Part 1: Biological Networks 1.Protein-protein interaction networks 2.Regulatory networks...

Part 1: Biological Networks

1. Protein-protein interaction networks

2. Regulatory networks

3. Expression networks

4. Metabolic networks

5. … more biological networks

6. Other types of networks

[Qian, et al, J. Mol. Bio., 314:1053-1066]

Expression networks

Regulatory networks

[Horak, et al, Genes & Development, 16:3017-3033]

Expression networksRegulatory networks


Interaction networks

Metabolic networks

[DeRisi, Iyer, and Brown, Science, 278:680-686]


Interaction networks

Metabolic networks

... more biological networks

All of SCOP entries

1Oxido-

reductases

3Hydrolases

1.1Acting on CH-OH

1.1.1.1 Alcohol dehydrogenase

ENZYME

1.1.1NAD and

NADP acceptor

NON-ENZYME

3.1Acting on

ester bonds

1 Meta-bolism

1.1 Carb.

metab.

3.8 Extracel.

matrix

3.8.2 Extracel.

matrixglyco-protein

1.1.1 Polysach.

metab.

3.8.2.1 Fibro-nectin

General similarity Functional class similarityPrecise functional similarity

3 Cell

structure

1.5Acting on

CH-NH

3.4Acting on

peptide bonds

1.1.1.3Homoserine

dehydrogenase

1.2Nucleotide

metab.

3.1 Nucleus

3.8.2.2Tenascin

1.1.1.1 Glycogenmetab.

1.1.1.2 Starchmetab.

3.1.1.1 Carboxylesterase

3.1.1Carboxylic

ester hydro-lases

3.1.1.8 Cholineesterase

Hierarchies & DAGs [Enzyme, Bairoch; GO, Ashburner; MIPS, Mewes, Frishman]

Neural networks[Cajal]

Gene order networks Genetic interaction networks[Boone]

... more biological networks

Other types of networks

Disease Spread

[Krebs]

Social Network

Food Web

ElectronicCircuit

Internet[Burch & Cheswick]

Part 2: Graphs, Networks

• Graph definition• Topological properties of graphs

- Degree of a node

- Clustering coefficient

- Characteristic path length

• Random networks• Small World networks• Scale Free networks

• Graph: a pair of sets G={P,E} where P is a set of nodes, and E is a set of edges that connect 2 elements of P.

• Directed, undirected graphs

• Large, complex networks are ubiquitous in the world:

- Genetic networks- Nervous system- Social interactions- World Wide Web

• Degree of a node: the number of edges incident on the node

i

Degree of node i = 5

Clustering coefficient LOCAL property

• The clustering coefficient of node i is the ratio of the number of edges that exist among its neighbours, over the number of edges that could exist

iE

Clustering coefficient of node i = 1/6

• The clustering coefficient for the entire network C is the average of all the

iC

Characteristic path length GLOBAL property

• is the number of edges in the shortest path between vertices i and j

( , )i jL

• The characteristic path length L of a graph is the average of the for every possible pair (i,j)

( , )i jL

( , ) 2i jL

i

j

Networks with small values of L are said to have the “small world property”

Models for networks of complex topology

• Erdos-Renyi (1960)• Watts-Strogatz (1998)• Barabasi-Albert (1999)

The Erdős-Rényi [ER] model (1960)

• Start with N vertices and no edges• Connect each pair of vertices with probability PER

Important result: many properties in these graphs appear quite suddenly, at a threshold value of PER(N)

-If PER~c/N with c<1, then almost all vertices belong to isolated trees-Cycles of all orders appear at PER ~ 1/N

The Watts-Strogatz [WS] model (1998)

• Start with a regular network with N vertices• Rewire each edge with probability p

For p=0 (Regular Networks): •high clustering coefficient •high characteristic path length

For p=1 (Random Networks): •low clustering coefficient•low characteristic path length

QUESTION: What happens for intermediate values of p?

1) There is a broad interval of p for which L is small but C remains large

2) Small world networks are common :

The Barabási-Albert [BA] model (1999)

ER Model

Look at the distribution of degrees

ER Model WS Model actors power grid www

The probability of finding a highly connected node decreases exponentially with k

( ) ~P K K

• GROWTH: starting with a small number of vertices m0 at every timestep add a new vertex with m ≤ m0

• PREFERENTIAL ATTACHMENT: the probability Π that a new vertex will be connected to vertex i depends on the connectivity of that vertex:

● two problems with the previous models:

1. N does not vary

2. the probability that two vertices are connected is uniform

( ) ii

jj

kk

k

a) Connectivity distribution with N = m0+t=300000 and m0=m=1(circles), m0=m=3 (squares), and m0=m=5 (diamons) and m0=m=7 (triangles)

b) P(k) for m0=m=5 and system size N=100000 (circles), N=150000 (squares) and N=200000 (diamonds)

Scale Free Networks

Part 3: Machine Learning

• Artificial Intelligence/Machine Learning• Definition of Learning• 3 types of learning

1. Supervised learning2. Unsupervised learning3. Reinforcement Learning

• Classification problems, regression problems• Occam’s razor• Estimating generalization • Some important topics:

1. Naïve Bayes2. Probability density estimation3. Linear discriminants4. Non-linear discriminants (Decision Trees, Support Vector

Machines)

Bayes’ Rule: minimum classification error is achieved by selecting the class with largest posterior probability

PROBLEM: we are given and we have to decide whether it is an a or a b

1( , , )Tdx xx

Classification Problems

Regression Problems

PROBLEM: we are only given the red points, and we would like approximate the blue curve (e.g. with polynomial functions)

QUESTION: which solution should I pick? And why?

Naïve Bayes

F 1 F 2 F 3 … F n TARGET

Gene 1 1 1.34 1 … 2.23 1

Gene 2 0 4.24 44 … 2.3 1

Gene 3 1 3.59 34 … 34.42 0

Gene 4 1 0.001 64 … 24.3 0

Gene 5 0 6.87 6 … 6.5 0

… … … … … … …

Gene n 1 4.56 72 … 5.3 1

Example: given a set of features for each gene, predict whether it is essential

( | )( | ) k k

k

p C P CP C

p

xx

x

Bayes Rule: select the class with the highest posterior probability

For a problem with two classes this becomes:

0Cotherwise, choose class

1 1

0 0

( | )1

( | )

p C P C

p C P C

x

x 1C then choose class if

1

0

( | )

( | )i

ii

p x CL

p x Cwhere and are called Likelihood Ratio for feature i.

1 2 1 2| , ,..., | | | |i n i i i n ip C p x x x C p x C p x C p x C x

Naïve Bayes approximation:

1 1 2 1 1 1

1 0 2 0 0 0

01 2

1

( | ) ( | ) ( | )1

( | ) ( | ) ( | )n

n

n

p x C p x C p x C P C

p x C p x C p x C P C

P CL L L

P C

For a two classes problem:

Probability density estimation

• Assume a certain probabilistic model for each class • Learn the parameters for each model (EM algorithm)

Linear discriminants

• assume a specific functional form for the discriminant function• learn its parameters

Decision Trees (C4.5, CART)

ISSUES: • how to choose the “best” attribute• how to prune the tree

Trees can be converted into rules !

Part 4: Networks Predictions

• Naïve Bayes for inferring Protein-Protein Interactions

Network Gold-Standards

The data

[Jansen, Yu, et al., Science; Yu, et al., Genome Res.]

Feature 2, e.g. same functionFeature 1, e.g. co-expression

Gold-standard +Gold-standard –


( | )

( | )i

ii

p xL

p x

Likelihood Ratio for Feature i:






L1 = (4/4)/(3/6) =2( | )

( | )i

ii

p xL

p x





( | )

( | )i

ii

p xL

p x

Likelihood Ratio for Feature i: L1 = (4/4)/(3/6) =2



Network Gold-StandardsL1 = (4/4)/(3/6) =2L2 = (3/4)/(3/6) =1.5

For each protein pair:LR = L1 L2log(LR) = log(L1) + log(L2)

( | )

( | )i

ii

p xL

p x


1. Individual features are weak predictors,

LR ~ 10;

2. Bayesian integration is much more powerful,

LRcutoff = 600 ~9000 interactions

1 3.632 0.583 0.53

1 124.932 85.503 87.974 67.365 26.466 9.877 4.338 1.679 0.70

10 0.2511 0.1412 0.1113 0.0514 0.0415 0.0916 0.0017 0.0018 0.0019 0.00

1 25.502 22.533 8.634 21.285 0.06

1 9.222 14.363 4.384 3.055 0.76

Functional similarity: MIPS

Functional similarity: GO

Essentiality

Co-expression

Part 1: Biological Networks 1.Protein-protein interaction networks 2.Regulatory networks...

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