$k_{m}=$ transcription rate of mRNA

$\delta([mRNA],[siRNA])=$ rate of mRNA degraded by siRNA

$\frac{d[mRNA]}{dt}=k_{m}-\delta([mRNA],[siRNA])$

$k_{T}=$ rate of protein translation

$d_{p}=$ decay rate of protein

$\frac{d[protein]}{dt}=k_{T}[mRNA]-d_{p}[protein]$

Finding $\delta([mRNA],[siRNA])$

$\lambda=$ affinity of siRNA target site for siRNA guided RISC complex

$\kappa=$ mRNA cleave + dissociation rate of bound siRNA RISC

$siRNA^{*}=$ siRNA loaded (active) RISC complex

$siRNA^{*}_{bound}=$ mRNA bound siRNA^{*} complex

Rate of mRNA degradation/decay

$mRNA\xrightarrow[]{deg_{nonspec}}degmRNA$

$\frac{d[degmRNA]}{dt}=deg_{mRNA}[mRNA]$

Rate of formation of mRNA-siRNA* complexes (takes into account siRNA* recycling)

$\frac{d[mRNA-siRNA^{*}]}{dt}=0$

Assuming that RISC* complex is replenished rapidly given ATP.

Rate of formation of mRNA

$siRNA^{*}+mRNA\xrightarrow[]{\lambda}mRNA-siRNA^{*}$

$\frac{d[mRNA]}{dt}=-\lambda[siRNA^{*}[mRNA]-\frac{d[degmRNA]}{dt}]$

Rate of mRNA cleavage

$mRNA-siRNA^{*}\xrightarrow[]{\kappa}cleavedmRNA+siRNA^{*}$

$\frac{d[cleavedmRNA]}{dt}=\kappa[mRNA-siRNA^{*}]=\kappa[siRNA^{*}_{bound}]$

By Conservation of Rates

$0=\frac{d[mRNA]}{dt}+\frac{d[mRNA-siRNA^{*}]}{dt}+\frac{cleavedmRNA}{dt}+\frac{d[degmRNA]}{dt}-\frac{d[mRNA]}{dt}$

$=0+\frac{d[cleavedmRNA]}{dt}+\frac{d[mRNA]}{dt}$

$\lambda[siRNA^{*}][mRNA]+\frac{d[degmRNA]}{dt}=\frac{d[cleavedmRNA]}{dt}+\frac{d[degmRNA]}{dt}$

$\lambda[siRNA^{*}][mRNA]=\kappa[siRNA^{*}_{bound}]$

$[siRNA^{*}]=[siRNA^{*}_{total}]-[siRNA^{*}_{bound}]$

Substituting and solving for $[siRNA^{*}_{bound}]$

$[siRNA^{*}_{bound}]=\frac{\lambda[siRNA^{*}_{total}][mRNA]}{\kappa+\lambda[mRNA]}$

$\frac{d[cleavedmRNA]}{dt}=\kappa[siRNA^{*}_{bound}]=\frac{\kappa\lambda[siRNA^{*}_{total}][mRNA]}{\kappa+\lambda[mRNA]}$

Therefore the relationship between total concentration of siRNA and rate of degradation of mRNA due to gene knockdown can be expressed in an enzymatic kinetic reaction (Michaelis-Menten)

$\therefore deg_{siRNA \rightarrow mRNA}=\frac{\kappa\lambda[siRNA^{*}_{total}][mRNA]}{\kappa+\lambda[mRNA]}$

Model Network

Network Elements

Model Network

Network Elements

Normal bacteria growth, used as a comparison to growth curve with IPTG induction.

The two graphs above are a comparison of bacteria growth without IPTG induction (without shRNA production) and with IPTG induction (with shRNA production). The shRNA produced is designed to inhibit eGFP. Since the two growth curves are virtually identical, this shows that production of the designed shRNA is not toxic to the bacteria.