Construction of a large scale integrated map of macrophage pathogen recognition and effector systems

Background

Mathematical models provide abstract representations of the information gained from experimental observations on the structure and function of a particular biological system. Conferring a predictive character on a given mathematical formulation often relies on determining a number of non-measurable parameters that largely condition the model's response. These parameters can be identified by fitting the model to experimental data. However, this fit can only be accomplished when identifiability can be guaranteed.

Results

We propose a novel iterative identification procedure for detecting and dealing with the lack of identifiability. The procedure involves the following steps: 1) performing a structural identifiability analysis to detect identifiable parameters; 2) globally ranking the parameters to assist in the selection of the most relevant parameters; 3) calibrating the model using global optimization methods; 4) conducting a practical identifiability analysis consisting of two (a priori and a posteriori) phases aimed at evaluating the quality of given experimental designs and of the parameter estimates, respectively and 5) optimal experimental design so as to compute the scheme of experiments that maximizes the quality and quantity of information for fitting the model.

Conclusions

The presented procedure was used to iteratively identify a mathematical model that describes the NF-κB regulatory module involving several unknown parameters. We demonstrated the lack of identifiability of the model under typical experimental conditions and computed optimal dynamic experiments that largely improved identifiability properties.     

Modeling and identification of endocrine-metabolic systems. theoretical aspects and their importance in practice

Some guidelines for modeling, identification, and validation in endocrinology and metabolism are presented. Fundamentals based on recent theoretical developments are given, and their essential role for solving physiological and clinical problems is shown. Concepts and issues discussed include a priori and a posteriori identifiability; optimal experiment design; assessment of model validity; simple and complex models; compartmental and noncompartmental models. Recent studies on the glucose-insulin system and on ketone body metabolism are used to illustrate the methodological line of thought discussed in the paper.     

Calibration uncertainty in molecular dating analyses: there is no substitute for the prior evaluation of time priors

Calibration is the rate-determining step in every molecular clock analysis and, hence, considerable effort has been expended in the development of approaches to distinguish good from bad calibrations. These can be categorized into a priori evaluation of the intrinsic fossil evidence, and a posteriori evaluation of congruence through cross-validation. We contrasted these competing approaches and explored the impact of different interpretations of the fossil evidence upon Bayesian divergence time estimation. The results demonstrate that a posteriori approaches can lead to the selection of erroneous calibrations. Bayesian posterior estimates are also shown to be extremely sensitive to the probabilistic interpretation of temporal constraints. Furthermore, the effective time priors implemented within an analysis differ for individual calibrations when employed alone and in differing combination with others. This compromises the implicit assumption of all calibration consistency methods, that the impact of an individual calibration is the same when used alone or in unison with others. Thus, the most effective means of establishing the quality of fossil-based calibrations is through a priori evaluation of the intrinsic palaeontological, stratigraphic, geochronological and phylogenetic data. However, effort expended in establishing calibrations will not be rewarded unless they are implemented faithfully in divergence time analyses.     

Cladistic and phylogenetic biogeography: the art and the science of discovery

All methods used in historical biogeographical analysis aim to obtain resolved area cladograms that represent historical relationships among areas in which monophyletic groups of taxa are distributed. When neither widespread nor sympatric taxa are present in the distribution of a monophyletic group, all methods obtain the same resolved area cladogram that conforms to a simple vicariance scenario. In most cases, however, the distribution of monophyletic groups of taxa is not that simple. A priori and a posteriori methods of historical biogeography differ in the way in which they deal with widespread and sympatric taxa. A posteriori methods are empirically superior to a priori methods, as they provide a more parsimonious accounting of the input data, do not eliminate or modify input data, and do not suffer from internal inconsistencies in implementation. When factual errors are corrected, the exemplar presented by M.C. Ebach & C.J. Humphries (Journal of Biogeography, 2002, 29 , 427) purporting to show inconsistencies in implementation by a posteriori methods actually corroborates the opposite. The rationale for preferring a priori methods thus corresponds to ontological rather than to epistemological considerations. We herein identify two different research programmes, cladistic biogeography (associated with a priori methods) and phylogenetic biogeography (associated with a posteriori methods). The aim of cladistic biogeography is to fit all elements of all taxon–area cladograms to a single set of area relationships, maintaining historical singularity of areas. The aim of phylogenetic biogeography is to document, most parsimoniously, the geographical context of speciation events. The recent contribution by M.C. Ebach & C.J. Humphries (Journal of Biogeography, 2002, 29 , 427) makes it clear that cladistic biogeography using a priori methods is an inductivist/verificationist research programme, whereas phylogenetic biogeography is hypothetico‐deductivist/falsificationist. Cladistic biogeography can become hypothetic‐deductive by using a posteriori methods of analysis.     

A priori and a posteriori approaches for finding genes of evolutionary interest in non-model species: Osmoregulatory genes in the kidney transcriptome of the desert rodent Dipodomys spectabilis (banner-tailed kangaroo rat)

One common goal in evolutionary biology is the identification of genes underlying adaptive traits of evolutionary interest. Recently next-generation sequencing techniques have greatly facilitated such evolutionary studies in species otherwise depauperate of genomic resources. Kangaroo rats (Dipodomys sp.) serve as exemplars of adaptation in that they inhabit extremely arid environments, yet require no drinking water because of ultra-efficient kidney function and osmoregulation. As a basis for identifying water conservation genes in kangaroo rats, we conducted a priori bioinformatics searches in model rodents (Mus musculus and Rattus norvegicus) to identify candidate genes with known or suspected osmoregulatory function. We then obtained 446,758 reads via 454 pyrosequencing to characterize genes expressed in the kidney of banner-tailed kangaroo rats (Dipodomys spectabilis). We also determined candidates a posteriori by identifying genes that were overexpressed in the kidney. The kangaroo rat sequences revealed nine different a priori candidate genes predicted from our Mus and Rattus searches, as well as 32 a posteriori candidate genes that were overexpressed in kidney. Mutations in two of these genes, Slc12a1 and Slc12a3, cause human renal diseases that result in the inability to concentrate urine. These genes are likely key determinants of physiological water conservation in desert rodents.     

Parameter Identifiability and Redundancy: Theoretical Considerations

Background

Models for complex biological systems may involve a large number of parameters. It may well be that some of these parameters cannot be derived from observed data via regression techniques. Such parameters are said to be unidentifiable, the remaining parameters being identifiable. Closely related to this idea is that of redundancy, that a set of parameters can be expressed in terms of some smaller set. Before data is analysed it is critical to determine which model parameters are identifiable or redundant to avoid ill-defined and poorly convergent regression.

Methodology/Principal Findings

In this paper we outline general considerations on parameter identifiability, and introduce the notion of weak local identifiability and gradient weak local identifiability. These are based on local properties of the likelihood, in particular the rank of the Hessian matrix. We relate these to the notions of parameter identifiability and redundancy previously introduced by Rothenberg (Econometrica 39 (1971) 577–591) and Catchpole and Morgan (Biometrika 84 (1997) 187–196). Within the widely used exponential family, parameter irredundancy, local identifiability, gradient weak local identifiability and weak local identifiability are shown to be largely equivalent. We consider applications to a recently developed class of cancer models of Little and Wright (Math Biosciences 183 (2003) 111–134) and Little et al. (J Theoret Biol 254 (2008) 229–238) that generalize a large number of other recently used quasi-biological cancer models.

Conclusions/Significance

We have shown that the previously developed concepts of parameter local identifiability and redundancy are closely related to the apparently weaker properties of weak local identifiability and gradient weak local identifiability—within the widely used exponential family these concepts largely coincide.     

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.     

On the identifiability of metabolic network models

A major problem for the identification of metabolic network models is parameter identifiability, that is, the possibility to unambiguously infer the parameter values from the data. Identifiability problems may be due to the structure of the model, in particular implicit dependencies between the parameters, or to limitations in the quantity and quality of the available data. We address the detection and resolution of identifiability problems for a class of pseudo-linear models of metabolism, so-called linlog models. Linlog models have the advantage that parameter estimation reduces to linear or orthogonal regression, which facilitates the analysis of identifiability. We develop precise definitions of structural and practical identifiability, and clarify the fundamental relations between these concepts. In addition, we use singular value decomposition to detect identifiability problems and reduce the model to an identifiable approximation by a principal component analysis approach. The criterion is adapted to real data, which are frequently scarce, incomplete, and noisy. The test of the criterion on a model with simulated data shows that it is capable of correctly identifying the principal components of the data vector. The application to a state-of-the-art dataset on central carbon metabolism in Escherichia coli yields the surprising result that only $4$ out of $31$ reactions, and $37$ out of $100$ parameters, are identifiable. This underlines the practical importance of identifiability analysis and model reduction in the modeling of large-scale metabolic networks. Although our approach has been developed in the context of linlog models, it carries over to other pseudo-linear models, such as generalized mass-action (power-law) models. Moreover, it provides useful hints for the identifiability analysis of more general classes of nonlinear models of metabolism.     

Easy parameter identifiability analysis with COPASI

Background and scope

Differential equation systems modeling biochemical reaction networks can only give quantitative predictions, when they are in accordance with experimental data. However, even if a model can well recapitulate given data, it is often the case that some of its kinetic parameters can be arbitrarily chosen without significantly affecting the simulation results. This indicates a lack of appropriate data to determine those parameters. In this case, the parameter is called to be practically non-identifiable. Well-identified parameters are paramount for reliable quantitative predictions and, therefore, identifiability analysis is an important topic in modeling of biochemical reaction networks. Here, we describe a hidden feature of the free modeling software COPASI, which can be exploited to easily and quickly conduct a parameter identifiability analysis of differential equation systems by calculating likelihood profiles. The proposed combination of an established method for parameter identifiability analysis with the user-friendly features of COPASI offers an easy and rapid access to parameter identifiability analysis even for non-experts.

Availability

COPASI is freely available for academic use at http://www.copasi.org.     

On the identifiability of linear compartmental systems: a revisited transfer function approach based on topological properties

This work deals with the problem of the a priori identifiability of compartmental systems from input-output experiments. A new approach is presented, in which, having associated a directed graph with the matrix to be identified, a set of “forms” is defined which are functions of the elements of matrix itself. It is shown how, by exploiting the topological properties of the graph and its subgraphs, the problem can be simplified into one of smaller dimensions. Examples are provided to illustrate this new approach.     

Comparison and testing of least-squares time domain inverse solutions in electrocardiography

The use of several mathematical methods for estimating epicardial ECG potentials from arrays of body surface potentials has been reported in the literature; most of these methods are based on least-squares reconstruction principles and operate in the time-space domain. In this paper we introduce a general Bayesian maximum a posteriori (MAP) framework for time domain inverse solutions in the presence of noise. The two most popular previously applied least-squares methods, constrained (regularized) least-squares and low-rank approximation through the singular value decomposition, are placed in this framework, each of them requiring the a priori knowledge of a ‘regularization parameter’, which defines the degree of smoothing to be applied to the inversion. Results of simulations using these two methods are presented; they compare the ability of each method to reconstruct epicardial potentials. We used the geometric configuration of the torso and internal organs of an individual subject as reconstructed from CT scans. The accuracy of each method at each epicardial location was tested as a function of measurement noise, the size and shape of the subarray of torso sensors, and the regularization parameter. We paid particular attention to an assessment of the potential of these methods for clinical use by testing the effect of using compact, small-size subarrays of torso potentials while maintaining a high degree of resolution on the epicardium.     

Parameter Identifiability and Redundancy in a General Class of Stochastic Carcinogenesis Models

Background

Heidenreich et al. (Risk Anal 1997 17 391–399) considered parameter identifiability in the context of the two-mutation cancer model and demonstrated that combinations of all but two of the model parameters are identifiable. We consider the problem of identifiability in the recently developed carcinogenesis models of Little and Wright (Math Biosci 2003 183 111–134) and Little et al. (J Theoret Biol 2008 254 229–238). These models, which incorporate genomic instability, generalize a large number of other quasi-biological cancer models, in particular those of Armitage and Doll (Br J Cancer 1954 8 1–12), the two-mutation model (Moolgavkar et al. Math Biosci 1979 47 55–77), the generalized multistage model of Little (Biometrics 1995 51 1278–1291), and a recently developed cancer model of Nowak et al. (PNAS 2002 99 16226–16231).

Methodology/Principal Findings

We show that in the simpler model proposed by Little and Wright (Math Biosci 2003 183 111–134) the number of identifiable combinations of parameters is at most two less than the number of biological parameters, thereby generalizing previous results of Heidenreich et al. (Risk Anal 1997 17 391–399) for the two-mutation model. For the more general model of Little et al. (J Theoret Biol 2008 254 229–238) the number of identifiable combinations of parameters is at most less than the number of biological parameters, where is the number of destabilization types, thereby also generalizing all these results. Numerical evaluations suggest that these bounds are sharp. We also identify particular combinations of identifiable parameters.

Conclusions/Significance

We have shown that the previous results on parameter identifiability can be generalized to much larger classes of quasi-biological carcinogenesis model, and also identify particular combinations of identifiable parameters. These results are of theoretical interest, but also of practical significance to anyone attempting to estimate parameters for this large class of cancer models.     

Finding identifiable parameter combinations in nonlinear ODE models and the rational reparameterization of their input-output equations

When examining the structural identifiability properties of dynamic system models, some parameters can take on an infinite number of values and yet yield identical input-output data. These parameters and the model are then said to be unidentifiable. Finding identifiable combinations of parameters with which to reparameterize the model provides a means for quantitatively analyzing the model and computing solutions in terms of the combinations. In this paper, we revisit and explore the properties of an algorithm for finding identifiable parameter combinations using Gröbner Bases and prove useful theoretical properties of these parameter combinations. We prove a set of M algebraically independent identifiable parameter combinations can be found using this algorithm and that there exists a unique rational reparameterization of the input-output equations over these parameter combinations. We also demonstrate application of the procedure to a nonlinear biomodel.     

Convergence and parallelism: is a new life ahead of old concepts?

In comparative biology, character observations initially separate similar and dissimilar characters. Only similar characters are considered for phylogeny reconstruction; their homology is attested in a two‐step process, firstly a priori of phylogeny reconstruction by accurate similarity statements, and secondly a posteriori of phylogeny analysis by congruence with other characters. Any pattern of non‐homology is then a homoplasy, commonly, but vaguely, associated with “convergence”. In this logical scheme, there is no way to analyze characters which look similar, but cannot meet usual criteria for homology statements, i.e., false similarity detected a priori of phylogenetic analysis, even though such characters may represent evolutionarily significant patterns of character transformations. Because phylogenies are not only patterns of taxa relationships but also references for evolutionary studies, we propose to redefine the traditional concepts of parallelism and convergence to associate patterns of non‐homology with explicit theoretical contexts: homoplasy is restricted to non‐similarity detected a posteriori of phylogeny analysis and related to parallelism; non‐similarity detected a priori of phylogenetic analysis and necessarily described by different characters would then correspond to a convergence event s. str. We propose to characterize these characters as heterologous (heterology). Heterology and homoplasy correspond to different non‐similarity patterns and processes; they are also associated with different patterns of taxa relationships: homoplasy can occur only in non‐sister group taxa; no such limit exists for heterology. The usefulness of these terms and concepts is illustrated with patterns of acoustic evolution in ensiferan insects. © The Willi Hennig Society 2005.     

A PDMP model of the epithelial cell turn-over in the intestinal crypt including microbiota-derived regulations

Mechanistic models are a powerful tool to gain insights into biological processes. The parameters of such models, e.g. kinetic rate constants, usually cannot be measured directly but need to be inferred from experimental data. In this article, we study dynamical models of the translation kinetics after mRNA transfection and analyze their parameter identifiability. That is, whether parameters can be uniquely determined from perfect or realistic data in theory and practice. Previous studies have considered ordinary differential equation (ODE) models of the process, and here we formulate a stochastic differential equation (SDE) model. For both model types, we consider structural identifiability based on the model equations and practical identifiability based on simulated as well as experimental data and find that the SDE model provides better parameter identifiability than the ODE model. Moreover, our analysis shows that even for those parameters of the ODE model that are considered to be identifiable, the obtained estimates are sometimes unreliable. Overall, our study clearly demonstrates the relevance of considering different modeling approaches and that stochastic models can provide more reliable and informative results.

     

Critique of parsimony analysis of endemicity as a method of historical biogeography

Aim Assess the value of parsimony analysis of endemism as either an a priori (cladistic) and an a posteriori (phylogenetic) method of historical biogeography. Location World‐wide. Methods Parsimony analysis of endemicity (PAE) and Brooks parsimony analysis (BPA). Results Parsimony analysis of endemicity is capable of finding correct and unambiguous area relationships only under scenarios of vicariance in combination with non‐response to vicariance or extinction. An empirical comparison between PAE and BPA, using the poeciliid fish genera Heterandria and Xiphophorus, demonstrates that PAE fails to document much of the historical complexity in this relatively simple system. Main conclusions The a priori assumptions of PAE are far more restrictive than those made by other a priori methods, limiting its utility as a method of cladistic biogeography. The inability of PAE to detect perfect vicariance or biogeographical histories involving dispersal, renders it unsuitable as a method of phylogenetic biogeography.     

Detecting Specific Populations in Mixtures

Mixed stock analysis (MSA) estimates the relative contributions of distinct populations in a mixture of organisms. Increasingly, MSA is used to judge the presence or absence of specific populations in specific mixture samples. This is commonly done by inspecting the bootstrap confidence interval of the contribution of interest. This method has a number of statistical deficiencies, including almost zero power to detect small contributions even if the population has perfect identifiability. We introduce a more powerful method based on the likelihood ratio test and compare both methods in a simulation demonstration using a 17 population baseline of sockeye salmon, Oncorhynchus nerka, from the Kenai River, Alaska, watershed. Power to detect a nonzero contribution will vary with the population(s) identifiability relative to the rest of the baseline, the contribution size, mixture sample size, and analysis method. The demonstration shows that the likelihood ratio method is always more powerful than the bootstrap method, the two methods only being equal when both display 100% power. Power declines for both methods as contribution declines, but it declines faster and goes to zero for the bootstrap method. Power declines quickly for both methods as population identifiability declines, though the likelihood ratio test is able to capitalize on the presence of 'perfect identification' characteristics, such as private alleles. Given the baseline-specific nature of detection power, researchers are encouraged to conduct a priori power analyses similar to the current demonstration when planning their applications.     

Parsimonious mixtures of multivariate contaminated normal distributions

A mixture of multivariate contaminated normal distributions is developed for model‐based clustering. In addition to the parameters of the classical normal mixture, our contaminated mixture has, for each cluster, a parameter controlling the proportion of mild outliers and one specifying the degree of contamination. Crucially, these parameters do not have to be specified a priori, adding a flexibility to our approach. Parsimony is introduced via eigen‐decomposition of the component covariance matrices, and sufficient conditions for the identifiability of all the members of the resulting family are provided. An expectation‐conditional maximization algorithm is outlined for parameter estimation and various implementation issues are discussed. Using a large‐scale simulation study, the behavior of the proposed approach is investigated and comparison with well‐established finite mixtures is provided. The performance of this novel family of models is also illustrated on artificial and real data.     

Spatial variability in macroinvertebrate assemblages: comparing regional and multivariate approaches for classifying reference sites in South Africa

Spatial variability in macroinvertebrate assemblages was examined with the aim of evaluating the utility of regional classification systems in aquatic bioassessment. Sampling was undertaken at reference sites in the Western Cape and Mpumalanga, South Africa, using the rapid bioassessment method SASS4 (South African Scoring System Version 4). Multivariate analysis of macroinvertebrate assemblages showed that assemblages varied regionally with differences most apparent in upland areas, i.e. mountain streams and foothill-cobble beds, with lowland areas less regionally distinct. Within regions, longitudinal zonation into upland and lowland areas was important, with sites grouping on the basis of broad geomorphological zones or subregions. Of the upland sites, differentiation into mountain streams and foothill-cobble beds was not evident, although overall variability of assemblages within upland areas, in particular the Western Cape, was very high. In general, a priori regional classification of sites using the hierarchical spatial framework developed in South Africa provided a useful framework for preliminary classification of reference sites. Groups of sites based on a posteriori analysis of macroinvertebrate data, however, provided a more robust classification than any regional classification. Spatial classifications therefore offer geographic partitions within which to expect somewhat similar conditions, and regional reference sites selected within the context of the spatial framework are likely to be more representative of specific river types than those selected without using the framework. Classification of sites needs to be an iterative process that allows for subjective a priori regional classifications to be modified on the basis of independent, objective a posteriori classification of biological assemblages.     

Classification of microarray data using gene networks

Background  

Microarrays have become extremely useful for analysing genetic phenomena, but establishing a relation between microarray analysis results (typically a list of genes) and their biological significance is often difficult. Currently, the standard approach is to map a posteriori the results onto gene networks in order to elucidate the functions perturbed at the level of pathways. However, integrating a priori knowledge of the gene networks could help in the statistical analysis of gene expression data and in their biological interpretation.