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KnowledgeKnowledge--Based Study of Protein Based Study of Protein StructureStructure--Function Correlations Function Correlations using Computational Geometry using Computational Geometry

Iosif Vaisman, George Mason University

email: [email protected]: proteins.gmu.edu

BIBM-2009 Tutorial

Protein representation (Crambin)

Protein representation (Crambin)

4 neighbors7 neighbors2 neighbors

10 neighbors

Neighbor identification in proteins

Voronoi Tessellation

Delaunay simplex isdefined by points,whose Voronoi polyhedra have common vertex

Delaunay simplex isalways a triangle ina 2D space and a tetrahedron in a 3Dspace

Delaunay Tessellation

Neighbor identification in proteins:Voronoi/Delaunay Tessellation in 2D


Voronoi Tessellation Delaunay Tessellation


67

6

Voronoi tessellation of Che Y protein

Adopted from Angelov et al., Proteins, 2002, 49(4):446

Delaunay tessellation of Crambin

Delaunay tessellation of Crambin

Dealunay simplices classification

d ij

d jk

dkl

j k

i

l

ii+1i+2

i+3j

ii+2 i+1

j+1

j

ii+1

k

j

ii+1l

k

j

i

{1-1-1-1} {2-1-1} {2-2} {3-1} {4}

Classification of Delaunay simplicesby sequential proximity

{4}(11,12,13,14)

{2,2}(3,4,32,33)

{2,1,1}(6,7,42,45)

{3,1}(38,39,40,46)

{1,1,1,1}(2,23,27,34)

Types of Delaunay simplices in Crambin

l

( ) /l l lii j

j>∑ − 2 215T =

Tetrahedrality of Delaunay simplices

Tetrahedrality distribution of Delaunay simplices

00.05

0.10.15

0.20.25

0.30.35

0.40.45

0.5

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4

Tetrahedrality

Freq

uenc

y Class {1,1,1,1}Class {2,1,1}Class {2,2}Class {3,1}Class {4}

Correlations between protein structure family assignment and relative content of classes of

Delaunay simplices

0

0.1

0.2

0.3

0.4

0.5

10 20 30 40 50 60 70 80 90 100

Proteins

−α −α/β −β

Compositional propensities of Delaunay simplices

qijkl = logfijkl

pijkl

C =4!

i

n

Π (ti!)

f- observed quadruplet frequency,

pijkl = Caiajakal, a - residue frequency

il

k

j

AAAA: C = 4! / 4! = 1

AAAV: C = 4! / (3! x 1!) = 4

AAVV: C = 4! / (2! x 2!) = 6AAVR: C = 4! / (2! x 1! x 1!) = 12

AVRS: C = 4! / (1! x 1! x 1! x 1!) ) = 24

Counting Quadruplets• assuming order independence among residues comprising

Delaunay simplices, the maximum number of all possible combinations of quadruplets forming such simplices is 8855

D F E C

C C D E

C C D D

C C C D

C C C C

204

⎛ ⎞⎜ ⎟⎝ ⎠

1920

2⎛ ⎞

⋅ ⎜ ⎟⎝ ⎠

202

⎛ ⎞⎜ ⎟⎝ ⎠

20 19⋅

20

4845

3420

190

380

20

8855

Delaunay simplices with distinct composition

Log-

likel

yhoo

d ra

tio

2000 4000 6000 8000

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.51 CCCC 3.0810032 CCCY 2.130043 CCHH 1.9608144 CCCG 1.7822675 CCCH 1.7427596 CCCW 1.7242757 CCCS 1.7242758 CCCQ 1.6573299 CCCF 1.621613... ... ...

... ... ...8343 CDDL -0.901668344 IRRV -0.902178345 AEYY -0.905358346 KKRV -0.950818347 CKRS -0.961338348 CEKP -0.984338349 HKKS -0.984728350 CGLR -1.147378351 ACKN -1.16297

Log-likelihood of amino acid quadruplets with different compositions

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

FFFF

PPPP

FFFP

FFFL

FPPP

FFPP

FFLL

LPPP

FFLP

FLPP

FLLL

FLLP

LLPP

LLLP

LLLL

tz-7{2,1,1}{2,2}{3,1}{4}

(F) Ala, Val, Phe, Ile, Leu, Pro, Met(L) Asp, Glu, Lys, Arg(P) Ser, Thr, Tyr, Cys, Asn, Gln, His, Trp

Log-likelihood of amino acid quadruplets(reduced alphabet)

Log-likelihood of amino acid quadruplets

Log-likelihood of amino acid quadruplets

Protein backbone (Cα only)Delaunay tessellation

3

45

6

7

10

14

152

9

8

12

13

1

11

Simplex a : d1=1, d2=1Simplex b : d1=2, d2=4Simplex c : d1=2, d2=1

a

bc

a

b

cd1

d2

1 2 3 4

1

2

3

4

2D analogy for topological matrices

beads on a chain

• enumerate the beads• define a sequence distance

dij = j − i −1

• apply the transformation, T

T : d a

0 if d = 01 if d =12 if d = 23 if d = 34 if 4 ≤ d ≤ 65 if 7 ≤ d ≤116 if 12 ≤ d ≤ 207 if 21≤ d ≤ 498 if 50 ≤ d ≤1009 if d ≥101

⎧

⎨

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

Topological similarity

djk=0

dkl=2

dlm= 35

j

k=j+1

l=j+4

m=j+40

T:djk→0T:dkl→2T:dlm→7

map the protein onto a 3-D array, M,where the elements, Mnpr are thenumber of simplices whose edgessatisfy:

(a) The euclidean length of any one simplex edge is less than 10 Å(partial tessellation).

(b) djk=n(c) dkl=p(d) dlm=r


create a 1000-tuple vector from the the elements, Mnpr

M = {M000, M001, …, M010, M011, …, …, M999}

compare the topology by computing the supremum norm (supnorm)between two proteins’ 1000-tuple vectors:

supnorm(M,M’) = Σ ⎪Mi - Mi’ ⎪


Ns = Mii= 0

999

∑inherent length dependence:

power law trend:~Cx1/2

average correlationcoefficient

0.99


geometry changes with local structure.topology does not.


removing intrinsic score length dependence:

it’s simple … normalize the vectors before comparing proteins.

ˆ M =v M v M

Score = ˆ M i − ˆ ′ M ii

∑


all-to-all comparison from WHATIF set of non-redundant proteins:

1424 chains with R-factor < 0.25, resolution < 2.5, Sequence ID < 30%


insensitivity to secondary structure content

Correlation coefficient:0.89


example from FSSP: 2 proteins close in RMSD but far in topology

1thg_


1alv A 1thg _

topological score: 4.77RMSD: 3.0 Angstroms


example from FSSP: 2 proteins close in topology but far in RMSD

1qlg A


1qlg A 2bbk H

topological score: 4.17RMSD: 4.2 Angstroms


example from CATH: a protein classified as α-β by topological comparisonbut as “zero” by CATH.

each structure has 8 secondary structure elements

1hip _ 1gua B

CATH: neither alpha nor beta Neighbor by topology … however,CATH: alpha-beta


computational cost:average CPU time per comparison in an all-to-all comparisonwithin a set of 100 proteins.

CE: 7.86s per pair-wise comparison

TOPOLOGICAL SCORE: 0.39s per pair-wise comparison

topological scoring is 20 times faster than alignment.


Cath Class f f/frandomα 0.957 7.48β 0.914 7.09α-β 0.826 3.72neither 0.333 15.9Scop Class f f/frandomα 0.840 3.36β 0.760 3.04α+β 0.610 2.44α/β 0.730 2.92

classificatory capability (major classes)

for each protein in the database

1) compare protein to all other proteins2) the lowest score is the closest neighbor3) the SCOP/CATH classification of the closest

neighbor is the classification of the protein


rmsd

0 1 2 3 4 5

TOPO

LOG

ICA

L SI

MIL

ARIT

Y SC

OR

E

0

300

600

900

1200

1cex

1cuc1cuwB1cuz 1ffb

1xzj1xzkB

1bs9

1brt

3chy

3tgl

1ido1wab

2cmd 1amp

Cutinase (1cex)

Topological similarity score between structures within FSSP families

Topological similarity score between conformations from MD simulation

Acylphosphatase (2acy)

rmsd

0 1 2 3

TOPO

LOG

ICAL

SIM

ILAR

ITY

SCO

RE

0

50

100

150

200

250

300

350

Protein backbone (Cα only)Delaunay tessellation

3

3 3

3

222

22

2

3 333

11 1

a

b

Residues a and bparticipate in 3 simplices of class 3each, i.e. they belong to a helix.

In 3D the assignmentto a helix requires 4 simplices of class 4

2D analogy for topological secondary structure assignment

0 50 100 150 200 250

1cex

01234

0 50 100 150 200 250

1arb

01234

0 50 100 150 200 250

1akz

01234

0 50 100 150 200 250

1ah7

01234

DSSP

DSSP

DSSP

DSSP

Topological profiles and DSSP assignment of helices

Secondary structure assignment

−100 0 100

−10

00

100

DSSP

−100 0 100

−10

00

100

C4.5

−100 0 100

−10

00

100

t4_motif

−100 0 100

−10

00

100

DEFINE

−100 0 100

−10

00

100

P−SEA

−100 0 100

−10

00

100

secstr

−100 0 100

−10

00

100

STRIDE

−100 0 100

−10

00

100

XTLsstr

Residue number

t4 v

alue

t4 profile and DSSP helix call for 2mnr

Residue and mutant score

q1q6

q5

q4

q3

q2 qres= Σqi

Qmut= Σ(qres – qres)wt mut

Residue Environment Scores

• assuming order independence among residues comprising Delaunay simplices, the maximum number of all possible combinations of quadruplets forming such simplices is 8855

• log-likelihood:– = frequency of quadruplets containing residues i,j,k,l in a

representative training set of high-resolution protein structures with low primary sequence identity

– = frequency of random occurrence of the quadruplet• total statistical potential (aka topological score or sequence-structure

compatibility score) of protein: sum the log-likelihoods of all quadruplets forming the Delaunay simplices of the protein tessellation

• individual residue potentials (aka residue environment scores): sum the log-likelihoods of only those quadruplets in which the given residue participates as one of the four vertices– yields a 3D-1D potential profile

( )logijkl ijkl ijklq f p=

ijklf

ijklp

DNA binding residues in HMG1

Residue number10 20 30 40 50 60 70

Scor

e

0

KPRGKMSSYAFFVQTCREEHKKKHPDASVNFSEFSKKCSERWKTMSAKEKGKFEDMAKADKARYEREMKTY

Coordinate file 1ckt: Ohndorf U-M et al. Nature 399:708

F37A

Protein-protein and protein-DNA interfaces (HMG-D)

Residue number0 10 20 30 40 50 60 70

Sco

re

Coordinate file 1qrv: Murphy F V et al. EMBO Journal 18:6610

A

A`

B`

A-A`

A`-B`

Computational mutagenesis of T4 lysozymeReversibility of mutations

R2 = 0.9886

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3

Protein Mutation Score change

1l63 T26E -2.49180l E26T 2.01

1l63 A82S 1.49123l S82A -1.49

1l63 V87M -0.281cu3 M87V 0.22

1l63 A93C -1.98138l C93A 1.78

1l63 T152S -1.081goj S152T 1.12

Reversibility of mutations

Reverse Mutation Potential Difference Score

-15 -10 -5 0 5 10 15

Forw

ard

Mut

atio

n P

oten

tial D

iffer

ence

Sco

re

-15

-10

-5

0

5

10

15

HH (22)PH (33)HP (40)PP (112)

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