MAX_mouse ADKRAHHNALERKRRDHIKDSFHSLRDSVPSLQG-EKASRAQILDKATEYIQYMRRKNHTDMAX_MOUSE ----------------------EQPRFQsa-------ASRAQILDKATEYIQYMRRKNHTMAX3_HUMAN ADKRAHHNALERKRRDHIKDSFHSLRDSVPSLQG-EKASRAQILDKATEYIQYMRRKNHTMAX_RAT -DKRAHHNALERKRRDHIKDSFHSLRDSVPSLQG-EKASRAQILDKATEYIQYMRRKNHTMAX_CAT ADKRAHHNALERKRRDHIKDSFHSLRDSVPSLQG-EKASRAQILDKATEYIQYMRRKNHTMAX_CHICK ADKRAHHNALERKRRDHIKDSFHSLRDSVPSLQG-EKASRAQILDKATEYIQYMRRKNHTMAX_XENOPUS ADKRAHHNALERKRRDHIKDSFHGLRDSVPSLQG-EKASRAQILDKATEYIQYMRRKNHTMAX_ZFISH ADKRAHHNALERKRRDHIKDSFHSLRDSVPALQG-EKASRAQILDKATEYIQYMRRKNHTMYCX_CARP ADKRAHHNALERKRRDHIKDSFHSLRDSVPALQG-EKASRAQILDKATEYIQYMRRKNHTzMax_Zfish ADKRAHHNALERKRRDHIKDSFHSLRDSVPALQG-EKASRAQILDKATEYIQYMRRKNHTXMax2_Xpus ADKRAHHNALERKRRDHIKDSFHGLRDSVPSLQG-EKASRAQILDKATEYIQYMRRKNHTDMAX_FLY AEKRAHHNALERRRRDHIKESFTNLREAVPTLKG-EKASRAQILKKTTECIQTMRRKISEF46G10_WORM DDRRAHHNELERRRRDHIKDHFTILKDAIPLLDG-EK-SRALILKRAVEFIHVMQTKLSSMAX1_WORM RHAREQHNALERRRRDNIKDMYTSLREVVPDANG-ERASRAVILKKAIESIEKGQSDSATMAD_MOUSE TSSRSTHNEMEKNRRAHLRLCLEKLKGLVP-L-GPESHTTLSLLTKAKLHIKKLEDCDRKMAD3_MOUSE NSGRSVHNELEKRRRAQLKRCLEQLRQQMP-L-GVDCYTTLSLL-RARVHIQKLEEQEQQMAD4_XENLA QNNRSSHNELEKHRRAKLRLYLEQLKQLVP-L-GPDSHTTLSLLKRAKMHIKKLEEQDRKMAD4_XENLA TVGRSSHNELEKHRRAKLRLYLEQLKQLVP-L-GPDSHTTLSLLKRAKMHIKKLEEQDRKMAD4_HUMAN PNNRSSHNELEKHRRAKLRLYLEQLKQLVP-L-GPDSHTTLSLLKRAKVHIKKLEEQDRRMAD4_MOUSE PNNRSSHNELEKHRRAKLRLYLEQLKQLGP-L-GPDSHTTLSLLKRAKMHIKKLEEQDRRMADL1H_WORM KHSRTAHNELEKTRRANLRGCLETLKMLVPCVSDA--NTTLALLTRARDHIIELQDSNAAMXI1_HUMAN TANRSTHNELEKNRRAHLRLCLERLKVLIP-L-GPDCHTTLGLLNKAKAHIKKLEEAERKMXI1_MOUSE TANRSTHNELEKNRRAHLRLCLERLKVLIP-L-GPDCHTTLGLLNKAKAHIKKLEEAERK

Typical Aligned Biological Sequence Data

Sequence data are highly dense, alphabetic, systematic missing data, often highly conserved (low variability), little replication, little within protein variability, usually more amino acids than sequences, etc. Very difficult to analyze statistically in a meaningful way

A major goal of biological research is to provide general models or systematic

principals to explain complex phenomena. In proteins, this involves

the molecular architecture of their component parts their origins,

regulation, interrelationships and evolution.

MDALQLANSAFAVDLFKQLCEKEPLGNVLFSPICLSTSLSLAQVGAKGDTANEIGQVLHFENVKDIPFGFQTVTSDVNKLSSFYSLKLIKRLYVDKSLNLSTEFISSTKRPYAKELETVDFKDKLEETKGQINNSIKDLTDGHFENILADNSVNDQTKILVVNAAYFVGKWMKKFPESETKECPFRLNKTDTKPVQMMNMEATFCMGNIDSINCKIIELPFQNKHLSMFILLPKDVEDESTGLEKIEKQLNSESLSQWTNPSTMANAKVKLSIPKFKVEKMIDPKACLENLGLKHIFSEDTSDFSGMSETKGVALSNVIHKVCLEITEDGGDSIEVPGARILQHKDELNADHPFIYIIRHNKTRNIIFFGKFCSP

Serine proteinase inhibition

(Ovalbumin)

Computational Biology Holy GrailSequence → Structure → Function → Evolution

O

BC

D

E

O

B

C

D

E

Interrelationships with other proteins –

the “Network” Problem

Evolution of the network

Are phylogenetic trees the best

way to describe

protein sequence

variability and evolution?

Phylogenetic Trees: Pros

Trees are good for• describing hierarchical patterns• clustering taxa• describing extent of lineage divergence• estimating ancestral relationships• summarizing overall changes in data.

Phylogenetic Trees: cons

Trees are NOT very good for:• understanding dimensionality• analyzing covariation• describing the biological basis of evolutionary divergence• elucidating the components upon which selection might operate

Significance of Covariation

• Understanding covariation is fundamental to modeling protein structure and evolution• Evolutionary and structural change constrained by covariation among amino acids • Accurate prediction of protein structure requires knowledge of covariance structure• Covariation reduces the dimensionality of phylogenetic information• Analytical procedures (like ML) make strong assumptions about covariances

Structural constraints

Decomposition of Covariance Among Amino Acid sites i

and jCij = CP + CS + CF + CI + e

Phylogenetic

constraints

Functionalconstraints

Interactions between model

components

Unexplainedeffects

abcdef

abcDeF

aBcdeF

abCDeF abCDeF

AbCdeF

abcDeF

abcDef

aBCdeF

ABcdeF

aBCdEF

ABcdeF

ABcdeF

Geneduplication(paralogy)

orthology

evolutionary time

Phylogeneticconstraints

(Taxa related by a common evolutionary history)

Compensatoryinteractions

7

1

2

34

5

6

Proximityeffects

Constraints dueto folding

Structural constraints – associationsamong amino acids arise from the

3-dimensional geometry or “folding” pattern of proteins.

Some Structural Associations in Proteins

Hydrophobic interactions: associations among amino acids with non-polar side chains

Salt bridges: correlations between charged residues

Hydrogen bonds: correlations between electron donors and receptors

Size constraints: correlations between large and small side chains

bHLH transcriptional regulators control a diverse array of developmental processes.

Basic (B) region binds to hexanucleotide E-box and controls gene expression.

Helix-loop-helix (HLH) region involved in protein – protein interactions (dimerization)

At least 5 different DNA binding groups in bHLH proteins based on how basic region interacts with E-Box

bHLH – Leucine Zipper StructureBasic Helix-1 Loop Helix-2 Zipper

COOH

Hydrophobic residues

Monomer

Dimer

DNA binding domain

helical domains

NH

Many bHLH proteins lack

LZ

i j k . n

i Ei

j Mij Ej

k Mik Mjk Ek

. . . . .

n Min Mjn Mkn . En 

Generalized Covariance MatrixAmino acid sites

E reflects amino acid diversity at each siteM describes mutual information between sites

Entropy Dynamics in bHLH Domain

B-2

B-6

B-7

B-8

B-11

B-12

H1-14H1-21

H1-25

H1-26

H2-64

H2-63

H2-62

H2-61

H2-60

H2-59H2-56

H2-55

H2-52

H2-57

H2-54H2-50

H2-53

H2-51

L-49L-48

L-47

L-31

L-46

L-45

L-29 L-30

H1-28

H1-24

H1-27

H1-20

B-4

B-5

H1-22

H1-23

H1-17

H1-18

H1-15

B-9

B-10B-13

H1-16B-1

B-3

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

B-1

B-2

B-3

B-4

B-5

B-6

B-7

B-8

B-9

B-1

0B

-11

B-1

2B

-13

H1-

14H

1-15

H1-

16H

1-17

H1-

18H

1-19

H1-

20H

1-21

H1-

22H

1-23

H1-

24H

1-25

H1-

26H

1-27

H1-

28L

-29

L-3

0L

-31

L-4

5L

-46

L-4

7L

-48

L-4

9H

2-50

H2-

53H

2-54

H2-

55H

2-56

H2-

57H

2-58

H2-

59H

2-60

H2-

61H

2-62

H2-

63H

2-64

H2-

65H

2-66

Amino Acids

No

rma

lize

d E

ntr

op

y

Dynamic pattern indicative of amphipathic α-helix with highly variable hydrophilic face alternating with conserved hydrophobic core. Spectral analysis confirms periodicity of ~3.6.

Zhi Wang

Packed sites in Max defining hydrophobic core are in red Packed sites in Max defining hydrophobic core are in red

Amino acid composition defines DNA binding groups

C3L

A2L

C1L

G4R

G1R

A4L

G3R

T2R

C5L G5L

T4L

G3L

T2L

G1L

C1R

A2R

C3R

C4R3'

5' 3'5'

5'

Glu 9A

Arg 13B

His 5B

His 5A

Arg 13A

Glu 9B

Arg 12A

Ser 41B

Lys 42B

Lys 6B

Lys 6A

Glu 10B

Glu 10A

Ser 41A

Lys 42B

Arg 15BC5RG5R

From Shimizu et al. (1997)

E-box Phosphates

Base pair recognitions

Phosphate recognitions

PHO4 - DNA Base Contacts

Group B bHLH

Packed sites in Max are underlined

Entropy Dynamics in bHLH Domain

B-2

B-6

B-7

B-8

B-11

B-12

H1-14H1-21

H1-25

H1-26

H2-64

H2-63

H2-62

H2-61

H2-60

H2-59H2-56

H2-55

H2-52

H2-57

H2-54H2-50

H2-53

H2-51

L-49L-48

L-47

L-31

L-46

L-45

L-29 L-30

H1-28

H1-24

H1-27

H1-20

B-4

B-5

H1-22

H1-23

H1-17

H1-18H1-15

B-9

B-10B-13

H1-16B-1

B-3

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00B

-1B

-2B

-3B

-4B

-5B

-6B

-7B

-8B

-9B

-10

B-1

1B

-12

B-1

3H

1-14

H1-

15H

1-16

H1-

17H

1-18

H1-

19H

1-20

H1-

21H

1-22

H1-

23H

1-24

H1-

25H

1-26

H1-

27H

1-28

L-2

9L

-30

L-3

1L

-45

L-4

6L

-47

L-4

8L

-49

H2-

50H

2-53

H2-

54H

2-55

H2-

56H

2-57

H2-

58H

2-59

H2-

60H

2-61

H2-

62H

2-63

H2-

64H

2-65

H2-

66

Amino Acids

No

rma

lize

d E

ntr

op

y

Contacts backbone in some groups

Contacts base

Contacts both

Contacts phosphate backbone

Dotted lines describe packing interactions

Solid arrows contacts with phosphate backbone

Dotted arrows contact with backbone and DNA

Intersite interactions in bHLH Domain

B-2

B-6

B-7

B-8

B-11

B-12

H1-14H1-21

H1-25

H1-26

H2-64

H2-63

H2-62

H2-61

H2-60

H2-59H2-56

H2-55

H2-52

H2-57

H2-54H2-50

H2-53

H2-51

L-49L-48

L-47

L-31

L-46

L-45

L-29 L-30

H1-28

H1-24

H1-27

H1-20

B-4

B-5

H1-22

H1-23

H1-17

H1-18

H1-15

B-9

B-10B-13

H1-16B-1

B-3

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

B-1

B-2

B-3

B-4

B-5

B-6

B-7

B-8

B-9

B-1

0B

-11

B-1

2B

-13

H1-

14H

1-15

H1-

16H

1-17

H1-

18H

1-19

H1-

20H

1-21

H1-

22H

1-23

H1-

24H

1-25

H1-

26H

1-27

H1-

28L

-29

L-3

0L

-31

L-4

5L

-46

L-4

7L

-48

L-4

9H

2-50

H2-

53H

2-54

H2-

55H

2-56

H2-

57H

2-58

H2-

59H

2-60

H2-

61H

2-62

H2-

63H

2-64

H2-

65H

2-66

No

rma

lize

d E

ntr

op

y

H1-16 packs against H1-20

H1-20 has many van der Waal contacts with H2 side chains

H1-23 packs against H2-53, H2-57

H-27 packs against H2-60, H2-61

H-28 terminates Helix 1

L-47 stabilizes loop path

H2-50 starts Helix 2, packs with H1-20, anchors Helix 2 to DNA

H2-53 packs against H2-50 and H2-54’

H2-54 packs against H2-50 and H2-53

H2-61 packs against symmetry mate 61 and 60

H2-64 interacts with symmetry mate and terminates helix 2

Max protein

HLH x HLH interaction

region

Factor Analysis of Mutual Information Matrix of Amino Acids

• 64 amino acids of bHLH domain, 288 sequences• 64 x 64 matrix of standardized MI matrix elements• Maximum Likelihood factor analysis used• 7 factors extracted that accounted for all of the common

information• Multivariate patterns of amino acid covariation described

and then related to known 3-D structure of bHLH domain from crystal structure studies

Factor Analysis of bHLH Domain Covariances

• Analyses involving covariances among 49 amino acid sites in bHLH domain• 288 separate bHLH domain sequences• Normalized mutual information values used• Seven significant eigenvectors • Each reflected significant multivariate components of covariation• Each eigenvector represented important phylogenetic, structural and functional information

Michael Buck

Flow of Statistical Analyses

Multiple alignment of sequences

Compute E, MI matrix for sequence elements

Factor analysis of MI matrix. patterns of covariation

Project factor coefficients onto RasMol models.

Interpret

Factor Analysis of 500 amino acid

attributes. Compute factor

scores

Transform alphabetic amino acid

codes to numericalfactor scores.

(5 datasets)

Compute R matrices.of each data set. Factor analysis on

each dataset

Determine patterns of

physiochemicalvariation within

proteins

Model underlyingcauses of

multidimensional protein variation

Model underlyingcauses of

multidimensional protein variation

I II IIISite1 X1 I X1 II X1 IIISite2 X2 I X2 II X2 IIISite3 X3 I X3 II X3 IIISite4 Xn I Xn II Xn III

Eigen-Structure

E1M12 E2M13 M23 E3M1n M2n M3n En

Association matrixN x N Amino acid sites

“Mutual Information” Factor Analysis

Magnitude of coefficientsfor amino acid sites and number

of factors estimatescomplexity and dimensions of

phylogenetic information

Site Domain Factor1 Factor2 Factor3 Factor4 Factor5 Factor6 Factor7 Comm

site30 L-2 0.646 0.161 0.112 0.254 0.066 0.138 0.079 0.549site31 L-3 0.646 0.136 0.108 0.042 0.344 -0.131 -0.137 0.604site29 L-1 0.599 0.113 0.054 0.069 0.162 0.077 0.063 0.415site63 H2-14 0.548 0.190 0.108 0.152 0.039 0.085 0.099 0.389site4 B-4 0.541 0.173 0.078 0.081 0.120 0.135 0.189 0.404

site56 H2-7 0.524 0.237 0.259 0.287 0.004 -0.001 0.171 0.509site21 H1-8 0.502 0.129 0.233 0.210 0.081 0.205 0.158 0.440site58 H2-9 0.501 0.120 0.323 0.062 0.065 -0.009 0.041 0.380site3 B-3 0.491 0.174 0.160 0.190 0.091 0.162 0.144 0.389

site45 L-4 0.488 0.100 0.081 0.153 0.233 0.052 0.102 0.345site15 H1-2 0.486 0.136 0.234 0.122 0.216 0.063 0.151 0.398site11 B-11 0.479 0.196 0.171 0.198 0.026 0.178 0.143 0.389site14 H1-1 0.472 0.252 0.344 0.135 0.133 0.107 0.121 0.467site62 H2-13 0.471 0.123 0.207 0.103 0.138 0.024 0.129 0.327site18 H1-5 0.470 0.123 0.009 0.111 0.037 0.072 0.143 0.275site59 H2-10 0.453 0.186 0.163 0.106 0.019 0.025 0.135 0.297site25 H1-12 0.450 0.104 0.091 0.115 0.159 0.143 0.182 0.314site7 B-7 0.446 0.167 0.158 0.207 0.149 0.177 0.083 0.355

site52 H2-3 0.443 0.056 0.338 0.190 0.254 0.141 0.130 0.451site46 L-5 0.422 0.100 0.161 0.134 0.323 0.100 0.180 0.378site26 H1-13 0.416 0.063 0.404 0.153 0.076 0.064 0.126 0.389site51 H2-2 0.402 0.271 0.362 0.197 0.245 -0.009 -0.029 0.466site55 H2-6 0.392 0.228 0.310 0.260 0.155 -0.016 0.164 0.421site5 B-5 0.391 0.276 0.333 0.190 0.265 0.202 0.122 0.502

site22 H1-9 0.386 0.213 0.129 0.175 0.069 0.030 0.048 0.250site19 H1-6 0.379 0.273 0.334 0.078 0.238 0.237 0.140 0.469site8 B-8 0.375 0.405 0.296 0.031 0.255 0.223 0.198 0.547

site13 B-13 0.370 0.267 0.452 0.051 0.163 0.042 0.030 0.445site48 L-7 0.337 0.174 0.266 0.256 0.062 -0.022 0.374 0.424site27 H1-14 0.305 0.115 0.091 0.396 0.016 0.084 0.051 0.281site6 B-6 0.299 0.495 0.193 0.314 0.223 0.244 0.011 0.579

site16 H1-3 0.294 0.344 0.345 0.190 0.062 0.228 -0.025 0.417site1 B-1 0.285 0.262 0.053 0.058 0.046 0.040 0.292 0.245

site49 L-8 0.263 0.148 0.300 0.314 0.466 0.113 0.062 0.514site20 H1-7 0.262 0.440 0.287 0.060 0.128 -0.077 0.015 0.371site47 L-6 0.238 0.050 0.100 0.407 0.031 0.272 0.076 0.315site28 H1-15 0.234 0.078 0.037 0.552 0.219 -0.054 0.068 0.423site24 H1-11 0.226 0.088 0.275 0.340 0.155 0.234 -0.019 0.329site64 H2-15 0.204 0.096 0.055 0.074 0.297 0.350 0.209 0.313

Eigenvalue 14.582 1.982 0.993 0.841 0.647 0.591 0.523

Cumulat % 0.723 0.822 0.871 0.913 0.945 0.974 1.000

Portion of Factor matrix of MI values for 64 amino acid sites of bHLH domain. Varimax rotation of ML factor solution. High coefficients on each vector shown in yellow.

Factor analyses describe:

• Dimensionality of shared and unique covariation

• Major patterns of amino acid covariation among all major bHLH lineages• A model for structural and sequence evolution in bHLH• An understanding of the biological bases of simultaneous changes among amino acid sites

DNA

Factor 1

> 0.5> 0.4> 0.3< 0.3

Factor Coefficients

bHLH monom

er

Accounts for 72% of sequence common covariance in 288 proteins

22 of 49 sites with factor coefficients > 0.4

Most sites with high coefficients occur on exposed or hydrophilic face of helices

High correlation between factor coefficients and site by site entropy values, clade membership and loop length

Sequence variation reflected by Factor 1 has strong phylogenetic signal

Factor 1Showing the orientation of the side-chains of the amino acids on the

hydrophilic surface and awayfrom DNA

Use dummy variables for classification codes

Estimate phylogenetic tree from well-aligned sequences

Define clades (monophyletic lineages)

Delimiting clades uses both biological and

statistical information -- clade definition can be somewhat subjective

Assign dummy variable to all sequences in each clade

Covariance between given site and dummy variable measures phylogenetic signal

Prediction of clade membership by multivariate statistics used to define “sequence signatures”

“Group membership” approach useful for structural and functional variables also

?

A

C

D

E

F

B

G

H

Estimating phylogenetic signal in any amino acid?

  Fact1 Fact2 Fact3 Fact4 Fact5 Fact6 Fact7

clade 0.705 0.078 0.725 0.244 0.219 -0.039 -0.131

group 0.168 0.414 0.584 0.092 0.424 0.161 -0.070

loop-len 0.741 0.146 0.531 0.277 0.233 -0.147 -0.319

comm 0.276 0.598 0.234 0.086 0.164 0.150 -0.180

entropy 0.938 -0.149 0.298 0.153 0.030 -0.088 -0.038

Correlations of Factor Coefficients of Pair-wise Mutual Information Values with Extrinsic Variables

Clade = monophyletic lineages of proteins with equivalent functions

Group = DNA Binding Groups based upon E-Box binding patterns

Loop-length = number of residues in the loop region separating helices 1 and 2

Comm = Communality from factor analysis; amount of variability at site explained by 7 factors

Entropy = extent of uncertainty (variability) at each site in the bHLH sequence domain

Critical correlation coefficient to reject Ho at P < 0.01 = 0.43

B-2

B2-57

B-9

H1-12

B-10

B-9

B-6

H1-20

H1-16

B-8

B-6

H1-16

Factor 2

10% of sequence variability Large factor coefficients for 8 sites:

6 from DNA binding region, 1 from each helix

B2, B6, B8, B10 and B12 involved in protein side-chain – phosphate backbone contacts. B9 also contacts DNA base

Site H1-20 buried site with many van der Waal contacts with Helix 2. H2-57 important structurally.

Sites important to maintain structural “geometry”

All sites with high coefficients occur at nadirs of entropy dynamics plots. Highly conserved but intrinsic variability covarys among these 8 sites

7 of 8 sites components of “sequence signature” that identifies all bHLH proteins

Packed sites in Max are underlined

Entropy Dynamics in bHLH Domain

B-2

B-6

B-7

B-8

B-11

B-12

H1-14H1-21

H1-25

H1-26

H2-64

H2-63

H2-62

H2-61

H2-60

H2-59H2-56

H2-55

H2-52

H2-57

H2-54H2-50

H2-53

H2-51

L-49L-48

L-47

L-31

L-46

L-45

L-29L-30

H1-28

H1-24

H1-27

H1-20

B-4

B-5

H1-22

H1-23

H1-17

H1-18

H1-15

B-9

B-10B-13

H1-16B-1

B-3

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00B

-1B

-2B

-3B

-4B

-5B

-6B

-7B

-8B

-9B

-10

B-1

1B

-12

B-1

3H

1-14

H1-

15H

1-16

H1-

17H

1-18

H1-

19H

1-20

H1-

21H

1-22

H1-

23H

1-24

H1-

25H

1-26

H1-

27H

1-28

L-2

9L

-30

L-3

1L

-45

L-4

6L

-47

L-4

8L

-49

H2-

50H

2-53

H2-

54H

2-55

H2-

56H

2-57

H2-

58H

2-59

H2-

60H

2-61

H2-

62H

2-63

H2-

64H

2-65

H2-

66

Amino Acids

No

rma

lize

d E

ntr

op

y

High coefficients – Factor 1

High coefficients – Factor 2

H1-13

H1-14

B-5 Factor 3

H1-26

H1-14

H1-26

H2-49H1-16

H1-19 H2-51

H2-55

H2-58

B-5

H2-58

8 sites with large factor coefficients

Sites involved with interrelationships between variable and conserved sites. Each site adjacent to highly conserved “packed” site.

Suggests role in compensatory variation

Potentially important to maintain geometry of hydrophobic core

Strong phylogenetic content.

Factor 4H1-28

B-6 B-6

L-47

L-49L-49

L-47

H1-24

H1-27

H2-60

H1-28 – P in 75% of sequences, initiates loop

H1-27 – packs against H2-60, H2-61

L-47 – stabilizes loop path back into groove

H2-60 – interaction with helix 1 (H1-27)

Definition of the Loop Region

MyoD 3-D Structure

DNA Binding Group - A

SREBP

Max

USF

MyoD

Models of bHLH-DNA Structure

PHO4

• Structures available for 6 proteins

• Good fit of all to canonical structure

• Function of bHLH domain well- understood

• Simple domain structure

• Phylogeny well-documented