MATLAB Licenses

MATLAB resources:
Instruction in MATLAB is done throughout Week 1, but the following videos can be used as a supplement and as reference on specific topics.
MATLAB Basics and Navigation:
What is MATLAB?
The MATLAB Environment
MATLAB Variables
MATLAB as a Calculator
Mathematical Functions
Adding Files to the Search Path

Vectors and Matrices:
Creating Vectors
Creating Matrices
Creating Uniformly Spaced Vectors
Vector Arithmetic

Data and Visualization:
Importing Data
Statistical Functions with Matrices
Making Histograms
Line Plots
Multiple Plots
Annotating Graphs

Finding Data that Meet Criteria:
Accessing Elements of an Array
Logical Operators
Conditional Data Selection

Differential Equations and Functions:
Numerically Solving First Order ODEs
Solving Systems of 1st Order ODEs
Writing Functions
Passing Functions as Inputs

Further Training
If you'd like to go further, you can learn MATLAB interactively, at your own pace, with MATLAB Academy. The two hour MALTAB Onramp is free and gives you more practice using MATLAB. To access MATLAB Academy you will need to first use the MATLAB License link for this course to associate your MathWorks account with the license. Then use this link to go to MATLAB Academy: https://matlabacademy.mathworks.com/

Lecture 6 - Introduction to Dynamical Systems - Part 1

Enzyme-catalyzed reactions

具体推导过程见Unit 2: Structural Biochemistry-0826

将Vmax带入V0表达式可得下图所示，是建模中最常用的公式。

Lecture 7 - Introduction to Dynamical Systems - Part 2

Euler’s method for solving ODE systems

设定固定的dt,初始值xo,算法f (x)，可以模拟预测后面的值。这便是model模拟的基础。

一个实例如下：

对该实例的具体解释。

Lecture 8 - Introduction to Dynamical Systems - Part 3

Yeast glycolytic oscillations-Phase-plane techniques for 2D systems

左图中形成了一个圈，是欠稳定系统；右图最终集中于一个点，属于稳定系统

看左图右上角箭头处，是斜向下的，为什么呢？因为此时GLU和ATP浓度都很高，代入dG/dt，为负数，所以GLU应该是减少的方向。同理，代入dATP/dt，为正数，所以ATP应该是增加的方向。

Lecture 9 - Introduction to Dynamical Systems - Part 4

nullclines:

This is the set of points for which d[G]/dt = 0; d[ATP]/dt = 0. Where the nullclines intersect, both derivatives are zero. This is a “fixed point”

利用nullclines判断方向

在右上角ATP和G的值都很大，代入公式中dATP/dt>0而dG/dt<0;当箭头下行穿过dATP/dt时，dATP/dt变为<0，此时ATP减少，而dG/dt<0保持不变，所以箭头方向指向左下方；以此类推。

在Km=13时，左图G和ATP震荡加剧；右图圆圈由中间的虚心点逐渐扩散

在Km=20时，正好相反

How can we understand stable and unstable fixed points mathematically?

以下图为例，左边的特征值实部是0.004，说明是unstable状态，用虚点表示；而右边的特征值实部是-0.0125，说明是stable状态，用实点表示。

Bifurcations

A bifurcation is where the system qualitatively changes its behavior
有两种方式可以判断已经达到了Bifurcations：第一是Gmin和Gmax趋于一致；第二实部由正变为负的转折点。