On the Relationship of Steady States of Continuous and Discrete Models Arising from Biology

For many biological systems that have been modeled using continuous and discrete models, it has been shown that such models have similar dynamical properties. In this paper, we prove that this happens in more general cases. We show that under some conditions there is a bijection between the steady states of continuous and discrete models arising from biological systems. Our results also provide a novel method to analyze certain classes of nonlinear models using discrete mathematics.     

Uniform Sampling of Steady States in Metabolic Networks: Heterogeneous Scales and Rounding

The uniform sampling of convex polytopes is an interesting computational problem with many applications in inference from linear constraints, but the performances of sampling algorithms can be affected by ill-conditioning. This is the case of inferring the feasible steady states in models of metabolic networks, since they can show heterogeneous time scales. In this work we focus on rounding procedures based on building an ellipsoid that closely matches the sampling space, that can be used to define an efficient hit-and-run (HR) Markov Chain Monte Carlo. In this way the uniformity of the sampling of the convex space of interest is rigorously guaranteed, at odds with non markovian methods. We analyze and compare three rounding methods in order to sample the feasible steady states of metabolic networks of three models of growing size up to genomic scale. The first is based on principal component analysis (PCA), the second on linear programming (LP) and finally we employ the Lovazs ellipsoid method (LEM). Our results show that a rounding procedure dramatically improves the performances of the HR in these inference problems and suggest that a combination of LEM or LP with a subsequent PCA perform the best. We finally compare the distributions of the HR with that of two heuristics based on the Artificially Centered hit-and-run (ACHR), gpSampler and optGpSampler. They show a good agreement with the results of the HR for the small network, while on genome scale models present inconsistencies.     

Scalable Steady State Analysis of Boolean Biological Regulatory Networks

Background

Computing the long term behavior of regulatory and signaling networks is critical in understanding how biological functions take place in organisms. Steady states of these networks determine the activity levels of individual entities in the long run. Identifying all the steady states of these networks is difficult due to the state space explosion problem.

Methodology

In this paper, we propose a method for identifying all the steady states of Boolean regulatory and signaling networks accurately and efficiently. We build a mathematical model that allows pruning a large portion of the state space quickly without causing any false dismissals. For the remaining state space, which is typically very small compared to the whole state space, we develop a randomized traversal method that extracts the steady states. We estimate the number of steady states, and the expected behavior of individual genes and gene pairs in steady states in an online fashion. Also, we formulate a stopping criterion that terminates the traversal as soon as user supplied percentage of the results are returned with high confidence.

Conclusions

This method identifies the observed steady states of boolean biological networks computationally. Our algorithm successfully reported the G1 phases of both budding and fission yeast cell cycles. Besides, the experiments suggest that this method is useful in identifying co-expressed genes as well. By analyzing the steady state profile of Hedgehog network, we were able to find the highly co-expressed gene pair GL1-SMO together with other such pairs.

Availability

Source code of this work is available at http://bioinformatics.cise.ufl.edu/palSteady.html twocolumnfalse]     

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

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

Discrete Logic Modelling Optimization to Contextualize Prior Knowledge Networks Using PRUNET

High-throughput technologies have led to the generation of an increasing amount of data in different areas of biology. Datasets capturing the cell’s response to its intra- and extra-cellular microenvironment allows such data to be incorporated as signed and directed graphs or influence networks. These prior knowledge networks (PKNs) represent our current knowledge of the causality of cellular signal transduction. New signalling data is often examined and interpreted in conjunction with PKNs. However, different biological contexts, such as cell type or disease states, may have distinct variants of signalling pathways, resulting in the misinterpretation of new data. The identification of inconsistencies between measured data and signalling topologies, as well as the training of PKNs using context specific datasets (PKN contextualization), are necessary conditions to construct reliable, predictive models, which are current challenges in the systems biology of cell signalling. Here we present PRUNET, a user-friendly software tool designed to address the contextualization of a PKNs to specific experimental conditions. As the input, the algorithm takes a PKN and the expression profile of two given stable steady states or cellular phenotypes. The PKN is iteratively pruned using an evolutionary algorithm to perform an optimization process. This optimization rests in a match between predicted attractors in a discrete logic model (Boolean) and a Booleanized representation of the phenotypes, within a population of alternative subnetworks that evolves iteratively. We validated the algorithm applying PRUNET to four biological examples and using the resulting contextualized networks to predict missing expression values and to simulate well-characterized perturbations. PRUNET constitutes a tool for the automatic curation of a PKN to make it suitable for describing biological processes under particular experimental conditions. The general applicability of the implemented algorithm makes PRUNET suitable for a variety of biological processes, for instance cellular reprogramming or transitions between healthy and disease states.     

Identifying Discrete States of a Biological System Using a Novel Step Detection Algorithm

Identification of discrete states is a common task when studying biological systems on microscopic scales. Here, we present a novel step detection algorithm that is ideally suited to locate steplike features separating adjacent plateaus, even if they are smooth and hidden by noise. It can be adjusted to detect very low or narrow steps that cannot be recognized by conventional methods. We demonstrate the applicability of the technique on various experimental data and show strong evidence of sub-10-pN steps in atomic force spectroscopy measurements performed with living lymphocytes.     

Dynamical properties of Discrete Reaction Networks

Reaction networks are commonly used to model the dynamics of populations subject to transformations that follow an imposed stoichiometry. This paper focuses on the efficient characterisation of dynamical properties of Discrete Reaction Networks (DRNs). DRNs can be seen as modeling the underlying discrete nondeterministic transitions of stochastic models of reaction networks. In that sense, a proof of non-reachability in a given DRN has immediate implications for any concrete stochastic model based on that DRN, independent of the choice of kinetic laws and constants. Moreover, if we assume that stochastic kinetic rates are given by the mass-action law (or any other kinetic law that gives non-vanishing probability to each reaction if the required number of interacting substrates is present), then reachability properties are equivalent in the two settings. The analysis of two types of global dynamical properties of DRNs is addressed: irreducibility, i.e., the ability to reach any discrete state from any other state; and recurrence, i.e., the ability to return to any initial state. Our results consider both the verification of such properties when species are present in a large copy number, and in the general case. The necessary and sufficient conditions obtained involve algebraic conditions on the network reactions which in most cases can be verified using linear programming. Finally, the relationship of DRN irreducibility and recurrence with dynamical properties of stochastic and continuous models of reaction networks is discussed.     

Chemical Reaction Systems with Toric Steady States

Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.     

Optimal Control of Transitions between Nonequilibrium Steady States

Biological systems fundamentally exist out of equilibrium in order to preserve organized structures and processes. Many changing cellular conditions can be represented as transitions between nonequilibrium steady states, and organisms have an interest in optimizing such transitions. Using the Hatano-Sasa Y-value, we extend a recently developed geometrical framework for determining optimal protocols so that it can be applied to systems driven from nonequilibrium steady states. We calculate and numerically verify optimal protocols for a colloidal particle dragged through solution by a translating optical trap with two controllable parameters. We offer experimental predictions, specifically that optimal protocols are significantly less costly than naive ones. Optimal protocols similar to these may ultimately point to design principles for biological energy transduction systems and guide the design of artificial molecular machines.     

Multiple Quantitative Trait Analysis Using Bayesian Networks

Models for genome-wide prediction and association studies usually target a single phenotypic trait. However, in animal and plant genetics it is common to record information on multiple phenotypes for each individual that will be genotyped. Modeling traits individually disregards the fact that they are most likely associated due to pleiotropy and shared biological basis, thus providing only a partial, confounded view of genetic effects and phenotypic interactions. In this article we use data from a Multiparent Advanced Generation Inter-Cross (MAGIC) winter wheat population to explore Bayesian networks as a convenient and interpretable framework for the simultaneous modeling of multiple quantitative traits. We show that they are equivalent to multivariate genetic best linear unbiased prediction (GBLUP) and that they are competitive with single-trait elastic net and single-trait GBLUP in predictive performance. Finally, we discuss their relationship with other additive-effects models and their advantages in inference and interpretation. MAGIC populations provide an ideal setting for this kind of investigation because the very low population structure and large sample size result in predictive models with good power and limited confounding due to relatedness.     

Dynamical Behaviour of Viral Cycle and Identification of Steady States

The molecular biology of viruses can be effectively described by kinetic logic as several feedback loops are implicated in all viral cycles and as viral proteins generally display several functions. We applied this method to the study of the rhabdovirus cycle.Formally, the dynamics of the model are explored on the basis of a discrete caricature (kinetic logic), with special emphasis on the role of the constitutive feedback loops to determine the essential dynamical behaviour of the viral cycle. From a biological point of view, our model accounts for several stable regimes or attractors: healthy cells, acute infection and different kinds of persistent infections, a multistationarity in good agreement with the existence of several positive feedback loops in our system.     

N-site Phosphorylation Systems with 2N-1 Steady States

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of \(n\) -site sequential distributive phosphorylation are therefore studied frequently. In particular, in Wang and Sontag (J Math Biol 57:29–52, 2008), it is shown that models of \(n\) -site sequential distributive phosphorylation admit at most \(2n-1\) steady states. Wang and Sontag furthermore conjecture that for odd \(n\) , there are at most \(n\) and that, for even \(n\) , there are at most \(n+1\) steady states. This, however, is not true: building on earlier work in Holstein et al. (Bull Math Biol 75(11):2028–2058, 2013), we present a scalar determining equation for multistationarity which will lead to parameter values where a \(3\) -site system has \(5\) steady states and parameter values where a \(4\) -site system has \(7\) steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag. We furthermore study the inherent geometric properties of multistationarity in \(n\) -site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein.     

Jerarca: Efficient Analysis of Complex Networks Using Hierarchical Clustering

Background

How to extract useful information from complex biological networks is a major goal in many fields, especially in genomics and proteomics. We have shown in several works that iterative hierarchical clustering, as implemented in the UVCluster program, is a powerful tool to analyze many of those networks. However, the amount of computation time required to perform UVCluster analyses imposed significant limitations to its use.

Methodology/Principal Findings

We describe the suite Jerarca, designed to efficiently convert networks of interacting units into dendrograms by means of iterative hierarchical clustering. Jerarca is divided into three main sections. First, weighted distances among units are computed using up to three different approaches: a more efficient version of UVCluster and two new, related algorithms called RCluster and SCluster. Second, Jerarca builds dendrograms based on those distances, using well-known phylogenetic algorithms, such as UPGMA or Neighbor-Joining. Finally, Jerarca provides optimal partitions of the trees using statistical criteria based on the distribution of intra- and intercluster connections. Outputs compatible with the phylogenetic software MEGA and the Cytoscape package are generated, allowing the results to be easily visualized.

Conclusions/Significance

The four main advantages of Jerarca in respect to UVCluster are: 1) Improved speed of a novel UVCluster algorithm; 2) Additional, alternative strategies to perform iterative hierarchical clustering; 3) Automatic evaluation of the hierarchical trees to obtain optimal partitions; and, 4) Outputs compatible with popular software such as MEGA and Cytoscape.     

Identifying Movement States From Location Data Using Cluster Analysis

ABSTRACT Animal movement studies regularly use movement states (e.g., slow and fast) derived from remotely sensed locations to make inferences about strategies of resource use. However, the number of movement state categories used is often arbitrary and rarely inferred from the data. Identifying groups with similar movement characteristics is a statistical problem. We present a framework based on k-means clustering and gap statistic for evaluating the number of movement states without making a priori assumptions about the number of clusters. This allowed us to distinguish 4 movement states using turning angle and step length derived from Global Positioning System locations and head movements derived from tip switches in a neck collar of free-ranging elk (Cervus elaphus) in west central Alberta, Canada. Based on movement characteristics and on the linkage between each state and landscape features, we were able to identify inter-patch movements, intra-patch foraging, rest, and inter-patch foraging movements. Linking behavior to environment (e.g., state-dependent habitat use) can inform decisions on landscape management for wildlife.     

Quantum Transport in Networks and Photosynthetic Complexes at the Steady State

Recently, several works have analysed the efficiency of photosynthetic complexes in a transient scenario and how that efficiency is affected by environmental noise. Here, following a quantum master equation approach, we study the energy and excitation transport in fully connected networks both in general and in the particular case of the Fenna–Matthew–Olson complex. The analysis is carried out for the steady state of the system where the excitation energy is constantly “flowing” through the system. Steady state transport scenarios are particularly relevant if the evolution of the quantum system is not conditioned on the arrival of individual excitations. By adding dephasing to the system, we analyse the possibility of noise-enhancement of the quantum transport.     

Cross-Scale Numerical Simulations Using Discrete Particle Models

Abstract

We propose a concept for a homogenous computational model in carrying out cross-scale numerical experiments on liquids. The model employs the particle paradigm and comprises three types of simulation techniques: molecular dynamics (MD), dissipative particle dynamics (DPD) and smoothed particle hydrodynamics (SPH). With respect to the definition of the collision operator, this model may work in different hierarchical spatial and time scales as: MD in the atomistic scale, DPD in the mesoscale and SPH in the macroscale. The optimal computational efficiency of the three types of cross-scale experiments are estimated in dependence on: the system size N-where N is the number of particles-and the number of processors P employed for computer simulation. For the three-hierarchical-stage, as embodied in the MD-DPD-SPH model, the efficiency is proportional to N ^8/7 but its dependence on P is different for each of the three types of cross-scale experiments. The problem of matching the different scales is discussed.     

Prediction of Mitochondrial Proteins Using Discrete Wavelet Transform

A new method was proposed for prediction of mitochondrial proteins by the discrete wavelet transform, based on the sequence–scale similarity measurement. This sequence–scale similarity, revealing more information than other conventional methods, does not rely on subcellular location information and can directly predict protein sequences with different length. In our experiments, 499 mitochondrial protein sequences, constituting a mitochondria database, were used as training dataset, and 681 non-mitochondrial protein sequences were tested. The system can predict these sequences with sensitivity, specificity, accuracy and MCC of 50.30%, 95.74%, 76.53% and 0.54, respectively. Source code of the new program is available on request from the authors.