^{1}and Bela Novak

^{1,a)}

### Abstract

Bistable switches have important roles in cellular decision-making processes. Bistability can be the consequence of positive or double-negative feedback loops. Although necessary, such feedback is not sufficient for bistability, which also requires nonlinearity. Nonlinearity can be provided by synergy of multiple feedback loops or by an ultrasensitive response within a single feedback loop. However, these two possibilities are not mutually exclusive; a combination of them is also possible. Here we analyze a biochemical regulatory network that controls a crucial cell cycle transition in all eukaryotic cells and contains multiple redundant feedback loops and nonlinearity. We show in this realistic biological example that two redundant feedback loops have different effects on the position of one of the saddle-node bifurcations of the system, which determines where the system switches. This illustrates that even though the roles of positive and double-negative feedbacks have been regarded as equivalent, the difference in their architectures can lead to differences in their effects on the system. We speculate that this conclusion could be general for other bistable systems with redundant feedback loops.

We develop a realistic model for an evolutionarily conserved cell cycle control network. The model explains the bistability of the network by multisite phosphorylation of the regulatory enzymes involved in positive and double-negative feedback loops. The two redundant feedback loops have different effects on the positions of saddle-node bifurcation points at the boundary of the bistable regime. We propose that this conclusion could be characteristic for other biological regulatory networks with redundant feedbacks.

We are grateful to the members of the Novak group for discussion. We thank John J. Tyson and Bernhard Schmierer for critically reading the manuscript. Our research is supported by BBSRC, EC (Unicellsys and Mitosys), and an Oxford Clarendon Fund Scholarship.

I. INTRODUCTION

II. CONTROL OF MITOTIC ENTRY

III. THE MPF BISTABLE SWITCH

IV. MPF SWITCH BASED ON MULTISITE PHOSPHORYLATION

V. THE ROLE OF THE INDIVIDUAL FEEDBACK LOOPS IN THE MPF BISTABLE SWITCH

VI. MUTATION ANALYSIS WITH DIFFERENT FEEDBACK STRENGTHS ON THE TWO LOOPS

VII. AN EXPLANATION FOR THE DIFFERENT EFFECTS OF THE WEE1 AND CDC25 FEEDBACK LOOPS

VIII. DISCUSSION

## Figures

The MPF bistable switch controlling mitotic entry. (a) Wiring diagram: MPF activates its activator (Cdc25) and inhibits its own inhibitor (Wee1), thereby creating a positive and a double-negative feedback loop. An unregulated phosphatase, PP, reverses MPF-dependent phosphorylation. Unphosphorylated Wee1 and Cdk1/CycB (MPF) and phosphorylated Cdc25 are the active forms of each species. Phosphorylated Wee1 and Cdk1/CycB (preMPF) as well as unphosphorylated Cdc25 are inacrive forms. (b) One-parameter bifurcation diagram: active MPF as a function of total cyclin level for the NT model [Eqs. (1)–(5)]. The solid lines represent stable steady states and the dashed line unstable steady states. The point labeled A shows the saddle-node bifurcation which represents the cyclin threshold for MPF activation. Point I shows the cyclin threshold for MPF inactivation. Parameter values: ; ; ; ; ; ; ; ; ; ; ; , ; .

The MPF bistable switch controlling mitotic entry. (a) Wiring diagram: MPF activates its activator (Cdc25) and inhibits its own inhibitor (Wee1), thereby creating a positive and a double-negative feedback loop. An unregulated phosphatase, PP, reverses MPF-dependent phosphorylation. Unphosphorylated Wee1 and Cdk1/CycB (MPF) and phosphorylated Cdc25 are the active forms of each species. Phosphorylated Wee1 and Cdk1/CycB (preMPF) as well as unphosphorylated Cdc25 are inacrive forms. (b) One-parameter bifurcation diagram: active MPF as a function of total cyclin level for the NT model [Eqs. (1)–(5)]. The solid lines represent stable steady states and the dashed line unstable steady states. The point labeled A shows the saddle-node bifurcation which represents the cyclin threshold for MPF activation. Point I shows the cyclin threshold for MPF inactivation. Parameter values: ; ; ; ; ; ; ; ; ; ; ; , ; .

The multisite phosphorylation model of the MPF bistable switch. (a) Wiring diagram: MPF phosphorylates both Wee1 and Cdc25 at five sites with an ordered and distributive mechanism (the opposing phosphatase PP also acts in an ordered and distributive manner). Phosphorylation inhibits Wee1 and activates Cdc25. We assume that Wee1 forms with 0 to 2 phosphorylations are active and those with 3–5 phosphorylations are inactive. The opposite is true for Cdc25. Active Wee1 phosphorylates MPF to yield its inactive form, preMPF. Active Cdc25 dephosphorylates preMPF to yield active MPF. (b) Steady state distribution of different phosphoforms as a function of the kinase/phosphatase (*k/h*) ratio for a protein with five phosphorylation sites, according to Eq. (9). (c) Concentration of active Cdc25 (Cdc25P) and Wee1, and cumulative activities of Cdc25 and Wee1 ( and , respectively) as a function of MPF activity, calculated using Eqs. (4), (5), (10), and (11). (d) One-parameter bifurcation diagram. Active MPF as a function of total cyclin . The solid lines represent stable steady states and the dashed line unstable steady states. Parameter values can be found in Table I.

The multisite phosphorylation model of the MPF bistable switch. (a) Wiring diagram: MPF phosphorylates both Wee1 and Cdc25 at five sites with an ordered and distributive mechanism (the opposing phosphatase PP also acts in an ordered and distributive manner). Phosphorylation inhibits Wee1 and activates Cdc25. We assume that Wee1 forms with 0 to 2 phosphorylations are active and those with 3–5 phosphorylations are inactive. The opposite is true for Cdc25. Active Wee1 phosphorylates MPF to yield its inactive form, preMPF. Active Cdc25 dephosphorylates preMPF to yield active MPF. (b) Steady state distribution of different phosphoforms as a function of the kinase/phosphatase (*k/h*) ratio for a protein with five phosphorylation sites, according to Eq. (9). (c) Concentration of active Cdc25 (Cdc25P) and Wee1, and cumulative activities of Cdc25 and Wee1 ( and , respectively) as a function of MPF activity, calculated using Eqs. (4), (5), (10), and (11). (d) One-parameter bifurcation diagram. Active MPF as a function of total cyclin . The solid lines represent stable steady states and the dashed line unstable steady states. Parameter values can be found in Table I.

Effect of mutations of all the MPF phosphorylation sites on Cdc25 or Wee1. (a) Elimination of sites: one-parameter bifurcation diagrams for the WT or normal case [same as Fig. 2(d)] and nonphosphorylatable Cdc25 (NP-Cdc25) or Wee1 (NP-Wee1). For NP-Wee1 only the Cdc25 positive feedback is functional. For NP-Cdc25 only the double-negative feedback operates. (b) Phospho-mimicking of all phosphorylation sites: one-parameter bifurcation diagrams for the WT or normal case [same as Fig. 2(d)], and phosphomimetic Cdc25 (PM-Cdc25) or Wee1 (PM-Wee1). This also results in a single functional feedback loop in the mutant cases but the activities of the mutant proteins are the opposite of (a). Parameters as in Table I except: , for NP-Cdc25; , for NP-Wee1; , for PM-Cdc25; , for PM-Wee1.

Effect of mutations of all the MPF phosphorylation sites on Cdc25 or Wee1. (a) Elimination of sites: one-parameter bifurcation diagrams for the WT or normal case [same as Fig. 2(d)] and nonphosphorylatable Cdc25 (NP-Cdc25) or Wee1 (NP-Wee1). For NP-Wee1 only the Cdc25 positive feedback is functional. For NP-Cdc25 only the double-negative feedback operates. (b) Phospho-mimicking of all phosphorylation sites: one-parameter bifurcation diagrams for the WT or normal case [same as Fig. 2(d)], and phosphomimetic Cdc25 (PM-Cdc25) or Wee1 (PM-Wee1). This also results in a single functional feedback loop in the mutant cases but the activities of the mutant proteins are the opposite of (a). Parameters as in Table I except: , for NP-Cdc25; , for NP-Wee1; , for PM-Cdc25; , for PM-Wee1.

Two-parameter bifurcation diagrams for the rates of activation and inactivation of Cdc25 and Wee1. The loci of saddle-node bifurcations (cyclin thresholds for MPF activation and inactivation) are plotted as a function of kinetic parameters influencing the feedback loops. (a) , (b) , (c) , and (d) . The dashed line in each case shows the value of the parameter used in the standard model, specified in Table I.

Two-parameter bifurcation diagrams for the rates of activation and inactivation of Cdc25 and Wee1. The loci of saddle-node bifurcations (cyclin thresholds for MPF activation and inactivation) are plotted as a function of kinetic parameters influencing the feedback loops. (a) , (b) , (c) , and (d) . The dashed line in each case shows the value of the parameter used in the standard model, specified in Table I.

Effect of elimination of all the MPF phosphorylations sites on Cdc25 or Wee1 with unequal strength of the two feedback loops. The plots on the right [(a), (c),and (e)] show the concentrations of Wee1 and Cdc25 with decreasing values of the rate constant for Cdc25 activation by MPF and the plots on the left [(b), (d), and (f)] the corresponding one-parameter bifurcation diagrams for the WT with normal phosphorylation sites on both Wee1 and Cdc25 and the nonphosphorylatable mutants NP-Cdc25 and NP-Wee1. In (d) and (f), the fourth curve also shows the original wild-type curve, which corresponds to the wild-type in (b). Therefore, (a) and (b) are the same as Figs. 2(c) and 3(a), respectively. (c) and (d) are the same as (a) and (b) except that . (e) and (f) are the same as (a) and (b) except that .

Effect of elimination of all the MPF phosphorylations sites on Cdc25 or Wee1 with unequal strength of the two feedback loops. The plots on the right [(a), (c),and (e)] show the concentrations of Wee1 and Cdc25 with decreasing values of the rate constant for Cdc25 activation by MPF and the plots on the left [(b), (d), and (f)] the corresponding one-parameter bifurcation diagrams for the WT with normal phosphorylation sites on both Wee1 and Cdc25 and the nonphosphorylatable mutants NP-Cdc25 and NP-Wee1. In (d) and (f), the fourth curve also shows the original wild-type curve, which corresponds to the wild-type in (b). Therefore, (a) and (b) are the same as Figs. 2(c) and 3(a), respectively. (c) and (d) are the same as (a) and (b) except that . (e) and (f) are the same as (a) and (b) except that .

Rates of MPF activation and inactivation as a function of MPF. The rate of MPF activation, , is plotted at three different levels. The rate of MPF inactivation, , does not change with cyclin level. Wherever the rate curves intersect with each other, the system is in steady state since the rates are equal. The steady state is stable (filled circles) if the activation rate is higher than the inactivation rate to the left of the steady state and it is unstable (open circle) if the opposite is true. The rates were calculated with Eqs. (12) and (13). Parameter values can be found in Table I and values are specified for the relevant curves.

Rates of MPF activation and inactivation as a function of MPF. The rate of MPF activation, , is plotted at three different levels. The rate of MPF inactivation, , does not change with cyclin level. Wherever the rate curves intersect with each other, the system is in steady state since the rates are equal. The steady state is stable (filled circles) if the activation rate is higher than the inactivation rate to the left of the steady state and it is unstable (open circle) if the opposite is true. The rates were calculated with Eqs. (12) and (13). Parameter values can be found in Table I and values are specified for the relevant curves.

## Tables

Parameter values for the MPF switch model based on multisite phosphorylation. All rate constants (’s) have a dimension of , while all the others are dimensionless. These parameter values were used for the standard model, unless specified otherwise.

Parameter values for the MPF switch model based on multisite phosphorylation. All rate constants (’s) have a dimension of , while all the others are dimensionless. These parameter values were used for the standard model, unless specified otherwise.

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