Intro to modeling

April 22 2011

Part of the course:Introduction to biological modeling

1. Intro to modeling

2. Intro to biological modeling (Floor)

3. Modeling oscillators (Rob)

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

Example: Heating of water

Set the model

• dTw/dt = Rate of change of temperature of the water [°C s-1]

• Tw = Temperature of the water [°C]

• Th = Temperature of the heater [°C]

• Ta = Temperature of the air [°C]

• c1 = Heat transfer coefficient 1 [s-1]

• c2 = Heat transfer coefficent 2 [s-1]

• We will measure Tw

Initial parameter values

• From scientific literature.

• Decide which one(s) will be estimated from the experimental data.

Fit the experimental data with the model

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

inpu

t u

p = [0.00029 0.00023]

0 50 100 150 200 250 30023.5

24

24.5

25

25.5

26

26.5

27

time

outp

ut y

and

z (

--)

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

inpu

t u

p = [0.000137 -0.00102]

0 50 100 150 200 250 30023.5

24

24.5

25

25.5

26

26.5

27

time

outp

ut y

and

z (

--)

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

inpu

t u

p = [0.00015 0.0001]

0 50 100 150 200 250 30023.5

24

24.5

25

25.5

26

26.5

27

time

outp

ut y

and

z (

--) P1 = 1

P2 = 1SS = 350

P1 = 1.1P2 = 3SS = 55

P1 = 1.13P2 = 4SS = 12

Manual estimation of the parameters

How precise are the estimated parameters?

2.42.5

2.62.7

2.82.9

x 10-4 -2.4

-2.35-2.3

-2.25-2.2

-2.15-2.1

-2.05-2

x 10-4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

p2

sum of squares surface

p1

We can look at the sum of squares surface:

Why modeling?

• Simulate the system

• Estimate parameters

• Control the system

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)

What’s a model?

Types of models

• Deterministic

• Non-deterministic

• Probabilistic

• Discrete

• Continuous

(in time)

(in variable values)

Grey box

How do we build a model?

White box:First principles

Black box:Measurements

Modeling techniques in biology

• Boolean

• Bayesian

• Differential Equations– Ordinary (ODE)

– Partial (PDE)

– Delay (DDE)

• …

ODEs

• Rate of change

One independent variable

One or more dependent variablesOrder

True or false?

TRUE

Only valid when x is the independent variable.

Are these ODEs?

YES

Independent

Dependent; First order

YES

Independent

Dependent; Third order

How to solve ODEs?

General Solution Particular Solution

How to solve ODEs?

• Analytical or

• Numerical solution

– Euler

– ode45

Analytical solutionSolve:

Numerical solution

The basis is Taylor’s series expansion:

Taylor’s series example 1

Taylor’s series example 2

Euler’s method

Under which condition is this valid?

Euler’s method

Solution in multiple time-steps. Iterate:

Solution in one time-step:

Example

# Time steps 1 5

f(x)

What does it have to do with my ODE model?

Model

Initial value

Step

Time step

ode45

• ODE solver in MATLAB

• Uses information from the fourth and fifth derivative

• How to use it:

[t,x] = ode45(@function, 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.

What do we need to use ode45?

• Model

• Parameter(s) values

• Input(s) values

• Independent variable (time) values = to, tf and Δt

• Initial values of dependant variable(s)

Model

Re-write in state-space form:

2.State variables(x): terms with derivative

3.Parameters(p): constants

4.Input(u): what‘s left

From paper to MATLAB

Identify state variables, parameters, inputs.

Define equations and parameter values.

Define initial values, call functions, plot

results.

Example: an irreversible reaction

Now the language details…

20x0 = 30 0

0time = 0.1 0.2 … 1

0.5p = 0.02

Magic word: Function

20x0 = 30 0

0time = 0.1 0.2 … 1

dxdt =

3x11

t0 t1 t2

x1

x2

x3

t3 … t10

20x0 = 30 0

0time = 0.1 0.2 … 1

IN

OUTdxdt =

11x3

x1 x2 x3

t0

t1

t2

t3

… t10

Summary: model function data

Row vector

Row vector

Matrix

0t = 0.1 0.2 … 1

x =

11x3

x1 x2 x3

t0

t1

t2

t3

… t10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

Summary: script data

20x0 = 30 0

0time = 0.1 0.2 … 1

IN

OUTx =

11x3

x1 x2 x3

t0

t1

t2

t3

… t10

0t = 0.1 0.2 … 1

Row vector

Row vector

Matrix

Row vector

From paper to MATLAB

Identify state variables, parameters, inputs.

Define equations and parameter values.

Define initial values, call functions, plot

results.

Break

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.

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 (;)

Hands-on Tutorial

Your turn!

I. Biological system: Irreversible reaction (15 min)

Your turn!

I. Physical system: Greenhouse temperature (30 min)

HomeworkIII. Biological system: Bioreactor

What’s next

Learn/Teach:Introduction to biological modeling (Floor)Modeling oscillators (Rob)

Reproduce modeling by Danino et al. 2010:MATLAB 7.0Team: Brendan, Floor, Rob, Dorett, Mariana, ...

May 6th presentation