Rhythmic and non-rhythmic attractors in asynchronous random Boolean networks

In multi-component, discrete systems, such as Boolean networks and cellular automata, the scheme of updating of the individual elements plays a crucial role in determining their dynamic properties and their suitability as models of complex phenomena. Many interesting properties of these systems rely heavily on the use of synchronous updating of the individual elements. Considerations of parsimony have motivated the claim that, if the natural systems being modelled lack any clear evidence of synchronously driven elements, then random asynchronous updating should be used by default. The introduction of a random element precludes the possibility of strictly cyclic behaviour. In principle, this poses the question of whether asynchronously driven Boolean networks, cellular automata, etc., are inherently bad choices at the time of modelling rhythmic phenomena. This paper focuses on this subsidiary issue for the case of Asynchronous Random Boolean Networks (ARBNs). It defines measures of pseudo-periodicity between states and sufficiently relaxed statistical constraints. These measures are used to guide a genetic algorithm to find appropriate examples. Success in this search for a number of cases, and the subsequent statistical analysis lead to the conclusion that ARBNs can indeed be used as models of co-ordinated rhythmic phenomena, which may be stronger precisely because of their in-built asynchrony. The same technique is used to find non-stationary attractors that show no rhythm. Evidence suggests that the latter are more abundant than rhythmic attractor. The methodology is flexible, and allows for more demanding statistical conditions for defining pseudo-periodicity, and constraining the evolutionary search.     

Inference of S-system models of genetic networks by solving one-dimensional function optimization problems

Voit and Almeida have proposed the decoupling approach as a method for inferring the S-system models of genetic networks. The decoupling approach defines the inference of a genetic network as a problem requiring the solutions of sets of algebraic equations. The computation can be accomplished in a very short time, as the approach estimates S-system parameters without solving any of the differential equations. Yet the defined algebraic equations are non-linear, which sometimes prevents us from finding reasonable S-system parameters. In this study, we propose a new technique to overcome this drawback of the decoupling approach. This technique transforms the problem of solving each set of algebraic equations into a one-dimensional function optimization problem. The computation can still be accomplished in a relatively short time, as the problem is transformed by solving a linear programming problem. We confirm the effectiveness of the proposed approach through numerical experiments.     

A semiparametric estimate of treatment effects with censored data

A semiparametric estimate of an average regression effect with right-censored failure time data has recently been proposed under the Cox-type model where the regression effect beta(t) is allowed to vary with time. In this article, we derive a simple algebraic relationship between this average regression effect and a measurement of group differences in k-sample transformation models when the random error belongs to the G(rho) family of Harrington and Fleming (1982, Biometrika 69, 553-566), the latter being equivalent to the conditional regression effect in a gamma frailty model. The models considered here are suitable for the attenuating hazard ratios that often arise in practice. The results reveal an interesting connection among the above three classes of models as alternatives to the proportional hazards assumption and add to our understanding of the behavior of the partial likelihood estimate under nonproportional hazards. The algebraic relationship provides a simple estimator under the transformation model. We develop a variance estimator based on the empirical influence function that is much easier to compute than the previously suggested resampling methods. When there is truncation in the right tail of the failure times, we propose a method of bias correction to improve the coverage properties of the confidence intervals. The estimate, its estimated variance, and the bias correction term can all be calculated with minor modifications to standard software for proportional hazards regression.     

Can a simple algebraic analysis predict markers-genome heterozygosity correlations?

A current algebraic analysis on genome-wide heterozygosity estimates suggests that correlations between molecular markers and genome-wide heterozygosity, rho, depend on the ratio between the number of markers used, r, and the number of genome loci, n; that is: rho approximately square root r/n. Hence, it is unfeasible to obtain reliable estimates of genome-wide heterozygosity in species of large genome using a few markers. We cast some doubts about this analysis as it assumed that the probability that an individual was heterozygous at a locus is equal to the average heterozygosity of this locus in the population. However, we believe that individual heterozygosity at a given locus depends on individual pedigree. Because the pedigree is common for all loci of an individual, their probabilities of heterozygosity are not independent within the genome. We first performed simulations generating random genomes for 100 individuals. Among these individuals, markers and genome-wide heterozygosities correlated as expected from the above equation. However, when we simulated random mating among these individuals and in successive generations including their descendents, as occur in real populations, the correlations between markers and genome-wide heterozygosity were much higher than those predicted from algebraic analyses, and estimates of genome-wide heterozygosity improved slightly with the increment of the number of loci in the genome.     

MIVQUE and REML Estimators of Variance Components under Proportionality Condition

In this study, a one-way random effect model with unequal cell variances is considered, and the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and Restricted Maximum Likelihood (REML) estimator of the variance components are studied. The algebraic inversion of the variance matrix of the observation vector is obtained to achieve some computational convenience. Using the proportionality condition described by Talukder (1992) that the cell sizes are proportional to the cell variances, MIVQUE and REML estimators are shown to be the same as the ANOVA estimators.     

Ionic charge conservation and long-term steady state in the Luo-Rudy dynamic cell model.

It has been postulated that cardiac cell models accounting for changes in intracellular ion concentrations violate a conservation principle, and, as a result, computed parameters (e.g., ion concentrations and transmembrane potential, V(m)) drift in time, never attaining steady state. To address this issue, models have been proposed that invoke the charge conservation principle to calculate V(m) from ion concentrations ("algebraic" method), rather than from transmembrane current ("differential" method). The aims of this study are to compare model behavior during prolonged periods of pacing using the algebraic and differential methods, and to address the issue of model drift. We pace the Luo-Rudy dynamic model of a cardiac ventricular cell and compare the time-dependent behavior of computed parameters using the algebraic and differential methods. When ions carried by the stimulus current are taken into account, the algebraic and differential methods yield identical results and neither shows drift in computed parameters. The present study establishes the proper pacing protocol for simulation studies of cellular behavior during long periods of rapid pacing. Such studies are essential for mechanistic understanding of arrhythmogenesis, since cells are subjected to rapid periodic stimulation during many arrhythmias.     

Maximum likelihood estimation for a special type of grouped data with an application to a dose-response problem.

A short-cut method is given for calculating grouped maximum likelihood (ML) estimates when the data are relatively coarsely grouped in some directions, but more finely grouped in others. The algebraic details are then worked out for a dose-response problem that generates data of this kind. The situation envisaged is a variation on the usual quantal response problem in that dosage levels are taken to be random but grouped. Finally, the method is applied both to real and simulated response data conforming to this pattern and shown to work well in practice.     

Towards a calculus of biomolecular complexes at equilibrium

An overview is presented of the construction and use of algebraic partition functions to represent the equilibrium statistical mechanics of multimolecular complexes and their action within a larger regulatory network. Unlike many applications of equilibrium statistical mechanics, multimolecular complexes may operate with various subsets of their components present and connected to the others, the rest remaining in solution. Thus they are variable-structure systems. This aspect of their behavior may be accounted for by the use of 'fugacity' variables as a representation within the partition functions. Four principles are proposed by which the combinatorics of molecular complex construction can be reflected in the construction of their partition functions. The corresponding algebraic operations on partition functions are multiplication, addition, function composition and a less commonly used operation called contraction. Each has a natural interpretation in terms of probability distributions on multimolecular structures. Possible generalizations to nonequilibrium statistical mechanics are briefly discussed.     

Indirect Estimates of Total Fertility Rate Using Child Woman/Ratio: A Comparison with the Bogue-Palmore Method

Indirect estimation methodologies of the total fertility rate (TFR) have a long history within demography and have provided important techniques applied demographers can use when data is sparse or lacking. However new methodologies for approximating the total fertility rate have not been proposed in nearly 30 years. This study presents a novel method for indirectly approximating the total fertility rate using an algebraic rearrangement of the general fertility rate (GFR) through the known relationship between GFR and TFR. It then compares the proposed method to the well-known Bogue-Palmore method. These methods are compared in 196 countries and include overall errors as well as characteristics of the countries that contribute to fertility behavior. Additionally, these methods were compared geographically to find any geographical patterns. We find this novel method is not only simpler than the Bogue-Palmore method, requiring fewer data inputs, but also has reduced algebraic and absolute errors when compared with the Bogue-Palmore method and specifically outperforms the Bogue-Palmore method in developing countries. We find that our novel method may be useful estimation procedure for demographers.     

Calculation of steady-state rate equations and the fluxes between substrates and products in enzyme reactions.

1. Two methods are described for deriving the steady-state velocity of an enzyme reaction from a consideration of fluxes between enzyme intermediates. The equivalent-reaction technique, in which enzyme intermediates are systematically eliminated and replaced by equivalent reactions, appears the most generally useful. The methods are applicable to all enzyme mechanisms, including three-substrate and random Bi Bi Ping Pong mechanisms. Solutions are obtained in algebraic form and these are presented for the common random Bi Bi mechanisms. The steady-state quantities of the enzyme intermediates may also be calculated. Additional steps may be introduced into enzyme mechanisms for which the steady-state velocity equation is already known. 2. The calculation of fluxes between substrates and products in three-substrate and random Bi Bi Ping Pong mechanisms is described. 3. It is concluded that the new methods may offer advantages in ease of calculation and in the analysis of the effects of individual steps on the overall reaction. The methods are used to show that an ordered addition of two substrates to an enzyme which is activated by another ligand will not necessarily give hyperbolic steady-state-velocity kinetics or the flux ratios characteristic of an ordered addition, if the dissociation of the ligand from the enzyme is random.     

A modified theory of the potential and inhomogeneous ion distribution near charged surfaces

A new theory is proposed to calculate the surface potential. The potential is derived from a modified force equilibrium resulting in an algebraic equation. Special boundary conditions are introduced to handle the modified forces. Numeric examples are given based on erythrocyte data and are compared with the Gouy-Chapman theory.     

Modeling multivariate survival data by a semiparametric random effects proportional odds model

In this article, the focus is on the analysis of multivariate survival time data with various types of dependence structures. Examples of multivariate survival data include clustered data and repeated measurements from the same subject, such as the interrecurrence times of cancer tumors. A random effect semiparametric proportional odds model is proposed as an alternative to the proportional hazards model. The distribution of the random effects is assumed to be multivariate normal and the random effect is assumed to act additively to the baseline log-odds function. This class of models, which includes the usual shared random effects model, the additive variance components model, and the dynamic random effects model as special cases, is highly flexible and is capable of modeling a wide range of multivariate survival data. A unified estimation procedure is proposed to estimate the regression and dependence parameters simultaneously by means of a marginal-likelihood approach. Unlike the fully parametric case, the regression parameter estimate is not sensitive to the choice of correlation structure of the random effects. The marginal likelihood is approximated by the Monte Carlo method. Simulation studies are carried out to investigate the performance of the proposed method. The proposed method is applied to two well-known data sets, including clustered data and recurrent event times data.     

Order-restricted inference for clustered ROC data with application to fingerprint matching accuracy

Receiver operating characteristic (ROC) curve is commonly used to evaluate and compare the accuracy of classification methods or markers. Estimating ROC curves has been an important problem in various fields including biometric recognition and diagnostic medicine. In real applications, classification markers are often developed under two or more ordered conditions, such that a natural stochastic ordering exists among the observations. Incorporating such a stochastic ordering into estimation can improve statistical efficiency (Davidov and Herman, 2012). In addition, clustered and correlated data arise when multiple measurements are gleaned from the same subject, making estimation of ROC curves complicated due to within-cluster correlations. In this article, we propose to model the ROC curve using a weighted empirical process to jointly account for the order constraint and within-cluster correlation structure. The algebraic properties of resulting summary statistics of the ROC curve such as its area and partial area are also studied. The algebraic expressions reduce to the ones by Davidov and Herman (2012) for independent observations. We derive asymptotic properties of the proposed order-restricted estimators and show that they have smaller mean-squared errors than the existing estimators. Simulation studies also demonstrate better performance of the newly proposed estimators over existing methods for finite samples. The proposed method is further exemplified with the fingerprint matching data from the National Institute of Standards and Technology Special Database 4.     

The probability density of the total IBD length over a single autosome in unilineal relationships

Several authors have studied identity by descent (IBD) by way of a continuous recombination process along a chromosome. Despite its potential uses in, for example, gene mapping or delineation of biological relationships there has been no exact algebraic result given for the probability density function of the IBD proportion in any familial relationship. Other authors have derived algebraic approximations in the case of half-sibs by way of the Poisson clumping heuristic and used computational methods to compute the distribution function of the IBD sharing for unilineal relationships. Here we provide a general numerical method for finding the density of IBD sharing that could be applied to any unilineal relationship and more importantly we derive algebraically an expression for the density for a grandparent-grandchild relationship. Initially we assume that recombination events occur at random along a chromosome, then go on to show how the method could be extended to incorporate a form of genetic interference.     

Analyzing incomplete longitudinal clinical trial data

Using standard missing data taxonomy, due to Rubin and co-workers, and simple algebraic derivations, it is argued that some simple but commonly used methods to handle incomplete longitudinal clinical trial data, such as complete case analyses and methods based on last observation carried forward, require restrictive assumptions and stand on a weaker theoretical foundation than likelihood-based methods developed under the missing at random (MAR) framework. Given the availability of flexible software for analyzing longitudinal sequences of unequal length, implementation of likelihood-based MAR analyses is not limited by computational considerations. While such analyses are valid under the comparatively weak assumption of MAR, the possibility of data missing not at random (MNAR) is difficult to rule out. It is argued, however, that MNAR analyses are, themselves, surrounded with problems and therefore, rather than ignoring MNAR analyses altogether or blindly shifting to them, their optimal place is within sensitivity analysis. The concepts developed here are illustrated using data from three clinical trials, where it is shown that the analysis method may have an impact on the conclusions of the study.     

Testing random effects in linear mixed‐effects models with serially correlated errors

In linear mixed‐effects models, random effects are used to capture the heterogeneity and variability between individuals due to unmeasured covariates or unknown biological differences. Testing for the need of random effects is a nonstandard problem because it requires testing on the boundary of parameter space where the asymptotic chi‐squared distribution of the classical tests such as likelihood ratio and score tests is incorrect. In the literature several tests have been proposed to overcome this difficulty, however all of these tests rely on the restrictive assumption of i.i.d. measurement errors. The presence of correlated errors, which often happens in practice, makes testing random effects much more difficult. In this paper, we propose a permutation test for random effects in the presence of serially correlated errors. The proposed test not only avoids issues with the boundary of parameter space, but also can be used for testing multiple random effects and any subset of them. Our permutation procedure includes the permutation procedure in Drikvandi, Verbeke, Khodadadi, and Partovi Nia (2013) as a special case when errors are i.i.d., though the test statistics are different. We use simulations and a real data analysis to evaluate the performance of the proposed permutation test. We have found that random slopes for linear and quadratic time effects may not be significant when measurement errors are serially correlated.     

Analysis of Ter-Ter enzyme kinetic mechanisms by computer simulation of isotope exchange at chemical equilibrium: development and application of ISOTER, a personal-computer-based program

A convenient, personal-computer-based program has been developed that allows simulation of isotopic exchange kinetics at chemical equilibrium catalyzed by a three reactant-three product (TerTer) enzyme system: A + B + C integral of P + Q + R. This program, ISOTER, utilizes a rapid algebraic method to calculate the exchange rate between any reactant-product pair as a function of the substrate concentration and avoids altogether the necessity of deriving an explicit (but cumbersome and impractical) equation for exchange rate. ISOTER was used to generate model saturation patterns for 16 different TerTer kinetic mechanisms, varying different combinations of reactant-product pairs in constant ratio at equilibrium: [all substrates], [A, P], [B, Q], and [C, R], while holding the nonvaried components constant. These model studies indicate that virtually every one of these mechanisms can be distinguished from the others. In addition, ISOTER has been used to fit multiple sets of experimental data for Escherichia coli glutamine synthetase, which produced a set of rate constants consistent with the previously proposed "preferred order random" kinetic mechanism.     

Profile Parsimony (PP): an analysis under Implied Weights (IW)

Faith and Trueman [Syst. Biol. (2001) 331] recently proposed “Profile Parsimony” (PP) as a new form of phylogenetic analysis, but it is equivalent to the Implied Weights (IW) method when certain functions are used. The PP method cannot accommodate missing/inapplicable cells and/or multistate characters. Finally we demonstrate that, as defined, PP scores are not random, and there is no need to generate random trees as proposed by Faith and Trueman.