Bioinformatics Advance Access originally published online on November 12, 2005
Bioinformatics 2006 22(2):226-232; doi:10.1093/bioinformatics/bti765
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Intervention in a family of Boolean networks
1Department of Electrical Engineering, Texas A&M University College Station, TX 77843, USA
2Translational Genomics Research Institute 400 North Fifth Street, Suite 1600, Phoenix, AZ 85004, USA
*To whom correspondence should be adderessed.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONSYSTEMS AND METHODSALGORITHMIMPLEMENTATIONDISCUSSIONCONCLUDING REMARKSREFERENCES |
|---|
Motivation: Intervention in a gene regulatory network is used to avoid undesirable states, such as those associated with a disease. Several types of intervention have been studied in the framework of a probabilistic Boolean network (PBN), which is a collection of Boolean networks in which the gene state vector transitions according to the rules of one of the constituent networks and where network choice is governed by a selection distribution. The theory of automatic control has been applied to find optimal strategies for manipulating external control variables that affect the transition probabilities to desirably affect dynamic evolution over a finite time horizon. In this paper we treat a case in which we lack the governing probability structure for Boolean network selection, so we simply have a family of Boolean networks, but where these networks possess a common attractor structure. This corresponds to the situation in which network construction is treated as an ill-posed inverse problem in which there are many Boolean networks created from the data under the constraint that they all possess attractor structures matching the data states, which are assumed to arise from sampling the steady state of the real biological network.
Results: Given a family of Boolean networks possessing a common attractor structure composed of singleton attractors, a control algorithm is derived by minimizing a composite finite-horizon cost function that is a weighted average over all the individual networks, the idea being that we desire a control policy that on average suits the networks because these are viewed as equivalent relative to the data. The weighting for each network at any time point is taken to be proportional to the instantaneous estimated probability of that network being the underlying network governing the state transition. The results are applied to a family of Boolean networks derived from gene-expression data collected in a study of metastatic melanoma, the intent being to devise a control strategy that reduces the WNT5A gene's action in affecting biological regulation.
Availability: The software is available on request.
Contact: edward{at}ee.tamu.edu
Supplementary information: The supplementary Information is available at http://ee.tamu.edu/~edward/tree
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONSYSTEMS AND METHODSALGORITHMIMPLEMENTATIONDISCUSSIONCONCLUDING REMARKSREFERENCES |
|---|
Numerous gene regulatory models have been proposed. For the most part these have been developed for descriptive purposes, by which we mean that their purpose is to characterize gene interaction. From a translational perspective, a salient objective is to base diagnosis and treatment for disease upon these models. For treatment, this constitutes the derivation of intervention strategies that affect the network in beneficial ways. To date, the largest effort in this direction has been in the context of probabilistic Boolean networks (PBNs). A PBN is essentially a collection of Boolean networks in which at any discrete time point the gene state vector transitions according to the rules of one of the Boolean networks (Shmulevich et al., 2002a). The earliest effort at intervention involved resetting the state of a PBN to a more desirable initial state and letting the network evolve from there (Shmulevich et al., 2002b). More sophisticated approaches have since been proposed that use the methods of automatic control (Datta et al., 2003, 2004; Pal et al., 2005a). These employ external control variables that can be manipulated in progression to affect the dynamic evolution of a network over a finite time horizon.
Given a dataset consisting of gene-expression measurements, PBN design constitutes an ill-posed inverse problem that is treated using a design algorithm to generate a solution. Inference can be formalized by postulating criteria that constitute a solution space for the inverse problem. The criteria come in two forms: (1) the constraint criteria are composed of restrictions on the form of the network and (2) the operational criteria are composed of relations that must be satisfied between the model and the data. The solution space consists of all PBNs that satisfy the two sets of criteria. Recognizing that PBNs are composed of Boolean networks, and since it is difficult to infer the probabilistic structure among the constituent Boolean networks from the steady-state data typically used for design, a more general view may be taken in which the inverse problem is restricted to determining a solution space of Boolean networks and then finding networks in that space (Pal et al., 2005b). Without a probabilistic structure between the Boolean networks, we have a family of Boolean networks satisfying both the constraint and operational criteria. If desired, one can then go further and construct a PBN using networks from the family, or one can simply treat the family as a collection of solutions to the Boolean-network inverse problem.
In this paper, we derive a control algorithm that can be applied to a family of Boolean networks. This is accomplished by minimizing a composite cost function that is a weighted average cost over the entire family. Ideally, the weighting for each member of the family at any time point would be proportional to the instantaneous probability of a particular network being the governing network. Although these instantaneous probabilities are not known, we adaptively estimate them from the available data and the estimate is used to implement the control algorithm.
|
SYSTEMS AND METHODS |
|---|
|
TOPABSTRACTINTRODUCTIONSYSTEMS AND METHODSALGORITHMIMPLEMENTATIONDISCUSSIONCONCLUDING REMARKSREFERENCES |
|---|
Boolean networks
A Boolean network (BN) consists of a set of nodes (genes) in which each gene can take on one of two binary values, 0 or 1 (Kauffman, 1969, 1993). Given n genes, the activity level of gene i at time step k is denoted by xi(k), where xi(k) = 0 indicates that gene i is not expressed and xi(k) = 1 indicates that it is expressed. The overall expression levels of all the genes in the network at time step k is given by the state (row) vector x(k) = [x1(k), x2(k), ... , xn(k)], also called the gene activity profile (GAP) of the network at time k. Gene i evolves from time k to k + 1 according to the Boolean function fi(x1(k), x2(k), ... , xn(k)). Usually the value of fi does not depend on the entire set {x1, x2, ... , xn} of n gene values but only on a finite subset
of it. This set is called the predictor set for the i-th gene. Specifying the truth table for the functions f1, f2, ... , fn along with the associated predictor sets 
supplies all the information necessary to determine the time evolution of the states of the BN.
The binary n-digit state vector x(k) can be mapped to positive integers z(k) so that as x(k) ranges from 00 
0 to 11 1, z(k) goes from 1 to 2n. Here we employ the decimal representation z(k) and the set 
= {1, 2, ... , 2n} constitutes the state space for the Boolean network.
Attractors play a key role in Boolean networks. Given a starting state, within a finite number of steps, the network will transition into a cycle of states, called an attractor cycle, and will continue to cycle thereafter. Non-attractor states are transient and are visited at most once on any network trajectory. The level of a state is the number of transitions required for the network to transition from the state into an attractor cycle. Attractors are often identified with phenotypes (Kauffman, 1993). Real biological systems are typically assumed to have short attractor cycles. Singleton attractors are a key interest since these are associated with phenomena such as cell proliferation and apoptosis (Huang, 1999). We focus on Boolean networks possessing only singleton attractors.
In most cases we lack time-course gene-expression measurements corresponding to the temporal evolution of the network, and our assumption is that the measurements (or almost all of them) are taken in the steady state (Zhou et al., 2004; Pal et al., 2005a, b)—see Pal et al., 2005b for a discussion of the biological considerations concerning the steady-state assumption. Under this assumption data states are, with probability near one, attractor states. Thus, we would like them to be attractors in the model and their inclusion or lack of inclusion in a designed network can be used to support or not support network validity. If we take the view that there is no reason to believe that data states are not attractor states, then we may wish to require that the attractor states of a designed network exactly match the data states. To achieve this end, an algorithm has been developed to generate Boolean networks with a prescribed attractor structure (Pal et al., 2005b). To look for biologically meaningful networks in the space of desired networks, the number of predictors for a gene and the number of levels for a transient state are bounded. To avoid the inclusion of non-regulating genes, each gene must occur in the predictor set of at least one other gene. The algorithm can function in two modes. In one, it begins with genes, attractor states, predictor sets and a maximum number of levels, and generates all possible networks having these; in a more general mode, it begins with genes, attractor states, a maximum predictor-set size and a maximum level, and generates networks having these. For instance, in the first mode suppose there are three genes x1, x2, x3, attractor states 000 and 101, predictor sets 
and 
, and maximum level 3. Three 3-gene networks with the given singleton attractors, predictor sets and maximum level are shown in Figure 1 using decimal representation.
|
and attractors 1(000) and 6(101).Control for a probabilistic Boolean network
A PBN consists of a finite collection of BNs over a fixed set of genes, where each BN is defined by a fixed network function. At any given moment of discrete time there is a probability p of randomly switching the state of the PBN, so that each constituent BN is a BN with random perturbation. Moreover, at each moment of time there is a probability q of switching to a different constituent BN, where, given a switch, each BN composing the network has a probability of being selected. If q = 1, then a new network function is randomly selected at each time point; if q < 1, then the PBN remains in a given constituent BN until the random binary variable governed by q calls for a network switch. If q = 1, the PBN is said to be instantaneously random, the idea being to model uncertainty in model selection; if q < 1, it is said to be context-sensitive, the idea being to model the situation where the model is affected by latent variables outside the model. To motivate and to set the background for the control policy proposed in this paper, we briefly review the results of the original analysis for an instantaneously random PBN (Datta et al., 2003)—see Pal et al., 2005a for control in a context-senstive PBN.
Consider the problem of external control in an instantaneously random PBN with n genes and m control inputs, u1, u2, ... , um, each of which can take on only the binary values 0 or 1. At any time k, the row vector 
describes the complete status of all the control inputs. u(k) can take on all binary values from 00 0 to 11 1. One can equivalently represent the control input status using a decimal number v(k) ranging from 0 to 2m – 1, so that 
is the set of possible control actions. This set could be a function of the state, because not all control alternatives may be available from all states. As shown in Datta et al. (2003), the one-step evolution of the probability distribution vector in the case of a PBN containing 2n states with control inputs takes place according to the equation
 | (1) |
We define the following quantities:
- aij(v) is the i-th row, j-th column entry of the stochastic matrix A(v),
.
- M represents the treatment/intervention window; control actions are taken at steps 0, 1, ... , M – 1.
- For any
is the cost of applying the control v in state i at the k-th time step.
- For any
is the terminal cost associated with the state i, i.e. the cost of ending up in state i at the M-th time step, when no more control steps are remaining.
For a given initial state z(0), the optimal cost for the finite horizon optimal control problem is given by J0(z(0)), where for k = 0, 1, 2, ... , M – 1, Jk(i) represents the cost-to-go function at the k-th time step starting from state i (Datta et al., 2003) The Jks can be found using the following dynamic programming algorithm (Bellman, 1957):
 | (2) |
 | (3) |
Equation (2) states that the optimal cost to go from state i at the k-th time step is the sum of the cost of the optimal control action at state i and the expected value of the cost to go at the (k + 1)-th time step. Since there is no control action in the terminal time step, (3) simply formalizes the fact that the cost to go at the terminal time step equals the penalty associated with the terminal state.
The optimal control obtained as a solution to (2), (3) can be represented as a table 
, where 
is the discrete time variable. To set up such a table, we first tabulate JM(i) for any 
using (3). JM–1(i) and the corresponding minimizing control v can be calculated and stored for all using (2) and making use of the JM(j) values tabulated earlier. By repeating these steps we can fill up the table for k = M – 2, ... , 0.
The optimal control solution of (2) and (3) is from a very general setting; however, for genetic regulatory networks, the class of allowable controls is severely constrained since the control action consists of forcibly altering the expression status of only one, or perhaps, two genes. This limited control objective is dictated primarily by limitations on the kind of interventions that appear to be within the current realm of biological possibility.
|
ALGORITHM |
|---|
|
TOPABSTRACTINTRODUCTIONSYSTEMS AND METHODSALGORITHMIMPLEMENTATIONDISCUSSIONCONCLUDING REMARKSREFERENCES |
|---|
If a family of BNs is designed whose attractors match the data, assuming the family is not too small we have the expectation that the underlying biological phenomena are closely modeled by at least some of the BNs in the family. In the absence of perfect knowledge as to which BNs are capable of better representing the underlying phenomena, we develop a control policy that optimizes a composite cost function over the entire family of BNs.
Let 
be a set of L Boolean networks N1, N2, ... , NL possessing identical sets of singleton attractors, all sharing the same state space and the same control space 
. Associated with each network is an initial probability of it representing the underlying phenomenon. Since this information is not available, we will adaptively estimate these probabilities as more transitions are observed. For each network Nl, l = 1, 2, ... , L define
to be the i-th row, j-th column entry of the matrix Al(v) of the network Nl;
to be the cost of applying the control v at the k-th time step in state i in network Nl;
to be the terminal cost associated with state i in network Nl.
We define the belief vector 
, where 
is the probability of network Nl being the underlying network at the k-th time step. 
k is the probability distribution vector for the family of networks at the k-th time step. Since k is unknown, we will make an initial guess for it and update it as more information becomes available. The use of this vector is inspired by the information vector in Smallwood et al. (1973).
Suppose i is the current state at step k, is the current estimate of the belief vector, and upon application of control v we observe state j at the next time step. Then the new belief vector is ' = T(, i | j,v), where the transformation T can be obtained by use of Bayes' theorem and the theorem of total probability,
 | (4) |
Dynamic programming over a family of networks
Suppose we are given an initial belief vector 0 and an initial state z(0). The initial belief vector is based on our prior knowledge of the system. It could be a function of likelihood or Bayesian scores of networks, or it could be uniform to reflect no prior knowledge. Our objective is to find controls v(0), v(1), ... , v(k), ... , v(M – 1) to minimize the expectation of the cost-to-go function over all networks in . The cost to go at the k-th time step (0 
k < M) is a function of the current state z(k) and the updated belief vector k. Motivated by (2) for the single PBN case, we define the average optimal cost-to-go function by
 | (5) |
The inner summation is the expectation over all 
of the cost to go at the (k + 1)-th step in the l-th network on observing j. We then add to it the cost of control at the k-th step and average over all the networks in the family. Finally we take the minimum over all control actions in to obtain the optimal policy and the cost to go at the k-th step.
The terminal cost for a state i is trivially defined to be the average terminal cost over the entire family:
 | (6) |
In Datta et al. (2003) terminal penalties were assigned to states based on the expression level of certain key genes; however, as discussed in Choudhary et al. (2005), it may be more reasonable to assign terminal penalties based on the long-term prospective behavior of the system in the absence of control. For any Markov chain with closed classes the procedure is summarized by the following points:
- Partition the states of the Markov chain into transient and persistent states.
- For singleton attractors the penalty
is set according to the status of the penalty gene(s). A penalty gene is a gene for which certain expression statuses are known to be undesirable.
- For a closed class the penalty is based on the fraction of time spent in states having a penalty gene in an undesirable profile.
- For a transient state j, the terminal penalty
where the summation variable i corresponds to a closed class or a singleton attractor. Prob(z(
(7) 
) = i | z(t) = j, Network = Nl) is the long-term probability of getting absorbed in i starting from state j, in network Nl.
|
IMPLEMENTATION |
|---|
|
TOPABSTRACTINTRODUCTIONSYSTEMS AND METHODSALGORITHMIMPLEMENTATIONDISCUSSIONCONCLUDING REMARKSREFERENCES |
|---|
The solution to the dynamic programming problem (5), (6) will now be presented as a policy tree that is optimal specific to a particular initial state and an initial belief vector. An M-step policy tree has an optimal action as its root with branches for each possible observation (in our case states) followed by M – 1-step policy trees. A detailed exposition on construction and pruning of such trees can be found in Kaelbling et al. (1998). Here we state an algorithm that is close to exhaustive enumeration and subsequent pruning. We use a data structure node with five components STATE, BELIEF-VECTOR, OPTIMAL-COST, OPTIMAL-CONTROL and DEPTH. The algorithm involves the following steps:
- Compute the M step control and the corresponding J0(i), J1(i), ... , JM(i)
i 
S for each of the networks N1, ... , NL as a table. Table 1 is such a table of controls for the example to be presented later in the next section.
- Initialize the tree's root node with the first observed state STATE = z(0), BELIEF-VECTOR = 0 and DEPTH = 0.
- Expand the root node and all the subsequently generated nodes; while BELIEF-VECTOR
el (i.e. the network Nl is not uniquely identified to be the underlying network) and DEPTH M. To expand a particular node in the tree with STATE = i, BELIEF-VECTOR = k and DEPTH = k, we consider all possible states that could be observed next. In other words, a child node is created for any j, such that 
with 
. Such a node has STATE = j, BELIEF-VECTOR = T(k, i | j, v) and DEPTH = k + 1.
- Now consider all the leaf nodes. For the nodes with DEPTH = M we use (6) to obtain OPTIMAL-COST. For leaf nodes with DEPTH = k M and BELIEF-VECTOR = el and some STATE = i, OPTIMAL-COST is set to Jk(i) from the table for network Nl.
- Now use (5) for all nodes with DEPTH = M – 1, ... 0 (in that order) to obtain OPTIMAL-COST and the minimizing v as the OPTIMAL-CONTROL.
- Prune the subtrees generated with non-optimal actions to obtain PolTR, the policy tree. The optimal policy follows the table for network Nl onwards from a node which has BELIEF-VECTOR = el.
|
At first glance, generating the tree may seem to be a formidable task due to the potentially large branching factor which can be as high as
. However, in the case of a family of BNs this is a much more manageable task due to the following mitigating factors: (1) the branching factor is small since not all states would be observed following an action due to similarities in the different BN transitions—for instance, in Figure 1 state 3 goes to state 1 in all the three networks; (2) from a given node, if more than one node is generated for some v, then the BELIEF-VECTORs for the children would be more sparse than the parent, and in some cases it would become a leaf node with BELIEF-VECTOR = el; and (3) the set of possible control actions is not large owing to the limited number of genes for intervention. |
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONSYSTEMS AND METHODSALGORITHMIMPLEMENTATIONDISCUSSIONCONCLUDING REMARKSREFERENCES |
|---|
Illustrative example
To illustrate the algorithmic details, we consider the 3-gene network introduced previously. Suppose state 1 (000) is a desirable attractor state with terminal penalty 0 and state 6 (101) is an undesirable state with terminal penalty +5. The terminal cost of any other non-attractor state is the cost of the attractor whose basin it is in. For instance for the network in Figure 1 (1.2), the non-attractor states 2, 3, 4 and 5 have terminal penalty 0, while states 7 and 8 have terminal penalty +5. Let x1 be the control gene and suppose that the control action is to forcibly flip this gene: for v(k) = 1, flip gene x1 at the k-th time step and for v(k) = 0 leave it as is. Let the cost of control
. Transitions take place according to the network transition rule—e.g. in the network in Figure 1 (1.2), if z(k) = 6 (101) and v(k) = 1, then z(k + 1) = 1, corresponding to a jump from state 6 (101) to state 2 (001) and then evolution to the state 1 (000). We show the evaluation of the policy tree PolTR starting from an initial state z(0) = 4 and 
in Figure 2. Figure 3 shows the corresponding policy tree obtained after pruning.
|
|
We now proceed to compare the performance of an M = 2 step policy PolTR and policies obtained using two other methods to control this family of networks.
Single network. We calculate the optimal policy Poll for each network Nl in the family. We obtain the control policy as a table with M rows and 
columns. Each element v(m, i) is the control alternative to be used when the state is i at the m-th time step. Since a single-network policy does not apply to the entire family, to implement a policy tree we follow one of the possible state observations after each action. It may happen that some of the possible states observed may not be listed as an option in a single-network policy tree. For a single BN the policy tree is a tree with a branching factor of 1, i.e. a path. Single network optimal policies for each network are listed in Table 1.
Context switching. The context-sensitive PBN design of Pal et al. (2005a) is more general than the method proposed here because there is no requirement that the constituent BNs possess identical attractor structures; however, it is more constrained in the sense that it assumes knowledge of the PBN switching structure. If, as is assumed here, we lack knowledge of the probabilistic structure governing BN selection so that we do not view the family of BNs as composing a single PBN and if the attractor structures are identical, as with a design strategy in which the attractors of each BN match the data states, then it may well be that the policy proposed here could outperform the context-sensitive-PBN method. If, for the present 3-gene example, we assume that the BNs compose a PBN in which each has equal probability of being selected, then the method of Pal et al. (2005a) yields the optimal control policy, PolSW, presented in the last row of Table 1.
To assess the performance of a particular policy Pol, we apply it to all the networks in the family starting from each initial state i and obtain 
. We then compute
 | (8) |
0 = 
Table 2 shows the results of applying various policies for all possible initial states. As measured by the value of the optimal cost, the policy PolTR of this work is superior. More examples appear on the companion website.
|
Melanoma application
We now apply the methodology of this paper to derive an optimal intervention strategy for a family of gene regulatory networks. These networks were developed from data collected in a study of metastatic melanoma in which the abundance of messenger RNA for the gene WNT5A was found to be highly discriminating between cells with properties typically associated with high metastatic competence versus those with low metastatic competence (Bittner et al., 2000). These findings were validated and expanded in a second study in which experimentally increasing the levels of the Wnt5a protein secreted by a melanoma cell line via genetic engineering methods directly altered the metastatic competence of that cell as measured by the standard in vitro assays for metastasis (Weeraratna et al., 2000). A further finding was that an intervention that blocked the Wnt5a protein from activating its receptor by the use of an antibody that binds the Wnt5a protein could substantially reduce Wnt5a's ability to induce a metastatic phenotype. This suggests a study of control based on interventions that alter the contribution of the WNT5A gene's action to biological regulation, since disruption of this influence could reduce the chance of a melanoma metastasizing a desirable outcome.
The original study used 587 genes with 31 gene expression patterns with varying stress conditions. A network with 587 genes would be intractable to design and subsequently control. Consequently the network was reduced to seven genes believed to be the most closely knit: PIRIN, S100P, RET1, MART1, HADHB, STC2 and WNT5A (Datta et al., 2004). Since all 31 data points correspond to steady-state behavior, they should be considered as attractors in the networks. However, out of the 31 samples only 18 were distinct. To reduce the number of attractors, we formed seven clusters from the data points and treated the cluster centers as attractors. These attractors are shown in Table 3. The first column is used to classify them into two categories, GOOD and BAD, depending on the status of the WNT5A gene.
|
Using the procedure of Pal et al. (2005b), we obtained four distinct BNs (N1, N2, N3, N4) with the same set of seven attractors. These networks are available on the companion website. We assigned a penalty of 5 to all states in the basin of the undesirable attractors (WNT5A = 1) and 0 to all the other states. We used PIRIN as the control gene. A forcible alteration in the expression level of PIRIN is associated with v = 1 while v = 0 represents no control. In a reasoning similar to our previous work (Datta et al., 2003, 2004), a terminal penalty of 5 for bad states versus 0 for good states and a control cost of 1 for intervention versus 0 for no intervention is our attempt to capture the intuitive notions of the relative costs of ending up in desirable versus undesirable state and the cost of intervention.
A pruned policy tree for M = 3 with initial belief vector 
and initial state z(0) = 3 is shown in Figure 4. The expected cost is 0.75 when we control using PolTR, 1.5 when using PolSW, and 1.75, 2.5, 1.5 and 1.75 when using Pol1, Pol2, Pol3 and Pol4, respectively. The expected uncontrolled cost is 2.5. For all horizons M and all initial states 
the method of this paper is superior to those discussed in the earlier section. Out of the 128 states in the network, 89 states needed to be controlled in at least one of the 4 networks. In particular for M = 5, starting from such states PolTR was more effective than PolSW in reducing the cost by 0.1152 on average. In terms of absolute probabilities PolTR was able to take the system to a desirable attractor starting from all initial states and all networks with a probability 1.0, except for states 4, 36, 68 and 100 in network N2, which are uncontrollable from PIRIN. For PolSW, states 4, 8, 24, 36, 68 and 100 are not taken to a desirable attractor in N2. In the event of N2 being the underlying network, starting from states 4, 36, 68 100, PolTR recognizes this and gives up promptly, while PolSW keeps on applying control, incurring extra costs, without any extra benefit.
|
Policy trees for initial state z(0) = 93,
and M = 2, 3 and 4 are shown in Figure 5. The expected cost with M = 2 is 1.0, which can be further reduced to 0.25 if M 
4. This is reasonable because the algorithm has more time steps to identify and control the system. For M = 4, the policy computation took 0.28 s on a 2.4 GHz, P4 processor system. More examples appear on the website.
|
|
CONCLUDING REMARKS |
|---|
|
TOPABSTRACTINTRODUCTIONSYSTEMS AND METHODSALGORITHMIMPLEMENTATIONDISCUSSIONCONCLUDING REMARKSREFERENCES |
|---|
In conclusion, we have developed a method to optimally control a family of BNs that share a common attractor structure. Such a family arises naturally from the steady-state data obtained from gene expression microarrays. The control algorithm is presented as a policy tree depending on an initial belief vector that is updated in an adaptive fashion. At every stage of the evolution, the estimated belief vector is used to appropriately weight the individual networks in the construction of the composite cost function to be minimized.
|
Acknowledgments |
|---|
This work was supported in part by the National Science Foundation under grants ECS-0355227 and CCF-0514644, and by the Translational Genomics Research Institute.
Conflict of Interest: none declared.
|
FOOTNOTES |
|---|
Associate Editor: Steen Knudsen
Received on September 7, 2005; revised on October 22, 2005; accepted on November 3, 2005
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONSYSTEMS AND METHODSALGORITHMIMPLEMENTATIONDISCUSSIONCONCLUDING REMARKSREFERENCES |
|---|
Bittner, M., et al. (2000) Molecular classification of cutaneous malignant melanoma by gene expression profiling. Nature, 406, 536–540[CrossRef][Medline].
Bellman, R. Dynamic Programming, (1957) , Princeton, NJ Princeton University Press.
Choudhary, A., et al. (2005) Assignment of terminal penalties in gene regulatory networks. Proceedings of the American Control ConferencePortland , pp. , pp. 417–422.
Datta, A., et al. (2003) External control in Markovian genetic regulatory networks. Mach. Learning, 52, 169–191[CrossRef].
Datta, A., et al. (2004) External control in Markovian genetic regulatory networks: the imperfect information case. Bioinformatics, 20, 924–930
Huang, S. (1999) Gene expression profiling, genetic networks, and cellular states: an integrating concept for tumorigenesis and drug discovery. Mol. Med, . 77, 469–480.
Kaelbling, L., et al. (1998) Planning and acting in partially observable stochastic domains. Artif. Intell, . 101, 99–134[CrossRef].
Kauffman, S. (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. Theor. Biol, . 22, 437–467.
Kauffman, S. The Origins of Order: Self-organization and Selection in Evolution, (1993) , NY Oxford University Press.
Pal, R., et al. (2005a) Intervention in context-sensitive probabilistic Boolean networks. Bioinformatics, 21, 1211–1218
Pal, R., et al. (2005b) Generating Boolean networks with a prescribed attractor structure. Bioinformatics, 21, 4021–4025
Smallwood, R.D. and Sondik, E.J. (1973) Optimal control of partially observable Markov processes over a finite horizon. Operations Res, . 21, 1071–1088
Shmulevich, I., et al. (2002a) Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics, 18, 261–274
Shmulevich, I., et al. (2002b) Gene perturbation and intervention in probabilistic Boolean networks. Bioinformatics, 18, 1319–1331
Weeraratna, A., et al. (2002) Wnt5a signaling directly affects cell motility and invasion of metastatic melanoma. Cancer Cell, 1, 279–288[CrossRef][Web of Science][Medline].
Zhou, X., et al. (2004) A Bayesian connectivity-based approach to constructing probabilistic gene regulatory networks. Bioinformatics, 20, 2918–2927
CiteULike 
Connotea 
Del.icio.us What's this?
This article has been cited by other articles:
|
|
 |
|
  R. Layek, A. Datta, R. Pal, and E. R. Dougherty Adaptive intervention in probabilistic boolean networks Bioinformatics, August 15, 2009; 25(16): 2042 - 2048. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||