Odes and Stochastics with Matlab

Odes and Stochastics with Matlab - Analyzing mathematical...

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Unformatted text preview: Analyzing mathematical models with MATLAB: numerically integrating ordinary di ff erential equations and simulating stochastic systems Contents 1 Motivation 3 2 A Hello World! di ff erential equation 3 2.1 Absolute and relative tolerance . . . . . . . . . . . . . . . . . . 5 3 Bacterial growth curves 6 3.1 Di ff erential equation . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.1 Physics analogy . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Parameter estimation by data fitting . . . . . . . . . . . . . . . 10 4 Polymer growth model / prion dynamics 13 5 Translation of mRNA 14 5.1 Fixed point and response time . . . . . . . . . . . . . . . . . . . 16 5.2 Phase plane analysis . . . . . . . . . . . . . . . . . . . . . . . . 17 6 Mutual inhibition 17 7 Discrete stochastic systems 20 7.1 A memoryless elementary chemical reaction leads to a random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 8 Bacterial growth 22 8.1 Chemical master equation . . . . . . . . . . . . . . . . . . . . . 22 8.2 Exact simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 CONTENTS 2 9 Translation of mRNA 25 10 The prion model a nonlinear system with feedback 27 11 Microtubule dynamics 28 CONTENTS 3 1. Motivation In order to get a quantitative understanding of a given biological process, it is necessary to come up with a mathematical description of the process we have in mind. Give such a mathematical model, we have to analyze it in order to gain some understanding. Using a computer, even complicated looking things like multidimensional di ff erential equations or stochastic systems can be analyzed numerically, in a surprisingly straightforward manner. This tutorial will be all about getting a handle on mathematical models by analyzing them using MATLAB. Boxed expressions indicate stu ff for you to do. Bridging the language barrier from mathematics to biology and vice versa can be hard. Its possibly the biggest obstacle to overcome, but the rewards can be staggering and make it all worth it. When spending a lot of much time on one side of the fence its useful to remind ourselves that not one side trumps the other. Clearly its all hand in hand. Dont ever get involved in trying to settle whats more important! The grand goal is clear for this kind of course which is often hailed as bringing together physics and biology: were not aiming for inter-disciplinarity but for non-disciplinarity! This course is part of that aim and will introduce some of the basic concepts that can be used to gain insights into mathematical descriptions of biological processes. It will be a starting point to get your hands dirty with MATLAB....
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This document was uploaded on 01/03/2012.

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Odes and Stochastics with Matlab - Analyzing mathematical...

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