User:Jaroslaw Karcz/Modelling Sandbox: Difference between revisions
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==THE KEY== | |||
<math>\nabla \times \nabla \times v = ...</math><br><br> | |||
<math>\nabla \times v = \epsilon_{spq}v_{q,p}</math><br> | |||
<math>\nabla \times \nabla \times v = \epsilon_{irs}(\epsilon_{spq}v_{q,p})_{,r}</math><br><br> | |||
<math>\Rightarrow \epsilon_{irs} \epsilon_{spq} (v_{q,p})_{r}</math><br> | |||
<math>\Rightarrow \epsilon_{sir}\epsilon_{spq}v_{q,pr}</math>....cyclic permutation<br> | |||
<math>\Rightarrow (\delta_{ip}\delta_{rq} - \delta_{iq}\delta_{rp})v_{q,pr}</math><br> | |||
<math>\Rightarrow \delta_{ip}\delta_{rq}v_{q,pr} - \delta_{iq}\delta_{rp}v_{q,pr}</math><br> | |||
<math>\Rightarrow \delta_{ip}v_{r,pr} - \delta_{iq}v_{q,rr}</math><br> | |||
<math>\Rightarrow v_{r,ir} - v_{i,rr}</math><br> | |||
<math>\Rightarrow v_{r,ri} - v_{i,rr} ......since..... v_{r,ri} = v_{r,ir}</math> ...continuous function <br> | |||
====Convert to vector notation==== | |||
<math>\Rightarrow\nabla (\nabla \cdot v) - \nabla^{2}v</math><br> | |||
==Model Development== | ==Model Development== | ||
The process of modelling consists of a number of layers; the following is a description of the modelling workflow: | The process of modelling consists of a number of layers; the following is a description of the modelling workflow: | ||
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<br clear="all"> | <br clear="all"> | ||
<hr class=divider2> | <hr class=divider2> | ||
<div class="noprint" style="padding: 10px; color: #ffffff; background-color: #EEEEFF; width: 660px; align: center"> | |||
<center> | |||
[[Part:BBa_F2620 |Part Main Page]] | |||
| |||
[[Part:BBa_F2620:Transfer Function |Transfer Function]] | |||
[[Part:BBa_F2620:Specificity|Specificity]] | |||
[[Part:BBa_F2620:Response time |Response time]] | |||
[[Part:BBa_F2620:Stability |Stability]] | |||
[[Part:BBa_F2620:Experience |Add Data]] | |||
</center> | |||
</div><br> | |||
==Introduction== | |||
The real world is dominated by complexity, especially biological systems | |||
<br> | |||
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 == | |||
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;" | |||
! Parameter | |||
! Value | |||
! Description | |||
! Comment (literature, derived?) | |||
|- | |||
| k<sub>1</sub> | |||
| x [units] | |||
| max. transcription rate of constitutive promoter (pTET) | |||
| Estimate | |||
|- | |||
| k<sub>2</sub> | |||
| | |||
| | |||
| | |||
|- | |||
| k<sub>3</sub> | |||
| | |||
| | |||
| | |||
|- | |||
| k<sub>4</sub> | |||
| | |||
| | |||
| | |||
|- | |||
| k<sub>5</sub> | |||
| | |||
| | |||
| | |||
|- | |||
| k<sub>6</sub> | |||
| x [units] | |||
| | |||
| | |||
|- | |||
| <math> \delta_{GFP} </math> | |||
| 0.029 hrs <sup>-1</sup> | |||
| degradation rate of GFP | |||
| Literature <font color = red> ~~ give reference </font> | |||
|} | |||
<br> | |||
==Tabs== |
Latest revision as of 19:41, 6 December 2007
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 |