7. Modeling Mutations with a Boolean network approach
7.3 Results of Boolean simulation of the ﬁssion yeast cell cycle
A consistent model should be able to describe not only the wild-type but also the mutants of the ﬁssion yeast cell cycle. There are at least three possible types of muta-tions: temperature-sensitive, loss-of-function and overexpressing mutants. The ﬁrst type corresponds to reduced activity, the second type to zero-activity and the third type to an overproduced activity of a protein. For modeling the temperature-sensitive mutants in [191, 154, 205] the appropriate kinetic constants are reduced by 10%, for loss-of-function mutants these constants are set to zero, and for overexpressing mu-tants they are increased by a factor of several.
In terms of the Boolean approach one cannot model temperature-sensitive muta-tions when the activity of proteins changes slightly. For this reason we model mostly loss-of-function and overexpressing mutations. In order to model loss-of-function mutations, we delete from the network the mutation node and run the updated model.
Below we describe the dynamical properties and biological explanations of modeled mutations. Table 7.3 summarizes the properties of mutations.
Parent node Daughter node Rule of activation (comments)
Rule of inhibition (comments)
Start node Kinases (SK):
Cdc2/Cig1, Cdc2/Cig2, Cdc2/Puc1
Start node works as an indicator of mass of the cell and acti-vates Starter Kinases (SK): Cdc2/Cig1, Cdc2/Cig2,
Cdc2/Puc1, +1 .
SK Ste9, Rum1 Phosphorylate,
thereby inactivate, -1 [191, 205]
Cdc2/Cdc13 Cdc25 Cdc25 is
phosphory-lated thereby acti-vated, +1 .
Wee1, Mik1 Tyr15 Phosphorylate,
inacti-vating, -1 
Rum1 Cdc2/Cdc13 Binds and inhibits
ac-tivity, -1 Cdc2/Cdc13 .
Cdc2/Cdc13 Rum1 Phosphorylates and
thereby targets Rum1 for degradation. -1 [191, 205]
Ste9 Cdc2/Cdc13 Labels Cdc13 for
degradation [205, 191], -1.
Tyr15, Cdc2/Cdc13 Slp1 Highly activated
Cdc2/Cdc13 acti-vates Slp1, Tyr15 has to be active, too [154, 191]+1.
Slp1 Cdc2/Cdc13 Promotes degradation
of Cdc13, thereby the activity of Cdc2/Cdc13 drops -1 
Slp1 PP Activates, +1 
Ste9, Rum1, Wee1, Mik1
Activates Rum1, Ste9, and the tyrosine-modifying enzymes (Wee1, Mik1, , +1
Cdc25 Tyr15 Cdc25 reverses
phos-phorylation of Cdc2, thereby Tyr15 becomes active, +1 [191, 154]
Cdc2/Cdc13 Ste9 inhibits -1 
PP Cdc25 inhibits -1
Cdc2/Cdc13 Wee1, Mik1 inhibits -1 
Tab. 7.1: The rules of interaction of the main elements involved in the ﬁssion yeast cell cycle regulation.
TimeStartCig1/Cdc2 Cig2/Cdc2 Puc1/Cdc2 Cdc2/Cdc13 Ste9Rum1Slp1Tyr15Wee1Cdc25PPPhasecomments
! " #
! " # ! " #
Fig. 7.3: The temporal evolution of protein state of Wee and Cdc25 mutant cells. The black/white color responds to active/inactive state of a protein correspondingly.
Wee and Cdc25mutants
The duration of S and G2 phases are controlled by down-regulation of Wee by Cdc2/Cdc13. If Wee is absent (Wee), then the cell enters mitosis with a smaller size, but it stays viable . The modeling of Wee conﬁrms this. The temporal evolution of protein states stays the same as in wild-type. The system has one ﬁxed point which corresponds to the G1 stable state (Fig. 7.3 a)
However, if some other antagonists of Cdc2/Cdc13 are also mutated, e.g. Rum1 - Wee or Ste9Wee, then the cells divide too fast and do not have enough time
Start Cig1/Cdc2 Cig2/Cdc2 Puc1/Cdc2 Cdc2/Cdc13 Ste9 Rum1 Slp1 Cdc2_Tyr15 Wee Cdc25 PP
basin size 788 136 33 28 11 8 6 4 3 2 1 1 1 1 1
Fig. 7.4: All attractors of the dynamics of the network model for the wild-type ﬁssion yeast cell cycle regulation
to grow . With every division cells get smaller and smaller until they die. In our model Starter kinases – Cig1/Cdc2, Cig2/Cdc2 and Puc1/Cdc2 are not inﬂu-enced by Rum1 and Ste9 for simplicity. In fact Cig2 is partly inhibited by Rum1 and possibly by Ste9 [154, 191]. For this reason one cannot separate Wee and Rum1Wee, Ste9Wee mutations. However, our model reproduces the triple mutation Rum1Ste9Wee. In this case the system shows oscillations and is not viable. The cell divides uncontrollably and the temporal evolution of protein states is shown in Fig.7. 3 b – step 10 repeats step 6, that is the system goes periodically through the same sequence of states.
In order to understand the WeeCdc25 mutation, one has to take into account that Cdc25 has a back-up enzyme, called Pyp3. Pyp3 is a tyrosine-phosphate with a much lower activity, which means that dephosphorylation of Cdc2 is non zero, when Pyp3 is present. Therefore one can model WeeCdc25 mutation as follows: Node Wee is deleted and the weight of the link connecting Cdc25 to Cdc2 Tyr15 is set to 0.5 instead of the usual 1. This results in a vital mutation, when the cell goes through all phases, and the temporal evolution of proteins is the same as for a wild cell, except
Fig. 7.5: The temporal evolution of protein state of Ste9, Rum1 and Slp1 mutant cells. The black/white color responds to active/inactive state of a protein correspondingly.
the Wee, which is for this mutation OFF (Fig 7. 3 c). This is conﬁrmed by experi-mental data . The removal of the nodes Cdc25 and Wee corresponds to a triple mutant WeeCdc25Pyp3, when Tyr15 stays phosphorylated. This mutation is not viable. The cell cannot enter mitosis, since Tyr15 stays phosphorylated, thereby preventing Cdc2/Cdc13 to reach high activity. Our model reproduces this, one can see in Fig. 7. 3 d that Cdc2 Tyr15 stays inactive and the cell cannot enter mitosis.
Overexpression of proteins can be modeled as following. Overexpression means that the activity is signiﬁcantly increased. In the frames of our model overexpression is interpreted in a such a way that overexpressed protein has a constant positive input, which corresponds to a negative theta in (6.1). Here and further we choose theta = -0.5 for all overexpressed mutants. Therefore for modelingW eets Cdc25opsince there
! " # $ % ! " # $ %
Fig. 7.6: The temporal evolution of protein state of Start kinases mutant cells. Part 1. The black/white color responds to active/inactive state of a protein correspondingly.
is no way to distinguish between reduced activity and no-activity, mutation W eets was substituted with Wee and the threshold of activation was changed to -0.5 for Cdc25. In Fig. 7. 3e one sees that mitosis happens very quickly without having appropriate G2 phase, which means that mitosis is initiated before the replication of DNA was completed. In case of overexpression of Wee the cell remains viable (Fig 7.
9 b). Experimental observations conﬁrm these results .
Mutations of Cdc2/Cdc13 antagonists: Ste9, Rum1, Slp1 mutants Fission yeast survives in the absence of Ste9 or Rum1  and grows normally. Our observations of temporal protein evolution conﬁrm this fact (Fig. 7.5 a, b). The system has one ﬁxed point G1 that is reached after the evolution through all G1-S-G2-M phases. However, the absence of the other Cdc2/Cdc13 antagonist – Slp1 – has a lethal eﬀect. The recent studies  show that Slp1 is a lethal mutation that prevents the mitosis. The dynamical behavior of the model for Slp1 shows that the system reaches a ﬁxed point, which corresponds to G2 late phase, right before entering mitosis. The evolution of the proteins ﬁrst coincides with wild-type, but then stays freezing at step 6 (Fig. 7. 5 c).
Mutations of cyclins: Cig1, Cig2, Puc1, Cdc2 and cyclin-dependent kinase Cdc13
The only cyclin which makes ﬁssion yeast to die is Cdc13. The presence of Cdc13 is essential for normal progression through the cell cycle . In the absence of Cdc13 the cells elongate abnormally and cannot enter mitosis. Our simulation conﬁrm this (Fig. 7.8 a): The Start Kinases Cig1Cdc2 , Cig2Cdc2, Puc1Cdc2 switch oﬀ the Cdc2/Cdc13 antagonists during G1-S phase, but in the absence of Cdc13 the cell cycle cannot evolve further. The system remains on the fourth step of wild-type cell cycle evolution (Fig. 7.6 a,b,f,g).
Start Kinase Cig1/Cdc2, Cig2/Cdc2 and Puc1/Cdc2 are responsible for deacti-vation of Cdc2/Cdc13 antagonists. Mutations of cyclins of Start Kinase inﬂuence only the duration of the G1 phase, which becomes longer, when they are mutated.
Thereby mutants Cig1, Cig2, Puc1 as well as their double mutants and triple mutants Cig1Cig2, Cig1Puc1, Cig2Puc1, Cig1Cig2Puc1are viable.
Due to simpliﬁcations of the Start kinase interactions we made, our model is able to reproduce only single and double mutations and not triple mutations (Fig.7.6 b, c, d, e, f, g, h, i, j). Due to the fact that the time in a Boolean model is discrete, one cannot distinguish wild-type and Start Kinase mutants. The temporal evolution is similar to the wild-type. Double mutations and triple mutations Cig1W eets, Cig2W eets, Cig1Cig2W eets, Puc1 W eets, Cig2 - Rum1, Ste9Cig1, Ste9Puc1, Cig1Rum1, WeeCdc2, WeeCig2 Cig1, Ste9Cig2, Cig2Rum1, Puc1Rum1, WeeCig1, WeePuc1 are also viable (Fig 7.7 a-j).
Overexpressions of Cdc13 and Cdc2 are non-lethal, that is also proven by our model: The cell evolves through all phases as in a wild-type. The knock-out mutations (Cdc13, Cdc2) bring the cell to the lethal mutation and block it in G1 and G2 (Fig. 7.9).