Toggle switches, pulses and oscillations are intrinsic properties of the Src activation/deactivation cycle

Abstract

Src-family kinases (SFKs) play a pivotal role in growth factor signaling, mitosis, cell motility and invasiveness. In their basal state, SFKs maintain a closed autoinhibited conformation, where the Src homology 2 domain interacts with an inhibitory phosphotyrosine in the C-terminus. Activation involves dephosphorylation of this inhibitory phosphotyrosine, followed by intermolecular autophosphorylation of a specific tyrosine residue in the activation loop. The spatiotemporal dynamics of SFK activation controls cell behavior, yet these dynamics remain largely uninvestigated. In the present study, we show that the basic properties of the Src activation/deactivation cycle can bring about complex signaling dynamics, including oscillations, toggle switches and excitable behavior. These intricate dynamics do not require imposed external feedback loops and occur at constant activities of Src inhibitors and activators, such as C-terminal Src kinase and receptor-type protein tyrosine phosphatases. We demonstrate that C-terminal Src kinase and receptor-type protein tyrosine phosphatase underexpression or their simultaneous overexpression can transform Src response patterns into oscillatory or bistable responses, respectively. Similarly, Src overexpression leads to dysregulation of Src activity, promoting sustained self-perpetuating oscillations. Distinct types of responses can allow SFKs to trigger different cell-fate decisions, where cellular outcomes are determined by the stimulation threshold and history. Our mathematical model helps to understand the puzzling experimental observations and suggests conditions where these different kinetic behaviors of SFKs can be tested experimentally.

Fig. 1. Kinetic scheme of the Src activation/deactivation cycle

Four possible forms of the Src molecule are shown. S_i is the autoinhibited conformation where the inhibitory tyrosine residue is phosphorylated and the activatory residue is dephosphorylated, S is the partially active form where both the inhibitory and activatory residues are dephosphorylated, S_a1 is the fully active conformation where the inhibitory tyrosine residue is dephosphorylated and the activatory residue is phosphorylated, and S_a2 is the fully active form where both the inhibitory and activatory residues are phosphorylated. The solid lines with arrows present the Src cycle reactions catalyzed by the indicated enzymes. The dotted green lines specify intermolecular autophosphorylation reactions.

Fig. 2. Different types of QSS curve intersections determine the Src cycle steady states and dynamics

One stable steady state (O) or three steady states (stable O₁ and O₃ and unstable O₂) exist for both positive (panels A, C) and negative (B) slopes of the linear (blue) QSS curve (Eq.11), which intersects the Z-shaped (black) QSS curve (Eq.10). The parameter values are (A) k₁=0.2 s⁻¹ (line 1), 0.34 s⁻¹ (2) and 0.6 s⁻¹ (3), k₂=0.3 s⁻¹. (B) k₁=0.5 (1), 0.8 (2) and 1.5 (3), k₂=1 (all rate constants are in s⁻¹). C. A single unstable steady state (O) surrounded by a limit cycle (red), which corresponds to stable oscillatory pattern of Src activity, k₁=0.1, k₂=0.01, k₅=2, k₆=1 (s⁻¹). The resting state in vivo (s_i = 0.916, s₁ = s₂ = 7.32·10⁻⁵) was taken as the initial condition (“rest”), the movement direction is shown by arrows. For all curves in A – C the remaining parameters are, k₃=20, k₄=1, k₇=1 (s⁻¹), β=0.01, δ=0.05, ξ=1.

Fig. 3. Bistability and oscillations in the Src cycle

A. Hysteresis in steady-state responses of active Src fraction (s₁) to changes in the active RPTP concentration ([RPTP]). Dotted line corresponds to unstable steady states located at the intermediate branch of the curve between turning points P₁ and P₂ (marked bold). B. The time dependence of s₁ responses to stepwise changes in active [RPTP]; these changes are conditionally taken as 9 nM variations. Arrows in B show the time point of step changes in [RPTP]. The corresponding [RPTP] values, 117.5, 126.5 and 135.5 (nM), are indicated by dashed lines 1 - 3 in A and shown by upper line in B. The catalytic efficiency of RPTP (steps 1 and 6) is k^cat / K_M = 3.6·10⁻³ and 0.02 (nM⁻¹s⁻¹); the first-order rate constants, k₁ and k₆ are calculated as k^cat[RPTP]/K_M (Eq. 3); k₂=0.5, k₅=10 (s⁻¹). C. Sustained oscillations of Src fractions (s₁ – black, s₂ – red, s_i – black, s - blue). The time behavior corresponds to the limit cycle trajectory shown in Fig. 2C, arrows indicate the onset of stimulation, k = 0.1; k₂= 0.01; k₅=2, k₆=1 (s⁻¹). For all curves in A – C, the remaining parameters are given in the legend to Fig. 2.

Fig. 4. Src excitable behavior in response to rectangular pulse inputs (A, B) and perturbations to the initial concentrations (C, D)

Initially Src resides in a stable, but excitable steady state. For sub-threshold or over threshold stimuli, responses of the active Src fractions, s₁ and s₂, remain small or undergo large excursions generating high-amplitude responses, before returning to the same basal steady state. A. At time t₀ = 5 s (marked by arrow), the rate constant k₁ was increased from the basal level of 0.001 to 0.1 s⁻¹ (from point 1 on panel B to the level that corresponds to the unstable steady state, point 2). After time t₁ = t₀ + 9 s (bold line 1) or t₂ = t₀ + 10 s (bold line 2), k₁ was decreased to the basal level. The time-dependent responses of the active Src fractions, s₁ (black) and s₂ (blue) are shown by dashed and solid lines for 9 and 10 s stimulation periods, respectively. B. The trajectories (red) that correspond to the time-dependent responses in A and the QSS curves (black and blue) are shown in the plane of s₁ and s₂. C. At time t₀ = 5 s, a perturbation (Δs₁) to the steady state increased s₁ from 0.0082 to 0.03 (point 1) or 0.04 (point 2). Accordingly, the following equation was used for the total of the normalized concentrations, s_i + s + s₁ + s₂ = 1 + Δs₁ . The time-dependent responses to a sub-threshold perturbation (starting from point 1) and to a perturbation over threshold (starting from point 2) are shown by dashed and solid lines, respectively. D. The trajectories (red) that correspond to the time-dependent responses in C and the QSS curves (black and blue) are shown in the plane of s_i and s₁. k₁=0.03 s⁻¹. For all plots shown in A - D, the remaining parameters are given in the legend to Fig. 2C.

Fig. 5. Bifurcation diagrams unveil different Src dynamics

A. In the plane of active RPTP and Csk concentrations, bifurcation boundaries separate regions of different types of Src dynamics, determined by the Hopf (red lines) and saddle-node (black lines) bifurcations. These regions are numbered as follows, 1 – a single stable steady state, 2 – bistability domain, two stable states separated by a saddle, 3 – oscillations, a single unstable steady state, 4 – oscillations, three unstable steady states, 5 – one stable and two unstable steady states. Dashed line parallel to the [RPTP] axis crosses the plane at 25 nM [Csk]. Insert shows the zoomed-in region 4. B. One parameter bifurcation diagrams represent steady-state dependencies of Src active and inactive fractions s₁ and s_i on [RPTP] at four different constant [Csk] values, indicated near each curve (curves have different colors). Closed circles are turning points; dotted lines correspond to unstable steady states. Csk catalytic efficiency is, k^cat / K_M = 0.002 and 0.04 (nM⁻¹s⁻¹) for steps 2 and 5; the first-order rate constants, k₂ and k₅ are calculated as k^cat[Csk]/K_M (Eq. 3). The remaining parameters are the same as in the legend to Fig. 3.

Fig 6. Control of the period and amplitude of Src oscillations by the activities of the activatory phosphatase RPTP and inhibitory kinase Csk

Dependence of the oscillation amplitude (A) and period (B) on the active RPTP concentration at constant Csk concentration (25 nM). The amplitude is the difference between maximal (s_1max) and minimal (s_1min) values of the relative active Src fraction (red curves). Black solid line indicates stable steady states, whereas the dotted black line shows unstable steady states (steady state values are designated as s_1SS). Dependence of the oscillation amplitude (C) and period (D) on the active Csk concentration at constant RPTP concentration (30 nM). The parameter values are indicated in the legend to Fig. 5.

Fig. 7. Bifurcation diagram in the plane of the rate constant k₁ and total Src abundance

k₁ is the rate constant of dephosphorylation of inhibitory tyrosine in the Src C-terminus. Types of bifurcation boundaries and the numbering of regions with different Src dynamics are the same as in Fig. 5. Src autocatalytic efficiency is $k_{a1}^{\mathit{cat}}∕K_{a1}=0.05{\mathrm{nM}}^{-1}{\mathrm{s}}^{-1}$, $V_4^{\max }=400\mathrm{nM}∕s$, k₄=4 nM. The remaining parameters are the same as in the legend to Fig. 3. Insert shows the zoomed-in region 4.