Team:NTU-Taida/Modeling/Plasmid-Stability

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Modeling-Plasmid

Plasmid-Instability-Model

Contents

How to model plasmid instability:

We use Cooper's model (Cooper, N.S., M.E. Brown, and C.A. Caulcott, A ) to model plasmid instability, and set a protocol to suggest users which modules can be used to prove their system stability.

NTU-Taida-Negative du.jpg

Cooper’s model: Under steady state, population distribution of bacteria follows underlying equation:

NTU-Taida-Stability-eq1.png
NTU-Taida-Stability-eq2.png

Growth rate and dilution rate have such relation:

NTU-Taida-Stability-eq3.png


Substitute in

NTU-Taida-Stability-eq4.png

produce

NTU-Taida-Stability-eq5.png

This equation belongs to the Bernoulli form of Differential equation and can be solved as:

NTU-Taida-Stability-eq6.png

Initially, plasmid loss usually is 0. NTU-Taida-Stability-eq7.png So the equation can be simplified as:

NTU-Taida-Stability-eq8.png

There can be three kinds of plasmid instability condition:

1.Growth rate difference >> segregation instability
2.Growth rate difference =< segregation instability
3.Negative growth rate difference >> segregation instability

Knowing the condition of your expression system is important, if growth rate difference is much greater than segregation instability, partition system cannot help stabilize such system. We will show how to discriminate between three conditions and how to use our parts to solve the condition.

1.Growth rate difference >> segregation instabilityNTU-Taida-Stability-eq9.png
The equation can be further simplified.

NTU-Taida-Stability-eq10.png

NTU-Taida-du bigger than R.jpg

R=10^-5 x-axis:generation y-axis:p- ratio

NTU-Taida-du bigger than R with init001.jpg

R=10^-5 init=0.01 x-axis:generation y-axis:p- ratio

The simulation shows that initial condition greatly affect the result, and the curve is sigmoid. The segregation instability shows minor effect on the plasmid loss rate (not shown).

2.Growth rate difference =< segregation instabilityNTU-Taida-Stability-eq11.png

NTU-Taida-Stability-eq12.png
NTU-Taida-du smaller or equal to R.jpg

Under this condition, plasmid-bearing cell shows exponential decay. Segregation instability greatly affect the stability.

3.Negative growth rate difference >> segregation instabilityNTU-Taida-Stability-eq13.png

NTU-Taida-Stability-eq14.png