Browsing by Author "Gedeon, Tomas"

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The chemostat with lateral gene transfer
(2010-11) De Leenheer, Patrick; Dockery, Jack D.; Gedeon, Tomas; Young, T.
We investigate the standard chemostat model when lateral gene transfer is taken into account. We will show that when the different genotypes have growth rate functions that are sufficiently close to a common growth rate function, and when the yields of the genotypes are sufficiently close to a common value, then the population evolves to a globally stable steady state, at which all genotypes coexist. These results can explain why the antibiotic-resistant strains persist in the pathogen population.
Combinatorial Representation of Parameter Space for Switching Networks
(2016-11) Cummins, Bree; Harker, Shaun; Mischaikow, Konstantin; Mok, Kafung; Gedeon, Tomas
We 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.
Comparison of Combinatorial Signatures of Global Network Dynamics Generated by Two Classes of ODE Models
(2019-04) Crawford-Kahrl, Peter; Cummins, Bree; Gedeon, Tomas
Modeling 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.
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.
Convergence Properties of Posttranslationally Modified Protein-Protein Switching Networks with Fast Decay Rates
(2016-06) Fan, G.; Cummins, Bree; Gedeon, Tomas
A significant conceptual difficulty in the use of switching systems to model regulatory networks is the presence of so-called "black walls" co-dimension 1 regions of phase space with a vector field pointing inward on both sides of the hyperplane. Black walls result from the existence of direct negative self-regulation in the system. One biologically inspired way of removing black walls is the introduction of intermediate variables that mediate the negative self-regulation. In this paper, we study such a perturbation. We replace a switching system with a higher-dimensional switching system with rapidly decaying intermediate proteins, and compare the dynamics between the two systems. We find that the while the individual solutions of the original system can be approximated for a finite time by solutions of a sufficiently close perturbed system, there are always solutions that are not well approximated for any fixed perturbation. We also study a particular example, where global basins of attraction of the perturbed system have a strikingly different form than those of the original system. We perform this analysis using techniques that are adapted to dealing with non-smooth systems.
Creating Soft Clusterings of Data Via the Information Bottleneck Method
(2013-03) Ricker, Russell; Parker, Albert; Gedeon, Tomas
Information-based distortion methods have been used successfully to analyze the relationship between stimulus and reaction spaces. Distortion methods make few assumptions concerning the correspondence between the two spaces, providing maximally informative relationships between them. I used the Information Bottleneck technique to create soft clustering of a synthetic data set with 50 stimuli and 50 neural responses with a multivariate Gaussian (either with 4-blobs or 10-blobs) describing their hypothetical relationship. The algorithm utilized an annealing method to solve the high-dimensional non-linear problem, and was implemented using Matlab. As the annealing parameter increased, the solution to the problem underwent a series of phase transitions, or bifurcations, that eventually stabilized to a nearly deterministic clustering. By calculating the matrix of second derivatives (Hessian), we are able to determine when the bifurcations occur. By calculating the third and fourth derivatives we are able to determine whether the bifurcations are subcritical or supercritical. The existence of subcritical branching implies that several solutions not found by the method of annealing exist. Because the method of annealing is guaranteed to converge, the subcritical branch must turn at a later bifurcation and become optimal.
Cyclic Feedback Systems with Quorum Sensing Coupling
(2016-06) Gedeon, Tomas; Pernarowski, Mark; Wilander, Adam
Synchronization and desynchronization is of great interest in the study of circadian rhythms, metabolic oscillations and time-dependent cell aggregate behaviors. Several recent studies examine synchronization and other dynamics in models of repressilators coupled by a quorum sensing mechanism that uses a diffusive signal. Their numerical simulations have shown the complexity of the collective behavior depends sensitively on which protein upregulates diffusive signal. In this paper, we rigorously prove that the collective dynamics indeed strongly depends on how the signaling network integrates into the repressilator network. In fact we prove a general result for a class of negative cyclic feedback systems with signaling of which the repressilator is but one example. We show that if the feedback along the signaling loop is also negative, the resulting negative feedback, negative signaling (Nf-Ns) system admits either unique stable equilibrium, or a stable oscillation. When a positive signaling feedback is included, the system is no longer (Nf-Ns) and numerically exhibits multistable dynamics (Ullner etÂ al. in Phys Rev Lett 99:148103, 2007; Phys Rev E 78:031904, 2008). We demonstrate that this multistability emerges through saddle node bifurcations of a sole cubic curve-as in generic bistable models.
Delayed protein synthesis reduces the correlation between mRNA and protein fluctuations
(2012-08) Gedeon, Tomas; Bokes, Pavol
Recent experimental results indicate that, in single Escherichia coli cells, the fluctuations in mRNA level are uncorrelated with those of protein. However, a basic two-stage model for prokaryotic gene expression suggests that there ought to be a degree of correlation between the two. Therefore, it is important to investigate realistic modifications of the basic model that have the potential to reduce the theoretical level of the correlation. In this work, we focus on translational and reporter maturation delay, reporting that its introduction into the two-stage model reduces the cross correlation between instantaneous mRNA and protein levels. Our results indicate that the experimentally observed sample correlation coefficient between mRNA and protein levels may increase if the protein measurements are shifted back in time by the value of the delay.
Discontinuous Galerkin calculations for a nonlinear PDE model of DNA transcription with short, transient and frequent pausing
(2014-09) Davis, Lisa; Gedeon, Tomas; Thorenson, Jennifer R.
A discontinuous Galerkin finite element method is used to approximate solutions to a classical traffic flow PDE. This PDE is used to model the biological process of transcription; the process of transferring genetic information from DNA either to mRNA or to rRNA. The transcription process is punctuated by short, frequent RNAP pauses which are incorporated into the model as traffic lights. These pauses cause a delay in the average transcription process. The DG solution of the nonlinear model is used to calculate the delay and to determine the effect of the pauses on the average transcription time. Numerical error measurements between the DG solution and the true solution (derived by the method of characteristics) are given for a simple model problem. It shows an excellent agreement in a neighborhood away from the shocks as well as O(Δx) convergence for the delay calculation. Preliminary parameter studies indicate that in a system with multiple pauses both the location and time duration of the pauses can significantly affect the average delay experienced by an RNAP.
DSGRN: Examining the Dynamics of Families of Logical Models
(2018-06) Cummins, Bree; Gedeon, Tomas; Harker, Shaun; Mischaikow, Konstantin
We 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.
Dynamics of Simple Food Webs
(2015-09) Gedeon, Tomas; Murphy, Patrick
We consider a simple food web with commensal relationship, where organisms utilize both external resources and resources produced by other organisms. We show that in such a community with no competition, there is at most one possible equilibrium for each fixed set of surviving species, and develop a natural condition that determines which species survive based on available resource. Our main result shows that among all possible communities described by equilibria, the one which is stable has the largest number of surviving species and largest combined biomass and hence maximizes utilization of available resources.
Effect of model selection on prediction of periodic behavior in gene regulatory networks
(2012-08) Gedeon, Tomas; Cummins, G.; Heys, Jeffrey J.
One of the current challenges for cell biology is understanding of the system level cellular behavior from the knowledge of a network of the individual subcellular agents. We address a question of how the model selection affects the predicted dynamic behavior of a gene network. In particular, for a fixed network structure, we compare protein-only models with models in which each transcriptional activation is represented both by mRNA and protein concentrations. We compare linear behavior near equilibria for both cyclic feedback systems and a general system. We show that, in general, explicit inclusion of the mRNA in the model weakens the stability of equilibria. We also study numerically dynamics of a particular gene network and show significant differences in global dynamics between the two types of models.
The effects of interleukin-2 on immune response regulation
(2017-02) Waters, Ryan S.; Perry, Justin S. A.; Han, SunPil; Bielekova, Bibiana; Gedeon, Tomas
The 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.
Effects of Spatial Localization on Microbial Consortia Growth
(2017-01) Venters, Michael; Carlson, Ross P.; Gedeon, Tomas; Heys, Jeffrey J.
Microbial consortia are commonly observed in natural and synthetic systems, and these consortia frequently result in higher biomass production relative to monocultures. The focus here is on the impact of initial spatial localization and substrate diffusivity on the growth of a model microbial consortium consisting of a producer strain that consumes glucose and produces acetate and a scavenger strain that consumes the acetate. The mathematical model is based on an individual cell model where growth is described by Monod kinetics, and substrate transport is described by a continuum-based, non-equilibrium reaction-diffusion model where convective transport is negligible (e.g., in a biofilm). The first set of results focus on a single producer cell at the center of the domain and surrounded by an initial population of scavenger cells. The impact of the initial population density and substrate diffusivity is examined. A transition is observed from the highest initial density resulting in the greatest cell growth to cell growth being independent of initial density. A high initial density minimizes diffusive transport time and is typically expected to result in the highest growth, but this expected behavior is not predicted in environments with lower diffusivity or larger length scales. When the producer cells are placed on the bottom of the domain with the scavenger cells above in a layered biofilm arrangement, a similar critical transition is observed. For the highest diffusivity values examined, a thin, dense initial scavenger layer is optimal for cell growth. However, for smaller diffusivity values, a thicker, less dense initial scavenger layer provides maximal growth. The overall conclusion is that high density clustering of members of a food chain is optimal under most common transport conditions, but under some slow transport conditions, high density clustering may not be optimal for microbial growth.
Environment Constrains Fitness Advantages of Division of Labor in Microbial Consortia Engineered for Metabolite Push or Pull Interactions
(American Society for Microbiology, 2022-08) Beck, Ashely E.; Pintar, Kathryn; Schepens, Diana; Schrammeck, Ashely; Johnson, Timothy; Bleem, Alissa; Du, Martina; Harcombe, William R.; Bernstein, Hans C.; Heys, Jeffrey J.; Gedeon, Tomas; Carlson, Ross P.
Fitness benefits from division of labor are well documented in microbial consortia, but the dependency of the benefits on environmental context is poorly understood. Two synthetic Escherichia coli consortia were built to test the relationships between exchanged organic acid, local environment, and opportunity costs of different metabolic strategies. Opportunity costs quantify benefits not realized due to selecting one phenotype over another. The consortia catabolized glucose and exchanged either acetic or lactic acid to create producer-consumer food webs. The organic acids had different inhibitory properties and different opportunity costs associated with their positions in central metabolism. The exchanged metabolites modulated different consortial dynamics. The acetic acid-exchanging (AAE) consortium had a “push” interaction motif where acetic acid was secreted faster by the producer than the consumer imported it, while the lactic acid-exchanging (LAE) consortium had a “pull” interaction motif where the consumer imported lactic acid at a comparable rate to its production. The LAE consortium outperformed wild-type (WT) batch cultures under the environmental context of weakly buffered conditions, achieving a 55% increase in biomass titer, a 51% increase in biomass per proton yield, an 86% increase in substrate conversion, and the complete elimination of by-product accumulation all relative to the WT. However, the LAE consortium had the trade-off of a 42% lower specific growth rate. The AAE consortium did not outperform the WT in any considered performance metric. Performance advantages of the LAE consortium were sensitive to environment; increasing the medium buffering capacity negated the performance advantages compared to WT.
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, Lun
Understanding 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.
Genetic networks encode secrets of their past
(Elsevier BV, 2022-03) Crawford-Kahrl, Peter; Nerem, Robert R.; Cummins, Bree; Gedeon, Tomas
Research 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.
Global dynamics for steep nonlinearities in two dimensions
(2017-01) Gedeon, Tomas; Harker, Shaun; Kokubu, Hiroshi; Mischaikow, Konstantin; Oka, Hiroe
This 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.
Global dynamics for switching systems and their extensions by linear differential equations
(2018-11) Huttinga, Zane; Cummins, Bree; Gedeon, Tomas; Mischaikow, Konstantin
Switching 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.
Hantavirus transmission in sylvan and peridomestic environment
(2010-04) Gedeon, Tomas; Bodelon, Clara; Kuenzi, Amy
We developed a compartmental model for hantavirus infection in deer mice (Peromyscus maniculatus) with the goal of comparing relative importance of direct and indirect transmission in sylvan and peridomestic environments. A direct transmission occurs when the infection is mediated by the contact of an infected and an uninfected mouse, while an indirect transmission occurs when the infection is mediated by the contact of an uninfected mouse with, for instance, infected soil. Based on population dynamics data and estimates of hantavirus decay in the two types of environments, our model predicts that direct transmission dominates in the sylvan environment, while both pathways are important in peridomestic environments. The model allows us to compute a basic reproduction number R0, which indicates whether the virus will be endemic or eradicated from the mouse population, in both an autonomous and a time-periodic model. Our analysis can be used to evaluate various eradication strategies.