Endy:Chassis engineering/VM2.0

VM2.0 regulation design considerations 

Reduced Model
$$ \dot{[u]} = \frac{a_{0}+a_{1}[u]}{1+[u]+[v]^{n}}-[u]\qquad(1) $$ $$ \dot{[v]} = \frac{b_{0}+b_{1}[u]}{1+[u]+[v]^{n}}-[v]\qquad(2) $$
 * Two species, RNAP (activator) and repressor
 * Continuous differential equations
 * MATLAB
 * Dimensionless variables, lumped parameters.
 * Parameterized for T7 RNAP, "typical" repressor

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

$$\displaystyle [v]$$ = 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 -

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

Big questions to answer

 * 1) What are the steady state levels of RNAP/Repressor as a function of parameters?
 * 2) *Setting the LHS of Equation 3 to 0 and solving for the steady state level, $$\displaystyle u_{ss}$$ with $$\scriptstyle n=2$$ and ignoring small terms, the (single) fixed point, is $$u_{ss} = \frac{\sqrt{a_{1}}}{r}$$
 * 3) What is the material usage like?
 * 4) What happens when RNAP level drops suddenly (e.g. when another T7 reporter in the cell is derepressed.)

Reduced model results
 Species {{hide|1=
 * 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
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 * A cooperative autogene network can exhibit bistability or monostability depending on parameter values (7.81). Does this apply if there is no cooperativity?