Detecting small attractors of large Boolean networks by function‐reduction‐based strategy

Abstract

Boolean networks (BNs) are widely used to model gene regulatory networks and to design therapeutic intervention strategies to affect the long‐term behaviour of systems. A central aim of Boolean‐network analysis is to find attractors that correspond to various cellular states, such as cell types or the stage of cell differentiation. This problem is NP‐hard and various algorithms have been used to tackle it with considerable success. The idea is that a singleton attractor corresponds to n consistent subsequences in the truth table. To find these subsequences, the authors gradually reduce the entire truth table of Boolean functions by extending a partial gene activity profile (GAP). Not only does this process delete inconsistent subsequences in truth tables, it also directly determines values for some nodes not extended, which means it can abandon the partial GAPs that cannot lead to an attractor as early as possible. The results of simulation show that the proposed algorithm can detect small attractors with length p = 4 in BNs of up to 200 nodes with average indegree K = 2.

1 Introduction

From a translational perspective, modelling gene regulatory networks provides a mathematical basis for system‐based optimal therapeutic strategies. Boolean networks (BNs), and the more general class of probabilistic BNs, are one of the most popular models for gene regulatory networks. Although the BN model is very simple, its dynamic process is complex and can yield insight into the global behaviour of genetic regulatory networks [1 ]. From any initial condition, the system eventually evolves into a limited set of stable states called attractors. The set of states leading the system to a specific attractor is called the basin of attraction. An attractor containing only one state is called a singleton attractor; otherwise, it is called a cyclic attractor. The function of attractors is subject to two different interpretations: one interpretation, which follows Kauffman [2 ], is that each attractor should correspond to a cell type. The other interpretation of attractors is that they correspond to the states of cell growth, differentiation and apoptosis [3 ]. Under the latter interpretation, cyclic attractors should correspond to cell cycles (growth) and singleton attractors should correspond to differentiated or apoptosis states. These two interpretations are complementary because one cell type can have several neighbouring attractors and each of them corresponds to different states of cellular function [4 ].

The goal of systems biology is to design strategies to intervene in the system. In external intervention, we can design a policy for each state x so that the system evolves towards the expected attractor as quickly as possible. In structure intervention, we can either flip outputs of Boolean functions on some input states or delete some regulatory relationships between gene pairs so that the basin of undesirable attractors becomes small. Thus, identifying attractors and their corresponding basin is a fundamental problem in system intervention.

The total number of states for a BN with n genes is 2 ⁿ . Akutsu et al. [5 ] proved that finding a singleton attractor is NP‐hard. That is to say, the singleton attractor problem cannot be solved in polynomial time unless P = NP. The naïve method is to make an exhaustive search by examining all 2 ⁿ possible states of a network. However, this approach would be too time consuming and impractical for networks with hundreds of genes. Various tools have thus been developed to identify attractors for networks with few nodes. For example, Garg et al. [6 ] introduced the concept of binary decision diagrams (BDDs) and developed genYsis. Gonzalez et al. [7 ] developed a tool called GINsim that implements Thomas's program. Hinkelmann et al. [8 ] used algebraic methods to develop ADAM. Helikar and Rogers [9 ] developed the simulation platform ChemicalChains and Müssel et al. [10 ] developed BoolNet. Zheng et al. [11, 12 ] combined iterative methods with reduced order BDDs to identify attractors in synchronous BNs. In addition, the attractors of synchronous BNs are used in asynchronous Boolean translation functions to derive attractors of asynchronous scenarios. The ROBBD structure provides a compact representation of Boolean functions so that they can be efficiently stored. However, this method constitutes, in essence, an exhaustive search that must visit the entire state space.

To study the dynamics of BNs with hundreds of nodes, some researchers focused on simplifying the underlying networks. Naldi et al. [13 ] proposed a reduction method based on iteratively removing nodes without a self‐loop from the network. This method is proven to preserve the fixed points of a system, but it may introduce spurious oscillations into the reduced model. Subsequently, Veliz‐Cuba [14 ] adapted this method by further simplifying the redundant Boolean functions. Saadatpour et al. [15 ] proposed a two‐step network reduction method that iteratively merges simple mediator nodes (i.e. nodes with unity in‐degree and out‐degree).

Another direction is to use some heuristic pruning strategy to avoid extending partial gene activity profiles (GAPs) that do not lead to an attraction state. To reduce the search space, Zhang et al. [16 ] proposed an algorithm based on feedback vertex sets (FVSs) to identify singleton attractors and an outdegree‐based algorithm to identify cyclic attractors. The main idea is to stop extend those partial GAPs that violate the definition of attractors. To accelerate the pruning process, Berntenis and Ebeling [17 ] proposed constructing all possible partial GAPs from the first ten genes with the largest outdegree and then extending nodes whose parent nodes are already in the previously determined gene set. They developed a software package lnet to detect the fixed states in asynchronous BNs. The general process is similar to the outdegree‐based algorithm proposed in. [16 ].

In this paper, we propose a new method to accelerate the pruning process of identifying singleton attractors based on a function‐reduction strategy. In addition, by applying this method to high‐order BNs, small attractors can also be identified. The results on synthetic and biological networks indicate that the proposed algorithm performs better than the algorithms proposed in [16 ] and can be applied to networks with larger size. This paper is structured as follows: Background information and necessary definitions are provided in Section 2. The motivation behind our ideas and their implementation are presented in Sections 3 and 4, and the results of using the proposed algorithm on simulated networks and five biological networks are presented in Section 5. Finally, concluding remarks are given in Section 6.

2 Definitions

2.1 Boolean networks

A BN G (V, F ) is defined by a set of nodes V = {x ₁, …, x_n }, x_i ∈ {0, 1} and a set of Boolean functions F = {f ₁, …, f_n }, $f_i:\{0,1\}{\hspace{0.17em}}^{k_i}\to \{0,1\}$ . Each node x_i represents the expression state of a gene, where x_i = 0 means that the gene is off and x_i = 1 means that it is on. To update its value, each node x_i is assigned a Boolean function $x_i(t+1)=f_i\mathbf{(}{x}_{i1}(t),\dots ,x_{i{k}_i}(t)\mathbf{)}$ with k_i specific input nodes. Under the synchronous‐updating scheme, all genes are updated simultaneously according to their corresponding update functions. The network's state at time t is a GAP, which is denoted by a binary vector x (t ) = (x ₁ (t ), …, x_n (t )). In the absence of noise, the state of the system at the next time step is

$$x(t+1)=F\mathbf{(}{x}_1(t),\dots ,x_n(t)\mathbf{)}$$ (1)

Given an initial state, the network will eventually settle down and cycle endlessly through a set of states called an ‘attractor cycle’. The set of all initial states that reach a particular attractor cycle forms the ‘basin of attraction’ for the cycle. A consecutive sequence of GAPs is called an ‘attractor’ with period p if

$$x(t+p)=x(t)$$ (2)

An attractor with period 1 is called a singleton attractor and an attractor with period p > 1 is called a ‘cyclic attractor’. Fig. 1 and 2 shows a BN consisting of five genes and the corresponding truth table of Boolean functions. This BN contains the four singleton attractors 00000, 00100, 10011 and 11111 and one attractor cycle 11110 ⇄ 11010.

Fig 1.

A simple BN with five genes

Fig 2.

State‐transition diagram for BN defined in Fig. 1

2.2 p th‐order BNs

Given the BN G (V, F ), the state transition of its p th BN G (V, F ^{(p )} ) updates as follows

$$x(t+1)=F^{(p)}(x_1(t-p+1),\dots ,x_n(t-p+1))$$ (3)

The conventional BN is of order p = 1. The p th‐order parent nodes of gene x_i can be obtained by backtracking p steps. An interesting fact is that the states in the attractors of BN G (V, F ) are the same as those in BN G (V, F ^{(p )} ). An attractor A with period q in G (V, F ) will break into l attractors A ₁, …, A_l with period q /l in G (V, F ^{(p )} ), where l is the greatest common divisor of q and p. If p = q, then the attractor A will become p singleton attractors in G (V, F ^{(p )} ). Fig. 3 presents the state‐transition diagram of the second order BN defined in Fig. 1. The four original singleton attractors remain singleton attractors, whereas the cyclic attractor 11110 ⇄ breaks into two singleton attractors 11110 and 11010.

Fig 3.

State‐transition diagram for 2th‐order BN defined in Fig. 1

3 Algorithm

Although finding a singleton attractor is NP hard, developing practical algorithms to identify small attractors for networks with hundreds of nodes is possible. In this section, we first introduce two recursive algorithms proposed by Zhang et al. in [16 ] and then present our motivation and the corresponding algorithm. The core process is to extend a partial GAP according to a given gene ordering. If a partial GAP cannot be extended to an attractor, it is deleted and the next partial GAP is examined.

3.1 Two recursive algorithms

A FVS V ^FVS is a subset of nodes of V = {x ₁, …, x_n } whose state can completely determine the states of the other nodes x_i ∉ V ^FVS. If a FVS $V^{\mathrm{FVS}}=\left\{x_1^{\mathrm{FVS}},x_2^{\mathrm{FVS}},\dots ,x_M^{\mathrm{FVS}}\right\}$ exists for network G (V, F ), then checking the states of these nodes to search all attractors is sufficient. In addition, the search space is reduced from 2 ⁿ to 2 ^M (M < n ). The problem of finding a minimum‐feedback vertex set is known to be NP hard [18 ]. Zhang et al. [16 ] simply uses a greedy algorithm to find a (not necessarily minimum) ‘feedback vertex set’. For the BN in Fig. 1, V ^FVS = {x ₄, x ₅, x ₃, x ₁ } is a FVS.

Since the state of nodes with large outdegree can determine the state of more genes, they sort the nodes in V ^FVS according to their outdegree (nodes are ordered from larger to smaller outdegree). If we extend the nodes in V ^FVS according to this order, then these partial GAPs that cannot lead to attraction states will be deleted early in the recursive process. The FVS‐based algorithm extends a partial GAP by one node in V ^FVS at a time. At the m th recursive step, the states of the first m −1 genes are determined. This algorithm first extends the partial GAP by adding $x_m^{\mathrm{FVS}}(t)=0$ . If $x_j^{\mathrm{FVS}}(t)=x_j^{\mathrm{FVS}}(t+1)$ or $x_j^{\mathrm{FVS}}(t+1)$ is not determined for all for j = 1, …, m, the algorithm proceeds to the next recursive step. In other words, if a possibility exists that the current partial GAP can be extended to a singleton attractor, the algorithm proceeds to the next recursive step. Next, it extends the partial GAP by adding $x_m^{\mathrm{FVS}}(t)=1$ and executes a similar procedure. Finally, the algorithm returns to the previous step. Since the parent nodes of some nodes in V ^FVS may be the nodes x_i ∉ V ^FVS, the current values of nodes $x_j^{\mathrm{FVS}}(t)$ must be propagated to nodes x_i ∉ V ^FVS, which are needed to check if $x_j^{\mathrm{FVS}}(t)=x_j^{\mathrm{FVS}}(t+1)$ holds. The pseudocode of the FVS‐based procedure to determine singleton attractors is summarised in Algorithm 1 (see Fig. 4 ).

Fig 4.

FVS‐based algorithm to determine singleton attractors

To find attractors with any period p requires checking whether x_j (t + p ) = x_j (t ). Zhang et al. proposed a simple outdegree‐based algorithm which extends the entire node set V instead of the FVS V ^FVS. The pseudocode describing the general procedure to determine cyclic attractors of period p is summarised in Algorithm 2 (see Fig. 5 ).

Fig 5.

Outdegree‐based algorithm to determine attractors with any period p

3.2 Function‐reduction‐based algorithm

Here we focus on determining the singleton attractors. Equation (2 ) shows that the state of each gene in a singleton attractor satisfies x_i (t ) = x_i (t + 1) (1 ≤ i ≤ n ). Our motivation is that the problem of identifying a singleton attractor can be considered as the problem of finding n consistent binary subsequences in the truth table of Boolean functions. A subsequence is defined as the binary value $x_{i_1}(t)\cdots x_{i_{k_i}}(t)x_i(t+1)$ in the Boolean function of gene x_i . Consistence means that the value of x_i (t ) in any subsequence satisfies x_i (t ) = x_i (t + 1). For example, 0111, 0110, 1010, 11 and 1111 in Table 1 are five consistent subsequences that can be assembled into a singleton attractor 10011. In these subsequences, the value of x ₅ (t ) is 1 in the input states 011, 011, 101 and 111 for genes x ₁, x ₂, x ₃ and x ₅ and the value of x ₅ (t + 1) corresponding to the input state 111 is also 1.

Table 1.

Truth tables for partial GAP x ₄ = 1 for BN in Fig. 1

	x 1		x 2		x 3		x 4		x 5
	245	1	345	2	135	3	4	4	145	5
1	010	1	010	1	000	0	1	1	010	0
2	011	1	011	0	010	1			110	0
3	110	1	110	1	011	1			111	1
4	111	1	111	1	101	0
5					111	1

A partial GAP is actually a constraint on the truth table, which can be used to delete subsequences that violate it in each function. Given a partial GAP x ₁, …, x_m = a ₁, …, a_m (a_j = {0, 1},1 ≤ j ≤ m ), the original Boolean‐function set F will be reduced to a smaller conditioned Boolean‐function set $F_{x_1,\dots ,x_m=a_1,\dots ,a_m}$ . If the current partial GAP is x ₄ = 1, the original Boolean‐function set F becomes $F_{x_{{}_4}=1}$, which corresponds to the truth tables shown in Table 1. We see that x ₁ (t + 1) ≡ 1, which means that the state of gene x ₁ is frozen to 1, provided x ₄ (t ) = 1. This indicates that x ₁ = 1 holds if there is a singleton attractor with x ₄ = 1. Thus, we need only extend the partial GAP x ₄ = 1 to x ₄ x ₁ = 11 without considering the partial GAP x ₄ x ₁ = 10. The corresponding truth tables for $F_{x_4{x}_1=11}$ are shown in Table 2. We find x ₅ (t )≡1, which means that x ₅ (t ) is 1 in all input states of gene x ₃ provided x ₄ (t )x ₁ (t ) = 11. If a singleton attractor exists that satisfies x ₄ x ₁ = 11, then x ₅ = 1. Similarly, we need only extend the partial GAP x ₄ x ₁ = 11 to x ₄ x ₁ x ₅ = 111 without considering the partial GAP x ₄ x ₁ x ₅ = 110. The truth table for $F_{x_4{x}_1{x}_5=111}$ is shown in Table 3, where we observe that neither x ₂ (t ) and x ₃ (t ) nor x ₂ (t + 1) and x ₃ (t + 1) are constant for any function. We decide to extend x ₄ x ₁ x ₅ = 111 to x ₄ x ₁ x ₅ x ₃ = 1110 and x ₄ x ₁ x ₅ x ₃ = 1111 because the outdegree of gene x ₃ is larger than that of gene x ₂. Considering now the partial GAP x ₄ x ₁ x ₅ x ₃ = 1110, we see from Table 4 that x ₂ (t + 1)≡0, which reaches to a singleton attractor 10011. Applying the same procedure to the partial GAP x ₄ x ₁ x ₅ x ₃ = 1111 can reach the singleton 11111.

Table 2.

Truth tables obtained from partial GAP x ₄ x ₁ = 11 in Table 1

	x 1		x 2		x 3		x 4		x 5
	245	1	345	2	135	3	4	4	145	5
1	010	1	010	1	101	0	1	1	110	0
2	011	1	011	0	111	1			111	1
3	110	1	110	1
4	111	1	111	1

Table 3.

Truth tables obtained from partial GAP x ₄ x ₁ x ₅ = 111 in Table 1

	x 1		x 2		x 3		x 4		x 5
	245	1	345	2	135	3	4	4	145	5
1	011	1	011	0	101	0	1	1	111	1
2	111	1	111	1	111	1

Table 4.

Truth tables obtained from partial GAP x ₄ x ₁ x ₅ x ₃ = 1110 in Table 1

	x 1		x 2		x 3		x 4		x 5
	245	1	345	2	135	3	4	4	145	5
1	011	1	011	0	101	0	1	1	111	1
2	111	1

In above procedure, if a partial GAP does not lead to a singleton attractor (i.e. x ₄ x ₅ = 01), it will result at some step in at least one null function (i.e. f ₅ ) in the conditioned truth table, and then we should stop to extend this partial GAP. To delete more inconsistent subsequences in the truth table as early as possible, we also rank the nodes x_i based on their outdegree. Specifically, r_i = d_i + 1 if x_i is self‐regulated; otherwise, it is defined as the outdegree d_i . The intuition is that the constraint x_i (t ) = x_i (t + 1) can directly delete those inconsistent subsequences in the truth table of the self‐regulated genes (i.e. x ₃, x ₄, x ₅ ). The pseudocode of the proposed FR‐based algorithm is summarised in Algorithm 3 (see Fig. 6 ).

Fig 6.

FR‐based algorithm to determine singleton attractors

Fig. 7 shows the search tree of both the FVS‐based algorithm with V ^FVS = {x ₄, x ₅, x ₁, x ₃ } and the proposed FR‐based algorithm for the BN from Fig. 1. The proposed algorithm also starts with the node with the largest rank, which in this case is node x ₄, which has a rank of five. The search tree for our algorithm is much smaller than that for the FVS‐based algorithm, which we attribute to the fact that, given a specific partial GAP, our algorithm can directly determine the value of some nodes whereas the FVS‐based algorithm must extend it sufficiently so that the state of some extended nodes can be determined at the next time step. For example, only when the partial GAP x ₄ = 0 is extended to the longer partial GAPs x ₄ x ₅ x ₁ = 010, x ₄ x ₅ x ₁ = 011, x ₄ x ₅ x ₁ x ₃ = 0010 and x ₄ x ₅ x ₁ x ₃ = 0011 can the FVS‐based algorithm detect the conflict x ₅ (t )≠x ₅ (t + 1) for the first two partial GAPs and x ₃ (t )≠x ₃ (t + 1) for the last two partial GAPs. However, the proposed algorithm directly determines the value of nodes x ₅, x ₂ and x ₁ to be 0, 1 and 0, respectively, so that it requires less iterations than the FVS‐based algorithm. In the extending process, both algorithms have to scan the truth table. The difference is that the FVS‐based algorithm may not scan the full table in each iteration, whereas the proposed algorithm scans the entire current conditioned truth table. As the scan of the truth table can be a constant, the average time of both algorithms is dominated by the number of iterations. Thus, the proposed FR‐based algorithm should be less time consuming than the FVS‐based algorithm.

Fig 7.

Search tree of Algorithms 1 and 3 (Figs. 4 and 6 ) for identifying singleton attractors

a FVS‐based algorithm with V ^FVS = {x ₄, x ₅, x ₁, x ₃ }

b FR‐based algorithm

4 Implementation

A relevant node is one that eventually influences its own state. A set including all relevant nodes can be obtained by iteratively removing nodes that have either frozen and become part of the frozen core or are non‐frozen, in which case they influence only irrelevant nodes. The algorithm to determine the relevant node set is very complicated. Kaufman et al. [19 ] proposed a stochastic process that removes frozen nodes and non‐frozen irrelevant nodes from a network and ends when there are only potentially relevant nodes left. In this work, we first delete the frozen nodes and then iteratively remove nodes that do not regulate other nodes.

We used Visual C++ to code a single‐threaded program to identify the attractors of BN. The only hardware requirement is the available random access memory. To compare the performance of the three algorithms, we also coded the two algorithms proposed by Zhang et al. The network can be loaded from a text file, where the first line corresponds to the gene number n in the network. The second line is the indegree k_i of each gene. It then follows that, for each gene, n rows correspond to the indices of the input genes. Finally, the Boolean functions are followed by one gene per line. The source code is freely available from the authors.

5 Results

5.1 Simulation on synthetic networks

In this section, we present the results of the three algorithms on random BNs. All of the numerical experiments were done on a windows computer with a Core 2.33 GHz processor and 2 GB of physical memory. For each gene number n (n = 20, 30, 40, 50) and each indegree K (K = 2, 3,…, 10), we randomly generate 200 BNs and take the average values. Fig. 8 shows the average time for the three methods to identify the singleton attractors for each n and K. Given a specific n, the average time increases exponentially with the average indegree K. The larger the indegree K, the more complex the regulation relationship. It is easy to understand that the average time to reach a conflict x_i (t )≠ x_i (t + 1) increases with the average indegree K.

Fig 8.

Average elapsed time as a function of the average indegree K for the three algorithms applied to random networks for gene numbers n = 20, 30, 40, 50

Concerning the performance of the three algorithms, the simulation results show that the FVS‐based algorithm performs better than the general outdegree‐based algorithm and that the proposed FR‐based algorithm performs best in most cases. Fig. 9 shows the average number of iterations required by the three algorithms. It shows that the proposed FR‐based algorithm requires less partial GAPs to be extended than the other two algorithms, which is the main reason why it performs best.

Fig 9.

Number of iterations required by the three algorithms to identify singleton attractors for average indegree K = 2

To show the scalability of the proposed FR‐based algorithm, we generate 200 networks with n = 50 ∼ 450 and K = 1 ∼ 5. Fig. 10 shows the average time required to identify singleton attractors. When the average indegree K is very small (i.e. K = 1, 2), the proposed FR‐based algorithm can quickly identify singleton attractors for networks with thousands of nodes. However if the average indegree K > 3, the average time will increase dramatically with the size of the networks. When the average indegree K = 5, the proposed FR‐based algorithm can identify all singleton attractors for networks with about 120 nodes in about 2 h.

Fig 10.

Average elapsed time as a function of number of nodes of the function‐reduction‐based algorithm for relatively large gene numbers and for indegrees K = 1, 2, 3, 4, 5

Fig. 11 shows the average time of the proposed FR‐based algorithm and the outdegree‐based algorithm to identify attractors with period p = 1 and p = 2 on networks with K = 2. We can see that the proposed FR‐based algorithm performs significantly better than the general outdegree‐based algorithm. In addition, it generally takes more time to identify attractors with period p = 2 than those with p = 1. For the outdegree‐based algorithm, it has to extend more partial GAPs to reach a conflict x_i (t + 2)≠x_i (t ). For the FR‐based algorithm, the average indegree K in the second‐order network is generally greater than that in the first‐order network. For example, the first‐, second‐ and third‐order parent nodes of node x ₁ in Fig. 1 are {x ₂, x ₄, x ₅ }, {x ₁, x ₃, x ₄, x ₅ } and {x ₁, x ₂, x ₃, x ₄, x ₅ }, respectively. According to Fig. 10, the increase of the average indegree K in the second‐order network results in the increase of the average time of the FR‐based algorithm. The main limitation of the proposed FR‐based algorithm is the available random access computer memory. According our numerical experiment, for networks with n = 200 and K = 2, the proposed FR‐based algorithm can identify attractors with period p ≤ 4 in an acceptable time. Genetic regulatory networks are commonly believed to have sparse connectivity topology; in other words, most genes are regulated by a few genes. The proposed FR‐based algorithm should identify attractors with period p = 4 in larger biological networks. In the present work, it still remains challenging to apply the current algorithm to networks with larger K because the average indegree K in high‐order BNs generally increases the order p.

Fig 11.

Average elapsed time to identify attractors with period p = 1 and p = 2 for indegree K = 2

5.2 Benchmark networks

Because the FVS‐based algorithm only identifies singleton attractors, we applied the FR‐based algorithm and the outdegree‐based algorithm to five benchmark networks: mammalian cell [20 ], protein‐ex [21 ], T‐helper [22 ], T‐cell receptor [23 ] and dendritic cell [6 ]. Table 5 presents the time for both algorithms for the parameters p = 1, 2, 3, 4. The performance of the FR‐based algorithm is clearly better in general than that of the outdegree‐based algorithm.

Table 5.

Average time for FR‐based algorithm and outdegree‐based algorithm on five benchmark networks (time in milliseconds)

Benchmark	n	K	Gene FAtt	FR‐based				Outdegree‐based
				p = 1	p = 2	p = 3	p = 4	p = 1	p = 2	p = 3	p = 4
Mammalian Cell	10	3.5	4	0.2	1.4	4.2	6. 6	0.2	0.5	1.4	4.7
Protein‐ex	16	1	710	0.2	0.2	0.3	0.6	0.2	0.3	0.6	1.1
T‐helper	23	1.7	6	0.8	2.0	4.1	32.2	11.4	42.8	132	915
T‐cell Receptor	40	1.4	10 800	0.3	0.5	1.3	4.8	2	11.6	30.9	251.4
Dendritic Cell	111	1.4	2	0.3	0.3	0.8	2.0	39.7	59.4	78.6	97.4

We also show the time required by the software package geneFAtt, which was developed by Zheng et al. [11 ]. Of the five networks, the FR‐based algorithm takes much less time to identify small attractors for protein‐ex and T‐cell receptor than geneFAtt, whereas it takes longer than geneFAtt for T‐helper. They take about the same time for the other two networks, mammalian cell and dendritic cell. This further demonstrates the effectiveness of the proposed FR‐based algorithm for identifying small attractors. The main difference between geneFAtt and the FR‐based algorithm is that the former is essentially an exhaustive method whereas the latter adopts a heuristic strategy. This allows it to be applied to larger networks.

6 Conclusion

A salient problem in translational genomics is to design therapeutic intervention strategies for gene regulatory networks. Identifying attractors in BNs is thus a key issue for designing effective intervention policies. Although much research has focused on addressing this problem, its application to large networks still remains a problem. Since the state space of BNs increases exponentially with the number of nodes, an heuristic search is a potential strategy to solve this problem.

In [16 ], Zhang et al. first proposed several heuristic searching algorithms which delete the partial GAPs that violate the definition of attractors. Based on this method, we propose herein a FR‐based algorithm to identify singleton attractors. The main idea is to use reduced Boolean functions on an extended partial GAPs to determine whether it violates the definition of singleton attractors and to find if frozen nodes exist that have not been extended at the same time. The earlier the determination of the value of some frozen nodes, the more we can delete states that cannot lead to a singleton attractor. Applying this algorithm to the corresponding p th‐order Boolean network also allows cyclic attractors with small period p to be identified. Simulation results show that the FR‐based algorithm is more efficient than those proposed by Zhang et al. in [16 ]. Given BNs with n = 200 and K = 2, the FR‐based algorithm can quickly and reliably identify small attractors with p ≤ 4.

In addition, the authors in [13 –15 ] use the reduction methods in an attempt to delete nodes or relations that have no effect on the dynamics of the network. In this case, the size of the resulting networks and their state space becomes small. For the proposed FR‐based algorithm, the reduction of the function is based on a specific partial GAP. The goal is to find whether a conflict exists or whether some frozen nodes exist whose values have not been extended. The size of the state space of the system does not change. In fact, the aforementioned reduction methods can be used on any algorithm in preprocessing.

Identifying small attractors is just the first step for studying dynamics of a system; the size of their basin is also important for understanding the dynamic behaviour of systems and for designing potential control strategies. Given a network with 100 nodes, the size of its state space is 2¹⁰⁰. Determining the size of each basin of attraction is an incredible feat. Developing methods to estimate the size of the basin of attraction is another urgent problem to be studied.

7 Acknowledgments

This work was funded in part by the National Science Foundation of China (grant nos. 61272018, 60970065 and 61174162) and the Zhejiang Provincial Natural Science Foundation of China (grant Nos. R1110261 and LY13F010007).