
MBW:Modelling the Tryptophan OperonFrom MathBioContentsSummaryThis is a summary and extension of the model of the tryptophan operon presented by Moises Santillan and Michael C. Mackey ^{[1]}. First, the tryptophan operon is described, followed by a discussion of the Santillan and Mackey model. We then reproduce the results of Santillan and Mackey, extending them to include sensitivity analysis. The Tryptophan OperonUnderstanding of the molecular biology of gene expression has improved significantly over the years. However, few mathematical models have been derived for these systems. One well characterized mechanism of gene regulation involves operons. An operon is "a set of genes transcribed under the control of an operator gene" ^{[2]}. In more detail, an operon is composed of a promoter, an operator and a sequence of genes. Transcription factors can bind to the operator to either positively or negatively influence transcription of the entire operon. For example, if the gene needs to be turned off, a repressor protein can bind to the operator region and prevent the binding of RNA polymerase, thus preventing the transcription. In many cases, proteins produced by the operon affect the regulator proteins. There are two operons that have been studied extensively by molecular biologists, the Lac operon and the tryptophan (Trp) operon. The paper by Santillian and Mackey suggests an improved model for the dynamics of the tryptophan operon regulation ^{[1]}. Tryptophan is an essential amino acid. This is studied in microorganisms because it is a fairly simple pathway but is also necessary for life if not provided tryptophan from the environment (which would not occur in nature). The dynamics of operon regulation is important to study because many microorganisms have proteins that exhibit operon control and these dynamics may be utilized to control/destroy a microorganism. Santillan and Mackey ModelTable 1: Model Variables and symbols ^{[1]}. The Model variables are shown in Table 1 where free operon concentration, free mRNA concentration, total enzyme concentration and tryptophan concentrations are varying with time.
This model assumes that the only source of operons is through growth and the second term in the equation is rate of mRNA polymerase binding to operons and then the rate at which the mRNA polymerases leave the binding region. The amount of mRNA polymerases that are leaving the binding region is proportional to the amount of polymerase that bound a distance in time ago, thus this needs to be modeled with a delay term. This is all proportional to the fraction of operons that are not being actively repressed. The term describes the number of active repressor molecules which must be done through two tryptophan proteins binding noncooperatively to the activator molecule.
For all of these equations, the is the amount of growth that occurred over the delay time, . (For a project solving a 2time delay difference equation, see: MBW:Extensions to a 2Delay GlucoseInsulin Regulatory Model) Physiological Meaning of ParametersNow that each of the equations have been described a description of what each parameter means physically in order to be able to the discuss the relevance of the sensitivity findings.
ResultsReplication of ResultsSantillan and Mackey used their model and fit the parameters to data given by Yanofsky and Horn. ^{[3]}. The experiments done by Yanofsky and Horn took E. coli and let them achieve a steady state in minimal media + tryptophan then transferred them to just minimal media. They did this for both the wild type of "E. coli" bacteria as well as two mutant strains, "trpL29" and "trpL75". The values that Santillan and Mackey approximated for the wild type are shown in the thumbnail to the right. The figure below shows the the Santillan and Mackey plots with the model and data where the wild type is on the top of both plots, (with + and x data points). The left plot has the "trpL29" mutant strain with o data points and the lower line and the right plot has the "trpL75" mutant strain with o data points and the lower line. Figure 1: Santillan and Mackey Results ^{[1]}. We have replicated the results that are presented by Santillan and Mackey using code presented in the appendix and the figures can be seen below ^{[1]}. Unsurprisingly, these results agree perfectly with the Santillan and Mackey Results seen above in Figure 1. Figure 2: Numerical Replication of Santillan and Mackey Results ^{[1]}. Sensitivity AnalysisNext we examined then sensitivity of the solution to each of the variables. This was done by doing a change on each variable, and then numerically calculating the derivative using a central difference approximation. Each sensitivity was then normalized for comparison. The sensitivities are plotted vs. time in three separate categories: Low, Medium, and High. Figure 3: Time sensitivities for tryptophan operon for each variable. Grouped into similar affect levels.
Figure 4: General Variable Sensitivity As one can see g has the largest overall effect on the solution. This is because g is directly related to the rate of removal of tryptophan from the system, allowing for more (or less) tryptophan to be produced. K and Ki are also large because they have a direct affect on the activity of the enzyme. b has a large affect because it is related to the intrinsic transcriptional efficiency. DiscussionThis report examined the paper Dynamic regulation of the tryptophan operon: A modeling study and comparison with experimental data ^{[1]}. This paper developed a detailed mathematical model of tryptophan operon and fitted the parameters using experimental data. The numerical results from Santillian and Mackey were reproduced. A sensitivity analysis was then performed on the parameters to examine which has the greatest effects. It was found that the parameter g has the largest overall effect on the solution due to its direct relation with the removal rate of tryptophan from the system. On a side note, the parameters chosen by Santillan and Mackey did not do a very good job of fitting the data, in the authors opinion, because the lines in Figure 1 do not match the data well. The experimental data is initially large for the wild type E. coli and slowly decays towards steady state, but the model increases rapidly in the first minutes but then continues to slowly towards the steady state values. It appears that the wild type should have a solution of the shape that the L75 mutation has. An improvement to this model would be to start with parameters that better fit the data, and then repeat the sensitivity analysis around these points. Since the sensitivity analysis shows the solution has the greatest sensitivity to most of the parameters in the first 20 minutes, more data points need to be taken during this time. This would allow for a better fit of the data initially, giving researchers more insight into the dynamics of the tryptophan operon. Project AdditionsMathematics UsedThe original paper used a system of time delay differential equations. This equations were solved numerically and compared to experimental results. The authors of the review added sensitivity analysis. Type of ModelThe model is on the molecular scale. It aims to describe the interactions of a small set of molecules in the nucleus, mainly with respect to various rate constants. Biological SystemThe paper aims to model the Trp operon. The Trp operon is a set of genes that are necessary for the production of the amino acid tryptophan. Since the operon is repressed by tryptophan, when tryptophan is not present, the genes are transcribed and translated into enzymes used for tryptophan synthesis. This is one form of a gene regulator network. Discussion of a Recent PaperTwo years after the paper discussed in this article, M.C. Mackey published a paper modelling the lac operon. ^{[1]} Unsurprisingly, the model is remarkable similar to the one developed here for the Trp operon; they develop a system of time delay differential equations relating various relevant rates, with 22 parameters. Many of the parameters values are taken form previous studies, however, a couple of them are determined by fitting their model to experimental data. After determining parameter values, they solve the system numerically and compare it two different sets of data. Qualitatively, the model seems to better describe the experimental data than the model discussed above for the Trp operon. This could be in part because they had more data available describing the lac operon. Through their analysis, they determine that there is potentially a realistic set of parameters that would result in bistable behavior, corresponding to a cusp bifurcation. References
AppendixBelow are the list of files used in MATLAB to produce results. 