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:

1. Definition of the problem
2. Verification of information available
3. Selection of model structure
4. Establishing a simple model
5. Sensitivity analysis
6. Experimental tests of the model predictions
7. Stating the agreements and divergences between experimental and modelling results, including any emergent behaviour
8. Iterative refinement of model

[math]\displaystyle{ f_{obj}(k) = \sum_{i=1}^q (f_{obs}(i) - f_{per}(i,k))^2 }[/math]

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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