Bhartiya et al. (2003), Tryptophan Operon
July 2007, model of the month by Antonia Mayer*
Original model: BIOMD0000000062
Although molecular biology has made tremendous progress in the elucidation of individual components of cells, providing detailed descriptions of their structures and qualitative descriptions of their interactions, a broader, more quantitative picture of biochemical interactions still eludes biologists (cf. Fraser and Harland 2000, in ). One prominent category of such interactions is that of genetic regulatory networks, many of which are complex. Consequently, systems biology began with creating mathematical models of the dynamic behavior of simple genetic networks, as a necessary prerequisite to understanding more elaborate systems . Aspects of these models are being incorporated into more complex models (e.g. ).
One of the best studied of these simpler systems is the tryptophan operon, an array of structural genes regulated by the same locus. (Another is the lac operon, see model BIOMD0000000065 and its description.) The trp operon is a negative-feedback mechanism which controls tryptophan synthesis in prokaryotes (for review, see ). The amino acid tryptophan is required for protein synthesis in cells, but the cost of its synthesis is high , and hence the production of tryptophan in a bacterium is carefully regulated. In qualitative terms, the operon behaves in this manner (Fig.1):
In the absence of a complete repressor protein (a repressor protein with two bound tryptophan molecules), the trp operator is unobstructed, the genes encoding anthranilate synthase are transcribed, and tryptophan synthesis proceeds. Note that the repressor protein possesses two tryptophan binding sites, and becomes capable of repressing the trp operator only when these sites are filled (for review, see ). Clearly, this is a regulatory strategy that seems capable of maintaining tryptophan concentration at some stable value, in principle. But how sensitive is this system? How well can it damp oscillations in tryptophan concentration? And is the system optimized through two tryptophan binding sites on the trp repressor, rather than one or three ...?
Systems biology is helpful here: mathematical models can illuminate the forces underlying the dynamics of systems like these, and even provide evidence regarding the adaptive value of their components. One such model (BIOMD0000000062), by Bhartiya et al. , suggests that the design of the trp operon is not only functional but is in fact structurally optimal for its task.
Figure 1: Regulation of intracellular concentration by the tryptophan operon, from . The concentration of tryptophan (T) is affected by (1) synthesis of the enzyme anthranilate synthase (E), which proceeds with rate constant k1; (2) the action of anthranilate synthase in production of tryptophan from a nitrogenous substrate (NDS), which proceeds with rate constant k2; and (3) take-in of tryptophan from the environment (T ext), which is instantaneous. Tryptophan binds with the repressor (R), enabling it to bind with the operator (O) and thereby inhibit the synthesis of E. These last three steps are considered to be fast and therefore at equilibrium.
Building on the model of Santillan and Mackey , Bhartiya et al. considered tryptophan synthesis, extracellular import, consumption, repression, enzyme synthesis, cell growth, and the number of tryptophan binding sites on the repressor.
Bhartiya et al. derive their argument for the optimality of the trp operon regulatory system from comparisons to controller design for industrial systems (, ). For example, the transport of extracellular tryptophan into the cell is nearly instantaneous, much like an optimized industrial controller system converging quickly to a desired operating level. Another feature of the trp operon which appears to be optimal is that of the regulator gain of the system—the speed and strength with which it corrects for a divergence from the desired state—in this case, the change in the concentration of the regulating enzyme in response to changes in the concentration of extracellular tryptophan. Thus, when the system is saturated with tryptophan (Fig2, iii), about 10% of enzyme is transcribed by the operon, although no tryptophan is synthesized. For lower extracellular tryptophan concentration (Fig.2, ii), the enzyme level is higher and the system response is overdamped. When no extracellular tryptophan is present, the enzyme profile (Fig.2, i) corresponds to that of intracellular tryptophan, signifying an underdamped response from the trp operon system. Notably, both the initial oscillations and the final steady-state behavior are suggested by the data points in experimental results . However, this modeling study pointed to the necessity of further experimental work, "to verify reproducibility of the transient oscillations [of enzyme concentration]" .
Figure 2: Model of dynamic synthase levels for extracellular tryptophan concentrations of 0 μm(i), 0.14 μm(ii), and 1.4 μm(iii), from . Synthase concentration is normalized to 1 μm. Experimental data from  is represented as crosses.
Figure 3: Dynamic behavior of normalized tryptophan concentration in absence of extracellular tryptophan, from . Modeled to parameters corresponding to only one tryptophan molecule (solid line) and of two (dotted line) binding to the repressor.
A main result of this model was to suggest that the sensitivity of the system in response to tryptophan results from the stoichiometry of the repressor's requirement to be bound to two molecules in order to stably bind to the operator. Fig.3 shows steady-state intracellular tryptophan concentrations to differ widely between a theoretical single-binding model (solid line) and the double-binding model (dotted line). The steady-state concentration for the former is only about one-fourth of that reached in E. coli cells in absence of extracellular tryptophan (16 μm), and even less than the maximum concentration reached in doubly bound tryptophan in the model (37 μm). There is another interesting similarity between industrial controllers and the trp operon; the former are often designed to ensure a quarter decay ratio in the response overshoot , and the latter appears to exhibit this property. Models of a theoretical repressor containing only one tryptophan binding site do not show this behavior, and models of repressors with more than two tryptophan binding sites show no further gains in performance. Although, as noted above, further experimental evidence is needed to establish the transient fluctuations in enzyme concentration predicted by this model (and consequently the sensitivity of the regulatory mechanism), this last result hints at a possible evolutionary rationale of the existing structure for the trp operon repressor.
* Tristan Eversole is gratefully acknowledged for his helpful comments.
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