School of Molecular Biosciences, Washington State University, PO Box 647520, Pullman, WA 99164, USA

Biological Systems Engineering, Washington State University, Pullman, WA 99164, USA

Center for Reproductive Biology, Washington State University, Pullman, WA 99164, USA

Abstract

Background

Meiosis is the sexual reproduction process common to eukaryotes. The diploid yeast

Results

The mathematical model is capable of simulating the orderly and transient dynamics of meiotic proteins including Ime1, the master regulator of meiotic initiation, and Ime2, a kinase encoded by an early gene. The model is validated by quantitative sporulation phenotypes of single-gene knockouts. Thus, we can use the model to make novel predictions on the cooperation between proteins in the signaling pathway. Virtual perturbations on feedback loops suggest that both positive and negative feedback loops are required to terminate expression of early meiotic proteins. Bifurcation analyses on feedback loops indicate that multiple feedback loops are coordinated to modulate sporulation efficiency. In particular, positive auto-regulation of Ime2 produces a bistable system with a normal meiotic state and a more efficient meiotic state.

Conclusions

By systematically scanning through feedback loops in the mathematical model, we demonstrate that, in yeast, the decisions to terminate protein expression and to sporulate at different efficiencies stem from feedback signals toward the master regulator Ime1 and the early meiotic protein Ime2. We argue that the architecture of meiotic initiation pathway generates a robust mechanism that assures a rapid and complete transition into meiosis. This type of systems-level regulation is a commonly used mechanism controlling developmental programs in yeast and other organisms. Our mathematical model uncovers key regulations that can be manipulated to enhance sporulation efficiency, an important first step in the development of new strategies for producing gametes with high quality and quantity.

Background

The diploid yeast

Many key players and their interactions that control yeast meiotic initiation have now been identified (see Figure

A signaling pathway that controls yeast meiotic initiation

**A signaling pathway that controls yeast meiotic initiation.** Proteins enclosed in an oval are model variables. Phosphorylated proteins are labeled with the letter P in a grey circle. Solid lines represent phosphorylation, dephosphorylation, or degradation; dashed lines indicate regulatory interactions between proteins. The arrow at the end of a dashed line depicts activation; the bar at the end of a dashed line shows repression.

Inactivation of PKA under meiotic conditions leads to enhanced activity of Rim11, a kinase that phosphorylates Ime1 and Ume6

Because of complex feedback regulation on the meiotic initiation pathway, mathematical modeling becomes an important tool to understand dynamic behaviors of signaling molecules and how their interactions ensure different degrees of sporulation efficiency. Feedback controls, which link the output of a circuit back to its input, are a key mechanism to stabilize cell-fate decision. Both experimental data and computational modeling suggest important roles of feedback loops in regulating mitotic entry and exit, cell growth, cell cycle, and pheromone pathways

Boolean network models—a discrete method—have been developed to simulate the dynamics of meiotic initiation pathways. One study focuses on predicting sporulation efficiency upon gene deletions, and the other explains transient transcription of

Results

A signaling pathway that controls yeast meiotic initiation

We construct a signaling pathway to describe the initial phase of yeast meiosis based on the literature (Figure

**Feedback**

**Type**

Ime1 —| pSok2 —| Ime1

Double-negative feedback

Ime1 — > pIme1 — > Ime2 —| Ime1

Negative feedback

Ime2 — > Ime2

Positive feedback

The constructed model is an abstract of real pathways, incorporating major players and events. The effects of other molecules are reflected indirectly in the model. For example, we assume PKA remains constant at a low level under meiotic conditions

Using mitotic initial conditions (all variables are 0 except pSok2 with a value of 1) and baseline parameter values (Table

**Parameter**

**Definition**

**Value**

**Reference**

Synthesis, dimension =/hour

_{
ime1
}

Synthesis rate of Ime1

10

_{
ime2
}

Synthesis rate of Ime2

10

^{
’
}
_{
ime2
}

The maximum rate of auto-regulation-dependent Ime2 synthesis

3

Estimated

Degradation, dimension =/hour

_{
ime1
}

Degradation rate of Ime1

1

^{
’
}
_{
ime1
}

The maximum rate of Ime2-activated Ime1 degradation

1

_{
pime1
}

Degradation rate of pIme1

1

_{
ime2
}

Degradation rate of Ime2

8

Phosphorylation, dimension =/hour

_{
rim11
}

Phosphorylation rate of Rim11

0.01

_{
rim11
}

Dephosphorylation rate of Rim11

0.1

_{
ume6
}

Phosphorylation rate of Ume6

0.3

_{
ume6
}

Dephosphorylation rate of Ume6

0.01

_{
sok2
}

Phosphorylation rate of Sok2

0.7

_{
sok2
}

Dephosphorylation rate of Sok2

1

_{
ime1
}

Phosphorylation rate of Ime1

2

Estimated

Constant, dimensionless

_{
sok2
}

Constant measuring half-maximum inhibition of Sok2 phosphorylation by Ime1

0.05

Estimated

_{
ime1
}

Constant measuring half-maximum inhibition of Ime1 synthesis by pSok2

0.01

Estimated

_{
1
}

Constant measuring half-maximum activation of Ime1 degradation by Ime2

0.01

Estimated

_{
2
}

Constant measuring half-maximum activation of Ime2 synthesis through auto-regulation

1.4

Estimated

_{
3
}

Constant measuring half-maximum activation of Ime2 degradation

2

Estimated

Numerical simulations of protein levels in wild type

**Numerical simulations of protein levels in wild type.** The mathematical model includes Equations 1, 2, 3, 4, 5 and 6. Initial condition of all variables is 0 except for pSok2 with a value of 1. Parameter values are listed in Table

**Variable**

**Steady state value**

Rim11

0.91

pUme6

0.96

pSok2

0.25

Ime1

0.05

pIme1

0.10

Ime2

0.27

**InitialConditions.xls.** Randomization of initial conditions.

Click here for file

Model validation by sporulation-deficient and proficient genes

High-throughput screens of ~4,000 yeast deletion strains have identified 267 genes required for sporulation (sporulation-deficient genes) and 102 genes whose disruption enhances sporulation efficiency (sporulation-proficient genes)

**Debjit031213S.pdf.** Tables S1,S2, S3, S4, S5, S6, S7 and S8, Figures S1, S2, S3, S4, S5, S6, S7, S8 and S9.

Click here for file

Ime2 is used as the model readout for sporulation phenotypes because it is the most downstream protein that reflects changes in all others in the pathway. We find that Ime2 levels remain at zero in the knockout models of sporulation-deficient genes

Model validation by simulating single-gene knockouts

**Model validation by simulating single-gene knockouts. ****A**) Ime2 dynamics when deleting genes individually in the model. Red dashed line: **B**) Pearson correlation between experimental sporulation/pre-sporulation ratios and simulated Ime2 values at steady state for five single-gene knockouts and wild type.

A global analysis of parameter sensitivity

When

Analyses of global parameter sensitivity

**Analyses of global parameter sensitivity.** Global effects of parameters on model variables are investigated by simultaneously varying all 19 parameters within a range of one order of magnitude larger and smaller than baseline values. A random sample of each parameter is generated from the range with a uniform distribution; the sampling is performed 5,000 times to calculate parameter sensitivities. Sensitivity value is between 0 and 1; the larger the value, the more important a parameter is to the output of a variable.

The overall pattern indicates that early meiotic proteins are sensitive to parameters that directly regulate their homeostasis. Levels of Rim11, pUme6, and pSok2 are mainly affected by phosphorylation and dephosphorylation. Rim11 and pSok2 are more sensitive to dephosphorylation than phosphorylation (_{
rim11
}, _{
sok2
}), but the opposite is true for pUme6 (_{
ume6
}). The findings are consistent with the active forms of these proteins in meiosis. Synthesis and phosphorylation are most important to alter the Ime1 level (_{
ime1
}, _{
ime1
}); both processes directly determine the gain or loss of Ime1. The level of pIme1 is primarily modulated by the synthesis of Ime1 and degradation of Ime2 (_{
ime1
}, _{
ime2
}). Ime1 synthesis indirectly controls pIme1 through regulating Ime1; Ime2 degradation indirectly influences pIme1 through Ime2-activated Ime1 degradation. Parameters that control Ime2 auto-regulation and degradation have the greatest influence on Ime2 variations (^{
’
}
_{
ime2
}, _{
ime2
}).

Feedback loops control transient expression of signaling molecules

Feedback regulations are important for coordinated and transient behaviors of developmental systems

**
Double-negative feedback between pSok2 and Ime1
**

Phosphorylated Sok2 is an upstream repressor of Ime1, and, conversely, Ime1 inhibits Sok2 phosphorylation, forming a double-negative feedback loop. We first evaluate the effect of inhibition from pSok2 to Ime1, by varying _{
ime1
}, the constant measuring half-maximum inhibition of Ime1 synthesis by pSok2 (Figure _{
sok2
}, the constant measuring half-maximum inhibition of Sok2 phosphorylation by Ime1 (Figure _{
ime1
} and _{
sok2
} (Figure

Simulation analyses of double-negative feedback loop between pSok2 and Ime1

**Simulation analyses of double-negative feedback loop between pSok2 and Ime1. ****A**) Ime1 and Ime2 dynamics when varying _{ime1}, the constant measuring half-maximum inhibition of Ime1 synthesis by pSok2. Red curve: _{ime1} = ∞ (deleting the inhibition); cyan curve: _{ime1} = 0.1 (decreasing the inhibition); blue curve: _{ime1} = 0.001 (increasing the inhibition); green curve: _{ime1} = 0.01 (baseline value). **B**) Ime1 and Ime2 dynamics when varying _{sok2}, the constant measuring half-maximum inhibition of Sok2 phosphorylation by Ime1. Red curve: _{sok2} = ∞ (deleting the inhibition); cyan curve: _{sok2} = 0.5 (decreasing the inhibition); blue curve: _{sok2} = 0.005 (increasing the inhibition); green curve: _{sok2} = 0.05 (baseline value). **C)** Ime1 and Ime2 dynamics when varying _{ime1} and _{sok2} simultaneously. Red curve: _{ime1} = ∞ and _{sok2} = ∞ (deleting the feedback loop); cyan curve: _{ime1} = 0.1 and _{sok2} = 0.5 (decreasing the feedback loop); blue curve: _{ime1} = 0.001 and _{sok2} = 0.005 (increasing the feedback loop); green curve: _{ime1} = 0.01 and _{sok2} = 0.05 (baseline value). The mathematical model of the feedback knockout is described in Additional file

**
Negative feedback from Ime2 to Ime1
**

Protein destruction is a commonly used mechanism controlling cell cycle transitions _{
ime1
}, the maximum rate of Ime2-activated Ime1 degradation, the negative feedback is enhanced (blue curve). Both amplitude and duration of Ime1 peak decrease, which lead to no expression of Ime2 (Figure _{
ime1
} (cyan and red curves), we observe not only increased peak height but also increased peak width for both Ime1 and Ime2. Ime2 rises to infinity in the feedback knockout model (Figure _{
1
}, the constant measuring half-maximum activation of Ime1 degradation by Ime2 (Figure _{
ime1
} and _{
1
} simultaneously (Figure _{
ime1
}, suggesting that the negative feedback ensures transient expression of both Ime1 and Ime2.

Simulation analyses of negative feedback loop from Ime2 to Ime1

**Simulation analyses of negative feedback loop from Ime2 to Ime1. ****A**) Ime1 and Ime2 dynamics when varying _{ime1}, the maximum rate of Ime2-activated Ime1 degradation. Red curve: _{ime1} = 0 (deleting the feedback loop); cyan curve: _{ime1} = 0.1 (decreasing the feedback loop); blue curve: _{ime1} = 10 (increasing the feedback loop); green curve: _{ime1} = 1 (baseline value). **B**) Ime1 and Ime2 dynamics when varying _{1}, the constant measuring half-maximum activation of Ime1 degradation by Ime2. Red curve: _{1} = ∞ (deleting the feedback loop); cyan curve: _{1} = 0.1 (decreasing the feedback loop); blue curve: _{1} = 0.001 (increasing the feedback loop); green curve: _{1} = 0.01 (baseline value). **C**) Ime1 and Ime2 dynamics when varying _{ime1} and _{1} simultaneously. Red curve: _{ime1} = 0 and _{1} = ∞ (deleting the feedback loop); cyan curve: _{ime1} = 0.1 and _{1} = 0.1 (decreasing the feedback loop); blue curve: _{ime1} = 10 and _{1} = 0.001 (increasing the feedback loop); green curve: _{ime1} = 1 and _{1} = 0.01 (baseline value). The mathematical model of the feedback knockout is described in Additional file

**
Auto-positive feedback of Ime2
**

Multiple lines of evidence support positive auto-regulation of Ime2: transcriptional activation of its own expression _{
ime2
}, the maximum rate of auto-regulation-dependent Ime2 synthesis (Figure _{
2
}, the constant measuring half-maximum activation of Ime2 synthesis through auto-regulation (Figure _{
ime2
} than in _{
2
}, consistent with the global analysis of parameter sensitivity (Figure

Simulation analyses of positive feedback loop of Ime2

**Simulation analyses of positive feedback loop of Ime2. ****A**) Ime1 and Ime2 dynamics when varying _{ime2}, the maximum rate of auto-regulation-dependent Ime2 synthesis. Red curve: _{ime2} = 0 (deleting the feedback loop); cyan curve: _{ime2} = 0.3 (decreasing the feedback loop); blue curve: _{ime2} = 30 (increasing the feedback loop); green curve: _{ime2} = 3 (baseline value). **B**) Ime1 and Ime2 dynamics when varying _{2}, the constant measuring half- maximum activation of Ime2 synthesis through auto-regulation. Red curve: _{2} = ∞ (deleting the feedback loop); cyan curve: _{2} = 14 (decreasing the feedback loop); blue curve: _{2} = 0.14 (increasing the feedback loop); green curve: _{2} = 1.4 (baseline value). **C**) Ime1 and Ime2 dynamics when varying _{ime2} and _{2} simultaneously. Red curve: _{ime2} = 0 and _{2} = ∞ (deleting the feedback loop); cyan curve: _{ime2} = 0.3 and _{2} = 14 (decreasing the feedback loop); blue curve: _{ime2} = 30 and _{2} = 0.14 (increasing the feedback loop); green curve: _{ime2} = 3 and _{2} = 1.4 (baseline value). The mathematical model of the feedback knockout is described in Additional file

Feedback loops control sporulation efficiency

Feedback regulations are known to control cell fate decision

**
Double-negative feedback between pSok2 and Ime1
**

Mutual antagonism between pSok2 and Ime1 is controlled by _{
ime1
} and _{
sok2
}, half-maximum inhibition constants. Varying either _{
ime1
} or _{
sok2
} produces two stable steady states separated by an unstable steady state (Figure _{
ime1
} = 0.01, _{
sok2
} = 0.05), the default equilibrium value of Ime2 is obtained (Ime2 = 0.27). When _{
ime1
} increases or _{
sok2
} decreases, implying that Ime1 wins over pSok2, Ime2 reaches a higher stable state. This higher state indicates that sporulation efficiency can be improved by manipulating double-negative feedback loop between pSok2 and Ime1.

Bifurcation analyses of double-negative feedback loop between pSok2 and Ime1 and negative feedback loop from Ime2 to Ime1

**Bifurcation analyses of double-negative feedback loop between pSok2 and Ime1 and negative feedback loop from Ime2 to Ime1. ****A**) Steady state value of Ime2 as a function of _{ime1}, the constant measuring half-maximum inhibition of Ime1 synthesis by pSok2. **B**) Steady state value of Ime2 as a function of _{sok2}, the constant measuring half-maximum inhibition of Sok2 phosphorylation by Ime1. **C**) Steady state value of Ime2 as a function of _{1}, the constant measuring half-maximum activation of Ime1 degradation by Ime2. Red segments represent stable steady states, whereas black segments trace unstable steady states.

PKA mediates phosphorylation of Sok2 and Rim11. To examine whether PKA is also a bifurcation parameter, we vary the phosphorylation rates of Sok2 and Rim11 simultaneously (Additional file

**
Negative feedback from Ime2 to Ime1
**

Parameter _{
1
} is the half-maximum constant of Ime2 inhibition on Ime1. Changing _{
1
} results in two stable steady states, separated by a very short segment of unstable steady state (Figure _{
1
} = 0.01) leads to the default Ime2 equilibrium (Ime2 = 0.27). The inhibition from Ime2 to Ime1 decreases with the increase of _{
1
}, producing enhanced Ime2 level. This indicates that sporulation efficiency can be improved by repressing negative feedback from Ime2 to Ime1.

**
Auto-positive feedback of Ime2
**

Auto-regulation of Ime2 is approximated by a Hill function with the coefficient of 5. We use high Hill coefficient to define cooperative and ultrasensitive regulatory processes because this feedback loop represents not only auto-regulation but also multiple interactions between Ime2 and other molecules (e.g., Cln3/Cdc28, Ndt80). Parameter _{
2
} is the half-maximum constant of Ime2 auto-regulation. Plotting Ime2 as a function of _{
2
} (Figure _{
2
} = 1.4). When _{
2
} is less than 0.5, the auto-regulation is enhanced, which leads to a higher steady state response of Ime2. With _{
2
} in the region of 0.5-0.7, the system becomes bistable. Ime2 can take two different values, characterizing states of default and higher sporulation efficiency. These two stable states can be reached for the same set of parameters depending on initial conditions.

Bifurcation analyses of positive feedback loop of Ime2

**Bifurcation analyses of positive feedback loop of Ime2.** Steady state value of Ime2 as a function of _{2}, the constant measuring half-maximum activation of Ime2 synthesis through auto-regulation. The use of different Hill coefficients,

The Hill coefficient determines the switch-like behavior of Ime2 equilibrium. We find that the range of _{
2
} in which the system exhibits bistability is sensitive to the Hill coefficient (Figure _{
2
} corresponds to a single value of Ime2. Higher coefficients result in the transition from a monostable to a bistable system. A Hill coefficient of 7 expands the region of bistability across a broad range of parameter space, making the cell fate more robust with respect to perturbations in the feedback loop. This result indicates that cooperativity of Ime2 molecules is essential for producing bistable sporulation outcomes.

Discussion

Precise regulation of a gene cascade in a coordinated manner is required for initiating a developmental program at the right time. This is often achieved through the activation of an upstream master regulator, which is controlled by multiple input signals and further regulates expression of downstream genes. Downstream genes, in turn, feed back to the regulator to modulate the entire pathway activity. The combinational nature of feedback loops ensures correct temporal dynamics of a developmental program

The goal of this study is to understand and predict the effect of the control structure, i.e., feedback loops, on transient expression of early meiotic proteins and on distinct sporulation efficiencies observed in budding yeast. We construct a meiotic initiation pathway using an ODE-based model that includes regulation of Ime1, the master regulator, and five other early-meiotic proteins. We consider three feedback loops that control expression of these proteins: double-negative feedback between pSok2 and Ime1, negative feedback from Ime2 to Ime1, and auto-positive feedback of Ime2. In particular, Ime1 is controlled by an upstream inhibitor, Sok2, and a downstream inhibitor, Ime2.

The model is capable of simulating orderly and transient expression of meiotic proteins, without relying on putative repressors to shut down gene expression

The new insights gained from this study are two fold. First, we conclude that feedback loops play important roles in terminating expression of early meiotic proteins. Negative feedback from Ime2 to Ime1 is responsible for transient expression of both Ime1 and Ime2, in agreement with previous finding

More importantly, the second new insight from exploration of the model is that feedback loops are responsible for tuning the efficiency of meiotic pathways. We perform bifurcation analyses on feedback loops using the equilibrium value of Ime2 as the pathway readout. We find that, by adjusting each of the two arms of mutual inhibition between pSok2 and Ime1, the system is able to move from a default meiotic state to a more efficient meiotic state. Similarly, by manipulating the strength of negative feedback loop from Ime2 to Ime1, the model readily produces a default meiotic state and a more efficient meiotic state. Auto-positive regulation of Ime2 is characterized by the Hill function with a high coefficient, providing a simple, reasonably accurate approximation for multiple regulations occurring on Ime2. This positive feedback generates a bistable pathway with two alternative stable steady states—the default meiotic state and a more efficient meiotic state. The robustness of bistability is sensitive to the Hill coefficient, indicating a strong cooperativity and nonlinearity in the response of Ime2 to the feedback. We propose that the combinational feedback regulation controls sporulation efficiency and guarantees that meiotic initiation proceeds in an accurate temporal scale.

Our mathematical model constitutes physical interactions of early meiotic proteins and provides mechanistic insights into ordered appearance of key regulators and sporulation efficiency. Such a model illustrates how different feedback regulations are integrated in the signaling pathway to cause changes in protein expression and meiotic outcome. The model is a reduced system of differential equations, including only Rim11, Ume6, Sok2, Ime1, and Ime2. Other proteins and/or links involved in meiotic initiation are traced indirectly. Validation using deletion mutants of meiotic genes suggests that major regulatory interactions have been captured. We demonstrate that the ordinary differential equation method can depict the most prominent features of signaling pathway during yeast meiotic initiation. Our mathematical model allows for uncovering key regulations that can lead to manipulation of the pathway to enhance sporulation efficiency. This represents an important first step in designing new strategies for producing gametes with high quality and quantity.

Conclusions

We develop a dynamic model to describe signaling pathways that operate during yeast meiotic initiation. Our study suggests that both positive and negative feedback loops control transient expression of early meiotic proteins, and multiple feedback loops regulate the efficiency of meiotic progression. Thus, yeast meiotic initiation is the consequence of systems-level feedback that leads cells into distinct sporulation states.

Methods

An ODE model

We formulate a mathematical model to describe the temporal dynamics of a signaling pathway that controls yeast meiotic initiation. The kinetics is based on SK1, a strain commonly used for studying yeast meiosis. Six proteins in either phosphorylated or unphosphorylated form are model variables. All variables are dimensionless and represent relative protein levels within the range of zero to one unit. The rate of change of protein levels is captured by ordinary differential equations, which include terms describing protein synthesis, degradation, phosphorylation, dephosphorylation, as well as regulatory activation and repression.

Phosphorylated or unphosphorylated Rim11, Ume6, and Sok2 are variables in Equations 1, 2 and 3. Because these three proteins exhibit uniform expression levels over the entire course of sporulation

Unphosphorylated and phosphorylated Ime1 are variables in Equations 4 and 5. Ime1 synthesis is inhibited by Sok2, as described using an inhibitory Hill function

Equation 6 describes the rate of change of Ime2, the most downstream protein in the model. The synthesis of Ime2 depends on phosphorylated forms of Ime1 and Ume6

The ODE model in the Systems Biology Markup Language format is provided as Additional file

**YeastMeioticInitiation.xml.** The odel in the Systems Biology Markup Language format.

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Initial conditions of variables

We consider a mitotic initial state. All variables except for pSok2 are set to zero because a very low level of meiosis-specific proteins could be detected during vegetative growth

Parameter values

Parameters are either rate coefficients with a dimension of per hour or dimensionless constants (Table

Synthesis rates of genome-wide proteins have been calculated from experiments when yeast cells grow mitotically in glucose medium

The half-life of Ime1 is 0.5 hour in the presence of Ime2 and 1 hour in the absence of Ime2

Although no kinetic data exist for phosphorylation, the dephosphorylation rate is estimated to be higher than the phosphorylation rate for Rim11 and Sok2 because the unphosphorylated forms of proteins are active during meiosis

Numerical simulations

The ODE model is implemented in MATLAB. Numerical simulation of the model is performed with MATLAB using a non-stiff solver ode45. Numerical results are confirmed with other non-stiff or stiff solvers: ode23, ode113, ode15s, ode23s, and ode23t.

Parameter sensitivity analysis

We perform global sensitivity analyses using the software package SBML-SAT

Bifurcation analysis

We use the software XPPAUT (

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

DR, YS and PY carried out the studies. PY conceived of the project. DR, YS and PY drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgments

The authors would like to acknowledge Leanne Whitmore for critical reading of the manuscript. This study was supported by the March of Dimes Basil O’Connor Starter Scholar Research Award #5-FY10-485 to PY.