Cell-cycle transitions: a common role for stoichiometric inhibitors

Abstract

The cell division cycle is the process by which eukaryotic cells replicate their chromosomes and partition them to two daughter cells. To maintain the integrity of the genome, proliferating cells must be able to block progression through the division cycle at key transition points (called "checkpoints") if there have been problems in the replication of the chromosomes or their biorientation on the mitotic spindle. These checkpoints are governed by protein-interaction networks, composed of phase-specific cell-cycle activators and inhibitors. Examples include Cdk1:Clb5 and its inhibitor Sic1 at the G1/S checkpoint in budding yeast, APC:Cdc20 and its inhibitor MCC at the mitotic checkpoint, and PP2A:B55 and its inhibitor, alpha-endosulfine, at the mitotic-exit checkpoint. Each of these inhibitors is a substrate as well as a stoichiometric inhibitor of the cell-cycle activator. Because the production of each inhibitor is promoted by a regulatory protein that is itself inhibited by the cell-cycle activator, their interaction network presents a regulatory motif characteristic of a "feedback-amplified domineering substrate" (FADS). We describe how the FADS motif responds to signals in the manner of a bistable toggle switch, and then we discuss how this toggle switch accounts for the abrupt and irreversible nature of three specific cell-cycle checkpoints.

FIGURE 1:

A generic network motif for the bistable switches discussed in this paper. (A) Influence diagram. In an “influence diagram,” barbed arrows (e.g., A → S) mean that “substance A activates substance S,” whereas blunt connectors (e.g., I ⊣ A) mean that “I inhibits A,” without any implications about the molecular mechanism of activation or inhibition. Such influence diagrams are often called “network motifs” because they indicate the “topology” of network interactions without specifying the biochemical mechanism. In this particular influence diagram, A activates S, which promotes a cell-cycle transition (CCT). The CCT is held off by I, which inhibits A. I, A, and are locked in a positive-feedback amplification loop (the red interactions), which is responsible for the bistable switching properties of the motif. The double-negative feedback loop between A and I is responsible for the nonlinear activation of A by a mechanism of “stoichiometric inhibitor ultrasensitivity,” as will be described later. The double-negative feedback loop by itself is not capable of generating bistability, which we indicate by using a dashed connector from A to I (A – – –| I). By activating , the “wait” signal (W) holds off the CCT and by driving the production of A, the “go” signal (G) promotes the CCT. The motif as a whole is a “signal processing” system (enclosed in the dotted box), which receives “wait” and “go” signals and determines whether the cell will pass the next cell-cycle transition (a binary decision). (B) Phase plane for the network motif in A. In Supplemental Text S1, we propose a simple mathematical model for the influence diagram in A. In the model, the inhibitor I has three forms: I_free, I_modified, and A:I complex, with [I_free] + [I_modified] + [A:I] = [I]_total = constant. Similarly, [A_F] + [A:I] = [A]_T = constant and [_F] + [_M] = []_T = constant, where we have introduced the abbreviations F for “free,” M for “modified,” and T for “total.” The mathematical model is described by a pair of ordinary differential equations for d[I_M]/dt and d[_F]/dt. In the “phase plane” (the Cartesian coordinate system spanned by [I_M] and [_F]), we plot the curves (called “nullclines”), where d[I_M]/dt = 0 (black curve) and where d[_F]/dt = 0 (blue curve), for a particular choice of parameter values (see Supplemental Text S1). The small arrows indicate the directions of the vector field along the nullclines. The nullclines intersect in three places (steady states): two stable steady states (•) separated by an unstable steady state (⚬). (C) Signal–response curve. In the simple mathematical model underlying the phase plane in B, the go-signal is the total concentration of A, and the response variable is the steady-state concentration of I_M (thinking of I_M as representative of all the substrates modified by A). In this diagram, solid green lines represent stable steady states of the bistable switch, and the black dashed line is the locus of unstable steady states. When the signal is small, 0 < [A]_T < 1.5, the concentration of free activator is very small, [A_F] ≈ 0, and substrates of the activator are sparsely modified (e.g., [I_M]/[I]_T < 0.25 in this figure). Hence, the lower branch of green curves represents the pretransition state. When the signal is large, [A]_T > 1.2, the concentration of free activator is large, [A_F] ≈ [A]_T, and substrates of the activator are heavily modified (e.g., [I_M]/[I]_T > 0.75), which represents the posttransition state. The system is “bistable” for 1.2 < [A]_T < 1.5. As [A]_T (the go-signal) increases from 0 toward a final value of 2 (blue curve), the system makes an abrupt transition from the pretransition state to the posttransition state at [A]_T ≈ 1.5. See Supplemental Text S1 for details of this calculation.

FIGURE 2:

Properties of reaction mechanisms that lack the positive-feedback amplification loop. (A) The reaction mechanism for stoichiometric inhibition of A by strong binding to I. In a reaction mechanism, as opposed to an influence diagram, solid arrows indicate chemical reactions (reactants at one end and products at the other), and dashed arrows indicate catalytic influences (enzyme at one end and catalyzed reaction at the other). In some instances, a dashed arrow points to a catalyzed “process” (e.g., CCT) rather than a specific chemical reaction. (B) The reaction mechanism by which a domineering substrate I is shut off by enzyme A, because A catalyzes the modification of I to an ineffectual form, I_M. (C) The reaction mechanism for regenerating the domineering substrate I by the action of a demodifying enzyme . (D) Ultrasensitive “signal–response” curve for S_M as a function of A_T, for the reaction mechanism in A. The dashed line is the limiting case of the signal–response curve derived in the main text, for J_A = 0.5. The solid curve is the ultrasensitive response curve for realistic values of rate constants, as derived in Supplemental Text S2. (E) For the reaction mechanism in B, I is steadily converted to I_M over the course of time, and eventually the beleaguered enzyme A is able to modify its intended substrate (S → S_M) and induce the cell-cycle transition. See Supplemental Text S3 for details.

FIGURE 3:

Bistability in the full network, with the positive-feedback amplification loop. (A) Reaction mechanism for a feedback-amplified domineering substrate. This reaction mechanism is a particular realization of the FADS motif illustrated in Figure 1A. The enzyme, W, that demodifies _M is a “transition wait” signal. (B) Phase plane diagrams for the bistable switch. For A_T = 0.5, there is a unique, stable steady state (•) with _F ≈ _T = 1, I_M ≈ 0.3 << I_T = 3, A_F ≈ 0. For A_T = 2, there is a unique, stable steady state (•) with _F ≈ 0, I_M ≈ I_T = 3, A_F ≈ 2. For A_T = 1, the network is bistable: there are two stable steady states (•) separated by an unstable steady state (⚬). See Supplemental Text S4 for details of the calculations in B, C, and D. (C) Signal–response curve for a bistable switch. The go-signal is the total concentration of A; the response is the concentration of active (free) A. Solid lines plot the loci of stable steady states as functions of [A]_T; dashed line plots the locus of unstable steady states. The network is bistable over the interval 0.70 < [A]_T < 1.74. (D) The toggle-switch domain in parameter space. The FADS motif is bistable within the wedge-shaped region bounded by the green curves. [A]_T is the go-signal for the cell-cycle transition; k_modR is the rate constant for the modification of by A. For k_modR = 0, the positive feedback amplification loop is broken, and the FADS motif is mono-stable.

FIGURE 4:

Cell-cycle transitions governed by FADS motifs. For each of the following three cases, the modeling details (reaction mechanisms, differential equations, parameter values, and phase plane diagrams) are given in Supplemental Texts S5–S7 and Supplemental Figures S1–S3. (A, B) The G1/S transition in budding yeast: motif and signal-response curve, respectively. The domineering substrate is Sic1, the beleaguered enzyme is Cdk1:Clb5, and the regenerating factor is Swi5. The steady-state activity of Cdk1:Clb5 is plotted as a function of the activity of Cdk1:Cln2, the go-signal for the G1/S transition in budding yeast. A newborn cell, with [Cdk1:Cln2] = 0, starts in the lower steady state (the G1 phase of the cell cycle; i.e., the pretransition state), with active Swi5, plenty of Sic1, and Cdk1:Clb5 silenced by binding to Sic1. The rise of Cln2-dependent kinase late in G1 phase triggers the irreversible transition into S phase (red curve), when Swi5 is silenced, Sic1 is degraded, and Clb5-dependent kinase activity is high. (C, D) The spindle assembly checkpoint (SAC): motif and signal–response curve, respectively. The domineering substrate is the MCC, the beleaguered enzyme is APC:Cdc20, and the regenerating factor is the mitotic Cdk activity (Cdk1:CycB). The steady-state activity of APC:Cdc20 is plotted as a function of the fraction of kinetochores that are unattached to the mitotic spindle (uKT, the wait-signal). As the cell enters prometaphase, it starts in the lower right corner of the diagram, with all kinetochores unattached, Cdk1:CycB activity high, MCC active, and APC:Cdc20 activity low (the SAC is on). As the cell proceeds through prometaphase, more and more kinetochores become correctly attached to the bipolar spindle, and uKT drops close to 0. For uKT small enough, the control system leaves the pretransition steady state and flips to the posttransition steady state (red curve), with APC:Cdc20 active, securin degraded, and cyclin B level dropping. The transition period comprises metaphase and early anaphase of the classical mitotic sequence; during this time it is possible to reverse the transition and force a cell back into a prometaphase-like state. However, at some point in late anaphase/telophase, the transition becomes irreversible, because Cdk1:CycB activity is so low that detached kinetochores can no longer activate the MCC. (E, F) Exit from mitosis: motif and signal–response curve, respectively. The domineering substrate is ENSA-P, the beleaguered enzyme is PP2A:B55, and the regenerating enzyme is Gwl-P. The steady-state activity of PP2A:B55 is plotted as a function of the activity of the mitotic cyclin-dependent kinase, Cdk1:CycB, the wait-signal for exit-from-mitosis. The late anaphase/telophase cell is waiting for [Cdk1:CycB] to be reduced sufficiently to allow PP2A:B55 to dephosphorylate Gwl-P, the regenerating enzyme. Then the control system fully activates PP2A:B55 (red curve), which dephosphorylates mitotic substrates and thereby returns the cell to G1 phase of the cell cycle.