Computational Biology: An overview

Shrish Tiwari

CCMB, Hyderabad

Mathematics, Computers & Biology

“The book of nature is written in the language of mathematics…”

- Galileo What about biology?

Changing scenario due to the development of Biological sequence data Chaos theory Game theory

Computer Applications in Biology

Pattern recognition

Pattern formation and characterisation

Structural modeling of bio-molecules

Modeling of macro-systems

Image processing

Data management and warehousing

Statistical analysis Next

Pattern recognition

Predicting protein-coding genes (GenScan)

Motif search (MotifScan, promoter search) Finding repeats (TRF, Reputer) Predicting secondary structure (PHDsec,

nnpredict) Classification of proteins (SCOP) Prediction of active/functional sites in

proteins (PDBsitescan)Back

Patterns in nature

Simulated Patterns

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Structural modeling

Protein folding: homology modeling, threading, ab initio methods

Protein interaction networks, biochemical pathways

Cellular membrane dynamics

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Macro-system modeling

Modeling of dynamics of organs like brain and heart

Modeling of environmental dynamics, interacting species

Modeling of population growth and expansion

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Image processing

Gridding of spots in the image Removing background intensity

(usually not uniform across the array)

Computing the ratio of intensities in case of two colour probes

Comparison of slides from different arrays Back

Computational Tools

Dynamic programming algorithm

Markov Model, Hidden Markov Model, Artificial Neural Network, Fourier Transform

Molecular dynamics, Monte Carlo, Genetic Algorithm simulations

Cellular Automata

Game theory

Statistical tools

Dynamic Programming

An optimisation tool that works on problems which can be broken down to sub-problems

Used widely in sequence alignment algorithms in bioinformatics

Other applications: speech, vocabulary, grammar recognition

Back

Pattern recognition tools

Markov model: state of system at time t depends on its state at time t-1, transition probabilities between states are defined. Example: gene finding

Artificial neural networks: attempt to simulate the learning process of real neural network system

Fourier transform: measure correlations between states at different time/space points Back

Optimisation tools

Molecular dynamics: apply Newton’s equation of motion to follow the dynamics of a system

Monte Carlo simulation: randomly hop from one state to another until you find the optimal state Back

Genetic algorithm: attempt to simulate evolutionary mechanism of mutations and recombination to find the optimal solution

Cellular Automata

Components: 1) a lattice, 2) finite number of states at each node, 3) rule defining the evolution of a state in time

Example: game of life _ 1) on a 2-d lattice each cell represents an individual, 2) states 0 (dead) or 1 (live), 3) a cell dies if it has less than 2 or more than 3 live neighbours, a dead cell becomes live if 3 of its neighbours are live

Simple “life” patterns

Still lives

Oscillator

Glider

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Game theory Game: 1) involves 2 or more players, 2)

one or more outcomes, 3) outcome depends on strategy adopted by each player

Components: 1) 2 or more players, 2) set of all possible actions, 3) information available to players before deciding on an action, 4) payoff consequences, 5) description of player’s preference over payoffs

Game theory: an example

Traffic as a game: The commuters are players Traffic rules define the set of possible actions

(including disobeying traffic rules) Payoff consequences: fined if you violate

traffic rules, you may suffer injury in accidents or die

Information available: Players preferences: safe driving, dangerous

driving etc. Back

Statistical tools

Expectation value computation to assess the significance of alignment

Clustering methods: UPGMA, WPGMA, k-means etc.

Assessing significance of genotype-phenotype association: chi-square test, Fisher’s exact test etc.

Chaos Theory: An Introduction

One of the behaviours of a non-linear dynamical system

Deterministic yet unpredictable!!

Sensitive to initial conditions/small perturbations

First discovered by Lorenz when he was simulating the weather dynamics using simplified hydro-dynamics model

The Lorenz attractor

Simplified model of convections in the atmosphere

dx / dt = a (y - x)

dy / dt = x (b - z) - y

dz / dt = xy - c z

a = 10, b = 28, c = 8/3

The Bernoulli shift

Map: f:x (2x mod 1), 0 ≤ x ≤ 1.

t = 0 1 2 3 4 5 6 7 8

x = .2 .4 .8 .6 .2 .4 .8 .6 .2

.21 .42 .84 .68 .36 .72 .44 .88 .76 Binary representation:

0.2: 0.001100110011…

0.21: 0.001101011100…

Chaotic dynamics: An example

Simplest system exhibiting chaos, the logistic map: xn+1 = rxn(1 – xn ), 0 < xn < 1

This simple equation exhibits a rich dynamical behaviour, ranging from stationary state to chaotic dynamics, as the parameter r varies from 0-4

This system models the population dynamics of a species whose generations do not overlap

Stationary state

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Xn Series1

Acrobat Document

2-periodicity

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1 13 25 37 49 61 73 85 97Time

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4-periodicity

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1 13 25 37 49 61 73 85 97

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Chaos

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1 14 27 40 53 66 79 92

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Logistic map bifurcation diagram

First return map

Plot of xn+1 against xn for discrete systems, and xt+T against xt for continuous dynamics, where T is some fixed interval

Return map of a periodic orbit is a finite set of points

Return map of a stochastic system a scatter of infinite number of points

Return map of a chaotic system an infinite number of points in a structure

Return map: Logistic map

Return map: Lorenz attractor

Controlling chaos

Different kinds of control are possible: Suppression of chaos, I.e. bring the system out

of chaotic behaviour into some regular dynamics: e.g. adaptive control

Remain in the chaotic dynamics, but force the system to remain in one of the unstable periodic orbits: e.g. OGY (Ott, Grebogi & Yorke) method

Sustain or enhance chaos: desirable for example in combustion where homogeneous mixing of gas and air improves the combustion

Synchronisation: confidential communication

Control of cardiac chaos

A. Garfinkel et al. applied the OGY method of control to arrest arrhythmia in a rabbit’s heart (Science 257, 1230-35 (1992) )

Arrhythmia was induced in the rabbit heart by injecting the animal with the drug ouabain

The first return map In-1 vs. In, the interbeat interval, identified periodic orbits with saddle instability

When the heart dynamics approached one of these points, small electrical pulses were used to force the system on the unstable periodic orbit

Prey-Predator Model

Simplest description of prey-predator interactions is given by the Lotka-Volterra equations:

dH/dt = rH – aHPdP/dt = bHP – mP

H: density of prey P: denstiy of predatorsr: intrinsic prey growth rate a: predation

rateb: reproduction rate of predator per prey

eatenm: predator mortality rate

Game theory Deals with situations involving:

2 or more players Choice of action depends on some strategy One or more outcomes Outcome depends on strategy adopted by all

players: strategic interaction Elements of a game:

Players Set of all possible actions Information available to players The payoff consequences A description of players’ preferences over

payoffs

Prisoners’ dilemma: An example

Players: 2 prisoners A and B Two possible actions for each prisoner:

Prisoner A: Confess, Don’t confess Prisoner B: Confess, Don’t confess

Prisoners choose simultaneously, without knowing what the other choses

Payoff quantified by years in prison: fewer years greater payoff

Outcomes: 1) both don’t confess: 1 year in prison for both, 2) 1 confesses other does not: the one who confesses is free, other gets 15 years, 3) both confess: both get 5 years

Prey-predator model with predators using hawk and dove tactics

P. Auger et al. recently studied a prey-predator model with the predators using a mix of hawk and dove strategies (Mathematical Sciences 177&178, 185-200 (2002) )

A classical Lotka-Volterra model was used to describe the prey-predator interaction

Predators use two behavioural tactics when they contest a prey with another predator: hawk or dove

Prey-predator model with predators using hawk and dove tactics

Assumptions: Gain depends on the prey density, which

modifies predator behaviour The prey-predator interaction acts at a

slow time scale The behavioural change of predator works

on fast time scale

Aim: effects of individual predator behaviour on the dynamics of the prey-predator system

Study carried out for different prey densities

Prey-predator model with predators using hawk and dove tactics

Conclusions: There is a relationship between

behaviour and prey density Aggressive (or hawk) behaviour prevails

in high prey density A mix of hawk and dove strategy

observed for low prey density A change of view: aggressive

behaviour is not advantageous when prey (resources) are rare and collaboration should be favoured

This is just the beginning …

Mathematics and computers are playing an increasingly important role in biology

We have just begun to scratch the surface of biological discoveries

The field is vast and largely untapped so we need young minds to be fascinated by these problems

References

A. Garfinkel, M.L. Spano, W.L. Ditto and J.N. Weiss “Controlling cardiac chaos” Science 257, 1230-1235 (1992).

P. Auger, R.B. de la Parra, S. Morand and E. Sanchez “A prey-predator model with predators using a hawk and dove tactics” Math. Biosci. 177&178, 185-200 (2002)