Endy:Chassis engineering/VM2.0

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VM2.0 regulation design considerations

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

Reduced network for VM2.0
  • Two species, RNAP (activator) and repressor
  • Continuous differential equations
  • 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

  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, [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]
  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.)

Reduced model results

Time Course for RNAP and repressor species in reduced model
Nullclines and fixed point for plausible parameter values


  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?