Introduction to Mathematical Modeling

Transcription

1 Introduction to Mathematical Modeling - Systems Theory in the Toolbox for Systems Biology The 5 th International Course in Yeast Systems Biology 2011 June 6, 2011, PhD, Assoc Prof Head of Department Systems Biology and Bioimaging Systems Biology and Bioimaging 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. Research Topics: 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 1

2 Outline What is Systems Theory? Description of Dynamic Systems Linear Systems Non-linear Systems Stability Linearization Qualitative Behavior Phase plane Bifurcations Feedback Summary 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... 2

3 What is Systems Theory? We will focus on Dynamic Systems! Dynamic Systems - Examples 3

4 Dynamic Systems - Definition Inputs Outputs 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. Dynamic Systems - Linear vs Nonlinear Linear system: u 1 +u 2 y 1 +y 2 α u α y Large toolbox of analytical results Often good approximation for small perturbations around a steady state Try simple things first! 4

5 Dynamic Systems - Some Other Aspects Dynamic Static Deterministic Stochastic Continuous-time Discrete-time Lumped Distributed Change oriented Discrete event Description of Dynamic Systems - Differential Equations, Linear Systems Linear differential equations where The State-space Form: - inital state - state - input - output The state at time t is the minimal amount of information needed to determine future output given future input. 5

6 Description of Dynamic Systems - A multi-compartment example Matrix form: Notebook example Description of Dynamic Systems - Differential Equations, Non-linear Systems The State-space Form: - inital state - state - input - output Now, and are non-linear vector valued functions 6

7 Description of Dynamic Systems - A Pathway Example S Mutual activation* Notation: R E-P E * Sniffers, buzzers, toggles, and blinkers: dynamics of regulatory and signaling pathways in the cell, Tyson et al. Notebook example 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. 7

8 Stability - Linear Systems System: Stationary point: The stability properties of is determined by the eigenvalues of A. eig(a) are solutions det(λi-a)=0 (an algebraic equation in λ). Asymptotic stability iff. Ex Stability of the compartmental model Notebook example Stability - Stationary State System: Stationary state: for all t>0 How to find it? Solutions to! Ex Stationary state of the pathway model Notebook example The stationary state is the operating point for many systems! Definitely not for all! 8

9 Linearization The system: Taylor expansion of f around x 0 and u 0 gives: New variables x and u evaluated in x 0 and u 0. Ex Linearization and stability of the pathway model Notebook example Qualitative Behavior - Phase-plane A phase plane of a dynamic system is an xygraph 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 9

10 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. Feedback Engineering Biology r e y G - Sloppy: There is feedback in the system if a signal is injected/transmitted elsewhere in such a way that it influence itself. Ex A pathway example 10

11 Feedback - Why? Decreases sensitivity to disturbances Increases robustness (makes it possible to build systems with exact behavior from inexact parts) Feedback - Biology The feedback is very often of non-additive character! Ex Phosphorylation cascades 11

12 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 Feedback - Biology, PI control Integral feedback: E1 E2 Y v3 A v4 (saturation) Stationary value of Y is independent of everything except parameters in v 3 and v 4 PWL 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,

13 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 13