The quality of zero bounds for complex polynomials

In this paper, we evaluate the quality of zero bounds on the moduli of univariate complex polynomials. We select classical and recently developed bounds and evaluate their quality by using several sets of complex polynomials. As the quality of priori bounds has not been investigated thoroughly, our results can be useful to find optimal bounds to locate the zeros of complex polynomials.     

The relationship between zeros and factors of binding polynomials and cooperativity in protein-ligand binding

Cooperativity in the protein-ligand binding process is discussed in terms of the zeros of the binding polynomial and the corresponding possible factorizations of the binding polynomial into polynomials having non-negative coefficients. Particular attention is paid to the case in which the real parts of all zeros are negative (Hurwitz polynomial) and the case in which the binding polynomial admits no positive factorization (positive irreducible polynomial). Such factorizations are then interpreted as site linkage patterns and related to cooperativity. The possible combinations of zeros of the binding polynomials for the MWC and KNF tetrahedral, square and linear models are determined and the corresponding factorization and linkage patterns analyzed. An application and interpretation are then made for data obtained from Trout I hemoglobin.     

Structural Differentiation of Graphs Using Hosoya-Based Indices

In this paper, we introduce the Hosoya-Spectral indices and the Hosoya information content of a graph. The first measure combines structural information captured by partial Hosoya polynomials and graph spectra. The latter is a graph entropy measure which is based on blocks consisting of vertices with the same partial Hosoya polynomial. We evaluate the discrimination power of these quantities by interpreting numerical results.     

A probabilistic model for fitting MWC polynomials in protein-ligand binding

Given a binding polynomial in Adair form, A(x) = 1 + beta 1 x + ... + beta n x n, beta i greater than or equal to 0, a basic problem is to determine a method of fitting a model polynomial to A(x) and a quantitative measure of the goodness of fit. This paper presents such a method for fitting Monod-Wyman-Changeux (MWC) model polynomials when A(x) is of degree three or four. The method of fitting is based on the property that the zeros of an MWC polynomial of any degree lie on a circle in the complex plane. The parameters in the MWC model are determined so that if possible this circle coincides with the circle on which lie the zeros of A(x). The measure of goodness of fit is provided by a probabilistic model which gives the probability that a binding polynomial has its zeros on a circle on which lie the zeros of an MWC polynomial and if so, the probability that the juxtaposition of the two sets of zeros can occur by chance alone.     

Algorithmic computation of knot polynomials of secondary structure elements of proteins.

The classification of protein structures is an important and still outstanding problem. The purpose of this paper is threefold. First, we utilize a relation between the Tutte and homfly polynomial to show that the Alexander-Conway polynomial can be algorithmically computed for a given planar graph. Second, as special cases of planar graphs, we use polymer graphs of protein structures. More precisely, we use three building blocks of the three-dimensional protein structure--alpha-helix, antiparallel beta-sheet, and parallel beta-sheet--and calculate, for their corresponding polymer graphs, the Tutte polynomials analytically by providing recurrence equations for all three secondary structure elements. Third, we present numerical results comparing the results from our analytical calculations with the numerical results of our algorithm-not only to test consistency, but also to demonstrate that all assigned polynomials are unique labels of the secondary structure elements. This paves the way for an automatic classification of protein structures.     

A General Method for Computing the Homfly Polynomial of DNA Double Crossover 3-Regular Links

In the last 20 years or so, chemists and molecular biologists have synthesized some novel DNA polyhedra. Polyhedral links were introduced to model DNA polyhedra and study topological properties of DNA polyhedra. As a very powerful invariant of oriented links, the Homfly polynomial of some of such polyhedral links with small number of crossings has been obtained. However, it is a challenge to compute Homfly polynomials of polyhedral links with large number of crossings such as double crossover 3-regular links considered here. In this paper, a general method is given for computing the chain polynomial of the truncated cubic graph with two different labels from the chain polynomial of the original labeled cubic graph by substitutions. As a result, we can obtain the Homfly polynomial of the double crossover 3-regular link which has relatively large number of crossings.     

Protein-ligand binding with a missing species

Considerable experimental evidence has been produced recently that shows that in the binding of oxygen or carbon monoxide to certain tetrameric hemoglobins, the triply-ligated species is virtually non-existent. The binding polynomial representing this phenomenon for the general case is P(x) = 1 + beta 1x + ... + beta n-1xn-1 + beta nxn, where beta n-1 is nearly zero. The zeros, factorization and associated Hill plots of such binding polynomials with beta n-1 = 0 are investigated for the general case, and are analyzed in detail for n = 3 and n = 4. These results are then compared with the results obtained from experimental data on a number of tetrameric hemoglobins for which beta 3 is small. One concludes that, apart from the slope of the high-saturation asymptote of the Hill plot, a small perturbation of beta 3 from zero produces small changes in other properties associated with the binding process, such as fractional saturation, maximum Hill slope, and zeros and factorization of the binding polynomial.     

Characteristic and counting polynomials: modelling nonane isomers properties

The major goal of this study was to investigate the broad application of graph polynomials to the analysis of Henry's law constants (solubility) of nonane isomers. In this context, Henry's law constants of nonane isomers were modelled using characteristic and counting polynomials. The characteristic and counting polynomials on the distance matrix (CDi), on the maximal fragments matrix (CMx), on the complement of maximal fragments matrix (CcM) and on the Szeged matrix (CSz) were calculated for each compound. One of the nonane isomers, 4-methyloctane, was identified as an outlier and was withdrawn from further analysis. This report describes the performance and characteristics of most significant models. The results showed that Henry's law constants of nonane isomers could be modelled by using characteristic polynomial and counting polynomial on the distance matrix.     

Genetic programming of polynomial harmonic networks using the discrete Fourier transform

This paper presents a genetic programming system that evolves polynomial harmonic networks. These are multilayer feed-forward neural networks with polynomial activation functions. The novel hybrids assume that harmonics with non-multiple frequencies may enter as inputs the activation polynomials. The harmonics with non-multiple, irregular frequencies are derived analytically using the discrete Fourier transform. The polynomial harmonic networks have tree-structured topology which makes them especially suitable for evolutionary structural search. Empirical results show that this hybrid genetic programming system outperforms an evolutionary system manipulating polynomials, the traditional Koza-style genetic programming, and the harmonic GMDH network algorithm on processing time series.     

Description of the topological entanglement of DNA catenanes and knots by a powerful method involving strand passage and recombination

We utilize a recently discovered, powerful method to classify the topological state of knots and catenanes. In this method, each such form is associated with a unique polynomial. These polynomials allow a rigorous determination of whether knotted or catenated DNA molecules that appear distinct actually are, and indicate the structure of related molecules. A tabulation is given of the polynomials for all possible stereoisomers of many of the knotted and catenated forms that are found in DNA. The polynomials for a substrate DNA molecule and the products obtained from it by either recombination or strand passage by a topoisomerase are related by a simple theorem. This theorem affords natural applications of the polynomial method to these processes. Examples are presented involving site-specific recombination by the transposon Tn3-encoded resolvase and the phage lambda integrase, in which product structure is predicted as a function of crossover mechanism.     

An analogue of the binding polynomial for the case of ligands binding to an aggregating macromolecule

It is known that when a macromolecule exists as a mixture of forms with different molecular weights in equilibrium in dilute solution with ligands then the saturation functions cannot be derived by differentiating a binding polynomial. However, we demonstrate that in such circumstances a saturation function can be expressed in terms of the binding polynomials for each of the polymeric forms of macromolecule and integration of this using the conservation equation affords a function, Q, which is the analogue of the binding polynomial for a non-aggregating system. The existence of Q leads to heterotropic linkage relationships.     

Explaining variation in tropical plant community composition: influence of environmental and spatial data quality

The degree to which variation in plant community composition (beta-diversity) is predictable from environmental variation, relative to other spatial processes, is of considerable current interest. We addressed this question in Costa Rican rain forest pteridophytes (1,045 plots, 127 species). We also tested the effect of data quality on the results, which has largely been overlooked in earlier studies. To do so, we compared two alternative spatial models [polynomial vs. principal coordinates of neighbour matrices (PCNM)] and ten alternative environmental models (all available environmental variables vs. four subsets, and including their polynomials vs. not). Of the environmental data types, soil chemistry contributed most to explaining pteridophyte community variation, followed in decreasing order of contribution by topography, soil type and forest structure. Environmentally explained variation increased moderately when polynomials of the environmental variables were included. Spatially explained variation increased substantially when the multi-scale PCNM spatial model was used instead of the traditional, broad-scale polynomial spatial model. The best model combination (PCNM spatial model and full environmental model including polynomials) explained 32% of pteridophyte community variation, after correcting for the number of sampling sites and explanatory variables. Overall evidence for environmental control of beta-diversity was strong, and the main floristic gradients detected were correlated with environmental variation at all scales encompassed by the study (c. 100–2,000 m). Depending on model choice, however, total explained variation differed more than fourfold, and the apparent relative importance of space and environment could be reversed. Therefore, we advocate a broader recognition of the impacts that data quality has on analysis results. A general understanding of the relative contributions of spatial and environmental processes to species distributions and beta-diversity requires that methodological artefacts are separated from real ecological differences.     

Testing inflated zeros in binomial regression models

Binomial regression models are commonly applied to proportion data such as those relating to the mortality and infection rates of diseases. However, it is often the case that the responses may exhibit excessive zeros; in such cases a zero‐inflated binomial (ZIB) regression model can be applied instead. In practice, it is essential to test if there are excessive zeros in the outcome to help choose an appropriate model. The binomial models can yield biased inference if there are excessive zeros, while ZIB models may be unnecessarily complex and hard to interpret, and even face convergence issues, if there are no excessive zeros. In this paper, we develop a new test for testing zero inflation in binomial regression models by directly comparing the amount of observed zeros with what would be expected under the binomial regression model. A closed form of the test statistic, as well as the asymptotic properties of the test, is derived based on estimating equations. Our systematic simulation studies show that the new test performs very well in most cases, and outperforms the classical Wald, likelihood ratio, and score tests, especially in controlling type I errors. Two real data examples are also included for illustrative purpose.     

Testing differences in relative growth rate: A method avoiding curve fitting and pairing

A method is discussed to test differences in relative growth rates. This method is based on an analysis of variance, with In-transformed plant weight as dependent variable. A significant Group × Time interaction indicates differences in relative growth rates between groups. The advantages over the "classical" and "functional" growth analyses are: (1) No pairing procedure is required. (2) More than two groups may be evaluated in one analysis. (3) No decision is required about the polynomial used to fit the data. (4) By partitioning the interaction effect using orthogonal polynomials insight is gained into the nature of differences in relative growth rate. (5) By concentrating attention on the lower order terms of the polynomials, the influence of extraneous variation on conclusions may be minimized.     

Clustering gene expression data using graph separators

Recent work has used graphs to modelize expression data from microarray experiments, in view of partitioning the genes into clusters. In this paper, we introduce the use of a decomposition by clique separators. Our aim is to improve the classical clustering methods in two ways: first we want to allow an overlap between clusters, as this seems biologically sound, and second we want to be guided by the structure of the graph to define the number of clusters. We test this approach with a well-known yeast database (Saccharomyces cerevisiae). Our results are good, as the expression profiles of the clusters we find are very coherent. Moreover, we are able to organize into another graph the clusters we find, and order them in a fashion which turns out to respect the chronological order defined by the the sporulation process.     

A new measure of cooperativity in protein-ligand binding

An allosteric binding system consisting of a single ligand and a nondissociating macromolecule having multiple binding sites can be represented by a binding polynomial. Various properties of the binding process can be obtained by analyzing the coefficients of the binding polynomial and such functions as the binding curve and the Hill plot. The Hill plot has an asymptote of unit slope at each end and the departure of the slope from unity at any point can be used to measure the effective interaction free energy at that point. Of particular interest in detecting and measuring cooperativity are extrema of the Hill slope and its value at the half-saturation point. If the binding polynomial is symmetric, then there is an extremum of the Hill slope at the half-saturation point. This value, the Hill coefficient, is a convenient measure of cooperativity. The purpose of this paper is to express the Hill coefficient for symmetric binding polynomials in terms of the roots of the polynomial and to give an interpretation of cooperativity in terms of the geometric pattern of the roots in the complex plane. This interpretation is then applied to the binding polynomials for the MWC (Monod-Wyman-Changeux) and KNF (Koshland-Nemethy-Filmer) models.     

Steady State Detection of Chemical Reaction Networks Using a Simplified Analytical Method

Chemical reaction networks (CRNs) are susceptible to mathematical modelling. The dynamic behavior of CRNs can be investigated by solving the polynomial equations derived from its structure. However, simple CRN give rise to non-linear polynomials that are difficult to resolve. Here we propose a procedure to locate the steady states of CRNs from a formula derived through algebraic geometry methods. We have applied this procedure to define the steady states of a classic CRN that exhibits instability, and to a model of programmed cell death.     

A polynomial based model for cell fate prediction in human diseases

Background

Cell fate regulation directly affects tissue homeostasis and human health. Research on cell fate decision sheds light on key regulators, facilitates understanding the mechanisms, and suggests novel strategies to treat human diseases that are related to abnormal cell development.

Results

In this study, we proposed a polynomial based model to predict cell fate. This model was derived from Taylor series. As a case study, gene expression data of pancreatic cells were adopted to test and verify the model. As numerous features (genes) are available, we employed two kinds of feature selection methods, i.e. correlation based and apoptosis pathway based. Then polynomials of different degrees were used to refine the cell fate prediction function. 10-fold cross-validation was carried out to evaluate the performance of our model. In addition, we analyzed the stability of the resultant cell fate prediction model by evaluating the ranges of the parameters, as well as assessing the variances of the predicted values at randomly selected points. Results show that, within both the two considered gene selection methods, the prediction accuracies of polynomials of different degrees show little differences. Interestingly, the linear polynomial (degree 1 polynomial) is more stable than others. When comparing the linear polynomials based on the two gene selection methods, it shows that although the accuracy of the linear polynomial that uses correlation analysis outcomes is a little higher (achieves 86.62%), the one within genes of the apoptosis pathway is much more stable.

Conclusions

Considering both the prediction accuracy and the stability of polynomial models of different degrees, the linear model is a preferred choice for cell fate prediction with gene expression data of pancreatic cells. The presented cell fate prediction model can be extended to other cells, which may be important for basic research as well as clinical study of cell development related diseases.