Fig 1.
Interplay between steady state and activation time.
(A) Schematic of input-output responses. An input ligand with concentration x (left) is processed by a system from which we measure a molecular readout, from which the readout’s steady-state level or activation time can be quantified (right). Here, “activation time” is defined as the time required for the readout to increase by one molecule after the input has been introduced. (B) Schematic of coupled and decoupled input-output responses for the steady-state level and activation time, given that the input is an activator. In the former, the steady-state level increases while the activation time decreases with input concentration; in the latter, the steady-state level increases with the input concentration, while the activation time remains constant. (C) Measurements of RNA polymerase loading rate (top), average transcription onset time (bottom, purple), and fraction of reporter-expressing nuclei (bottom, orange) for a reporter MS2 construct with the wild-type hunchback promoter, along the antero-posterior axis of the Drosophila melanogaster blastoderm (nuclear cycle 13), reproduced from [9, Fig 4]. Low “A/P position” corresponds to the anterior end of the embryo, where the Bicoid concentration is high; high values correspond to the posterior end of the embryo, where the Bicoid concentration is low. (D–G) The models used in this paper, where M is the molecular readout and x denotes the input concentration. See text for more details. (D–E) Chain models with implicit ligand binding. The ligand’s regulatory effect is captured in the edge labels as arbitrary functions of x. The graph, , without the terminal state M is used to calculate the steady-state level; the augmented graph,
, is used to calculate the activation time. See text for more details. (F–G) Ladder models that explicitly incorporate ligand binding. The vertical edges represent ligand binding and unbinding, with rates kon x and koff, respectively. As with the chain models, the graph,
, without the terminal state M is used to calculate the steady-state level, while the augmented graph,
, is used to calculate the activation time. See text for more details.
Fig 2.
Using spanning trees and forests to calculate steady-state responses and activation times.
(A) An example three-vertex graph, , and a corresponding augmented graph,
, as discussed in the text. Here,
. (B) The spanning trees of
rooted at 1, 2 and 3. Roots are shown in orange. These spanning trees contribute to the calculation of the steady-state response, through Eq. 9; see the text for details. (C) The spanning forests of
rooted at {j,4}, for each j = 1, 2, 3, in which there is a path from vertex 1 to vertex j (top three rows); and the spanning trees of
rooted at 4 (bottom row). Roots are shown in orange. These spanning forests contribute to the calculation of the activation time, through Eq. 10; see the text for details.
Table 1.
Dynamic ranges of and
in
, for regulatory regimes that promote readout production through the regulation of one transition.
Fig 3.
Decoupling under rate scale separation in the ladder model, , for case 1.III in Table 1 (A–E) and a generalization of case 2.I in Table 2 (F–J).
(A) Schematic of with regulation of
(case 1.III). (B) Distribution of coupling scores after termination of the PSO. Each PSO run was terminated whenever f < 1 for more than 5 consecutive generations. For almost all the runs (n = 99) f < 0.1. (C) Evolution of f over each PSO run. The red curve represents the “best” optimized parameter set with the smallest value of f. (D) Input-output responses of optimized parameter sets for which f < 1 and the steady-state level increases monotonically with x. (E) Distributions of parameter values corresponding to the curves in panel D, with
and
. The green curve represents the best parameter set (red curve in C). (F) Schematic of
with regulation of
and
(generalization of case 2.I). (G) Distribution of coupling scores after termination of the PSO. Each PSO run was terminated after 23 hours of computation time. For almost all the runs (n = 99) f < 0.1. (H) Evolution of f over each PSO run. The red curve represents the best parameter set. (I) Distributions of optimized parameter values for which f < 1 and the steady-state level increases monotonically with x. The green curve corresponds to the best parameter set (red curve in H). (J) Normalized dynamic ranges for two families of parameter sets, with the parameters set as follows:
,
,
,
,
,
or 10, and
varied over a logarithmic range. The dots represent numerical computations (Materials and Methods), and the dashed lines represent the formulas in Table 2 (case 2.I).
Table 2.
Dynamic ranges of and
in
, for regulatory regimes that promote readout production through the regulation of two transitions.
Fig 4.
Decoupling under rate scale constraint arises under incoherent regulation.
(A) Distributions of the coupling score, f, obtained from optimization with various rate scale constraints (RSC). The lower the RSC, the more similar the horizontal transition rates are forced to be. Optimisations were terminated after a predefined compute time (Materials and Methods). Only the parameter sets for which f < 1 are shown, and their number for each RSC value is given underneath the corresponding set of points. (B) Input-output responses corresponding to the parameter sets obtained from optimisation with RSC = 0.005, for which f < 1 and increases with x. (C) Values of
and
in the parameter sets corresponding to the responses in B (RSC = 0.005). (D) Parameter values corresponding to the responses in B (RSC = 0.005). The green line represents the best parameter set (minimum f). Each parameter is plotted in units of
, except for kon, which is in units of
. (E) Heatmap of the coupling score, f, with respect to
and
, with the other parameters set to the most optimal parameter set (green curve in D), along with select choices of
and
(crosses) whose corresponding input-output curves are shown in F. (F) Input-output curves corresponding to the parameter sets indicated in E. (G) Overlap between the concentration ranges over which the input-output curves in B change by 90%. Given two intervals [a,b] and [c,d], the overlap is computed as
. The closer this value is to zero, the less overlap there is between the concentration ranges over which
and
exhibit the greatest change. (H) Values of
and
in the best parameter set for each choice of RSC.
Fig 5.
Decoupling from an equilibrium of initial states.
(A) Schematic of the definition of activation time. This definition assumes that, before the ligand is introduced, the system has reached a steady state over the unbound states. That is, at the moment immediately prior to the introduction of ligand (t = 0 and x = 0), the system may occupy any of the unbound states (red vertices), each according to its steady-state probability. (B) Distributions of the coupling score, f, obtained from optimization with various rate scale constraints (RSC). Only the parameter sets for which f < 1 are shown, and their number for each RSC value is given underneath the corresponding set of points. (C) Input-output responses corresponding to the parameter sets obtained from optimization with RSC = 0.005, for which f < 1 and
increases with x. (D) Values of
and
in the parameter sets corresponding to the responses in C (RSC = 0.005). (E) Parameter values corresponding to the responses in C (RSC = 0.005). The green line represents the best parameter set. Each parameter is plotted in units of
, except for kon, which is in units of
. (F) Heatmap of the coupling score, f, with respect to
and
, with the other parameters set to the most optimal parameter set (green curve in E), along with select choices of
and
(crosses) whose corresponding input-output curves are shown in G. (G) Input-output curves corresponding to the parameter sets indicated in F. (H) Values of
and
in the best parameter set for each choice of RSC.
Fig 6.
Sensitivity of decoupling via the rate scale separation (A–C) and incoherent regulation (D–I) mechanisms to single-parameter perturbations.
(A) The best parameter set, , obtained from the optimization in Fig 3A–E (see also Fig 3E, green line). (B–C) Distributions of
(B) and
(C) arising from perturbations in each parameter, starting from the choice of
in A. The colormap represents the log-ratio of the perturbed parameter value,
, with respect to the optimal parameter value,
. (D) The best parameter set,
, obtained from the optimization in Fig 4 (see also Fig 4D, green line). (E–F) Distributions of
(E) and
(F) arising from perturbations in each parameter, starting from the choice of
in D. (G) A modified version of the parameter set in D, in which all horizontal rates have been set to the same value,
, as described in the text. (H–I) Distributions of
(H) and
(I) arising from perturbations in each parameter, starting from the choice of
in G. Here, the horizontal rates were perturbed as a single parameter, ℓ, as described in the text.