Schematic plot of cell cycle dynamics. The x-axis represents the controlling parameter p (such as cyclin synthesis rate k₁) that controls the dynamics of the cell cycle. The y-axis represents the activated Cyclin:CDK complex driving the cell cycle. The unfilled circle is the first Hopf bifurcation point, which acts as the checkpoint in the cell cycle. See text for further details.

(A) Signaling networks for cyclin and CDK regulation, (B) SKP2 regulation, and (C) APC/CDC20 regulation. The solid lines with arrowheads indicate synthesis or degradation of a protein, or a biochemical reaction step. The dashed lines indicate that the reaction is catalyzed by the protein or protein complex connected to the line. The dotted lines indicate the phosphorylation, and thus dephosphorylation may involve multiple steps, as described in the text. Small gray spheres represent degraded protein molecules.

Simulation of the R point experiment. At t = 1300 (vertical dotted line with arrow), one cell (solid line) passed the R point, but the other (dashed line) had not. Growth for both cells was halted for a time period of 50 time units to simulate growth arrest after brief serum starvation. (A) Cell size (s) versus time (t). Note that the first mitosis is delayed for the cell which did not pass the R point, but the second mitosis is not delayed. (B) Active Cyclin:CDK (x) versus time. Numbers indicate the division times of the two cells.

Cell size (above) and active Cyclin:CDK activity (below, both arbitrary units) versus time. (A) Control. (B) The wee1 synthesis rate k₁₀ was reduced from 10 to 1 to simulate the wee1 mutation. Note the smaller cell size. (C) Cycle time versus birth size for the control case with k₁₀ = 10 (unfilled circles) and for the wee1 mutation with k₁₀ = 1 (solid circles). Inset shows how cell size s affects the value of the cyclin synthesis rate k₁ at the time of cell division, as in Eq. 2.

(A) Steady states (lines) and limit cycle maxima/minima (circles) of active Cyclin:CDK vs. cyclin synthesis rate k₁ without the negative feedback (gray lines) and with the negative feedback (black line and circles). Solid lines are stable steady states; dashed lines are unstable steady states. SN1 and SN2 mark the saddle-node bifurcations without negative feedback present, and H1 and H2 mark the two Hopf bifurcations with negative feedback present. (B) Active Cyclin:CDK (thick solid line), free cyclin (dashed line), and total cyclin (thin solid line) versus time for k₁ = 300 in the limit cycle regime shown in A. (C) Active Cyclin:CDK versus time for k₁ = 150 in the excitable regime in A. At external stimulus was applied at t = 600 for a duration of 0.5 time units.

Saddle-node and Hopf bifurcations leading to bistability and limit cycles, respectively. (A) Schematic illustration of a stable node (solid circle) and saddle point (unfilled circle) in a two-variable space, such as a space of active Cyclin:CDK and free cyclin. Arrows represent trajectories. (B) Illustration of bistability and hysteresis in a variable-parameter space. Solid lines and solid circles are for stable nodes. Dashed line and unfilled circle are for saddle points. (C) Schematic illustration of a stable focus (solid circle), unstable focus (unfilled circle), and limit cycle (thick large circle). A Hopf bifurcation occurs when the steady state changes from a stable focus to an unstable focus. (D) Hopf bifurcation and limit cycle in a variable-parameter space. H1 is the first Hopf bifurcation point and H2 is the second. Solid circles and solid lines indicate the stable focus, and unfilled circle and dashed line the unstable focus. Unfilled squares are the maxima of the variable during limit cycle oscillation, and solid squares the minima.

Phase diagrams in two parameter planes, with other parameters fixed. LOW is stably low kinase activity, HIGH is stably high kinase activity, LC is limit cycle oscillation, and BS is bistability. The oscillation period of limit cycle is color-coded in the limit cycle region. (A) In the plane of cyclin synthesis rate (k₁) and cyclin degradation rate constant (k₂) without the negative feedback. (B) In the plane of cyclin synthesis rate (k₁) and cyclin degradation rate constant (k₂) with negative feedback exerted only on free cyclin; k_2u = 50 and k_7u = 0. (C) In the plane of cyclin synthesis rate (k₁) and cyclin degradation rate constant (k₂) with the negative feedback exerted on both free cyclin and active Cyclin:CDK complex; k_2u = 25 and k_7u = 50. (D) In the plane of cyclin synthesis rate (k₁) and rate constant of negative feedback (k_2u). (E) In the plane of cyclin synthesis rate (k₁) and time delay (τ) for negative feedback. (F) In the plane of cyclin synthesis rate (k₁) and CDC25 synthesis and degradation ratio (k₈/k₉), by either changing CDC25 synthesis rate (k₈) (LC in the colored area) or by changing CDC25 degradation rate constant (k₉) (LC between the two black lines). (G) In the plane of cyclin synthesis rate (k₁) and wee1 synthesis rate (k₁₀). (H) In the plane of cyclin synthesis rate (k₁) and CKI synthesis rate (k₁₂). (D–G) With negative feedback exerted only on free cyclin; k_2u = 50 and k_7u = 0. A–G. No CKI (k₁₂ = 0). (H) With CKI (k₁₂ > 0). The color coding for the period of the limit cycle is shown as the color bar.