Difference between revisions of "Team:Freiburg/Modeling"

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Modeling

Mathematical models and computer simulations provide a great way to describe the function and operation of BioBrick Parts and Devices. Synthetic Biology is an engineering discipline, and part of engineering is simulation and modeling to determine the behavior of your design before you build it. Designing and simulating can be iterated many times in a computer before moving to the lab. This award is for teams who build a model of their system and use it to inform system design or simulate expected behavior in conjunction with experiments in the wetlab.

Here are a few examples from previous teams:

• ETH Zurich 2014
• Waterloo 2014

Detailed System

Transcription

ODE System

\[ tc0(1): \;\;\; \frac{dc^{RNAP}_{free}[t]}{dt}\: =\: c^{RNAP}_{bound}[t] \cdot k^{RNAP}_{gain} - c^{RNAP}_{free}[t] \cdot k^{RNAP}_{loss} \] \[ \hphantom{tc0(1): \;\;\; \frac{dc^{RNAP}_{free}[t]}{dt}\: =\: } + c^{RNAP}_{sigma}[t] \cdot k^{sigma}_{off} - c^{RNAP}_{free}[t] \cdot c^{sigma}[t] \cdot k^{sigma}_{on} \] \[ \hphantom{tc0(1): \;\;\; \frac{dc^{RNAP}_{free}[t]}{dt}\: =\: } + c^{RNAP}_{elongter}[-1][t] \cdot k^{RNAP}_{diss} \]
\[ tc0(2): \;\;\; \frac{dc^{sigma}[t]}{dt}\: =\: c^{sigma}_{bound}[t] \cdot k^{sigma}_{gain} - c^{sigma}[t] \cdot k^{sigma}_{loss} + c^{RNAPsigma}_{bound}[t] \cdot k^{sigma}_{off} \] \[ \hphantom{tc0(2): \;\;\; \frac{dc^{sigma}[t]}{dt}\: =\: } + c^{RNAP}_{sigma}[t] \cdot k^{sigma}_{off} - c^{RNAP}_{free}[t] \cdot c^{sigma}[t] \cdot k^{sigma}_{on} \] \[ \hphantom{tc0(2): \;\;\; \frac{dc^{sigma}[t]}{dt}\: =\: } + c^{RNAP}_{ini}[-1][t] \cdot k^{tc}_{prel} \]
\[ tc0(3.1): \;\;\; \frac{dc^{RNAP}_{sigmaint}[t]}{dt}\: =\: c^{RNAP}_{free}[t] \cdot c^{sigma}[t] \cdot k^{sigma}_{on} - c^{RNAP}_{sigmaint}[t] \cdot k^{sigma}_{off} \] \[ \hphantom{tc0(3.1): \;\;\; \frac{dc^{RNAP}_{sigmaint}[t]}{dt}\: =\: } + c^{RNAP}_{sigma}[t] \cdot k^{RNAPsigma}_{isore} - c^{RNAP}_{sigmaint}[t] \cdot k^{RNAPsigma}_{iso} \] \[ tc0(3.2): \;\;\; \frac{dc^{RNAP}_{sigma}[t]}{dt}\: =\: c^{RNAPsigma}_{bound}[t] \cdot k^{RNAP}_{gain} \cdot c^{RNAP}_{sigma}[t] \cdot k^{RNAP}_{loss} \] \[ \hphantom{tc0(3.2): \;\;\; \frac{dc^{RNAP}_{sigma}[t]}{dt}\: =\: } + \sum \limits_{i=0}^n c^{RNAP}_{on}[i][t] \cdot k^{RNAP}_{off} - c^{RNAP}_{sigma}[t] \cdot p^{DNA} \cdot l^{DNA} \cdot k^{RNAP}_{on} \] \[ \hphantom {tc0(3.2): \;\;\; \frac{dc^{RNAP}_{sigma}[t]}{dt}\: =\: } + c^{RNAP}_{sigmaint}[t] \cdot k^{sigma}_{iso} - c^{RNAP}_{sigma}[t] \cdot k^{RNAPsigma}_{isore} \]
\[ tc0(4): \;\;\; \frac{dc^{RNAP}_{on}[i][t]}{dt}\: =\: c^{RNAP}_{sigma}[t] \cdot p^{DNA} \cdot k^{RNAP}_{on} \] \[ \hphantom{tc0(4): \;\;\; \frac{dc^{RNAP}_{on}[i][t]}{dt}\: =\: } + c^{RNAP}_{on}[i\:-\:v^{RNAP}_{move} \cdot dt][t] \cdot (1 - k^{RNAP}_{off}) - c^{RNAP}_{on}[i][t] \]
\[ tc0(5): \;\;\; \frac{dc^{RNAP}_{prom}[t]}{dt}\: =\: \sum \limits_{i\:=\:n - v^{RNAP}_{move} \cdot dt}^n c^{RNAP}_{on}[i][t] \cdot (1 - k^{RNAP}_{off}) \] \[ \hphantom{tc0(5): \;\;\; \frac{dc^{RNAP}_{prom}[t]}{dt}\: =\: } + c^{RNAP}_{open}[t] \cdot k^{tc}_{closed} - c^{RNAP}_{prom}[t] \cdot k^{tc}_{open} \]
\[ tc0(6): \;\;\; \frac{dc^{RNAP}_{open}[t]}{dt}\: =\: c^{RNAP}_{prom}[t] \cdot k^{tc}_{open} - c^{RNAP}_{open}[t] \cdot k^{tc}_{closed} \] \[ \hphantom{tc0(6): \;\;\; \frac{dc^{RNAP}_{open}[t]}{dt}\: =\: } + c^{RNAP}_{ini}[-1][t] \cdot k^{tc}_{iniab} - c^{RNAP}_{open}[t] \cdot c^{ATP}[t] \cdot c^{X_1 TP}[t] \cdot k^{tc}_{ini1} \]
\[ tc0(7): \;\;\; \frac{dc^{RNAP}_{ini1}[t]}{dt}\: =\: c^{RNAP}_{open}[t] \cdot c^{ATP}[t] \cdot c^{X_1 TP}[t] \cdot k^{tc}_{ini1} - c^{RNAP}_{ini1}[t] \cdot c^{X_2 TP}[t] \cdot k^{tc}_{inix} \]
\[ tc0(8.1): \;\;\; \frac{dc^{RNAP}_{ini}[i][t]}{dt}\: =\: c^{RNAP}_{ini}[i-1][t] \cdot c^{X_i TP}[t] \cdot k^{tc}_{inix} - c^{RNAP}_{ini}[i][t] \cdot c^{X_i+1 TP}[t] \cdot k^{tc}_{inix}, \] \[ \hphantom{tc0(8.1): \;\;\; \frac{dc^{RNAP}_{ini}[i][t]}{dt}\: =\: } (i = 2, ..., l^{ini-1}) \] \[ tc0(8.2): \;\;\; \frac{dc^{RNAP}_{ini}[1][t]}{dt}\: =\: \frac{dc^{RNAP}_{ini1}[t]}{dt} \] \[ tc0(8.3): \;\;\; \frac{dc^{RNAP}_{ini}[-1][t]}{dt}\: =\: c^{RNAP}_{ini}[-2][t] \cdot c^{X_-1 TP}[t] \cdot k^{tc}_{inix} - c^{RNAP}_{ini}[-1][t] \cdot (k^{tc}_{iniab} + k^{tc}_{prel}) \]
\[ tc0(9): \;\;\; \frac{dc^{RNAP}_{prel}[t]}{dt}\: =\: c^{RNAP}_{ini}[-1][t] \cdot k^{tc}_{prel} - c^{RNAP}_{prel}[t] \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} \]
\[ tc0(10.1): \;\;\; \frac{dc^{RNAP}_{elong}[i][t]}{dt}\: =\: c^{RNAP}_{elong}[i-1][t] \cdot (1 - prob^{tc}_{mm}) \cdot c^{X_i TP}[t] \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(10.1): \;\;\; \frac{dc^{RNAP}_{elong}[i][t]}{dt}\: =\: } - c^{RNAP}_{elong}[i][t] \cdot ((1 - prob^{tc}_{mm}) \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} + prob^{tc}_{mm} \cdot (c^{NTPs}[t] - c^{X_1 TP}[t]) \cdot k^{tc}_{elong}) \] \[ \hphantom{tc0(10.1): \;\;\; \frac{dc^{RNAP}_{elong}[i][t]}{dt}\: =\: } + c^{RNAP}_{elongGreAB}[j + l^{mRNA}_{cl}][t] \cdot k^{GreAB}_{cat}, \] \[ \hphantom{tc0(10.1): \;\;\; \frac{dc^{RNAP}_{elong}[i][t]}{dt}\: =\: } (i = 2, ..., l^{elong-1}),\: (j = i\: and\: j = 2, ..., l^{elong} - l^{mRNA}_{cl}) \] \[ tc0(10.2): \;\;\; \frac{dc^{RNAP}_{elong}[1][t]}{dt}\: =\: c^{RNAP}_{prel}[t] \cdot (1 - prob^{tc}_{mm}) \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(10.2): \;\;\; \frac{dc^{RNAP}_{elong}[1][t]}{dt}\: =\: } - c^{RNAP}_{elong}[1][t] \cdot ((1 - prob^{tc}_{mm}) \cdot c^{X_1 TP}[t] \cdot k^{tc}_{elong} + prob^{tc}_{mm} \cdot (c^{NTPs}[t] - c^{X_1 TP}[t]) \cdot k^{tc}_{elong}) \] \[ \hphantom{tc0(10.2): \;\;\; \frac{dc^{RNAP}_{elong}[1][t]}{dt}\: =\: } + c^{RNAP}_{elongGreAB}[l^{mRNA}_{cl}][t] \cdot k^{GreAB}_{cat} \] \[ tc0(10.3): \;\;\; \frac{dc^{RNAP}_{elong}[-1][t]}{dt}\: =\: c^{RNAP}_{elong}[-2][t] \cdot (1 - prob^{tc}_{mm}) \cdot c^{X_{-1} TP}[t] \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(10.3): \;\;\; \frac{dc^{RNAP}_{elong}[-1][t]}{dt}\: =\: } - c^{RNAP}_{elong}[-1][t] \cdot l^{mRNA} \cdot c^{pprot} * k^{pprot}_{on} \]
\[ tc0(11.1): \;\;\; \frac{dc^{RNAP}_{elongter}[i][t]}{dt}\: =\: (c^{RNAP}_{elongter}[i-1][t] - c^{RNAP}_{elongter}[i][t]) \cdot c^{ATP}[t] \cdot k^{pprot}_{cat} \] \[ \hphantom{tc0(11.1): \;\;\; \frac{dc^{RNAP}_{elongter}[i][t]}{dt}\: =\: } + c^{RNAP}_{elong}[i] \cdot c^{pprot} \cdot k^{pprot}_{on}, \] \[ \hphantom{tc0(11.1): \;\;\; \frac{dc^{RNAP}_{elongter}[i][t]}{dt}\: =\: } (i = 2, ..., l^{mRNA}-1) \] \[ tc0(11.2): \;\;\; \frac{dc^{RNAP}_{elongter}[-1][t]}{dt}\: =\: c^{RNAP}_{elongter}[-2][t] \cdot c^{ATP}[t] \cdot k^{pprot}_{cat} - c^{RNAP}_{elongter}[-1][t] * k^{RNAP}_{diss} \] \[ \hphantom{tc0(11.2): \;\;\; \frac{dc^{RNAP}_{elongter}[-1][t]}{dt}\: =\: } + c^{RNAP}_{elong}[-1][t] \cdot c^{pprot}[t] \cdot k^{pprot}_{on} \]
\[ tc0(12): \;\;\; \frac{dc^{mRNA}[t]}{dt}\: =\: c^{RNAP}_{elongter}[-1][t] \cdot k^{RNAP}_{diss} - c^{RNAse}_{onmRNA}[t] \cdot k^{RNAse}_{cat} \]
\[ tc0(13): \;\;\; \frac{dc^{RNAP}_{elongmm}[i][t]}{dt}\: =\: c^{RNAP}_{elong}[i-1][t] \cdot prob^{tc}_{mm} \cdot (c^{NTPs}[t] - c^{X_1 TP}[t]) \cdot k^{tc}_{elong} \] \[ \hphantom{tc0(13): \;\;\; \frac{dc^{RNAP}_{elongmm}[i][t]}{dt}\: =\: } - c^{RNAP}_{elongmm}[i][t] \cdot c^{GreAB}[t] \cdot k^{GreAB}_{on} \]
\[ tc0(14): \;\;\; \frac{dc^{RNAP}_{elongGreAB}[i][t]}{dt}\: =\: c^{RNAP}_{elongmm}[i][t] \cdot c^{GreAB}[t] \cdot k^{GreAB}_{on} - c^{RNAP}_{elongGreAB}[i][t] \cdot k^{GreAB}_{cat} \]
\[ tc0(15): \;\;\; \frac{dc^{RNAse}[t]}{dt}\: =\: c^{RNAse}_{bound}[t] \cdot k^{RNAse}_{gain} - c^{RNAse}[t] \cdot k^{RNAse}_{loss} \] \[ \hphantom{tc0(15): \;\;\; \frac{dc^{RNAse}[t]}{dt}\: =\: } + (c^{RNAse}_{onmRNAc{l}[t] + c^{RNAse}_{onmRNAab}[t] + c^{RNAse}_{onmRNA}[t]) \cdot k^{RNAse}_{cat} \] \[ \hphantom{tc0(15): \;\;\; \frac{dc^{RNAse}[t]}{dt}\: =\: } - (c^{mRNAcl}[t] + c^{mRNAab}[t] + c^{mRNA}[t]) \cdot c^{RNAse}[t] \cdot k^{RNAse}_{on} \]
\[ tc0(16): \;\;\; (\frac{dc^{RNAse}_{onmRNA}[t]}{dt} ,\: \frac{dc^{RNAse}_{onmRNAab}[t]}{dt} ,\: \frac{dc^{RNAse}_{onmRNAcl}[t]}{dt}) \] \[ \hphantom{tc0(16): \;\;\; } = c^R{NAse}[t] \cdot (c^{mRNA}[t],\: c^{mRNAab}[t],\: c^{mRNAcl}[t]) \cdot k^{RNAse}_{on} - (c^{RNAse}_{onmRNA}[t],\: c^{RNAse}_{onmRNAab}[t],\: c^{RNAse}_{onmRNAcl}[t]) \cdot k^{RNAse}_{cat} \]
tc0(17): \(dc^mRNAab[t]/dt = c^RNAP_ini[-1][t] * k^tc_iniab - c^RNAse_onmRNAab[t] * k^RNAse_cat \) tc0(18): \(dc^mRNAcl[t]/dt = sum \limits_{i=1}^n c^RNAP_elongGreAB[i][t] * k^GreAB_cat + 2 * c^RNAse_onmRNA[t] * k^RNAse_cat - c^RNAse_onmRNAcl[t] * k^RNAse_cat \) tc0(19.1): \(dc^entity_bound[t]/dt = c^entity[t] * k^entity_loss - c^entity_bound[t] * k^entity_gain, (entity not in {RNAP, RNAPsigma}) \) tc0(19.2): \((dc^RNAP_bound[t]/dt, dc^RNAPsigma_bound[t]/dt) = c^RNAP[t] * k^RNAP_loss - c^RNAP_bound[t] * k^RNAP_gain + (c^RNAPsigma_bound[t], -c^RNAPsigma_bound[t]) * k^sigma_off \) tc0(20): \(dc^pprot[t]/dt = c^pprot_bound[t] * k^pprot_gain - c^pprot[t] * k^pprot_loss + c^RNAP_elongter[-1][t] * k^RNAP_diss - c^RNAP_elong[-1][t] * l^mRNA * c^pprot[t] * k^pprot_on \) tc0(21): \(dc^GreAB[t]/dt = c^GreAB_bound[t] * k^GreAB_gain - c^GreAB[t] * k^GreAB_loss + sum \limits_{i=1}^n c^RNAP_elongGreAB[i][t] * k^GreAB_cat - sum \limits_{i=1}^n c^RNAP_elongmm[i][t] * c^GreAB[t] * k^GreAB_on \) tc0(22): \(dc^NTP[t]/dt = - sum \limits_{i=2, X_i=N}^n c^RNAP_ini[i-1][t] * c^{X_i TP}[t] * k^tc_inix - c^RNAP_prel[t] * (1-prob^tc_mm) * c^{X_1 TP}[t] * k^tc_elong - sum \limits_{i=2, X_i=N}^n-1 c^RNAP_elong[i-1][t] * (1-prob^tc_mm) * c^{X_i TP}[t] * k^tc_elong [- c^RNAP_open[t] * c^ATP[t] * c^X_1 TP[t] * k^tc_ini1]_{for X_1 = N} [-c^RNAP_open[t] * c^ATP[t] * c^NTP[t] * k^tc_ini1 - sum \limits_{i=1}^n-1 c^RNAP_elongter[i][t] * c^ATP[t] * k^pprot_cat]_{for N = A} \) tc0(23): \(dc^NTPs[t]/dt = dc^ATP[t]/dt + dc^TTP[t]/dt + dc^GTP[t]/dt + dc^CTP[t]/dt \)