Design of versatile biochemical switches that respond to amplitude, duration, and spatial cues

Abstract

Cells often mount ultrasensitive (switch-like) responses to stimuli. The design principles underlying many switches are not known. We computationally studied the switching behavior of GTPases, and found that this first-order kinetic system can show ultrasensitivity. Analytical solutions indicate that ultrasensitive first-order reactions can yield switches that respond to signal amplitude or duration. The three-component GTPase system is analogous to the physical fermion gas. This analogy allows for an analytical understanding of the functional capabilities of first-order ultrasensitive systems. Experiments show amplitude- and time-dependent Rap GTPase switching in response to Cannabinoid-1 receptor signal. This first-order switch arises from relative reaction rates and the concentrations ratios of the activator and deactivator of Rap. First-order ultrasensitivity is applicable to many systems where threshold for transition between states is dependent on the duration, amplitude, or location of a distal signal. We conclude that the emergence of ultrasensitivity from coupled first-order reactions provides a versatile mechanism for the design of biochemical switches.

Fig. 1.

Numerical simulations of Rap regulation. A detailed simulation of the Rap1 pathways was performed by using Virtual Cell (see Fig. S1 for the pathways). α2AR were stimulated for fixed duration and with various amplitudes, evenly distributed on a logarithmic scale. Then, Rap was activated by a βAR stimulus. (A) The activation level is clustered into two groups of low and high activation. (B and C) α2AR-stimulated steady state Rap activity is ultrasensitive with respect to concentration and duration. (D) βAR stimulation of Rap is subsensitive with respect to signal amplitude. (E) βAR activation of Rap is ultrasensitive with respect to signal duration.

Fig. 2.

Ultrasensitive response in GTPase activation. (A) Schematic diagram of the small GTPase cycle and its regulation. (B) Illustration of the mathematical reasoning ultrasensitivity. First-order GAP* deactivation yields exponential dependence of the GAP on the signal (BI). The dependence of the GTPase activation on the GAP* (BII) is hyperbolic. However, dependence of the activation level on the upstream signal (BIII) is ultrasensitive. The straight lines show how different signals (one order of magnitude apart each other) are clustered into two groups of high and low activation level.

Fig. 3.

Spatial properties of coupled switches. (A) A general scheme where two GTPases are regulated by a common GAP. (B) Simulation of spatial domain formation. (BI) Due to diffusion mechanism, the spatial distribution of GAP is exponential. (BII) Because each GTPase has its own GEF, the activation level of each GTPase has a different dependence on the local GAP concentration. (BIII) This difference yields different switching points, resulting in distinct compartments.

Fig. 4.

Simulations of time-dependent switch under various conditions. Steady-state activation level of the GTPase are plotted as a function of signal duration for A₀(k_GAP/k_GEF) = 1, 10, or 100 (Left, Middle, and Right, respectively), and b = 0.01, 0.1, and 1 (Bottom, Middle, and Top). All curves are semilog plots, and two examples in linear scale are in Insets for [τ] < 10.

Fig. 5.

Experimentally observed ultrasensitivity. In Neuro 2A cells, Rap1 was activated for different times by the cannabinoid receptor-1 agonist HU-210 (A) or different amounts of HU-210 (B). The shaded lines are the results of individual experiments, and the black line is their average. Hill coefficients were calculated as described in SI Text.

Fig. 6.

Ultrasensitivity by activation of GEF. Activation of Rho-family GTPase by GDI phosphorylation was simulated. Activation exhibits ultrasensitive response with respect to level of protein kinase signal.