Fig 1.
Distinct qualitative behaviors for two models of a double PTM.
This is illustrated by the time series plots for the double phosphorylated substrate with randomized initial conditions for fixed total substrate and enzyme concentrations. (a) the processive mechanism exhibits a unique global attractor, (b) a distributive mechanism exhibits multistability for some parameters. See networks (11), (12) and the accompanying discussion. The parameters are given in S1 Text-§6.
Table 1.
Comparison with other methods in the literature.
The row that corresponds to “admissible kinetics” asks about the functional form of the reaction rates for which the method is applicable. “Global attractor” asks whether the method is able to provide guarantees for the global convergence to an attractor. “Uniqueness with i/o perturbations” asks whether the method can guarantee uniqueness of steady states with respect to arbitrary addition of inflows and outflows to the network (i.e., “homogeneous CFSTR” in the terminology of [45]). Rows that correspond to “PTM cycle” and “Kinetic proofreading” ask whether the method can tackle the networks (9) and (15), respectively. We have picked these two networks as non-trivial examples that are pertinent to systems biology. The question of a global attractor for HJF-type networks is marked by an asterisk (*) since a proof has been proposed in a preprint [46] but is not formally published yet. (See [47] also).
Fig 2.
Illustration of a post-translational modification reaction network.
(a) The list of reactions with six species. A kinase E interacts with a substrate S to form a complex C1 which transforms into a phosphorylated substrate Y. Similarly, a phosphatase F dephosphorylates Y back to S via an intermediate complex C2. (b) The ODE equation description of the time-evolution of the concentration of the species. (c) The graphical representation of the network as a Petri-net. A circle represents a species and a rectangle represents a reaction, (d) The Jacobian matrix of the reaction rate vector. This is the only information we assume to be known about R(x).
Fig 3.
(a) Simple binding. (b) Simple binding with enzyme inflow-outflow. (c) Cooperative binding. (d) Competitive binding.
Fig 4.
Gray-colored species are intermediates.
Fig 5.
Transcription and translation.
Gray-colored species are intermediates. (a) Transcription. (b) Translation with a leak.
Fig 6.
Gray colored species are intermediates. (a) Basic enzymatic motif. (b) Enzymatic cycle. (c) Full PTM cycle.
Fig 7.
Energy-constrained PTM cycles.
(a) Phosphorylation is modeled only. The black-colored component is the basic motif proposed in [63] (b) A full phosphorylation-dephosphorylation cycle with energy expenditures modeled. The gray species are intermediates.
Fig 8.
(a) A multisite PTM with distinct enzymes. (b) A multiple PTM with a processive mechanism. (c) The “all-encompassing” processive PTM mechanism. (d) Double PTM Cycle with a distributive mechanism.
Fig 9.
Phosphotransfer and phosphorelay networks.
(a) Phosphotransfer network. (b) Phosphotransfer with phosphorylation/dephosphorylation. (c) A phosphorelay network.
Fig 10.
(a) McKeithan’s T-Cell kinetic proofreading network. (b) ERK signaling Pathway With RKIP Regulation.
Fig 11.
(a) Schematic representation. λ0 is the initiation rate, λi is the elongation rate from site i to site i + 1, and λn is the production rate. The state variable xi ∈ [0, 1] is the occupancy level of the site i. (b) Reaction network representation. Xi corresponds to the occupancy level, while Yi corresponds to the vacancy level.
Fig 12.
Safety sets computed via RLFs.
(a),(b), Safety sets for the PTM cycle (Fig 6c). (a) The safety set corresponding to the rate-dependent RLF for the PTM cycle. It is the α-level set of V where α has been chosen such that the concentration of S does not exceed 2.5. (b) The safety set corresponding to the concentration-dependent RLF. The safety set has been chosen similarly to satisfy the same condition. (c),(d), Sub-levels sets for the safety sets corresponding to the rate-dependent RLF (7) for the double processive PTM cycle (Fig 8b). (c) The sublevel set (with [X2] = 0) of the α-level set of V where α has been chosen such that the concentration of E does not exceed 2.5 on the sublevel set. (d) Another sublevel set of the same set in (c) with [X2] = 0.5AU.
Fig 13.
Flux analysis for McKeithan’s T-cell kinetic proofreading network.
The plot depicts an upper bound on the input flux versus the maximum allowed concentration of the end product with Michaelis-Menten kinetics R6(c2) = c2/(0.1c2 + 1) and Mass-Action kinetics R6(c2) = c2.