Molecular network control through boolean canalization

Boolean networks are an important class of computational models for molecular interaction networks. Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been extensively studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. Recently, dynamic network control approaches have been used for the design of new therapeutic interventions and for other applications such as stem cell reprogramming. This work studies the role of canalization in the control of Boolean molecular networks. It provides a method for identifying the potential edges to control in the wiring diagram of a network for avoiding undesirable state transitions. The method is based on identifying appropriate input-output combinations on undesirable transitions that can be modified using the edges in the wiring diagram of the network. Moreover, a method for estimating the number of changed transitions in the state space of the system as a result of an edge deletion in the wiring diagram is presented. The control methods of this paper were applied to a mutated cell-cycle model and to a p53-mdm2 model to identify potential control targets.

practical applications in cancer systems biology such as the identification of new therapeutic targets has stimulated the development of computational tools that can help to identify new intervention targets.Experimentally, the interventions are realized by manipulating the wiring diagram of a system with the use of drugs or by gene knockouts to impact the dynamics of the system so that it is directed towards a desired state [5,7,19,32,33].From the modeling perspective, the identification of intervention targets amounts to finding a set of relevant nodes and edges that can be used for performing interventions in silico.
Many dynamic systems theory approaches have been used over the last decades to develop computational tools for analyzing the dynamics of GRNs.As a result, a large variety of models exists today.Boolean networks is a class of computational models in which genes can only be in one of two states: ON or OFF.BNs, and more general discrete models in which genes can take on more than two states, have been effectively used to model biological systems such as the yeast cell cycle network [20], the Th regulatory network [22], the lac operon [30], the p53-mdm2 complex [1,5,25], A. thaliana [3], and for many other systems [2,6,10,11,27,34].
Boolean networks (BNs) as models for GRNs were introduced by S. Kauffmann [16] and R. Thomas [29].BNs have been proposed as a framework that does not rely on kinetic constants and therefore requires fewer parameters to estimate, which simplifies analysis.Boolean canalizing rules were introduced by S. Kauffman and collaborators [14] and reflect the concept of canalization in evolutionary biology that Waddington pioneered in 1942 [31].Boolean canalization has been intensively studied from the network dynamic perspective [13,15,21,24].It has been shown that networks that use only nested canalizing rules exhibit more stable dynamics compare to network using random rules [15,23].Furthermore, it has shown that each additional layer of canalization provides a degree of stability [18,21].The Boolean functions in published models tend to have many canalizing variables [14,23].

Boolean Networks
A Boolean network can be defined as a dynamical system that is discrete in time as well as in variable states.More formally, consider a collection x 1 , . . ., x n of variables, each of which can take on values in the binary set {0, 1}.A Boolean network in the variables x 1 , . . ., x n is a function where each coordinate function f i is a Boolean function on a subset of {x 1 , . . ., x n } which represents how the future value of the i-th variable depends on the present values of the variables.
The dynamical properties of a Boolean network are given by the difference equation x(t + 1) = F(x(t)); that is, the dynamics is generated by iteration of F.More precisely, the dynamics of F is given by the state space graph S, defined as the graph with vertices in K n = {0, 1} n which has an edge from x ∈ {0, 1} n to y ∈ {0, 1} n if and only if y = F(x).In this context, the problem of finding the states x ∈ {0, 1} n where the system will get stabilized is of particular importance.These special points of the state space are called attractors of a Boolean network and these may include steady states (fixed points), where F(x) = x, and cycles, where F r (x) = x for some integer number r > 1. Attractors in Boolean network modeling might represent cell types [16] or cellular states such as apoptosis, proliferation, or cell senescence [12,28].

Canalizing functions
A Boolean function f (x 1 , . . ., x n ) : {0, 1} n → {0, 1} is canalizing in the variable x i with canalizing input value a and canalizing output value b if f (x 1 , . . ., x i = a, . . ., x n ) = b.That is, once x i gets its canalizing input, it by itself determines the output of the function regardless of the value of the other variables.The variable x i is called a canalizing variable.

Partially nested canalizing functions
Any Boolean function f (x 1 , . . ., x n ) : {0, 1} n → {0, 1} can be represented in the form: where either n = d, where x d is a terminal canalizing variable and g is a constant, or n < d where g(x σ(d+1) , . . ., x σ(n) ) is a non-constant function and none of the variables x σ(d+1) , . . ., x σ(n) are canalizing for g.The integer d is called the nested canalizing depth of f .Such Boolean functions are called partially nested canalizing functions (PNCFs), see [18] for more details.

Layers of Canalization
A Boolean function can be represented in different forms as a nested canalizing function.A unique representation of the function is obtained by grouping the variables in layers of canalization [21].Every Boolean function can be uniquely written as where M i = k i i=1 (x i j + a i j ), P c is a polynomial with no canalizing variables, and k = k 1 + • • • + k r is the canalizing depth.Each variable x i appears in exactly one of the M 1 , M 2 , . . ., M r , P c .The proof of this property is given in [9].The number r in Equation 1 is called the layer number of f .Example 1.1.Consider the Boolean functions f 1 , f 2 , and f 3 with truth tables given at Table 1.The layers representation for f 1 is and for f 3 is Thus f 1 has layer number equal to 1, f 2 has layer number equal to 2, and f 3 has layer number 1.The polynomial P c does not have canalizing variables.

Definition of control actions
This paper considers two types of control action: deletion of edges and constant expression of edges.An edge deletion represents the experimental intervention that prevents a regulation from happening.This action can be achieved by the use of therapeutic drugs that target a specific gene interaction, see reference [5] where this type of control has been experimentally applied.A node deletion can be represented by the deletion of all of its outgoing edges.A constant expression or a constitutive activation of a node might result in aberrant cell proliferation and cancer, see [5] where the constant expression of cyclin G in the signaling pathway of p53 is reported as a signature of abnormal gene expression that leads to cancer.But constant expressions could also help to drive the system into a more desirable state, see [27] where constant expression of nodes have been proposed as potential controls.As a proof of principle, this paper will consider the constant expression of an edge as a potential control action.
Definition 1.2 (Edge Control).Consider the edge x i → x j in the wiring diagram W. For u i,j ∈ F 2 , the control of the edge x i → x j consists of manipulating the input variable x i for f j in the following way: For each value of u i,j we have the following control settings: • When u i,j = 0, F j (x, u i,j ) = f j (x j 1 , . . ., x i , . . ., x jm ).That is, the control is not active.
• When u i,j = 1, F j (x, u i,j ) = f j (x j 1 , . . ., x i = 0, . . ., x jm ).This is the case when the control is active and the action represents the removal of the edge x i → x j .
For simplicity, in Definition 1.2 we considered only edge deletions.To include both the deletion and constant expression of an edge we could consider the following control function Then for each combination of u − i,j and u + i,j we have the following control settings: • For u − i = 0, u + i = 0, F j (x, 0, 0) = f j (x j 1 , . . ., x i , . . ., x jm ).That is, the control is not active.
. ., x i = 0, . . ., x jm ).This action represents the knock out of the node x j .
. This action represents the constant expression of the node x j .
. This action changes the variable x i to its negative value and might not be a relevant case of control.

Eliminating state transitions through edge deletion and constant expression
We avoid undesirable state transitions in the state space graph of a system of canalizing functions by means of edge deletion in the system's wiring diagram.
Let F = (f 1 , . . ., f n ) : {0, 1} n → {0, 1} n be a Boolean network and S = (V s , E s ) be the state space graph of F, where V s ⊆ {0, 1} n is the vertex set of S and E s ⊆ {0, 1} n ×{0, 1} n is its edge set.Suppose for u, v ∈ V s there is a directed edge {u, v} ∈ E s which represents an undesirable transition.We eliminate the transition by deleting appropriate edges from the wiring diagram of the system, W = (V w , E w ), where The following is a sufficient condition for eliminating a transition from S through deleting an edge in E w .
Method 2.1.Suppose x t ∈ V w which takes input from x k ∈ V w , i.e. {x k , x t } ∈ E w (we will also use the notation x k → x t ).Let also x k be a canalizing variable in f t , the functions that determines the state of x t in S.
If the following four conditions are met, then deleting the edge {x k , x t } from E w results in eliminating the transition {u, v} from E s : 1.No variable in a more dominant layer assumes its canalizing input in u.
3. The k-th entry of u is 1, i.e.

4.
x k has canalizing output that is the negation of the t-th entry of v, The reason behind the first condition is that if any variable whose layer is more or equally dominant than x k 's layer has assumed its canalizing input in u, then replacing x k with 0 will have no effect on f t 's output.Deleting an edge has to impose change on the network for control and so the second requirement is needed since if [u] k = 0 already, then deleting the edge x k → x t will have no effect on the network.The third condition has a similar explanation.
Similar sufficient conditions can be stated for eliminating a state space transition through constant expression of an edge, simply by replacing 0 with 1 and 1 with 0.
Node deletion can also be used for control through canalization.In that case, node deletion corresponds to deleting the outgoing edges from the deleted node and Method 2.1 can be applied to each one of them.

Effect of edge deletion and constant expression on the state space
We count the maximum number of state space transitions that can be changed as a result of deleting a single edge.
changes in the state space if k ≤ d and x k is in the r-th layer of f t , where 1 , . . .r are the numbers of variables in layers 1, . . ., r, respectively; that is, the probability that any transition will be removed from the state space upon deletion of x k is not canalizing; thus the probability that a particular transition will be removed from the state space upon deletion of x k → x t is at most ( 1 2 ) d−1 .To see how the bound is calculated, notice that when x k → x t is deleted, half of the transition table can potentially change (the other half had x k = 0 already).Of the remaining half, half contains the canalizing input of a variable in the most dominant layer and so x k cannot cause change.Now half of the half only can possibly change but half of that has the canalizing input of another variable from a more or equally dominant layer to the one where x k is, thus preventing x k from causing change, etc.
This upper bound remains the same when instead of deleting an edge, an edge is constantly expressed.

General procedure for identifying control edges
Below we provide a general procedure for identifying control edges based on the two methods we developed.Figure 1 further illustrates it.
Given a Boolean network model for a biological system: 1. Formulate a goal in terms of the part of the state space you wish to avoid, e.g. a fixed point or a cycle.If it contains more than one state, consider all states that are part of it.Choose to begin with one of them, v.
2. For control via edge deletion, identify all 1's in the state u preceding v; for control via constant expression, identify all 0's.

Results
We apply the control methods we developed to the Boolean models of two networks: a model of the human tumor suppressor gene p53 pathways [17] and a mammalian cell-cycle network [8].

p53-mdm2 model
In [17], a Boolean model, Eq. ( 2), of the widely studied p53 pathway is built, where the external signal is dna dsb, the DNA damage input.
ATM next = Wip1(ATM + dna dsb) The other variables are ATM, p53, Wip1, and Mdm2.When dna dsb = 0, the state space has a single fixed point, (0000), corresponding to no stress.However, when dna dsb = 1, i.e. the DNA damage input turns on, the state space contains a single cycle of length seven (Fig. 3) and no fixed points.The cycle represents cyclic variation in the expression patterns of all the four genes.We want to prevent this cycle from taking place through removing one or multiple transitions from it.
The other variables are ATM, p53, Wip1, and Mdm2.When dna dsb = 0, the state space has single fixed point, (0000), corresponding to no stress.However, when dna dsb = 1, i.e. the DNA damage input turns on, the state space contains a single cycle of length seven (Fig. 1) and no fixed points.The cycle represents cyclic variation in the expression patterns of all the four genes.It is undesirable because all four proteins should be inactive in a no stress situation [2].We want to prevent this cycle from taking place through removing one or multiple transitions from it.

ATM p53
Mdm2 Wip1 dna dsb The wiring diagram of the model is presented on Fig. 2. By Method 1, we identify that deleting edge p53!Wip1 (which also happens to correspond to deleting the node Wip1) has the e↵ect of removing the The wiring diagram of the model i Fig. 2. By Method 1, we identify tha p53!Wip1 (which also happens to deleting the node Wip1) has the e↵ect The bold entries in the states correspon tries where conditions 3 and 4 of Method a result, the system has only a single st (1100).Unfortunately, not all four protein tive in it but we will see that this can through constant expression of the edge W as it will be discussed later.
We can also count the number of changes ing p53!Wip1 from the wiring diagram ind state space transitions: by Method 2, ther

Gene p53 pathways model
In [?], a Boolean model, Eq. (??), of the widely studied p53 pathway is built, where the external signal is dna dsb, the DNA damage input.
The other variables are ATM, p53, Wip1, and Mdm2.When dna dsb = 0, the state space has single fixed point, (0000), corresponding to stress.However, when dna dsb = 1, i.e. the DNA damage input turns on, the state space contains a single cycle of length seven (Fig. ??) and no fixed points.The cycle represents cyclic variation in the expression patterns of all the four genes.It is undesirable because all four proteins should be inactive in a no stress situation [?].We want to prevent this cycle from taking place through removing one or multiple transitions from it.The wiring diagram of the model is presented on Fig. ??.By Method ??, we identify that deleting edge p53!Wip1 (which also happens to correspond tries where conditions 3 a result, the system ha (1100).Unfortunately, tive in it but we will s through constant expres as it will be discussed la We can also count the ing p53!Wip1 from the state space transitions: The bold entries in tries where condition As a result, the syste (1100).Unfortunately tive in it but we wi through constant exp as it will be discussed We can also count t ing p53!Wip1 from the state space trans be up to 2 4 1 = 8 exactly as many, dem Method ?? is sharp: space, eight contain p inal update rules, the remains 1 (and so Wi ing p53!Wip1, Wip1 of its update rule is state space.The wiring diagram of the model is presented on Fig. 2. By Method 2.1, we identify that deleting edge p53→Wip1 (which also happens to correspond to deleting the node Wip1) has the effect of removing the following four transitions from the undesirable cycle in Fig. 3: (1100) → (1110), The bold entries in the states correspond to the entries where conditions 3 and 4 of Method 2.1 apply.As a result, the system has only a single steady state, (1100).Unfortunately, not all four proteins are inactive in it but we will see that this can be achieved through constant expression of the edge Wip1→ATM, as it will be discussed later.

Constant expression for removal of transiti
We can also count the number of changes that deleting p53→Wip1 from the wiring diagram induces on the state space transitions: by Method 2.2, there can be up to 2 4−1 = 8 changes and in fact we observe exactly as many, demonstrating that the bound from Method 2.2 is sharp: out of the 16 states in the state space, eight contain p53 = 1.When applying the original update rules, the value of p53 in these eight states remains 1 (and so Wip1 becomes 1), while after deleting p53→Wip1, Wip1 becomes 0 since now the output of its update rule is 0, thus causing a change in the state space.
Constant expression of an edge is another strategy for removal of transitions in a state space graph.There are analogous conditions to Method 2.1 for constant expression, obtained by simply replacing 0 with 1 and vice versa.For example, Wip1 is a canalizing variable in the function of ATM with canalizing input 1 and canalizing output 0. Therefore, we can set the edge Wip1→ATM to constant expression in order to remove the following transitions from the undesirable cycle: (10001) → (11001), (11001) → (11101), and (00011) → (10001).The result is a state space with fixed point (0000), corresponding to no stress as when dna dsb = 0. Another option for control via constant expression is the edge Mdm2→p53 which also results in a single steady state, although this time it is (1000).

Mutated cell-cycle network model
As a second application, we consider Fauré et al. [8] who proposed a Boolean model of the cell cycle progression.We focus on the scenario when the tumor suppressor retinoblastoma protein Rb is absent as reflected in Eq. ( 3).The wiring diagram for that case is given in Fig. 4. Fauré et al. [8] assume that the expression of CycD changes independently of the cell's content and reflects the state of the growth factor.According to their model, the mammalian cell cycle with a mutated phenotype will cycle through the eight states (Fig. 5) even when CycD is inactive.
urrugarra and Dimitrova Page 3 of 5 o 2 4 1 = 8 changes and in fact we observe exactly as any, demonstrating that the bound from Method 2 is harp: out of the 16 states in the state space, eight conain p53 = 1.When applying the original update rules, he value of p53 in these eight states remains 1 (and o Wip1 becomes 1), while after deleting p53!Wip1, ip1 becomes 0 since now the output of its update ule is 0, thus causing a change in the state space.Constant expression of an edge is another strategy or removal of transitions in a state space graph.There re analogous conditions to Method 1 for constant exression, obtained by simply replacing 0 with 1 and ice versa.For example, Wip1 is a canalizing variable n the function of ATM with canalizing input 1 and analizing output 0. Therefore, we can set the edge ip1!ATM to constant expression in order to reove the following transitions from the undesirable ycle: (10001) !(11001), ( 11001) !(11101), and 00011) !(10001).The result is a state space with xed point (0000), corresponding to no stress as when na dsb = 0. Another option for control via constant xpression is the edge Mdm2!p53 which also results n a single steady state, although this time it is (1000).utated cell-cycle network model s a second application, we consider Fauré et al. [3] ho proposed a Boolean model of the cell cycle proression.We focus on the scenario when the tumor uppressor retinoblastoma protein Rb is absent as reected in Eq. ( 2).The wiring diagram for that case s given in Fig. 3. Fauré et al. [3] assume that the exression of CycD changes independently of the cell's ontent and reflects the state of the growth factor.ccording to their model, the mammalian cell cycle ith a mutated phenotype will cycle through the eight tates (Fig. 4) even when CycD is inactive.E2F = (CycA ^CycB) _ (p27 ^CycB) / / (11110100) We find that there are three edges in the wiring diagram that can possibly be used as control for avoiding the cycle in (??).These edges, given in red in Fig. ??, were identified following Method ?? applied on transitions in the 7-cycle.We note that avoiding transition(s) from the cycle through edge deletion in the wiring diagram does not guarantee that there will be no cycles in the resulting state space; it only guarantees the removal of the particular cycle that contains the targeted transition.For example, deleting Cdh1!UbcH10 results in a single fixed point, (101100110).Deleting the other two edges, on the other hand, removes cycle (??) but creates other cycles: the deletion of E2F!CycE results in another 7-cycle; deleting CycB!Cdc20 has the consequence of introducing two fixed points and a two-cycle.Comb ing the last two edge deletions also creates two fix points and a two-cycle.Therefore, the only edge de tion with a desirable outcome is Cdh1!UbcH10. [

Discussion
We notice that the bound from Method ?? is sha This was demonstrated using the model of the hum tumor suppressor gene p53 pathways.In general, exact bound from part (a) of Method ?? is achiev when f µ t = x k or x k (as a function), where f µ t is function obtained from f t by by plugging in the can izing input of the variables that are more or equa dominant to x k .
It is important to point out that Method ?? o guarantees that a certain transition will be avoid and one may be able to use this to remove a cy from a state space as we did for the p53 pathw model.However, the method does not guarantee t the system will not contain any cycles since remov transitions from a cycle destroys the cycle but m create a di↵erent one; nor does it guarantee that resulting fixed point will be exactly the desirable o as it was also the case with the p53 pathways mod

Conclusions
Method ?? provides a measure of the impact of ed and node deletion onto the state space of a model establishes that this impact di↵ers significantly ba on the canalizing properties of the nodes involved: edge coming from a node with stronger canalizat (represented in the model by a variable in a more do inant layer), has exponentially higher probability change the state space than an edge from a node w weaker canalization (represented by a variable in a l dominant layer).

Section title
Text for this section . . .

Sub-heading for section
Text for this sub-heading . . .

Sub-sub heading for section
Text for this sub-sub-heading . . .We propose four edges from the wiring diagram in Fig. 4 that can be used for control in order to avoid the cycle in Fig. 5.These edges were identified nes.It is undesirable because all four proteins should e inactive in a no stress situation [2].We want to prent this cycle from taking place through removing one multiple transitions from it.By Method 1, we identify that deleting edge p53!Wip1 hich also happens to correspond to deleting the node ip1) has the e↵ect of removing four transitions from e undesirable cycle: (1100) !(1110) (1110) !(0110) (0110) !(0111) (0111) !(0011) The bold entries in the states correspond to the enies where conditions 3 and 4 of Method 1 apply.As result, the system has only a single steady state, 100).Unfortunately, not all four proteins are inacve in it but we will see that this can be achieved rough constant expression of the edge Wip1!ATM, it will be discussed later.We can also count the number of changes that deletg p53!Wip1 from the wiring diagram induces on the ate space transitions: by Method 2, there can be up 2 4 1 = 8 changes and in fact we observe exactly as any, demonstrating that the bound from Method 2 is arp: out of the 16 states in the state space, eight conin p53 = 1.When applying the original update rules, e value of p53 in these eight states remains 1 (and Wip1 becomes 1), while after deleting p53!Wip1, ip1 becomes 0 since now the output of its update le is 0, thus causing a change in the state space.Constant expression of an edge is another strategy r removal of transitions in a state space graph.There gression.We focus on the scenario when the tumor suppressor retinoblastoma protein Rb is absent.The wiring diagram for that case is given in Fig. 1.Fauré et al. [3] assume that the expression of CycD changes independently of the cell's content and reflects the state of the growth factor.According to their model, Eq. (?), the mammalian cell cycle with a mutated phenotype will cycle through the eight states given in (2) even when CycD is inactive.
Variables E2F, CycE, CycA, p27, Cdc20, Cdh1, UbcH10, CycB.o o [CHANGE ACCORDINGLY; We find that there are three edges in the wiring diagram that can possibly be used as control for avoiding the cycle in (2).These edges, given in red in Fig. 1, were identified following Method 1 applied on transitions in the 7-cycle.We note that avoiding transition(s) from the cycle through edge deletion in the wiring diagram does not guarantee that there will be no cycles in the resulting state space; it only guarantees the removal of the particular cycle that contains the targeted transition.For example, deleting Cdh1!UbcH10 results in a single fixed point, (101100110).Deleting the other two edges, on the other hand, removes cycle (2) but creates other cycles: the deletion of E2F!CycE results in another 7-cycle; deleting CycB!Cdc20 has the consequence of Figure 5: The cycle which persists in the absence of a growth factor the mutated phenotype (Rb inactive).The state space is defined as (E2F CycE CycA p27 Cdc20 Cdh1 Ubc CycB).
following Method 2.1 applied on transitions in the cycle with the objective of eliminating the cycle and also leading the system to fixed point(s) where p27 = Cdh1 = 1 as in a normal cell.The results are summarized in Table 2. Other attempts at control produced fixed points and/or cycles where p27 = Cdh1 = 0.

Discussion
In the Methods section, we noticed that the bound from Method 2.2 is sharp.This was demonstrated using the p53 model.In general, the exact bound from part (a) of Method 2.2 is achieved when F t (x, u t,k ) = x k or x k (as a function), where F t (x, u t,k ) is the function obtained from f t by plugging in the canalizing input of the variables that are more or equally dominant to x k .The bound from Method 2.2 can also help choose which edge to delete or constantly express when there is more than one option with the purpose of controlling the side effects resulting from an edge manipulation.If it is desirable to minimize the impact on the state space, thus avoiding possible negative side effects on the system, one should choose for control an edge whose input variable is in the least dominant layer possible in the target function.That is, if x i and x j are both canalizing variables in f t and x j is in a less dominant layer than x i , then one should choose to delete or constantly express the edge x j → x t since according to the bound of Method 2.2, maximum impact of this control is smaller than if x i → x t is manipulated.
It is important to point out that Method 2.1 only guarantees that a certain transition will be and one may be able to use this to remove a cycle from a state space as we did for the p53 model.However, the method does not guarantee that the system will not contain other cycles since removing transitions from a cycle destroys the cycle but may also create a different one; nor does it guarantee that the resulting fixed point will be exactly the desired one as it was observed in both applications.To find controllers that give the desired fixed points one could use the algebraic methods described in [26].
This paper considers edge manipulations as potential control actions to avoid undesirable attractors.Control through edge manipulations in the wiring diagram of a network has been previously considered in [4].Although the authors of [4] consider edge additions in the wiring diagram as control actions, that is, by adding new regulators to the existing set of regulators that help the system to transition into a desirable attractor.

Conclusions
The structure of the canalizing variables in a biologically relevant Boolean rule plays an important role in the control of Boolean networks.Special combinations of canalizing inputs can help identify network controllers and the canalizing structure of a Boolean function allows to estimate the number of transitions that change after using the type of controllers proposed in this paper.Moreover, the hierarchy of the canalizing variables can be used for assessing the impact on the network dynamics as a result of a given control.This paper exploits the canalizing properties of Boolean rules to derive a method that can be useful for identifying control targets for avoiding undesirable states.Additionally, it provides a method for assessing the impact of the controllers on the dynamics of the uncontrolled network.Thus these two complementary methods can help in the selection of appropriate controllers.Method 2.1 gives a practical way for identifying the potential edges to control in the wiring diagram of a network for avoiding undesirable state transitions.Method 2.2, on the other hand, provides a measure of the impact of an edge deletion onto the state space of a model and establishes that this impact differs significantly based on the canalizing properties of the nodes involved: an edge coming from a node with stronger canalization, represented in the model by a variable in a more dominant layer, has exponentially higher probability to change the state space than an edge from a node with weaker canalization, represented by a variable in a less dominant layer.Therefore, Method 2.2 is a useful tool for assessing the impact of the controllers identified by Method 2.1 on the dynamics of the system providing a way for selecting desirable controllers.

Figure 1 :
Figure 1: Flowchart of the procedure for identifying edges as control candidates.

Figure 2
Figure 2 Wiring diagram of the p53 pathway fro Pointed arrows correspond to positive regulatory and blunt arrows to negative regulatory interacti edge deletion control candidate is given as a dot and the two constant expression control candida denoted by thicker edges.

Figure 2 :
Figure 2: Wiring diagram of the p53-mdm2 model from[17].Pointed arrows correspond to positive regulatory interactions and blunt arrows to negative regulatory interactions.The edge deletion control candidate is given as a dotted edge and the two constant expression control candidates are denoted by thicker edges.

Figure 1
Figure 1 Wiring diagram Pointed arrows correspo and blunt arrows to neg edge deletion control ca and the two constant e denoted by thicker edge

Figure 3 :
Figure 3: The cycle for the Boolean network of the p53 pathways in the presence of DNA damage.The state space is defined as (ATM p53 Wip1 Mdm2).
WHEN I TRY TO REMOVE THE CYCLE THR CONSTANT EXPRESSION, THE ONLY CAN DATE IS Cdc20!CycA BECAUSE Rd IS ALWA 0 WHEN CycD=1.HOWEVER, THIS DELETIO RESULTS IN ANOTHER 7-CYCLE.NOT TA ABOUT CONSTANT EXPRESSION IN THIS E AMPLE?]

Figure 3 Figure 4 :
Figure3Wiring diagram of the mutated (CycD and Rb inactive) mammalian cell-cycle network.Pointed arrows correspond to positive regulatory interactions and blunt arrows to negative regulatory interactions[3].The deletion of the dotted edge results in removing the undesirable cycle from the state space of the system and introducing a single fixed point with p27 and Cdh1 both active.

0 Table 1 :
Truth table for the Boolean functions in Example 1.1.
If you wish to avoid a cycle or other trajectory that contains several states, you can repeat the above steps on all states in order to find all control options.6.If you find multiple edges as candidates for control, you may want to choose to delete the ones that have the smallest impact on the state space thus minimizing the side effects of edge manipulation.
3. Begin with, say, the leftmost 1 (or 0) in u.Observe the canalizing structure of the functions in your model.Then check if the conditions of Method 2.1 are satisfied.4.If some of the conditions of Method 2.1 are not met or you are looking for other control options, proceed to the next 1 (or 0) in u.5.

Table 2 :
Summary of control results for the mutated cell-cycle network.