Detecting and Removing Inconsistencies between Experimental Data and Signaling Network Topologies Using Integer Linear Programming on Interaction Graphs

Cross-referencing experimental data with our current knowledge of signaling network topologies is one central goal of mathematical modeling of cellular signal transduction networks. We present a new methodology for data-driven interrogation and training of signaling networks. While most published methods for signaling network inference operate on Bayesian, Boolean, or ODE models, our approach uses integer linear programming (ILP) on interaction graphs to encode constraints on the qualitative behavior of the nodes. These constraints are posed by the network topology and their formulation as ILP allows us to predict the possible qualitative changes (up, down, no effect) of the activation levels of the nodes for a given stimulus. We provide four basic operations to detect and remove inconsistencies between measurements and predicted behavior: (i) find a topology-consistent explanation for responses of signaling nodes measured in a stimulus-response experiment (if none exists, find the closest explanation); (ii) determine a minimal set of nodes that need to be corrected to make an inconsistent scenario consistent; (iii) determine the optimal subgraph of the given network topology which can best reflect measurements from a set of experimental scenarios; (iv) find possibly missing edges that would improve the consistency of the graph with respect to a set of experimental scenarios the most. We demonstrate the applicability of the proposed approach by interrogating a manually curated interaction graph model of EGFR/ErbB signaling against a library of high-throughput phosphoproteomic data measured in primary hepatocytes. Our methods detect interactions that are likely to be inactive in hepatocytes and provide suggestions for new interactions that, if included, would significantly improve the goodness of fit. Our framework is highly flexible and the underlying model requires only easily accessible biological knowledge. All related algorithms were implemented in a freely available toolbox SigNetTrainer making it an appealing approach for various applications.     

Predicting shifts in dynamics of cannibalistic field populations using individual-based models

The occurrence of qualitative shifts in population dynamical regimes has long been the focus of population biologists. Nonlinear ecological models predict that these shifts in dynamical regimes may occur as a result of parameter shifts, but unambiguous empirical evidence is largely restricted to laboratory populations. We used an individual-based modelling approach to predict dynamical shifts in field fish populations where the capacity to cannibalize differed between species. Model-generated individual growth trajectories that reflect different population dynamics were confronted with empirically observed growth trajectories, showing that our ordering and quantitative estimates of the different cannibalistic species in terms of life-history characteristics led to correct qualitative predictions of their dynamics.     

Analysis of models of irregular shape by solution X-ray scattering: the case of the 50S ribosomal subunit from E. coli

Structural models of biological macromolecules can be tested by comparing calculated and experimental solution scattering curves. We have developed an approach for computing scattering shape functions at medium resolution from models proposed on the basis of other techniques such as electron microscopy. We present the results obtained with the 50S ribosomal subunit from Escherichia coli; two models are considered, one proposed by Lake (1976), the other one by Tischendorf et al. (1975). Although the two models are similar in many respects, their scattering shape functions are significantly different. The comparison with the experimental scattering curve allows us to check the scale of the models and, after scaling, to quantitate the agreement between the observed and the calculated curves. Finally, it can provide a starting point for the structural interpretation of the X-ray data.     

Lévy Flights and Self-Similar Exploratory Behaviour of Termite Workers: Beyond Model Fitting

Animal movements have been related to optimal foraging strategies where self-similar trajectories are central. Most of the experimental studies done so far have focused mainly on fitting statistical models to data in order to test for movement patterns described by power-laws. Here we show by analyzing over half a million movement displacements that isolated termite workers actually exhibit a range of very interesting dynamical properties –including Lévy flights– in their exploratory behaviour. Going beyond the current trend of statistical model fitting alone, our study analyses anomalous diffusion and structure functions to estimate values of the scaling exponents describing displacement statistics. We evince the fractal nature of the movement patterns and show how the scaling exponents describing termite space exploration intriguingly comply with mathematical relations found in the physics of transport phenomena. By doing this, we rescue a rich variety of physical and biological phenomenology that can be potentially important and meaningful for the study of complex animal behavior and, in particular, for the study of how patterns of exploratory behaviour of individual social insects may impact not only their feeding demands but also nestmate encounter patterns and, hence, their dynamics at the social scale.     

Qualitative and Quantitative Analyses of the Echolocation Strategies of Bats on the Basis of Mathematical Modelling and Laboratory Experiments

Prey pursuit by an echolocating bat was studied theoretically and experimentally. First, a mathematical model was proposed to describe the flight dynamics of a bat and a single prey. In this model, the flight angle of the bat was affected by angles related to the flight path of the single moving prey, that is, the angle from the bat to the prey and the flight angle of the prey. Numerical simulation showed that the success rate of prey capture was high, when the bat mainly used the angle to the prey to minimize the distance to the prey, and also used the flight angle of the prey to minimize the difference in flight directions of itself and the prey. Second, parameters in the model were estimated according to experimental data obtained from video recordings taken while a Japanese horseshoe bat (Rhinolphus derrumequinum nippon) pursued a moving moth (Goniocraspidum pryeri) in a flight chamber. One of the estimated parameter values, which represents the ratio in the use of the angles, was consistent with the optimal value of the numerical simulation. This agreement between the numerical simulation and parameter estimation suggests that a bat chooses an effective flight path for successful prey capture by using the angles. Finally, the mathematical model was extended to include a bat and prey. Parameter estimation of the extended model based on laboratory experiments revealed the existence of bat’s dynamical attention towards prey, that is, simultaneous pursuit of prey and selective pursuit of respective prey. Thus, our mathematical model contributes not only to quantitative analysis of effective foraging, but also to qualitative evaluation of a bat’s dynamical flight strategy during multiple prey pursuit.     

Knowledge discovery and system biology in molecular medicine: an application on neurodegenerative diseases

The possibility to study an organism in terms of system theory has been proposed in the past, but only the advancement of molecular biology techniques allow us to investigate the dynamical properties of a biological system in a more quantitative and rational way than before . These new techniques can gave only the basic level view of an organisms functionality. The comprehension of its dynamical behaviour depends on the possibility to perform a multiple level analysis. Functional genomics has stimulated the interest in the investigation the dynamical behaviour of an organism as a whole. These activities are commonly known as System Biology, and its interests ranges from molecules to organs. One of the more promising applications is the 'disease modeling'. The use of experimental models is a common procedure in pharmacological and clinical researches; today this approach is supported by 'in silico' predictive methods. This investigation can be improved by a combination of experimental and computational tools. The Machine Learning (ML) tools are able to process different heterogeneous data sources, taking into account this peculiarity, they could be fruitfully applied to support a multilevel data processing (molecular, cellular and morphological) that is the prerequisite for the formal model design; these techniques can allow us to extract the knowledge for mathematical model development. The aim of our work is the development and implementation of a system that combines ML and dynamical models simulations. The program is addressed to the virtual analysis of the pathways involved in neurodegenerative diseases. These pathologies are multifactorial diseases and the relevance of the different factors has not yet been well elucidated. This is a very complex task; in order to test the integrative approach our program has been limited to the analysis of the effects of a specific protein, the Cyclin dependent kinase 5 (CDK5) which relies on the induction of neuronal apoptosis. The system has a modular structure centred on a textual knowledge discovery approach. The text mining is the only way to enhance the capability to extract ,from multiple data sources, the information required for the dynamical simulator. The user may access the publically available modules through the following site: http://biocomp.ge.ismac.cnr.it.     

Hamiltonian Monte Carlo methods for efficient parameter estimation in steady state dynamical systems

Background

Parameter estimation for differential equation models of intracellular processes is a highly relevant bu challenging task. The available experimental data do not usually contain enough information to identify all parameters uniquely, resulting in ill-posed estimation problems with often highly correlated parameters. Sampling-based Bayesian statistical approaches are appropriate for tackling this problem. The samples are typically generated via Markov chain Monte Carlo, however such methods are computationally expensive and their convergence may be slow, especially if there are strong correlations between parameters. Monte Carlo methods based on Euclidean or Riemannian Hamiltonian dynamics have been shown to outperform other samplers by making proposal moves that take the local sensitivities of the system’s states into account and accepting these moves with high probability. However, the high computational cost involved with calculating the Hamiltonian trajectories prevents their widespread use for all but the smallest differential equation models. The further development of efficient sampling algorithms is therefore an important step towards improving the statistical analysis of predictive models of intracellular processes.

Results

We show how state of the art Hamiltonian Monte Carlo methods may be significantly improved for steady state dynamical models. We present a novel approach for efficiently calculating the required geometric quantities by tracking steady states across the Hamiltonian trajectories using a Newton-Raphson method and employing local sensitivity information. Using our approach, we compare both Euclidean and Riemannian versions of Hamiltonian Monte Carlo on three models for intracellular processes with real data and demonstrate at least an order of magnitude improvement in the effective sampling speed. We further demonstrate the wider applicability of our approach to other gradient based MCMC methods, such as those based on Langevin diffusions.

Conclusion

Our approach is strictly benefitial in all test cases. The Matlab sources implementing our MCMC methodology is available from https://github.com/a-kramer/ode_rmhmc.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-253) contains supplementary material, which is available to authorized users.     

A unified association analysis approach for family and unrelated samples correcting for stratification

There are two common designs for association mapping of complex diseases: case-control and family-based designs. A case-control sample is more powerful to detect genetic effects than a family-based sample that contains the same numbers of affected and unaffected persons, although additional markers may be required to control for spurious association. When family and unrelated samples are available, statistical analyses are often performed in the family and unrelated samples separately, conditioning on parental information for the former, thus resulting in reduced power. In this report, we propose a unified approach that can incorporate both family and case-control samples and, provided the additional markers are available, at the same time corrects for population stratification. We apply the principal components of a marker matrix to adjust for the effect of population stratification. This unified approach makes it unnecessary to perform a conditional analysis of the family data and is more powerful than the separate analyses of unrelated and family samples, or a meta-analysis performed by combining the results of the usual separate analyses. This property is demonstrated in both a variety of simulation models and empirical data. The proposed approach can be equally applied to the analysis of both qualitative and quantitative traits.     

Predicting trophic relations in ecological networks: A test of the Allometric Diet Breadth Model

Few food web theory hypotheses/predictions can be readily tested using likelihoods of reproducing the data. Simple probabilistic models for food web structure, however, are an exception as their likelihoods were recently derived. Here I test the performance of a more complex model for food web structure that is grounded in the allometric scaling of interactions with body size and the theory of optimal foraging (Allometric Diet Breadth Model—ADBM). This deterministic model has been evaluated by measuring the fraction of trophic relations it correctly predicts. I contrasted this value with that produced by simpler models based on body sizes and found that the quantitative information on allometric scaling and optimal foraging does not significantly increase model fit. Also, I present a method to compute the p-value for the fraction of trophic interactions correctly predicted by the ADBM, or any other model, with respect to three probabilistic models. I find that the ADBM predicts significantly more links than random graphs, but other models can outperform it. Although optimal foraging and allometric scaling may improve our understanding of food webs, the ADBM needs to be modified or replaced to find support in the data.     

Functional methods for quantifying agonists and antagonists.

The development of occupancy theory has allowed the formulation of a series of mathematical models that describe the interaction of agonists and antagonists with their receptors, in terms of affinity and efficacy. These models provide a framework for the analysis and interpretation of E/[A] curve data and have proved to be useful tools in quantitative pharmacology. Unfortunately, despite the proven utility of this approach and the widespread availability of powerful computer-based curve-fitting programs [BMDP (41), Microsoft Excel. etc.], which greatly facilitate analysis, the application of mathematical modeling remains the exception rather than the rule in pharmacological studies.     

An ecological framework linking scales across space and time based on self-thinning

Scaling up from measurements made at small spatial and short temporal scales is a central challenge in the ecological and related sciences, where predictions at larger scales and over long time periods are required. It involves two quite distinct aspects: a formulation of a theoretical framework for calculating space-time averages, and an acquisition of data to support that framework. In this paper, we address the theoretical part of the question, and although our primary motivation was an understanding of carbon accounting our formulation is more general. To that end, we adopt a dynamical systems approach, and incorporate a new dynamical formulation of self-thinning. We show how to calculate rates of change for total (and average) plant dry mass, volume, and carbon, in terms of the properties of the individual plants. The results emphasize how local scale statistics (such as, variation in the size of individuals) lead to nonlinear variation at larger scales. Further, we describe how regular and stochastic disturbance can be readily incorporated into this framework. It is shown that stochastic disturbance at patch-scales, results in (to first approximation) regular disturbance at ecosystem scales, and hence can be formulated as such. We conclude that a dynamical formulation of self-thinning can be used as a generic framework for scaling ecological processes in space and time.     

Mechanochemical couplings of kinesin motors

Kinesins are molecular motors capable of moving processively along microtubule in a stepwise manner by hydrolyzing ATP. Numerous experimental results on various aspects of their dynamical behaviours are available in literature. Although a number of models of tightly coordinated mechanism have been proposed to explain some experimental results, up to now no good explanation has been given to all these experimental results by using a single model. We have recently proposed such a model of partially coordinated hand-over-hand moving mechanism. In this paper, we use this model to study in detail various aspects of the dynamical properties of single kinesin molecules. We show that kinesin dimers walk hand-over-hand along microtubules in a partially coordinated rather than a tightly coordinated manner. The degree of coordination depends on the ratio of the two heads' ATPase rates that are in turn determined by both internal elastic force and external load. We have tested this model using various available experimental results on different samples and obtained a good agreement between the theory and the experiments.     

Multiple imputation for missing values through conditional Semiparametric odds ratio models

Multiple imputation is a practically useful approach to handling incompletely observed data in statistical analysis. Parameter estimation and inference based on imputed full data have been made easy by Rubin's rule for result combination. However, creating proper imputation that accommodates flexible models for statistical analysis in practice can be very challenging. We propose an imputation framework that uses conditional semiparametric odds ratio models to impute the missing values. The proposed imputation framework is more flexible and robust than the imputation approach based on the normal model. It is a compatible framework in comparison to the approach based on fully conditionally specified models. The proposed algorithms for multiple imputation through the Markov chain Monte Carlo sampling approach can be straightforwardly carried out. Simulation studies demonstrate that the proposed approach performs better than existing, commonly used imputation approaches. The proposed approach is applied to imputing missing values in bone fracture data.