Team:Harvard/Model/Diffeqs
Differential Equations
Transcripts
$$\frac{d[mRNA_{csgA}]}{dt} = \alpha_{1}[g_{csgA}][RNA_{pol}] - \zeta_{1}[mRNA_{csgA}]$$ $$\frac{d[mRNA_{csgBCEFG}]}{dt} = \alpha_{2}[g_{csgBCEFG}][RNA_{pol}] - \zeta_{2}[mRNA_{csgBCEFG}]$$
Cytoplasmic proteins
$$\frac{d[csgA_{cyt}]}{dt} = \beta_{1}[mRNA_{csgA}] - \gamma_{1}[csgA_{cyt}][SecB]^4 + \gamma_{-1}[secB:csgA] - \epsilon_{1}[csgA_{cyt}]^2 + \epsilon_{-1}[F_{cyt}] - \epsilon_{2}[F_{cyt}][csgA_{cyt}] + \epsilon_{-2}[F_{cyt}]$$ $$\frac{d[SecB:csgA]}{dt} = \gamma_{1}[csgA_{cyt}][SecB]^4 - \gamma_{-1}[SecB:csgA] - \gamma_{2}[SecB:csgA][SecA]^2[SecYEG]^2$$ $$\frac{d[Sec:csgA]}{dt} = \gamma_{2}[SecB:csgA][SecA]^2[SecYEG]^2 - \gamma_{3}[Sec:csgA]$$ $$\frac{d[F_{cyt}]}{dt} = \epsilon_{1}[csgA_{cyt}]^2 - \epsilon_{-1}[F_{cyt}]$$ $$\frac{d[csgB_{cyt}]}{dt} = \beta_{2}[mRNA_{csgBCEFG}] - \gamma_{1}[csgB_{cyt}][SecB]^4 + \gamma_{-1}[secB:csgB]$$ $$\frac{d[SecB:csgB]}{dt} = \gamma_{1}[csgB_{cyt}][SecB]^4 - \gamma_{-1}[SecB:csgB] - \gamma_{2}[SecB:csgB][SecA]^2[SecYEG]^2$$ $$\frac{d[Sec:csgB]}{dt} = \gamma_{2}[SecB:csgB][SecA]^2[SecYEG]^2 - \gamma_{3}[Sec:csgB]$$ $$\frac{d[csgC_{cyt}]}{dt} = \beta_{3}[mRNA_{csgBCEFG}] - \gamma_{1}[csgC_{cyt}][SecB]^4 + \gamma_{-1}[secB:csgC]$$ $$\frac{d[SecB:csgC]}{dt} = \gamma_{1}[csgC_{cyt}][SecB]^4 - \gamma_{-1}[SecB:csgC] - \gamma_{2}[SecB:csgC][SecA]^2[SecYEG]^2$$ $$\frac{d[Sec:csgC]}{dt} = \gamma_{2}[SecB:csgC][SecA]^2[SecYEG]^2 - \gamma_{3}[Sec:csgC]$$ $$\frac{d[csgE_{cyt}]}{dt} = \beta_{4}[mRNA_{csgBCEFG}] - \gamma_{1}[csgE_{cyt}][SecB]^4 + \gamma_{-1}[secB:csgE]$$ $$\frac{d[SecB:csgE]}{dt} = \gamma_{1}[csgE_{cyt}][SecB]^4 - \gamma_{-1}[SecB:csgE] - \gamma_{2}[SecB:csgE][SecA]^2[SecYEG]^2$$ $$\frac{d[Sec:csgE]}{dt} = \gamma_{2}[SecB:csgE][SecA]^2[SecYEG]^2 - \gamma_{3}[Sec:csgE]$$ $$\frac{d[csgF_{cyt}]}{dt} = \beta_{5}[mRNA_{csgBCEFG}] - \gamma_{1}[csgF_{cyt}][SecB]^4 + \gamma_{-1}[secB:csgF]$$ $$\frac{d[SecB:csgF]}{dt} = \gamma_{1}[csgF_{cyt}][SecB]^4 - \gamma_{-1}[SecB:csgF] - \gamma_{2}[SecB:csgF][SecA]^2[SecYEG]^2$$ $$\frac{d[Sec:csgF]}{dt} = \gamma_{2}[SecB:csgF][SecA]^2[SecYEG]^2 - \gamma_{3}[Sec:csgF]$$ $$\frac{d[csgG_{cyt}]}{dt} = \beta_{6}[mRNA_{csgBCEFG}] - \gamma_{1}[csgG_{cyt}][SecB]^4 + \gamma_{-1}[secB:csgG]$$ $$\frac{d[SecB:csgG]}{dt} = \gamma_{1}[csgG_{cyt}][SecB]^4 - \gamma_{-1}[SecB:csgG] - \gamma_{2}[SecB:csgG][SecA]^2[SecYEG]^2$$ $$\frac{d[Sec:csgG]}{dt} = \gamma_{2}[SecB:csgG][SecA]^2[SecYEG]^2 - \gamma_{3}[Sec:csgG]$$ $$\begin{align} \frac{d[SecB]}{dt} & = \gamma_{3} \{[Sec:csgA] + [Sec:csgB] + [Sec:csgC] + [Sec:csgE] + [Sec:csgF] + [Sec:csgG]\} \\ & - \gamma_{1}[SecB]^4\{[csgA_{cyt}] + [csgB_{cyt}] + [csgC_{cyt}] + [csgE_{cyt}] + [csgF_{cyt}] + [csgG_{cyt}]\} \\ & + \gamma_{-1} \{[SecB:csgA] + [SecB:csgB] + [SecB:csgC] + [SecB:csgE] + [SecB:csgF] + [SecB:csgG]\} \end{align}$$ $$\begin{align} \frac{d[SecA]}{dt} & = \gamma_{3} \{[Sec:csgA] + [Sec:csgB] + [Sec:csgC] + [Sec:csgE] + [Sec:csgF] + [Sec:csgG]\} \\ & - \gamma_{2}[SecA]^2[SecYEG]^2\{[SecB:csgA] + [SecB:csgB] + [SecB:csgC] + [SecB:csgE] + [SecB:csgF] + [SecB:csgG]\} \end{align}$$ $$\begin{align} \frac{d[SecYEG]}{dt} & = \gamma_{3} \{[Sec:csgA] + [Sec:csgB] + [Sec:csgC] + [Sec:csgE] + [Sec:csgF] + [Sec:csgG]\} \\ & - \gamma_{2}[SecA]^2[SecYEG]^2\{[SecB:csgA] + [SecB:csgB] + [SecB:csgC] + [SecB:csgE] + [SecB:csgF] + [SecB:csgG]\} \end{align}$$
Periplasmic Proteins
$$\frac{d[csgE_{9}]}{dt} = \delta_{1}[csgE_{per}]^9 - \delta_{-1}[csgE_{9}]$$ $$\frac{d[csgG_{9}]}{dt} = \delta_{3}[csgG_{per}]^9 - \delta_{-3}[csgG_{9}] - \delta_{4}[csgE_{9}][csgF_{ECM}]^2[csgG_{9}] + \delta_{-4}[csgGEF]$$ $$\frac{d[csgGEF]}{dt} = \delta_{4}[csgE_{9}][csgF_{ECM}]^2[csgG_{9}] - \delta_{-4}[csgGEF]$$ $$\frac{d[csgA_{per}]}{dt} = \gamma_{3}[Sec:csgA] - \delta_{5}[csgA_{per}][csgC_{per}] - \epsilon_{1}[csgA_{per}]^2 + \epsilon_{-1}[F_{per}] - \epsilon_{2}[F_{per}][csgA_{per}] + \epsilon_{-2}[F_{per}]$$ $$\frac{d[F_{per}]}{dt} = \epsilon_{1}[csgA_{per}]^2 - \epsilon_{-1}[F_{per}]$$ $$\frac{d[csgB_{per}]}{dt} = \gamma_{3}[Sec:csgB] - \delta_{5}[csgB_{per}][csgC_{per}] $$ $$\frac{d[csgC_{per}]}{dt} = \gamma_{3}[Sec:csgC] - \delta_{5}[csgA_{per}][csgC_{per}] - \delta_{5}[csgA_{per}][csgC_{per}] + D \frac{[csgC:csgA] - [csgA_{ECM}]}{\omega} S N_A + D \frac{[csgC:csgB] - [csgB_{ECM}]}{\omega} S N_A$$ $$\frac{d[csgC:csgA]}{dt} = \delta_{5}[csgA_{per}][csgC_{per}] - D \frac{[csgC:csgA] - [csgA_{ECM}]}{\omega} S N_A $$ $$\frac{d[csgC:csgB]}{dt} = \delta_{5}[csgB_{per}][csgC_{per}] - D \frac{[csgC:csgB] - [csgB_{ECM}]}{\omega} S N_A $$ $$\frac{d[csgE_{per}]}{dt} = \gamma_{3}[Sec:csgE] $$ $$\frac{d[csgF_{per}]}{dt} = \gamma_{3}[Sec:csgF] - \delta_{2}[csgF_{per}]$$ $$\frac{d[csgG_{per}]}{dt} = \gamma_{3}[Sec:csgG] - \delta_{3}[csgG_{per}]^9$$
Extracellular Secretion
$$\frac{d[csgF_{ECM}]}{dt} = D \frac{[csgF_{per}] - [csgF_{ECM}]}{\omega} S N_A$$ $$\frac{d[csgA_{ECM}]}{dt} = D \frac{[csgC:csgA] - [csgA_{ECM}]}{\omega} S N_A - \epsilon_{1}[csgA_{ECM}]^2 + \epsilon_{-1}[F_{ECM}] - \epsilon_{2}[F_{ECM}][csgA_{ECM}] + \epsilon_{-2}[F_{ECM}]$$ $$\frac{d[F_{ECM}]}{dt} = \epsilon_{1}[csgA_{ECM}]^2 - \epsilon_{-1}[F_{ECM}]$$ $$\frac{d[csgB_{ECM}]}{dt} = D \frac{[csgC:csgB] - [csgB_{ECM}]}{\omega} S N_A$$
Rate constants
| Symbol | Definition | Value | Units | Reference | \(\alpha_1\) | Rate of transcription of csgA | 0.0921 | \(sec^{-1}\) | Proshkin, Sergey, et al. 2010 |
|---|---|---|---|---|
| \(\alpha_2\) | Rate of transcription of csgB-G | 0.0214 | \(sec^{-1}\) | Proshkin, Sergey, et al. |
| \(\beta_0\) | Basal protein translation rate | 0.1 | \(sec^{-1}\) | Proshkin, Sergey, et al. |
| \(\gamma_1\) | Rate of binding of SecB to csgX | 0.25 | \(uM^{-1}sec^{-1}\) | Agarwal 2010 |
| \(\gamma_{-1}\) | Rate of dissociation of SecB:csgX | 0.025 | \(sec^{-1}\) | Agarwal 2010 |
| \(\gamma_2\) | Rate of formation of SecABYEG:csgX complex | 0.0085 | \(uM^{-1}sec^{-1}\) | Agarwal 2010 |
| \(\gamma_3\) | Rate of translocation of csgX from cytoplasm to periplasm | 1.00 | \(sec^{-1}\) | Agarwal 2010 |
| \(\delta_1\) | Rate of formation of \(csgE_9\) nonamer | 0.76 | \(uM^{-8}sec^{-1}\) | Lee et. al 2015 |
| \(\delta_{-1}\) | Rate of dissociation of \(csgE_9\) nonamer | \(28 \times 10^{-3}\) | \(sec^{-1}\) | Lee et. al 2015 |
| \(\delta_{3}\) | Rate of formation of \(csgG_9\) nonamer | 0.76 | \(uM^{-8}sec^{-1}\) | Lee et. al 2015 |
| \(\delta_{-3}\) | Rate of dissociation of \(csgG_9\) nonamer | \(28 \times 10^{-3}\) | \(sec^{-1}\) | Lee et. al 2015 |
| \(\delta_{4}\) | Rate of formation of csgGEF complex | 0.0384 | \(uM^{-1}sec^{-1}\) | Agarwal 2010 |
| \(\delta_{-4}\) | Rate of dissociation of csgGEF complex | TBD | \(sec^{-1}\) | Agarwal 2010 |
| \(\delta_{5}\) | Rate of formation of csgC:csgX complex | 0.25 | \(uM^{-1}sec^{-1}\) | Agarwal 2010 |
| \(D\) | Diffusion coefficient | \(1.44 \times 10^{-11}\) | \(cm^2s^{-1}\) | Lardner 1977 |
| \(\omega\) | E. coli outer membrane thickness | \(10 \times 10^{-8}\) | \(m\) | HMS BioNumbers Database |
| \(SA\) | E. coli outer membrane surface area | \(4.42 \times 10^{-12}\) | \(m^2\) | HMS BioNumbers Database |
