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Computational Biology: An overview Shrish Tiwari CCMB, Hyderabad

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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

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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 Next

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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

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Patterns in nature

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Simulated Patterns Back

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Structural modeling Protein folding: homology modeling, threading, ab initio methods Protein interaction networks, biochemical pathways Cellular membrane dynamics Back

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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 Back

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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 Back

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Computational Tools Dynamic programming algorithm Markov Model, Hidden Markov Model, Artificial Neural Network, Fourier Transform Markov Model, Hidden Markov Model, Artificial Neural Network, Fourier Transform Molecular dynamics, Monte Carlo, Genetic Algorithm simulations Molecular dynamics, Monte Carlo, Genetic Algorithm simulations Cellular Automata Game theory Statistical tools

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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

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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 BackBack

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

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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

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Simple “life” patterns Still lives Oscillator Glider Back

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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

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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

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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.

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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

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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

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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…

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Chaotic dynamics: An example Simplest system exhibiting chaos, the logistic map: x n+1 = rx n (1 – x n ), 0 < x n < 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

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Logistic map bifurcation diagram

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First return map Plot of x n+1 against x n for discrete systems, and x t+T against x t 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

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Return map: Logistic map

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Return map: Lorenz attractor

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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

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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 I n-1 vs. I n, 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

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Prey-Predator Model Simplest description of prey-predator interactions is given by the Lotka- Volterra equations: dH/dt = rH – aHP dP/dt = bHP – mP H: density of preyP: denstiy of predators r: intrinsic prey growth rate a: predation rate b: reproduction rate of predator per prey eaten m: predator mortality rate

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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

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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

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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

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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

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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

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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

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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)

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