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An Empirical Free Energy Function that Explains
and Predicts Binding Affinities Joseph Audie CMD
Bioscience Meriden, CT 064540
Introduction A free energy function is defined as
a mathematical expression that relates
macroscopic free energy changes to molecular
properties. Free energy functions can be used to
explain and predict receptor-ligand binding
affinities and discriminate native from
non-native binding modes. Free energy functions
play a vital role in modern drug design. Here,
we present a novel physics-based free energy
function (Affinity). The function was
parameterized by regression analysis on a giving
R2 0.97 and Q2 0.91. We used the function to
blindly and successfully predict binding
affinities for 31 protein-protein complexes (R2
0.79, rmsd 1.2 kcalmol-1). The results show
that Affinity can be used to accurately and
quickly predict binding affinities for a range of
ligands, from dipeptides to large proteins.
Affinity also performed well in direct comparison
with a recently described statistical potential.
The coefficients assigned by regression analysis
are physically realistic. Importantly, Affinity
can calculate binding free energies in a matter
of seconds on a single workstation. The data
indicates that Affinity is well suited for
solving a wide range of protein and peptide
design and discovery problems.
Conclusions Using regression analysis we have
derived and tested a novel physics-based free
energy function, Affinity. Statistical and
physical testing suggests each term in the
function accurately estimates its thermodynamic
correlate. The available data indicates that
Affinity can be used to explain and accurately
predict binding affinities for a large and
diverse set of proteins. Affinity performed well
in direct comparison with a recently described,
state-of-the-art, knowledge-based function
(Dcomplex). Because of its low computational
cost, Affinity is well-suited to serve as a
scoring function for a number of protein design
and discovery problems .

Results
Fig 2 to evaluate the regression coefficients for
the charge and hydrophobic desolvation terms, we
compared experimental host-guest tripeptide
water-octanol transfer free energies with
desolvation free energies estimated using our
regression equation. The results indicate
excellent agreement, further suggesting the
physical and theoretical validity of our model.
Importantly, these results (and results not
shown) seem to justify our novel assumption that
polar groups do not suffer a net desolvation
penalty. More recently, similar results were
obtained for 17 host-gust pentapeptides (data not
shown).
Literature cited Zhang, C., S. Liu, Q.
Zhu, andY. Zhou. 2005. A knowledge-based energy
function for protein-ligand, protein-protein, and
protein-DNA complexes. J Med Chem
48(7)2325-2335. Guerois, R., J.E.
Nielsen, andL. Serrano. 2002. Predicting changes
in the stability of proteins and protein
complexes a study of more than 1000 mutations. J
Mol Biol 320(2)369-387 Hu, X., andB.
Kuhlman. 2006. Protein design simulations suggest
that side-chain conformational entropy is not a
strong determinant of amino acid environmental
preferences. Proteins 62(3)739-748.
Baldwin, R.L. 2003. In search of the energetic
role of peptide hydrogen bonds. J Biol Chem
278(20)17581-17588 Ma, X.H., C.X. Wang,
C.H. Li, andW.Z. Chen. 2002. A fast empirical
approach to binding free energy calculations
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Biol 285(5)2177-2198 Laskowski, R.A.
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Fig 1 Results from the regression analysis using
our equation for ?Gbind. The graph and
supporting table indicate an excellent fit.
These results show that our regression model can
account for the training set binding affinities.
Materials and methods Starting from first
principles we derived a novel master
thermodynamic equation. Importantly, our
innovative derivation allows us to neglect
several contributions that complicate other
functions . Thus, the master equation simply
includes terms for charge and hydrophobic group
desolvation, changes in conformational entropy,
receptor-ligand salt bridges and hydrogen bonds,
and water mediated interactions at the interface.
Importantly, our function, to the best of our
knowledge, is the first to treat
receptor-water-ligand interactions
implicitly. We associated each thermodynamic
term with a physical descriptor and assigned
parameters with regression analysis on a training
set of 24 high quality and internally consistent
structures. The equation was then evaluated for
its statistical and physical validity. We also
tested the equation in blind prediction on a 35
member test set and, finally, compared its
predictive performance with a statistical
potential (Dcomplex)
Table 1 Statistical analysis of the regression
model showing that each regressor is a
statistically significant estimator of the
binding affinity. This analysis allows us to
rule out the possibility that the observed
associations in our equation are a product of
random chance.
Fig 3 We tested our equation in blind prediction
on a test set of 35 protein complexes and
obtained excellent results for 31 of them. The 4
outliers are represented as open circles. This
analysis clearly establishes our regression
equation as predictive. Importantly, the test
set included several Sh2-pTYR complexes with
interface contacts that are totally absent from
the training set. This result suggests Affinity
is transferable.
Acknowledgments Id like to thank Dr. Suzanne
Scarlata, Dr. Robert Rizzo, and Dr. David Greene
for input and feedback. Id also like to thank
Steven Ness for writing the current version of
Affinity.
Table 2 Physical evaluation of each regression
coefficient. The results indicate good agreement
between the coefficients assigned by regression
analysis and those inferred from experiment and
theory. This analysis shows that, in addition to
statistical legitimacy, our model enjoys physical
legitimacy as well.
For further information Please contact
audiej_at_sacredheart.edu or joseph.audie_at_cmdbioscien
ce.com. More information on this and related
projects can be obtained at www.cmdbioscience.com

Fig 4 Using the 31 successful predictions from
the test set, we compared the predictive
performance of Affinity with a recently
described statistical potential (Dcomplex).