Operating regimes of signaling cycles: statics, dynamics, and noise filtering

Abstract

A ubiquitous building block of signaling pathways is a cycle of covalent modification (e.g., phosphorylation and dephosphorylation in MAPK cascades). Our paper explores the kind of information processing and filtering that can be accomplished by this simple biochemical circuit. Signaling cycles are particularly known for exhibiting a highly sigmoidal (ultrasensitive) input-output characteristic in a certain steady-state regime. Here, we systematically study the cycle's steady-state behavior and its response to time-varying stimuli. We demonstrate that the cycle can actually operate in four different regimes, each with its specific input-output characteristics. These results are obtained using the total quasi-steady-state approximation, which is more generally valid than the typically used Michaelis-Menten approximation for enzymatic reactions. We invoke experimental data that suggest the possibility of signaling cycles operating in one of the new regimes. We then consider the cycle's dynamic behavior, which has so far been relatively neglected. We demonstrate that the intrinsic architecture of the cycles makes them act--in all four regimes--as tunable low-pass filters, filtering out high-frequency fluctuations or noise in signals and environmental cues. Moreover, the cutoff frequency can be adjusted by the cell. Numerical simulations show that our analytical results hold well even for noise of large amplitude. We suggest that noise filtering and tunability make signaling cycles versatile components of more elaborate cell-signaling pathways.

Figure 1. Diagram of the Signaling Cycle

The cycle consists of a protein that can be in an inactive (I) or active (A) form. It is activated and deactivated by two enzymatic species, termed kinase (E ₁) and phosphatase (E ₂), respectively. The reactions and reaction rates that describe the cycle are shown on the right.

Figure 2. Steady-State Behavior of the Four Cycle Regimes

(A) When both enzymes are unsaturated, the steady-state response is hyperbolic. The parameters used for this cycle are , a ₁ = 1, K ₁ = 10,000, a ₂ = 1, , K ₂ = 10,000, k ₁ = 1, and k ₂ = 1, where all reaction rates are in units of 1/s, concentrations and Michaelis constants are in nanomoles, and second-order reaction rates (a ₁ and a ₂) are in 1/nM/s. (B) When the kinase is saturated and the phosphatase unsaturated, a linear response results. The parameters here are , a ₁ = 100, K ₁ = 10, a ₂ = 1, , K ₂ = 10,000, k ₁ = 500, and k ₂ = 10,000. (C) When the kinase is unsaturated and the phosphatase saturated, a threshold-hyperbolic response results. The parameters for this cycle are , a ₁ = 100, K ₁ = 10,000, a ₂ = 100, , K ₂ = 1, k ₁ = 25, and k ₂ = 1. (D) When both enzymes are saturated, an ultrasensitive response results. The parameters used for this cycle are , a ₁ = 100, K ₁ = 10, a ₂ = 100, , K ₂ = 10, k ₁ = 1, and k ₂ = 1. The parameters for the four cycles were chosen to be comparable in magnitude to values found in the literature (see [11,62], for example).

Figure 3. Relative Error

The relative error between the steady-state characteristic of the hyperbolic (A), signal-transducing (B), threshold-hyperbolic (C), and ultrasensitive (D) regimes, and that of the tQSSA in Equation 3 are shown. To compute the error for a regime, we first approximated the average squared difference between the regime's steady state and that of Equation 3 and then divided its square root by the total substrate S ₁. A relative error of 0.1 then corresponds to an average absolute difference between the steady-state characteristic of the regime and that of Equation 3 of 0.1S_t (see Text S5). The figures here show that the relative error for each regime is small for a wide region of the K ₁ versus K ₂ space, demonstrating that the four regimes cover almost the full space. The parameters used for this cycle are the same as those in Figure 2D, except K ₁ and K ₂, which were varied in the range of values shown in the x and y axes in this figure. The dashed lines enclose the regions where each regime is expected to describe the system well.

Figure 4. Magnitude of the Response of the Cycle O (Normalized by the Steady-State Saturation Value) versus the Input Frequency ω, for Three Different Input Amplitudes a

The traces in (A), (B), (C), and (D) show the response of the hyperbolic, signal-transducing, threshold-hyperbolic, and ultrasensitive switches, respectively, as shown in Figure 2. The solid lines are the analytical approximation (Equation 4). The dotted lines are obtained from numerical simulation of the full system.

Figure 5. Dynamic Response of the Cycles to Fast and Slow Inputs

The cycle has a characteristic response time τ_c that is a function of its parameters (see “Dynamic Response” in Results), and which is different for all four regimes. This plot shows the response of all four regimes to (1) a slow input that has a period equal to twice the characteristic response time of the cycle, followed by (2) a fast input with a period equal to one-fifth of the cycle's response time. For clarity, time was normalized by dividing by the characteristic time of each cycle. The signal in red represents the input kinase levels (for the threshold-hyperbolic switch, the input used is actually twice the red signal), whereas the blue traces in (A), (B), (C), and (D) show the response of the hyperbolic, signal-transducing, threshold-hyperbolic, and ultrasensitive switches, respectively, as shown in Figure 2.

Figure 6. Response of the Four Cycles to the Input Buried in Noise

The input is a sum of a slow signal (same as in Figure 4) and a Gaussian uncorrelated noise. The resulting input signals are shown in red. The blue traces in (A), (B), (C), and (D) show the response of the hyperbolic, signal-transducing, threshold-hyperbolic, and ultrasensitive switches, respectively, as shown in Figure 2. The response shows that the cycles respond to the signal only and ignore or filter out the noise in the input. Time was normalized by the characteristic time of each cycle to facilitate comparison among cycles.