Computational Biophysics and Bioinformatics, Department of Physics, Clemson University, Clemson, SC 29634, USA

James Byrnes High School, Duncan, SC 29334, USA

South Carolina Governor School for Science and Mathematics, Hartsville, SC 29550, USA

Center for Bioinformatics, The University of Kansas, Lawrence, KS 66047, USA

Abstract

Calculations of electrostatic properties of protein-protein complexes are usually done within framework of a model with a certain set of parameters. In this paper we present a comprehensive statistical analysis of the sensitivity of the electrostatic component of binding free energy (ΔΔG_{el}) with respect with different force fields (Charmm, Amber, and OPLS), different values of the internal dielectric constant, and different presentations of molecular surface (different values of the probe radius). The study was done using the largest so far set of entries comprising 260 hetero and 2148 homo protein-protein complexes extracted from a previously developed database of protein complexes (_{el}) is very sensitive to the force field parameters, the minimization procedure, the values of the internal dielectric constant, and the probe radius. Nevertheless our results indicate that certain trends in ΔΔG_{el }behavior are much less sensitive to the calculation parameters. For instance, the fraction of the homo-complexes, for which the electrostatics was found to oppose binding, is 80% regardless of the force fields and parameters used. For the hetero-complexes, however, the percentage of the cases for which electrostatics opposed binding varied from 43% to 85%, depending on the protocol and parameters employed. A significant correlation was found between the effects caused by raising the internal dielectric constant and decreasing the probe radius. Correlations were also found among the results obtained with different force fields. However, despite of the correlations found, the absolute ΔΔG_{el }calculated with different force field parameters could differ more than tens of kcal/mol in some cases. Set of rules of obtaining confident predictions of absolute ΔΔG_{el }and ΔΔG_{el }sign are provided in the conclusion section.

PACS codes: 87.15.A-,

Introduction

Proteins are essential components of the living cell

The internal dielectric constant is an important characteristic of protein's polarizability

In this study we test the sensitivity of the electrostatic component of the binding free energy with respect to different force field parameters, structural relaxation, values of the internal dielectric constant, and probe radii. The study will be done on a large set of protein-protein complexes comprised of 260 hetero- and 2148 homo-complexes. Such a large dataset has never been investigated before, especially with energy minimized structures.

Methods

Protein-protein complexes used in the study and their parameters

Protein-protein hetero-complexes subjected to the study were extracted from the

This resulted in an initial set of 260 hetero- and 2617 homo-protein complexes, which after removing structures for which some of the computational procedures could not produce the desired accuracy was reduced to 260 hetero- and 2148 homo-protein complexes. The interfacial area was calculated with the

Hydrogen placement and energy minimization

Some of the structures in our initial data set had structural defects, and thus all of the structures were subjected to the

A common approach in computing biophysical quantities using the 3D structures of biological macromolecules is to refine these structures by performing energy minimization with a particular force field. Following this strategy, we created another set of protonated structures (hereafter referred to as the minimized set) for all of the PDB files described above by running the

Calculation of the electrostatic component of the binding free energy

The electrostatic component of the binding free energy (ΔΔG_{el}) was calculated as the difference of the electrostatic free energies of the complex and of the free molecules

_{el }= Δ_{el}(_{el }(_{el}(

where Δ_{el}(

The calculations were performed assuming that all of the Arg, Asp, Glu, and Lys residues were ionized in both free and bound states, while the His residues were considered neutral. In order to reduce the complexity of the problem, all ionizable residues were kept in their default charge state. Thus, ionization changes induced by complex formation (suggested either by experimental data

Parameters that will be varied

Force field parameters

Three force field parameters were used in this study: Charmm27

Internal dielectric constant

What the value of the internal dielectric constant (ε(in)) should be for a given protein is the subject of much debate in the scientific community

Probe radius

The molecular surface in FDPB algorithms is determined by rolling a water probe with a particular radius to find the molecular surface according to Richard's algorithm

Results

Two types of complexes, hetero- and homo-complexes are the subjects of our study. We will explore how similar and how different are they with respect to their macroscopic characteristics. Since the physical process of binding is governed by the same forces, it may be expected that their physico-chemical properties should be quite similar. Table

Macroscopic parameters of hetero- and homo-complexes

Hetero-complexes

Homo-complexes

Max interfacial area (A**2)

8223.6

9595.2

Min interfacial area (A**2)

275.7

252.5

Mean interfacial area (A**2)

1440.2

1260.7

Max interfacial residues

408

461

Min interfacial residues

15

12

Mean interfacial residues

77

73

Max/min net charge

+15/-48

+23/-53

Max/min interfacial charge

+13/-14

+10/-16

Percentage having opposite net charge of the monomers

40%

0%

Percentage having opposite interfacial change of the monomers (in parentheses are the cases for which at least one of the interfaces had zero net charge)

58% (23% zero)

8% (33% zero)

The files are taken from 40% sequence identity subset of the ProtCom database.

Effects of the internal dielectric constant

In this study, the value of the dielectric constant was varied from one, which is the value usually used in molecular dynamics simulations with explicit water model, to twenty, which was introduced by Gilson and co-workers_{el}. For ε(in) = 1 and ε(in) = 2 in 94% of the non-minimized complex cases, the electrostatics is calculated to oppose binding. Minimization of the structures slightly lowers this percentage to 87%; however, in the vast majority of the cases, the calculated electrostatic component of the binding free energy still opposes binding. The corresponding numbers for homo-complexes are 91% and 85% for non-minimized and minimized complexes, respectively. These percentages are quite similar and show no significant difference between the electrostatic components of the binding free energy for the hetero- and homo-complexes. Further increasing the value of the internal dielectric constant makes ΔΔG_{el }smaller and smaller, which results in a narrower distribution, but the majority of the energies remain positive. At the largest value used in this study (ε(in) = 20), in 62% of the non-minimized hetero-complex cases, the electrostatic component of the binding free energy was calculated to oppose binding. This percentage drops to 43% in case of minimized hetero-complexes. Quite different percentages were calculated in case of the homo-complexes. Raising the internal dielectric constant to twenty only minimally changed the percentage of the ΔΔG_{el }calculated to favor binding; however, in at least 91% of the cases, ΔΔG_{el }still opposes binding. Energy minimization had a significant effect, lowering the percentage of cases in which the electrostatic component of the binding free energy was calculated to be positive number to 77%. However, electrostatics was still calculated to oppose binding in the vast majority of the homo-complexes.

Distribution of the electrostatic component of the binding energy (ΔΔG_{el}) calculated with dielectric constant ε(in) = 2.0, probe radius of 1.4A and Charmm27 force field

Distribution of the electrostatic component of the binding energy (ΔΔG_{el}) calculated with dielectric constant ε(in) = 2.0, probe radius of 1.4A and Charmm27 force field. Percentage is the count of ΔΔG_{el }normalized in respect with the total number of complexes. The results are presented in case of: (a) non-minimized hetero-complexes. (b) minimized hetero-complexes. (c) non-minimized homo-complexes. (d)minimized homo-complexes.

Why were the calculated distributions of ΔΔG_{el }using the low internal dielectric constant quite similar for the hetero- and homo-complexes but significantly different using high values of the internal dielectric constant? A possible explanation is offered by the numbers reported in the last two rows of Table _{el }for homo-complexes compared with the ΔΔG_{el }calculated for the hetero-complexes. Increasing the internal dielectric constant combined with minimizing the energy of the structures could turn some of the small positive ΔΔG_{el }calculated for the hetero-complexes into small negative ΔΔG_{el}. However, ΔΔG_{el }are large positive numbers in the homo-complexes set, and an increase of the internal dielectric constant combined with the refinement of the structures cannot dramatically change the percentage of the cases in which electrostatics was calculated to oppose binding.

The observation made for the hetero-complexes, that their structural refinement through the energy minimization of corresponding complexes results in a change of the sign of the calculated ΔΔG_{el}, deserves attention. The electrostatics is one of the components of the total energy and the total energy is minimized in the minimization protocol. There is no reason to expect that minimizing the total energy should result in the minimization of the energy components. However, apparently the electrostatic component was actually optimized in the energy minimization protocol. Perhaps this indicates that if a given energy component favors a particular energy state, the minimization of the total energy will most likely result in the enhancement of this energy component as well.

The results of this section are summarized in Fig. _{el }distributions are plotted against the internal dielectric constant. The calculations were done with the probe radius R = 1.4A. It can be seen that variations of the internal dielectric constant within the range 1.0–8.0 cause dramatic changes in the mean of the energy distributions for all types of complexes. In contrast, an increasing the magnitude of the internal dielectric constant above 8.0 does not cause much change in the calculated ΔΔG_{el}. Despite the changes of the distributions' mean magnitudes as the internal dielectric constant varies, the mean is always above zero for the homo-complexes and for the non-minimized hetero-complexes. The mean drops below zero only in minimized hetero-complex set when the internal dielectric constant is above 8.0.

The mean of the ΔΔG_{el }distributions calculated with at probe radius R = 1.4 A plotted as a function of the internal dielectric constant ε(in)

The mean of the ΔΔG_{el }distributions calculated with at probe radius R = 1.4 A plotted as a function of the internal dielectric constant ε(in).

Effects of the probe radius

The probe radius in finite-difference algorithm plays an important role in determining both the molecular surface and the internal dielectric space of molecules_{el}, with and without minimized structures are offset towards positive energies, .i.e., in the vast majority (over 85%) of cases, the calculated energies oppose binding. As the probe radius magnitude decreases and reaches 0.5A and 0.0A, the distributions move to the left, towards negative ΔΔG_{el}. However, the distributions of ΔΔG_{el }for non-minimized hetero- and homo-complexes and for minimized homo-complexes remain mostly at positive energies. At a probe radius R = 0.0A, for 81% of the non-minimized hereto-complexes, the electrostatics was calculated to oppose binding. The percentage of the homo-complexes with positive ΔΔG_{el }was 94% and 73%, for non-minimized and minimized complexes, respectively. The only exceptions are the distributions of ΔΔG_{el }for minimized hetero-complexes at probe radii of 0.5A and 0.0A. At a probe radius of 0.0A, for 40% of the minimized hetero-complexes, the electrostatics was calculated to oppose binding. It is interesting to note that similar percentage were calculated for minimized hetero-complexes using a "standard" probe radius of 1.4A, but assigning a high internal dielectric constant of 20, resulted in 43% of the calculated energies being positive.

Distribution of the electrostatic component of the binding energy (ΔΔG_{el}) calculated with dielectric constant ε(in) = 2.0, probe radius of 0.0A and Charmm27 force field

Distribution of the electrostatic component of the binding energy (ΔΔG_{el}) calculated with dielectric constant ε(in) = 2.0, probe radius of 0.0A and Charmm27 force field. Percentage is the count of ΔΔG_{el }normalized in respect with the total number of complexes. The results are presented in case of: (a) non-minimized hetero-complexes. (b) minimized hetero-complexes. (c) non-minimized homo-complexes. (d)minimized homo-complexes.

The observed similarities of the effects caused by increasing the internal dielectric constant and decreasing the magnitude of the probe radius inspired us to investigate a possible correlation between the ΔΔG_{el }calculated with a "standard" probe radius value but with a high internal dielectric constant and the ΔΔG_{el }calculated with a small probe radius and a "standard" internal dielectric constant. Here by "standard" we mean values that are most frequently reported in the literature. To address such a possibility, the corresponding ΔΔG_{el }for each type of dataset, hetero- and homo-complexes, non-minimized and minimized, were subjected to the following procedure: ΔΔG_{el }calculated with probe radius 0.0A and 'standard" dielectric constant of 2.0 were plotted against ΔΔG_{el }calculated with probe radius 1.4A and ε(in) = 1, 2, 4, 8 and 20. The plot resulting to largest correlation coefficient and a slope close to 1.0 was considered as the best fit. In case of non-minimized hetero-complexes (Fig. _{el }calculated with slightly larger probe radius of 0.5A and 'standard" dielectric constant of 2.0 versus ΔΔG_{el }calculated with probe radius 1.4A and ε(in) = 1, 2, 4, 8 and 20. The best fits in terms of correlation coefficient and slope are reported in Table

Slopes of the fitting lines and the corresponding correlation coefficients

Type of complex

Probe radius R = 1.4A and different ε(in)

ε(in) = 2 and different probe radius R

Slope of the fitting line

Correlation coefficient

Hetero, non-minimized

8.0

0.0

1.63

0.67

Hetero, minimized

8.0

0.0

1.31

0.59

Homo, non-minimized

4.0

0.0

1.22

0.72

Homo, minimized

4.0

0.0

1.23

0.85

Hetero, non-minimized

4.0

0.5

0.69

0.52

Hetero, minimized

4.0

0.5

0.92

0.79

Homo, non-minimized

4.0

0.5

1.44

0.87

Homo, minimized

4.0

0.5

1.34

0.85

The results are presented for the _{el }calculated with "standard" probe radius of 1.4A versus small probe radius of 0.0 and 0.5A. The _{el }calculations with "standard" probe radius of 1.4A.

ΔΔG_{el }calculated with probe radius 0.0A and ε(in) = 2.0 versus ΔΔG_{el }calculated with "standard" probe radius 1.4A and different ε(in): (a) non-minimized hetero-complexes, ε(in) = 8.0

ΔΔG_{el }calculated with probe radius 0.0A and ε(in) = 2.0 versus ΔΔG_{el }calculated with "standard" probe radius 1.4A and different ε(in): (a) non-minimized hetero-complexes, ε(in) = 8.0. (b) non-minimized homo-complexes, ε(in) = 4.0.

The effect of the probe radius on the calculated ΔΔG_{el }is summarized in Fig. _{el }distributions are shown as a function of the probe radius. It can be seen that decreasing the magnitude of the probe radius results in a decrease of the mean of the ΔΔG_{el }distributions. A similar trend was discussed in the previous paragraph regarding increasing the internal dielectric constant. The decrease of the mean is almost the same for all cases: for the hetero- and homo-complexes and for the non-minimized and minimized. However, the mean of the distribution of homo-complexes at R = 1.4A is much more positive than that of the hetero-complexes, and the decrease caused by lowering the radius is not enough to make it a negative number. In contrast, the change is enough to turn the sign of the mean of the minimized hetero-complexes and to make it negative.

The mean of the ΔΔG_{el }distributions calculated with internal dielectric constant ε(in) = 2.0 as a function of the probe radius

The mean of the ΔΔG_{el }distributions calculated with internal dielectric constant ε(in) = 2.0 as a function of the probe radius.

The effects of the force field

The above results were calculated using Charmm27 force filed parameters for both calculations of the electrostatic energies and of the minimization protocol. To test the sensitivity of the results with respect to the force fields, we repeated the calculations with Amber98 and OPLS force fields. This required energy minimization of both the hetero- and homo-complexes with Amber and OPLS force fields, respectively. The results are reported in the Supplementary Materials section, where the ΔΔG_{el }calculated with Charmm27 are plotted against the corresponding ΔΔG_{el }calculated with Amber98 and OPLS, respectively. The data points were fitted with straight lines and the scopes and correlation coefficients recorded. These slopes and correlation coefficients are provided in Tables _{el }calculated with different force fields as

The parameters of the distributions of ΔΔΔG_{el }calculated for hetero-complexes.

Amber-Charmm

Amber-OPLS

Charmm-OPLS

Min

-36.6 (-33.8)

-41.5 (-75.9)

-33.7 (-183.5)

Max

54.3 (107.6)

75.4 (100.1)

56.7 (76.4)

Mean

5.8 (10.3)

13.0 (19.0)

7.3 (8.8)

Median

5.9 (8.6)

10.8 (16.3)

6.0 (7.3)

RMS

15.0 (17.0)

21.5 (27.7)

15.5 (23.6)

Variance

194 (183.4)

295 (412)

188 (481)

R

0.95 (0.94)

0.92 (0.90)

0.95 (0.89)

slope

0.96 (0.83)

0.89 (1.08)

0.91 (1.21)

The numbers without parentheses are for non-minimized and the numbers in parentheses are for minimized complexes. The last two rows report the correlation coefficient and the slope of the fitting line of the graphs provided in Additional file

The parameters of the distributions of ΔΔΔG_{el }calculated for homo-complexes.

Amber-Charmm

Amber-OPLS

Charmm-OPLS

Min

-226.8 (-274.9)

-644.7 (-673.3)

-300.1 (-327.5)

Max

262.3 (166.4)

572.8 (438.8)

329.1 (269.4)

Mean

15.3 (-2.63)

-30.9 (-15.2)

2.6 (4.8)

Median

9.5 (-2.12)

-28.1 (-13.6)

0.7 (1.6)

RMS

32.6 (30.7)

70.4 (63.6)

27.7 (21.8)

Variance

829 (940)

4003 (3267)

768 (871)

R

0.92 (0.93)

0.78 (0.85)

0.96 (0.95)

slope

0.85 (0.81)

0.88 (0.89)

0.94 (0.93)

The numbers without parentheses are for non-minimized and the numbers in parentheses are for minimized complexes. The last two rows report the correlation coefficient and the slope of the fitting line of the graphs provided in Additional file

_{el }(_{el}(_{el}(

where X, Y stand for either the Charmm27, Amber98 or OPLS force fields. These differences were calculated for all types of complexes, both non-minimized and minimized. The parameters of the resulting distributions are reported in Tables

Correlation plots of the electrostatic component of the binding free energy. The plots show the correlations of the electrostatic component of the binding free energy calculated with different force fields.

Click here for file

It can be seen that both the correlation coefficients and the slope of the fitting lines are close to 1.0, indicating that ΔΔG_{el}s calculated with different force fields are quite similar. However, a closer look at Tables _{el }calculated with Charmm27 is 5.8 and is 10.3 kcal/mol less positive than that of the corresponding means calculated with Amber98 in the case of non-minimized and minimized hereto-complexes, respectively. The same offset is even more pronounced when comparing the results obtained with the Amber and OPLS force fields. In the hetero-complex cases, the means differ by 13.0 and 19.0 kcal/mol for non-minimized and minimized complexes, respectively. Lastly, comparing the results obtained with the Charmm27 and OPLS force fields, we observed that the mean of OPLS is less positive as compared to the mean of the energy distribution calculated with Charmm27. Thus, these results suggest that the OPLS force field resulted in less positive ΔΔG_{el }(less unfavorable energies), and the Charmm27 and Amber98 force fields resulted in the most unfavorable ΔΔG_{el }in the hetero-complex cases.

In addition to the above observations, in some exceptional cases the ΔΔG_{el }calculated with different force fields differed by more than 100 kcal/mol. Particular example is tail-associated lysozyme bound to baseplate structural protein GP27 (PDB ID _{el }calculated with Charmm27, e(in) = 1.0 and probe radius 1.4A was 261.7 kcal/mol while it was calculated to be 391.1 kcal/mol with Amber98 force field. We have not investigated these exceptions in detail; however, the differences reported in Tables

Discussion

We have done extensive testing of the sensitivity of the electrostatic component of the binding free energy with respect to internal dielectric constant values, probe radius, and several commonly used force fields. The goal was also to reveal the general trends and to elaborate on the role of electrostatics on the binding for hetero- and homo-complexes. The study was done on a very large set of protein-protein complexes: 260 hetero-complexes and 2148 homo-complexes. Energy minimization and full-scale electrostatic calculations have never been done on such a large dataset before. This ensures that the obtained results and conclusions are statistically significant.

A common practice is to validate the results of numerical simulations against experimental data. However, it is impossible to experimentally determine the electrostatic component of the binding free energy. The binding free energy is made of a variety of energy terms whose interplay results in the observed affinity. Even the pH-dependence of the binding is not a pure electrostatic event

Despite the fact that electrostatics is only one of the many forces contributing to the binding, in many cases, its involvement could be related to biologically important effects. For this reason, many researchers employ electrostatic calculations to find out the electrostatic component of the binding free energy. However, the value of the dielectric constant, the method of determining the molecular surface (probe radius), and the force field parameters are still a personal preference. Our study demonstrated that the absolute value of the calculated ΔΔG_{el }is very sensitive to all of the above-mentioned parameters. While this is expected to be the case for different internal dielectric constants and to some extent for different probe radii, the observation that the results are quite sensitive to the force field parameters deserves attention. The average difference of the ΔΔG_{el }calculated with different force fields can be as large as 20 kcal/mol and more. It seems to us that the energy minimization with the corresponding force field does not make much difference. Since the minimization protocol minimizes the total energy and not just the electrostatic component, the outcome for electrostatic energy will vary case by case. All of these observations indicate that the absolute values of the ΔΔG_{el }should be considered with precaution.

This study revealed a major difference between hetero- and homo-complexes with respect to the calculated ΔΔG_{el}. The calculated electrostatic component of the binding free energy opposes binding for approximately 80% of the homo-complex cases, despite the internal dielectric constant, probe radius, and force field. In contrast, the role of electrostatics on the binding of hetero-complexes depends on all of the factors above. Using a low internal dielectric constant and a probe radius of 1.4A, despite the force field used, most of the calculated ΔΔG_{el }opposes binding. However, increasing the internal dielectric constant to 20 or decreasing the probe radius to R = 0.0A results in 60% of the cases having a ΔΔG_{el }favoring the binding. These findings offer a pragmatic prescription for assessing the confidence of the predicted role of ΔΔG_{el }on the binding affinity. It seems that despite the differences between the homo- and hetero-complexes, if the binding free energy is calculated to be a large positive quantity, then the conclusions could be considered independent from the choice of the parameters. However, cases in which the ΔΔG_{el }are calculated with a particular set of parameters and corresponding magnitude is a small number close to zero are more complicated. Our study indicates that the calculated ΔΔG_{el }could easily change sign upon changing the parameters of the protocol or upon switching the force field. Therefore, the conclusions made for the role of electrostatics on the binding in case of a small ΔΔG_{el }should be carefully investigated by varying the force fields and the parameters of the computational protocol.

Our study, being applied on 2408 protein-protein complexes, offers the possibility to statistically address the question of what role electrostatics play in protein-protein binding. A similar question of what role salt bridges play on protein stability was extensively studied by Tidor and co-workers

The results of our study indicate that the role electrostatics have on protein-protein binding in hetero-complexes cases is the same as the role salt bridges have on protein stability; i.e., in some cases, it will favor the binding, but in other cases it will oppose the binding. In our previous study

In this work we have not considered possible protonation changes induced by the binding. It was done in order to reduce computational demand of the numerical protocol. However, protonation changes may be common phenomena in protein binding as demonstrated recently by Jensen and co-workers _{el }more favorable (or less unfavorable) compared with calculations done with default charge states. However, most of the predicted ionization changes are fractional numbers and modeling of non-integer charge changes requires ensemble approach in computing ΔΔG_{el}, a task which is quite difficult to accomplish on a set of 2408 protein complexes.

The study revealed that for a significant fraction of hetero-complexes and in vast majority of homo-complexes the electrostatics opposes binding. Since most of the protein-protein complexes in the cell are homo-complexes, the overall role of electrostatics is predicted to oppose binding. Instead, electrostatics provides the necessary specificity and steering, processes equally important for protein-protein association.

Conclusion

The analysis of the sensitivity of the ΔΔG_{el}, calculated with different force fields, internal dielectric constants, probe radius value and minimization protocols, gives us the opportunity to suggest a set of rules of calculating (a) the absolute value of ΔΔG_{el }and (b) the sign of ΔΔG_{el}: (a1) if there is no prior knowledge what the effective value of both internal dielectric constant and probe radius are for a given protein complex and a particular protocol of calculating the ΔΔG_{el}, then the absolute value of calculated ΔΔG_{el }is meaningless; (a2) provided that the effective value of both internal dielectric constant and probe radius are known for a given protein complex and a particular protocol of calculating the ΔΔG_{el}, then the absolute value of ΔΔG_{el }should be obtained as an average of ΔΔG_{el }calculated with different force fields; (a3) energy minimization does not help in obtaining consistent ΔΔG_{el}'s, since it minimizes the total energy, not just the electrostatic component. Therefore, the absolute value of ΔΔG_{el }should be obtained as an average of ΔΔG_{el }calculated with different force fields, provided that other parameters are known; (b1) if there is no prior knowledge what the effective value of both internal dielectric constant and probe radius are for a given protein complex and a particular protocol of calculating the ΔΔG_{el}, then the sign of ΔΔG_{el }(determining the role of electrostatics on binding) is meaningful only if the absolute ΔΔG_{el }calculated with either high internal dielectric constant or zero probe radius is larger than 1 kcal/mol. In case of hetero-complexes, if ΔΔG_{el }is calculated to be positive and larger than 1 kcal/mol with ε(in) = 20 and probe radius 1.4A with given force field, then the probability of remaining positive in the calculations performed with different force fields is 0.99 (or 0.97 with ε(in) = 2). In case of homo-complexes, this probability is 0.95 (or 0.94 with ε(in) = 2); (b2) if there is a prior knowledge of the effective value of internal dielectric constant and probe radius, then calculations with a particular force field with and without minimization provide a good estimate of ΔΔG_{el }sign in about 90% of the cases. In case of hetero-complexes, if ΔΔG_{el }is calculated to be positive and larger than 1 kcal/mol with ε(in) = 20 and probe radius 1.4A with given force field, then the probability of remaining positive in the calculations performed with different dielectric constant is 0.98. Probability of change of the sign ΔΔG_{el }using different probe radii is about 0.1. In case of homo-complexes, these probabilities are 0.96 and 0.87.

Acknowledgements

The minimization of the structures used in this work was made possible by a Condor pool deployed and maintained by Clemson Computing and Information Technology. The authors would like to acknowledge the support of the staff from the Cyber Infrastructure Technology Integration group, especially Barr Von Oehsen. This research was supported by an award to Clemson University from the Howard Hughes Medical Institute Undergraduate Science Education Program.