Computational analysis reveals the coupling between bistability and the sign of a feedback loop in a TGF-β1 activation model

Abstract

Background: Bistable behaviors are prevalent in cell signaling and can be modeled by ordinary differential equations (ODEs) with kinetic parameters. A bistable switch has recently been found to regulate the activation of transforming growth factor-β1 (TGF-β1) in the context of liver fibrosis, and an ordinary differential equation (ODE) model was published showing that the net activation of TGF-β1 depends on the balance between two antagonistic sub-pathways.

Results: Through modeling the effects of perturbations that affect both sub-pathways, we revealed that bistability is coupled with the signs of feedback loops in the model. We extended the model to include calcium and Krüppel-like factor 2 (KLF2), both regulators of Thrombospondin-1 (TSP1) and Plasmin (PLS). Increased levels of extracellular calcium, which alters the TSP1-PLS balance, would cause high levels of TGF-β1, resembling a fibrotic state. KLF2, which suppresses production of TSP1 and plasminogen activator inhibitor-1 (PAI1), would eradicate bistability and preclude the fibrotic steady-state. Finally, the loop PLS - TGF-β1 - PAI1 had previously been reported as negative feedback, but the model suggested a stronger indirect effect of PLS down-regulating PAI1 to produce positive (double-negative) feedback in a fibrotic state. Further simulations showed that activation of KLF2 was able to restore negative feedback in the PLS - TGF-β1 - PAI1 loop.

Conclusions: Using the TGF-β1 activation model as a case study, we showed that external factors such as calcium or KLF2 can induce or eradicate bistability, accompanied by a switch in the sign of a feedback loop (PLS - TGF-β1 - PAI1) in the model. The coupling between bistability and positive/negative feedback suggests an alternative way of characterizing a dynamical system and its biological implications.

Keywords: Bifurcation analysis; Biochemical network; Bistability; Computational modelling; Dynamical systems; ODEs; Positive feedback; TGF-β1.

Fig. 1

TGF-β1 bistable activation model. Black arrows represent the reactions from [5]. Red arrows represent the effects of calcium on the PLS-TSP1 interaction. Blue arrows represent the effects of KLF2 on PAI1 and TSP1 production. uPA is urokinase plasminogen activator, and PLG is plasminogen

Fig. 2

Calcium and KLF2 have potential influence on the steady state of TGF-β1 activation. a-b TGF-β1 abundance is plotted over time, after initializing the system from a given set of initial concentrations with (a) low calcium or (b) high calcium. In both cases, curves have been colored blue if they converge to a steady state with low TGF-β1 activation (ssP), and colored red if they converge to a steady state with high TGF-β1 activation (ssT). c Calcium causes a shift in the separatrix between steady states. Low-calcium and high-calcium simulations were performed using various initial concentrations of PLS and TSP1. After observing in Fig. 2a-b that initial conditions with 0.5ssP and 0.5ssT were usually in the basin of convergence for the ssP state (for the low calcium model), we decided to shift the initial conditions. For studying the behavior of the separatrix, the initial concentrations were set to .25*ssP and .75*ssT. The steady state outcomes are shown by colors, with red indicating the steady state with high TGF-β1 (ssT), and blue indicating low TGF-β1 (ssP). For each combination of PLS and TSP1, the low calcium result is indicated by the color of the small inner dot, and the high-calcium result is indicated by the color of the outer circle. d Two steady state (s.s.) levels of TGF-β1 and PAI1 under different levels of KLF2. Red squares represent steady states with high TGF-β1 activation (ssT), while blue dots represent steady states with low TGF-β1 activation (ssP). The y-axis is the log of the steady state concentration of TGF-β1. e Bifurcation analysis under KLF2 low and KLF2 high conditions

Fig. 3

The bistability of the system correlates with the sign of the PLS-PAI1 feedback loop. a KLF2 affects the sign of PLS-PAI1 feedback loop. We designed an exogenous addition of PLS into the system using a step function for the level of PLS over time (top panel, black curve). Stimulating the TGF-β1 activation model with exogenous PLS caused two different effects in silico, depending on the KLF2 status. In the absence of KLF2 (red curve on left), the stimulus caused positive (double-negative) feedback between PAI1 and PLS, which can occur via the red arrows shown. In the presence of KLF2 (blue curve on right), exogenous PLS treatment caused a positive effect on PAI1 and the negative feedback loop (blue arrows) was restored. b Methods to calculate the 2d bistable region. We did bifurcation analysis of kp2 for a series of kp1 value (100 equally spaced values between kp1 = 1.2 and kp1 = 0) for the TGF-β1 activation model. The stable s.s. on the bifurcation curve were plotted as solid blue line, while the unstable s.s. were plotted as dotted red line. The Limit Points (LP), which tell the bistable interval of the bifurcation parameter (kp2) were denoted by black dots on the bifurcation curve. c The series of KLF2 levels cross the boundary of the bistable region in kp1-kp2 phase plane. The 2d bistable region (dark gray) in kp1-kp1 phase plane can be constructed based on the kp2 coordinates of LPs in (b) and their corresponding kp1 values. KLF2 levels are represented as diamond dots. d Overlap between bistable region and PLS-PAI1 positive feedback region in the kp1-kp2 phase plane. Bistable region is denoted by solid dark gray, while PLS-PAI1 positive feedback region is denoted by dotted light gray. The x and y axis are all in log scale. e Empirically, PLS can down-regulate PAI1 gene expression in a co-culture of hepatocytes and HSC-T6 cells. Hepatic stellate cells (HSC-T6) were cultured with primary rat hepatocytes in a 7:1 ratio overnight, creating a fibrosis-like state with high TGF-β1 level. The next day, the medium was changed to non-serum medium with different doses of PLS. Cells were collected 24 h later