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Modeling gene regulation is an important problem in genomic research. Boolean networks (BN) and its generalization probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. BN is a deterministic model while PBN is a stochastic model. In a PBN, on one hand, its stationary distribution gives important information about the long-run behavior of the network. On the other hand, one may be interested in system synthesis which requires the construction of networks from the observed stationary distribution. This results in an inverse problem which is ill-posed and challenging. Because there may be many networks or no network having the given properties and the size of the inverse problem is huge. In this paper, we consider the problem of constructing PBNs from a given stationary distribution and a set of given Boolean Networks (BNs). We first formulate the inverse problem as a constrained least squares problem. We then propose a heuristic method based on Conjugate Gradient (CG) algorithm, an iterative method, to solve the resulting least squares problem. We also introduce an estimation method for the parameters of the PBNs. Numerical examples are then given to demonstrate the effectiveness of the proposed methods.

Objective: Re-construction of a genetic regulatory network from a given time-series gene expression data is an important research topic in systems biology. One of the main difficulties in building a genetic regulatory network lies in the fact that practical data set has a huge number of genes vs. a small number of sampling time points. In this paper, we propose a new linear regression model that may overcome this difficulty for uncovering the regulatory relationship in a genetic network. Methods: The proposed multiple regression model makes use of the scale-free property of a real biological network. In particular, a filter is constructed by using this scale-free property and some appropriate statistical tests to remove redundant interactions among the genes. A model is then constructed by minimizing the gap between the observed and the predicted data. Results: Numerical examples based on yeast gene expression data are given to demonstrate that the proposed model fits the practical data very well. Some interesting properties of the genes and the underlying network are also observed. Conclusions: In conclusion, we propose a new multiple regression model based on the scale-free property of real biological network for genetic regulatory network inference. Numerical results using yeast cell cycle gene expression dataset show the effectiveness of our method. We expect that the proposed method can be widely used for genetic network inference using high-throughput gene expression data from various species for systems biology discovery.

Let V be a real or complex finite-dimensional vector space, and let N be a set of norms on V. The norm hull of a vector x ∈ V with respect to N is the set of vectors y ∈ V that satisfy ∥y∥ ≤ ∥x∥ for all ∥ · ∥ ∈ N. We study and give characterization of the norm hull for some sets of well-known norms on general vector spaces, and for the set of algebra norms and the set of induced norms on the algebra of n × n real or complex matrices.

We give two simple sufficient conditions under which the multiaffine image in the complex plane of an m-dimensional cube is a convex polygon. A third condition which, in some generic sense, is necessary and sufficient is then obtained. Our conditions involve checking the locations of the image of the vertices of the cube. These results help determine whether a parameterized family of polynomials th stable.

Linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions, subsets, relations, etc., invariant. The earliest papers on linear preserver problems date back to 1897, and a great deal of effort has been devoted to the study of this type of question since then. We present a brief picture of the subject, aiming at giving a gentle introduction to the reader. Then we describe some techniques used in our recent papers on this type of problem.

Let F denote either the complex field C or the real field R. Let V be Sn(F) or Kn(F), the vector spaces of all n × n symmetric and skew-symmetric matrices, respectively, over F. For c=(c1,…,cn)≠0 with c1⩾ ⋯ ⩾cn⩾0, the c-spectral norm of a matrix A∈V is the quantity ‖A‖ c = ∑ i=l n c i σ i (A) , where σ1(A)⩾ ⋯ ⩾σn(A) are the singular values of A. Let d=(d1,…,dn)≠0 with d1⩾ ⋯ ⩾dn⩾0. We study the linear isometries between the normed spaces (V,‖·‖c) and (V,‖·‖d), by using the fact that they are dual transformations of the linear operators which map ∑(d) onto ∑(c), where ∑(c) = {X∈ V :X has singular values c 1 ,…,c n } . It is shown that such isometries (and hence their dual transformations) exist if and only if c and d are scalar multiples of each other. In such case, we completely determine the structure of such isometries, and prove that they and their dual transformations belong to a same class of operators. In the proof, we obtain characterizations of the extreme points of the unit ball in V (for different cases) with respect to ‖·‖c, which is of independent interest.

Let Fm×n (m⩽n) denote the linear space of all m × n complex or real matrices according as F=C or R. Let c=(c1,…,cm)≠0 be such that c1⩾⋯⩾cm⩾0. The c-spectral norm of a matrix AϵFm×n is the quantity ‖A‖c∑i=Imciσi(A). where σ1(A)⩾⋯⩾σm(A) are the singular values of A. Let d=(d1,…,dm)≠0, where d1⩾⋯⩾dm⩾0. We consider the linear isometries between the normed spaces (Fm×n,∥·∥c) and (Fm×n,∥·∥d), and prove that they are dual transformations of the linear operators which map L(d) onto L(c), where L(c)= {XϵFm×n:X has singular values c1,…,cm}.

Let V be an n-dimentional unitary space with inner product (·,·) and S the set {x∈V:(x, x)=1}. For any A∈Hom(V, V) and q∈ C with ∣q∣⩽1, we define W(A:q)={(Ax, y):x, y∈S, (x, y)=q} . If q=1, then W(A:q) is just the classical numerical range {(Ax, x):x∈S}, the convexity of which is well known. Another generalization of the numerical range is the C-numerical range, which is defined to be the set W C (A)={ tr (CU ∗ AU):U unitary } where C∈Hom(V, V). In this note, we prove that W(A:q) is always convex and that WC(A) is convex for all A if rank C=1 or n=2.