Simple Biochemical Pathways far from Steady State Can Provide Switchlike and Integrated Responses

Abstract

Covalent modification cycles (systems in which the activity of a substrate is regulated by the action of two opposing enzymes) and ligand/receptor interactions are ubiquitous in signaling systems and their steady-state properties are well understood. However, the behavior of such systems far from steady state remains unclear. Here, we analyze the properties of covalent modification cycles and ligand/receptor interactions driven by the accumulation of the activating enzyme and the ligand, respectively. We show that for a large range of parameters both systems produce sharp switchlike response and yet allow for temporal integration of the signal, two desirable signaling properties. Ultrasensitivity is observed also in a region of parameters where the steady-state response is hyperbolic. The temporal integration properties are tunable by regulating the levels of the deactivating enzyme and receptor, as well as by adjusting the rate of accumulation of the activating enzyme and ligand. We propose that this tunability is used to generate precise responses in signaling systems.

Main Text

Ligand/receptor systems and covalent modification cycles are the basic units of signaling systems (1). Extracellular ligands and membrane receptors are used as a mean of cell-cell communication or communication between cells and their environment. Covalent modification cycles operate intracellularly to process a large variety of biochemical signals (1). These systems have been extensively studied. However, their properties far from steady state remain poorly understood, despite their significance for many biological applications. We have studied the behavior of both ligand/receptor systems and covalent modification cycles in response to accumulation of the ligand or the activating enzyme (e.g., due to transcription).

Covalent modification cycles

Covalent modification cycles consist of a substrate that is activated by an enzyme through a posttranslational modification and deactivated by another enzyme through the reverse modification (2). At steady state, they produce switchlike response when both enzymes operate at saturation (zero-order ultrasensitivity) and a hyperbolic response when they operate in the first-order regime (2). Interesting signaling regimes are observed when one enzyme operates in one regime and the other in the opposite regime (3). One interesting feature of covalent modification cycles is a potential for time-based averaging that filters signals of frequency higher than a characteristic frequency determined by the properties of the regulating enzymes (3,4). It is not known where these features extend to regimes in which cycles operate far from steady state.

We have studied the behavior of covalent modification cycles in response to accumulation of the activating enzyme. We assume that the activating enzyme, E_a, has zero activity until time t = 0, when it starts to increase, while the activity of the deactivating enzyme, E_d, is constant (Fig. 1). The substrate can be in an inactive conformation, S, and an active one, S^∗. The total amount of substrate S_T = S + S^∗ is assumed to be constant. We limit our analysis to conditions that are well described by the first-order Michaelis-Menten approximation (S, S^∗ << K_m,a, K_m,d), because this is a good assumption for most signaling systems. In this regime the system is described by the equation

$$\frac{d{S}^{\ast }}{dt}=k_a{E}_a\left(S_T-S^{\ast }\right)-k_d{E}_d{S}^{\ast },$$ (1)

where we have defined k_a = k_cat,a/K_m,a and k_d = k_cat,d/K_m,d. This equation can be cast in dimensionless form by defining x = S^∗/S_T and τ = t/t₀ (where t₀ = 1/k_dE_d), yielding

$$\frac{dx}{d\tau }=\epsilon \left(\tau \right)-\left(1+\epsilon \left(\tau \right)\right)x,$$ (2)

where ε(τ) is the ratio between the activity of E_a and E_d and must be expressed as a function of the dimensionless time τ. The cycle is characterized by a response time which in nondimensional units is (1 + ε(τ))⁻¹. For slow varying signals, i.e., dε/dτ << 1, the cycle equilibrates to its steady-state value before significant changes in ε and the dynamic of x is essentially given by the steady-state response: x_ss(τ) = ε(τ)/(1 + ε(τ)). On the other hand, for dε/dτ >> 1, ε accumulates to values much larger than 1 before the cycle has time to respond. The value x changes very slowly until the response time of the cycle (which decreases as ε increases) becomes of order dε/dτ. At this point, the cycle starts responding rapidly. Because ε has accumulated to values much greater than 1 before the cycle starts responding and because the response of x is bound to 1, the response eventually becomes very rapid. We, therefore, predict that covalent modification cycles operating far from steady state act as molecular switches, rapidly undergoing a transition from an off-state to an on-state.

Figure 1.

Covalent modification cycle driven by accumulation of the activating enzyme. (A) Diagram of the enzymatic reaction controlling the cycle and plot of the dynamics of E_a and E_d for linear accumulation of E_a. Dependency of x on τ for different values of A for n = 1 (B) and n = 7 (C). (D) Dependency of the Hill coefficient on A for n = 1–7. The value at A >> 1 represents n_H,max, the maximum Hill coefficient that can be achieved for each n.

To test this prediction, we studied cases in which the accumulation of E_a(t) is described by a power law: E_a(t) = rtⁿ, a scenario relevant for certain biological conditions. For example, although constant production of a stable protein would result in E_a ∼ rt, accumulating mRNA and stable protein would give E_a(t) ∼ rt². Equation 2 becomes

$$\frac{dx}{d\tau }=A{\tau }^n-\left(1+A{\tau }^n\right)x.$$ (3)

We observe that A = k_ar/(k_dE_d)ⁿ⁺¹ is the sole parameter controlling the dynamics of the system. The dynamical behavior of the covalent modification cycles is described by the solution of Eq. 3:

$$x\left(\tau \right)=1-e^{-\left(\tau +\frac{A{\tau }^{n+1}}{n+1}\right)}\left(1+\underset{0}{\overset{\tau }{\int }}{e}^{\left({\tau }^{\prime }+\frac{A{\tau }^{\prime n+1}}{n+1}\right)}d{\tau }^{\prime }\right).$$ (4)

We approximate the dependency of x on τ with a Hill function:

$${\tau }^{n_H}/\left(K^{n_H}+{\tau }^{n_H}\right)$$

(from Ferrell (5)). At steady state, the Hill coefficients are equal to n. We found that far from steady state there is a large range of conditions in which Hill coefficients were greater than 1, indicative of a sigmoidal (ultrasensitive) response. In the case of linear accumulation Hill coefficients up to almost 3 could be achieved, which are significantly higher than the Hill coefficient of the steady-state response. For n > 1, the dynamical response of x also displayed Hill coefficients higher than n. However, the ratio between n_H/n was reduced for higher n. These results confirm that cells can achieve an ultrasensitive response as a function of time using covalent modification cycles operating far from steady state. We observe that the sigmoidal response observed in response to rapid linear accumulation is purely an out-of-equilibrium property, while the ultrasensitivity observed for n > 1 is in part the result of nonlinear accumulation of the activating enzyme.

Ultrasensitivity increases monotonically with A. To determine the maximum Hill coefficient that can be achieved by the cycle for each n, we analyzed the behavior in the limit in which A >> 1. In this limit, the cycle is effectively described by

$$\frac{dx}{d\tau }\approx A{\tau }^n\left(1-x\right).$$ (5)

The solution of this equation is

$$x\left(\tau \right)=1-e^{-\frac{A{\tau }^{n+1}}{n+1}}.$$

From this expression, we derive the maximum Hill coefficient (n_H,max) for any value of n. This value can be significantly higher than n, confirming that rapid accumulation of the activating enzyme results in increased ultrasensitivity.

The dynamical response of covalent modification cycles to small perturbations is well described in the frequency space by a low-pass filter with the cutoff frequency set by the response time (3,4). We tested whether this behavior holds in response to rapid accumulation of the activating enzyme. Using dimensional analysis, we deduce that the system is characterized by two frequencies: γ = k_dE_d and γ_a = (k_ar)^1/(n+1). Because the amount of substrate is constant, we expect the relaxation rate of S^∗ to be γ_d + γ_a, which in dimensionless units becomes K ≈ (γ_d + γ_a)t₀ = 1 + A^1/(n+1). Stochastic simulations (6) confirm that the system behaves effectively as a low-pass filter (7) with cutoff frequency regulated by K (Fig. 2). This noise-filtering ability could be useful to generate reliable responses to stochastic signals.

Figure 2.

(A) An amplitude Bode plot shows that a covalent modification cycle driven by accumulation of the activating enzyme behaves as a low-pass filter. The values |x(ω)|² and |N(ω)|² indicate the power spectrum of x and the noise, respectively. The data shown are for A = 4, n = 1, uncorrelated white-noise ξ, i.e., 〈ξ(t) ξ(t′)〉 = σ²δ(t–t′), where σ = 0.3 and δ(t–t′) indicates Dirac’s delta. The fit (red line) is to the equation describing a low-pass filter controlled by an exponential kernel, i.e., x(ω)/N(ω) = A/(ω₀² + ω²). (B) The cutoff frequency ω₀ scales proportionally to K. Note that ω₀ is dimensionless (expressed in 1/t₀ unit).

Receptor/ligand systems

We consider the case in which a ligand L and a receptor R interact to form an activated complex LR. The total amount of receptor is assumed to be constant, R_T, while the total amount of ligand accumulates over time, L_T(t). The association and dissociation are described simply by k_on and k_off parameters. In this situation the system is described by the following differential equation:

$$\frac{dLR}{dt}=k_{\text{on}}\left(L_T\left(t\right)-LR\right)\left(R_T-LR\right)-k_{\text{off}}LR.$$ (6)

This equation can be cast in dimensionless form by defining φ = LR/R_T and τ = t(k_onR_T + k_off), yielding

$$\frac{d\varphi }{d\tau }=\epsilon \left(\tau \right)-\left(1+\epsilon \left(\tau \right)\right)\varphi +\delta {\varphi }^2,$$ (7)

where ε(τ) is the dimensionless ligand dynamics expressed as a function of τ and δ = k_onR_T/(k_onR_T + k_off). For δ << 1, the behavior of the receptor/ligand system is very similar to the behavior of covalent modification cycles and will display similar ultrasensitivity and temporal integration properties. Numerical simulations demonstrate that Hill coefficients increase proportionally to the rate of accumulation of the ligand and to δ, although these increases are very small for δ ≈ 1 (Fig. 3). In fact, for δ ≈ 1, ultrasensitivity is observed in the linear case also for A < 1. These results indicate the existence of a large range of parameters in which a simple receptor/ligand system can generate ultrasensitivity.

Figure 3.

Ligand/receptor systems far from steady state can generate ultrasensitivity. Hill coefficient as a function of A and δ for n = 1 (A) and n = 7 (B).

Conclusions

We have shown that cells can achieve nonlinear responses and temporal integration of biochemical signals by driving covalent modification cycles (ligand/receptor) through the accumulation of the activating enzyme (ligand). These circuits are readily tunable, because ultrasensitivity and integration time can be controlled by adjusting the rate of accumulation of the activating enzyme and the concentration of the deactivating enzyme (receptor), as shown in our recent work on the control of mitosis during Drosophila gastrulation (8). Mitosis during Drosophila gastrulation is controlled by a covalent modification cycle regulating the activity of Cdk1, the master regulator of the cell cycle (8). Entry into mitosis is driven by accumulation of the phosphatase Cdc25^string, which activates Cdk1, while the kinase Wee1 inhibits it. Feedbacks are dispensable for the switchlike nature of mitosis and for setting the integration time (8). Instead, the inferred integration time is inversely proportional to the amount of Wee1 (8).

These observations suggest that mitosis during Drosophila gastrulation is controlled by an out-of-equilibrium covalent modification cycle and that integration time might be used to filter noise in the dynamics of the circuit components to increase precision. It must be observed that measuring the integration time of biochemical pathways in vivo is a difficult task due to the limited methodology for measuring the dynamics of enzyme activity and protein phosphorylation. In systems in which the dynamics of the covalent cycle drives a cellular transition, the integration time can be inferred statistically by determining the variable that provides the best fit of the timing of the transition (8). However, this analysis requires precise measurements of the timing of the cellular transition, because delays in such detection will appear statistically indistinguishable from integration time on similar timescales (8). On the other hand, time delays should not scale with changes in the concentration of deactivating enzyme as the integration time.

Measuring such scaling is the best experimental strategy to determine whether cells use covalent modification cycles to integrate signals over time. Future analysis of signaling pathways in cellular control systems will reveal the importance of temporal integration through covalent modification cycles and ligand/receptor.