Polynomial-Time Algorithm for Controllability Test of a Class of Boolean Biological Networks
© Koichi Kobayashi et al. 2010
Received: 12 April 2010
Accepted: 17 June 2010
Published: 8 July 2010
In recent years, Boolean-network-model-based approaches to dynamical analysis of complex biological networks such as gene regulatory networks have been extensively studied. One of the fundamental problems in control theory of such networks is the problem of determining whether a given substance quantity can be arbitrarily controlled by operating the other substance quantities, which we call the controllability problem. This paper proposes a polynomial-time algorithm for solving this problem. Although the algorithm is based on a sufficient condition for controllability, it is easily computable for a wider class of large-scale biological networks compared with the existing approaches. A key to this success in our approach is to give up computing Boolean operations in a rigorous way and to exploit an adjacency matrix of a directed graph induced by a Boolean network. By applying the proposed approach to a neurotransmitter signaling pathway, it is shown that it is effective.
Various approaches to modeling, analysis, and control synthesis of biological networks such as gene regulatory networks and metabolic networks have been recently developed in the control community as well as the theoretical biology community . In these approaches, it is one of the final goals to develop systematic drug discovery and cancer treatment [2, 3]. Biological networks in general can be expressed by ordinary/partial differential equations with high nonlinearity and high dimensionality. Since such complexities cause difficulties in analysis and control design, various simpler models such as Petri nets, Bayesian networks, Boolean networks, and hybrid systems have been proposed for dealing with complex and large-scale biological networks at the expense of rigorous analysis (see e.g., [4, 5]).
This paper discusses the controllability problem of biological networks. In gene regulatory networks, for example, the controllability problem is defined as the problem of determining whether expressions of genes of interest can be arbitrarily controlled by expressions of a specified set of the other genes. As far as we know, two approaches to the controllability analysis of such biological networks have been developed so far: a piecewise affine model-based approach and a Boolean network model-based approach. However, the former approach can be applied to only the class of relatively low-dimensional systems [6, 7].
On the other hand, a Boolean network model, where binary state variables are assigned to nodes and the transition rules of the state are given by Boolean functions [8, 9], will be more practical for analysis of large-scale biological networks thanks to its bold simplification. Akutsu et al. have recently discussed the controllability problem of Boolean networks with control nodes and controlled nodes and have proven that this problem is NP-hard in a general setting . Furthermore, they have proposed a polynomial-time algorithm for the classes of networks including a tree structure or at most one loop, and an exponential-time algorithm for the other classes. Indeed there is a criticism that a Boolean network model is too simple as a model of biological networks, but for large-scale networks it will be able to provide some indication or clue towards further detailed analysis. Thus various approaches based on this model have been well-studied so far (see e.g., [11–19]).
Motivated by the theoretical results in , this paper also focuses on the controllability problem of Boolean networks with control nodes and controlled nodes and proposes a sufficient condition for the Boolean network to be controllable, which can be easily verified by a polynomial-time algorithm. Our standing point is to give up computing complex Boolean operations in a rigorous way and to focus on deriving an easily-checkable sufficient condition for controllability so as to be applied to large-scale networks. The obtained algorithm is based on simple operations on an adjacency matrix of a directed graph induced by a Boolean network. This is a remarkable point of our approach, different from the method in , and enables us to apply our approach to a wider class of Boolean networks including nontree structures.
First, after the definition of controllability of Boolean network models with control nodes and controlled nodes is described, a sufficient condition for the controllability is derived in the form of an algorithm. Next, the computational complexity for the algorithm is discussed to show that it is a polynomial-time algorithm. In addition, PC-based numerical experiments show that the obtained algorithm is applicable to a class of Boolean networks with at least 1000 nodes. Finally, as an illustrative example, the proposed algorithm is applied to the Boolean network model of a neurotransmitter signaling pathway , which expresses an interaction pathway between the glutamatergic and dopaminergic receptors. Note that the polynomial-time algorithm proposed in  cannot be always applied to this problem. This Boolean network model consists of 16 nodes, and the problem of simultaneously controlling two important nodes among them, that is, concentration of exocytosis and phospholipase C, is discussed based on the proposed algorithm. As a result, we show that for example, they can be simultaneously controlled by keeping substance concentration at the other 4 nodes constant with appropriate values.
Notation 1. Let denote the set of nonnegative integers and the set of matrices consisting of elements and . We also denote by and the identity matrix and the zero matrix, respectively. For simplicity of notation, we sometimes use the symbol instead of and the symbol instead of . Let express the transpose of the matrix .
2. Boolean Network Models
This section provides a brief review on a Boolean network model [8, 9]. A Boolean network model consists of a set of nodes and a set of regulation rules for nodes, where each node expresses a gene, a molecule, or an event in the genetic network. The state variable at node takes a Boolean value of or representing "inactive" or "active" status of the node, respectively. A regulation rule for each node is given in terms of a Boolean function, and each node state changes synchronously.
Since the caspase C3a is responsible for cleaving or breaking many other proteins, a high-level of the C3a concentration, that is, implies cell near-death; otherwise, cell survival. As seen in (1), if the concentration of IAP is high ( ) or the concentration of the caspase C8a is low ( ), then the concentration of C3a gets low, that is, . On the other hand, and at the next time depend on the value of as well as . In this way, some dynamical interactions exist. See [21, 22] for further details.
where is the state vector at time , and is a Boolean function, where logical operators consist of AND ( ), OR ( ), NOT ( ), and XOR ( ).
3. Problem Formulation
In a Boolean network model (2), the state is uniquely determined by giving the initial state , which implies that (2) is an autonomous system and has no control nodes.
Furthermore, the product of and in expresses a product operation on matrices/vectors of the real number field. Thus the above condition on guarantees that the output is the state variable itself, that is, for each there exists such that holds. The case of is also included here. This condition on will not be restrictive in analyzing controllability of biological networks such as gene regulatory networks, since the relation on regulation among genes/molecules will be mainly discussed there.
For the system of (3), the notion of output-controllability is defined as follows.
Definition 1. Suppose that for the system of (3), the finite time and the initial state are given. Then the system is said to be -output-controllable at if for every , there exists a control input sequence , , such that . Furthermore, the system is said to be -output-controllable if it is -output-controllable at every .
The above notion of controllability comes from the fact that, for example, in control of genetic networks we often would like to determine if expressions of certain gene of interest (corresponding to ) will be able to be inhibited (or activated) by means of appropriately adjusting the expressions of a given set of genes (corresponding to ). It is remarked that we assume that the control time is explicitly specified in the above definition.
can be treated by assumption. Then let us verify the -output controllability of the system (5). As discussed in Section 2, expresses cell near-death, and expresses cell survival. So we would like to know if the system is -output-controllable with respect to the output . Suppose that (i.e., the initial states of IAP, C3a and C8a are all low-level), (i.e., ), and . Then since holds independently of by simple calculation, we see that system (5) is not -output-controllable at , which implies that we cannot control the system from the state "cell survival" within 2 time steps no matter how the control value of is given.
(i.e., ). Then since and are obtained, we see that system (5) is -output-controllable at , for example, (a) for , , (b) for , , (c) for , , and (d) for , . This implies we can simultaneously control the value of and at . In this way, the proposed controllability enables us to verify the existence of a control input sequence such that the output has the desired value in a given finite time, and the obtained result indicates how to give the value of a control input sequence.
which is given by . Although this model is very simple, it provides significant clues to address this problem. For the Boolean network model (9), we can consider three possible specifications, choosing either , , or to be the control input for the system.
So if , holds irrespective of the value of , similarly for the case of . Therefore, we see that system (10) is not -output-controllable. In the same way, we see that system (10) is not -output-controllable in every case of , , and for .
Thus we see that the system is not -output-controllable for , while that the system is -output-controllable with for both cases of and .
which implies that the system (14) is not -output-controllable.
Note that for (11) with , we see that the controllability property does not hold due to the fact that directly depends on . On the other hand, for (13) with , is adjacent to and in the Boolean network, which implies that is arbitrarily given by and . In a similar way, is adjacent to . However, cannot be realized by and because always holds when . These examples are very important in discussing the controllability in a Boolean network, that is, if the Boolean function of includes an initial state , or includes the same input in the outputs at the same time, then the system in question is not -output-controllable. In the following section, by motivating the above discussion, we will consider to derive a controllability condition.
Remark 1. In the above example, we assume that when some genes are identified as control inputs, the original dynamics of the corresponding genes can be ignored. However, in the case that the corresponding gene has a strong interaction with other genes, this assumption may not be suitable. One of methods for coping with such a case is to add a new gene (node) that works as the control input , where it is called an external control node. Our approach below can be also applied to this case.
4. Output-Controllability Condition
This section presents a sufficient condition for the system (3) to be -output-controllable in the form of an algorithm.
where is any Boolean function of a vector of binary variables. Obviously such identities on or affect the controllability in a Boolean network (note that even if , holds irrespective of ).
So in Boolean biological networks, there exists the case that identities are appeared. However, identities may not be appeared in the real biological relevance. The reasons why such identities are appeared are that the state is binarized and that a time-delay of the state is ignored. To overcome the latter point, a temporal Boolean network model has been proposed in . However, identities may appear even in a temporal Boolean network. The output-controllability condition proposed below can be similarly applied to a temporal Boolean network model.
Thus first of all, we will focus on finding such identities in before discussing a kind of initial condition and a kind of input-independency. This will require the introduction for several symbols.
The following assumption is made.
Assumption 1. The Boolean function in (3) has no redundant variables.
subject to .
For the system (21), expresses whether there exist paths between and , and , or and for any given . In the case of (26), we see that is adjacent to and . In other words, expresses which elements of , , and are variables of a Boolean function representing . However, note here that from , we cannot specify an explicit form of the Boolean function in question.
where , , and . Let also , , and denote each element of , , , respectively. If holds, then there exist paths between and . For the state , , of the system (3), let express the index set of elements of operated by the logical NOT as . In a similar way, for the control input , , of the system (3), let express the index set of elements of operated by the logical NOT as . Here, holds. In addition, there is a one-to-one correspondence between each element of , and the index of . Let and express the index of corresponding to and , respectively. In the case of the system (23), , hold, and for , holds.
4.2. Proposed Algorithm
Now we are in a position to propose a -output-controllability test algorithm. Since this kind of problem is NP-hard , we pay our attention on deriving a sufficient condition for the controllability. Although this sufficient condition is given in the form of an algorithm, it is somewhat complex. Thus before describing an algorithm, we describe the outline of the algorithm.
First, we consider a necessary condition for to include identical equations. From Figure 2 of the example (23), we see that in (18), which has no identities, has two paths from , and that is connected to some node on the paths. In this way, if some identical equation exists in , there always exist more than 2 paths from to some state and also the logical-NOT operations exist on the paths, which is a necessary condition and not necessarily a sufficient condition. Since it will spend huge time to rigorously specify the existence of identities for a large network, we consider here to exclude the cases satisfying the above necessary condition, that is, we do not determine here the controllability in such cases.
as a result of recursive calculation (see Section 6 for such an example), where , are some Boolean functions. This system is obviously -output controllable because each is expressed by different and no exists in . From the viewpoint of adjacency relation, this implies that there exists no path between and , there exists at least one path from each to some , and each has a path with only one or has no path to any . This can be easily found from the adjacency matrix, although it is a sufficient condition for the controllability. This is a rough story of our approach.
The proposed algorithm is given as follows.
Algorithm 1 (T-output-controllability test algorithm).
Part A: Check of the Existence of Identical Equations.
Step 1. Set . Compute , , and .
Step 2. If , go to Step 6. Otherwise set . Compute , , and .
Step 3. If there exists such that or such that , denote them by or , respectively, and go to Step 4. Otherwise, go to Step 2 if and go to Step 6 if .
Step 4. If there exists such that or such that , and 1 or 1 holds, go to Step 8. Otherwise, go to Step 5.
Substep 5.1. Set .
Substep 5.2. If any element of -th column or -th column in is greater than or equal to , go to Step 8. Otherwise, go to Substep 5.3.
Substep 5.3. If , set and go to Substep 5.2, or else go to Step 2.
Part B: Check of the Independence of Each .
each column vector of is a nonzero vector;
each row vector of is a zero vector, or has only one element with a nonzero value.
For , there exists no satisfying .
Step 8. This algorithm cannot determine whether the system (3) is -output-controllable or not (at ).
The above algorithm allows us to determine the -output-controllability of the system as follows.
and , are some subsets of the index set . Then using the forms of (31) and (32), the following lemma on Part A of Algorithm 1 is obtained.
Lemma 1. In Step 6, includes neither identities of (31) nor identities of (32).
Proof. In Step 3, from for some , , and , we see that more than 2 paths from to exist, which is necessary for the identity on to exist (similarly for the case of ). Thus we next focus on the existence of logical NOT (i.e., ) in these paths in Step 4 and Step 5.
Consider the case that the logical NOT (i.e., ) corresponding to or obtained in Step 3 exists in (21), in other words, either or holds. Then the condition implies that the term is included in the paths in question, which is a necessary condition for the existence of the identity in . Thus we exclude this case (Step 4). (similarly for the case ).
In the other case, from (31), (32), for , some , to exist in the paths in question is necessary for the existence of identities. If any element of the -column or the -column of is greater than or equal to 1, some element of exists in the paths in question. Thus we exclude this case (Substep 5.2). Therefore, it follows that includes no identities in Step 6.
From Lemma 1, we see that the case that includes the identities that have the form of (31) or (32) is excluded from the viewpoint of a necessary condition for the identity to exist in . Thus we obtain the following theorem.
the system (3) is -output-controllable if conditions (i), (ii), and (iii) in Step 6 hold subject to Part A of Algorithm 1,
for a given , the system (3) is -output-controllable at if condition (iv) in Step 7 holds subject to Part A and Step 6.
Proof. First, the statement (i) is proven for the system satisfying the condition that includes neither identities of (31) nor identities of (32). From Lemma 1, this condition is satisfied in Step 6. Then condition (i) in Step 6 implies that there exists no path between each element of and each element of , since the -th element of expresses if a path from to exists or not. On the other hand, note that -th element of expresses if a path from to exists or not . Thus condition (ii) in Step 6 implies that there exists at least one path from each element of to some .
Furthermore, condition (iii) in Step 6 means that the input for each and has a path connected to only one element of or has no path to any element of . From these conditions, it follows that each affects at most one and not the other , . Hence the value of can be independently specified by the corresponding , which implies that system (21) is -output-controllable.
Next, the statement (ii) is proven. Since condition (i) in Step 6 does not hold, in this case, there exists a path between some element of and some element of . On the other hand, condition (iv) in Step 7 guarantees that there exists no path between constant elements of and elements of . Thus is not affected by the value of . Therefore, from (ii)–(iv), it follows that system (21) is -output-controllable at . This completes the proof.
In Step 3, from , we obtain and . In Step 4, from and , we have and . So go to Step 8, that is, it is impossible to determine if system (17) is -output-controllable. In fact, from (18), includes the identity . Thus we see that there exists an identical equation.
From Step 1 Step 2 Step 3 Step 6, we can see that the system (17) is -output-controllable. In fact, by simple calculation, the Boolean function of is derived as .
This system is -output-controllable for . However, the proposed algorithm cannot determine whether this system is -output-controllable or not; thus there exists a class of systems such that the proposed algorithm cannot determine the controllability. Needless to say, it will not be so easy to cope with various cases stated above due to high nonlinearity of Boolean functions.
While the proposed algorithm includes such disadvantages, one of the main advantages of the algorithm is that the computational complexity of the above algorithm is very small. This will be discussed in the following section.
5. Computational Complexity Analysis
In this section, we discuss the computational complexity of the algorithm proposed in the previous section.
First, let us recall the definition of the symbols used here. The number of the state, the control input, and the output in (3) are denoted by , , and , respectively. The number of the logical NOT appeared in (3) is expressed by . In addition, expresses the control time. Then the following result is obtained.
Lemma 2. The computational complexity of the proposed algorithm is for , .
Proof. The computation of the proposed algorithm consists of (a) checking each condition of Part A, and (b) checking whether conditions (i) to (iv) hold or not.
First, (b) is considered. The computational complexity to compute is . So the computational complexity to compute and is given by both . Further, the computational complexity to compute the product of and is . So by simple calculation, the computational complexity of is obtained as . The computational complexity of generating is obviously less than the case of . Therefore, the computational complexity to compute and is , which also includes the computational complexity to check conditions (i) to (iv) in Steps 6 and 7 for given and .
Next, (a) is considered. The matrices , , are obtained directly from , and the computational complexity of Step 5 is . As a result, since the computational complexity of each checking in Part A is , the computational complexity of Part A is less than .
Therefore, the computational complexity of the proposed algorithm is given by .
Computational time of the proposed algorithm ( )
Worst comp. time [sec]
6. Application to Neurotransmitter Signaling Pathway
From Figure 3, we see that this Boolean network includes at least four loops, for example, the loop of , , , and , the loop of , , , , and , and so forth. In synaptic plasticity, it is required that the binary value of expressing exocytosis can be arbitrarily controlled. Furthermore, phospholipase C ( ) is a kind of enzymes that cleaves phospholipids and as a result protein kinase C as well as calcium are activated. The former, protein kinase C, which works outside of the network in Figure 3, is one of key enzymes in signal transduction pathways. Thus since phospholipase C affects the other significant network, it will be important to simultaneously control the value of and the value of . Therefore and are regarded as the output, that is, . In addition, we assume that and cannot be directly controlled.
For a fixed dimension of and the fixed output , all combinations of , , are considered as the control inputs, which we call the input-combinations. Then for a given , the proposed algorithm is applied to the system of the form (3) obtained for each input-combination of . It is remarked that depending on the choice of the kind of control inputs, there exist several cases to which the polynomial-time algorithm proposed in  cannot be applied due to the graph-structure constraints. Furthermore, it is also remarked that even for fixed control inputs, the controllability problem is NP-hard. So the problem of finding efficient control inputs that make the system controllable is further harder than this problem.
By applying our algorithm to the case of each input-combination of and each fixed , we obtain, for example, the following results. In the case of and , we can find that among input-combinations, there exist at least 6 input-combinations of that make the system -output-controllable. In this way, since the proposed algorithm for each input-combination is very efficient, for example, the computation time via the proposed algorithm is about 10 [sec] for Boolean networks with 600 nodes (see Table 1) and , it enables us to verify the controllability condition for a certain number of input-combinations within a practical time; for example, about 3 [hours] will be required for 1000 input-combinations of a Boolean network with 600 nodes.
which implies that the value of can be freely given by control inputs, for example, (a) for , , , , , and (b) for , , , , .
Finally, we discuss the control input sequence realizing the desired output values. One of criticisms in control of Boolean networks is to assume that the value of the control input can be arbitrarily given at each time. In many biological systems, this assumption is not always satisfied, and input constraints are frequently imposed. One of input constraints is that the value of the control input is given as a constant within a certain sufficiently long time period. Although it is one of future works to explicitly deal with such an input constraint, based on the proposed algorithm, we may also find a constant-valued sequence of control inputs for which the desired values of outputs are obtained. For example, in (40) and (41), let us consider to find a control input sequence satisfying and . Since , , , , and are obtained as one of solutions, it is remarked that the following control inputs are given as any binary value: , , , , and , , , , , and , , , , , and , , , , . This allows us to give the values of the control input sequences as a constant, that is, , , , , . Thus the proposed algorithm helps us to find a practically useful control input sequence. Furthermore, this kind of degree of freedom in control inputs may be used for the optimal control problem. Once we can determine control input variables by our algorithm, we can use a tool for finding optimal control input sequences, which have been developed in hybrid control theory, for example, [25, 26]. It is expected that such an analysis will provide one of guidelines in experimental approaches to the control problem of biological networks.
In this paper, the controllability analysis for biological networks expressed by a Boolean network model with control nodes (inputs) and controlled nodes (outputs) has been discussed. First, a sufficient condition for the Boolean network model to be output-controllable has been derived by exploiting an adjacency matrix of its network graph. The obtained condition, which is given in the form of an algorithm, can be checked in polynomial time with respect to the state/input dimensions and the control time period; it will be one of the powerful tools that can provide some clues for finding effective control inputs to control a large-scale biological network. Next, by PC-based numerical experiments, it has been shown that the proposed method is applicable to large-scale Boolean networks with at least 1000 nodes. Finally, the proposed method has been applied to the Boolean network model expressing a neurotransmitter signaling pathway, and has shown that it is controllable with respect to both exocytosis and phospholipase C when appropriate control inputs are used.
There are many interesting open problems to be addressed in the future. It is one of the most important issues to characterize a class of Boolean networks to which our algorithm can be applied as a necessary and sufficient condition. In addition, extensions to the case of systems with input constraints and uncertainty are also one of the significant topics.
The authors would like to thank Professor Tatsuya Akutsu, Kyoto University for fruitful discussions and valuable comments.
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