Competitive inhibition can linearize dose-response and generate a linear rectifier

Abstract

Many biological responses require a dynamic range that is larger than standard bi-molecular interactions allow, yet the also ability to remain off at low input. Here we mathematically show that an enzyme reaction system involving a combination of competitive inhibition, conservation of the total level of substrate and inhibitor, and positive feedback can behave like a linear rectifier-that is, a network motif with an input-output relationship that is linearly sensitive to substrate above a threshold but unresponsive below the threshold. We propose that the evolutionarily conserved yeast SAGA histone acetylation complex may possess the proper physiological response characteristics and molecular interactions needed to perform as a linear rectifier, and we suggest potential experiments to test this hypothesis. One implication of this work is that linear responses and linear rectifiers might be easier to evolve or synthetically construct than is currently appreciated.

Keywords: Competitive inhibition; Enzyme kinetics; Linear rectifiers; SAGA; Transcriptional regulation.

Figure 1

Competitive inhibition combined with mass conservation leads to a linear relationship between the substrate concentration and reaction rate. (A–D) Typical response functions and their sensitivities. (A) Michaelian response (black) and a linear response (red), and (B) their derivatives. The derivative of the linear response highlights that it is sensitive over the entire input range while the Michaelian response saturates at high input values. Neither of these responses have a threshold. (C) Several example rectifiers. Rectifiers are unresponsive below a threshold (dashed horizontal line). Ideal Michaelian rectifier (black), ideal linear rectifier (red), and a Hill function with n = 10 (green) are given along with their derivatives (D). The Hill function and the Michaelian rectifier are akin to a step-like response; each has an ‘on’ and ‘off’ regime with a clear threshold and low sensitivity in each regime. However, the linear rectifier has a threshold that defines an ‘off’ regime while being sensitive to the input across the whole ‘on’ regime; the switch is not sensitive to changes in input concentration within the ‘on’ regime. (E) Kinetic schema of a general enzymatic reaction that is prone to competitive inhibition. Where K_M is the Michaelis-Menten constant of the substrate and K_I is the inhibitor dissociation constant. (F) When the inhibitor levels are constant (black and green lines), the response is Michaelian. When the total substrate and inhibitor levels are constant, i.e. [S] + [I] = T, the response is linear throughout the entire range of the substrate (red line). (G) A contour plot of the normalized acetylation rate. The dependence of the reaction rate on the substrate and inhibitor levels can be represented as trajectories in the S - I plane. When inhibitor levels are constant (black and green lines) these trajectories are parallel to the substrate axis. When the total level of the substrate and inhibitor is constant (green line), the trajectory has a slope of -1; this slope results in a linearized response.

Figure 2

Deviations from ideal linearization. (A) The case in which the substrate affinity is higher than the inhibitor affinity, i.e. K_I/K_M is larger than one. (B) The case in which the total concentration of the substrate and the inhibitor is not constant but rather increases with the substrate levels. As a result, trajectories in the S - I plane might have a slope other than -1. (C) The kinetics constant that lead a linear (I) and Michaelian (II) reaction to behave equivalently in the physiological range, where K_I/K_M = 1 and T/K_M = 10 (similar to the case for the Ac-CoA/CoA system (Biology Box), see text for general relation). A Michaelian scheme (II) needs an affinity that is about 100 times larger and a maximal velocity that is about 10 times larger than the competitive inhibition scheme (I).

Figure 3

Conversion of a linear response to a linear rectifier. (A) A competitive inhibition linearization motif that includes positive feedback. (B) Relationship between substrate and reaction rate with (red) and without (black) positive feedback and with (dashed) or without (solid) linearization. (C) The rate derivate shows how introducing competitive binding with substrate-inhibitor interplay leads to a response that isnot sensitive below a threshold and does not saturate above it. (D) Reaction scheme with auto-activation and competitive inhibition linearization motif. (E) Reaction rate and (F) its derivative with (dashed) and without (solid) competitive binding and with (red) and without (black) multi-site auto-acetylation.

Figure 4

Histone acetylation as a linear rectifier: a hypothetical case. (A) Acetylation of histones involves the serial sequential formation of a ternary complex between SAGA, Ac-CoA, and histone. (B) CoA competes with Ac-CoA (top) and multisite auto-acetylation can provide switch-like character (bottom) to the formation of catalytically competent SAGA. Together, this provides a response that has both a sharp threshold (C) and a linear regime (D).