# VM2.0 regulation design considerations

1. 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)
2. Self-booting/controlled
• Ability to turn on or off
• Portability
3. Tunable
• Pros and cons of DNA copy number, promoter strength, repressor affinities etc.
4. Efficient
• Minimizing levels of repressor needed
• Minimizing consumption of small molecules
What are the metrics for each of the design considerations?

## Reduced Model

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

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

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

$\displaystyle{ \displaystyle [v] }$ = 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 -

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

1. 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, $\displaystyle{ \displaystyle u_{ss} }$ with $\displaystyle{ n=2 }$ and ignoring small terms, the (single) fixed point, is $\displaystyle{ u_{ss} = \frac{\sqrt{a_{1}}}{r} }$
2. What is the material usage like?
3. What happens when RNAP level drops suddenly (e.g. when another T7 reporter in the cell is derepressed.)

## Species

1. T7 RNAP
2. Repressor
3. Ribosomes
4. Repressible T7 promoter
5. T7RNAP-promoter complex
6. Repressor-promoter complex
7. T7 RNAP mRNA
8. Repressor mRNA
9. Elongating T7 RNAP
10. Elongating Ribosomes
11. 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?