Scholarly Work - Mathematical Sciences
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/8719
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Item Extending Combinatorial Regulatory Network Modeling to Include Activity Control and Decay Modulation(Society for Industrial & Applied Mathematics, 2022-09) Cummins, Bree; Gameiro, Marcio; Gedeon, Tomas; Kepley, Shane; Mischaikow, Konstantin; Zhang, LunUnderstanding how the structure of within-system interactions affects the dynamics of the system is important in many areas of science. We extend a network dynamics modeling platform DSGRN, which combinatorializes both dynamics and parameter space to construct finite but accurate summaries of network dynamics, to new types of interactions. While the standard DSGRN assumes that each network edge controls the rate of abundance of the target node, the new edges may control either activity level or a decay rate of its target. While motivated by processes of post-transcriptional modification and ubiquitination in systems biology, our extension is applicable to the dynamics of any signed directed network.Item Extremal event graphs: A (stable) tool for analyzing noisy time series data(American Institute of Mathematical Sciences, 2023-01) Belton, Robin; Cummins, Bree; Gedeon, Tomáš; Fasy, Brittany TereseLocal maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.Item Modeling Transport Regulation in Gene Regulatory Networks(Springer Science and Business Media LLC, 2022-07) Fox, Erika; Cummins, Bree; Duncan, William; Gedeon, TomášA gene regulatory network summarizes the interactions between a set of genes and regulatory gene products. These interactions include transcriptional regulation, protein activity regulation, and regulation of the transport of proteins between cellular compartments. DSGRN is a network modeling approach that builds on traditions of discrete-time Boolean models and continuous-time switching system models. When all interactions are transcriptional, DSGRN uses a combinatorial approximation to describe the entire range of dynamics that is compatible with network structure. Here we present an extension of the DGSRN approach to transport regulation across a boundary between compartments, such as a cellular membrane. We illustrate our approach by searching a model of the p53-Mdm2 network for the potential to admit two experimentally observed distinct stable periodic cycles.Item Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline(MyJove Corporation, 2021-12) Moseley, Robert C.; Campione, Sophia; Cummins, Bree; Motta, Francis; Haase, Steven B.Developing gene regulatory network models is a major challenge in systems biology. Several computational tools and pipelines have been developed to tackle this challenge, including the newly developed Inherent Dynamics Pipeline. The Inherent Dynamics Pipeline consists of several previously published tools that work synergistically and are connected in a linear fashion, where the output of one tool is then used as input for the following tool. As with most computational techniques, each step of the Inherent Dynamics Pipeline requires the user to make choices about parameters that don’t have a precise biological definition. These choices can substantially impact gene regulatory network models produced by the analysis. For this reason, the ability to visualize and explore the consequences of various parameter choices at each step can help increase confidence in the choices and the results.The Inherent Dynamics Visualizer is a comprehensive visualization package that streamlines the process of evaluating parameter choices through an interactive interface within a web browser. The user can separately examine the output of each step of the pipeline, make intuitive changes based on visual information, and benefit from the automatic production of necessary input files for the Inherent Dynamics Pipeline. The Inherent Dynamics Visualizer provides an unparalleled level of access to a highly intricate tool for the discovery of gene regulatory networks from time series transcriptomic data.Item Joint realizability of monotone Boolean functions(Elsevier BV, 2022-06) Crawford-Kahrl, Peter; Cummins, Bree; Gedeon, TomášThe study of monotone Boolean functions (MBFs) has a long history. We explore a connection between MBFs and ordinary differential equation (ODE) models of gene regulation, and, in particular, a problem of the realization of an MBF as a function describing the state transition graph of an ODE. We formulate a problem of joint realizability of finite collections of MBFs by establishing a connection between the parameterized dynamics of a class of ODEs and a collection of MBFs. We pose a question of what collections of MBFs can be realized by ODEs that belong to nested classes defined by increased algebraic complexity of their right-hand sides. As we progressively restrict the algebraic form of the ODE, we show by a combination of theory and explicit examples that the class of jointly realizable functions strictly decreases. Our results impact the study of regulatory network dynamics, as well as the classical area of MBFs. We conclude with a series of potential extensions and conjectures.Item Genetic networks encode secrets of their past(Elsevier BV, 2022-03) Crawford-Kahrl, Peter; Nerem, Robert R.; Cummins, Bree; Gedeon, TomasResearch shows that gene duplication followed by either repurposing or removal of duplicated genes is an important contributor to evolution of gene and protein interaction networks. We aim to identify which characteristics of a network can arise through this process, and which must have been produced in a different way. To model the network evolution, we postulate vertex duplication and edge deletion as evolutionary operations on graphs. Using the novel concept of an ancestrally distinguished subgraph, we show how features of present-day networks require certain features of their ancestors. In particular, ancestrally distinguished subgraphs cannot be introduced by vertex duplication. Additionally, if vertex duplication and edge deletion are the only evolutionary mechanisms, then a graph’s ancestrally distinguished subgraphs must be contained in all of the graph’s ancestors. We analyze two experimentally derived genetic networks and show that our results accurately predict lack of large ancestrally distinguished subgraphs, despite this feature being statistically improbable in associated random networks. This observation is consistent with the hypothesis that these networks evolved primarily via vertex duplication. The tools we provide open the door for analyzing ancestral networks using current networks. Our results apply to edge-labeled (e.g. signed) graphs which are either undirected or directed.Item Using extremal events to characterize noisy time series(2020-02) Berry, Eric; Cummins, Bree; Nerem, Robert R.; Smith, Lauren M.; Haase, Steven B.; Gedeon, TomasExperimental time series provide an informative window into the underlying dynamical system, and the timing of the extrema of a time series (or its derivative) contains information about its structure. However, the time series often contain significant measurement errors. We describe a method for characterizing a time series for any assumed level of measurement error 𝜀 by a sequence of intervals, each of which is guaranteed to contain an extremum for any function that 𝜀-approximates the time series. Based on the merge tree of a continuous function, we define a new object called the normalized branch decomposition, which allows us to compute intervals for any level 𝜀. We show that there is a well-defined total order on these intervals for a single time series, and that it is naturally extended to a partial order across a collection of time series comprising a dataset. We use the order of the extracted intervals in two applications. First, the partial order describing a single dataset can be used to pattern match against switching model output (Cummins et al. in SIAM J Appl Dyn Syst 17(2):1589–1616, 2018), which allows the rejection of a network model. Second, the comparison between graph distances of the partial orders of different datasets can be used to quantify similarity between biological replicates.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.