College of Letters & Science
Permanent URI for this communityhttps://scholarworks.montana.edu/handle/1/37
The College of Letters and Science, the largest center for learning, teaching and research at Montana State University, offers students an excellent liberal arts and sciences education in nearly 50 majors, 25 minors and over 25 graduate degrees within the four areas of the humanities, natural sciences, mathematics and social sciences.
Browse
33 results
Search Results
Item Comparison of Combinatorial Signatures of Global Network Dynamics Generated by Two Classes of ODE Models(2019-04) Crawford-Kahrl, Peter; Cummins, Bree; Gedeon, TomasModeling the dynamics of biological networks introduces many challenges, among them the lack of first principle models, the size of the networks, and difficulties with parameterization. Discrete time Boolean networks and related continuous time switching systems provide a computationally accessible way to translate the structure of the network to predictions about the dynamics. Recent work has shown that the parameterized dynamics of switching systems can be captured by a combinatorial object, called a Dynamic Signatures Generated by Regulatory Networks (DSGRN) database, that consists of a parameter graph characterizing a finite parameter space decomposition, whose nodes are assigned a Morse graph that captures global dynamics for all corresponding parameters. We show that for a given network there is a way to associate the same type of object by considering a continuous time ODE system with a continuous right-hand side, which we call an L-system. The main goal of this paper is to compare the two DSGRN databases for the same network. Since the L-systems can be thought of as perturbations (not necessarily small) of the switching systems, our results address the correspondence between global parameterized dynamics of switching systems and their perturbations. We show that, at corresponding parameters, there is an order preserving map from the Morse graph of the switching system to that of the L-system that is surjective on the set of attractors and bijective on the set of fixed-point attractors. We provide important examples showing why this correspondence cannot be strengthened.Item DSGRN: Examining the Dynamics of Families of Logical Models(2018-06) Cummins, Bree; Gedeon, Tomas; Harker, Shaun; Mischaikow, KonstantinWe present a computational tool DSGRN for exploring the dynamics of a network by computing summaries of the dynamics of switching models compatible with the network across all parameters. The network can arise directly from a biological problem, or indirectly as the interaction graph of a Boolean model. This tool computes a finite decomposition of parameter space such that for each region, the state transition graph that describes the coarse dynamical behavior of a network is the same. Each of these parameter regions corresponds to a different logical description of the network dynamics. The comparison of dynamics across parameters with experimental data allows the rejection of parameter regimes or entire networks as viable models for representing the underlying regulatory mechanisms. This in turn allows a search through the space of perturbations of a given network for networks that robustly fit the data. These are the first steps toward discovering a network that optimally matches the observed dynamics by searching through the space of networks.Item Global dynamics for switching systems and their extensions by linear differential equations(2018-11) Huttinga, Zane; Cummins, Bree; Gedeon, Tomas; Mischaikow, KonstantinSwitching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.Item Competitive resource allocation to metabolic pathways contributes to overflow metabolisms and emergent properties in cross-feeding microbial consortia(2018-04) Carlson, Ross P.; Beck, Ashley E.; Phalak, Poonam; Fields, Matthew W.; Gedeon, Tomas; Hanley, Luke; Harcombe, W. R.; Henson, Michael A.; Heys, Jeffrey J.Resource scarcity is a common stress in nature and has a major impact on microbial physiology. This review highlights microbial acclimations to resource scarcity, focusing on resource investment strategies for chemoheterotrophs from the molecular level to the pathway level. Competitive resource allocation strategies often lead to a phenotype known as overflow metabolism; the resulting overflow byproducts can stabilize cooperative interactions in microbial communities and can lead to cross-feeding consortia. These consortia can exhibit emergent properties such as enhanced resource usage and biomass productivity. The literature distilled here draws parallels between in silico and laboratory studies and ties them together with ecological theories to better understand microbial stress responses and mutualistic consortia functioning.Item The effects of interleukin-2 on immune response regulation(2017-02) Waters, Ryan S.; Perry, Justin S. A.; Han, SunPil; Bielekova, Bibiana; Gedeon, TomasThe immune system has many adaptive and dynamic components that are regulated to ensure appropriate, precise and rapid response to a foreign pathogen. A delayed or inadequate immune response can lead to prolonged disease, while an excessive or under-regulated response can lead to autoimmunity. The cytokine, interleukin-2 (IL-2) and its receptor IL-2R play an important role in maintaining this balance.The IL-2 receptor transduces pSTAT5 signal through both the intermediate and high affinity receptors, which differ from each other by the presence of CD25 chain in IL-2 receptor. We present experimental data on the kinetics of pSTAT5 signalling through both of the receptors and develop a model that captures this kinetics. We then use this model to parameterize key aspects of two additional models in which we propose and study two different mechanisms by which IL-2 receptor can transduce distinct signals leading to either an activated or a non-activated cell state. We speculate that this initial state differentiation, perhaps enhanced by downstream feedbacks, may eventually lead to differential cell fates.Our result shows that non-linear dynamical models can suggest resolution of a puzzling array of seemingly contradictory experimental results on IL-2 effect on proliferation and differentiation of T-cells.Item A Mechanistic Model for Cooperative Behavior of Co-transcribing RNA Polymerases(2016-08) Davis, Lisa; Gedeon, Tomas; Heberling, Tamra; Gedeon, Jakub; Morgan, CharlesIn fast-transcribing prokaryotic genes, such as an rrn gene in Escherichia coli, many RNA polymerases (RNAPs) transcribe the DNA simultaneously. Active elongation of RNAPs is often interrupted by pauses, which has been observed to cause RNAP traffic jams; yet some studies indicate that elongation seems to be faster in the presence of multiple RNAPs than elongation by a single RNAP. We propose that an interaction between RNAPs via the torque produced by RNAP motion on helically twisted DNA can explain this apparent paradox. We have incorporated the torque mechanism into a stochastic model and simulated transcription both with and without torque. Simulation results illustrate that the torque causes shorter pause durations and fewer collisions between polymerases. Our results suggest that the torsional interaction of RNAPs is an important mechanism in maintaining fast transcription times, and that transcription should be viewed as a cooperative group effort by multiple polymerases.Item Combinatorial Representation of Parameter Space for Switching Networks(2016-11) Cummins, Bree; Harker, Shaun; Mischaikow, Konstantin; Mok, Kafung; Gedeon, TomasWe describe the theoretical and computational framework for the Dynamic Signatures Generated by Regulatory Networks (DSGRN) database. The motivation stems from an urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and have decades of parameters. The input to the database computations is a regulatory network, i.e., a directed graph with edges indicating up or down regulation. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate) and three parameters for each edge (low level of expression, high level of expression, and threshold at which expression levels change). Since the nonlinearities of switching systems are piecewise constant, there is a natural decomposition of phase space into cells from which the dynamics can be described combinatorially in terms of a state transition graph. This in turn leads to a compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent dynamics. The focus of this paper is on the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant. We use this decomposition to construct an SQL database that can be effectively searched for dynamical signatures such as bistability, stable or unstable oscillations, and stable equilibria. We include two simple 3-node networks to provide small explicit examples of the type of information stored in the DSGRN database. To demonstrate the computational capabilities of this system we consider a simple network associated with p53 that involves 5 nodes and a 29-dimensional parameter space.Item Global dynamics for steep nonlinearities in two dimensions(2017-01) Gedeon, Tomas; Harker, Shaun; Kokubu, Hiroshi; Mischaikow, Konstantin; Oka, HiroeThis paper discusses a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. We study switching models of regulatory networks. To each switching network we associate a Morse graph, a computable object that describes a Morse decomposition of the dynamics. In this paper we show that all smooth perturbations of the switching system share the same Morse graph and we compute explicit bounds on the size of the allowable perturbation. This shows that computationally tractable switching systems can be used to characterize dynamics of smooth systems with steep nonlinearities.Item Symmetry breaking clusters in soft clustering decoding of neural codes(2010-02) Parker, Albert E.; Dimitrov, Alexander G.; Gedeon, TomasInformation-based distortion methods have been used successfully in the analysis of neural coding problems. These approaches allow the discovery of neural symbols and the corresponding stimulus space of a neuron or neural ensemble quantitatively, while making few assumptions about the nature of either the code or of relevant stimulus features. The neural codebook is derived by quantizing sensory stimuli and neural responses into a small set of clusters, and optimizing the quantization to minimize an information distortion function. The method of annealing has been used to solve the corresponding high-dimensional nonlinear optimization problem. The annealing solutions undergo a series of bifurcations, which we study using bifurcation theory in the presence of symmetries. In this contribution we describe these symmetry breaking bifurcations in detail, and indicate some of the consequences of the form of the bifurcations. In particular, we show that the annealing solutions break symmetry at pitchfork bifurcations, and that subcritical branches can exist. Thus, at a subcritical bifurcation, there are local information distortion solutions which are not found by the method of annealing. Since the annealing procedure is guaranteed to converge to a local solution eventually, the subcritical branch must turn and become optimal at some later saddle-node bifurcation, which we have shown occur generically for this class of problems. This implies that the rate distortion curve, while convex for noninformation-based distortion measures, is not convex for information-based distortion methods.Item The mathematical structure of information bottleneck methods(2012-03) Gedeon, Tomas; Parker, Albert E.; Dimitrov, Alexander G.Information Bottleneck-based methods use mutual information as a distortion function in order to extract relevant details about the structure of a complex system by compression. One of the approaches used to generate optimal compressed representations is by annealing a parameter. In this manuscript we present a common framework for the study of annealing in information distortion problems. We identify features that should be common to any annealing optimization problem. The main mathematical tools that we use come from the analysis of dynamical systems in the presence of symmetry (equivariant bifurcation theory). Through the compression problem, we make connections to the world of combinatorial optimization and pattern recognition. The two approaches use very different vocabularies and consider different problems to be “interesting†. We provide an initial link, through the Normalized Cut Problem, where the two disciplines can exchange tools and ideas.