# Endy:Chassis engineering/VM2.0

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# VM2.0 regulation design considerations

- Stability
- Noise
- analytical stability analysis on very simple model or Routh-Hurwitz analysis for full model
- parameter sensitivity analysis

- Response time
- Better to have this fast or slow (slow response time averages out short time scale fluctuations)

- Noise
- Self-booting/controlled
- Ability to turn on or off
- Portability

- Tunable
- Pros and cons of DNA copy number, promoter strength, repressor affinities etc.

- Efficient
- Minimizing levels of repressor needed
- Minimizing consumption of small molecules

## Reduced Model

- Two species, RNAP (activator) and repressor
- Continuous differential equations
- MATLAB
- Dimensionless variables, lumped parameters.
- Parameterized for T7 RNAP, "typical" repressor

[math]\displaystyle{
\dot{[u]} = \frac{a_{0}+a_{1}[u]}{1+[u]+[v]^{n}}-[u]\qquad(1)
}[/math]

[math]\displaystyle{
\dot{[v]} = \frac{b_{0}+b_{1}[u]}{1+[u]+[v]^{n}}-[v]\qquad(2)
}[/math]

[math]\displaystyle{ \displaystyle [u] }[/math] = dimensionless concentration of T7 RNAP

[math]\displaystyle{ \displaystyle [v] }[/math] = dimensionless concentration of repressor

If I assume that the two species are expressed in a constant ratio (i.e polycistronic expression or under promoters of proportional strength and have similar degradation rates) then the two equations can be reduced to one -

[math]\displaystyle{ \dot{[u]} = \frac{a_{0}+a_{1}[u]}{1+[u]+r[u]^{n}}-[u]\qquad(3) }[/math]

### Big questions to answer

- What are the steady state levels of RNAP/Repressor as a function of parameters?
- Setting the LHS of Equation 3 to 0 and solving for the steady state level, [math]\displaystyle{ \displaystyle u_{ss} }[/math] with [math]\displaystyle{ \scriptstyle n=2 }[/math] and ignoring small terms, the (single) fixed point, is [math]\displaystyle{ u_{ss} = \frac{\sqrt{a_{1}}}{r} }[/math]

- What is the material usage like?
- What happens when RNAP level drops suddenly (e.g. when another T7 reporter in the cell is derepressed.)

### Reduced model results

## Species

- T7 RNAP
- Repressor
- Ribosomes
- Repressible T7 promoter
- T7RNAP-promoter complex
- Repressor-promoter complex
- T7 RNAP mRNA
- Repressor mRNA
- Elongating T7 RNAP
- Elongating Ribosomes
- etc.

### Model analysis notes

- A cooperative autogene network can exhibit bistability or monostability depending on parameter values (7.81). Does this apply if there is no cooperativity?