Figure 1.
Anti-Stress Gene Regulatory Networks
(A–C) Schematic representations of the gene regulatory networks that mediate electrophilic stress response, heat shock response, and hypoxic response, respectively.
(D) Generalized negative feedback control scheme for the anti-stress gene regulatory networks in (A–C).
HIF, hypoxia-inducible factor.
Figure 2.
Analogy between the Anti-Stress Gene Regulatory Network and Proportional Negative Feedback Control System
(A) The generalized anti-stress gene regulatory system. Stressor S increases production of controlled variable Y, which is cleared by gene product G; Y activates transcription factor T, which upregulates gene expression of G; r0 − r3 are local gains.
(B) Proportional negative feedback system and its closed-loop gain. A is the open-loop gain, F is the feedback gain, and AF is the loop gain.
(C) The feedback system in (A) is rearranged so that S is the input, and the species of interest (Y, T, and G) is positioned as the output. The systems-level gain for each of the species can be generalized in terms of the open-loop gain (r0, r0r1, r0r1r2 for Y, T, G, respectively) and loop gain |r1r2r3|, conforming to the closed-loop gain in (B).
Figure 3.
Effects of Local Gains on Shape of Dose Response Curves in the Anti-Stress Gene Regulatory Network in Figure 2
(A) Enhancing local gain r1 through dimerization or trimerization of transcription factor T increases the superlinearity of Y versus S dose response curve, decreases the superlinearity of T versus S and G versus S dose response curves.
(B) Enhancing local gain r2 through T cooperative binding to the gene promoter or tetramerization of gene product G increases the superlinearity of Y versus S and T versus S dose response curves, decreases the superlinearity of G versus S dose response curve.
(C) Effects of combinatorial changes in r1 and r2; green line (r1 = 2, r2 = 4), blue line (r1 = 2, r2 = 2×1.5); note that high loop gains tend to linearize the G versus S dose response curve.
Default local gains: r1 = r2 = −r3 = 1. Solid line, simulation results; symbol, analytical results using Equations 8-10.
Figure 4.
Effect of Constitutive Activation on Systems-Level Gains and Dose Response Curves
Constitutive activation was modeled by implementing a basal-level expression of G in addition to T-driven expression.
(A) The lnY versus lnS curve transitions from a linear function lnY = lnS + lny0 to lnY = lnS + lny1 in the presence of constitutive activation.
(B) In the presence of constitutive activation, systems-level gain (dash-dotted line) decreases from unity to asymptotically approach
; in the absence of constitutive activation,
remains at
(unpublished data). Y responds to S in a more sensitive or less controlled manner in the presence of constitutive activation (solid line) than in its absence (dotted line).
(C) In the presence of constitutive activation, systems-level gain (dash-dotted line) increases from zero to asymptotically approach a maximum
; in the absence of constitutive activation,
remains at
(unpublished data). G responds to S in a more sluggish manner in the presence of constitutive activation (solid line) than in its absence (dotted line).
Figure 5.
Effect of Saturation of Gene Activation on Systems-Level Gains and Dose Response Curves
Saturation of gene activation was modeled by implementing saturable T binding to the gene promoter.
(A) In the presence of saturation of gene activation, the lnY versus lnS curve transitions from a linear function lnY = lnS + lny1 to lnY = lnS + lny2.
(B) Systems-level gain (dash-dotted line) increases from
to asymptotically approach unity. The Y versus S curve (solid line) transitions from a superlinear function
, through a sublinear segment, to a linear function Y = y2S.
(C) Systems-level gain (dash-dotted line) decreases from the maximum
to asymptotically approach zero. The G versus S curve (solid line) transitions from a function
to a horizontal line G = g2.
Figure 6.
Effect of Saturation of Enzyme G Activity by Controlled Variable Y on Systems-Level Gains and Dose Response Curves
(A) In the absence of feedback control, saturation of G by Y causes a sublinear catastrophic increase in Y in response to S (dotted line); in the presence of feedback control, the Y versus S dose response curve (solid line) grows in a much suppressed, superlinear fashion; the corresponding systems-level gain is low and varies only slightly as suggested by Equation 12.
(B) The enzyme G versus S dose response curve (solid line) grows in a nearly linear fashion, as the systems-level gain increases to approach unity according to Equation 14.
Figure 7.
Multi-Phasic Dose Response Relationships and Systems-Level Gains in the Presence of Constitutive Activation, Saturation of Gene Activation, and Saturation of Enzyme G by Y
Phase 1: superlinear with lesser control; phase 2: superlinear more highly controlled; phase 3: linear uncontrolled; phase 4: sublinear catastrophic.
Figure 8.
Variation of Saturation Terms Affects the Length of Superlinear Controlled and Linear Uncontrolled Phases
(A) Lowering the dissociation constant Kd for transcription factor T binding to the gene promoter shortens the superlinear highly controlled phase.
(B) Lowering the Michaelis–Menten constant Km of enzyme G for clearance of Y shortens the linear uncontrolled phase.
Figure 9.
Variations in the Pre-Transcriptional Local Gain (r1), Level of Constitutive Activation, Earliness of Saturation of Gene Activation, and Degree of Feedforward Activation Can Qualitatively Alter the Curvature of the Y versus Se (External Stress) Dose Response Curve in the Low-Dose Region
(A) Relatively small r1, high constitutive activation, and large Kd tend to retain the superlinear appearance in the low-dose region.
(B) Intermediate conditions lead to minimal curvature change.
(C) Relatively large r1, small constitutive activation, and small Kd render a sublinear appearance in the low-dose region.
(D) Introduction of feedforward activation can depress Y initially, giving rise to a J-shaped dose response.
Note: solid lines are simulation results, dashed straight lines were drawn to show how linear extrapolation from the high-dose region to the basal level can mis-estimate the response at low doses. Saturation of gene activation was modeled by changing Kd for transcription factor T binding to the gene promoter.
Figure 10.
Schematic Diagram of the Anti-Electrophilic Gene Regulatory Network Model
Refer to the main text for general description of the interactions, Text S4 and Tables S4 and S5 for kinetic details in the numbered reactions. Dashed lines with empty arrow head indicate the direction of logical control flowing between the transducer, controller, and plant. The diagram was generated in PathwayLab (InNetics, http://www.innetics.com).
Figure 11.
Simulation Results for the Anti-Electrophilic Gene Regulatory Network Model
(A) Temporal changes in the levels of electrophile X, conjugation product GSX, GSH, Nrf2, GCL, and GST, in response to different stressor doses.
(B) Steady-state dose response curves for the molecular species listed in (A). Dashed tangent lines originating from (−1, 0) were drawn to help visualizing curvature changes. One unit of external stressor level is equivalent to producing X at the basal production rate.
Figure 12.
Dose Response Curves of Electrophile X versus Stressor under Various Disrupted Conditions
(A) Effects of deregulation of Gclc, Gclm, Gs, and Gst genes by Nrf2. Deregulation was implemented by clamping mRNAs of respective genes at levels seen at the basal condition.
(B) Effects of lack of Nrf2 autoregulation and/or GST functioning as a homodimer. Removal of Nrf2 autoregulation was implemented similarly as in (A); de-dimerization of GST was implemented by replacing the quadratic term in Reaction 44 with a linear term that left GST concentration at the basal condition unchanged.
(C) Effects of product inhibition of GST-catalyzed reaction. For reduced GSX inhibition, Ki in Reaction 57 was increased to 850 μM from the default value 85 μM; for increased GSX inhibition, Ki was lowered to 8.5 μM. Deregulation and de-dimerization of MRP were similarly implemented as above for other enzymes.
(D) Effects of alteration in GSH levels via gene disruption or GCL activity inhibition. Genetic disruption of Gclc and Gclm genes was implemented by either setting the respective genes to half of the default values for heterozygous deficiency or to zero for homozygous deficiency. GCL activity inhibition was implemented by lowering kc in Reaction 54 to 2.5% of the default value.