Influence of parameter values on the oscillation sensitivities of two p53–Mdm2 models

Biomolecular networks that present oscillatory behavior are ubiquitous in nature. While some design principles for robust oscillations have been identified, it is not well understood how these oscillations are affected when the kinetic parameters are constantly changing or are not precisely known, as often occurs in cellular environments. Many models of diverse complexity level, for systems such as circadian rhythms, cell cycle or the p53 network, have been proposed. Here we assess the influence of hundreds of different parameter sets on the sensitivities of two configurations of a well-known oscillatory system, the p53 core network. We show that, for both models and all parameter sets, the parameter related to the p53 positive feedback, i.e. self-promotion, is the only one that presents sizeable sensitivities on extrema, periods and delay. Moreover, varying the parameter set values to change the dynamical characteristics of the response is more restricted in the simple model, whereas the complex model shows greater tunability. These results highlight the importance of the presence of specific network patterns, in addition to the role of parameter values, when we want to characterize oscillatory biochemical systems.

Electronic supplementary material

The online version of this article (doi:10.1007/s11693-015-9173-y) contains supplementary material, which is available to authorized users.     

Applications of the Repeatability of Quantitative Imaging Biomarkers: A Review of Statistical Analysis of Repeat Data Sets

Repeat imaging data sets performed on patients with cancer are becoming publicly available. The potential utility of these data sets for addressing important questions in imaging biomarker development is vast. In particular, these data sets may be useful to help characterize the variability of quantitative parameters derived from imaging. This article reviews statistical analysis that may be performed to use results of repeat imaging to 1) calculate the level of change in parameter value that may be seen in individual patients to confidently characterize that patient as showing true parameter change, 2) calculate the level of change in parameters value that may be seen in individual patients to confidently categorize that patient as showing true lack of parameter change, 3) determine if different imaging devices are interchangeable from the standpoint of repeatability, and 4) estimate the numbers of patients needed to precisely calculate repeatability. In addition, we recommend a set of statistical parameters that should be reported when the repeatability of continuous parameters is studied.     

A mixture model approach to the tests of concordance and discordance between two large-scale experiments with two-sample groups

MOTIVATION: Due to advances in experimental technologies, such as microarray, mass spectrometry and nuclear magnetic resonance, it is feasible to obtain large-scale data sets, in which measurements for a large number of features can be simultaneously collected. However, the sample sizes of these data sets are usually small due to their relatively high costs, which leads to the issue of concordance among different data sets collected for the same study: features should have consistent behavior in different data sets. There is a lack of rigorous statistical methods for evaluating this concordance or discordance. METHODS: Based on a three-component normal-mixture model, we propose two likelihood ratio tests for evaluating the concordance and discordance between two large-scale data sets with two sample groups. The parameter estimation is achieved through the expectation-maximization (E-M) algorithm. A normal-distribution-quantile-based method is used for data transformation. RESULTS: To evaluate the proposed tests, we conducted some simulation studies, which suggested their satisfactory performances. As applications, the proposed tests were applied to three SELDI-MS data sets with replicates. One data set has replicates from different platforms and the other two have replicates from the same platform. We found that data generated by SELDI-MS showed satisfactory concordance between replicates from the same platform but unsatisfactory concordance between replicates from different platforms. AVAILABILITY: The R codes are freely available at http://home.gwu.edu/~ylai/research/Concordance.     

Combining multiple data sets in a likelihood analysis: which models are the best?

Until recently, phylogenetic analyses have been routinely based on homologous sequences of a single gene. Given the vast number of gene sequences now available, phylogenetic studies are now based on the analysis of multiple genes. Thus, it has become necessary to devise statistical methods to combine multiple molecular data sets. Here, we compare several models for combining different genes for the purpose of evaluating the likelihood of tree topologies. Three methods of branch length estimation were studied: assuming all genes have the same branch lengths (concatenate model), assuming that branch lengths are proportional among genes (proportional model), or assuming that each gene has a separate set of branch lengths (separate model). We also compared three models of among-site rate variation: the homogenous model, a model that assumes one gamma parameter for all genes, and a model that assumes one gamma parameter for each gene. On the basis of two nuclear and one mitochondrial amino acid data sets, our results suggest that, depending on the data set chosen, either the separate model or the proportional model represents the most appropriate method for branch length analysis. For all the data sets examined, one gamma parameter for each gene represents the best model for among-site rate variation. Using these models we analyzed alternative mammalian tree topologies, and we describe the effect of the assumed model on the maximum likelihood tree. We show that the choice of the model has an impact on the best phylogeny obtained.     

Genetic variation maintained in multilocus models of additive quantitative traits under stabilizing selection.

Stabilizing selection for an intermediate optimum is generally considered to deplete genetic variation in quantitative traits. However, conflicting results from various types of models have been obtained. While classical analyses assuming a large number of independent additive loci with individually small effects indicated that no genetic variation is preserved under stabilizing selection, several analyses of two-locus models showed the contrary. We perform a complete analysis of a generalization of Wright's two-locus quadratic-optimum model and investigate numerically the ability of quadratic stabilizing selection to maintain genetic variation in additive quantitative traits controlled by up to five loci. A statistical approach is employed by choosing randomly 4000 parameter sets (allelic effects, recombination rates, and strength of selection) for a given number of loci. For each parameter set we iterate the recursion equations that describe the dynamics of gamete frequencies starting from 20 randomly chosen initial conditions until an equilibrium is reached, record the quantities of interest, and calculate their corresponding mean values. As the number of loci increases from two to five, the fraction of the genome expected to be polymorphic declines surprisingly rapidly, and the loci that are polymorphic increasingly are those with small effects on the trait. As a result, the genetic variance expected to be maintained under stabilizing selection decreases very rapidly with increased number of loci. The equilibrium structure expected under stabilizing selection on an additive trait differs markedly from that expected under selection with no constraints on genotypic fitness values. The expected genetic variance, the expected polymorphic fraction of the genome, as well as other quantities of interest, are only weakly dependent on the selection intensity and the level of recombination.     

Tree asymmetry—A sensitive and practical measure for binary topological trees

The topological structure of a binary tree is characterized by a measure called tree asymmetry, defined as the mean value of the asymmetry of its partitions. The statistical properties of this tree-asymmetry measure have been studied using a growth model for binary trees. The tree-asymmetry measure appears to be sensitive for topological differences and the tree-asymmetry expectation for the growth model that we used appears to be almost independent of the size of the trees. These properties and the simple definition make the measure suitable for practical use, for instance for characterizing, comparing and interpreting sets of branching patterns. Examples are given of the analysis of three sets of neuronal branching patterns. It is shown that the variance in tree-asymmetry values for these observed branching patterns corresponds perfectly with the variance predicted by the used growth model.     

Functional cohesion of gene sets determined by latent semantic indexing of PubMed abstracts

High-throughput genomic technologies enable researchers to identify genes that are co-regulated with respect to specific experimental conditions. Numerous statistical approaches have been developed to identify differentially expressed genes. Because each approach can produce distinct gene sets, it is difficult for biologists to determine which statistical approach yields biologically relevant gene sets and is appropriate for their study. To address this issue, we implemented Latent Semantic Indexing (LSI) to determine the functional coherence of gene sets. An LSI model was built using over 1 million Medline abstracts for over 20,000 mouse and human genes annotated in Entrez Gene. The gene-to-gene LSI-derived similarities were used to calculate a literature cohesion p-value (LPv) for a given gene set using a Fisher's exact test. We tested this method against genes in more than 6,000 functional pathways annotated in Gene Ontology (GO) and found that approximately 75% of gene sets in GO biological process category and 90% of the gene sets in GO molecular function and cellular component categories were functionally cohesive (LPv<0.05). These results indicate that the LPv methodology is both robust and accurate. Application of this method to previously published microarray datasets demonstrated that LPv can be helpful in selecting the appropriate feature extraction methods. To enable real-time calculation of LPv for mouse or human gene sets, we developed a web tool called Gene-set Cohesion Analysis Tool (GCAT). GCAT can complement other gene set enrichment approaches by determining the overall functional cohesion of data sets, taking into account both explicit and implicit gene interactions reported in the biomedical literature. Availability: GCAT is freely available at http://binf1.memphis.edu/gcat.     

Gene Ontology-based semantic alignment of biological pathways by evolutionary search

A large number of biological pathways have been elucidated recently, and there is a need for methods to analyze these pathways. One class of methods compares pathways semantically in order to discover parts that are evolutionarily conserved between species or to discover intraspecies similarities. Such methods usually require that the topologies of the pathways being compared are known, i.e. that a query pathway is being aligned to a model pathway. However, sometimes the query only consists of an unordered set of gene products. Previous methods for mapping sets of gene products onto known pathways have not been based on semantic comparison of gene products using ontologies or other abstraction hierarchies. Therefore, we here propose an approach that uses a similarity function defined in Gene Ontology (GO) terms to find semantic alignments when comparing paths in biological pathways where the nodes are gene products. A known pathway graph is used as a model, and an evolutionary algorithm (EA) is used to evolve putative paths from a set of experimentally determined gene products. The method uses a measure of GO term similarity to calculate a match score between gene products, and the fitness value of each candidate path alignment is derived from these match scores. A statistical test is used to assess the significance of evolved alignments. The performance of the method has been tested using regulatory pathways for S. cerevisiae and M. musculus.     

A nonlinear compartmental model of Sr metabolism. I. Non-steady-state kinetics and model building

A model of Sr metabolism was developed by using plasma and urinary Sr kinetic data obtained in groups of postmenopausal women who received four different oral doses of Sr and collected during the Sr administration period (25 days) and for 28 days after cessation of treatment. A nonlinear compartmental formalism that is appropriate for study of non-steady-state kinetics and allows dissociation of variables pertaining to Sr metabolism (system 1) from those indirectly operating on it (system 2) was used. At each stage of model development, the dose-dependent model response was fitted to the four sets of data considered simultaneously (1 set per dose). A seven-compartment model with internal Sr distribution and intestinal, urinary, and bone metabolic pathways was selected. It includes two kinds of nonlinearities: those accounting for saturable intestinal and bone processes, which behave as intrinsic nonlinearities because they are directly dependent on Sr, and extrinsic nonlinearities (dependent on system 2), which suggest the cooperative involvement of plasma Sr changes in modulating some intestinal and bone mineral metabolic pathways. With the set of identified parameter values, the initial steady-state model predictions are relevant to known physiology, and some peculiarities of model behavior for long-term Sr administration were simulated.     

Fuzzy C-means method for clustering microarray data

MOTIVATION: Clustering analysis of data from DNA microarray hybridization studies is essential for identifying biologically relevant groups of genes. Partitional clustering methods such as K-means or self-organizing maps assign each gene to a single cluster. However, these methods do not provide information about the influence of a given gene for the overall shape of clusters. Here we apply a fuzzy partitioning method, Fuzzy C-means (FCM), to attribute cluster membership values to genes. RESULTS: A major problem in applying the FCM method for clustering microarray data is the choice of the fuzziness parameter m. We show that the commonly used value m = 2 is not appropriate for some data sets, and that optimal values for m vary widely from one data set to another. We propose an empirical method, based on the distribution of distances between genes in a given data set, to determine an adequate value for m. By setting threshold levels for the membership values, genes which are tigthly associated to a given cluster can be selected. Using a yeast cell cycle data set as an example, we show that this selection increases the overall biological significance of the genes within the cluster. AVAILABILITY: Supplementary text and Matlab functions are available at http://www-igbmc.u-strasbg.fr/fcm/     

A Global Sensitivity Test for Evaluating Statistical Hypotheses with Nonidentifiable Models

Summary We consider the problem of evaluating a statistical hypothesis when some model characteristics are nonidentifiable from observed data. Such a scenario is common in meta‐analysis for assessing publication bias and in longitudinal studies for evaluating a covariate effect when dropouts are likely to be nonignorable. One possible approach to this problem is to fix a minimal set of sensitivity parameters conditional upon which hypothesized parameters are identifiable. Here, we extend this idea and show how to evaluate the hypothesis of interest using an infimum statistic over the whole support of the sensitivity parameter. We characterize the limiting distribution of the statistic as a process in the sensitivity parameter, which involves a careful theoretical analysis of its behavior under model misspecification. In practice, we suggest a nonparametric bootstrap procedure to implement this infimum test as well as to construct confidence bands for simultaneous pointwise tests across all values of the sensitivity parameter, adjusting for multiple testing. The methodology's practical utility is illustrated in an analysis of a longitudinal psychiatric study.     

Extending the method of mathematically controlled comparison to include numerical comparisons

MOTIVATION: The method of mathematically controlled comparison has been used for some time to determine which of two alternative regulatory designs is better according to specific quantitative criteria for functional effectiveness. In some cases, the results obtained using this technique are general and independent of parameter values and the answers are clear-cut. In others, the result might be general, but the demonstration is difficult and numerical results with specific parameter values can help to clarify the situation. In either case, numerical results with specific parameter values can also provide an answer to the question of how much larger the values might be. In contrast, a more ambiguous result is obtained when either of the alternatives can have the larger value for a given systemic property, depending on the specific values of the parameters. In any case, introduction of specific values for the parameters reduces the generality of the results. Therefore, we have been motivated to develop and apply statistical methods that would permit the use of numerical values for the parameters and yet retain some of the generality that makes mathematically controlled comparison so attractive. RESULTS: We illustrate this new numerical method in a step-by-step application using a very simple didactic example. We also validate the results by comparison with the corresponding results obtained using the previously developed analytical method. The analytical approach is briefly present for reference purposes, since some of the same key concepts are needed to understand the numerical method and the results are needed for comparison. The numerical method confirms the qualitative differences between the systemic behavior of alternative designs obtained from the analytical method. In addition, the numerical method allows for quantification of the differences and it provides results that are general in a statistical sense. For example, the older analytical method showed that overall feedback inhibition in an unbranched pathway makes the system more robust whereas it decreases the stability margin of the steady state. The numerical method shows that the magnitudes of these differences are not comparable. The differences in stability margins (1-2% on average) are small when compared to the differences in robustness (50-100% on average). Furthermore, the numerical method shows that the system with overall feedback responds more quickly to change than the otherwise equivalent system without overall feedback. These results suggest reasons why overall feedback inhibition is such a prevalent regulatory pattern in unbranched biosynthetic pathways.     

Interior-branch and bootstrap tests of phylogenetic trees

We have compared statistical properties of the interior-branch and bootstrap tests of phylogenetic trees when the neighbor-joining tree- building method is used. For each interior branch of a predetermined topology, the interior-branch and bootstrap tests provide the confidence values, PC and PB, respectively, that indicate the extent of statistical support of the sequence cluster generated by the branch. In phylogenetic analysis these two values are often interpreted in the same way, and if PC and PB are high (say, > or = 0.95), the sequence cluster is regarded as reliable. We have shown that PC is in fact the complement of the P-value used in the standard statistical test, but PB is not. Actually, the bootstrap test usually underestimates the extent of statistical support of species clusters. The relationship between the confidence values obtained by the two tests varies with both the topology and expected branch lengths of the true (model) tree. The most conspicuous difference between PC and PB is observed when the true tree is starlike, and there is a tendency for the difference to increase as the number of sequences in the tree increases. The reason for this is that the bootstrap test tends to become progressively more conservative as the number of sequences in the tree increases. Unlike the bootstrap, the interior-branch test has the same statistical properties irrespective of the number of sequences used when a predetermined tree is considered. Therefore, the interior-branch test appears to be preferable to the bootstrap test as long as unbiased estimators of evolutionary distances are used. However, when the interior-branch is applied to a tree estimated from a given data set, PC may give an overestimate of statistical confidence. For this case, we developed a method for computing a modified version (P'C) of the PC value and showed that this P'C tends to give a conservative estimate of statistical confidence, though it is not as conservative as PB. In this paper we have introduced a model in which evolutionary distances between sequences follow a multivariate normal distribution. This model allowed us to study the relationships between the two tests analytically.      

The stochastic logistic model with correlated carrying capacities reproduces beta-diversity metrics of microbial communities

The large taxonomic variability of microbial community composition is a consequence of the combination of environmental variability, mediated through ecological interactions, and stochasticity. Most of the analysis aiming to infer the biological factors determining this difference in community structure start by quantifying how much communities are similar in their composition, trough beta-diversity metrics. The central role that these metrics play in microbial ecology does not parallel with a quantitative understanding of their relationships and statistical properties. In particular, we lack a framework that reproduces the empirical statistical properties of beta-diversity metrics. Here we take a macroecological approach and introduce a model to reproduce the statistical properties of community similarity. The model is based on the statistical properties of individual communities and on a single tunable parameter, the correlation of species’ carrying capacities across communities, which sets the difference of two communities. The model reproduces quantitatively the empirical values of several commonly-used beta-diversity metrics, as well as the relationships between them. In particular, this modeling framework naturally reproduces the negative correlation between overlap and dissimilarity, which has been observed in both empirical and experimental communities and previously related to the existence of universal features of community dynamics. In this framework, such correlation naturally emerges due to the effect of random sampling.     

Does history repeat itself? Wavelets and the phylodynamics of influenza A

Unprecedented global surveillance of viruses will result in massive sequence data sets that require new statistical methods. These data sets press the limits of Bayesian phylogenetics as the high-dimensional parameters that comprise a phylogenetic tree increase the already sizable computational burden of these techniques. This burden often results in partitioning the data set, for example, by gene, and inferring the evolutionary dynamics of each partition independently, a compromise that results in stratified analyses that depend only on data within a given partition. However, parameter estimates inferred from these stratified models are likely strongly correlated, considering they rely on data from a single data set. To overcome this shortfall, we exploit the existing Monte Carlo realizations from stratified Bayesian analyses to efficiently estimate a nonparametric hierarchical wavelet-based model and learn about the time-varying parameters of effective population size that reflect levels of genetic diversity across all partitions simultaneously. Our methods are applied to complete genome influenza A sequences that span 13 years. We find that broad peaks and trends, as opposed to seasonal spikes, in the effective population size history distinguish individual segments from the complete genome. We also address hypotheses regarding intersegment dynamics within a formal statistical framework that accounts for correlation between segment-specific parameters.     

Estimation of the parameter uncertainty resulting from bounded-error data

A procedure is described for characterizing the set of all parameter vectors that are consistent with data corrupted by a bounded noise. The method applies to any parametric model that can be simulated on a computer when upper and lower bounds for the noise are known a priori. The convergence properties of the associated estimator are considered, as well as its behavior in the presence of outliers. To illustrate the versatility of the technique, problems are considered where (i) the set of the true values of the parameter vector does not reduce to a singleton, (ii) the model is not uniquely identifiable, (iii) the hypotheses on the noise bounds are not satisfied, and (iv) the data contain a majority of outliers.     

Sensitivity of stress and strain calculations to passive material parameters in cardiac mechanical models using unloaded geometries

Cardiac stress (load) and strain (stretch) are widely studied indicators of cardiac function and outcome, but are difficult or impossible to directly measure in relation to the cardiac microstructure. An alternative approach is to estimate these states using computer methods and image-based measurements, but this still requires knowledge of the tissue material properties and the unloaded state, both of which are difficult to determine. In this work, we tested the sensitivity of these two interdependent unknowns (reference geometry and material parameters) on stress and strain calculations in cardiac tissue. Our study used a finite element model of the human ventricle, with a hyperelastic passive material model, and was driven by a cell model mediated active contraction. We evaluated 21 different published parameter sets for the five parameters of the passive material model, and for each set we optimised the corresponding unloaded geometry and contractility parameter to model a single pressure-volume loop. The resulting mechanics were compared, and calculated systolic stresses were largely insensitive to the chosen parameter set when an unloading algorithm was used. Meanwhile, material strain calculations varied substantially depending on the choice of material parameters. These results indicate that determining the correct material and unloaded configuration may be highly important to understand strain driven processes, but less so for calculating stress estimates.     

Probing the dynamics of identified neurons with a data-driven modeling approach

In controlling animal behavior the nervous system has to perform within the operational limits set by the requirements of each specific behavior. The implications for the corresponding range of suitable network, single neuron, and ion channel properties have remained elusive. In this article we approach the question of how well-constrained properties of neuronal systems may be on the neuronal level. We used large data sets of the activity of isolated invertebrate identified cells and built an accurate conductance-based model for this cell type using customized automated parameter estimation techniques. By direct inspection of the data we found that the variability of the neurons is larger when they are isolated from the circuit than when in the intact system. Furthermore, the responses of the neurons to perturbations appear to be more consistent than their autonomous behavior under stationary conditions. In the developed model, the constraints on different parameters that enforce appropriate model dynamics vary widely from some very tightly controlled parameters to others that are almost arbitrary. The model also allows predictions for the effect of blocking selected ionic currents and to prove that the origin of irregular dynamics in the neuron model is proper chaoticity and that this chaoticity is typical in an appropriate sense. Our results indicate that data driven models are useful tools for the in-depth analysis of neuronal dynamics. The better consistency of responses to perturbations, in the real neurons as well as in the model, suggests a paradigm shift away from measuring autonomous dynamics alone towards protocols of controlled perturbations. Our predictions for the impact of channel blockers on the neuronal dynamics and the proof of chaoticity underscore the wide scope of our approach.