Computational Methods for Structural Bioinforamtics

and Computational Biology (3)

(Protein geometry and topology: volume and surfaces)

Jie Liang 梁 杰

Molecular and Systems Computational Bioengineering Lab (MoSCoBL)Department of Bioengineering

University of Illinois at Chicago上海交通大学系统医学研究院

上海生物信息技术研究中心

E-mail: jliang@uic.eduwww.uic.edu/~jliang

Dragon Star Short CourseSuzhou University, June 14 – June 18, 2009

Today’s Lecture

Space filling structures of proteins: volume and surface models,

Geometric constructs and algorithms: Voronoi diagram, Delaunaytriangulation, and alpha shapeAlgorithmsA bit of topology

Application in proteins packing and function prediciton

Today’s Lecture

Space filling structures of proteins: volume and surface models,

Geometric constructs and algorithms: Voronoi diagram, Delaunaytriangulation, and alpha shapeAlgorithmsA bit of topology

Application in proteins packing and function prediciton

Different structural models of proteins

Volumetric and surface models

Backbone centric viewSecondary structure, tertiary fold, side chain packing

But ligand and substrate sees differently!

We are interested in things like binding surfaces

Volumetric and surface modelsMuch more complicated, as there could be 10,000 atoms.

GDP Binding Pockets

Ras 21 Fts Z

Functional Voids and Pockets

Space-filling Model of Protein

The shape of a protein is complexProperties determined by distribution of charge density,

Chemical bonds transfer charges from one atom to anotherIsosurface of electron density depend on locations of atoms and interactions

X-ray scattering pattern are due to these distributions.

Space Filling model: Idealized modelAtom approximated by balls, difference between bonded and nonbondedregions ignored.

interlocking sphere model, fused ball model

Amenable for modeling and fast computation

Ball radius: many choices, eg. van der Waals radii

Mathematical Model: Union of ball model

For a molecule M of n atoms, the i-thatom is a ball bi, with center at zi ∈ R3

bi

≡

{x| x

∈

R3, |x-zi

| <= ri

},

parameterized by (zi

, ri

).

Molecuel M is formed by the union of a finite number n of such balls defining the set B:

M = U B = U i=1n

{bi

}

Creates a space-filling body corresponding to the volume of the union of the excluded volume

When taken vdw radii, the boundary ∂ U {B} is the van der Waals surface.

(Edelsbrunner, 1995; see also Liang

et al, 1998)

Solvent Accessible surface model

Solvent accessible surface (SA model):

Solvent: modeled as a ball

The surface generated by rolling the solvent ball along the van der Waals atoms.

Same as vdw model, but with inflated radii by that of the solvent radius

( Lee and Richards, 1974)

Molecular Surface Model

Molecular Surface Model (MS):

The surface rolled out by the front of the solvent ball.

Also called Connolly’s surface.

More on molecular surface model

( Michael Connolly, http://www.netsci.org/Science/Compchem/feature14 e.html )

Elementary Surface Pieces: SA

SA: the boundary surface is formed by three elementary pieces:

Convex sphereical surface pieces, arcs or curved line segments

Formed by two intersecting spheres

VertexIntersection point of three spheres

The whole surface: stitching of these three elementary pieces.

a

Vdw

surface: Shrunken version ofSA surface by 1.4 A

Elementary Surface Pieces: MS

MS: three different elementary pieces:

Convex sphereical surface pieces,

Concave toroidal surface pieces

Concave spheric surfaceThe latter two are also called “Re-entrant surface”

The whole surface: stitching of these three elementary pieces.

b

Relationship between different surface models

c

vdW and SA surfaces.

SA and MS surfaces:Shrink or expand atoms.

SA

MS

Vertex

concave spheric

surface piece

Arcsconcave toroidal

surface piece

Conv. surfade

Smaller conv

surface

SA and MS: Combinatorically equivalent

Homotopy equivalent

But, different metric properties!SA: void of 0-volume ---- MS: void of 4πr3/3

Today’s Lecture

Space filling structures of proteins: volume and surface models,

Geometric constructs and algorithms: Voronoi diagram, Delaunaytriangulation, and alpha shapeAlgorithmsA bit of topology

Application in proteins packing and function prediciton

Computing protein geometry

It is easy to conceptualize different surface models

But how to compute them?Topological properties

Metric properties (size measure)

Need:Geometric constructs

Mathematical structure

Algorithms

Geometric Constructs: Voronoi

Diagram

A point set S of atom centers in R3

The Voronoi region / Voronoi cell of an atom of an atom bi with center zi∈R3 :

Vi

= { x ∈

R3

| |x-zi

| <= |x-zj

|, zj

∈

S }

All points that are closer to (or as close as to) bi than any other balls bj

Alternative view:

Bisector plane has equal distance to both atoms, and forms a half space for bi.

Half space of bi with each of the other balls bj

Intersection of the half spaces forms the Voronoi cell, and is a convex region

Weighted Voronoi diagram

In reality, atoms have different radius

Cannot use Euclidean distance and bisector plane.

Use Power Distance:

πi

(x) ≡

||x-zi

||2

- ri2

which replaces the Euclidean distance

||x-zi

|| It is the length of thetangent line segment

Delaunay Triangulation

Convex hull of point set S:The smallest convex space contain all points of S.

It is formed by intersection of halfplanes, and is a convex polytope.

Delaunay triangulation:uniquely tesselate/tile up the space of the convex hull of a point set with tetrahedra, together with their triangles, edges, and vertices

(triangles instead of tetrahedra in 2D)

Dual relationship between Voronoi

and Delaunay

These two geometric constructs look very different!

In fact, they are dual to each other

Reflect the same combinatorial structures

The mapping process.

Delaunay triangulation:Formed by gluing 4 types of primitive elements:

Vertices: just atom centers

Edges: connecting atom centers zi and zj iff the intersection vi Å vj of their Voronoi regions vi and vj is not empty:

vi

Å

vj

≠

∅

Triangle: connecting atom centers zi , zj and zk iff

vi

Å

vj

Å

vk

≠

∅.

Tetrahedron: connecting atom centers zi , zj, zk and zl iff

vi

Å

vj

Å

vk

Å

vl

≠

∅.

They form Delaunay Complex.

What are vi Å vj ? vi Å vj Å vk ? and vi Å vj Å vk Å vl ?

Computing Delaunay Triangulation

Circumcircle of a Delaunaytriangle abc:

The unique circle passing through a, b, and c.Its center: the corresponding Voronoivertex vi Å vj Å vk

Empty circumcircle of triangle abc:

Contains no other points in S.

Circumcircle Claim (for triangles): Let S ⊂ R2 be finite, and let a, b, c ∈ S be three points. abc

is a Delaunay

triangle iff

the circumcircle

of abc

is empty

a

b

c

d

a

2 flip

a

b

c

d

acd

is

abc

is notan emptycircle

Supporting Circle

Supporting Circle Claim (for edges):

Let S ⊂

R2

be finite,

and let a, b ∈

S be two points.

ab

is a Delaunay

edge iff

there is an empty

circle that passes through a and b.

a

b

c

d

a

Not ab

here

Locally Delaunay

A triangulation K triangulates point set S if the triangles decompose the convex hull of S and the set of vertices is SAn edge ab ∈ K is locally Delaunay if

It belongs to only one triangle, and therefore bound the convex hullIt belongs to two triangles, abc and abd, and d lies outside the circumcircle of abc

Note: A locally Delaunay edge is not necessarily an edge of the Delaunaytriangulation.

Delaunay Lemma

If every edge is locally Delaunay, then all edges are Delaunya edges

Delaunay Lemma: If every edge of K is locally Delaunay, then K is the Delaunay triangulation of S

General idea: Take an arbitrary triangulation Go through each of the triangles,

make corrections using “flips" discussed below if a specific triangle contains an edge is not locally Delaunay.

Primitive Operation: Edge FlipIf ab is not locally Delaunay, then the union of the two triangles abc ∪ abd is a convex quadrangle acbd, and edge cd is locally Delaunay. We can replace edge ab by edge cd. This is called an edge-flip.

Also called 2-to-2 flip, as two old triangles are replaced by two new triangles.

Recursively check each boundary edge of the quadrangle abcd to see if it is also locally Delaunay after replacing ab by cd.

If not, recursively edge-flip it.

Data structure: use a stack.

2−to−2 flip

a

b

c

d

a

b

c

d

a

Incremental Algorithm

A finite set of points S = (z1, z2, …, zn)Start with a large auxiliary triangle containing all points.Insert the points one by one

Maintaining a Delaunay triangulation Di upon insertion of point zi

Search for the triangle τi-1 containing this new point.Add z_i and split τi-1 into 3 smaller triangles

1-to-3 flip, as it replaces one old triangle with three new triangles

Check if each of the three edges in τi-1 still satises the locally Delaunay requirement.

If not, perform a recursive edge-flip.

1−to−3 flip

b

Incremental Algorithm

Delaunay Tetrahedrization

More complicated in R3, but same basic idea:

Locate a tetrahedron that contains the newly inserted point.

“Locally Delaunay” replaced by “Locally convex”

Other flips than “2-to-2” flips

Sequentially adding points become important

Excellent expected performance.

(Edelsbrunner

and Shah, 1996)

Computing voronoi Diagram

Easy when Delaunay triangulation is available.Mathematical duality

Compute all of the Voronoi vertices, edges, and planar faces from the Delaunay tetrahedra, triangles, and edges. Because one point zi may be an vertex of many Delaunaytetrahedra, the Voronoi region of zi therefore may contain many Voronoi vertices, edges, and planar faces.

The efficient quad-edge data structure can be used for software implementation

A bit of topology: simplices

To encode the intersection relationship

0-simplices σ0: vertices, covex hull of 1-point

1-simplices σ1 : edgescovex hull of 2 points

2-simplices σ2 : trianglescovex hull of 3 points

3-simplices σ3 : tetrahedroncovex hull of 4 points

Tetrahedron is the most complicatedsimplex in R3

These can be glue tomodel arbitrailycomplicated molecule

Simplicial Complex

We can glue the simplicies together properly to form more complicated structuresSimplicial complex: the collection of faces of a finite number of simplicies

Any two are either disjoint, or meet in a common face

Formally, simplices form a simplicial complex K

K = {σ|I|-1

| Å i ∈

Ι

Vι

≠

∅ }

Simplicial complex from simplices iffσ

∈

K Æ

ι≤σ

→ ι

∈

K,

σ, ν

∈

K → σ

Å

ν

≤

σ, ν

Examples

Alpha Complex

Grow or shrink balls by a parameter α

For a ball bi of radius ri, its modified radius at a particular α is:

ri

(α) = (ri2+α)1/2

Shrunk when -ri<α <0,What happens if α < 0 and |α| > ri?

Simplex occurs when intersection appears.Collect simplices at different α as we increase α from -∞ to +∞

a b c

d e f

( Mucke

& Edelsbrunner, 1994, ACM Tran Graph )

Alpha complex and dual complex

At any specific α value, we have a dual simplicial complex or alpha complex:

When all atoms take incremented radius ri + rs, we have dual complex K0 of the protein molecule.

When α is sufficiently large, all simplicies are collected, and we have the Delaunay complex

A series of simplicial complex at different α values.

A series of 2D simplicial complexes (alpha shapes).

Each faithfully represents the geometric and topological property of the protein molecule at a particular resolution parametrized

by

the α

value

Equivalent View

For dual complex or alpha shape K0 at α=0,

Take a subset of the simplices iff the corresponding intersections of Voronoi cells overlap with the body of the union of the balls:

K0

= {σ|I|-1

| Åi

∈

I

Vi Å

∪ B ≠

∅

}

Alpha shape:The underlying space of alpha complex.

A Guide Map for Computing Geometric Properties

Combinatorial Information

A Guide Map for Computing Geometric Properties

Re-entrant surfaces in MS Model :

Concave spherical patch: boundary triangles in alpha shape

Delaunay

edge: vi

Å

vj

Å

vk

≠

∅

.Alpha edge: (vi

Å

vj

Å

vk

)

Å

∪

B ≠

∅

The line (segment) vi

Å

vj

Å

vk

intersect with SA of ∪ B:

at a surface vertex where three balls overlapCorresponds to a concave spherical surface patch in the molecular surface

c

A Guide Map for Computing Geometric Properties

Re-entrant surfaces in MS Model :

Concave toroidal surfaceboundary edges in alpha shape:

The Voronoi plane vi Å vjcoincides with the intersecting plane when two atoms meet

intersect with SA of ∪ B at an arc (line segment)Corresponds to a toroidal concave surface patch

Remaining part of the surface: convex

correspond to the vertices in alpha shape, That is, the atoms on the boundary of the alpha shape

c

A Guide Map for Computing Geometric Properties

Number of concave spheric pieces = number of boundary triangles in alpha shape

Number of toroidal pieces = number of boundary edges

Seem to scale roughly to O(n):due to constraints in bond lengths, bond angles, and excluded volume

Geometric structure from alpha shape

Mucke & Edelsbrunner, 1994, ACM Tran GraphEdelsbrunner, Facello, Liang, 1998, Disc Appl MathLiang, Edelsbrunner, Woodward, 1998, Protein Sci

Guide map for topological properties: Voids in proteins

Computing Metric Properties

Volume of a protein molecule in SA model

A grossly incorrect method: Sum the volume of individual atoms

Problem: over-exaggeration, because neglecting volume overlaps.

Explicit correction: Direct Inclusion-Exclusion

When two balls overlap, subtract intersection

When three balls overlap: Oversubtraction!

Need to add back volume of 3-intersection

Continues when thereare 4-

or 5-volume

overlaps.

(Edelsbrunner, 95)

Inclusion-Exclusion Principle

Corrected volume V(B) for a set of balls B is:

V(B) = ∑

vol(Å

T) > 0, T ⊂

{B} (-1)dim(T)-1

vol(Å

T),

Same as the Gauss-Bonnet Theorem combinatorically

Straightforward application of this does not work!Volume overlap can be 7-8 degrees,

Difficult to keep track, and difficult to compute volume overalps:

how large is the k-volume overlap of which one of the 7 \choose k or 8 \choose k overlapping atoms for all of k=2, …, 7?

(Edelsbrunner, 95)

( 1 )

How to avoid high-degree overlaps?

Insight: in R3, overlaps of five or more balls at a time can always be reduced to a ``+'' or a ``-'' signed combination of overlaps of four or fewer balls2-body, 3-body, and 4-body terms in Eqn 1 enter the formula iff there exist in the dual simplicial complex K0 of the molecule

the corresponding edge σij connecting the two balls (1-simplex), triangles σijk spanning the three balls (2-simplex), tetrahedron σijkl cornered on the four balls (3-simplex)

(Edelsbrunner, 95)

Simplified Inclusion-Exclusion Formula for Molecules

Volume for a set of balls B:

V(B) = ∑

σi ∈

K

vol

(bi

) -

∑

σij

∈

K

vol

(bi

Å

bj

) +

∑

σijk

∈

K

vol

(bi

Å

bj

Å

bk

) -

∑

σijkl

∈

K

vol

(biÅ

bj

Å

bk

Å

bl

)

Same idea for the calculation of surface area of molecules.

An Example in 2D

Total area of the union of disks (2D) is:

Atotal

= (A1

+ A2

+ A3

+ A4

)

- (A12

+ A23

+ A24

+ A34

)

+ A234

Here we:add the area of bi

if the corresponding vertex σ

i

∈

K0

,

subtract the area of bi

Å

bj

if σ

ij

∈

K0

add the area of bi

Å

bj

Å

bk

if σ

ijk

∈

K0

b1

b2

b3

b4

b1

b2

b3

b4

An Example in 2D

Without the guidance of the alpha complex:

Atotal

=

(A1

+ A2

+ A3

+ A4

)

- (A12

+ A13

+ A14

+ A23

+ A24

+ A34

)

+ ( A123 + A124 + A134 + A234 )

- A1234

But six of them are canceling redundant terms:

A13 = A123, A14 = A124, A134 = A1234

Computing them would be wasteful.

b1

b2

b3

b4

b1

b2

b3

b4

No Redundant Term

The redundant terms do not occur if we follow the alpha complex:

The part of the Voronoi regions contained in the respective balls for the redundant terms do not intersect: eg. A123 , A124 , and A134

The corresponding edges and triangles do not enter the alpha complex

In R2, we have terms of at most three disk intersections:

triangles in the alpha complex,

In R3, the most complicated terms are intersections of four balls:

tetrahedra in the alpha complex

b1

b2

b3

b4

Short Inclusion-Exclusion Term

Even faster computation with shorter expressions of area and volume

Use fractional coefficients in the formulas: Angles at the ball centers relative to its neighbor in the alphacomplex

Measured as fractions of circles or spheres and normalized between 0 and 1.

Atotal

= (1 · A1

+ φ2

· A2

+ φ3

· A3

+ φ4

· A4

)

− (1 · A12

+1/2 · A23

+ 1/2 · A24

+1/2 · A34

)

+ A

A: area of the triangles in alpha complex.b1

b2

b3

b4

Algorithm for Metric Size Calculation

Let V and A denote Volume and area of a molecule

Volume of ball intersections

Sector: solid angle x vol

of ball

Wedge: dihedral angle x

(Vcap

(i,j) + Vcap

(j,i) )

Pawn:

What about area calculations?

How to compute metric properties of MS models?

Hint: The MS model has the same structure as the SA model, but different primitive geometric objects

Concave spheric piece

Concave toroidal piece

Voids and pockets in proteins

Concave regions on protein surfaces

Shape complimentarity important for molecular recognition

Binding frequently occurs in p0ckets and voids

Eg. enzymes

Computing voids and pockets

Consider Delaunay tetrahedra not included in alpha complex.

Repeatedly merge those that share a 2-simplex

Obtain discrete set of empty tetrahedra

Voids: completely isolated

Pockets: connected to outside with a constricted mouth

Shallow depression: connected,

Size calculation: inclusion and exclusion.

Computing pockets in proteins

Edelsbrunner et al, 1998, Disc Appl MathLiang, Edelsbrunner, Woodward, 1998, Protein Sci

Today’s Lecture

Space filling structures of proteins: volume and surface models,

Geometric constructs and algorithms: Voronoi diagram, Delaunaytriangulation, and alpha shapeAlgorithmsA bit of topology

Application in proteins packing and function prediciton

Voids and Pockets in Soluble Proteins

“Protein interior is solid-like, tightly packed like a jig-saw puzzle”

High packing density (Richards, 1977)

Low compressibility (Gavish, Gratoon, and Harvey, 1983)

Many voids and pockets.At least 1 water molecule; 15/100 residues.

(Liang & Dill, 2001, Bioph J)

Proteins are not optimized by evolution to eliminate voids.

Protein dictated by generic compactness constraint related to nc.

Surfaces with unknown functional roles

http://cast.engr.uic.edu

Enzyme Functional Site Prediction

Where are the functional surfaces located ?

What are the key residues (active site residues) in the functional surfaces ?

Which mutations to make?

Our strategyGeometric

Computing( precomupted

in castPdatabase)

Locating

Key residues (red)4~5 residues

Proteinstructure~200 residues

FunctionalSurface (green)

Identifying

Structure is the only input!!

Validation study: large-scale prediction of functional surface

Data Set (~700 structures).

10-fold cross validation.

Average accuracy of predicting functional surfaces of proteins is 91.2%. Tseng and Liang, Annals of BME, 2007

The Universe of Protein Structures

Human genome: 3 billion nucleotides;

Number of genes: 20,000 –25,000; Protein families: 10,000; Number of folds: 1,000s

Currently in PDB: about 1,000 folds

Comparative modeling: needs a structural template with sequence identities > 30-35%

eg. ~50% of ORFs and ~18% of residues of S. cerevisiae genome

Structural Genomics: populating each fold with 4-5 structures

One for each superfamily at 30-35% sequence identities.

Fold of a novel gene can be identified

Its structure can then be interpolated by comparative modeling.

All β α/β

(from SCOP)

Predicting and characterizing protein functions

Important, but challenging tasks:Needs > 60-70% sequence identity.

Fold prediction: >20-30% sequence identity.

Proteins from structural genomics often are of unknown functions.

Sequence homologs are often hypothetical proteins.

(Rost, 02, JMB; Tian & Skolnick, 03, JMB)

Another Way: How to identify biologically important pockets and voids from random ones?

By Homology:

Assessing Local Sequence and Shape Similarity

(Binkowski, Adamian, Liang, 2003, JMB, 332:505-526)

Binding Site Pocket: Sparse Residues, Long Gaps

ATP Binding: cAMP

Dependet

Protein Kinase

(1cdk) and Tyr

Protein Kinase

c-src

(2src)

1cdk.A 49LGTGSFGRVMLVKHKETGNHFAMKILDKQKVVKLKQIEHTLNEKRILQAVNFPFLVKLEYSFKDNSNL YMVMEYVPGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIQVTDFG FAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVR FPSHFSSDLKDLLRNLLQVDLTKRFGNLKDGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSN F327

1cdk.A_p 49LGTGSFGRV A K V

MEYV E K EN L TD F

2src.m 273LGQGCFGEVWMGTWNGTTRVAIKTLKPGTMSPEAFLQEAQVMKKLRHEKLVQLYAVVSEEPIYIVTEYMSKGSLLDFLKGETGKYLRLPQLVDMAAQIASGMAYVERMNYVHRDLRAANILVGENLVCKVAD404

2src.m_p 273LGQGCFGEV A K V TEYM GS D D R AN L AD

Low overall sequence identity: 13 %

High Sequence Similarity of Pocket Residues

1cdk.A LGTGSFGRVAKVMEYV---EKENLTDF 24

2src.m LGQGCFGEVAKVTEYMGSDDRANLAD- 26

** *.**.**** **: :: **:*

1cdkcAMP Dependent Protein Kinase

2srcTyr Protein Kinase c-src

High sequence identity: 51 %

Sequence Similarity of Surface Pockets

Similarity detection:Dynamic programming SSEARCH (Pearson, 1998)

Order Dependent Sequence Pattern.

Statistics of Null Model:Gapless local alignment: Extreme Value Distribution

(Altschul & Karlin, 90)

Alignment with gaps: (Altschul, Bundschuh, Olsen & Hwa, 01)

Statistical Significance !

Approximation with EVD distribution (Pearson, 1998, JMB)

Kolmogorov-Smirnov Test:

Estimate K and λ parameters.

Estimation of E-value:

Estimate p value of observed Smith-Waterman score by EVD.

allall )( NpNNpE d ⋅≤−⋅=

)exp(1)'(

,ln'xexSp

KmnSS−−−=≥

−= λ

(Binkowski, Adamian, Liang, 2003, JMB, 332:505-526)

Shape Similarity Measure

cRMSD (coordinate root mean square distance)

oRMSD (Orientational RMSD):Place a unit sphere S2 at center of mass x0 ∈ R3

Map each residue x ∈ R3 to a unit vector on S2 :

f :

x

= (x, y, z)T 　

u

= (x -

x0

) / || x -

x0

||

Measuring RMSD between two sets of unit vectors.

(cf. uRMSD by Kedem and Chew, 2002)

Surprising Surface Surprising Surface SimilaritySimilarity

Retroviral proteaseRetroviral proteaseFamilyFamily

All All ββClassClass

Binds polyBinds poly--peptide substrate acetylpeptide substrate acetyl--pepstatinpepstatin

FoldFold

HIVHIV--1 Protease 1 Protease ((5hvp5hvp))

PocketPocket

Acid proteasesAcid proteases

CATHCATH

Hsp90Hsp90FamilyFamily

αα++ββClassClass

Binds protein segment geldanamycinBinds protein segment geldanamycin

FoldFold

Heat Shock Protein 90 Heat Shock Protein 90 ((1yes1yes))

PocketPocket

αα//β β sandwhichsandwhich

CATHCATH

••Conserved residues both important in Conserved residues both important in polypeptide binding polypeptide binding

•• Both pockets undergo conformational Both pockets undergo conformational changes upon binding changes upon binding

How to capture the evolutionary signal due to biological function?

(Tseng and Liang, 2006, Mol Biol Evo, 23:421; Tseng et al, 2009, J. Mol.Biol)

Strong selection pressures that increase the overall fitness of proteins.

But may not be related with biological function:

Structural constraints

Protein stability

Folding kinetics

Isolating selection pressure due to biological function:

Unsolved problem!

Deriving Scoring Matrix from Evolutionary History

A scoring matrix is critical:

Determines similarity between residues and hence statistical significance.

Derived from evolutionary history of proteins sharing the same function.

Existing approach: PAM and BLOSUM heuristics.

Position specific weight matrix.

Entropy/relative entropy for full proteins or domains.

Our approach: Evolution: Explicit phylogenetic tree.

Model: Continuous time Markov process.

Geometry: Evolution of only residues located in the binding region.

Bayesian Markov chain Monte Carlo.

A R N D C Q E G H I L K M F P S T W Y V

A

R

N

D

C

Q

E

G

H

I

L

K

M

F

P

S

T

W

Y

V

The Active Pocket [ValidPairs: 39]

(a)

A R N D C Q E G H I L K M F P S T W Y V

A

R

N

D

C

Q

E

G

H

I

L

K

M

F

P

S

T

W

Y

V

The rest of Surface [ValidPairs: 177]

(b)

A R N D C Q E G H I L K M F P S T W Y V

A

R

N

D

C

Q

E

G

H

I

L

K

M

F

P

S

T

W

Y

V

Interior [ValidPairs: 190]

(c)

A R N D C Q E G H I L K M F P S T W Y V

A

R

N

D

C

Q

E

G

H

I

L

K

M

F

P

S

T

W

Y

V

Surface [ValidPairs: 187]

(d)

Evolutionary rates of binding sites and other regions are different

Residues on protein functional surface experience different selection pressure.

Estimated substitution rate matrices of amylase:

• Functional surface residues.

• The remaining surface, • The interior residues• All surface residues.

Example 1: Finding alpha amylase by matching pocket surfaces

Challenging:– amylases often have low overall sequence identity (<25%).

–1bag, pocket 60; B. subtilis–14 sequences, none with structures, 2 are hypothetical

–1bg9; Barley–9 sequences, none with structures.

Criteria for declaring similar functional

surface to a matched surface

Search >2million surfaces with a template surface.

Shapes have to be very similar:p-value for cRMSD: < 10-3 .

Customized scoring matrices of 300 different time intervals.

The most similar surface has nmax of matrices capable of finding this homologous surface.

Declare a hit if >1/3 nmax of matrices give positive results.

Results for Amylase

• 1bag: found 58 PDB structures.

• 1bg9: found 48 PDB structures.

Altogether: 69

All belong to amylase (EC 3.2.1.1)

Query: B. subtilis Barley1bag 1bg9

Hits: human1b2y 1u2y22% 23%

0y)

False Positive Rate

Tru

ePosi

tive

Rate

(b)

Helmer-Citterich, M et al (BMC Bioinformat. 2005)Russell RB. (JMB 2003)Sternberg MJ Skolnick, JLichtarg, O (JMB2003)Ben-Tal, N and Pupko, T ( ConSurf )

• 110 protein families• Each points on the curve corresponds to p-values of various cRMSD cutoffs• Accuracy ~92% (EBI: 75%)

1

False positive rateTN FP

TN FP TN FPTrue positive rate

TPTP FN

≡

− =+ +

≡

+

Laskowski, Thornton, JMB, 05, 351:614-26

EBICATH domain, not function

Large Scale Prediction of Protein Functions

False Positive Rate (1-specificity)0.0 0.2 0.4 0.6 0.8 1.0

Tru

e Po

sitiv

e R

ate

(sen

sitiv

ity)

0.0

0.2

0.4

0.6

0.8

1.0

False Positive Rate

0.0 0.2 0.4 0.6 0.8 1.0

MC

C

0.78

0.80

0.82

0.84

0.86

0.88

(Tseng et al, 2009)

Orphan protein structures without functional annotation

Orphan proteins from Structural Genomics.

No known functions.

Often sequences homologs are hypothetical proteins.

Our tasks:Identify the functional pocket for the

structure.

Predict protein function.

Inferring biological functions of protein BioH

NP_991531

1A7U

1BRT

1HKH

1A88

1A8S

1A8Q

1IUN

1J1I

ZP_00217495

ZP_00221914

ZP_00194950

NP_822724

ZP_00031039

ZP_00032388

NP_069699

ZP_00081046

NP_273326

NP_618510

NP_871588

NP_778087

NP_249193

NP_790345

ZP_00086843

NP_904047

NP_298645

NP_796527

1M33

0.1 substitutions/site

99

100

94

68

100

93

91

60

62

70

85

44

63

82

97

66

99

53

93

98

100

49

85

90

85

The phylogenetic tree of 28 sequences related to BioH. Many are hypothetical genes.

The candidate binding pocket (CASTp id=35) of BioH (1m33) and a similar functional surface detected from (1p0p, E.C. 3.1.1.8)

Protein of unknownfunctions from structuralgenomics

A Probablistic Model of Enzyme Function

Enzymes have diverse reactivites:Some are very specific

Some can react with a range of substrates

An E.C. number alone is inadequate

Our approach: a probabilistic model

3.1.1.8 3.1.4.17 3.4.21.7 3.4.23.22

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Enzyme Classification Number

pro

bab

ility

BioH

Significant hits predicted for BioH:E.C. 3.1.1.8, p = 0.31, cholinesteraseE.C. 3.4.23.22, p = 0.26, aspartic endopeptidaseE.C. 3.4.21.7, p = 0.24, serine endopeptidaseE.C. 3.1.4.17, p = 0.20, phosphodiesterase

Experimental Results:Carboxyesterase E.C. 3.1.1.1 highLipase E.C. 3.1.1.3 lowThioesterase E.C. 3.1.1.5 lowAminopeptidease E.C. 3.4.11.5 low

Summary

Space filling structures of proteins: volume and surface models,

Geometric constructs and algorithms: Voronoi diagram, Delaunaytriangulation, and alpha shapeAlgorithmsA bit of topology

Application in proteins packing and function prediciton

References

H. Edelsbrunner and N. Shah. Incremental topological flipping works for regular triangulations. Algorithmica 15(1996), 223-241.

L. Guibas and J. Stolfi. Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Transactions on Graphiques, 4:74-123, 1985