Lecture1 - Computational Science and Engineering! Modeling...

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Computational Science and Engineering Modeling and Simulation in Computational Systems Biology Lecture 1: Ordinary Differential Equations and the Applications in Modeling Yang Cao Department of Computer Science
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Computational Science and Engineering Summary General Introduction for ODEs Modeling with ODEs
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Computational Science and Engineering A Chemically Reacting System Ω N M M R R , , 1 j R N S S , , 1 T else. something or else, something or , + j i i i S S S S A • Molecules of chemical species In a Volume , at temperature • Different conformation or excitation levels are considered different species if they behave differently elemental reaction channels Each describes a single instantaneous physical event which changes the population of at least one species. For example, RNAP Promoter + RNAP Promoter
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Computational Science and Engineering Ordinary Differential Equation For each species, assign a state variable, which describes its concentration or population. RNAP Promoter + RNAP Promoter 1 X 2 X 3 X Basic Deterministic Assumption: The state change is proportional to the state of the reactants and time t t X t kX t X Δ = Δ ) ( ) ( ) ( 2 1 1 ) ( ) ( ) ( 2 1 ' 1 t X t kX t X =
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Computational Science and Engineering Ordinary Differential Equations Many scientific applications result in the following system of equations which is called ordinary differential equations (ODEs). Two types: • Initial value problem (IVP) • Boundary value problem (BVP) F = m ˙ ˙ x dx dt = f ( t , x ) Example: Newton’s Motion Law
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Computational Science and Engineering Analytic Solution There are a few types of ODEs for which we are able to obtain analytic solution and analyze their properties. Equation with separable variables – example Total differential equations – example y ' = f ( x ) g ( y ) dy g ( y ) = f ( x ) dx + C y ' = λ y P ( x , y ) + Q ( x , y ) y ' = 0 y ' = y
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Computational Science and Engineering When we cannot find analytic solution Existence – Continuous function Uniqueness – Lipschitz condition Equilibrium State Stability – Example: linear system – Lyapunov function f ( t , y ) f ( t , z ) L y z y ' = Ay
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Computational Science and Engineering Numerical Solution Multistep Methods – Euler method – Adams method – BDF method Runge-Kutta Methods – Explicit RK – Implicit RK
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Computational Science and Engineering Softwares for Numerical Solution DASSL (BDF method) CVODE (BDF method) RADAU5 (RK method) MATLAB functions
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