Decoding of calcium oscillations by phosphorylation cycles: analytic results

Abstract

Experimental studies have demonstrated that Ca(2+)-regulated proteins are sensitive to the frequency of Ca(2+) oscillations, and several mathematical models for specific proteins have provided insight into the mechanisms involved. Because of the large number of Ca(2+)-regulated proteins in signal transduction, metabolism and gene expression, it is desirable to establish in general terms which molecular properties shape the response to oscillatory Ca(2+) signals. Here we address this question by analyzing in detail a model of a prototypical Ca(2+)-decoding module, consisting of a target protein whose activity is controlled by a Ca(2+)-activated kinase and the counteracting phosphatase. We show that this module can decode the frequency of Ca(2+) oscillations, at constant average Ca(2+) signal, provided that the Ca(2+) spikes are narrow and the oscillation frequency is sufficiently low--of the order of the phosphatase rate constant or below. Moreover, Ca(2+) oscillations activate the target more efficiently than a constant signal when Ca(2+) is bound cooperatively and with low affinity. Thus, the rate constants and the Ca(2+) affinities of the target-modifying enzymes can be tuned in such a way that the module responds optimally to Ca(2+) spikes of a certain amplitude and frequency. Frequency sensitivity is further enhanced when the limited duration of the external stimulus driving Ca(2+) signaling is accounted for. Thus, our study identifies molecular parameters that may be involved in establishing the specificity of cellular responses downstream of Ca(2+) oscillations.

FIGURE 1

Model for the decoding of Ca²⁺ oscillations by a target protein. (A) Model scheme. (B) Oscillations in the Ca²⁺ concentration S(t) have period T, amplitude S₀, spike width Δ, and average (C) The calcium signal induces oscillations in the phosphorylated (active) target protein. After an initial transient, the time-average target activity reaches the steady value

FIGURE 2

Target activity is sensitive to the frequency of narrow calcium spikes. (A) True frequency encoding occurs when the relative frequency ω changes while both amplitude S₀ and average stay constant. (B) An example of true frequency decoding is shown here. The mean target activity is plotted against ω using Eq. 7. The values of in the limit of very slow and very fast oscillations (Eqs. 13 and 14, respectively) are indicated by dashed lines. Parameters: σ = 10, γ = 0.25.

FIGURE 3

Critical Ca²⁺ sensitivity for an efficient decoding by oscillations. The critical Ca²⁺ dissociation constant K_S (relative to the spike amplitude S₀), above which Ca²⁺ oscillations are more potent than a constant Ca²⁺ signal, is plotted as a function of the duty ratio γ. Curves were obtained using Eq. 20. Parameters: n = 2 (dashed line), n = 4 (solid line).

FIGURE 4

Encoding and decoding of Ca²⁺ frequency and amplitude. (A) Biological frequency encoding: Oscillation period T changes, and therewith the calcium average while amplitude S₀ remains constant. (B and C) Frequency decoding: Target activity in response to oscillatory (solid lines) or constant (dashed line) Ca²⁺ signals. For the oscillatory signal, the average Ca²⁺ concentration is increased by reducing the oscillation period T. (B and C) Correspond to noncooperative (n = 1) and cooperative (n = 4) calcium binding, respectively. (D) Biological amplitude encoding: Oscillation amplitude S₀ changes, and therewith the calcium average while the period T remains constant. (E and F) Amplitude decoding: Target activity in response to oscillatory (solid lines) or constant (dashed line) Ca²⁺ signals. For the oscillatory signal, is changed by increasing the oscillation amplitude S₀. Panels E and F correspond to noncooperative (n = 1) and cooperative (n = 4) calcium binding, respectively. Parameters: K_S = 1 μM, S₀ = 1.5 μM, Δ = 10 s; in panels B and C, β = 0.01, 0.1, 1/s; in panels E and F, β = 0.1/s, and γ = 0.1, 0.25, 0.5.

FIGURE 5

Specificity of target activation. Protein 1 is less Ca²⁺-sensitive and responds slower than protein 2. (A) A constant Ca²⁺ signal preferentially activates protein 2. (B) When the average is kept constant, an optimal oscillatory signal exists for each target protein. (C) Amplitude-encoded oscillatory signals activate primarily protein 2 at low amplitudes and protein 1 at high amplitudes. (D) Frequency-encoded signals preferentially activate protein 2 at low amplitudes and protein 1 at high amplitudes, irrespective of the oscillation period. Protein parameters: signal parameters: Spike width Δ = 20 s; in panel C, γ = 0.5; in panel D, S₀ = 1,2,4 μM.

FIGURE 6

Kinetics of target activation by Ca²⁺ oscillations. (A) Shown is the time course of the active target X in response to an oscillatory signal (solid line). The activation time τ (Eq. 21) measures how fast the target protein becomes activated. Depicted also are the mean target activity during each oscillation i (dashed line) and the equivalent of the numerator Q in Eq. 21 (shadowed area). (B) The ratio τ/T is plotted against the relative frequency ω using the general expression (solid line) given in Eq. 35, and the simplified solution (dashed line) in Eq. 22. Parameters: γ = 0.3, σ = 2, 10.

FIGURE 7

Decoding of signals with limited amount of calcium. (A) Transient calcium signals: Ca²⁺ stimulation is a single continuous 0.5 min pulse (γ = 1) or 5 pulses of 0.1 min durations at 0.5 min (γ = 0.2), 1 min (γ = 0.1), and 2.5 min (γ = 0.04) intervals. (B) When the kinase responds fast to Ca²⁺ changes (Eq. 3 holds), the optimal signal is a single continuous Ca²⁺ pulse (γ_tr = 1). Shown are the time-courses of the target activity X in response to the Ca²⁺ signals depicted in panel A. (C) When the kinase is slowly activated (Eqs. 1 and 2 hold), the target activity becomes maximal by applying Ca²⁺ pulses at intervals of 0.5 min (γ_tr = 0.2). (D and E) Mean target protein activity for the last oscillation versus the duty ratio γ. A maximal activity is obtained for the optimal duty ratio γ_tr. Parameters: n = 1, S₀ = 2 μM, Δ = 0.1 min, m = 5 spikes; in panels B and D, β = 0.2/min, K_S = 0.2 μM; in panels C and E, α_X = 1/(μM min), β_X = 0.2/min, α_Y = 20/(μM min), β_Y = 4/min, Y_T = 1 μM .

FIGURE 8

Effect of kinetic parameters on the optimal transient signal. Optimal duty ratio γ_tr of a transient Ca²⁺ signal as a function of the dephosphorylation rate constant β_X of the target protein. The ratio α_X/β_X is kept constant. If the target protein is rapidly inactivated (β_X ≫ 1), the optimal Ca²⁺ signal is characterized by a high duty ratio. The optimal duty ratio will also increase by increasing the total duration of the Ca²⁺ spikes (upper line). The solid lines are calculated with the analytical approximation (Eq. 24); the solid squares are obtained after numerically solving the full system (Eqs. 2 and 3). Parameters: n = 1, S₀ = 2 μM, Δ = 0.2 and 0.4 min (lower and upper line, respectively), m = 5 spikes; α_X/β_X/μM, α_Y = 1/(μM min), β_Y = 1/min, Y_T = 1 μM.