Fig 1.
(A) Model of a linear pathway used for optimization. The substrate S is converted to a product P via five reactions catalyzed by enzymes with the concentration ej(t) with kinetic parameters kcat and Km and four intermediates Xi with concentration thresholds β that indicate their toxicity. The product P is diluted by vg (dilution rate). (B) Objective function of the optimization problem. The objective function minimizes the regulatory effort of e(t) (green areas) and the initial concentration of enzymes e(0).
Table 1.
Parameter and variable overview.
Fig 2.
Overview of data and its processing for validation of optimization results.
Rectangles depict the used databases and colors indicate, similar to the other figures, the considered pathway characteristics. A scheme of a 8-step linear pathway visualizes the binning into 5 intervals for the validation process.
Fig 3.
Relationship between regulatory effort and intermediate toxicity for low (A,B) and high enzyme costs (C,D).
(A,C) The fraction of regulation is displayed in blue for each enzyme. (B,D) The intermediate toxicity is depicted in yellow (white) for the case of a strongly (weakly) regulated enzyme and the toxicity threshold of intermediates before and after (lower values indicate higher toxicity). All plots show the distribution of values as a combination of violin- and box-plots indicating median and 25–75% percentile.
Fig 4.
Comparison of optimal regulatory programs.
(A) Optimal time-course of enzyme (in red) and intermediate (in blue) concentrations for σ = 1/30, uniform kinetics and no toxicity. (B) Optimal solution for the same parameters apart the low toxicity threshold β4 = 0.1 of intermediate X4. (C) Again the same parameters as in (A) apart the highly efficient third enzyme keff,3 = 10. Threshold are depicted as black lines and the arbitrary time horizon ranging from 0 to 30 is discarded for lucidity.
Fig 5.
Distribution of intermediate toxicity for each pathway interval of all compounds listed in metacyc (lower values indicate higher toxicity).
Fig 6.
Comparisons of intermediate toxicity distributions before (left), at (middle) and after strongly regulated enzymes for (A) transcriptional regulation and (B) post-translational regulation.
Yellow color indicates toxicity distributions for strongly regulated enzymes, and white color the corresponding toxicity distributions for direct downstream intermediates of weakly regulated enzymes (lower values indicate higher toxicity).
Fig 7.
Relation of kinetic efficiency and pathway position (A), regulation (B) and product toxicity (C, lower values indicate higher toxicity) in Escherichia coli.
Fig 8.
(A) Excerpt of metabolic context of acetate and lipid IVA biosynthesis. The green box highlights mechanisms of acetate toxicity. (B) lipid IVA biosynthetic pathway including key characteristics (lower log IC50 values indicate higher toxicity).