When a single dielectric constant has been used it has been chosen mostly larger than 1, e.g. 4 in MCCE , 8 in EGAD  and 11 in GB/IMC . A large value of the dielectric constant accounts in an empirical way for the missing degrees of freedom implied by a single conformer method (see the discussion by Schutz and Warshel ).
Monte Carlo simulation of the ionization state
The protonation state of the ionizable sidechains is changed according to a Monte Carlo procedure. Protonation or deprotonation are simulated by adding or subtracting a unit charge to an atom representing the ionizable group. The following atoms have been considered as representative for each ionizable group: CG for Asp, CD for Glu, NZ for Lys, CE for Arg, SG for Cys, OH for Tyr, N for the N-terminal ammine and C for the C-terminal carboxyl. For histidine protonation may occur alternatively at the atom NE2 and ND1.
Monte Carlo simulations are performed at 0.5 pH intervals between -2.0 and 18.0 and at each pH value average ionization values and average components of ionization free energy are computed. At each pH value the number of equilibration steps is 100 times the number of ionizable sites and the number of Monte Carlo steps is 1000 times the number of ionizable sites.
Based on Monte Carlo simulation the output of the program includes: i) the titration curve for each ionizable site; ii) the list of pKa values for each ionizable site; iii) the charge state of the protein; iv) the pH-dependent component of the free energy of folding. Compared to the scheme of Antosiewicz et al. we are using the solvent accessible surface (as detailed in the subsection "Solvent accessible surface") instead of the molecular surface because results are more stable.
Heuristic corrections (detailed hereafter) to the scheme of Antosewicz et al  have been done mainly to remove large contributions which are typically overestimated due to neglection of molecular flexibility. For instance for surface residues large unfavorable interactions may be accomodated by conformational relaxation. Also large charge-charge interactions are likely to lead to partial unfolding that relieves the strong unfavorable energy. These considerations are consistent with the idea of using different dielectric constants for buried and solvent exposed residues, as suggested many years ago by Demchuck and Wade .
Scaling self energies
Here we consider as solvent exposed all those sites that have generalized Born radius smaller than 4.0 Å and buried those that have generalized Born larger than 7.0 Å. Self-energy interactions (second right-hand term in Equation 32) are multiplied by a factor 2.0 for buried sites and by 0.25 for exposed residues. For all intermediate situations the multiplying factor is a linear combination of the two extreme values.
Scaling background interactions
For exposed residues unfavorable background interactions (third right-hand term in Equation 32) are weighted by the empirical function . The factor 2 in the Boltzmann-like function is purely empirical to downweigh a contribution that is likely to be relaxed by protein flexibility.
For each ionizable site the pKa is obtained as the midpoint of the titration curve. The value is further corrected by separating the shift in pKa due to the desolvation self-energy , background interactions and site-charge - site-charge interactions .
Final scaling of contributions
The site-charge - site-charge interactions shift is scaled by the function which sets 3 pKa units as the maximum contribution arising from interactions between titratable sites, consistent with the idea that large unfavorable interactions lead to partial or global unfolding. Similarly, background interaction shifts are scaled by the function , which sets 5 pKa units as the maximum contribution due to background interactions, in order to avoid few very large shifts.
Test sets and conditions
The 55 proteins selected by Tjong and Zhou  for testing the GBR6 model have been used here. Atoms of the PDB structures have been assigned charges and radii taken from the CHARMM forcefield, using the program PDB2PQR [61, 73]. The PQR file format is described at . The radius of hydrogen atoms was reset to 1.0 Å in order to avoid numerical inaccuracies in the analysis linked to the small radius of polar hydrogens in the CHARMM forcefield. The surface of the molecule was defined alternatively as the surface accessible to the center of a solvent probe sphere with radius 1.5 Å or as the surface accessible by the surface of a solvent probe sphere with radius 1.5 Å, except where noted. The first type of surface, here referred to as solvent accessible surface, is computed by default by the program while the second type of surface, here referred to as molecular surface, was computed using the program MSMS  and given as input to the program in the form of a list of vertices and normal vectors.
For the computation of self energies 1000 sites were randomly chosen in the 55 proteins providing an unbiased sample of different environments.
The inner dielectric constant was assumed to be 1.0, the outer dielectric constant was assumed to be 78.54, the temperature is 298.15 K, ionic strength 0.150 M, except where noted. For the computation of pKa of ionizable sites in proteins we used the database developed by Gunner and coworkers  available at . The test set includes pKa for structures with PDB id: 1a2p, 1a6k, 1beg, 1bf4, 1bhc, 1bus, 1bvi, 1bvv, 1cdc, 1coa, 1cvo, 1de3, 1dg9, 1dwr, 1egf, 1gb1, 1goa, 1h4g, 1ig5, 1igc, 1kf3, 1lni and 1lse.
Exact generalized Born radii for a conducting sphere
where α i is the generalized Born radius at point i.
The work of Mongan et al.  reports a compact form for the Born radii corresponding to the integrals above expressed in term of GBR6 multiplied by a correction factor.
List of abbreviations used
Conducting Surface Model
Root mean square deviation.
AJ and VY have been supported by Italian Ministry of Education and University - Borse giovani ricercatori indiani 2008. Dr. I.M. Lait is gratefully aknowledged.
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 4, 2012: Italian Bioinformatics Society (BITS): Annual Meeting 2011. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/13/S4.
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