The Energy Computation Paradox and ab initio Protein Folding
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Title: The Energy Computation Paradox and ab initio Protein Folding
Series Title: Faver JC, Benson ML, He X, Roberts BP, Wang B, et al. (2011) The Energy Computation Paradox and ab initio Protein Folding. PLoS ONE 6(4): e18868. doi:10.1371/journal.pone.0018868
Physical Description: Journal Article
Creator: Faver, John
Benson, Mark L.
He, Xiao
Roberts, Benjamin P.
Wang, Bing
Marshall, Michael S.
Sherrill, C. David
Merz, Kenneth M. Jr.
Publisher: PLoS ONE
Place of Publication: San Francisco, CA
Publication Date: 4/25/2011
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Abstract: The routine prediction of three-dimensional protein structure from sequence remains a challenge in computational biochemistry. It has been intuited that calculated energies from physics-based scoring functions are able to distinguish native from nonnative folds based on previous performance with small proteins and that conformational sampling is the fundamental bottleneck to successful folding. We demonstrate that as protein size increases, errors in the computed energies become a significant problem. We show, by using error probability density functions, that physics-based scores contain significant systematic and random errors relative to accurate reference energies. These errors propagate throughout an entire protein and distort its energy landscape to such an extent that modern scoring functions should have little chance of success in finding the free energy minima of large proteins. Nonetheless, by understanding errors in physics-based score functions, they can be reduced in a post-hoc manner, improving accuracy in energy computation and fold discrimination.
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The Energy Computation Paradox and ab initio Protein

Folding

John C. Faver1, Mark L. Benson1, Xiao He', Benjamin P. Roberts1, Bing Wang1, Michael S. Marshall2,
C. David Sherrill2, Kenneth M. Merz Jr.l*
1 Quantum Theory Project, University of Florida, Gainesville, Florida, United States of America, 2 Center for Computational Molecular Science and Technology, School of
Chemistry and Biochemistry, and School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia, United States of America


Abstract
The routine prediction of three-dimensional protein structure from sequence remains a challenge in computational
biochemistry. It has been intuited that calculated energies from physics-based scoring functions are able to distinguish
native from nonnative folds based on previous performance with small proteins and that conformational sampling is the
fundamental bottleneck to successful folding. We demonstrate that as protein size increases, errors in the computed
energies become a significant problem. We show, by using error probability density functions, that physics-based scores
contain significant systematic and random errors relative to accurate reference energies. These errors propagate throughout
an entire protein and distort its energy landscape to such an extent that modern scoring functions should have little chance
of success in finding the free energy minima of large proteins. Nonetheless, by understanding errors in physics-based score
functions, they can be reduced in a post-hoc manner, improving accuracy in energy computation and fold discrimination.

Citation: Faver JC, Benson ML, He X, Roberts BP, Wang B, et al. (2011) The Energy Computation Paradox and ab initio Protein Folding. PLoS ONE 6(4): e18868.
doi:1 0.1371/journal.pone.001 8868
Editor: Collin M. Stultz, Massachusetts Institute of Technology, United States of America
Received December 1, 2010; Accepted March 21, 2011; Published April 25, 2011
Copyright: � 2011 Faver et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was financed by National Institutes of Health (http://nih.gov/) GM044974 and GM066689; National Science Foundation (http://www.nsf.gov/)
CHE-1011360; and National Center for Computational Sciences at Oak Ridge National Laboratory (http://www.nccs.gov/) contract DE-AC05-00OR22725.
Publication of this article was funded in part by the University of Florida Open-Access Publishing Fund. The funders had no role in study design, data collection
and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: merz@qtp.ufl.edu


Introduction

A widely studied and yet largely unsolved problem in computational
biochemistry is the ab initio protein-folding problem the prediction of
three-dimensional protein structure from an amino acid sequence
[1,2]. In recent years physics-based methods (those that explicitly
model inter- and intramolecular interactions of a chemical system),
combined with extensive conformational searches and sampling, have
been explored as a general solution to the problem. The basis of any
physics-based method used to study protein folding is the thermody-
namic hypothesis - that the biologically active (native) fold is a free
energy minimum [3]. This is the most widely used paradigm, i.hi.. .l1
there are a few known exceptions to the rule [4,5]. Molecular dynamics
(MD) simulations are commonly used to analyze the folding kinetics of
a protein using physics-based potentials, however the timescales needed
to fully simulate the folding processes of large proteins can be
prohibitively long [6,7,8,9]. Monte Carlo-based search and minimi-
zation techniques in conjunction with physics-based potentials are also
employed [10]. Unfortunately, these and other physics-based methods
have had difficulty in .- .1 1 I-,..1; . i; 1 . i .. .1. ;. C .l.1. of chains longer
than 100 amino acids [11,12].
One proposed explanation for the failure of current methods of
physics-based folding of large proteins is that "the primary obstacle to de
novo protein structure prediction is conformational sampling' [13]. Indeed, the
high number of degrees of freedom makes it difficult to find the free
energy minimum of a large protein. However, based on evidence
from folding kinetics, a protein's energy landscape may have some


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predictable features in its overall shape. Levinthal noted that even
1.... I 11 _. : proteins have access to vast amounts of conformational
space, they transition from denatured states to folded states
surprisingly quickly, as if the protein only selectively samples the
available conformational space [14]. Based on this, it has been
inferred that a protein's energy landscape is shaped like a high-
dimensional funnel with very many high-energy states surrounding a
deep global minimum (native state) [15,16]. Depending on the
protein, this funnel can have smooth or jagged slopes with variable
inclination, which determines the folding rate [17]. With this type of
folding landscape, the protein can easily "fall into" (perhaps after
overcoming local minima) the folded state from any of the unfolded
I ,. ;,1 ... 11 I ,. to sample the entire energy surface. If the funnel
concept is an accurate model of protein folding energy landscapes,
then exhaustive conformational sampling should not be necessary,
but adequate sampling remains an important component of a folding
.1 .;,1 due to the likely presence of local minima in the energy
surface, especially for slower folding proteins.
While sampling clearly plays a significant role in the ultimate
solution to the ab initio protein folding problem, it is important not
to overlook the role energy functions play. Physics-based ab initio
protein folding attempts to calculate the relative free energies of
protein conformations and energetically separate mis-folded
structures from native ones. The typical foundation of physics-
based potentials used in ab initio folding studies is the classical force
field and its derivatives [18]. These are generally built in a
piecewise manner based on model systems representing interac-


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Energy Computation Paradox and Protein Folding


tions found in proteins and are then extended to full protein
systems. It has been assumed that the ability to accurately
represent small model systems will yield an accurate representation
of a full protein. That is, uncertainties in the energies of the model
systems of - 1 kcal/mol are assumed to yield similar errors in the
energy of the much larger protein systems (see Figure 1) [19].
Moreover, it is generally 1 ....1,i that because force fields are
parameterized, they are largely subject to small random errors.
Paradoxically we show that (1) li..... 1, physics-based score
functions yield small uncertainties for small model systems, these
uncertainties increase dramatically with system size and that (2)
most computational methods, even those that have been
parameterized, contain large systematic and random errors when
applied to macromolecules. Hence, we conclude that current
energy functions introduce such significant uncertainties into
physics-based folding exercises that improving accuracy in energy
computation is just as important as sampling in solving the ab initio
protein folding problem.
This point is clearly illustrated in recent studies of the Pin 1 WW
domain carried out by two groups using two force fields of similar
construction, but with different parameter choices. Schulten and
co-workers attempted to fold Pinl i ;i,, ii,. CHARMM force field
and long MD simulations (10 pIs), but were unsuccessful [20]. This
was later shown to be due to issues with the force field utilized
[21]. However, recent long MD simulations (1 ms) of Shaw and
co-workers succeeded in folding Pini using a modified AMBER
force field (ff99sb) [22]. The forms of these two force fields both
trace their roots from the consistent force field of Lifson [18,23,24]
and co-workers and are similar in construction, but are
parameterized differently. This comparison shows the range of
uncertainty present in force fields, which can yield success or
failure, but the origin of this has not been well understood until the
present work.


The impact of our observation affects any method that attempts
to compute the total interaction energies of any macromolecular
process including: protein folding, protein-ligand docking, crystal
isomorph prediction, assembly of nanomaterials, and others. In a
previous study following our initial hypothesis [25], we applied
statistical error analysis to the problem of protein-ligand docking
[26]. In this study, it was observed that systematic and random
errors do indeed quickly accumulate 1, ..i. 1 ...i. large interacting
chemical systems. For several of the score functions examined, the
overall error estimates in the total interaction energy exceeded the
experimental free energy of ligand binding. In the case of protein
folding, error accumulation is also expected to be significant due to
the high number of chemical interactions involved in a protein fold
[19].

Analysis of Error Propagation in Protein Folds
Errors inherent in a calculation or measurement can be
described as either systematic or random. Systematic errors are
predictable in both sign and magnitude, while random errors are
not predictable in sign or magnitude. Because of this difference,
systematic errors propagate as a simple sum, while random errors
propagate as the square root of sum of squares [27]. Propagated
systematic error is correctable, since it describes an overall
predictable shift in the measured value. Propagated random
errors are not easily correctable, and are measured and reported as
the accumulation of error from all random error sources. Large
random error is a characteristic of a very imprecise method of
measurement. In searching the energy surface of a protein for the
global free energy minimum, total error from all sources should be
minimized.
To illustrate the effects of error propagation on the protein-
folding problem, imagine a protein's energy surface as a funnel


H H


H' "'. H..- "f H
H H
Nonpolar - Methane dimer


H

H )__0


>=�0-------- \
H-N H

H
Hydrogen bonding - formamide dimer


H
0------H-- N H



0 -----H--N ' H

H
Salt bridge - formate-guanidinium dimer

Figure 1. Example model systems used to build-up interactions in proteins. Accurate interaction energies for the model systems are
assumed to yield accurate global interaction energies for a folded protein.
doi:1 0.1 371/journal.pone.001 8868.g001


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Energy Computation Paradox and Protein Folding


surrounding the global minimum structure (folded state). In
general these surfaces are not smooth, but often contain many
local minima outside of the native state. If the folded protein has,
for example, 100 independent chemical contacts (e.g. van der
Waals interactions and hydrogen bonds between residues), and
each is computationally modeled to chemical accuracy (i.e., within
1 kcal/mol random error with respect to an experimental
measurement) [28], then random error propagation yields a total
error of -10 kcal/mol. This would imply that our hypothetical
computational model would have difficulty in distinguishing the
native state from any other state with overlapping error bars
within 10 kcal/mol. Thus our computational model may correctly
find several local minima, but if they differ in energy by less than
the error bar magnitudes, the position of the global minimum
(which is usually the native fold), could not be determined with
much certainty (see Figure 2). Dill realized the issue of error
propagation, leading him to suggest that a target of 0.1 kcal/mol
per amino acid would be an acceptable level of error for a protein of
100 amino acids yielding an overall random error bar of only
--1 kcal/mol [19]. Given that each amino acid can have several
intramolecular contacts, each with an associated error, achieving
this level of accuracy is indeed a very challenging endeavor.
In our attempt to estimate and correct for errors in physics-
based energy scores used in protein folding, we have taken an
approach previously described by Merz [25]. Intramolecular
interactions involved in protein folds are broken down into
chemical fragments and associated with reference interaction
energies obtained using converged quantum chemical calculations
or experimental measurements, if available. Energies from more
approximate theories are then compared to the reference
interaction energies to form fragment-based error estimates.
Estimating the error of a series of these fragment-based
interactions contained within a protein fold and then propagating
these errors 1.1. .11 .1 .I all fragments in the fold yields an estimate
of the total error associated with a computed total energy for the







-.calculate







actual en







minimum 1

minimum 3
native


protein. The error is broken down into a systematic portion which
can be corrected for and a random portion which cannot, but can
be reported as an error bar.
It is important to remember that the thermodynamic principle
of protein folding applies to the free energy, not the interaction
energy (differences of total electronic energies) that we use here.
The total electronic energy of folding, AEfolding, is just one part of
the folding free energy, AGfolding, obtained using the master
equation for the computation of the folding free energy from a
fully unfolded reference structure [19].


A, = AEfolding + AHcorrection - TASfolding + AA Gsolvation (1)

Here AHcorrection represents enthalpic corrections to the electronic
energy, ASfolding is the entropy change of folding and AAG.ovtion
is the difference in the solvation free energy of the folded and
unfolded states.
In order to reliably calculate the free energies of native and
nonnative protein folds, the errors associated with each term of
Equation 1 must be minimized. In view of this, our interaction
energy error estimates should be considered "best-case scenario"
or lower-limit free energy error estimates because they neglect the
error coming from the enthalpy, entropy, and solvation energy
terms. Nonetheless, if the potential energy surface is not well
modeled this will impact the quality of any estimate of entropy
because a poor quality potential energy surface can have effects on
entropy estimates that can be difficult to predict (e.g., sampling of
non-physical states). To obtain a reliable estimate of the free
energy it is essential that AE be well reproduced to ensure the
quality of the AS estimate. While it is true that systematic errors in
the three remaining terms of Equation 1 may cancel favorably
with systematic errors in AE, this effect has not yet been studied in
detail. Random error estimates, however, will only increase with
the addition of terms with nonzero uncertainties.


d energy landscape







lergy landscape


Figure 2. Distortions in computed energy landscapes due to error propagation. If each microstate of a protein under study contains a
significant amount of error in its calculated energy (shown here as error bars), computed folding surfaces become distorted with respect to the actual
folding surface. This effect introduces difficulty in distinguishing between local minima on the folding surface and in finding the native folds of
proteins. This effect is magnified for especially large proteins with many intramolecular contacts contributing to their stable protein folds.
doi:1 0.1 371/journal.pone.001 8868.g002


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Energy Computation Paradox and Protein Folding


Our method of error analysis assumes that electronic interaction
energies are additive, even 1 .... 1. free energies of interacting
fragments are not [19]. This approximation is supported by both
statistical mechanics [29] and isothermal calorimetry experiments
[30] involving protein-ligand interactions, but deviations from
additivity will impact the overall quality of our error estimates.
Nonetheless, it is very instructive to examine error estimates for
the AEfoldig term because, within our model, we can compare any
physics-based score function to chemically accurate quantum
chemical methods providing reliable estimates of the magnitudes
of energy errors and their contribution to AGfolding. If the AEfolding
errors are small, their impact on the uncertainty in AGfolding will
be small; otherwise they will have a significant impact on the
accuracy of AGfolding and the ability to predict native protein folds.

Methods

Interacting Fragment Database Generation
In order to generate a reference database of fragment-based
interacting systems involved in protein folds, we examined a native
fold of ubiquitin (PDBID: 1UBQ) [31] in detail. After adding and
optimizing hydrogens in AMBER [32] with the ff99sb force field
[33], the structure was viewed using Chimera [34], which was used
to highlight van der Waals contacts and hydrogen bonds resulting
in a total of 42 of the former and 50 of the latter. Each resulting
fragment interaction was evaluated in terms of gas-phase
interaction energy using a number of different methods. In
generating these fragments, hydrogen "link atoms" were used to
replace the severed bonds with the remainder of the protein.
Energies were evaluated with the ff99sb force field [33], the
Generalized Amber Force Field (GAFF) [35], ff03 [36], AMI [37],
PM3 . | PM6 .',I PDDG [40], PM6-DH2 [41], HF, MP2,
B97-D [42], M06, and M06-L [43]. The ff99sb and ff03 force field
methods underwent an atomic charge scaling procedure to
produce correct net charges on the database fragments. This
was necessary because the sum of parameterized force field
charges on one fragment often did not equal the total charge used
when calculating electronic energy with a QM reference method.
Unless the force field charges are scaled properly, additional errors
due to the lack of charge conservation are introduced [44].
The ab initio quantum-based methods employed several basis
sets and included the counterpoise correction for basis set
superposition error. Moller-Plesset perturbation theory 1.... 1
second order , Ii'_' with complete basis set extrapolations (CBS)
[45] from the aug-cc-pVTZ and aug-cc-pVQZ basis sets (hereafter
abbreviated as aXZ: X= D,T,Q) were used for most of the
reference values. Based on previous reports [46] and our
experience with error analysis on protein-ligand systems, the
coupled cluster method with single, double, and perturbative triple
excitations (CCSD(T)) CBS energies showed the largest improve-
ment from MP2/CBS for systems containing aromatic groups.
However, the present case contained no aromatic-aromatic
interactions and only eight total aromatic-nonpolar contacts.
Hence, CCSD(T)/CBS reference energies were computed (as
described in our previous work _'i, only for these fragments. The
addition of more protein molecules and specific interaction types
[47] to our reference database would further improve our ability to
estimate errors, but this is a time-consuming endeavor requiring a
large number of high-level quantum mechanical energy calcula-
tions [48].
The ff99sb and GAFF calculations were conducted with the
AMBER 11 suite of programs [32], and the ff03 energies were
calculated with the Schrodinger package [49]. AMI, PM3, PM6,
and PDDG energies were calculated with DivCon [50], and PM6-


DH2 energies were calculated with MOPAC2009 [51]. HF, MP2,
B97-D, M06, and M06-L energies were calculated with Gaussian
09 [52], and the CCSD(T)/CBS corrections used to generate
reference values were calculated with Molpro 2009 [53] and
NWChem 5.1 [54].

Results

A summary of the fragment-based interaction energy deviations
from reference energies is displayed in Table 1. The absolute
deviations from our reference data were fitted to Gaussian error
probability density functions with parameters li (mean error per
interaction) and c (standard deviation of the errors). The resulting
plots are presented in Table SI. The error distribution for B97-D/
TZVP is shown as an example in Figure 3. The fragments were
divided into two classes: nonpolar (van der Waals - blue) and polar
i, .i .._. .. bonding - red) interactions. In the case of B97-D/
TZVP, the mean error, representing the correctable systematic
error, is -0.29 kcal/mol and 0.59 kcal/mol per interaction for
nonpolar and polar interactions, respectively. The variance,
representing random error, is 0.02 (kcal/mol)2 for nonpolar and
0.158 (kcal/mol)2 for polar interactions. Thus this computational
model has a relatively precise description of van der Waals
interactions with only a slight offset, but it has a wider distribution
of errors for polar interactions.



Table 1. Interaction Energy Error Statistics of the 1 UBQ
Fragment Database.


Method tAIl F2AII pVdw (2VdW pPoIar F2polar R-factora

GAFF 0.36 3.26 0.25 0.36 0.46 5.64 0.127
FF99SBb 0.73 4.04 0.12 1.27 1.22 5.83 0.170
FF03b 0.83 6.61 0.18 0.81 1.36 10.86 0.259
AM1 3.15 9.50 1.04 0.70 4.85 10.28 0.373
PM3 2.65 7.89 0.14 0.77 4.67 4.59 0.352
PM6 1.67 2.24 0.84 0.32 2.34 2.82 0.211
PM6-DH2 0.30 1.23 0.09 0.10 0.62 1.95 0.071


PDDG
HF/6-31G*
HF/aDZ
HF/aTZ
HF/aQZ
MP2/6-31G*
MP2/aDZ
MP2/aTZ
B97-D/TZVP
M06/6-31G"
M06/aTZ
M06-L/6-31 G
M06-L/aTZ


3.21 16.23 0.62
1.94 1.32 2.27
2.14 1.22 2.29
2.10 1.17 2.28
2.08 1.16 2.28
1.24 0.64 1.12
0.48 0.16 0.21
0.16 0.02 0.05
0.20 1.06 0.29
0.75 0.42 0.63
0.73 0.16 0.57
0.71 0.43 0.40
0.75 0.14 0.55


Mean and variance of interaction energy deviations (kcal/mol) from reference
energies (a mix of MP2/CBS and CCSD(T)/CBS) for the interacting fragment
molecules present in ubiquitin. The set of fragments was divided into 42 van
der Waals interactions and 50 polar interactions. The related plots are presented
in Table S1. a) The calculated R-factor serves as an analogy to the residual
minimized in crystallographic structure refinement. A desirable value of R-factor
would be less than 0.1. b) The force field- based atomic charge parameters were
scaled to yield correct net charge on each fragment system.
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Energy Computation Paradox and Protein Folding


Error Distributions for 1 UBQ fragments with B97-D/TZVP


-4 -3 -2 -1 0 1 2 3 4
Deviation from Reference Energy (kcal/mol)


Figure 3. Histogram and probability density functions describing errors in B97-D/TZVP absolute electronic interaction energies of
molecular fragments built from the native fold of ubiquitin.
doi:10.1371/journal.pone.0018868.g003


Estimation of Errors in Protein Folds
With this database of interaction energy errors in place, lower-
limit estimates of both systematic and random errors can be
obtained for protein fold energies. Along with calculating the
energy of a protein fold, an analysis of its interacting fragment
components can be made. By determining the interaction type, an
estimate for a fragment's contribution to the overall error can be
retrieved from the database. The overall error is then propagated
as


Errorsystematic = Nii (2)





ErrorRandom = No) (3)


where i runs over all interaction types (e.g. polar, nonpolar) stored
in the database, Ni is the number of interactions of type i found in
the analyzed protein fold, and p; and of2 are the mean error per
interaction and variance about the mean error for interaction type
i. Note that total systematic error depends on the number of each
type of interaction and thus will not exactly cancel when
comparing different protein folds, since the folds may have
different numbers of interaction types. The evaluated total
systematic error should be subtracted from the evaluated energy
and the evaluated total random error can be reported along with
the corrected energy value.
In the case of B97-D/TZVP for ubiquitin (Figure 3), by using
the appropriate error probability density functions we estimate the
total systematic error to be 17.3 kcal/mol and the random error to
be 8.9 kcal/mol. Hence, the estimated systematic error is
comparable to a typical folding free energy of a protein, but this
error can be corrected. Unfortunately, the remaining random
error is still a significant portion of a typical folding free energy.


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Any other protein fold with a calculated energy within this
8.9 kcal/mol error bar should be considered indistinguishable
from the native fold by the computational method. The B97-D/
TZVP case represents a favorable example with small mean errors
and relatively tight error distributions (see Table 1 for other
examples), but it would be computationally intractable to use it to
study hundreds or thousands of decoys for a system of the size of
ubiquitin. More approximate and computationally accessible
methods yield higher estimated errors. For example, ff99sb yields
a systematic error of 66.0 kcal/mol and a random error of
18.4 kcal/mol. Such magnitudes of error bars are disturbing, since
any non-native structure lying within the 18.4 kcal/mol range
could not be distinguished from the native structure with ff99sb.
Furthermore, these error bars become even larger as larger
proteins with more molecular contacts are examined.
The magnitudes of these errors lead us to predict that current
physics-based scoring functions used in ab initio protein folding
studies can have total energy errors much larger than the folding
free energies of typical proteins. Hence, we conclude that accurate
energy computation and error reduction represents a major
stumbling block along with sampling in the achievement of an
ultimate solution to the ab initio protein-folding problem. However,
we are now in a position to correct for systematic errors, thereby
improving our computational outcomes.

Discrimination between Native and Decoy folds - the
Rosetta Decoy Set
In order to test our error hypothesis, we performed energy
calculations and error corrections on a portion of the Rosetta
decoy set, which contained 49 protein systems. Each one
comprised a crystal structure from the Protein Databank, 20
versions of the crystal structure that were relaxed with the Rosetta
score function, and 100 low energy decoys produced by Rosetta
[55]. The protein structures ranged from 50 to 146 residues in
chain 1. i, 11, so semiempirical QM calculations (utilizing modern
linear scaling methods) were feasible. Energies of all 5929 protein


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Energy Computation Paradox and Protein Folding


structures were calculated with ff99sb, PM6, and PM6-DH2. The
ff99sb calculations were performed with the generalized Born
implicit solvent model, and the PM6 and PM6-DH2 calculations
used the COSMO solvent model in MOPAC. Each structure was
then analyzed for the number of nonpolar and polar interactions
and the corresponding energy was corrected according to the
appropriate error probability density functions. To measure
improvement due to energy corrections, three values were
monitored: EGAP, z-score, and EBO. The EGAP (energy gap)
was defined as the difference between the energies of the lowest
energy decoy and the lowest native structure. The z-score is the
ratio of the difference between the lowest native fold energy and
the average energy of all folds to the standard deviation of all fold
energies. EBO (error bar overlap) yields true if the native structure
is found to lie within the lowest energy error bar; otherwise it is
false.
The results for the Rosetta decoy set analysis can be found in
Tables S2, S3, S4, and will only be summarized here. By
measuring EGAP, improvements due to error corrections were
observed in 27, 36, and 31 of the protein datasets for ff99sb, PM6,
and PM6-DH2, respectively. By measuring z-scores, improve-
ments were seen in 38, 41, and 39 of the systems. The native
structures were found within the lowest energy error bar in 45, 49,
and 49 of the systems. Overall, systematic error correction offers
some benefit, but improvement was not uniform. We observed
that, while the magnitudes of energy corrections varied over a set
of decoys, that variation was small. That is, both native and decoy
folds had significant systematic errors, but the changes in relative
energies after error correction were usually small. The decoy set
for PDBID: 1H6Z, for example, had an average energy correction
of 51.3 -5.2 kcal/mol for ff99sb, and the native structure had an
error correction of 56.9 kcal/mol. All ..-.l much of the
systematic error canceled when measuring EGAP, the improve-
ments due to error correction can still be significant compared to
folding free energies. This would especially be true if we had


included more non-native decoy folds in our analysis. The decoys
in this set are very native-like and have roughly the same number
of intramolecular contacts, leading to similar magnitudes of error
corrections. The difference in systematic errors between a native
structure and a partially unfolded structure is expected to be much
greater.
While systematic error can be estimated and removed, random
error in energy scores isn't easily correctable and represents poor
precision in scoring functions. After energy corrections, the native
structure of 1H6Z was not the lowest energy structure with ff99sb,
but its corrected energy was found to lie within the lowest energy
error bar. This result highlights a main disadvantage of using a
method with large random error since the native and decoy folds
could not be distinguished due to overlapping random error bars.
To highlight the dependence of the total random error on system
size, we estimated the total random error of the 5929 protein folds
with the three scoring functions. The relationship is shown in
Figure 4. As we examine larger proteins, their potential energy
surfaces should become increasingly more distorted due to
increased random error in the energy functions, which can have
unpredictable effects on the free energy landscape.

Discussion

Folded proteins are characterized by numerous van der Waals
and hydrogen bonding interactions that need to be accurately
accounted for when using physics-based score functions. Even
small errors in calculated energies between interacting partners
within a protein quickly accumulate to produce large overall
uncertainties in calculated total energies. This effect of error
propagation distorts the calculated potential energy surface of a
protein in a very complicated way, and therefore alters the shape
of the folding funnel in ways that are difficult (if possible) to
predict. One is only able to distinguish protein folds by energy
when energy differences are larger than their individual error bars.


Random Error Magnitudes vs. Chain Length in Rosetta Decoy Set


40 60 80 100
Chain Length


120 140 160


Figure 4. Dependence of random error estimates on chain length. Larger protein folds have more intramolecular interactions and thus larger
propagated random errors in evaluated total energies. This effect is expected to lead to difficulty in predicting the native folds of large proteins since
it leads to unpredictable distortions in the overall energy surface.
doi:1 0.1 371/journal.pone.001 8868.g004


PLoS ONE I www.plosone.org 6 April 2011 I Volume 6 1 Issue 4 1 e18868







Energy Computation Paradox and Protein Folding


Rather than having few native-like structures at the bottom of the
folding funnel, it now should be extended to include any fold
within the lowest energy error bar at the bottom. The bottom of
the folding funnel may even be populated with non-native states
predicted to be native by the scoring function, with the true native
states higher in calculated energy but perhaps with error bars
overlapping with the incorrectly predicted native states.
Since the free energy difference between a native and denatured
protein fold can be on the order of 10 20 kcal/mol, the errors in
interaction energies of the magnitude predicted herein suggest that
we are a long way from computing energies between native and
decoy folds at a level of accuracy necessary to generally solve the
ab initio protein folding problem, especially as larger proteins are
examined.
We have presented and demonstrated the use of a method to
estimate the magnitude of errors in computed energies of proteins
and shown that these can be corrected for in part, thereby
improving results obtained from physics-based scoring functions.
Systematic error correction can be applied as an endpoint
calculation or it can be computed on the fly, for example, in
interactive protein folding gaming exercises [56]. In addition, the
generation of error probability density functions provides a
ii ili.. ,i .method of analyzing and comparing different
score functions in terms of their ability to accurately model
molecular interactions. The research outlined herein brings a new
level of sophistication to energy computation that has largely been
lacking in computational biology and chemistry, opening the door
for novel ways to compare and improve modern scoring functions
used in studying complex systems ;,1. 1 ,i _.: numbers of inter- and
intramolecular interactions.


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Supporting Information

Table Si Table of error probability density functions
for each method studied. The blue curves represent the error
distributions of the nonpolar interactions, the red curves
correspond to polar interactions, and the black curves represent
all interactions. The numbers below each plot represent the
expected systematic and random error in the composite ubiquitin
system. Energy units are in kcal/mol.
(PDF)

Table S2 Table displaying statistics related to the
analysis of the Rosetta decoy set before and after energy
corrections for the ff99sb force field.
(PDF)

Table S3 Table displaying statistics related to the
analysis of the Rosetta decoy set before and after energy
corrections for PM6.
(PDF)

Table S4 Table displaying statistics related to the
analysis of the Rosetta decoy set before and after energy
corrections for PM6-DH2.
(PDF)


Author Contributions

Conceived and designed the experiments: KMM. Performed the
experiments: JCF MLB BW XH BPR MSM CDS. Analyzed the data:
JCF KMM. Wrote the paper: JCF KMM.


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