Optimal Performance of the Tryptophan Operon of E. coli: A stochastic, Dynamical, Mathematical-Modeling Approach

Abstract

In this work, we develop a detailed, stochastic, dynamical model for the tryptophan operon of E. coli, and estimate all of the model parameters from reported experimental data. We further employ the model to study the system performance, considering the amount of biochemical noise in the trp level, the system rise time after a nutritional shift, and the amount of repressor molecules necessary to maintain an adequate level of repression, as indicators of the system performance regime. We demonstrate that the level of cooperativity between repressor molecules bound to the first two operators in the trp promoter affects all of the above enlisted performance characteristics. Moreover, the cooperativity level found in the wild-type bacterial strain optimizes a cost-benefit function involving low biochemical noise in the tryptophan level, short rise time after a nutritional shift, and low number of regulatory molecules.

Appendix: Parameter Estimation

In this work, we consider a cell volume of 1 μm³ (Javelle et al. 2005). On the other hand, from experimental measures made by Bennett et al. (2009) on E. coli growing in minimal medium with glucose, the bacterial average doubling time is 77 min. Then

$$\begin{aligned} \mu\approx0.013~\text{min}^{-1}. \end{aligned}$$

Baker and Yanofsky (1972) calculated that during exponential growth E. coli possess around 1.8 copies of the trp operon. Therefore, the probabilities that a cell has 1 or 2 copies are 0.2 and 0.8, respectively. We developed an algorithm that use these probabilities to randomly choose the initial number of copies of the trp operon, and thus of the promoter. The initial state of the promoter(s) is selected by another algorithm that calculates and uses the probabilities of each of the 8 possible states at the initial levels of tryptophan.

From the work of Morse et al. (1968), the number of anthranilate synthase enzymes before de-repression is

$$\begin{aligned} E_0 \approx50~\text{molecules}. \end{aligned}$$

On the other hand, we have from the website E. coli Statistics ( http://ccdb.wishartlab.com/CCDB/cgi-bin/STAT_NEW.cgi ) that the number of tryptophan molecules is

$$\begin{aligned} T_0 \approx 80{,}000~\text{molecules}. \end{aligned}$$

The dissociation constant of the reaction through which a tryptophan binds one of its binding sites at the repressor, as obtained by Arvidson et al. (1986), is

$$\begin{aligned} K_T \approx44{,}160~\text{molecules}. \end{aligned}$$

From the experimental results of Gunsalus et al. (1986), in which the number of repressors in E. coli cultured in medium without tryptophan was calculated, we have that

$$\begin{aligned} R_{\mathrm{Tot}} \approx400~\text{molecules}. \end{aligned}$$

For the association propensities of the active repressor with the different operators ($k_{i}^{+}$) and the dissociation propensities of the repressor in its three different states ($k_{i,R}^{-}$, $k_{i,R_{T}}^{-}$, and $k_{i,R_{2T}}^{-}$), we used the values employed by Tabaka et al. (2008). These values, calculated from the experimental works of Grillo et al. (1999), Hurlburt and Yanofsky (1992), Zhang et al. (1994), and Jardetzky and Finucane (2007), are shown below:

$$\begin{aligned} \begin{array}{@{}l@{\qquad}l@{}} \displaystyle k_{1}^+ \approx8.1~\text{molecules}^{-2}\,\text{min}^{-1}, & \displaystyle k_{1,R_{2T}}^- \approx6.0~\text{molecules}^{-1}\,\text{min}^{-1}, \\ \displaystyle k_{1,R_T}^- \approx19.2~\text{molecules}^{-1}\,\text{min}^{-1}, &\displaystyle k_{1,R}^- \approx60.0~\text{molecules}^{-1}\,\text{min}^{-1}, \\ \displaystyle k_{2}^+ \approx0.312~\text{molecules}^{-2}\,\text{min}^{-1}, &\displaystyle k_{2,R_{2T}}^- \approx0.198~\text{molecules}^{-1}\,\text{min}^{-1}, \\ \displaystyle k_{2,R_T}^- \approx6.6~\text{molecules}^{-1}\,\text{min}^{-1}, & \displaystyle k_{2,R}^- \approx66.0~\text{molecules}^{-1}\,\text{min}^{-1}, \\ \displaystyle k_{3}^+ \approx0.3~\text{molecules}^{-2}\,\text{min}^{-1}, & \displaystyle k_{3,R_{2T}}^- \approx36.0~\text{molecules}^{-1}\,\text{min}^{-1}, \\ \displaystyle k_{3,R_T}^- \approx72.0~\text{molecules}^{-1}\,\text{min}^{-1}, & \displaystyle k_{3,R}^- \approx810.0~\text{molecules}^{-1}\,\text{min}^{-1}. \end{array} \end{aligned}$$

The dissociation constant of the reaction through which a tryptophan molecule binds one of its binding sites in a repressor bound to an operator, K _B , was recalculated from the work of Tabaka et al. (2008) using the K _T value obtained by Arvidson et al. (1986), instead of the one estimated by Schmitt et al. (1995):

$$\begin{aligned} K_B \approx1{,}980~\text{molecules}. \end{aligned}$$

The constant accounting for the cooperative interaction between repressors bound to operators O ₁ and O ₂ was experimentally estimated by Yang et al. (1996):

$$\begin{aligned} k_c \approx40. \end{aligned}$$

We computed the propensity for transcription initiation at a non-repressed promoter from the reported transcription initiation rate of the lac operon, and from the comparative strengths of the trp, lacUV5 and lac promoters (Kennell and Riezman 1977; De Boer et al. 1983; Deuschle et al. 1986). The trp promoter resulted to be around 1.43 times stronger than the lac promoter, which has a maximal transcription rate of 18.2 molecules/min. Thus,

$$\begin{aligned} k_M \approx26~\text{molecules/min}. \end{aligned}$$

Yanofsky et al. (1984) found that transcriptional attenuation is relieved only when the intracellular concentration of tryptophan is extremely low, and that the probability that transcriptional attenuation occurs at these conditions is 6 times smaller in comparison to instances in which tryptophan concentration is higher. The values of parameters K _G and α that allow Eq. (6) to represent this behavior are

$$\begin{aligned} K_G \approx1200~\text{molecules} \quad \text{and}\quad \alpha\approx18.8. \end{aligned}$$

To calculate the time between transcriptional initiation and the moment in which translation can start without being momentarily stopped by the RNA polymerase, τ _M , we need to remember that the enzyme anthranilate synthase is produced from trpE and trpD genes. Knowing that the distance between the site of transcriptional initiation and the start codon of trpD gene is 1724 nucleotides (nt) and that the RNA polymerase in bacteria with a doubling time of 60 min advance at a speed of a speed of 2700 nt/min, we obtain that it will take 0.64 min for the polymerase to start transcribing gene trpD (Yanofsky et al. 1981; Bremer and Dennis 1996). On the other hand, if we consider that the ribosome advances at a slightly higher rate (2880 nt/min) than the RNA polymerase, and that the length of trpD gene is 1596 nt, we obtain that

$$\begin{aligned} \tau_M \approx0.68~\mathrm{min}. \end{aligned}$$

Forchhammer et al. (1972) determined that the regions of the mRNA corresponding to the trpE and trpD genes have half-lives of 1 and 1.25 min, respectively. Using the shortest half-life, we have that the mRNA-degradation propensity is

$$\begin{aligned} \gamma_M \approx0.69~\text{min}^{-1}. \end{aligned}$$

To obtain the translation initiation propensity, k _E , we used two experimentally determined values. The first one is the number of ribosomes that translate each mRNA from the trp operon in all its life time, which is approximately 30 (Baker and Yanofsky 1972). The second one is the mRNA mean lifetime, which can be calculated from γ _M as 1.45 min. Thus,

$$\begin{aligned} k_E \approx20.7~\text{molecules/min}. \end{aligned}$$

Considering that the subunit of the enzyme anthranilate synthase that takes more time to be produced is TrpD, that the length of the trpD gene is 1,596 nt, and the ribosome elongation rate is 2,880 nt/min, we have

$$\begin{aligned} \tau_E \approx0.55~\text{min}. \end{aligned}$$

The propensity of tryptophan production by active enzyme was obtained from the experimental studies of Ito et al. (1969):

$$\begin{aligned} k_T \approx300~\text{molecules/min}. \end{aligned}$$

The values of the half saturation constant K _I and the corresponding Hill coefficient n, where calculated by Caligiuri and Bauerle (1991):

$$\begin{aligned} K_I \approx2500~\text{molecules}, \quad\text{and}\quad n \approx1.2. \end{aligned}$$

To compute the maximum propensity of tryptophan consumption, γ, we took the number of tryptophan molecules incorporated in all the proteins of the cell and we divided it by the time in which the cell, and thus the proteins, are doubled. From the website E. coli Statistics we obtain that each E. coli bacteria has around 2.6 million proteins, each one with an average length of 360 amino acids. With this information, and knowing that the relative abundance of tryptophan with respect to all the amino acids is 1.1 %, there are around 10.3 million tryptophan molecules present in proteins (Neidhardt et al. 1990). If the doubling time is 77 min, then

$$\begin{aligned} \gamma\approx134{,}000~\text{molecules/min}. \end{aligned}$$

To take into account that protein production, and consequently tryptophan consumption, decreases only when the number of tryptophan molecules in the cell is scarce we used

$$\begin{aligned} K_{\rho} \approx1{,}000~\text{molecules}. \end{aligned}$$

The system described in this work reaches stationary levels for the enzyme and tryptophan of around 1000 and 4100 molecules per bacterial cell, respectively. These values are in agreement with the experimental data of Bliss et al. (1982) and Bennett et al. (2009).