Team:Edinburgh UG/Model
Overview
We believe that mathematical modeling is fundamental to synthetic biology, and this is why we have decided to characterise recombination as comprehensively as possible. To achieve this, we split it into four parts. First, we compare how the deterministic and stochastic models simulate our recombinase-expressing E. coli strains. Second, we use our stochastic model to predict recombination efficiency and apply it to help witht the model of the Technion Israel iGEM team. Third, we combine the above two models together to perform in silico experiments for our logic gates and pulse generators. Finally, we have built tools based on literature to estimate the number of off-target recombination sites inside a genome, as well as to predict the effect of distance between two target sites on the rate of recombination.
Modeling the expression of recombinase in E. coli
In our project, we used the BL21 DE3 E. coli strain transformed with two plasmids, one being T7-LacO regulated recombinase expression construct in pET28b, a medium copy plasmid (10~15), another one being Promoter-LoxP-Terminator-LoxP-RFP in pSB3C5, a low copy plasmid (~5):
Temporary: recombination modules \begin{align} Cre + LL & \leftrightarrow LCL \\ Cre + LL & \leftrightarrow LLC \\ Cre + LCL & \leftrightarrow LC_2L\\ Cre + LCL & \leftrightarrow LCLC\\ Cre + LLC & \leftrightarrow LLC_2\\ Cre + LLC & \leftrightarrow LCLC\\ Cre + LC_2L & \leftrightarrow LC_2LC \\ Cre + LCLC & \leftrightarrow LC_2LC\\ Cre + LCLC & \leftrightarrow LCLC_2\\ Cre + LLC_2 & \leftrightarrow LCLC_2\\ Cre + LCLC_2 & \leftrightarrow S\\ Cre + LC_2LC & \leftrightarrow S\\ S & \leftrightarrow S^{star}\\ S^{star} & \leftrightarrow PC_2 + QC_2\\ PC_2 & \leftrightarrow PC + Cre\\ PC & \leftrightarrow P + Cre\\ QC_2 & \leftrightarrow QC + Cre\\ QC & \leftrightarrow Q + Cre\\ \end{align}As shown above, the LacI protein inhibitor is constitutively expressed, which dimerizes and binds to the operon regions controlling the expression of T7 polymerase and Cre recombinase. Without IPTG, we expect no T7 polymerase and Cre to be expressed. Without Cre recombinase being expressed, the terminator upstream of the RFP will not be excised, hence in the absence of IPTG we expect minimal expression of RFP. First, we run the model and determine the amount of LacI inhibitor, LacY permease, and T7 polymerase in equilibrium at its constitutive state. Using these values as the initial values, we then add the T7-LacO-Cre plasmid and the P-LTL-RFP plasmid into the model, and determine the amount of leaky RFP production in the absence of IPTG, and the amount of RFP production in the presence of IPTG at various concentrations and induction time.
Setting the parameters
We based our LacI-LacY induction model on the paper by Stamatakis and Mantzaris in 2009 (1). As both T7 and LacY in our model are regulated by LacI-LacO interaction, the parameters of T7 transcription and translation are treated as identical to LacY. Similarly, Cre transcription and translation is similar to that of the T7, except Cre transcription requires T7 polymerase, which we include in our list of equation. The second part of our model concerns the how Cre recombinase mediates recombination, which is based on the paper by Shoura et al. in 2012 (2). We have adopted the rate constants determined by Shoura et al. (2) and Ringrose et el. (3) in a series of in vitro experiments.
Differential Equations
\begin{equation} \frac{d[M_R]}{dt} = k_{sMR} - \lambda_{MR}[M_R] \end{equation} \begin{equation} \frac{d[R]}{dt} = k_{sR}[M_R] - 2k_{2R}[R]^2 + 2k_{-2R}[R_2] - \lambda_R[R] \end{equation} \begin{equation} \frac{d[R_2]}{dt} = k_{2R}[R]^2 - k_{-2R}[R_2] - k_{rT7}[R_2][O_{T7}] + k_{-rT7}([O_{T7}]_T-[O_{T7}]) - k_{rY}[R_2][O_Y] + k_{-rY}([O_Y]_T-[O_Y]) \\ - k_{rCre}[R_2][O_{Cre}] + k_{-rCre}([O_{Cre}]_T-[O_{Cre}]) - k_{dr1}[I]^2[R_2] + k_{-dr1}[I_2R_2] - \lambda_{R_2}[R_2]] \end{equation} \begin{equation} \frac{d[O_Y]}{dt} = - k_{rY}[R_2][O_Y] + k_{-rY}([O_Y]_T-[O_Y]) + k_{dr4}[I]^2([O_Y]_T-[O_Y]) - k_{-dr4}[I_2R_2][O_Y] \end{equation} \begin{equation} \frac{d[O_{T7}]}{dt} = - k_{rT7}[R_2][O_{T7}] + k_{-rT7}([O_{T7}]_T-[O_{T7}]) + k_{dr2}[I]^2([O_{T7}]_T-[O_{T7}]) - k_{-dr2}[I_2R_2][O_{T7}] \end{equation} \begin{equation} \frac{d[O_{Cre}]}{dt} = - k_{rCre}[R_2][O_{Cre}] + k_{-rCre}([O_{Cre}]_T-[O_{Cre}]) + k_{dr3}[I]^2([O_{Cre}]_T-[O_{Cre}]) - k_{-dr3}[I_2R_2][O_{Cre}] \end{equation} \begin{equation} \frac{d[I]}{dt} = -2k_{dr1}[I]^2[R_2] + 2k_{-dr1}[I_2R_2] - 2k_{dr2}[I]^2([O_{T7}]_T-[O_{T7}]) + 2k_{-dr2}[I_2R_2][O_{T7}] - 2k_{dr3}[I]^2([O_{Cre}]_T-[O_{Cre}]) \\ + 2k_{-dr3}[I_2R_2][O_{Cre}] - 2k_{dr4}[I]^2([O_Y]_T-[O_Y]) + 2k_{-dr4}[I_2R_2][O_Y] + k_{ft}[YI_{ex}] + k_t([I_{ex}]-[I]) + 2\lambda_{I_2R_2}[I_2R_2] + \lambda_{YI_{ex}}[YI_{ex}] \end{equation} \begin{equation} \frac{d[I_2R_2]}{dt} = k_{dr1}[I]^2[R_2] - k_{-dr1}[I_2R_2] + k_{dr2}[I]^2([O_{T7}]_T-[O_{T7}]) - k_{-dr2}[I_2R_2][O_{T7}] + k_{dr3}[I]^2([O_{Cre}]_T-[O_{Cre}]) \\ - k_{-dr3}[I_2R_2][O_{Cre}] + k_{dr4}[I]^2([O_Y]_T-[O_Y]) - k_{-dr4}[I_2R_2][O_Y] - \lambda_{I_2R_2}[I_2R_2] \end{equation} \begin{equation} \frac{d[M_Y]}{dt} = k_{s0MY}([O_Y]_T-[O_Y]) + k_{s1MY}[O_Y] - \lambda_{MY}[M_Y] \end{equation} \begin{equation} \frac{d[LacY]}{dt} = k_{sY}[M_Y] + (k_{ft} + k_{-p})[YI_{ex}] - k_p[LacY][I_{ex}] - \lambda_Y[LacY] \end{equation} \begin{equation} \frac{d[M_{T7}]}{dt} = k_{s0MT7}([O_{T7}]_T-[O_{T7}]) + k_{s1MT7}[O_{T7}] - \lambda_{M_{T7}}[M_{T7}] \end{equation} \begin{equation} \frac{d[T7]}{dt} = k_{sT7}[M_{T7}] - \lambda_{T7}[T_7] \end{equation} \begin{equation} \frac{d[M_{Cre}]}{dt} = k_{s0Mcre}([O_{Cre}]_T-[O_{Cre}])[T_7] + k_{s1Mcre}[O_{Cre}][T_7] - \lambda_{Mcre}[M_{Cre}] \end{equation} \begin{equation} \frac{d[Cre]}{dt} = k_{sCre}[M_{Cre}] - \lambda_{Cre}[Cre] - k_1[S][Cre] + k_{-1}[SCre] - k_1[SCre][Cre] + k_{-1}[SCre^{(a)}_2] - k_2[SCre][Cre] + k_{-2}[SCre^{(b)}_2] - \\ k_2[SCre^{(a)}_2][Cre] + k_{-2}[SCre_3] - k_1[SCre^{(b)}_2][Cre] + k_{-1}[SCre_3] - k_2[SCre_3][Cre] \\ + k_{-2}[SCre_4] + k_{-2}([QCre_2]+[PCre_2]) - k_2([QCre]+[PCre])[Cre] + k_{-1}([QCre]+[PCre]) - k_1([Q]+[P])[Cre] \end{equation} \begin{equation} \frac{d[YI_{ex}]}{dt} = - (k_{ft} + k_{-p})[YI_{ex}] + k_p[LacY][I_{ex}] - \lambda_{YI_{ex}}[YI_{ex}] \end{equation} \begin{equation} \frac{d[S]}{dt} = -k_1[S][Cre] + k_{-1}[SCre] \end{equation} \begin{equation} \frac{d[SCre]}{dt} = k_1[S][Cre] - k_{-1}[SCre] - (k_1 + k_2)[SCre][Cre] + k_{-1}[SCre^{(a)}_2] + k_{-2}[SCre^{(b)}_2] \end{equation} \begin{equation} \frac{d[SCre^{(a)}_2]}{dt} = k_1[SCre][Cre] - k_{-1}[SCre^{(a)}_2] - k_2[SCre^{(a)}_2][Cre] + k_{-2}[SCre_3] \end{equation} \begin{equation} \frac{d[SCre^{(b)}_2]}{dt} = k_2[SCre][Cre] - k_{-2}[SCre^{(b)}_2] - k_1[SCre^{(b)}_2][Cre] + k_{-1}[SCre_3] \end{equation} \begin{equation} \frac{d[SCre_3]}{dt} = k_2[SCre^{(a)}_2][Cre] - (k_{-2} + k_{-1})[SCre_3] + k_1[SCre^{(b)}_2][Cre] - k_2[SCre_3][Cre] + k_{-2}[SCre_4] \end{equation} \begin{equation} \frac{d[SCre_4]}{dt} = k_2[SCre_3][Cre] - (k_{-2} + k_3)[SCre_4] + k_{-3}[SC] \end{equation} \begin{equation} \frac{d[SC]}{dt} = k_3[SCre_4] - (k_{-3} + k_{-4})[SC] + k_4[QCre_2][PCre_2] \end{equation} \begin{equation} \frac{d[QCre_2]}{dt} = k_{-4}[SC] - k_4[QCre_2][PCre_2] - k_{-2}[QCre_2] + k_2[QCre][Cre] \end{equation} \begin{equation} \frac{d[PCre_2]}{dt} = k_{-4}[SC] - k_4[QCre_2][PCre_2] - k_{-2}[PCre_2] + k_2[PCre][Cre] \end{equation} \begin{equation} \frac{d[QCre]}{dt} = k_{-2}[QCre_2] - k_2[QCre][Cre] - k_{-1}[QCre] + k_1[Q][Cre] \end{equation} \begin{equation} \frac{d[PCre]}{dt} = k_{-2}[PCre_2] - k_2[PCre][Cre] - k_{-1}[PCre] + k_1[P][Cre] \end{equation} \begin{equation} \frac{d[Q]}{dt} = k_{-1}[QCre] - k_1[Q][Cre] \end{equation} \begin{equation} \frac{d[P]}{dt} = k_{-1}[PCre] - k_1[P][Cre] \end{equation} \begin{equation} \frac{d[M_{RFP}]}{dt} = k_{mRFP}([P] + [PCre] + [PCre_2]) - \lambda_{mRFP}[M_{RFP}] \end{equation} \begin{equation} \frac{d[RFP]}{dt} = k_{RFP}[M_{RFP}] - \lambda_{RFP}[RFP] \end{equation}
