1

Introduction to Mathematical Modeling - Systems Theory in the Toolbox for Systems Biology

Mats Jirstrand, PhD, Assoc Prof

Head of Department Systems Biology and Bioimaging

The 5th International Course in Yeast Systems Biology 2011June 6, 2011

Mats Jirstrand

Systems Biology and Bioimaging

� Pharmacokinetics-Pharmacodynamics

� Protein Synthesis and Secretion

� Arrhythmia- Atrial Fibrillation

- Ion-channel Modeling

� System Identification- Continuous-discrete Identification

- Nonlinear Mixed Effects Modeling

� Single Cell Image Analysis- Time-lapse, Tracking, Segmentation

� Medical Image Analysis- Shape Modeling, MR, PET, CT, SPEC

Research Topics:

The department conducts research, application and development of computational methods, software tools, and dynamic models of biological systems on different levels of abstraction utilizing time and spatially resolved measurement data.

2

Mats Jirstrand

Outline

� What is Systems Theory?

� Description of Dynamic Systems

� Linear Systems

� Non-linear Systems

� Stability

� Linearization

� Qualitative Behavior

� Phase plane

� Bifurcations

� Feedback

� Summary

Mats Jirstrand

What is Systems Theory?

Mathematical Systems Theory is concerned with

the description and understanding of systems by

the means of mathematics.

• Analysis

• Modeling

• Simulation

• Control

• Identification

• Filtering

• Communication

• Signal Processing

• ...

3

Mats Jirstrand

What is Systems Theory?

We will focus on Dynamic Systems!

Mats Jirstrand

Dynamic Systems- Examples

4

Mats Jirstrand

Dynamic Systems- Definition

A system is an object in which variables of different

kinds interact and produce observable signals.

In a dynamic system the current output value

depends not only on the current input value but

also on its earlier values.

Inputs Outputs

Mats Jirstrand

Dynamic Systems- Linear vs Nonlinear

Linear system:

u1+u2 → y1+y2

α u → α y

� Large toolbox of analytical results

� Often good approximation for small perturbations

around a steady state

� Try simple things first!

5

Mats Jirstrand

Dynamic Systems- Some Other Aspects

� Dynamic – Static

� Deterministic – Stochastic

� Continuous-time – Discrete-time

� Lumped – Distributed

� Change oriented – Discrete event

Mats Jirstrand

Description of Dynamic Systems- Differential Equations, Linear Systems

where

Linear differential equations

The state at time t is the minimal amount of information needed to determine future output given future input.

The State-space Form:

- state

- input

- output- inital state

6

Mats Jirstrand

Description of Dynamic Systems- A multi-compartment example

Matrix form:

Notebook example

Mats Jirstrand

Description of Dynamic Systems- Differential Equations, Non-linear Systems

The State-space Form:

- state

- input

- output- inital state

Now, and are non-linear vector valued functions

7

Mats Jirstrand

Description of Dynamic Systems- A Pathway Example

Notebook example

R

S

EE-P

Mutual activation*

* Sniffers, buzzers, toggles, and blinkers: dynamics of regulatory

and signaling pathways in the cell, Tyson et al.

Notation:

Mats Jirstrand

Stability- Stability of Solutions of Differential Equations

Let x*(t) be a solution given x*(0). The solution

is stable if for each ε there exists δ such that

|x(0)-x*(0)|<δ implies |x*(t)-x(t)|<ε for all t>0.

The solution is called asymptotically stable if

it is stable and and there exists δ such that

|x(0)-x*(0)|<δ implies |x*(t)-x(t)|→0, t→∞.

8

Mats Jirstrand

Stability- Linear Systems

The stability properties of is determined by the eigenvalues of A.

System: Stationary point: →

Asymptotic stability iff .

eig(A) are solutions det(λI-A)=0 (an algebraic equation in λ).

Ex Stability of the compartmental model Notebook example

Mats Jirstrand

How to find it? Solutions to !

Stability- Stationary State

Stationary state: → for all t>0

System:

The stationary state is the operating point for many systems!

Definitely not for all!

Ex Stationary state of the pathway model Notebook example

9

Mats Jirstrand

Linearization

The system:

Taylor expansion of f around x0 and u0 gives:

New variables ∆x and ∆u →

evaluated in x0 and u0. Ex Linearization and stability of the

pathway model Notebook example

Mats Jirstrand

Qualitative Behavior- Phase-plane

A phase plane of a dynamic system is an xy-

graph of how the state evolves over time.

� For 2D systems it’s a parametric plot with time as parameter

� For higher dimensional systems one may look at 2D-projections

� Stable systems: trajectories converging to a stationary state

� Oscillating systems: trajectories form closed orbits

Ex A phase plane of the pathway model

Notebook example

10

Mats Jirstrand

Qualitative Behavior- Bifurcations

A bifurcation is a qualitative change in the

behavior of a dynamic system when

changing a parameter.

Ex A stable stationary point becomes unstable →→

constant solution becomes an oscillatory limit cycle.This happens when the eigenvalues of the linearization crosses the

imaginary axis.

Mats Jirstrand

Feedback

Sloppy: There is feedback in the system if a

signal is injected/transmitted elsewhere in

such a way that it influence itself.

G-

r e y

Engineering Biology

Ex A pathway example

11

Mats Jirstrand

Feedback- Why?

� Decreases sensitivity to disturbances

� Increases robustness (makes it possible to build systems

with ”exact” behavior from inexact parts)

Mats Jirstrand

Feedback- Biology

The feedback is very often of non-additive character!

Ex Phosphorylation cascades

12

Mats Jirstrand

Feedback- Engineering, PI control

- error

- proportional control (P)

- proportional control

with integral action (PI)

When t→1, e(t) → 0. Otherwise u(t) will explode! Simulink example

Mats Jirstrand

Feedback- Biology, PI control

Y

A

v3

v4

E2

E1

(saturation)

Integral feedback:

Stationary value of Y is independent

of “everything” except parameters in

v3 and v4PWL simulation

Tau-Mu Yi, Yun Huang, Melvin I. Simon, and John Doyle. Robust perfect adaptation in bacterial chemotaxis through integral

feedback control. PNAS, April 25, 2000, vol 97, no 9, 4649-4653.

13

Mats Jirstrand

Summary

� Description of Dynamic Systems� Linear and non-linear ODEs and

� Stability� Linear systems

� Non-linear systems (stable linearization ) local stability)

� Stationary states →

� Linearization

� Qualitative Behavior� Phase plane

� Bifurcations

� Feedback� Biology: non-additive, (engineering: additive)

� Sensitivity

� Robustness