Heike Siebert

Temporal constraints in the logical analysis of regulatory networks

Starting from the logical description of gene regulatory networks developed by R. Thomas, we introduce an enhanced modeling approach based on timed automata. We obtain a refined qualitative description of the dynamical behavior by exploiting not only information on ratios of kinetic parameters related to synthesis and decay, but also constraints on the time delays associated with the operations of the system. We develop a formal framework for handling such temporal constraints using timed automata, discuss the relationship with the original Thomas formalism, and demonstrate the potential of our approach by analyzing an illustrative gene regulatory network of bacteriophage @l.

Incorporating time delays into the logical analysis of gene regulatory networks

Based on the logical description of gene regulatory networks developed by R. Thomas, we introduce an enhanced modelling approach that uses timed automata. It yields a refined qualitative description of the dynamics of the system incorporating information not only on ratios of kinetic constants related to synthesis and decay, but also on the time delays occurring in the operations of the system. We demonstrate the potential of our approach by analysing an illustrative gene regulatory network of bacteriophage λ.

Fixed points and normal families of quasiregular mappings

LetF be a family of mappingsK-quasiregular in some domainG. We show that if for eachf∈F, there existsk>1 such that thek-th iteratefk off has no fixed point, thenF is normal. Moreover, we examine to what extent this result holds if we consider only repelling fixed points, rather than fixed points in general. We also prove thatF is quasinormal, ifF contains only quasiregular mappings that do not have periodic points of some period greater than one inG. This implies that a quasiregular mappingf:ℝn with an essential singularity in ∞ has infinitely many periodic points of any period greater than one. These results generalize results of M. Essén, S. Wu, D. Bargmann and W. Bergweiler for holomorphic functions.

Parameter Identification and Model Ranking of Thomas Networks

We propose a new methodology for identification and analysis of discrete gene networks as defined by Rene Thomas, supported by a tool chain: (i) given a Thomas network with partially known kinetic parameters, we reduce the number of acceptable parametrizations to those that fit time-series measurements and reflect other known constraints by an improved technique of coloured LTL model checking performing efficiently on Thomas networks in distributed environment; (ii) we introduce classification of acceptable parametrizations to identify most optimal ones; (iii) we propose two ways of visualising parametrizations dynamics wrt time-series data. Finally, computational efficiency is evaluated and the methodology is validated on bacteriophage λ case study.

Deriving Behavior of Boolean Bioregulatory Networks from Subnetwork Dynamics

Abstract.In the well-known discrete modeling framework developed by R. Thomas, the structure of a biological regulatory network is captured in an interaction graph, which, together with a set of Boolean parameters, gives rise to a state transition graph describing all possible dynamical behaviors. For complex networks the analysis of the dynamics becomes more and more difficult, and efficient methods to carry out the analysis are needed. In this paper, we focus on identifying subnetworks of the system that govern the behavior of the system as a whole. We present methods to derive trajectories and attractors of the network from the dynamics suitable subnetworks display in isolation. In addition, we use these ideas to link the existence of certain structural motifs, namely circuits, in the interaction graph to the character and number of attractors in the state transition graph, generalizing and refining results presented in [10]. Lastly, we show for a specific class of networks that all possible asymptotic behaviors of networks in that class can be derived from the dynamics of easily identifiable subnetworks.

Computing maximal and minimal trap spaces of Boolean networks

AbstractAsymptotic behaviors are often of particular interest when analyzing Boolean networks that represent biological systems such as signal transduction or gene regulatory networks. Methods based on a generalization of the steady state notion, the so-called trap spaces, can be exploited to investigate attractor properties as well as for model reduction techniques. In this paper, we propose a novel optimization-based method for computing all minimal and maximal trap spaces and motivate their use. In particular, we add a new result yielding a lower bound for the number of cyclic attractors and illustrate the methods with a study of a MAPK pathway model. To test the efficiency and scalability of the method, we compare the performance of the ILP solver gurobi with the ASP solver potassco in a benchmark of random networks.

Time Series Dependent Analysis of Unparametrized Thomas Networks

This paper is concerned with the analysis of labeled Thomas networks using discrete time series. It focuses on refining the given edge labels and on assessing the data quality. The results are aimed at being exploitable for experimental design and include the prediction of new activatory or inhibitory effects of given interactions and yet unobserved oscillations of specific components in between specific sampling intervals. On the formal side, we generalize the concept of edge labels and introduce a discrete time series interpretation. This interpretation features two original concepts: 1) Incomplete measurements are admissible, and 2) it allows qualitative assumptions about the changes in gene expression by means of monotonicity. On the computational side, we provide a Python script, erda.py, that automates the suggested workflow by model checking and constraint satisfaction. We illustrate the workflow by investigating the yeast network IRMA.

Analysis of Discrete Bioregulatory Networks Using Symbolic Steady States

A discrete model of a biological regulatory network can be represented by a discrete function that contains all available information on interactions between network components and the rules governing the evolution of the network in a finite state space. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem. In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in some region of state space. We state rules to explicitly construct attractors of the system from subnetwork attractors. Using the results, we formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the graph representing the structure of the system. In addition, we discuss approaches to finding symbolic steady states. We focus both on dynamics derived via synchronous as well as asynchronous update rules. Lastly, we illustrate the results by analyzing a model of T helper cell differentiation.

Computing Symbolic Steady States of Boolean Networks

Asymptotic behavior is often of particular interest when analyzing asynchronous Boolean networks representing biological systems such as signal transduction or gene regulatory networks. Methods based on a generalization of the steady state notion, the so-called symbolic steady states, can be exploited to investigate attractor properties as well as for model reduction techniques conserving attractors. In this paper, we propose a novel optimization-based method for computing all maximal symbolic steady states and motivate their use. n particular, we add a new result yielding a lower bound for the number of cyclic attractors and illustrate the methods with a short study of a MAPK pathway model.

Logical-continuous modelling of post-translationally regulated bistability of curli fiber expression in Escherichia coli

BackgroundBacteria have developed a repertoire of signalling mechanisms that enable adaptive responses to fluctuating environmental conditions. The formation of biofilm, for example, allows persisting in times of external stresses, e.g. induced by antibiotics or a lack of nutrients. Adhesive curli fibers, the major extracellular matrix components in Escherichia coli biofilms, exhibit heterogeneous expression in isogenic cells exposed to identical external conditions. The dynamical mechanisms underlying this heterogeneity remain poorly understood. In this work, we elucidate the potential role of post-translational bistability as a source for this heterogeneity.ResultsWe introduce a structured modelling workflow combining logical network topology analysis with time-continuous deterministic and stochastic modelling. The aim is to evaluate the topological structure of the underlying signalling network and to identify and analyse model parameterisations that satisfy observations from a set of genetic knockout experiments. Our work supports the hypothesis that the phenotypic heterogeneity of curli expression in biofilm cells is induced by bistable regulation at the post-translational level. Stochastic modelling suggests diverse noise-induced switching behaviours between the stable states, depending on the expression levels of the c-di-GMP-producing (diguanylate cyclases, DGCs) and -degrading (phosphodiesterases, PDEs) enzymes and reveals the quantitative difference in stable c-di-GMP levels between distinct phenotypes. The most dominant type of behaviour is characterised by a fast switching from curli-off to curli-on with a slow switching in the reverse direction and the second most dominant type is a long-term differentiation into curli-on or curli-off cells. This behaviour may implicate an intrinsic feature of the system allowing for a fast adaptive response (curli-on) versus a slow transition to the curli-off state, in line with experimental observations.ConclusionThe combination of logical and continuous modelling enables a thorough analysis of different determinants of bistable regulation, i.e. network topology and biochemical kinetics, and allows for an incorporation of experimental data from heterogeneous sources. Our approach yields a mechanistic explanation for the phenotypic heterogeneity of curli fiber expression. Furthermore, the presented work provides a detailed insight into the interactions between the multiple DGC- and PDE-type enzymes and the role of c-di-GMP in dynamical regulation of cellular decisions.