Direct-coupling analysis of residue coevolution
captures native contacts across many protein families
Faruck Morcosa,1, Andrea Pagnanib,1, Bryan Lunta, Arianna Bertolinoc, Debora S. Marksd, Chris Sandere,
Riccardo Zecchinab,f, José N. Onuchica,g,2, Terence Hwaa,2, and Martin Weigtb,h,2
aCenter for Theoretical Biological Physics, University of California at San Diego, La Jolla, CA 92093-0374;
10126 Turin, Italy;
School, 20 Longwood Avenue, Boston, MA 02115;
York, NY 10065;
gCenter for Theoretical Biological Physics, Rice University, Houston, TX 77005-1827; and
Recherche 7238, Université Pierre et Marie Curie, 15 rue de l’École de Médecine, 75006 Paris, France
bHuman Genetics Foundation, Via Nizza 52,
dDepartment of Systems Biology, Harvard Medical
cInstitute for Scientific Interchange, Viale Settimio Severo 65, 10133 Turin, Italy;
eMemorial Sloan–Kettering Cancer Center, Computational Biology Center, 1275 York Avenue, New
fCenter for Computational Studies and Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy;
hLaboratoire de Génomique des Microorganismes, Unité Mixte de
Contributed by José N. Onuchic, October 12, 2011 (sent for review July 22, 2011)
The similarity in the three-dimensional structures of homologous
proteins imposes strong constraints on their sequence variability.
It has long been suggested that the resulting correlations among
amino acid compositions at different sequence positions can be
exploited to infer spatial contacts within the tertiary protein struc-
ture. Crucial to this inference is the ability to disentangle direct and
indirect correlations, as accomplished by the recently introduced
direct-coupling analysis (DCA). Here we develop a computationally
efficient implementation of DCA, which allows us to evaluate the
accuracyof contactpredictionby DCA for a largenumberof protein
domains, based purely on sequence information. DCA is shown to
yield a large number of correctly predicted contacts, recapitulating
the global structure of the contact map for the majority of the pro-
tein domains examined. Furthermore, our analysis captures clear
signals beyond intradomain residue contacts, arising, e.g., from
alternative protein conformations, ligand-mediated residue cou-
plings, and interdomain interactions in protein oligomers. Our
findings suggest that contacts predicted by DCA can be used as a
reliable guide to facilitate computational predictions of alternative
protein conformations, protein complex formation, and even the
de novo prediction of protein domain structures, contingent on the
existence of a large number of homologous sequences which are
being rapidly made available due to advances in genome sequen-
statistical sequence analysis ∣ residue–residue covariation ∣
contact map prediction ∣ maximum-entropy modeling
tures of proteins (1–10). However, such studies require a large
number (e.g., the order of 1,000) of homologous yet variable pro-
tein sequences. In the past, most studies of this type have there-
fore been limited to a few exemplary proteins for which a large
number of such sequences happened to be already available.
However, rapid advances in genome sequencing will soon be
able to generate this many sequences for the majority of common
bacterial proteins (11). Sequencing a large number of simple
eukaryotes such as yeast can in principle generate a similar num-
ber of common eukaryotic protein sequences. In this paper, we
provide a systematic evaluation of the information contained in
correlated substitution patterns for predicting residue contacts, a
first step toward a purely sequence-based approach to protein
The basic hypothesis connecting correlated substitution pat-
terns and residue–residue contacts is very simple: If two residues
of a protein or a pair of interacting proteins form a contact, a
destabilizing amino acid substitution at one position is expected
to be compensated by a substitution of the other position over the
evolutionary timescale, in order for the residue pair to maintain
attractive interaction. To test this hypothesis, the bacterial two-
component signaling (TCS) proteins (12) have been used because
orrelated substitution patterns between residues of a protein
family have been exploited to reveal information on the struc-
of the large number of TCS protein sequences, which already
numbered in the thousands 5-y ago (13). Simple covariance-based
analysis was first applied to characterize interactions between
residues belonging to partner proteins of the TCS pathways
(14, 15); it was found to partially predict correct interprotein
residue contacts, but also many residue pairs which are far apart.
A major shortcoming of covariance analysis is that correlations
between substitution patterns of interacting residues induce
secondary correlations between noninteracting residues. This
problem was subsequently overcome by the direct-coupling ana-
lysis (DCA) (16, 17), which aims at disentangling direct from
indirect correlations. The top 10 residue pairs identified by DCA
were all shown to be true contacts between the TCS proteins, and
they were used to guide the accurate prediction (3-Å rmsd) of the
interacting TCS protein complex (18, 19). Furthermore, DCA
was used to shed light on interaction specificity and interpathway
cross-talk in bacterial signal transduction (20).
Due to rapid advances in sequencing technology, there exists
by now a large number of bacterial genome projects, approxi-
mately 1,700 completed and 8,300 ongoing (11). These genome
sequences can be used to compute correlated substitution pat-
terns for a large number of common bacterial proteins and inter-
acting protein pairs, even if they are not duplicated (i.e., present
at one copy per genome on average). DCA can then be used in
principle to infer the interacting residues and eventually predict
tertiary and quaternary protein structures for the majority of bac-
terial proteins, as has been done so far for the TCS proteins. Here
we address a critical question for this line of pursuit—how well
does DCA identify native residue contacts in proteins other
Previously, a message-passing algorithm was used to imple-
ment DCA (16). This approach, here referred to as mpDCA, was
rather costly computationally because it is based on a slowly
converging iterative scheme. This cost makes it unfeasible to ap-
ply mpDCA to large-scale analysis across many protein families.
Here we will introduce mfDCA, an algorithm based on the mean-
field approximation of DCA. The mfDCA is 103to 104times fas-
ter than mpDCA, and hence can be used to analyze many long
protein sequences rapidly. By analyzing 131 large domain families
for which accurate structural information is available, we show
Author contributions: F.M., A.P., J.N.O., T.H., and M.W. designed research; F.M., A.P., B.L.,
A.B., D.S.M., R.Z., and M.W. performed research; A.P. and M.W. contributed new reagents/
analytic tools; F.M., B.L., A.B., D.S.M., C.S., J.N.O., T.H., and M.W. analyzed data; and F.M.,
A.P., C.S., J.N.O., T.H., and M.W. wrote the paper.
The authors declare no conflict of interest.
1F.M. and A.P. contributed equally to this work.
2To whom correspondence may be addressed. E-mail: email@example.com, firstname.lastname@example.org,
See Author Summary on page 19459.
This article contains supporting information online at www.pnas.org/lookup/suppl/
www.pnas.org/cgi/doi/10.1073/pnas.1111471108PNAS ∣ December 6, 2011 ∣ vol. 108 ∣ no. 49 ∣ E1293–E1301
that mfDCA captures a large number of intradomain contacts
across these domain families. Together, the predicted contacts
are able to recapitulate the global structure of the contact map.
Many cases, where mfDCA finds strong correlation between
distant residue pairs, have interesting biological reasons, includ-
ing interdomain contacts, alternative structures of the same
domain, and common interactions of residues with a ligand. The
mfDCA results are found to outperform those generated by sim-
ple covariance analysis as well as a recent approximate Bayesian
Results and Discussion
A Fast DCA Algorithm. In this study, we wish to characterize the
correlation between the amino acid occupancy of residue posi-
tions as a predictor of spatial proximity of these residues in folded
proteins. Starting with a multiple-sequence alignment (MSA) of a
large number of sequences of a given protein domain, extracted
using Pfam’s hidden Markov models (HMMs) (21, 22), the basic
quantities in this context are the frequency count fiðAÞ for a single
MSA column i, characterizing the relative frequency of finding
amino acid A in this column, and the frequency count fijðA;BÞ
for pairs of MSA columns i and j, characterizing the frequency
that amino acids A and B coappear in the same protein sequence
in MSA columns i and j. Alignment gaps are considered as the
21st amino acid. Mathematical definitions of these counts are
provided in Methods.
The raw statistical correlation obtained above suffers from a
sampling bias, resulting from phylogeny, multiple-strain sequen-
cing, and a biased selection of sequenced species. The problem
has been discussed extensively in the literature (10, 23–26). In this
study, we implemented a simple sampling correction, by counting
sequences with more than 80% identity and reweighting them in
the frequency counts. All the frequency calculations and results
reported below are obtained using this sampling correction; the
number of nonredundant sequences is measured as the effective
sequence number Meffafter reweighting (see Methods). The com-
parison to results without reweighting and to reweighting at 70%
in SI Appendix, Fig. S1 shows that reweighting systematically
improves the performance of DCA, but results are robust with
respect to precise value of reweighting.
A simple measure of correlation between these two columns
is the mutual information (MI), defined by Eq. 3 in Methods. As
we will show, the MI turns out to be an unreliable predictor of
spatial proximity. Central to our approach is the disentanglement
of direct and indirect correlations, which is attempted via DCA,
which takes the full set of fiðAÞ and fijðA;BÞ as inputs, and infers
“direct statistical couplings,” which generate the empirically
measured correlations. Their strength is quantified by the direct
information (DI) for each pair of MSA columns; see Eq. 12 in
Methods and ref. 16. However, the message-passing algorithm
used to implement DCA in ref. 16, mpDCA, was computationally
intensive, thus limiting its use in large-scale studies. Here we de-
veloped a much faster heuristic algorithm based on a mean-field
approach; see Methods. This algorithm, termed mfDCA, is able to
perform DCA for alignments of up to about 500 amino acids per
row, as compared to 60–70 amino acids in the message-passing
approach. For the same protein length, mfDCA is about 103to
104times faster, which results mainly from the fact that the costly
iterative parameter learning in mpDCA can be solved analytically
in a single step in mfDCA. This performance gain enabled us to
systematically analyze hundreds of protein domains and examine
the extent to which a high DI value is a predictor of spatial proxi-
mity in a folded protein. Many residue-position pairs, which
are close neighbors along the sequence, also show high MI
and/or DI. To evaluate nontrivial predictions, we therefore
restricted our analysis throughout the paper to pairs, which are
separated by at least five positions along the protein’s backbone.
Intradomain Contacts. We shall first illustrate the correlation
between the DI values and the spatial proximity of residue pairs
through a specific example, namely the domain family homolo-
gous to the DNA-recognition domain (region 2) of the bacterial
Sigma-70 factor (Pfam ID PF04542). The mfDCA was used to
compute the DI values using an Meffof approximately 3,700 non-
redundant sequences—i.e., below a threshold of 80% sequence
identity. The MSA columns with the 20 largest DI and MI values
are mapped to the sequence of the SigmaE factor of Escherichia
coli (encoded by rpoE) whose structure has been solved to 2-Å
resolution [Protein Data Bank (PDB) ID 1OR7; ref. 27]. The
residue pairs with the 20 highest ranked DI values are connected
by bonds of different colors in Fig. 1A. Those residue pairs with
minimum atomic distances <8 Å are defined as “contacts” and
are shown in red, the others in green.* Because only one out of
the top 20 DI pairs is green, DI is seen as a good predicator of
spatial contact, characterized by a true positive (TP) rate of 95%
for this protein. A similar analysis using the 20 highest MI values
(Fig. 1B) yielded 13 contacts (TP ¼ 65%), illustrating a reduced
predictive power by the simple covariance analysis. Furthermore,
we see that the DI predictions are more evenly distributed over
the entire domain, whereas many of the MI predictions are asso-
ciated with a few residues; this difference is significant for contact
map prediction and will be elaborated upon below.
In order to test the generality of the predictive power of DI
ranking as contacts, we applied the above analysis to 131 predo-
minantly bacterial domain families (with >90% of the sequences
belonging to bacterial organisms). These families were selected
according to the following two criteria (see Methods for details):
(i) The family contains Meff> 1;000 nonredundant sequences
after applying sampling correction for >80% identity, in order to
ensure statistical enrichment, and (ii) there exist at least two
available high-quality X-ray crystal structures (independent PDB
entries of resolution <3 Å), so that the degree of spatial proxi-
mity between each residue pair can be evaluated. The selected
domain families encompassed a total of 856 different PDB struc-
tures (see SI Appendix, Table S1). Note that Meffis found to
be typically in the range of one-third to one-half of the total
sequence number M (see SI Appendix, Fig. S2).
2 of the bacterial Sigma factor (Pfam ID PF04542) mapped to the sequence of
the SigmaE factor of E. coli (encoded by rpoE) (PDB ID 1OR7). A shows the top
20 DI predictions, and B shows the top 20 MI predictions for residue–residue
contacts, both with a minimum separation of five positions along the back-
bone. Each pair with distance <8 Å is connected by a red link, and the more
distant pairs are connected by the green links.
Contact predictions for the family of domains homologous to Region
*The choice of the relatively large value of 8-Å minimum atom distance as a cutoff value
for contacts is supported later in the discussion of Fig. 2B, where the distance distribution
of the top DI pairings is analyzed.
www.pnas.org/cgi/doi/10.1073/pnas.1111471108Morcos et al.
We computed the DI values for each residue pair of the 131
domain families and evaluated the degree to which high-ranking
DI pairs corresponded to actual contacts (minimum atomic dis-
tances <8 Å), based on the available structures for each domain.
The results are shown in Fig. 2A(black star). The x axis represents
the number of top-ranked DI pairs (separation >5 positions along
the sequence) considered and the y axis is the average fraction
of pairs up to this DI ranking that are true contacts. The latter
was calculated using the best-predicted structure†(i.e., the PDB
structure with the highest TP value) for each of the 131 families.
Similar results were obtained when considering all the available
structures; see below. In contrast, results computed using MI
ranking (red circle) gave significantly reduced TP rates.‡Also
shown in Fig. 2A are results generated by an approximate Baye-
sian approach, which has been established as the currently
best-performing algorithm in identifying contacts from sequence
correlation analysis (10). The Bayesian approach (yellow trian-
gle) is seen to perform better than the simple covariance analysis
(MI), but TP rates are not as high as the ones obtained by
mfDCA. Analogous results for the relative performance of these
methods are also observed for a collection of 25 eukaryotic pro-
teins analyzed (see SI Appendix, Fig. S3), suggesting that the
applicability of DCA is not restricted to bacterial proteins.
As seen in Fig. 2A, on average 84% of the top 20 DI pairs
found by mfDCA (black star, black solid curve) are true contacts.
The average TP rate is indicative of the TP of typical domain
families, as the individual TPs for the 131 families examined
are distributed mostly in the range of 0.7–1.0; see SI Appendix,
Fig. S4A evaluated using the best-predicted structure and SI
Appendix, Fig. S4B when all 856 structures are used. This figure
also shows little difference in the quality of the prediction using
the top 10, 20, or 30 DI pairs, and coherent results between
the best-predicted and all 856 structures, despite the somewhat
uneven distribution of available PDB structures over the 131 do-
main families. The distribution of the actual (minimum atomic)
intradomain distances between residue pairs with the top 10, 20,
and 30 DI ranking are shown in Fig. 2B, using the complete set of
856 PDB structures. The distribution exhibits a strong peak
around 3–5 Å with a weaker secondary peak around 7–8 Å, for
all three sets of DI rankings used. This double-peak structure is a
characteristic feature of the DCA results. It is not observed in the
background distribution of all residue pairs (see SI Appendix,
Fig. S5, which has a single maximum around 20–25 Å). In Fig. 2B,
this background is reflected by a small bump in the histograms
for the top 20 and 30 DI ranking pairs. The two short-distance
peaks are consistent with the biophysics of molecular contacts:
The first peak presumably arises from short-ranged interactions
like hydrogen bonding or pairings involved in secondary structure
formation, whereas the second peak likely corresponds to long-
ranged, possibly water-mediated contacts (28–30). The observa-
tion of this second, biologically reasonable peak in Fig. 2B also
motivates the choice of 8 Å as a cutoff distance for what is con-
sidered a residue–residue contact in Figs. 1 and 2A.
To understand how many sequences are actually needed for
mfDCA, we randomly generated subalignments for two protein
families; see SI Appendix, Fig. S6. For at least these two families,
an effective number of Meffof approximately 250 is already suffi-
cient to reach TP rates close to one for the top predicted residue
pairs, and the predictive power increases monotonously when
more sequences are available. These numbers are consistent with
but slightly larger than the sequence requirements reported in
ref. 31 for the statistical-coupling analysis originally proposed
in ref. 5.
Long-Distance High-DI Residue Pairs. The results from the previous
section illustrate the ability of mfDCA to identify intradomain
contacts with high sensitivity. However, a small fraction of
pairs showed high DI values (in the top 20–30 ranking) but were
located far away according to the available crystal structure. Here
we investigate various biological reasons for the appearance of
such long-distance direct correlations.
Interdomain Residue Contacts. Given the biological role of some
interdomain contacts (32), we studied if the appearance of
long-distance high-DI pairs may be due to interactions between
proteins which form oligomeric complexes, as described pre-
viously for the dimeric response regulators of the bacterial two-
component signaling system (16). To further investigate this
possibility, we examined members of the 131 proteins which
formed homodimers or higher-order oligomers according to the
corresponding X-ray crystal structures.
A first example is the ATPase domain of the family of the
nitrogen regulatory protein C (NtrC)-like sigma54-dependent
transcriptional activators (Pfam PF00158). Upon activation, dif-
ferent subunits of this domain are known to pack in the front-
for each of the 131 domains studied. DI results (★) clearly outperform the other two methods: MI (red ⦁) and an approximate Bayesian approach (yellow ▾)
developed by Burger and van Nimwegen (10). Their method aims at disentangling direct and indirect correlations by averaging over tree-shaped residue–
residue coupling networks, and it contains a phylogeny correction. The method can also reach length-400 multiple alignments as mfDCA does; our imple-
mentation follows closely the description in ref. 6. However, coupling trees do not allow for multiple coupling paths between two residues as DCA does,
possibly accounting for its lower TP rates compared to mfDCA. (B) The mfDCA predictions for the top 10, 20, and 30 residue pairs show a bimodal distribution
of intradomain distances with two frequency peaks around 3–5 and 7–8 Å.
(A) Mean TP rate for 131 domain families, as a function of the number of top-ranked contacts and histogram of the distances of all predicted structures
†The best-predicted structures were used due to the variance in the quality of PDB
structures. Also, for the number of cases where substantially different structures of the
same protein exist in the PDB, the existence of a single structure containing the predicted
contacts substantiates them as contacts of a native conformation of that protein.
‡Both DI and MI benefited modestly from sampling correction; see SI Appendix, Fig. S1 for
a comparison of the performance of these methods with/without sampling correction.
Morcos et al. PNAS
December 6, 2011
to-back orientation to form a heptameric ring, wrapping DNA
around the complex (33). We compared the DCA results to the
structure of NtrC1 of Aquifex aeolicus (PDB ID 1NY6; ref. 33).
Among the top 20 DI pairs, 17 were intradomain contacts. The
three remaining pairs were long-distance (>10 Å) within the
domain. Strikingly, all three were within 5 Å when paired with the
closest position in an adjacent subunit of the heptamer complex;
see Fig. 3. These pairs appear to have coevolved to maintain the
proper formation of the heptamer complex. A second example
of high-DI interdomain contact is shown in SI Appendix, Fig. S7
for the multidrug resistance protein MexA
aeruginosa, where nine subunits oligomerize to form a funnel-like
structure across the periplasmic space for antibiotic efflux (PDB
ID 1VF7; ref. 34).
We further tested the occurrence of interdomain contacts at a
global level. Out of the 131 studied domain families, 21 families
feature X-ray crystal structures involving oligomers with pre-
dicted interdomain contacts (see SI Appendix, Table S3). Among
the top 20 DI pairs that are not intradomain contacts, about half
of them turned out to be interdomain contacts as shown in
Alternative Domain Conformations. Another cause of long-distance
high-DI pairs is the occurrence of alternative conformations for
domains within the same family. As an illustration, we examine
the domain family GerE (Pfam PF00196), whose members in-
clude the DNA-binding domains of many response regulators
in two-component signaling systems.
Using the DNA-bound DNA-binding domain of the nitrate/
nitrite response regulator NarL of E. coli (PDB ID 1JE8; ref. 35)
as a structural template, we found that all of the top 20 DI pairs
are true contacts (red bonds in Fig. 4A). However, when mapping
the same DI pairs to the structure of the full-length transcrip-
tional regulatory protein DosR of Mycobacterium tuberculosis
(PDB ID 3C3W; ref. 36), seven pairs are found at distances >8 Å
(green bonds in Fig. 4B, with the response-regulator domain
shown in gray). Comparison of Fig. 4 A and B clearly shows that
all of the green bonds involve pairing with the C-terminal helix
(shown in light blue), which is significantly displaced in the
full-length structure, presumably due to interaction with the (un-
phosphorylated) regulatory domain. As proposed by Wisedchaisri
et al. (36), a likely scenario is that the DNA-binding domain of
DosR is broken up by the interdomain interaction in the absence
of phosphorylation, whereas phosphorylation of DosR restores
its DNA-binding domain into the active form represented by
the DNA-bound NarL structure.
It is difficult to estimate the extent to which alternative con-
formations may be responsible for the observed long-distance
high-DI contacts, for less characterized domains for which alter-
native conformations may not be known. However, the example
shown in Fig. 4 may motivate future studies to use these long-
(PDB ID 1NY6) out of the top 20 predicted couplets are multimerization contacts. Structures showing each of these three interdomain contacts which are
separated by less than 5 Å in a ring-like heptamer formed by Sigma54 interaction domains. (A) Residue pair GLU(174)-ARG(253), (B) residue pair PHE(226)-TYR
(261), and (C) residue pair ALA(197)-ALA(249). (D) Oligomerization contacts are found in 21 structures of the 131 families studied (see SI Appendix, Table S3).
These contacts represent a significant percentage of long-distance high-DI contacts observed in our predictions.
The only three long-distance high-DI predictions found out of the top 20 DI pairs in the Sigma54 interaction domain of protein NtrC1 of A. aeolicus
response-regulator DNA-binding domain (GerE, PF00196) (containing both
the dark- and light-blue colored regions). In A, the contacts are mapped
to the DNA-binding domain of E. coli NarL, bound to the DNA target
(PDB ID 1JE8). The TP rate for the top 20 DI pairs is 100%, and they are
all shown as red links. In B, the contacts are mapped to the full-length re-
sponse-regulator DosR of M. tuberculosis (PDB ID 3C3W), with the (unpho-
sphorylated) response-regulator domain shown in gray. The top 20 DI
pairings is only 65% in this case (13 red and 7 green links). The difference
in prediction quality for the two structures can be traced back to a major
reorientation of the C-terminal helix of the GerE domain (light blue) in B.
The figures show the top 20 contacts predicted by DI for the family of
www.pnas.org/cgi/doi/10.1073/pnas.1111471108Morcos et al.
distance high-DI contacts to explore possible alternative confor-
Ligand-Mediated Interactions. Another special case of interdomain
residue interactions and another cause of long-distance high-DI
pairing is shown in Fig. 5. Here, mfDCA found the metalloen-
zyme domain family (PF00903) to have a high-DI intradomain
residue pair which is separated by more than 14 Å when mapped
to the glutathione transferase FosA of P. aeruginosa (PDB ID
1NKI; ref. 37). FosA is a metalloglutathione transferase which
confers resistance to fosfomycin by catalyzing the addition of glu-
tathione to fosfomycin. It is a homodimeric enzyme whose activity
is dependent on Mn(II) and Kþ, and the Mn(II) center has been
proposed as part of the catalytic mechanism (37). We observed
that the two residues belonging to the different subunits of the
high-DI pair, Glu110 (pink) and His7 (yellow), are in direct con-
tact (3 Å residue pair and 1.5 Å residue-ligand separation) with
the Mn(II) ion (red) in the dimer configuration (Fig. 5). Thus, the
“direct interaction” between these residues found by mfDCA is
presumably mediated through their common interaction with a
third agent, the metal ion in this case. There may well be other
cases with interactions mediated by binding to other metabolites,
RNA, DNA, or proteins not captured in the available crystal
Contact Map Reconstruction. So far, we have focused on the top 20
DI pairs, which are largely intra- or interdomain contacts. How-
ever, one of the most striking features of the DI result in Fig. 2A is
how gradually the average TP rate declines with increasing DI
ranking. It is therefore possible to turn the question around:
How many residue pairs are predicted, when we require a given
minimum TP rate? For instance, one can go up to a DI ranking of
70 before the average TP rate declines to 70%, meaning that, if
one were to predict contacts using the top 70 DI pairs, one would
have obtained approximately 50 true contacts on average. This
feature may be exploited for sequence-based structure prediction
and deserves further analysis.
To become more quantitative, we define the number of accep-
table pairs NAPxas the (largest) number of DI-ranked pairs
where the specified TP rate (x%) is reached for a given protein.
NAPxcan be viewed as an index that characterizes the number of
contact predictions at a certain acceptable quality level (given
by x). We computed this index for every domain in all 856 struc-
tures in our database, for TP levels of 0.9, 0.8, and 0.7. The results
are shown as cumulative distributions in Fig. 6. A casual inspec-
tion of these distributions shows that there are many structures
with high NAP. Suppose the acceptable TP level is 0.7. The
median of NAP70is 52, meaning that, in half of the structures
examined, the number of high-ranking, predictive DI pairs is at
least 52. Furthermore, 70% of the structures have NAP70> 30
and 34% of the structures have NAP70> 100. A normalized ver-
sion of Fig. 6 with respect to the length of the domain L is shown
in SI Appendix, Fig. S8. In one extreme case involving the family
of bacterial tripartite tricarboxylate receptors (PF03401), NAP70
was 600—i.e., 70% of the top 600 DI pairs correspond to true
contacts when mapped to the best-predicted structure (PDB
ID 2QPQ; ref. 38); see SI Appendix, Fig. S9A. This domain has
a length of L ¼ 274 and has approximately 2,300 contacts. In an-
other example, the extracellular solute-binding family (PF00496)
mapped to the structure of the periplasmic oligopeptide-binding
protein OppA of Salmonella typhimurium (PDB ID 1JET; ref. 39)
has a NAP70of 497 (SI Appendix, Fig. S9B, L ¼ 372, and approxi-
mately 2,530 contacts).
We also computed the NAP70distribution using MI; see SI
Appendix, Fig. S10. The difference between DI and MI, about
10–20% in TP rate according to Fig. 2A, is seen much more sig-
nificantly when displayed according to the NAP index, with the
median NAP70being 5 for MI and 52 for DI, which shows that
DCA generates many more high-valued contact pair predictions.
We also compared the performance of DCA with the approxi-
mate Bayesian method (red dashed curve in SI Appendix,
Fig. S10), which gives a median NAP70of 25 that is halfway
between that of MI and mfDCA.
The large number of contacts correctly predicted by DCA
prompted us to explore the extent to which DCA may be used
to predict the contact maps of protein domains. For a domain
with L amino acids, we calculated the inferred maps by sorting
residue pairs according to their DI, and keeping the 2L high-
est-ranking pairs with minimum separation of five positions along
the sequence. For the contact map prediction, we included
further those pairings which have equal or larger DI than the ones
mentioned above, but with shorter separation along the sequence
because they may be informative about secondary structures.
Fig. 7 shows two examples of such contact map predictions,
for the prokaryotic promoter recognition domain of SigmaE
already shown in Fig. 1 (PDB ID 1OR7, L ¼ 71) and for the eu-
karyotic H-Ras protein (PDB ID 5P21; ref. 40, L ¼ 160). The
figure shows the native contact maps, together with the predic-
tions by MI (Fig. 7, Left) and DI (Fig. 7, Center). Correctly pre-
dicted native contacts (i.e., the TPs) are indicated in red. The
unpredicted native contacts taken from the X-ray crystal struc-
tures are shown in gray, and the incorrect predictions are shown
in green. It is evident that, for both proteins, DI works substan-
is an example of a case where long-distance high-DI pairs are in fact residue
pairs coordinating a ligand. The high-DI pair involving the residues Glu110
(pink) and His7 (yellow) coordinate a metal ion Mn(II) (red) in its dimer
configuration. Kþions are shown as larger spheres (gray and blue), each
coordinated by a monomer of the corresponding color.
The metalloenzyme domain (PF00903) of protein FosA (PDB ID 1NKI)
125 1020 50100 200 500
P(NAPx > n)
TP=0.7 (NAP70 DI)
TP=0.8 (NAP80 DI)
TP=0.9 (NAP90 DI)
a given TP rate x. The curves show the probability of NAPxto be larger than a
given number n for contacts at given TP rates of 0.9, 0.8, and 0.7. The curves
are computed for all 856 PDB structures in the dataset. We observe that the
probability of NAP70> 30 is 70% and NAP70> 100 is 34%, which implies that
a substantial number of protein domains can have accurate predictions that
go beyond the top 30 DI pairings. We also identify some exceptional cases
with NAP70> 600.
Cumulative distribution of the number of acceptable pairs (NAPx) for
Morcos et al.PNAS
December 6, 2011
tially better than MI, both in terms of the TP rate and the
representation of the native contact map. To become more quan-
titative, we have binned the predicted pairs according to their
separation along the primary amino acid sequence (Fig. 7, Right).
We observe that DI captures in particular a higher number and
more accurately those contacts between residues, which are very
distant along the sequence. Also, the DI predictions are more
evenly distributed, whereas MI predictions tend to cluster to-
We have shown the ability of DCA to identify with high-accuracy
residue pairs in domain families that might have coevolved to-
gether and hence are representative of physical proximity in the
three-dimensional fold of the domain. We have done an extensive
evaluation of these capabilities for a large number of families and
individual PDB structures. We found that DCA is not only able to
identify intradomain contacts but also interdomain residue pairs
that are part of oligomerization interfaces. Although we focused
on bacterial proteins, this methodology can be applied to the
ever-increasing number of eukaryotic sequences. Our initial re-
sults suggest that mfDCA performance is conserved for non-
bacterial proteins. One potential application is the identification
of interaction interfaces for homodimers that could ultimately
help in complex structure prediction, e.g., the cases in Fig. 3 and
SI Appendix, Fig. S7. Our results might open unexplored avenues
of research for which full contact maps could be estimated and
used as input data for de novo protein structure identification,
which is particularly interesting in the case of interdomain con-
tacts in multidomain proteins. Ultimately, this methodology can
be utilized with pairs of proteins rather than single proteins to
identify potential protein–protein interactions. An example of
this approach was introduced in ref. 16, however, the current
mathematical formulation of the method as well as its computa-
tional implementation allows an analysis to a much larger scale.
Despite the accuracy of the extracted signal, mfDCA cannot
be expected to extract all biological information contained in
the pair correlations. This idea can be illustrated by comparing
the mfDCA results to those of statistical-coupling analysis
(SCA), developed by Lockless and Ranganathan (5) and used
to identify “coevolving protein sectors” (41). We have applied
mfDCA to the data of ref. 41 for the Trypsin protein family
(Serine protease), where SCA identified three sectors related to
different functionalities of the protein, which cover almost 30%
of all residues. The mfDCA leads to an 83.3% TP rate for the top
30 contact predictions (PDB ID 3TGI; ref. 42)—i.e., to a perfor-
mance which is comparable to the other protein families analyzed
here. Out of the resulting 25 true contact pairs, only eight are
found within the identified sectors. Among them, three are
disulfide bonds (C42∶C58, C136∶C201, C191∶C220) and another
two are inside a catalytic triad crucial for the catalytic activity of
the protein family (H57∶S195, D102∶S195). The other 17 true
contacts predicted by mfDCA are distributed over the protein
fold, without obvious relation to the sectors (see SI Appendix,
Table S4). The difference in prediction can be traced back to
differences in the algorithmic approaches: SCA uses clustering
to identify larger groups of coevolving sites (sectors), whereas
DCA uses maximum-entropy modeling to extract pairs of directly
coupled residues. Thus, the two algorithms extract different and,
in both cases, biologically important information. It remains a
future challenge to develop techniques unifying SCA and DCA,
and to extract even more coevolutionary information from
Data Extraction. Sequence datasets were extracted primarily from Pfam
families with more than 1,000 nonredundant sequences. We decided to focus
on families that are predominantly bacterial (i.e., more than 90% of the
family sequences belong to bacterial organisms). Another requirement in
this dataset is that such families must have at least two known X-ray crystal
structures with a resolution of 3 Å or better. The PDB (43) was accessed
to obtain crystal structures of proteins. An additional criterion to improve
symbols the computational contact predictions using MI or DI ranking (red squares for TP and green squares for spatially distant pairs). The number of pairs is
determined such that there are 2L pairs with minimum separation five along the sequence, where L is the domain length. The right-most panels (C and F) bin
the predictions of MI (blue) and mfDCA (red) according to their separation along the protein sequence. The overall bars count all predictions, the shaded part
the TPs. Note in particular that mfDCA leads to a higher number of more accurate predictions for large separations. (A–C) The promoter recognition helix
domain of the SigmaE factor (PDB ID 1OR7). (D–F) The eukaryotic signaling protein Ras (PDB ID 1P21). For better comparability of native vs. predicted contacts,
the predictions are displayed only above the diagonal.
Two examples of contact map predictions using MI (A and D) and mfDCA (B and E). Gray symbols represent the native map with a cutoff of 8 Å, colored
www.pnas.org/cgi/doi/10.1073/pnas.1111471108Morcos et al.
statistical significance when picking sequences that belong to a particular
Pfam (22) family, was to use a stricter E-value threshold than the standard
used by the software package HMMER (21) to classify domain membership.
An in-house mapping application was developed to map domain family
alignments and predicted couplets to specific residues in PDB structures.
Some of the data extraction tools used in this study are described in more
detail in ref. 17. A total of 131 families were selected that complied with
all these criteria. A list of these Pfam families and the 856 PDB structures ana-
lyzed can be accessed in the SI Appendix, Tables S1 and S2).
For each family, the protein sequences are collected in one MSA denoted
(i.e., the length of the protein domains). Alignments are local alignments
to the Pfam HMM; because of the large number of proteins in each MSA,
we refrained from refinements using global alignment techniques.
LÞj a ¼ 1;:::;Mg, where L denotes the number of MSA columns
Sequence Statistics and Reweighting. As already mentioned in Results and
Discussion, the main inputs of DCA are reweighted frequency counts for
single MSA columns and column pairs:
In this equation, δA;Bdenotes the Kronecker symbol, which equals one if
A ¼ B, and zero otherwise. Furthermore, we have defined q ¼ 21 for the
number of different amino acids (also counting the gap), and a pseudocount
λ (44), whose value will be discussed below. The weighting of the influence of
a single sequence by the factor 1∕maaims at correcting for the sampling bias.
It is determined by the number
ma¼ jfb ∈ f1;…;MgjseqidðAa;AbÞ > 80%gj
of sequences Ab¼ ðAb
sequence identity (seqid) with Aa¼ ðAa
The same reweighting, but with a 100% sequence-identity threshold, would
remove multiple counts of repeated sequences. Reweighting systematically
improves the results (see SI Appendix, Fig. S1), with only a weak dependence
on the precise threshold value (in the range of 70–90%) and the specific
protein family. Last, we introduced the effective sequence number
for calculating the mutual information,
LÞ, b ∈ f1;:::;Mg, which have more than 80%
LÞ, where a itself is counted.
a¼11∕maas the sum over all sequence weights. These counts allow
which equals zero if and only if i and j are uncorrelated, and is positive
Maximum-Entropy Modeling. To disentangle direct and indirect couplings,
we aim at inferring a statistical model PðA1;:::;ALÞ for entire protein se-
quences ðA1;:::;ALÞ. To achieve coherence with data, we require this model
to generate the empirical frequency counts as marginals,
PðA1;…;ALÞ ≡ fiðAiÞ
PðA1;…;ALÞ ≡ fijðAi;AjÞ:
Besides this constraint, we aim at the most general, least-constrained model
PðA1;:::;ALÞ. This model can be achieved by applying the maximum-entropy
principle (45, 46), and it leads to an explicit mathematical form of PðA1;:::;ALÞ
as a Boltzmann distribution with pairwise couplings eijðA;BÞ and local biases
The model parameters have to be fitted such that  is satisfied. In this
fitting procedure, one has to consider that Eq. 5 contains more free para-
meters than there are independent conditions in , which allows one to
change couplings and fields together without changing the sum in the
exponent. Therefore, multiple but equivalent solutions for the fitting are
possible. To remove this freedom, we consider all couplings and fields
measured relative to the last amino acid A ¼ q, and set
∀ i;j;A: eijðA; qÞ ¼ eijðq;AÞ ¼ 0;hiðqÞ ¼ 0.
Details on the maximum-entropy approach are given in the SI Appendix.
Small-Coupling Expansion. Eq. 5 contains the normalization factor Z, in statis-
tical physics also called the partition function, which is defined as
and includes a sum of qLterms. Its direct calculation is infeasible for any rea-
listic protein length and approximations have to be used. In a prior paper
(16), several of us introduced a message-passing approach, which allows
the treatment of about 70 MSA columns simultaneously in about 2-d running
time on a standard desktop computer (larger MSAs need preprocessing to
decrease the number of columns before running message passing). Here
we introduce a much more efficient scheme, which for L ¼ 70 is about 3–4
orders of magnitude faster, and which allows one to directly analyze align-
ments with L ≤ 1;000 (L ≤ 500 on a standard computer because of limited
working memory). The total algorithmic complexity is Oðq3N3Þ. The major
speedup compared to the iterative message-passing solver results from the
fact that parameter inference can be done in a single computational step in
the new algorithm.
The approach is based on a small-coupling expansion (47, 48), which is
explained in detail in the SI Appendix: The exponential of Σi<jeijðAi;AjÞ in
Eq. 7 is expanded into a Taylor series. Keeping only the linear order of this
expansion, we obtain the well-known mean-field equations
containing the single-column counts, as well as a simple relation between
the coupling eijðA;BÞ and the pair counts fijðA;BÞ for all i;j ¼ 1;:::;L and
A;B ¼ 1;:::;q − 1
eijðA;BÞ ¼ −ðC−1ÞijðA;BÞ
CijðA;BÞ ¼ fijðA;BÞ − fiðAÞfjðBÞ:
Eqs. 6 and 9 completely determine the couplings in terms of the data.
Note that the connected-correlation matrix C defined in Eq. 10 is a ðq − 1ÞL ×
ðq − 1ÞL matrix; the pairs ði;AÞ and ðj;BÞ have to be understood as joint single
indices in the inversion in Eq. 9.
In general, when constructed without pseudocounts (λ ¼ 0), this matrix is
not invertible, and formally Eq. 9 leads to infinite couplings. Even introducing
site-specific reduced amino acid alphabets (only those actually observed in
the corresponding MSA column) is found to be not sufficient for invertibility.
The matrix can, however, be regularized by setting λ > 0. For small λ, ele-
ments diverging in the λ → 0 limit dominate the DI calculation discussed
in the next paragraph. To avoid such spurious high DI values, we have to
go to relatively large pseudocounts; λ ¼ Meffis found to be a reasonable
value throughout families and is used exclusively in this paper. SI Appendix,
Fig. S11 shows a sensitivity analysis for different values of the pseudocount
for two domain families. The mean TP rates are computed for pseudocount
values λ ¼ w · Meff, with the weights w ranging from 0.11 to 9. The optimum
value of λ is found for 1 ≤ w ≤ 1.5. Therefore, we used λ ¼ Meffthroughout
Because of the long run time of the message-passing approach (mpDCA),
we could not compare its performance for all proteins studied in this paper.
SI Appendix, Fig. S12 contains two examples: Trypsin (PF00089) and Trypsin
inhibitor (PF00014). In both cases, mfDCA outperforms mpDCA. Furthermore,
it is straightforward to include into DCA also the next order of the small-
Morcos et al.PNAS
December 6, 2011
coupling expansion beyond the mean-field approximation (which corre-
sponds to the so-called Thouless, Anderson, and Palmer (TAP) equations in
spin-glass physics; ref. 49). We do not find any systematic improvement of
the resulting algorithm, called tapDCA, when compared to mfDCA; see
SI Appendix, Fig. S12.
Direct Information. After having estimated the direct coupling eijðA;BÞ
through Eq. 8, we need a strategy for ranking the LðL − 1Þ possible interac-
tions according to their direct-coupling strength. Following the idea that MI
is a good measure for correlations, in ref. 16 we introduced a quantity called
direct information. It can be understood as the amount of MI between
columns i and j, which results from direct coupling alone.
To this end, we introduce for each column pair ði;jÞ an isolated two-site
expfeijðA;BÞ þ~hiðAÞ þ~hjðBÞg;
where the couplings eijðA;BÞ are computed using Eq. 8, and the auxiliary
fields~h are given implicitly by compatibility with the empirical single-residue
As before, in order to reduce the number of free parameters to the number
of independent constraints, these fields are required to fulfill~hiðqÞ ¼
~hjðqÞ ¼ 0. Note that the auxiliary fields have to be determined for each pair
ði;jÞ independently to fit Eq. 12. Finally, we define the DI as the MI of model
Algorithmic Implementation. The algorithmic implementation of the mean-
field approximation is sketched in the following steps:
1. Estimate the frequency counts fiðAÞ and fijðA;BÞ from the MSA, using
the pseudocount λ ¼ Meffin Eqs. 1 and 2.
2. Determine the empirical estimate of the connected-correlation matrix
3. Determine the couplings eijðA;BÞ according to the second of Eq. 9.
4. For each column pair i < j, estimate the direct information DIijby solving
Eqs. 11 and 12 for PðdirÞ
ðA;BÞ, and plug the result into Eq. 13.
An implementation of the code in Matlab is available upon request.
Note Added in Proof. Our direct-coupling analysis was recently used to infer
all-atom protein 3D structures, indicating that the high quality of contact
prediction reported here is capable of translating to good quality predicted
3D folds (50).
ACKNOWLEDGMENTS. We thank Hendrik Szurmant, Joanna Sulkowska, and
Lucy Colwell for useful discussions during the course of this work. This work
was supported by a European Commission Grant 267915 (to R.Z.), the Center
for Theoretical Biological Physics sponsored by the National Science Founda-
tion (NSF Grant PHY-0822283), and by NSF Grant MCB-1051438 (to J.N.O.).
1. Altschuh D, Lesk A, Bloomer A, Klug A (1987) Correlation of co-ordinated amino acid
substitutions with function in viruses related to tobacco mosaic virus. J Mol Biol
2. Göbel U, Sander C, Schneider R, Valencia A (1994) Correlated mutations and residue
contacts in proteins. Proteins Struct Funct Genet 18:309–317.
3. Neher E (1994) How frequent are correlated changes in families of protein sequences?
Proc Natl Acad Sci USA 91:98–102.
4. Shindyalov IN, Kolchanov NA, Sander C (1994) Can three-dimensional contacts in
protein structures be predicted by analysis of correlated mutations? Protein Eng
5. Lockless SW, Ranganathan R (1999) Evolutionarily conserved pathways of energetic
connectivity in protein families. Science 286:295–299.
6. Atchley WR, Wollenberg KR, Fitch WM, Terhalle W, Dress AW (2000) Correlations
among amino acid sites in bHLH protein domains: An information theoretic analysis.
Mol Biol Evol 17:164–178.
7. Fodor AA, Aldrich RW (2004) Influence of conservation on calculations of amino acid
covariance in multiple sequence alignments. Proteins Struct Func Bioinf 56:211–221.
8. Liu Z, Chen J, Thirumalai D (2009) On the accuracy of inferring energetic coupling
between distant sites in protein families from evolutionary imprints: Illustrations
using lattice model. Proteins Struct Func Bioinf 77:823–831.
9. Lashuel HA, Pappu R (2009) Amyloids go genomic: Insights regarding the sequence
determinants of prion formation from genome‐wide studies. Chembiochem
10. Burger L, van Nimwegen E (2010) Disentangling direct from indirect co-evolution of
residues in protein alignments. PLoS Comput Biol 6:e1000633.
11. Liolios K, et al. (2009) The Genomes On Line Database (GOLD) in 2009: Status of
genomic and metagenomic projects and their associated metadata. Nucleic Acids
12. Hoch JA (2000) Two-component and phosphorelay signal-transduction. Curr Opin
13. UlrichLE, ZhulinIB (2009)The MiST2database: A comprehensivegenomics resourceon
microbial signal transduction. Nucleic Acids Res 38:D401–D407.
14. White RA, Szurmant H, Hoch JA, Hwa T (2007) Features of protein-protein interactions
in two‐component signaling deduced from genomic libraries. Methods Enzymol
15. Skerker JM, et al. (2008) Rewiring the specificity of two-component signal transduc-
tion systems. Cell 133:1043–1054.
16. Weigt M, White RA, Szurmant H, Hoch JA, Hwa T (2009) Identificationof direct residue
contacts in protein–protein interaction by message passing. Proc Natl Acad Sci USA
17. Lunt B, et al. (2010) Inference of direct residue contacts in two-component signaling.
Methods Enzymol 471:17–41.
18. Schug A, Weigt M, Onuchic JN, Hwa T, Szurmant H (2009) High-resolution protein
complexes from integrating genomic information with molecular simulation. Proc
Natl Acad Sci USA 106:22124–22129.
19. Schug A, Weigt M, Hoch J, Onuchic J (2010) Computationalmodeling of phosphotrans-
fer complexes in two-component signaling. Methods Enzymol 471:43–58.
20. Procaccini A, Lunt B, Szurmant H, Hwa T, Weigt M (2011) Dissecting the specificity of
protein-protein interaction in bacterial two-component signaling: Orphans and
crosstalks. PLoS One 6:e19729.
21. Eddy SR (1998) Profile hidden Markov models. Bioinformatics 14:755–763.
22. Finn RD, et al. (2010) The Pfam protein families database. Nucleic Acids Res 38:
23. Wollenberg KR, Atchley WR (2000) Separation of phylogenetic and functional associa-
tionsin biologicalsequencesbyusing theparametricbootstrap.Proc NatlAcad SciUSA
24. Tillier ERM, Lui TWH (2003) Using multiple interdependency to separate functional
from phylogenetic correlations in protein alignments. Bioinformatics 19:750–755.
25. Gouveia-Oliveira R, Pedersen AG (2007) Finding coevolving amino acid residues using
row and column weighting of mutual information and multi-dimensional amino acid
representation. Algorithms Mol Biol 2:12–24.
26. Dunn SD, Wahl LM, Gloor GB (2007) Mutual information without the influence of phy-
logeny or entropy dramatically improves residue contact prediction. Bioinformatics
27. Campbell E, et al. (2003) Crystal structure of Escherichia coli sigmaE with the cytoplas-
mic domain of its anti-sigma RseA. Mol Cell 11:1067–1078.
28. Tanaka S, Scheraga HA (1976) Medium- and long-range interaction parameters
between amino acids for predicting three-dimensional structures of proteins. Macro-
29. Go N, Taketomi H (1978) Respective roles of short- and long-range interactions in
protein folding. Proc Natl Acad Sci USA 75:559–563.
30. Miyazawa S, Jernigan RL (2003) Long- and short-range interactions in native protein
structures are consistent/minimally frustrated in sequence space. Proteins Struct Funct
31. Dima RI, Thirumalai D (2006) Determination of network of residues that regulate
allostery in protein families using sequence analysis. Protein Sci 15:258–268.
32. Myers RS, Amaro RE, Luthey-Schulten ZA, Davisson VJ (2005) Reaction coupling
through interdomain contacts in imidazole glycerol phosphate synthase. Biochemistry
33. Lee S-Y (2003) Regulation of the transcriptional activator NtrC1: Structural studies of
the regulatory and AAA+ ATPase domains. Genes Dev 17:2552–2563.
34. Akama H, et al. (2004) Crystal structure of the membrane fusion protein, MexA, of
the multidrug transporter in Pseudomonas aeruginosa. J Biol Chem 279:25939–25942.
35. Maris AE, et al. (2002) Dimerization allows DNA target site recognition by the NarL
response regulator. Nat Struct Biol 9:771–778.
36. Wisedchaisri G, Wu M, Sherman DR, Hol WGJ (2008) Crystal structures of the response
regulator DosR from Mycobacterium tuberculosis suggest a helix rearrangement
mechanism for phosphorylation activation. J Mol Biol 378:227–242.
37. Rigsby RE, Rife CL, Fillgrove KL, Newcomer ME, Armstrong RN (2004) Phosphonofor-
mate: A minimal transition state analogue inhibitor of the fosfomycin resistance
protein, FosA. Biochemistry 43:13666–13673.
38. Herrou J, et al. (2007) Structure-based mechanism of ligand binding for periplasmic
solute-binding protein of the Bug family. J Mol Biol 373:954–964.
39. Tame JRH, Sleigh SH, Wilkinson AJ, Ladbury JE (1996) the role of water in sequence
independent ligand binding by an oligopeptide transporter protein. Nat Struct Biol
www.pnas.org/cgi/doi/10.1073/pnas.1111471108Morcos et al.
40. Pai EF, et al. (1990) Refined crystal structure of the triphosphate conformation of Download full-text
H-ras p21 at 1.35 A resolution: Implications for the mechanism of GTP hydrolysis.
EMBO J 9:2351–2359.
41. Halabi N, Rivoire O, Leibler S, Ranganathan R (2009) Protein sectors: Evolutionary units
of three-dimensional structure. Cell 138:774–786.
42. Pasternak A, Ringe D, Hedstrom L (1999) Comparison of anionic and cationic trypsino-
gens: The anionic activation domain is more flexible in solution and differs in its mode
of BPTI binding in the crystal structure. Protein Sci 8:253–258.
43. Berman HM, et al. (2000) The Protein Data Bank. Nucleic Acids Res 28:235–242.
44. Durbin R, Eddy S, Krogh A, Mitchison G (1998) Biological Sequence Analysis:
Probabilistic Models of Proteins and Nucleic Acids (Cambridge Univ Press, New York),
45. Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620–630.
46. Jaynes ET (1957) Information theory and statistical mechanics. II. Phys Rev
47. Plefka T (1982) Convergence condition of the TAP equation for the infinite-ranged
Ising spin glass model. J Phys A Math Gen 15:1971–1978.
48. Georges A, Yedidia J (1991) How to expand around mean-field theory using high-
temperature expansions. J Phys A Math Gen 24:2173–2192.
49. Thouless DJ, Anderson PW, Palmer RG (1977) Solution of “Solvable model of a spin
glass”. Philos Mag 35:593–601.
50. Marks DS, et al. 3D protein structure predicted from sequence., arXiv:1110.5091v2
Morcos et al.PNAS
December 6, 2011