User:Jaroslaw Karcz/Modelling Sandbox
THE KEY
[math]\displaystyle{ \nabla \times \nabla \times v = ... }[/math]
[math]\displaystyle{ \nabla \times v = \epsilon_{spq}v_{q,p} }[/math]
[math]\displaystyle{ \nabla \times \nabla \times v = \epsilon_{irs}(\epsilon_{spq}v_{q,p})_{,r} }[/math]
[math]\displaystyle{ \Rightarrow \epsilon_{irs} \epsilon_{spq} (v_{q,p})_{r} }[/math]
[math]\displaystyle{ \Rightarrow \epsilon_{sir}\epsilon_{spq}v_{q,pr} }[/math]....cyclic permutation
[math]\displaystyle{ \Rightarrow (\delta_{ip}\delta_{rq} - \delta_{iq}\delta_{rp})v_{q,pr} }[/math]
[math]\displaystyle{ \Rightarrow \delta_{ip}\delta_{rq}v_{q,pr} - \delta_{iq}\delta_{rp}v_{q,pr} }[/math]
[math]\displaystyle{ \Rightarrow \delta_{ip}v_{r,pr} - \delta_{iq}v_{q,rr} }[/math]
[math]\displaystyle{ \Rightarrow v_{r,ir} - v_{i,rr} }[/math]
[math]\displaystyle{ \Rightarrow v_{r,ri} - v_{i,rr} ......since..... v_{r,ri} = v_{r,ir} }[/math] ...continuous function
Convert to vector notation
[math]\displaystyle{ \Rightarrow\nabla (\nabla \cdot v) - \nabla^{2}v }[/math]
Model Development
The process of modelling consists of a number of layers; the following is a description of the modelling workflow:
- Definition of the problem
- Verification of information available
- Selection of model structure
- Establishing a simple model
- Sensitivity analysis
- Experimental tests of the model predictions
- Stating the agreements and divergences between experimental and modelling results, including any emergent behaviour
- Iterative refinement of model
[math]\displaystyle{ f_{obj}(k) = \sum_{i=1}^q (f_{obs}(i) - f_{per}(i,k))^2 }[/math]
Introduction
The real world is dominated by complexity, especially biological systems
Mathematical modelling and computer simulations provide a means of understanding the innate funtioning of system - dynamics, and to arrive at well-founded predictions about their future development and the effect of interactions with the environment.
So what is a model? A model is an abstract representation of objects and processes that explain the features/nature of these objects or processes. We present the model of our construct, as a system of differential equations to describe the dynamics of that network.
Model Parameters
Parameter | Value | Description | Comment (literature, derived?) |
---|---|---|---|
k1 | x [units] | max. transcription rate of constitutive promoter (pTET) | Estimate |
k2 | |||
k3 | |||
k4 | |||
k5 | |||
k6 | x [units] | ||
[math]\displaystyle{ \delta_{GFP} }[/math] | 0.029 hrs -1 | degradation rate of GFP | Literature ~~ give reference |