Fig 1.
Steady-state input-output characteristics.
Relationships for linear, Michaelian and ultrasensitive systems.
Fig 2.
Representations of the yeast osmosensing system.
(A) Schematic depiction of the osmosensing response. After an osmotic stress, the external osmotic pressure increases and water diffuses out of the cell, causing the turgor pressure and volume to decrease. Two parallel control paths are activated to regain volume and turgor pressure by adjusting the glycerol production: the activation of the Hog1 protein and all the corresponding mechanisms that promote glycerol production; the Fps1 channel, which regulates the outflow of glycerol and is immediately closed after the shock. (B) Engineering block diagram representation of a control model for the osmoregulation system.
Fig 3.
Best fit to osmoshocks for the UNF-UNF model.
Best fit to the experimental dataset for the cell volume (A) and the Hog1 (B) responses to three step osmoshocks of different magnitude; the experimental data for 0.2, 0.4, and 0.6 M of NaCl are indicated by black circles, red diamonds, and blue squares, respectively. The corresponding coloured solid lines represent the optimised model responses.
Table 1.
Different control schemes exploited for reproducing experimentally observed responses of yeast to different levels of osmoshock.
Fig 4.
Temporal dynamics of the controllers for the different models.
The output of the Fps1 channel (A), the glycerol production (B), the glycerol concentration (C), and the error (D) are shown for the different control schemes (PNF-PNF, PNF-INF, UNF-UNF as indicated in the legend), assuming an osmotic stress of 0.4 M of NaCl.
Fig 5.
Performance of the generic closed-loop feedback system.
(A) Block diagram representation of the generic control system model. (B) Steady state error for the generic closed-loop system, while varying the parameter a of the process, with the process parameter b, reference signal r and step disturbance amplitude aud held constant (b = r = 1, aud = 0.2). Intersections of the magenta line with the black dashed straight lines give steady state error values for the UNF controller (K = 0.01, n = 2 and kp = 1). Intersections of the green lines with the black dashed straight lines give steady state error values for the PNF controller (dashed-dotted line for kp = 1; dotted line for kp = 50). (C) Steady state error for the closed-loop system, while varying the amplitude aud of the step disturbance, ud, with different values of kp (red and magenta lines for kp = 1; orange line for kp = 2) and K (red line for K = 0.1; magenta and orange lines for K = 0.01) for the UNF controller (n is fixed at 2).
Fig 6.
Performance of the generic closed-loop feedback system using SMC, UNF, and its approximation by the piecewise function upw.
(A-C) Response dynamics obtained using: (A-B) SMC, a sliding mode controller (u(t) = kpsgn(e)) for different values of the gain kp and y(0) = y0; (C) UNF, its approximation by the piecewise function upw and the ideal SMC, with kp = 0.25, n = 2 and different values of K. The parameters a, b and the constant reference signal r are set equal to 1. The system output is initially equal to the desired constant reference value y(0) = y0 = r (A and C), whereas y(0) = y0 = 0 (B). A step disturbance, ud, with amplitude aud = 0.2 (A and C), aud = −0.2 (B), is applied at time t = 1. (D) Input-output (I-O) relationships for the UNF controller (solid magenta plot), its approximation by the saturation function sat(m ⋅ e) with m = n/(4K) (dashed blue plot), and the ideal SMC (i.e. the discontinuous nonlinearity sgn(e)—dashed-dotted green plot). When n = 2, then the saturation function sat(e/(2K)) is equal to the piecewise function upw that approximates UNF (see Eqs (A38)–(A39) in S1 Appendix).