Phenotypic repertoire of the FNR regulatory network in Escherichia coli

Abstract

The FNR protein in Escherichia coli is an O(2) sensor that modifies global gene expression to adapt the cell to anaerobic growth. Regulation of FNR involves continuous cycling of the protein between its active and inactive states under aerobic conditions without apparent function. This raises the question of what benefit to the overall life cycle might compensate for the cost of cycling and reveals that the role of this process is poorly understood. To address this problem, we introduce the concept of a 'system design space', which provides a rigorous definition of phenotype at the molecular level and a means of visualizing the phenotypic repertoire of the system. Our analysis reveals undesirable and desirable phenotypes with an optimal constellation of parameter values for the system. To facilitate a more concrete understanding of what the design space represents, we analyse mutations that alter the apparent dimerization rate constant of FNR. We show that our estimated wild-type value of this rate constant, which is difficult to measure in situ, is located within this constellation and that the behaviour of the system is compromised in mutants if the value of the apparent dimerization rate constant lies beyond the bounds of this optimal constellation.

Figure 1

A representation of the FNR (fumarate nitrate reduction) system in E. coli. FNR regulates the transition between aerobic/anaerobic growth. The monomer forms combine to produce the dimeric 4Fe-FNR, an active transcription factor that regulates the adaptation of the cell to O₂ limiting conditions. Aerobically, O₂ inactivates FNR, but the cell continues to produce and reactivate it. This results in a constant cycling of FNR between its three states apoFNR, 4Fe-FNR, and 2Fe-FNR. Aerobic cycling is tuned so that the inactive apoFNR predominates. Under anaerobic conditions, the absence of O₂ results in a rapid buildup of 4Fe-FNR. The 4Fe-FNR form dimerizes to produce an active transcription factor.

Figure 2

A diagram of the kinetic model for the FNR system. x₁ – fnr mRNA, x₂ – apoFNR and 2Fe-FNR, x₃ – 4Fe-FNR, x₆ – ClpXP protease, x₇ – iron sulfur cluster assembly proteins (Isc), x₈ – molecular O₂. The nucleotide and amino acid pools are assumed to be well regulated, and their nearly constant values are implicitly accounted for by the apparent rate constants for transcription and translation (a conventional assumption). The fate of material lost from the system by degradation and/or dilution is not shown. [See Tolla & Savageau (2010) for further details]

Figure 3

Standard curve for active FNR concentration as a function of O₂ saturation. The curve is calculated, both for the original piecewise model (Tolla & Savageau, 2010) (−) [Eqs. (11)-(13)] and the modified Hill model (- -) [Eqs. (1)-(5)], by setting the derivatives to zero and solving for active FNR (x₃) over a range of O₂ (x₈) concentrations.

Figure 4

Design space of the FNR regulatory network in E. coli. Regions (1, 2, 3) to the right of the critical O₂ threshold (K₂ = 10.4 μM) are associated with the aerobic environment, regions (5, 6, 8, 9, 13, 14) lying next to K₂ are associated with the microaerobic environment (see the expanded excerpt), and regions (4, 7, 10, 11, 12, 15) in which O₂ is very low or absent are associated with the fully anaerobic environment. The range of O₂ concentrations (0.001 to 220 μM) does not exceed the maximum dissolved O₂ concentration in the wild at 30° C. The nominal aerobic and fully anaerobic locations (○) are shown along with the path (- -) through the spectrum of phenotypes crossed during the transition between the aerobic/anaerobic growth states. The band of optimal performance (shaded area) prevents crossings into areas of poor performance (see text for discussion).

Figure 5

Classifying the regions of the FNR design space. (A) Dividing the design space into aerobic-like, microaerobic-like, or anaerobic-like behaviors based on the underlying subsystems (see main text for details), which gives rise to low (aerobic-like regions), medium (microaerobic-like regions), or high (anaerobic-like) levels of the active FNR protein. (B) Dividing the design space into aerobic, microaerobic, or anaerobic environments based on the dissolved O₂ content.

Figure 6

Comparison of the steady-state solutions (A) computed using the full system [Eqs (1)-(5)] and (B) computed for each phenotypic region. The coloring (z-axis) is the logarithm of the steady-state concentration of active FNR (μM) at each point in the design space, the y-axis is the logarithm of the apparent dimerization rate constant (min⁻¹ μM⁻²), and the x-axis is the logarithm of the O₂ concentration (μM). The shaded area marks the band of optimal performance. The nominal aerobic and anaerobic locations are shown (○).

Figure 7

(A) Dominant eigenvalue (z-axis) plotted over the design space. Dark red corresponds to systems whose local dynamics are faster. (B) Margin of stability as determined by the critical Routh criterion (z-axis) plotted over the design space (coloring is log scaled). Eigenvalues (A) and the margin of stability (B) are calculated for each point in the design space using the subsystem in the relevant region.

Figure 8

Global response time with the wild-type values of the parameters. In order to compare inactive FNR (- -) to the active FNR dimer (—), the concentration of dimeric active FNR is plotted as twice the concentration of the monomer. The fnr mRNA level (⋯) barely changes during either transition. (A) The aerobic-to-anaerobic transition: The system begins at the initial steady state (x₁ = 0.15, x₂ = 4.46, x₃ = 0.07 μM) with an O₂ level of 80 μM and shifts to the final steady state (x₁ = 0.11, x₂ = 0.17, x₃ = 1.73 μM) with a dissolved O₂ tension of 0.001 μM. (B) The anaerobic-to-aerobic transition: The system begins at the initial steady state (x₁ = 0.11, x₂ = 0.17, x₃ = 1.73 μM) with an O₂ level of 0.001 μM and shifts to the final steady state (x₁ = 0.15, x₂ = 4.46, x₃ = 0.07 μM) with a dissolved O₂ tension of 80 μM.

Figure 9

Global response times for transitions between aerobic and anaerobic environments. Data points to the right of K₂ = 10.4 μM (~1 in log space) represent a system starting at that operating point (various values for O₂ concentration and apparent dimerization rate constant) and ending at the nominal anaerobic O₂ concentration (white line at which [O₂] = 10⁻³ μM, −3 in log space). These correspond to an aerobic-to-anaerobic transition for a system with a specified apparent rate constant of dimerization and wild-type values for the other parameters of the system. Data points to the left of K₂ = 10.4 μM represent a system starting at that operating point (various values for O₂ concentration and apparent dimerization rate constant) and ending at the nominal aerobic O₂ concentration (black line at which [O₂] = 80 μM, ~1.9 in log space). These correspond to an anaerobic-to-aerobic transition for a system with a specified apparent rate constant of dimerization and wild-type values for the other parameters of the system. The global response times are plotted as a color (z-axis) at the operating point of origin in each case. (A) Time at which the concentration of active FNR reaches its peak (aerobic to anaerobic) or trough (anaerobic to aerobic). In the event that no peak/trough occurs for a specific set of values the time assigned is 500 minutes (maximum of the time window). (B) Time at which the concentration of active FNR settles to within ± 5% of its final steady-state value. The shaded area in (A) and (B) marks the band of optimal performance. Note that in (A) coloring corresponds to the logarithm of the peak time, whereas in (B) coloring corresponds directly to the settling time.

Figure 10

Illustration of the relationship between peak time and settling time. Curve a exhibits a rapid peak time followed by a slow settling time to reach its final steady state. Curve b exhibits a rapid settling time, but does have a peak. Curve c exhibits a slow settling time and has no peak.

Figure 11

Position in the design space of wild-type (○) and mutant strains from Moore et al. (2006) with increased () or decreased (●) apparent rate constants for dimerization of monomeric FNR into active FNR. The position of each strain is shown under both aerobic (right set of points) and fully anaerobic (left set of points) conditions (see text for discussion).