九大数理集中講義Comparison, Analysis, and Control

of Biological Networks (3)Domain-Based Mathematical Models for Protein Evolution

Tatsuya Akutsu

Bioinformatics Center

Institute for Chemical Research

Kyoto University

Contents A simple evolutionary model of protein

domains A domain-based model of protein-protein

interaction networks An evolutionary model of multi domain

proteins

Motivation of Our Studies Explaining observed distributions on

proteins and PPI networks PPI networks are scale-free [Jeong et al., 2001]

#(proteins having k domain families) follows exponential distribution 　　 [Koonin et al., 2002]

#(proteins having k domains) follows power-law [Koonin et al.,

2002]

#(domains appearing in k proteins) follows power-law [Wuchty, 2001]

Providing simple evolutionary models In real proteins, what evolve are not networks

but genes/proteins

An Evolutionary Model of Protein Domains

J.C. Nacher, M. Hayashida and T. Akutsu: Physica A, 367, 538-552, 2006

Protein Domain

Domain: Well-defined region within a protein that either performs a specific function or constitutes a stable unit

Protein consisting of3 domains

Evolutionary Model of Protein Domains

N proteins, each one consists of only one domain

(domains are different from each other)

We repeat T times the following steps:

a) With probability (1-a) we create a new protein with new domain(MUTATION)

b) Otherwise, we randomly select one protein and make a copy of it(PROTEIN DUPLICATION)

We assume that each protein consists of only one domain

Model(continued)

i : i-th kind of domain : number of proteins consisting of i-th domain : time when i-th domain was first created

t

ka

dt

dk ii a

ii t

tck

ik

1-aa

Duplication of Protein

T timesa ~ 1.0

Mutation

Q(k): number of domains each of which appears in k proteins

)]/1(1[)( akkQ

it

As in Barabasi &Albert 1999

Protein duplication

mutation

Prob.= aProb.= 1- a

Model of Protein Evolution

Exaplanation of Q(k)

1 2 3 4 5

Types of domains

6

Types of proteins

1,2,2,2,3,1 654321 kkkkkk

0)5()4(,)3(,)2(,)1( 61

63

62 QQQQQ

Our Model vs. Preferential Attachment Similarity

#(proteins with the i-th domain) ⇔ degree of the i-th node Duplication of protein with the i-th domain ⇔ Attachment of an edge to

the i-th node Mutation (creation of a protein with a new domain) ⇔ Addition of a new

vertex Difference:

new node

new edge

3)]/1(1[ vs. kk a

1-a aDuplication

a ~ 1.0

Mutation

PD(1)=3PD(2)=1PD(3)=1

A Domain-Based Model of Protein-Protein

Interaction Networks

J.C. Nacher, M. Hayashida and T. Akutsu: BioSystems, 95, 155-159, 2009

A Domain-Based Model of Protein-Protein Interactions [Sprinzak & Margalit 2001, Deng et al. 2002] Proteins interact ⇔ There exist interacting domain pair(s)

A B

C D

X

Y Z

Domain-DomainInteraction

Protein-ProteinInteraction

Combination of Domain Evolution Model and Domain-based Protein-Protein Interaction Model

Evolutional model of protein domains

Random interaction of domains

Domain-based protein-protein interactions Proteins interact ⇔ There exist interacting domain pair(s)

)]/1(1[)( aD kkP

) with interacts Pr( ji DD

Scale-free property of PPI (protein-protein interaction network) )]/1(1[)( a

PPI kkP

Mathematical Analysis

domainA

domainB

nA=x=3

nB=y=2

3 proteinswith

degree 2

However, if the number of domain-domain interactions is large,the distribution approaches to the normal distribution because of the central limit theorem

An Evolutionary Model of Multi Domain Proteins

J.C. Nacher, M. Hayashida and T. Akutsu: BioSystems, 101:127-135, 2010.

Domain Fusion and Internal Duplication (1) 1. Internal Duplication

Duplication of one or more domains inside one protein 2. Domain Fusion

Two proteins are merged

Domain Fusion

MutationProtein

DuplicationInternal Domain

duplication

Modeling of Duplication, Mutation and Fusion (1) Ni

(t) : #proteins having i domains at time t

pm : prob. mutation (creation of new protein) occurs

pd : prob. duplication occurs

pf : prob. fusion occurs

Modeling of Duplication, Mutation and Fusion (2)

By letting ni(t) =Ni

(t) /t and ni = ni

(t) for t→∞

Modeling of Duplication, Mutation and Fusion (3)Using generation function, we have exact solution

Using Stirling’s approximation

It shows nk follows almost exponential distribution

Modeling of Internal Duplication

By letting ni(t) =Ni

(t) /t and ni = ni

(t) for t→∞

nk follows

power-law

Combination of Mutation, Fusion, Internal/External Duplications

Difficult to solve

⇒ Computer simulation

Summary A simple (simplest? ) model of protein domain

evolution, which explains power-raw distribution A domain-based model of protein-protein

interaction network ⇒ Explains power-law property of PPI

⇒ Good agreement between simulation and real data

⇒ Simpler than existing models (e.g., duplication-divergence)

An evolutionary model of multi-domain proteins

⇒ #(proteins having k domain families) follows exponential

⇒ #(proteins having k domains) follows power-law

⇒ Good agreement between simulation and real data

⇒ Importance of role of internal duplications