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VM2.0 regulation design considerations

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Regulation scheme for VM2.0

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

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Reduced network for VM2.0

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{[u]} = \frac{a_{0}+a_{1}[u]}{1+[u]+[v]^{n}}-[u]\qquad(1)

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{[v]} = \frac{b_{0}+b_{1}[u]}{1+[u]+[v]^{n}}-[v]\qquad(2)

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle [u]

= dimensionless concentration of T7 RNAP

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 -

Failed to parse (Missing texvc executable; please see math/README to configure.): \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?
   • Setting the LHS of Equation 3 to 0 and solving for the steady state level, Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle u_{ss}

with Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle n=2
and ignoring small terms, the (single) fixed point, is Failed to parse (Missing texvc executable; please see math/README to configure.): u_{ss} = \frac{\sqrt{a_{1}}}{r}

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

Reduced model results

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Time Course for RNAP and repressor species in reduced model

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Nullclines and fixed point for plausible parameter values

Species