$k_{m}=$ 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