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Intro to modeling April

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Part of the course: Introduction to biological modeling 1.Intro to modeling 2.Intro to biological modeling (Floor) 3.Modeling oscillators (Rob)

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Today’s goal: Learn how to solve ODE models in MATLAB

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Example: Heating of water

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Set the model dT w /dt = Rate of change of temperature of the water [°C s -1 ] T w = Temperature of the water [°C] T h = Temperature of the heater [°C] T a = Temperature of the air [°C] c 1 = Heat transfer coefficient 1 [s -1 ] c 2 = Heat transfer coefficent 2 [s -1 ] We will measure T w

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Initial parameter values From scientific literature. Decide which one(s) will be estimated from the experimental data.

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Fit the experimental data with the model

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P1 = 1 P2 = 1 SS = 350 P1 = 1.1 P2 = 3 SS = 55 P1 = 1.13 P2 = 4 SS = 12 Manual estimation of the parameters

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How precise are the estimated parameters? We can look at the sum of squares surface:

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Why modeling? Simulate the system Estimate parameters Control the system

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Menu The math behind modeling (45 min) Break (5 min) A few words about programming (10 min) Hands-on tutorial (45 min) What’s next (5 min)

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What’s a model?

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Types of models Deterministic Non-deterministic Probabilistic Discrete Continuous (in time) (in variable values)

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Grey box How do we build a model? White box: First principles Black box: Measurements

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Modeling techniques in biology Boolean Bayesian Differential Equations – Ordinary (ODE) – Partial (PDE) – Delay (DDE) …

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ODEs Rate of change One independent variable One or more dependent variables Order

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True or false? TRUE Only valid when x is the independent variable.

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Are these ODEs? YES Independent Dependent; First order YES Independent Dependent; Third order

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How to solve ODEs? General Solution Particular Solution

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How to solve ODEs? Analytical or Numerical solution – Euler – ode45

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Analytical solution Solve:

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Numerical solution The basis is Taylor’s series expansion:

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Taylor’s series example 1

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Taylor’s series example 2

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Euler’s method Under which condition is this valid?

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Euler’s method Solution in multiple time-steps. Iterate: Solution in one time-step:

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Example # Time steps15 f(x)

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What does it have to do with my ODE model? Model Initial value Step Time step

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ode45 ODE solver in MATLAB Uses information from the fourth and fifth derivative How to use it: [t,x] = time, x0, [], *) *You can pass whatever, for example the parameters (p) and/or the inputs (u) Mandatory. Always put in the same order. Put it here if more info will be passed.

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What do we need to use ode45? i.Model ii.Parameter(s) values iii.Input(s) values iv.Independent variable (time) values = t o, t f and Δt v.Initial values of dependant variable(s)

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Model Re-write in state-space form: 1.State variables(x): terms with derivative 2.Parameters(p): constants 3.Input(u): what‘s left

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From paper to MATLAB Re-write model in state space form. Save the model as a MATLAB function. Write a script that calls ode45 and your model. Identify state variables, parameters, inputs. Define equations and parameter values. Define initial values, call functions, plot results.

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Example: an irreversible reaction

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Now the language details…

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20 x0 = time = …1

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0.5 p = 0.02 Magic word: Function 20 x0 = time = …1 dxdt = 3x11 t0t1t2 x1 x2 x3 t3… t10

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20 x0 = 3000 time = …1 IN OUT dxdt = 11x3 x1x2x3 t0 t1 t2 t3 … t10 Summary: model function data Row vector Matrix

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0 t = …1 x = 11x3 x1x2x3 t0 t1 t2 t3 … t10

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Summary: script data 20 x0 = 3000 time = …1 IN OUT x = 11x3 x1x2x3 t0 t1 t2 t3 … t10 0 t = …1 Row vector Matrix Row vector

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From paper to MATLAB Re-write model in state space form. Save the model as a MATLAB function. Write a script that calls ode45 and your model. Identify state variables, parameters, inputs. Define equations and parameter values. Define initial values, call functions, plot results.

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Break

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A few words about programming All programs run in a defined ‘vertical’ way. MATLAB is an interpreted language. Errors can be found at any time. Read the errors messages. Most errors are in manipulation of row/column vectors.

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A few words about programming Shell/command line = type commands interactively Scripts (.m) = save commands Functions (.m) = define a command. Give same name to the function and the file. Variables – Any word/letter that stores data – They remain after the program ends Semicolon (;)

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Hands-on Tutorial

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Your turn! I. Biological system: Irreversible reaction (15 min)

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Your turn! I. Physical system: Greenhouse temperature (30 min)

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Homework III. Biological system: Bioreactor

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What’s next Learn/Teach: Introduction to biological modeling (Floor) Modeling oscillators (Rob) Reproduce modeling by Danino et al. 2010: MATLAB 7.0 Team: Brendan, Floor, Rob, Dorett, Mariana,... May 6 th presentation

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