Xiaoqing Cheng

Integer programming-based method for observability of singleton attractors in Boolean networks

Boolean network (BN) is a popular mathematical model for revealing the behaviour of a genetic regulatory network. Furthermore, observability, an important network feature, plays a significant role in understanding the underlying network. Several studies have been done on analysis of observability of BNs and complex networks. However, the observability of attractor cycles, which can serve as biomarker detection, has not yet been addressed in the literature. This is an important, interesting and challenging problem that deserves a detailed study. In this study, a novel problem was first proposed on attractor observability in BNs. Identification of the minimum set of consecutive nodes can be used to discriminate different attractors. Furthermore, it can serve as a biomarker for different disease types (represented as different attractor cycles). Then a novel integer programming method was developed to identify the desired set of nodes. The proposed approach is demonstrated and verified by numerical examples. The computational results further illustrates that the proposed model is effective and efficient.

Sparse solution of nonnegative least squares problems with applications in the construction of probabilistic Boolean networks

In this paper, we consider finding a sparse solution of nonnegative least squares problems with a linear equality constraint. We propose a projection-based gradient descent method for solving huge size constrained least squares problems. Traditional Newton-based methods require solving a linear system. However, when the matrix is huge, it is neither practical to store it nor possible to solve it in a reasonable time. We therefore propose a matrix-free iterative scheme for solving the minimizer of the captured problem. This iterative scheme can be explained as a projection-based gradient descent method. In each iteration, a projection operation is performed to ensure the solution is feasible. The projection operation is equivalent to a shrinkage operator, which can guarantee the sparseness of the solution obtained. We show that the objective function is decreasing. We then apply the proposed method to the inverse problem of constructing a probabilistic Boolean network. Numerical examples are then given to illustrate both the efficiency and effectiveness of our proposed method. Copyright

Identifying a Probabilistic Boolean Threshold Network From Samples

This paper studies the problem of exactly identifying the structure of a probabilistic Boolean network (PBN) from a given set of samples, where PBNs are probabilistic extensions of Boolean networks. Cheng et al. studied the problem while focusing on PBNs consisting of pairs of AND/OR functions. This paper considers PBNs consisting of Boolean threshold functions while focusing on those threshold functions that have unit coefficients. The treatment of Boolean threshold functions, and triplets and

Exact Identification of the Structure of a Probabilistic Boolean Network from Samples

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On observability of attractors in Boolean Networks

-tuplets of such functions, necessitates a deepening of the theoretical analyses. It is shown that wide classes of PBNs with such threshold functions can be exactly identified from samples under reasonable constraints, which include: 1) PBNs in which any number of threshold functions can be assigned provided that all have the same number of input variables and 2) PBNs consisting of pairs of threshold functions with different numbers of input variables. It is also shown that the problem of deciding the equivalence of two Boolean threshold functions is solvable in pseudopolynomial time but remains co-NP complete.

Discovery of Boolean Metabolic Networks: Integer Linear Programming Based Approach

We study the number of samples required to uniquely determine the structure of a probabilistic Boolean network (PBN), where PBNs are probabilistic extensions of Boolean networks. We show via theoretical analysis and computational analysis that the structure of a PBN can be exactly identified with high probability from a relatively small number of samples for interesting classes of PBNs of bounded indegree. On the other hand, we also show that there exist classes of PBNs for which it is impossible to uniquely determine the structure of a PBN from samples.

Optimal Projection Method Determination by Logdet Divergence and Perturbed Von-Neumann Divergence

Boolean network (BN) is a popular mathematical model for revealing the behavior of a genetic regulatory network, and observability plays a vital role in understanding the underlying network feature. However, the observability of attractor cycles, which is an interesting and important problem, has not been addressed in the literature. In this paper, we first proposed a novel problem on attractor observability in BNs. Identification of the minimum set of consecutive nodes can be used to determine uniquely the attractor cycle from the others in the network. We then develop a linear-time algorithm to identify the desired set of nodes. The proposed approaches are demonstrated and verified by numerical examples. The computational results are given to illustrate both the efficiency and effectiveness of our proposed methods.

Discrimination of singleton and periodic attractors in Boolean networks

BackgroundTraditional drug discovery methods focused on the efficacy of drugs rather than their toxicity. However, toxicity and/or lack of efficacy are produced when unintended targets are affected in metabolic networks. Thus, identification of biological targets which can be manipulated to produce the desired effect with minimum side-effects has become an important and challenging topic. Efficient computational methods are required to identify the drug targets while incurring minimal side-effects.ResultsIn this paper, we propose a graph-based computational damage model that summarizes the impact of enzymes on compounds in metabolic networks. An efficient method based on Integer Linear Programming formalism is then developed to identify the optimal enzyme-combination so as to minimize the side-effects. The identified target enzymes for known successful drugs are then verified by comparing the results with those in the existing literature.ConclusionsSide-effects reduction plays a crucial role in the study of drug development. A graph-based computational damage model is proposed and the theoretical analysis states the captured problem is NP-completeness. The proposed approaches can therefore contribute to the discovery of drug targets. Our developed software is available at “http://hkumath.hku.hk/~wkc/APBC2018-metabolic-network.zip”.

On Eigen-matrix translation method for classification of biological data

BackgroundPositive semi-definiteness is a critical property in kernel methods for Support Vector Machine (SVM) by which efficient solutions can be guaranteed through convex quadratic programming. However, a lot of similarity functions in applications do not produce positive semi-definite kernels.MethodsWe propose projection method by constructing projection matrix on indefinite kernels. As a generalization of the spectrum method (denoising method and flipping method), the projection method shows better or comparable performance comparing to the corresponding indefinite kernel methods on a number of real world data sets. Under the Bregman matrix divergence theory, we can find suggested optimal λ in projection method using unconstrained optimization in kernel learning. In this paper we focus on optimal λ determination, in the pursuit of precise optimal λ determination method in unconstrained optimization framework. We developed a perturbed von-Neumann divergence to measure kernel relationships.ResultsWe compared optimal λ determination with Logdet Divergence and perturbed von-Neumann Divergence, aiming at finding better λ in projection method. Results on a number of real world data sets show that projection method with optimal λ by Logdet divergence demonstrate near optimal performance. And the perturbed von-Neumann Divergence can help determine a relatively better optimal projection method.ConclusionsProjection method ia easy to use for dealing with indefinite kernels. And the parameter embedded in the method can be determined through unconstrained optimization under Bregman matrix divergence theory. This may provide a new way in kernel SVMs for varied objectives.

A Semi-tensor Product Approach for Probabilistic Boolean Networks

Abstract Determining the minimum number of sensor nodes to observe the internal state of the whole system is important in analysis of complex networks. However, existing studies suggest that a large number of sensor nodes are needed to know the whole internal state. In this paper, we focus on identification of a small set of sensor nodes to discriminate statically and periodically steady states using the Boolean network model where steady states are often considered to correspond to cell types. In other words, we seek a minimum set of nodes to discriminate singleton and periodic attractors. We prove that one node is not necessarily enough but two nodes are always enough to discriminate two periodic attractors by using the Chinese remainder theorem. Based on this, we present an algorithm to determine the minimum number of nodes to discriminate all given attractors. We also present a much more efficient algorithm to discriminate singleton attractors. The results of computational experiments suggest that attractors in realistic Boolean networks can be discriminated by observing the states of only a small number of nodes.