Lecture 5 Mathematical Representations of Cell Biological Systems
5.A 使用数学表达式帮助我们解决生物学问题
- 数学表达式的两种策略：Analytical Solutions，指有确定的答案的；Numerical Solutions (numerical analysis):数值分析，只有近似的答案的，大部分都是数值分析。
- 最常用的是常微分方程ODE。其缺点是All reactants are presumed to have equal access to all other reactants without any hindrance阻力 The “well stirred” assumption works for processes that are solely cytoplasmic but the assumption is not valid when multiple compartments, such as cytoplasm and nucleus, are involved
Most often interactions between protein kinases and transcription factors involve multiple compartments
5.B 多分区ODE模型
- MAPK pathway的2分区模型为例
- 偏微分方程PDE。利用名为Virtual Cell 仿真细胞的程序
- 确定性Deterministic系统和随机性Stochastic系统
- 随机性Stochastic系统会使用Master Equation。描述系统相对于时间的进度的方程。
Lecture 6 Simulations of Cell Biological Systems
6.A
- Toy Models 玩具模型，使用随机参数，建立与模型相关的理论要点
Plausible Models 似真模型，Canonical and not cell type specific, hence may not be fully representative of every cell type Most common type of dynamical models动力学模型 in systems biology
Identifiable Models， 可识别模型Built to explain experimental data. System specific and fitted to experimental data Commonly used in drug action studies药物作用。Model parameters are often not connected to molecular details - 获取反应参数: initial concentrations and reaction rates are needed
These are often not easy to obtain as biochemical and cell biological experiments were often not geared towards getting rate measurements之前所做的实验并非获得反应速率 or absolute levels of components with the cell或细胞内组分含量的真实值
Sometimes parameters need to be guesstimated based on known values for similar components
Often rate parameters need to be estimated from indirect measurements such as time courses例如通过对ras激活过程进行监控 ,可以估算GAP和GEF蛋白的相对活性。
There are curve fitting programs for estimation of parameters such as COPASI - 建模的原则：
1） Do NOT over-simplify the model
2 ）Build models with enough detail to provide non-intuitive hypotheses非知觉假说 from simulations
3）Use available experimental data to obtain realistic (reasonable) parameters
4）Do NOT tweak (change) the parameters so that simulations show a desired behavior - 解ODE。使用Forward Euler Integration Method前向欧拉积分法 -- From a point on a curve, approximation of another nearby point on the curve can be made by moving a short distance along a line tangent to the curve.从一个点到另一个点的积分是沿着切线方向移动的很短的距离。这种方法已经用的不多。改进的方法Runge-Kutta Methods
- 解PDE。
Finite Element Method有限元法。Divide the continuous domain into smaller parts called elements by enforcing a mesh.通过生成网格而得到元素
Form a system of equations that governs flow of entities (e.g., proteins) between elements by discretizing PDEs Numerically solve the system of equations
Finite Volume Method有限体积法。使用已定义大小的网格。将表面积积分换算成体积积分。适合带有扩散的细胞生物模型。可以使用Virtual Cell - Software for Numerical Computation
Matlab -- Widely used commercial software suite for numerical computation
We teach our courses using Matlab
Mathematica -- Another widely used commercial software suite for numerical computation
GNU Octave -- Free software suite compatible with Matlab。免费的MATLAB
Virtual Cell -- Free modeling and analysis software that has unique capabilities for PDE models in conjunction with imaging experiments
6.B
- 处理实验中的误差。确保质量守恒、通过实验数据约束模型、不要依赖于心算。
