Modelling
Introduction
Mathematical modelling often gives useful insights into the behaviour of complex systems. Here, we model the synthetic gene oscillator circuit made by Stricker et al.(2008), where the protein product of the first gene (AraC) activates the expression of both the genes and the protein product of the second gene (LacI) inhibits expression of both the genes. It is expected that such a model would help test what modifications can be done in the mentioned oscillator, to couple it with other genes to achieve oscillations in their protein products, which are essential in the cell cycle.The oscillator described by Stricker et al. (2008) comprises of two genes AraC and LacI, placed downstream of a $p_{lac/ara-1} $hybrid promoter, which is activated by the binding of AraC protein in the presence of arabinose and repressed by LacI protein in the absence of IPTG. This forms a dual feedback loop that causes the levels of these proteins to oscillate periodically.
The model
The reactions in the lac ara system are as follows- \[P^{a/r}_{0,j} \ + \ a_2 \xrightleftharpoons{k_1} \ P^{a/r}_{1,j} \] \[P^{a/r}_{i,0} \ + \ r_4 \xrightleftharpoons{2k_2} \ P^{a/r}_{i,1} \] \[P^{a/r}_{i,1} \ + \ r_4 \xrightleftharpoons{0.5 k_2} \ P^{a/r}_{i,2} \] Where $a$ is the Concentration of AraC protein, $r$ is the Concentration of LacI protein. $P^{a/r}_{i,j}$ represent the states of promoters on the (a)ctivator/(r)epressor plasmids with i number of AraC dimers $(a_2)$ bound and $ j \ \ \epsilon \ (0, 1, 2) $ LacI tetramers $(r_4)$ bound. The equilibrium constants $k_1,k_2$ are dependent on IPTG and Arabinose concentration as given by- \[k_1 = \frac{[ara]^2}{[ara]^2 + (2.5^2)} .\frac{1}{1 + (\frac{[iptg]}{1.8})^2}\] \[k_2 = 2*(0.19 . \frac{1}{1 + (\frac{[iptg]}{0.035})^2} + 0.01)\] Transcription- \[\ \ \ \ \ \ P^{a/r}_{0,0} \xrightarrow{b_a} \ P^{a/r}_{0,0} + m_{a/r}\] \[ \ \ \ \ \ \ P^{a/r}_{0,1} \xrightarrow{\alpha b_a} \ P^{a/r}_{0,1} + m_{a/r}\] Translation and Protein folding- \[\ \ \ m_a \xrightarrow{t_a} m_a + a_{uf}\] \[ \ \ \ m_r \xrightarrow{t_r} m_r + r_{uf}\] \[a_{uf} \xrightarrow{k_{fa}} a\] \[r_{uf} \xrightarrow{k_{fa}} r\] where $m_{a/r}$ represents the number of activator/repressor transcripts; $a_{uf}$ and $r_{uf}$ are the unfolded monomeric versions of the activator and repressor; a and r are the folded monomeric versions of activator and repressor; $a_2$ and $r_2$ are the dimeric versions of activator and repressor; and $r_4$ represents the tetrameric version of the repressor. Dimerisation and Tetramerisation- \[a \ + \ a \xrightleftharpoons[k_{da}]{k_{da}} \ a_2 \] \[r \ + \ r \xrightleftharpoons[k_{dr}]{k_{dr}} \ r_2 \] \[r_2 \ + \ r_2 \xrightleftharpoons[k_{t}]{k_{t}} \ r_4 \] Degradation- \[a \xrightarrow{\lambda f(X)} \phi \] \[r \xrightarrow{f(X)} \phi \] \[m_a \xrightarrow{d_a} \phi \] \[m_r \xrightarrow{d_r} \phi \] \[a_{uf} \xrightarrow{{\lambda f(X)}} \phi \] \[r_{uf} \xrightarrow{{f(X)}} \phi \] We ignored protein looping, which is considered by Stricer et al. (2008) in their model. The reaction rates and other parameter values are given in the following table- $$ \begin{array}{|c|c|c|c|c|} \hline \text{Sr.} & \text{Parameter} & \text{Description} & \text{value} & \text{units} \\ \hline 1 & n_a & \text{copy number of Plasmid having AraC} & 50 & molecules \\\hline 2 & n_r & \text{copy number of Plasmid having LacI} & 25 & molecules \\ \hline 3 & \gamma_{ma}, \gamma_{mr} & \text{Degradation rate of AraC/LacI mRNA} & 0.54 & min^{-1} \\ \hline 4 & b_a,b_r & \text{Transcription rate of AraC, LacI genes} & 0.36 & min^{-1} \\ \hline 5 & \alpha & \text{Increased transcription rate Due to AraC binding} & 20 & \\ \hline 6 & t_{a} & \text{rate of transcription of AraC} & 90 & min^{-1} \\ \hline 7 & t_{r} & \text{rate of transcription of LacI} & 90 & min^{-1} \\ \hline 8 & k_{a} & \text{Rate of degradation of AraC protein} & 0.455 & min^{-1} \\ \hline 9 & k_{r} & \text{Rate of degradation of LacI protein} & 0.182 & min^{-1} \\ \hline 10 & k_{da} & \text{AraC dimerisation constant} & 100 & molecules^{-1} \\ \hline 11 & k_{dr1} & \text{LacI dimerisation constant} & 100 & molecules^{-1} \\ \hline 12 & k_{dr2} & \text{LacI tetramerisation constant} & 100 & molecules^{-1} \\ \hline 13 & k_{fa}, k_{fr} & \text{Rate of folding of proteins} & 0.13 & min^{-1} \\ \hline \end{array} $$When we solved the differential equations corresponding to each reaction using Simbiology in MATLAB, however, we were unable to reproduce the results reported. As a solution, a bottom-up approach of modelling was chosen, in which a simple model is built first and equations, parameters are added as required. The simple model, in this case, was chosen to be the cI-Lac oscillator described by Hasty et al.(2001). The two first order coupled differential equations were solved using Python 3.6 in Spyder 2.3.2 environment, a range of parameter values was scanned to obtain oscillations. The main difference between the two oscillators is that the promoter in cI-Lac oscillator has two binding sites for cI protein and one for LacI tetramer, whereas the promoter in Ara-Lac oscillator has one binding site for AraC protein and two sites for LacI binding. We then modified the model to fit the ara-lac system and incorporated IPTG, arabinose inducibility into the model. This model did not include the intermediate reactions of transcription and translation and also assumed that the reactions are dimerisation and tetramerisation are fast, and equilibrium is attained instantaneously. These assumptions were relaxed by adding four more differential equations, that of mRNA production and protein unfolding for both AraC and LacI; and also by adding a delay in the differential equation of mRNA.
The production of mRNA is governed by the differential equation- \begin{align} \frac{dm_a}{dt} = b_a(P^{a}_{0,0} + \alpha P^{a}_{1,0}) - \gamma_{ma} m_a \\ \frac{dm_r}{dt} = b_r(P^{r}_{0,0} + \alpha P^{r}_{1,0}) - \gamma_{mr} m_r \end{align}
Here, $P^{a/r}_{0,0}, P^{a/r}_{1,0}$ are also variables, and the differential equation above is coupled to the rate equation governing promoter binding reactions for LacI and AraC. We assume that these reactions are fast reactions and attain equilibrium very quickly. Thus, we can write- \begin{align} P^{a/r}_{1,0} = k_1 a_2 P^{a/r}_{0,0} \\ P^{a/r}_{i,1} = 2k_2 r_4 P^{a/r}_{i,0} \\ P^{a/r}_{i,2} = k_2 r_4 P^{a/r}_{i,1} \\ \end{align} The dimerisation and tetramerisation reactions give us- \begin{align} \begin{split} a_2 & = k_{da} a^2 \\ r_2 & = k_{dr1} r^2 \\ r_4 & = k_{dr2} r^2_2 \\ & = k_{dr2} k^2_{dr1} r^4\\ \end{split} \end{align} We define $c_a := k_1 k_{da}$ and $c_R := k_2 k_{dr2} k^2_{dr1}$. Which gives us - \begin{align} \begin{split} P^{a/r}_{1,0} & = k_1 k_{da} a^2 P^{a/r}_{0,0} \\ & = c_a a^2 P^{a/r}_{0,0} \\ P^{a/r}_{0,1} & = 2 k_1 k_{dr2} k^2_{dr1} r^4 P^{a/r}_{0,0} \\ & = 2c_r r^4 P^{a/r}_{0,0} \\ P^{a/r}_{0,2} & = k_1 k_{dr2} k^2_{dr1} r^4 P^{a/r}_{0,1}\\ & = 2 k^2_1 k^2_{dr2} k^4_{dr1} r^8 P^{a/r}_{0,0} \\ & = c^2_r r^8 P^{a/r}_{0,0} \\ P^{a/r}_{1,1} & = 2c_r r^4 P^{a/r}_{1,0} \\ & = 2c_a c_r a^2 r^4 P^{a/r}_{0,0} \\ P^{a/r}_{1,2} & = 2c_r r^4 P^{a/r}_{1,1} \\ & = c_a c^2_r a^2 r^8 P^{a/r}_{0,0} \\ \end{split} \end{align} Substituting this in above equations, we get \begin{align} \frac{dm_a}{dt} = b_a P^{r}_{0,0}(1 + \alpha c_a a^2 ) - \gamma_{ma} m_a \\ \frac{dm_r}{dt} = b_r P^{r}_{0,0}(1 + \alpha c_a a^2 ) - \gamma_{mr} m_r \\ \end{align} Also, The total number of promoters in all states remains constant in the cell as time progresses. So we impose the condition- \begin{align} n_a = \sum_{i =0,1} \sum_{j=0,1,2}P^{a}_{i,j} \\ n_r = \sum_{i =0,1} \sum_{j=0,1,2} P^{a}_{i,j} \\ \end{align} Rewriting $P^{a/r}_{i,j}$ in terms of $P^{a/r}_{0,0}$, factorising, simplifying and rearranging - \begin{align} P^{a}_{0,0} = \frac{n_a} {(1 + c_a a^2)(1+ 2c_r r^4 + {c_r}^2 r^8) }\\ P^{r}_{0,0} = \frac{n_r} {(1 + c_a a^2)(1+ 2c_r r^4 + {c_r}^2 r^8) } \\ \end{align}We substitue this in the above equations. So, In the end, we have six coupled first order delay differential equations-
\begin{align} \frac{dm_a}{dt} &= \frac{n_a b_a (1+\alpha c_a a^2)}{(1 + c_a a^2)(1+ 2c_r r^4 + {c_r}^2 r^8)} - \gamma_{m_a} m_a \\ \frac{dm_r}{dt} &= \frac{n_r b_r(1+\alpha c_a a^2)}{(1 + c_a a^2)(1+2c_rr^4 + c_r^2 r^8)} - \gamma_{m_r} m_r \\ \frac{da_{uf}}{dt} &= t_a m_a - k_{fa} a_{uf} - \gamma_{a_{uf}} a_{uf} \\ \frac{dr_{uf}}{dt} &= t_r m_r - k_{fr} r_{uf} - \gamma_{r_{uf}} r_{uf} \\ \frac{da}{dt} &= k_{fa} a_{uf} - k_a a \\ \frac{dr}{dt} &= k_{fr} r_{uf} - k_r r \\ \end{align} We solved these equations using PyDDE package in Python3.6 and scanned the IPTG and Arabinose concentrations.Results
As can be seen from fig.1, our model gives stable oscillations in the concentration of AraC and LacI proteins. The period of the oscillations follows a curve given by fig.2, which matches well with the curve reported by Stricker et al.(2008).
Figure 1: Oscillations in AraC concentrations.
Figure 2: change in time period vs IPTG concentrations.
References
- Hasty, J., Dolnik, M., Rottschäfer, V., and Collins, J. J. (2002). Syn-thetic gene network for entraining and amplifying cellular os-cillations.Phys. Rev. Lett., 88:148101
- Stricker, J., Cookson, S., Bennett, M. R., Mather, W. H., Tsimring,L. S., and Hasty, J. (2008). A fast, robust and tunable syntheticgene oscillator.Nature, 456(7221):516–519
