Physics Department, Computational Biophysics and Bioinformatics, Clemson University, Clemson, SC, 29642, USA

Department of Computer Science, Clemson University, Clemson, SC, 29642, USA

Abstract

Background

Accurate modeling of electrostatic potential and corresponding energies becomes increasingly important for understanding properties of biological macromolecules and their complexes. However, this is not an easy task due to the irregular shape of biological entities and the presence of water and mobile ions.

Results

Here we report a comprehensive suite for the well-known Poisson-Boltzmann solver, DelPhi, enriched with additional features to facilitate DelPhi usage. The suite allows for easy download of both DelPhi executable files and source code along with a makefile for local installations. The users can obtain the DelPhi manual and parameter files required for the corresponding investigation. Non-experienced researchers can download examples containing all necessary data to carry out DelPhi runs on a set of selected examples illustrating various DelPhi features and demonstrating DelPhi’s accuracy against analytical solutions.

Conclusions

DelPhi suite offers not only the DelPhi executable and sources files, examples and parameter files, but also provides links to third party developed resources either utilizing DelPhi or providing plugins for DelPhi. In addition, the users and developers are offered a forum to share ideas, resolve issues, report bugs and seek help with respect to the DelPhi package. The resource is available free of charge for academic users from URL:

Background

Electrostatic interactions play an important role in biological systems

Biomolecules function in water, which makes the calculation of electrostatic potential a challenge due to the complexity of the water environment

In addition to the DelPhi source code and executable files which are free for academic users, several other resources are provided

Implementation

The PBE model treats solvent as a continuum medium with high dielectric constant. Biomolecules are considered as low dielectric cavities made of charged atoms. Ions in the water phase are modeled as non-interacting point charges and their distribution obeys the Boltzmann law. Utilizing the Gauss-Seidel method, DelPhi solves both linear and nonlinear PBE in a cube of N × N × N grid points

The overall architecture of DelPhi is shown in the flowchart in Figure

Flowchart of the DelPhi program

**Flowchart of the DelPhi program.**

In addition to the routines described above, DelPhi also has some unique functionalities, such as handling multiple dielectric constants and mixed multivalent ions, rapid constructing molecular surface, generating geometric objects and performing calculations. The multiple dielectric constant method divides the bio-molecular system into different parts, and assigns each part a specific dielectric constant allowing the difference in conformational flexibility to be modeled by different dielectric constants as illustrated in Ref.

Results and discussion

There are several important characteristics used to classify methods and software packages: accuracy, rate of convergence and speed of calculations. In the next several subsections, we performed several tests on DelPhi with respect to these features.

Accuracy test

Accuracy is one major concern of any numerical solver. In order to measure the solver’s accuracy and demonstrate that it solves the exact problem, the solver is usually tested on simple examples for which analytical solutions exist. In this subsection, three simple examples with regular geometry were selected. We compared their analytical solutions with the numerical ones obtained by DelPhi. Since the computational algorithm does not distinguish simple geometrical objects from real biological macromolecules with more complex shape, it indicates that DelPhi solves the PB equation and produces close numerical approximations to the real solutions. The following constants were fixed in all examples: elementary charge _{,} vacuum permittivity _{.}

The first example presents a charged atom with a lower dielectric constant _{int} immersed in a continuum media with a higher dielectric constant _{ext}. In this example, the electrostatic component of solvation energy Δ^{sol} can be obtained by the Born formula and is explicitly given by

where

**(A) Schematic** illustration **of example 1: A charged sphere with low dielectric constant is inside a media with a high dielectric constant**

**(A) Schematic illustration of example 1: A charged sphere with low dielectric constant is inside a media with a high dielectric constant.** (**B**) Electrostatic component of the solvation energies calculated by DelPhi against the analytical solution.

Setting _{int} = 4.0, _{ext} = 80.0 and ^{sol} obtained by Equation (1) are −6673.71kT, -3336.86kT and −2224.57kT (rounded to two decimals) for radii

Two charges in a protein

The next example, shown in Figure _{int} and _{ext} again. Two atoms with radii _{i} and _{j} are centered at points with polar coordinates _{i} and _{j}, respectively. This example has been studied by Barry Honig and co-workers

where

where

**(A) Schematic illustration of example 2: a cavity with low dielectric constant is inside a media with high dielectric constant**

**(A) Schematic illustration of example 2: a cavity with low dielectric constant is inside a media with high dielectric constant.** Two charged atoms are located inside the cavity. (**B**) Electrostatic component of solvation energy obtained from DelPhi compared with analytical solution.

Substituting

A sphere in semi-infinite dielectric region

The third example considers a space split into two regions with different dielectric constants, as shown in Figure _{1} and that in the right region is _{2} (_{1} is initially positioned in the right region. The distance between the center of the sphere and the boundary of two regions is denoted by

**(A) Schematic illustration of example3: the space is divided by media with two different dielectric constants**

**(A) Schematic illustration of example3: the space is divided by media with two different dielectric constants.** A sphere with low dielectric constant _{1} is initially positioned in region with high dielectric constant _{2} and moves into the region with low dielectric constant _{1}. (**B**) Electrostatic component of solvation energy derived from DelPhi compared with analytical solutions. Blue solid lines represent analytical solutions, red circles represent numerical results from DelPhi.

The blue curve in Figure

Rate of convergence

The rate of convergence is another major concern from the numerical point of view. DelPhi utilizes the Gauss-Seidel iteration method, along with the optimized Successive Over-Relaxation method

These tests were performed by varying the value of scale from 0.5 points/Å to 6.5 points/Å at step size 0.1 points/Å. Noticing that scale is the reciprocal of grid spacing, this means larger scale results in finer discretization of the cube. The filling percentage of the cube,

Electrostatic solvation energies of 1ACB obtained from DelPhi with respect to different scale values

**Electrostatic solvation energies of 1ACB obtained from DelPhi with respect to different scale values.**

The energy calculations on the structure of 1ACB show that the approximate scale threshold is 1 points/Å. At scale larger than the threshold, the calculated electrostatic component of solvation energy is almost scale-independent and reaches steady value of −28089 kT. Achieving such steady value at scale 1 – 2 points/Å demonstrates the robustness of the algorithm that calculates the electrostatic component of solvation energy, so termed the corrected reaction field energy method

Speed of calculations

DelPhi utilizes various algorithms and modules to calculate electrostatic potential and energy. The basic modules include generating molecular surface, calculating electrostatic potential distribution and obtaining the corresponding electrostatic energy. The speed of calculations for each of these modules depends on various factors, such as scale, number of atoms/charges, shape/net charge of the molecule. In order to reveal their impact on the performance of DelPhi from the users’ point of view, we first tested DelPhi on a particular protein complex with fixed filling of the cube and increasing scale, and next, tested DelPhi on multiple proteins with fixed scale. All calculations were performed on the same type of CPU, Intel Xeon E5410 (2.33 G Hz), on the Palmetto cluster

Speed of calculations as a function of scale

In this case, we calculated the energy of

Electrostatic solvation energies calculation time of 1A19 with respect to different scale values

**Electrostatic solvation energies calculation time of 1A19 with respect to different scale values.**

Speed of calculations as a function of protein size

To evaluate the influence of protein size, 200 proteins from Zhang’s benchmark

Electrostatic solvation energies calculation time of 200 proteins with respect to (A) number of atoms and (B) number of charged groups of each protein

**Electrostatic solvation energies calculation time of 200 proteins with respect to (A) number of atoms and (B) number of charged groups of each protein.**

Electrostatic energy calculations on large biomolecules usually cost more CPU time primarily due to two factors: Firstly, large biomolecules need a large cube and consequently more grids to be represented. Secondly, larger biomolecules contain more charged atoms and require more time to calculate the energy terms. However, the curve of the 200 proteins is not smooth, because there are several other factors which influence the calculation time. The size of the modeling cube depends not only on the atom number, but also on the molecule’s shape. A narrow and long molecule may need a larger cube than a spherical molecule even if their atom numbers are the same. The irregularity of molecular surface also affects the iteration time. A molecule with an irregular surface requires more iterations to converge than a molecule with a regular, smooth surface. Finally, a molecule with a higher charge needs more iterations than a molecule with a lower charge. Due to above reasons, larger molecules usually (but not necessarily) cost more time than smaller molecules to calculate the corresponding potential and energies.

Effect of force field parameters (Charmm, Amber, OPLS and Parse)

It is well known that different force fields perform differently in protein folding

**Electrostatic solvation energy calculation of 1HVC using 4 different force fields**

**Electrostatic solvation energy calculation of 1HVC using 4 different force fields.** (**A**)-(**D**) Calculated electrostatic solvation energies using AMBER, CHARMM, OPLS, and PARSE force fields. (**E**) All of the 4 results are shown in one figure to show the differences. (**F**) Electrostatic potential surface of HIV-1 protease [PDB:1HVC], generated by using AMBER force field.

Results of the calculated electrostatic energies on 1HVC are shown in Figure

Conclusions

In this work, we described the DelPhi package and associated resources. DelPhi is a comprehensive suite including DelPhi website, web server, forum, DelPhi software and other tools. Several tests were performed on DelPhi in this work to demonstrate DelPhi’s capabilities in terms of accuracy, rate of convergence and speed of calculations. It was shown that DelPhi is a robust solver and capable of solving various biological applications. The benchmarks confirmed that DelPhi delivers energies that are almost grid independent, reaches convergence at scales equal to or larger than 1–2 grids/Å, and the speed of calculations is impressively fast. Finally, as shown in comparison with analytical solutions, the algorithm is, most importantly, capable of providing accurate energy calculations.

Availability and requirements

Project name: DelPhi

Project home page: e.g.

Operating system(s): Linux, Mac, Windows

Programming language: Fortran and C

Other requirements: no

License: free of charge license is required

Any restrictions to use by non-academics: Commercial users should contact Accelrys Inc.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LL analyzed the data, drafted the manuscript and maintains DelPhi package. CL maintains the DelPhi package and helped writing the manuscript. SS, JZ, SW, ZZ, LW and MP developed and maintain DelPhi web server and website and associated resources. EA supervised DelPhi development and maintenance and finally draft the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The work was supported by a grant from the Institute of General Medical Sciences, National Institutes of Health, award number 1R01GM093937.