Department of Computer Science, KAIST, Daejeon, Korea

Department of Bio and Brain Engineering, KAIST, Daejeon, Korea

Abstract

Background

Cell cycle process of budding yeast (

Results

To construct a timing-robust Boolean model that preserves checkpoint conditions of the budding yeast cell cycle, we used a model verification technique, ‘model checking’. By utilizing automatic and exhaustive verification of model checking, we found that previous models cannot properly capture essential checkpoint conditions in the presence of timing variations. In particular, such models violate the M phase checkpoint condition so that it allows a division of a budding yeast cell into two before the completion of its full DNA replication and synthesis. In this paper, we present a timing-robust model that preserves all the essential checkpoint conditions properly against timing variations. Our simulation results show that the proposed timing-robust model is more robust even against network perturbations and can better represent the nature of cell cycle than previous models.

Conclusions

To our knowledge this is the first work that rigorously examined the timing robustness of the cell cycle process of budding yeast with respect to checkpoint conditions using Boolean models. The proposed timing-robust model is the complete state-of-the-art model that guarantees no violation in terms of checkpoints known to date.

Background

A cell must undergo the process of duplicating all its components and separating them, more or less evenly, to two daughter cells such that each daughter has the information and dynamics necessary to repeat the process. Such cell cycle dynamics are known in more detail for the budding yeast,

Among several approaches

Timing robustness is the ability of a model to maintain its function in the presence of timing perturbations. Among a few ways to introduce various timing variations, Boolean modeling often uses an asynchronous updating of models. Unlike the synchronous update rule, the asynchronous update rule allows a maximum of one variable to be updated at each time instant, and if multiple variables are enabled to change, one of them is chosen arbitrarily. In this way, variations in reaction rates can be represented depending on the order in which the nodes update their values (i.e., some of the nodes update their values immediately while other nodes take longer). Here it is important to note that this way can generate a large number of distinct state transition trajectories, possibly as many as the number of all different order combinations. And such all different trajectories reflect every possible timing variation under dynamically changing environments. However, such a large number of trajectories to explore make it difficult to perform timing robustness analysis through biological simulation, since such simulation generally involves randomness for trajectory selection.

The main goal of this study is to construct a timing-robust Boolean model that properly preserves checkpoint conditions of the budding yeast cell cycle even in the presence of timing variations. Towards this goal, we used a model verification technique, ‘model checking’

In order to utilize model checkers, it needs to specify essential system properties in a form that model checkers can recognize, which is temporal logic

In this study, we present a timing-robust model that properly preserves all the essential checkpoint conditions against timing variations. The proposed model is the complete state-of-the-art model that guarantees no hazard in terms of checkpoints known to date. As a result, our model naturally eliminates the hazards contained in the previous models (i.e., Li,

Results and Discussion

In this study, model checkers, software tools of model checking, are used to examine whether or not a specified logical property holds on every possible state of a Boolean model. The inputs to the model checker consist of a Boolean model, described as a set of variables and rules that update their values, an initial state, and a logical property to check. In this study, previously published Boolean models of the budding yeast cell cycle (i.e., the Li,

We derived the logical properties to check from essential checkpoint conditions of the budding yeast cell cycle. Based on the comprehensive literature studies

**Essential ordered properties derived from checkpoints.** The PDF file contains a list of all the essential ordered properties derived from the up-to-date checkpoint conditions.

Click here for file

Timing robustness of the budding yeast cell cycle

The first hazard in the Mangla,

Budding yeast model

**Budding yeast model.** Nodes in the graph represent molecules. Lines with arrowhead represent the activation, and lines with flat ends indicate the inhibition. Thin arrows represent a low weight, a weight of 1/3, normal arrows indicate a medium weight, a weight of 1, and thick arrows represent a high weight, a weight of 3. (**A**) The model from Mangla, **B**) A subset of the model that highlights the first hazard. Nodes with values marked with * are enabled to change to the values. If Clb5 transitions to 1, both Clb2 and Mcm1 are eligible to change their values to 1 at the next time step. (**C**) If Mcm1 activates first, Cdc20 is enable to update its value as well as Clb2. (**D**) If Cdc20 activates before Clb2 transitions to 1, the first hazard occurs. (**E**) The hazard can be eliminated by replacing the weight of the reaction between Clb5 and Mcm1 to the one with the lower weight.

The Mangla,

The second hazard violating the properties 5 to 7 of the M-telophase checkpoint conditions (see Additional file

Budding yeast model

**Budding yeast model. **(**A**) A subset of the Mangla, **B**) The hazard can be eliminated by replacing the weights of reactions, from Cdc20 and Clb2 to Swi5, to stronger and weaker levels, respectively. Swi5 is also extended to have one of three eligible values like Clb2. (**C**) The final timing-robust model for the budding yeast cell cycle. Lines with blue are modified from the Mangla,

According to the study by Chen,

Figure

Temporal evolution of gene states for budding yeast cell cycle models

Mathematical modeling based on biochemical rate equations, provides a rigorous and reliable tool for unraveling the complexities of molecular regulatory networks. However, this approach is only suited for small and well-characterized systems with known kinetic parameters since there is a lack of detailed knowledge of quantitative reaction kinetics for most of the reactions in a cell

To investigate how closely Boolean models of the budding yeast cell cycle represent nature, we compared the temporal evolutions of gene states in Boolean models to that in the mathematical model. Note that the asynchronous update rule allows a maximum of one variable to be updated at each time instant, and if multiple variables are enabled to change, one of them is chosen in an arbitrary fashion. Thus, too many different state transitions can exist under the asynchronous update rule. When applying the synchronous update rule to Boolean models for simple comparison, we observed that the temporal evolution of gene states in the proposed model maintains a similar structure to those in the other Boolean models overall. However, we also found that the extension from the Mangla,

As shown in Figure

Temporal evolution of variables related to the first hazard in the budding yeast cell cycle model

**Temporal evolution of variables related to the first hazard in the budding yeast cell cycle model.** X-axis and y-axis represent the time and the level of activation concentration in simulation, respectively. (**A**) The low level activation of Clb2 through the time of 30 to 45 causes the transcription of Mcm1. Finally, Clb2 is activated to a high level by Mcm1. (**B**)-(**C**) At the time of 5, both Clb2 and Mcm1 are activated and do not form a positive feedback. (**D**) Clb2 is first activated at the time of 4, and then Mcm1 is activated. After the transcription of Mcm1, Clb2 is activated to value of 2, a high activation level, at the time of 7.

Moreover, the temporal evolution of Swi5 in the proposed model is closer to the dynamics of the gene in reality (Figure

Temporal evolution of variables related to the second hazard in the budding yeast cell cycle model

**Temporal evolution of variables related to the second hazard in the budding yeast cell cycle model.** X-axis and y-axis represent the time and the level of activation concentration in simulation, respectively. (**A**) Swi5 is activated to a low level by Mcm1, and then transcribed to a high level via the activation of Cdc20 at the time of 80. (**B**) Swi5 is inactive until Cdc20 is activated regardless of the activation of Mcm1. (**C**) Swi5 is deactivated until Cdc20 is activated regardless of the activation of Mcm1. (**D**) Swi5 is activated to a low level by the activation of Mcm1 at the time of 6, and then transcribed to a high level via the activation of Cdc20 at the time of 8.

From this simulation, it appears that the proposed model follows similar dynamics as those in the other Boolean models, even if the proposed model is extended from the others. In addition, in the proposed model, the temporal evolution of genes related to the extension shows better consistency with the one in the mathematical model of the budding yeast cell cycle. Therefore, the proposed model which is constructed by eliminating hazards in the previous models better reflects the nature of cell cycle than the previously published Boolean models.

Relative durations of each cell cycle phase

Many efforts have been made to discover checkpoints and key transcription factors responsible for phase transitions in cell cycles

Recent studies have assumed that every edge in the yeast cell cycle regulatory network proceeds with the same speed since transcriptions normally happen on similar time scales

**Phase**

**The Li, ****’s model**

**The Mangla, ****’s model**

**The proposed model**

Consistent with the experimental data

**Average**

**Variation**

**Average**

**Variation**

**Average**

**Variation**

S

9.30

1.55

9.30

1.55

7.47

0.41

G2/M

7.56

1.25

10.89

2.14

12.52

1.72

Attractor analysis

We used the proposed model to study the attractors of the network dynamics by starting from each of the 2^{9}×3^{2}=4,608 states in the 11-node proposed model with two 3-valued nodes. We found that all of the initial states eventually flow into one of the nine stationary states, also called attractors (Table

**Basin size**

**Cln3**

**MBF**

**SBF**

**Cln2**

**Cdh1**

**Swi5**

**Cdc20**

**Clb5**

**Sic1**

**Clb2**

**Mcm1**

Each attractor is represented in a row. The first column is the size of the basin of attraction for the attractor. The other 11 columns show the protein states of the attractor. In the proposed model, the biggest fixed point representing the stationary G1 phase attracts 4,323 or ≈ 94% from 4,608 initial states.

4323

0

0

0

0

1

0

0

0

1

0

0

87

0

1

1

1

0

2

1

0

1

0

0

68

0

0

1

1

0

0

0

0

0

0

0

46

0

1

0

0

1

0

0

0

1

0

0

44

0

0

0

0

0

0

0

0

0

0

0

16

0

0

1

1

0

2

1

0

1

0

0

12

0

0

0

0

1

0

0

0

0

0

0

10

0

0

0

0

0

0

0

0

1

0

0

2

0

1

0

0

0

0

0

0

1

0

0

**Basin size**

**Cln3**

**MBF**

**SBF**

**Cln2**

**Cdh1**

**Swi5**

**Cdc20**

**Clb5**

**Sic1**

**Clb2**

**Mcm1**

Each row and column represents the same as Table

1764

0

0

0

0

1

0

0

0

1

0

0

151

0

0

1

1

0

0

0

0

0

0

0

109

0

1

0

0

1

0

0

0

1

0

0

9

0

0

0

0

0

0

0

0

1

0

0

7

0

1

0

0

0

0

0

0

1

0

0

7

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

1

0

0

0

0

0

0

**Basin size**

**Cln3**

**MBF**

**SBF**

**Cln2**

**Cdh1**

**Swi5**

**Cdc20**

**Clb5**

**Sic1**

**Clb2**

**Mcm1**

Each row and column represents the same as Table

2769

0

0

0

0

1

0

0

0

1

0

0

159

0

1

1

1

0

1

1

0

1

0

0

52

0

0

1

1

0

0

0

0

0

0

0

44

0

1

0

0

1

0

0

0

1

0

0

18

0

0

1

1

0

1

1

0

1

0

0

17

0

0

0

0

0

0

0

0

1

0

0

7

0

0

0

0

0

0

0

0

0

0

0

5

0

1

0

0

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

0

0

0

0

Number of different state transitions and timing robustness

Table

**Phase**

**Li, **

**Mangla, **

**Proposed**

The major difference among the three models in terms of the number of different state transitions for each cell cycle phase is found in the G2 phase.

**Number of state transitions**

**Number of state transitions**

**Number of state transitions**

G1

51

51

51

S

6

6

1

G2

9

35

215

M

3

4

4

**Phase**

**Mutation**

**Li, **

**Mangla, **

**Proposed**

The second column represents the number of mutations applied to a model. To perturb only an interested cell cycle phase, we applied mutations to edges related to the corresponding phase by deleting, adding, or re-weighting them. In this way, 200 mutant networks were generated, and then checked whether they preserve the desired attractor, the G1 stationary state under the asynchronous update rule using NuSMV. Columns from third to fifth represent the fraction of timing-robust mutants in each model.

**distance**

**% of timing-robust**

**% of timing-robust**

**% of timing-robust**

1

0.00

0.31

0.31

G1

2

0.00

0.22

0.22

3

0.00

0.15

0.15

1

0.00

0.26

0.22

S

2

0.00

0.12

0.20

3

0.00

0.07

0.10

1

0.00

0.17

0.27

G2

2

0.00

0.09

0.21

3

0.00

0.08

0.12

1

0.00

0.22

0.28

M

2

0.00

0.14

0.16

3

0.00

0.13

0.13

We conjecture that the network robustness of a model is closely related to the number of different biological state transitions which satisfy properties of the model. This is because even if some state transitions are altered by mutations, the remaining transitions are likely to hold biological properties of the model, and our simulation results demonstrate this.

Limitations

The proposed model is the timing-robust model that properly preserves all the essential checkpoint conditions against timing variations. Although the proposed model is the complete state-of-the-art model that guarantees no hazard in terms of checkpoints known to date, it still has a gap with quantitative models. Our model is not yet directly applicable to explaining and predicting the quantitative outcome of biological experiments of the budding yeast cell cycle. Quantitative models can potentially describe molecular interactions with high precision and in quantitative terms that correspond to realistic laboratory measurements. However, Boolean models can still be used by a subset of researchers because of easy understanding of the dynamics of the budding yeast cell cycle. We expect that the proposed model can be used to help them by providing a more stable and timing-robust Boolean model.

The main obstacle in application of model checking in practice is the state space explosion problem

Conclusions

Timing robustness analysis is one of the most important and challenging problems in systems biology

Boolean network modeling, which is now a widely used modeling framework in systems biology, requires timing robustness analysis since reaction kinetic parameters inevitably vary over a certain range. However, most researchers paid little attention to timing robust analysis so far and assumed that Boolean models are updated in a synchronous manner, neglecting timing variations

A number of experimental studies on budding yeast,

Building upon the Li,

Boolean modeling and analysis of complex biological networks aim to provide a system-level understanding on complex biological phenomena

In systems biology, mathematical models are becoming too complicated to be validated by examining some essential dynamics in an ad hoc way. It becomes even more difficult to check manually whether a combination of dynamics (e.g., ordered dynamics) are met simultaneously in complex biological networks

Methods

NuSMV

Model checking

We used a symbolic model checker, NuSMV

A NuSMV model consists of one or more modules. Each module can declare variables and their update rules. Variables can be declared to have a range of discrete values. The rules specify how to initiate variables and update them at every time step from their current values. Update rules can be non-deterministic since they can result in different values of a variable under the same condition. NuSMV checks whether a given property holds over all different possibilities. In describing our budding yeast cell cycle model in the NuSMV input language, we assign a single variable to each node of the model and specify the update rules of every variable. By default, variables in the NuSMV model are updated in a synchronous manner. There is a global clock and all modules execute in parallel every time the global clock ticks. To apply the asynchronous update rule to the NuSMV model, we define an additional control variable. At each time step, the control variable indicates which variable to update in a non-deterministic manner such that only the update rules of the chosen variable can execute in the model. In addition, we add a

Programs in the language can be annotated by properties expressed in temporal logic, that is, computation tree logic (CTL)

Boolean model construction

In this paper, we presented Boolean networks of the budding yeast cell cycle. Boolean models include nodes and edges for different components and interactions of the system, respectively. Each node in the Boolean model has one of two values: 1 for ON (active) and 0 for OFF (inactive). A state _{i}(

where _{ij}represents the weight of an incoming edge to a node _{i}) is set to zero.

A synchronous Boolean model is one of the simplest implementations for the application of such Boolean update rules to nodes. In such a model, a Boolean update rule is applied to all the nodes simultaneously at each time instant. Synchronous models are deterministic since nodes are assumed to work in the same time scale, resulting in convergence to the same state from the same initial condition after the same number of time steps. As the result of applying the Boolean function described above synchronously, the first synchronous Boolean model for the budding yeast cell cycle, the Li, ^{11} initial states in the 11-node network model.

Although synchronous Boolean models have been widely used due to their simple nature and ease of implementation, they lack consideration of a variety of time scales in biological systems. To deal with this drawback, asynchronous models are suggested in which a maximum of one node is chosen to be updated at each time instant. Since it is usually unknown exactly how long specific biological processes take, most asynchronous algorithms are non-deterministic in a way that a single node is randomly chosen at each time unit. The Mangla,

In the model, the thresholds for the value of 1 (denoted by _{i,1}) and 2 (denoted by _{i,2}) for a node

Basically, the model proposed in this study extended a set of available values and weights for each node and edge, respectively. The proposed model also follows the same Boolean function as the Mangla,

Competing interests

The authors declare that they have no competing interests.

Author’s contributions

CH and IS jointly conceived the study, and wrote the manuscript. ML and DK (Dongsan Kim) interpreted simulation results and helped draft the manuscript. DK (Dongsup Kim) and KC critically reviewed the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MEST) (2009-0086964).