We provide a combinatorial formula for the expansion of immaculate noncommutative symmetric functions into complete homogeneous noncommutative symmetric functions. To do this, we introduce generalizations of Ferrers diagrams which we call GBPR diagrams. A GBPR diagram assigns a color (grey, blue, purple, or red) to each cell of the diagram. We define tunnel hooks, which play a role similar to that of the special rim hooks appearing in the Eğecioğlu-Remmel formula for the symmetric inverse Kostka matrix. We extend this interpretation to skew shapes and fully generalize to define immaculate functions indexed by integer sequences skewed by integer sequences. Finally, as an application of our combinatorial formula, we extend Campbell's results on ribbon decompositions of immaculate functions to a larger class of shapes.

Transcriptome studies that provide temporal information about transcript abundance facilitate identification of gene regulatory networks (GRNs). Inferring GRNs from time series data using computational modeling remains a central challenge in systems biology. Commonly employed clustering algorithms identify modules of like-responding genes but do not provide information on how these modules are interconnected. These methods also require users to specify parameters such as cluster number and size, adding complexity to the analysis. To address these challenges, we used a recently developed algorithm, partitioned local depth (PaLD), to generate cohesive networks for 4 time series transcriptome datasets (3 hormone and 1 abiotic stress dataset) from the model plant Arabidopsis thaliana. PaLD provided a cohesive network representation of the data, revealing networks with distinct structures and varying numbers of connections between transcripts. We utilized the networks to make predictions about GRNs by examining local neighborhoods of transcripts with highly similar temporal responses. We also partitioned the networks into groups of like-responding transcripts and identified enriched functional and regulatory features in them. Comparison of groups to clusters generated by commonly used approaches indicated that these methods identified modules of transcripts that have similar temporal and biological features, but also identified unique groups, suggesting that a PaLD-based approach (supplemented with a community detection algorithm) can complement existing methods. These results revealed that PaLD could sort like-responding transcripts into biologically meaningful neighborhoods and groups while requiring minimal user input and producing cohesive network structure, offering an additional tool to the systems biology community to predict GRNs.

Transcriptome studies that provide temporal information are valuable for identifying groups of similarly-behaving transcripts, giving insight into overarching gene regulatory networks. Nevertheless, inferring transcriptional networks from time series data is challenging, in part because it is difficult to holistically consider both local relationships and global structure of these complex and overlapping transcriptional responses. To address this need, we employed the Partitioned Local Depth (PaLD) method to examine four time series transcriptomic datasets generated using the model plant Arabidopsis thaliana. Here, we provide a self-contained description of the method and demonstrate how it can be used to make predictions about gene regulatory networks based on time series data. The analysis provides a global network representation of the data from which graph partitioning methods and neighborhood analysis can reveal smaller, more well-defined groups of like-responding transcripts. These groups of transcripts that change in response to hormone treatment (e.g., auxin or ethylene) or high salinity were demonstrated to be enriched in common biological function and/or binding of transcription factors that were not identified with prior analyses of this data using other clustering and inference methodologies. These results reveal the ability of PaLD to generate predictions about gene regulatory networks using time series transcriptomic data, which can be of value to the systems biology community.

Network models of gene interactions, using time course gene transcript abundance data, are computationally created using a genetic algorithm designed to incorporate hierarchical Bayesian methods with time series adjustments. The posterior probabilities of interaction between pairs of genes are based on likelihoods of directed acyclic graphs. This algorithm is applied to transcript abundance data collected from Arabidopsis thaliana genes. This study extends the underlying statistical and mathematical theory of the Norris-Patton likelihood by including time series adjustments.

Gene interaction network models from time course gene transcript abundance data are algorithmically created using a new aggressive genetic algorithm denoted by BCHC. The BCHC algorithm rigorously integrates probabilistic hierarchical likelihood and Bayesian methodology to produce accurate posterior probabilities of interactions between genes after observance of hierarchical gene transcript abundance data. Forbidden pairwise gene relationships are incorporated into the modeling process. This gene interaction model is compared to a previous gene interaction model utilizing the same data and Bayesian likelihood, however based upon an exponentially slower, less aggressive, and less adaptive Metropolis-Hasting search algorithm. In addition for a smaller data set, our gene interaction model is compared to less rigorous non-probabilistic Lasso estimated partial correlation models which do not fully incorporate the hierarchical structure. A comparison is also made between the smallest Bayesian model and tests for edges based on a restricted non-Bayesian hierarchical technique. The BCHC algorithm performs well when the number of genes is moderately increased, both in terms of execution time and model quality.

We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric functions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. Using this result, we give necessary and sufficient conditions for a dual immaculate quasisymmetric function to be symmetric. Moreover, we show that the product of a Schur function and a dual immaculate quasisymmetric function expands positively in the Young quasisymmetric Schur basis. We also discuss the decomposition of the Young noncommutative Schur functions into the immaculate functions. Finally, we provide a Remmel-Whitney-style rule to generate the coefficients of the decomposition of the dual immaculates into the Young quasisymmetric Schurs algorithmically and an analogous rule for the decomposition of the dual bases.

Oxidative stress is thought to play an important role in age-related disease including osteoarthritis (OA). IGF-1 is a growth factor in chondrocytes (cartilage cells) that plays an important role in promoting cartilage matrix synthesis and inhibiting cartilage degradation. In OA, chondrocytes show a reduced response to IGF-1 which may be due to an increased amount of oxidative stress in the cells as they age. This study investigated the effects of inducing oxidative stress with tert-butylhydroperoxide (tBHP) on the IGF-1 signaling networks of chondrocytes from three different tissue donors. Western blots were used to collect signaling data for various proteins which were later modeled using a heterogeneous computational algebraic method. It was found that the models were consistent under stimulation with IGF-1 alone, as indicated by a strong positive correlation between the replicate models. However, their response to IGF-1 under conditions of oxidative stress was inconsistent with replicate model correlations close to zero. These findings suggest that oxidative stress induces a chaotic response in the IGF-1 signaling pathway and demonstrates how some cell perturbations may affect the ability of the algorithm to produce consistent computational models from biological replicates.

Multiple approaches for reverse-engineering bio-logical networks from time-series data have been proposed in the computational biology literature. These approaches can be classified by their underlying mathematical algorithms, such as Bayesian or algebraic techniques, as well as by their time paradigm, which includes next-state and co-temporal modeling. The types of biological relationships, such as parent-child or siblings, discovered by these algorithms are quite varied. It is important to understand the strengths and weaknesses of the various algorithms and time paradigms on actual experimental data. We assess how well the co-temporal implementations of three algorithms, continuous Bayesian, discrete Bayesian, and computational algebraic, can 1) identify two types of entity relationships, parent and sibling, between biological entities, 2) deal with experimental sparse time course data, and 3) handle experimental noise seen in replicate data sets. These algorithms are evaluated, using the shuffle index metric, for how well the resulting models match literature models in terms of siblings and parent relationships. Results indicate that all three co-temporal algorithms perform well, at a statistically significant level, at finding sibling relationships, but perform relatively poorly in finding parent relationships.

Garsia–Haiman modules C[Xn,Yn]/Iγ are quotient rings in the variables Xn={x1,x2,…,xn} and Yn={y1,y2,…,yn} that generalize the quotient ring C[Xn]/I, where I is the ideal generated by the elementary symmetric polynomials ej(Xn) for 1⩽j⩽n. A bitableau basis for the Garsia–Haiman modules of hollow type is constructed. Applications of this basis to representation theory and other related polynomial spaces are considered.

The development of algorithms that conjecture proteomic networks from sparse time series laboratory data is an open problem with much current interest. The development of indices that measure how well the conjectured proteomic network matches a literature model is also an open problem. In this paper, we apply a computational algebra algorithm ([1, 2, 3]) to chondrocyte signaling data ([14]). In order to compare our model to the literature, we combine data from protein isoforms or from proteins that have been phosphorylated at different sites by summing the associated data measurements. The algorithm produces an ordered list of network edges. The resulting cotemporal model is compared to a composite next-state model derived from Signal Transduction Knowledge Environment (STKE) sources. A shuffle index is used to determine how these results from the computational algorithm compare to the composite network.

Signal transduction networks are crucial for inter- and intra-cellular signaling. Signals are often transmitted via covalent modification of protein structure, with phosphorylation/dephosphorylation as the primary example. In this paper, we apply a recently described method of computational algebra to the modeling of signaling networks, based on time-course protein modification data. Computational algebraic techniques are employed to construct next-state functions. A Monte Carlo method is used to approximate the Deegan–Packel Index of Power corresponding to the respective variables. The Deegan–Packel Index of Power is used to conjecture dependencies in the cellular signaling networks. We apply this method to two examples of protein modification time-course data available in the literature. These experiments identified protein carbonylation upon exposure of cells to sub-lethal concentrations of copper. We demonstrate that this method can identify protein dependencies that might correspond to regulatory mechanisms to shut down glycolysis in a reverse, step-wise fashion in response to copper-induced oxidative stress in yeast. These examples show that the computational algebra approach can identify dependencies that may outline signaling networks involved in the response of glycolytic enzymes to the oxidative stress caused by copper.

Reconstructing networks from time series data is a difficult inverse problem. We apply two methods to this problem using co-temporal functions. Co-temporal functions capture mathematical invariants over time series data. Two modeling techniques for co-temporal networks, one based on algebraic techniques and the other on Bayesian inference, are compared and contrasted on simulated biological network data.

This paper studies reciprocals of formal power series whose coefficients are monotone and bounded by a geometrically decaying sequence. Explicit and applicable, optimal decay rates are provided for the coefficients of the reciprocal series in terms of the parameters of the geometric bound. The results imply a best possible lower bound on the zeros of the series being considered.

A key issue in the study of protein signaling networks is understanding the relationships among proteins in the network. Understanding these relationships in the context of a network is one of the major challenges for modern biology [2, 6]. In the laboratory a time series of protein modification measurements is taken in order that relationships among the activations can be conjectured. Laubenbacher and Stigler [5] have developed an algorithm to make conjectures concerning gene expression. Their algorithm analyses the relations as variables in polynomials, using techniques based in computational algebra. This paper focuses on heuristics for applying their method to conjecture dependencies between proteins in signal transduction networks.

A two-variable analogue of the descents monomials is defined and is shown to form a basis for the dense Garsia-Haiman modules. A two-variable generalization of a decomposition of a P-partition is shown to give the algorithm for the expansion into this descent basis. Some examples of dense Garsia-Haiman modules include the coinvariant rings associated with certain complex reflection groups.

For certain subsets $S$ and $T$ of $${\cal A} =\bigl\{\cdots, (0,2), (0,1), (0,0), (1,0), (2,0), \cdots \bigr\}$$ and factor spaces ${\bf C}_{S}[X,Y]$, ${\bf C}^{+}_{S,T}[X,Y,Z,W]$ and ${\bf C}^{-}_{S,T}[X,Y,Z,W]$, bitableaux bases are constructed that are indexed by pairs of standard tableaux and sequences in the collections $\Upsilon_{\psi_S}$ and $\Upsilon_{\psi_T}$. These bases give combinatorial interpretations to the appropriate Hilbert series of these spaces as well as the graded character of ${\bf C}_{S}[X,Y]$. The factor space ${\bf C}_{S}[X,Y]$ is an analogue of the coinvariant ring of a polynomial ring in two sets of variables. ${\bf C}^{+}_{S,T}[X,Y,Z,W]$ and ${\bf C}^{-}_{S,T}[X,Y,Z,W]$ are analogues of coinvariant spaces in symmetric and skew-symmetric polynomial settings, respectively. The elements of the bitableaux bases are appropriately defined images in the polynomial spaces of bipermanents. The combinatorial interpretations of the respective Hilbert series and graded characters are given by statistics based on cocharge tableaux. Additionally, it is shown that the Hilbert series and graded characters factor nicely. One of these factors gives the Hilbert series of a collection of Schur functions $s_{\lambda/\mu}$ where $\mu$ varies in an appropriately defined $\lambda$.

Let C[X, Y] denote the ring of polynomials with complex coefficients in the variables X = {x1(, .)..., x(n)} and Y = {y1, ..., y(n)}, let S-n denote the symmetric group of order nl, let C-m denote the cyclic group C-m = {e2(pi ij/m) : 0 less than or equal to j less than or equal to m - 1} of order m, let H-k denote the subgroup of order k of C-m, and let G(n, m) = C-m ? S-n (the wreath product of C-m with S-n). Each element a phi of G(n, m) may be represented as a generalized permutation phi = [epsilon((1))sigma(1), ..., epsilon(n)sigma(n)] where epsilon(j) = e(2 pi ihj/m) where 0 less than or equal to hj less than or equal to m - 1 and sigma = [sigma(1), ..., sigma(n)] is an element of S-n. Let G(n,) (m,) (k) = {phi is an element of G(n, m) : Pi(i=1)(n) epsilon(i) is an element of H-k}. In this paper, alternants A(m, k, v) are defined in the variables X and Y and where n, m, and k are integers and v is a partition. Setting J(n, m, k, v)(X, Y) to be the ideal J(n, m, k, v)(X, Y) = {P is an element of C[X, Y] : P(partial derivative(xl), ..., partial derivative xn, partial derivative yl, ..., partial derivative(yn)) A(m, k, v) = 0}, where partial derivative(xi) (resp. partial derivative(yj)) is the partial differential operator with respect to x(i) (resp. y(j)), the action of G(n,) (m,) (k) on the quotient ring C-n,C- m,C- (k,) (v) = C[X, Y]/J(n, m, k, v) (X, Y) is isomorphic to the regular representation of G(n,) (m,) (k) when v = (1(p)), some v = (p) (and k divides in) or when v is a hook-shape and ic = i. Bases are constructed for these quotient rings that exhibit the decomposition of the regular representation into irreducibles. It should be noted that the alternants A,n,k,v are generalizations of the alternants defined by A. Garsia and M. Haiman (1996, Election. J. Combin. 3, No. 2). Thus the graded characters of C-n,C- (m,) (k,) (v) give generalizations of the q, t-Kostka coefficients. (C) 1999 Academic Press.

Let Q[X, Y] denote the ring of polynomials with rational coefficients in the variables X = {x1, x2,…, xn} and Y = {y1, y2,…, yn}. Garsia and Haiman in A remarkable q, tCatalan and q-Lagrange inversion (see [4]) define the diagonal harmonics as the solution space DHn = {P(X,Y) ∈ Q[X,Y] : ∑i=1n ∂xih∂yikP = 0 , they define the diagonal harmonic alternants DHAn as DHAn = {P(X, Y) ∈ DHn : σP = sign(σ)P} and conjecture that the dimension of DHAn is the Catalan number 1(n+1)2nn, they conjecture that the Hilbert series of DHAn is a q, t polynomial generalization of the Catalan numbers. In this paper, I conjecture that a collection of polynomials closely related to a collection of Schur functions is a basis for DHAn. This conjecture has been verified by computer up to n = 7. Furthermore, a ring EVn analogous to DHAn is studied and it is shown that in EVn the analogous conjecture is true. The Hilbert series of EVn gives a different q, t polynomial generalization of the Catalan numbers than that given by Garsia and Haiman.

LetR=Q[x1, x2, …, xn,y1, y2, …, yn,z1, …, zn,w1, …, wn], letRSn={P∈R:σP=P∀σ∈Sn} and letμandνbe hook shape partitions ofn. WithΔμ(X, Y) andΔν(Z, W) being appropriately defined determinants, ∂xibeing the partial derivative operator with respect toxiandP(∂)=P(∂x1, …, ∂xn, ∂y1, …, ∂wn), define Iμ, ν={P∈RSn:P(∂)Δμ(X, Y)Δν(Z, W)=0}. A basis is constructed for the polynomial quotient ringRSn/Iμ, νthat is indexed by pairs of standard tableaux. The Hilbert series ofRSn/Iμ, νis related to the Macdonaldq, t-Kostka coefficients.

Let R = Q[x1, x2, . . . , xn] be the ring of polynomials in the variables x1, x2, . . . , xn. Let WB be the finite reflection group of type Bn, let IB be a basic set of invariants of WB, and let R*B, denote the quotient of R by the ideal generated by IB. It is well known (see [Macdonald, 1991]) that the action of WB on the quotient ring R*B, viewed as a vector space over R, is isomorphic to the left regular representation of WB. Using methods similar to those in [Allen, 1992, 1993] we construct a basis PSC of R*B which exhibits the decomposition of R*B into its irreducible components. Now let WD be the finite reflection group of type Dn, let ID be a basic set of invariants for WD, and let R*D be the quotient of R with the ideal generated by ID. We will show that the basis PSC has the remarkable property that when restricted to R*D, exactly one-half of the elements of PSC are non-zero and the non-zero polynomials PSCD form a basis for R*D. The action of WD on the quotient ring R*D, viewed as a vector space of R, is also isomorphic to the left regular representation of WD (see [Macdonald, 1991]). This collection of polynomials PSCD gives the decomposition of R*D into its irreducible components when n is odd. A slight modification of PSCD gives a basis for the decomposition of R*D when n is even. We use these bases to construct the respective graded characters of R*B and R*D.