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Fox J, Cummins B, Moseley RC, Gameiro M, Haase SB. A yeast cell cycle pulse generator model shows consistency with multiple oscillatory and checkpoint mutant datasets. Math Biosci 2024; 367:109102. [PMID: 37939998 PMCID: PMC10842220 DOI: 10.1016/j.mbs.2023.109102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/06/2023] [Revised: 09/13/2023] [Accepted: 10/27/2023] [Indexed: 11/10/2023]
Abstract
Modeling biological systems holds great promise for speeding up the rate of discovery in systems biology by predicting experimental outcomes and suggesting targeted interventions. However, this process is dogged by an identifiability issue, in which network models and their parameters are not sufficiently constrained by coarse and noisy data to ensure unique solutions. In this work, we evaluated the capability of a simplified yeast cell-cycle network model to reproduce multiple observed transcriptomic behaviors under genomic mutations. We matched time-series data from both cycling and checkpoint arrested cells to model predictions using an asynchronous multi-level Boolean approach. We showed that this single network model, despite its simplicity, is capable of exhibiting dynamical behavior similar to the datasets in most cases, and we demonstrated the drop in severity of the identifiability issue that results from matching multiple datasets.
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Affiliation(s)
- Julian Fox
- Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA
| | - Breschine Cummins
- Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA.
| | | | - Marcio Gameiro
- Department of Mathematics, Rutgers University, New Brunswick, NJ, USA
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Rozum JC, Albert R. Identifying (un)controllable dynamical behavior in complex networks. PLoS Comput Biol 2018; 14:e1006630. [PMID: 30532150 PMCID: PMC6301693 DOI: 10.1371/journal.pcbi.1006630] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2018] [Revised: 12/20/2018] [Accepted: 11/11/2018] [Indexed: 12/13/2022] Open
Abstract
We present a technique applicable in any dynamical framework to identify control-robust subsets of an interacting system. These robust subsystems, which we call stable modules, are characterized by constraints on the variables that make up the subsystem. They are robust in the sense that if the defining constraints are satisfied at a given time, they remain satisfied for all later times, regardless of what happens in the rest of the system, and can only be broken if the constrained variables are externally manipulated. We identify stable modules as graph structures in an expanded network, which represents causal links between variable constraints. A stable module represents a system "decision point", or trap subspace. Using the expanded network, small stable modules can be composed sequentially to form larger stable modules that describe dynamics on the system level. Collections of large, mutually exclusive stable modules describe the system's repertoire of long-term behaviors. We implement this technique in a broad class of dynamical systems and illustrate its practical utility via examples and algorithmic analysis of two published biological network models. In the segment polarity gene network of Drosophila melanogaster, we obtain a state-space visualization that reproduces by novel means the four possible cell fates and predicts the outcome of cell transplant experiments. In the T-cell signaling network, we identify six signaling elements that determine the high-signal response and show that control of an element connected to them cannot disrupt this response.
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Affiliation(s)
- Jordan C Rozum
- Department of Physics, The Pennsylvania State University, University Park, PA, USA
| | - Réka Albert
- Department of Physics, The Pennsylvania State University, University Park, PA, USA
- Department of Biology, The Pennsylvania State University, University Park, PA, USA
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Mendes ND, Henriques R, Remy E, Carneiro J, Monteiro PT, Chaouiya C. Estimating Attractor Reachability in Asynchronous Logical Models. Front Physiol 2018; 9:1161. [PMID: 30245634 PMCID: PMC6137237 DOI: 10.3389/fphys.2018.01161] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2018] [Accepted: 08/02/2018] [Indexed: 12/12/2022] Open
Abstract
Logical models are well-suited to capture salient dynamical properties of regulatory networks. For networks controlling cell fate decisions, cell fates are associated with model attractors (stable states or cyclic attractors) whose identification and reachability properties are particularly relevant. While synchronous updates assume unlikely instantaneous or identical rates associated with component changes, the consideration of asynchronous updates is more realistic but, for large models, may hinder the analysis of the resulting non-deterministic concurrent dynamics. This complexity hampers the study of asymptotical behaviors, and most existing approaches suffer from efficiency bottlenecks, being generally unable to handle cyclical attractors and quantify attractor reachability. Here, we propose two algorithms providing probability estimates of attractor reachability in asynchronous dynamics. The first algorithm, named Firefront, exhaustively explores the state space from an initial state, and provides quasi-exact evaluations of the reachability probabilities of model attractors. The algorithm progresses in breadth, propagating the probabilities of each encountered state to its successors. Second, Avatar is an adapted Monte Carlo approach, better suited for models with large and intertwined transient and terminal cycles. Avatar iteratively explores the state space by randomly selecting trajectories and by using these random walks to estimate the likelihood of reaching an attractor. Unlike Monte Carlo simulations, Avatar is equipped to avoid getting trapped in transient cycles and to identify cyclic attractors. Firefront and Avatar are validated and compared to related methods, using as test cases logical models of synthetic and biological networks. Both algorithms are implemented as new functionalities of GINsim 3.0, a well-established software tool for logical modeling, providing executable GUI, Java API, and scripting facilities.
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Affiliation(s)
| | - Rui Henriques
- Department of Computer Science and Engineering, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal.,Instituto de Engenharia de Sistemas e Computadores Investigação e Desenvolvimento, Lisbon, Portugal
| | - Elisabeth Remy
- Aix Marseille University, CNRS, Centrale Marseille, I2M UMR 7373, Marseille, France
| | | | - Pedro T Monteiro
- Department of Computer Science and Engineering, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal.,Instituto de Engenharia de Sistemas e Computadores Investigação e Desenvolvimento, Lisbon, Portugal
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Hernansaiz-Ballesteros RD, Cardelli L, Csikász-Nagy A. Single molecules can operate as primitive biological sensors, switches and oscillators. BMC SYSTEMS BIOLOGY 2018; 12:70. [PMID: 29914480 PMCID: PMC6007071 DOI: 10.1186/s12918-018-0596-4] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/29/2018] [Accepted: 06/05/2018] [Indexed: 01/07/2023]
Abstract
Background Switch-like and oscillatory dynamical systems are widely observed in biology. We investigate the simplest biological switch that is composed of a single molecule that can be autocatalytically converted between two opposing activity forms. We test how this simple network can keep its switching behaviour under perturbations in the system. Results We show that this molecule can work as a robust bistable system, even for alterations in the reactions that drive the switching between various conformations. We propose that this single molecule system could work as a primitive biological sensor and show by steady state analysis of a mathematical model of the system that it could switch between possible states for changes in environmental signals. Particularly, we show that a single molecule phosphorylation-dephosphorylation switch could work as a nucleotide or energy sensor. We also notice that a given set of reductions in the reaction network can lead to the emergence of oscillatory behaviour. Conclusions We propose that evolution could have converted this switch into a single molecule oscillator, which could have been used as a primitive timekeeper. We discuss how the structure of the simplest known circadian clock regulatory system, found in cyanobacteria, resembles the proposed single molecule oscillator. Besides, we speculate if such minimal systems could have existed in an RNA world. Electronic supplementary material The online version of this article (10.1186/s12918-018-0596-4) contains supplementary material, which is available to authorized users.
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Affiliation(s)
- Rosa D Hernansaiz-Ballesteros
- Randall Centre for Cell and Molecular Biophysics and Institute for Mathematical and Molecular Biomedicine, King's College London, London, SE1 1UL, UK
| | - Luca Cardelli
- Microsoft Research, 21 Station Road, Cambridge, CB1 2FB, UK.,Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK
| | - Attila Csikász-Nagy
- Randall Centre for Cell and Molecular Biophysics and Institute for Mathematical and Molecular Biomedicine, King's College London, London, SE1 1UL, UK. .,Faculty of Information Technology and Bionics, Pázmány Péter Catholic University, Budapest, H-1083, Hungary.
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Figueiredo D, Martins MA, Chaves M. Applying differential dynamic logic to reconfigurable biological networks. Math Biosci 2017; 291:10-20. [PMID: 28610888 DOI: 10.1016/j.mbs.2017.05.012] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2017] [Accepted: 05/29/2017] [Indexed: 11/25/2022]
Abstract
Qualitative and quantitative modeling frameworks are widely used for analysis of biological regulatory networks, the former giving a preliminary overview of the system's global dynamics and the latter providing more detailed solutions. Another approach is to model biological regulatory networks as hybrid systems, i.e., systems which can display both continuous and discrete dynamic behaviors. Actually, the development of synthetic biology has shown that this is a suitable way to think about biological systems, which can often be constructed as networks with discrete controllers, and present hybrid behaviors. In this paper we discuss this approach as a special case of the reconfigurability paradigm, well studied in Computer Science (CS). In CS there are well developed computational tools to reason about hybrid systems. We argue that it is worth applying such tools in a biological context. One interesting tool is differential dynamic logic (dL), which has recently been developed by Platzer and applied to many case-studies. In this paper we discuss some simple examples of biological regulatory networks to illustrate how dL can be used as an alternative, or also as a complement to methods already used.
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Affiliation(s)
- Daniel Figueiredo
- CIDMA - Center for R&D Mathematics and Applications, Department of Mathematics, University of Aveiro, Portugal
| | - Manuel A Martins
- CIDMA - Center for R&D Mathematics and Applications, Department of Mathematics, University of Aveiro, Portugal.
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Saadatpour A, Albert R. A comparative study of qualitative and quantitative dynamic models of biological regulatory networks. ACTA ACUST UNITED AC 2016. [DOI: 10.1140/epjnbp/s40366-016-0031-y] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023]
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Stötzel C, Röblitz S, Siebert H. Complementing ODE-Based System Analysis Using Boolean Networks Derived from an Euler-Like Transformation. PLoS One 2015; 10:e0140954. [PMID: 26496494 PMCID: PMC4619740 DOI: 10.1371/journal.pone.0140954] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2015] [Accepted: 10/02/2015] [Indexed: 01/17/2023] Open
Abstract
In this paper, we present a systematic transition scheme for a large class of ordinary differential equations (ODEs) into Boolean networks. Our transition scheme can be applied to any system of ODEs whose right hand sides can be written as sums and products of monotone functions. It performs an Euler-like step which uses the signs of the right hand sides to obtain the Boolean update functions for every variable of the corresponding discrete model. The discrete model can, on one hand, be considered as another representation of the biological system or, alternatively, it can be used to further the analysis of the original ODE model. Since the generic transformation method does not guarantee any property conservation, a subsequent validation step is required. Depending on the purpose of the model this step can be based on experimental data or ODE simulations and characteristics. Analysis of the resulting Boolean model, both on its own and in comparison with the ODE model, then allows to investigate system properties not accessible in a purely continuous setting. The method is exemplarily applied to a previously published model of the bovine estrous cycle, which leads to new insights regarding the regulation among the components, and also indicates strongly that the system is tailored to generate stable oscillations.
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Affiliation(s)
- Claudia Stötzel
- Mathematics for Life and Materials Sciences, Zuse Institute Berlin, Berlin, Germany
| | - Susanna Röblitz
- Mathematics for Life and Materials Sciences, Zuse Institute Berlin, Berlin, Germany
- Dep. of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany
- * E-mail:
| | - Heike Siebert
- Dep. of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany
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Albert R, Thakar J. Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions. WILEY INTERDISCIPLINARY REVIEWS-SYSTEMS BIOLOGY AND MEDICINE 2015; 6:353-69. [PMID: 25269159 DOI: 10.1002/wsbm.1273] [Citation(s) in RCA: 69] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
The biomolecules inside or near cells form a complex interacting system. Cellular phenotypes and behaviors arise from the totality of interactions among the components of this system. A fruitful way of modeling interacting biomolecular systems is by network-based dynamic models that characterize each component by a state variable, and describe the change in the state variables due to the interactions in the system. Dynamic models can capture the stable state patterns of this interacting system and can connect them to different cell fates or behaviors. A Boolean or logic model characterizes each biomolecule by a binary state variable that relates the abundance of that molecule to a threshold abundance necessary for downstream processes. The regulation of this state variable is described in a parameter free manner, making Boolean modeling a practical choice for systems whose kinetic parameters have not been determined. Boolean models integrate the body of knowledge regarding the components and interactions of biomolecular systems, and capture the system's dynamic repertoire, for example the existence of multiple cell fates. These models were used for a variety of systems and led to important insights and predictions. Boolean models serve as an efficient exploratory model, a guide for follow-up experiments, and as a foundation for more quantitative models.
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Albert R, Collins JJ, Glass L. Introduction to focus issue: quantitative approaches to genetic networks. CHAOS (WOODBURY, N.Y.) 2013; 23:025001. [PMID: 23822498 DOI: 10.1063/1.4810923] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
All cells of living organisms contain similar genetic instructions encoded in the organism's DNA. In any particular cell, the control of the expression of each different gene is regulated, in part, by binding of molecular complexes to specific regions of the DNA. The molecular complexes are composed of protein molecules, called transcription factors, combined with various other molecules such as hormones and drugs. Since transcription factors are coded by genes, cellular function is partially determined by genetic networks. Recent research is making large strides to understand both the structure and the function of these networks. Further, the emerging discipline of synthetic biology is engineering novel gene circuits with specific dynamic properties to advance both basic science and potential practical applications. Although there is not yet a universally accepted mathematical framework for studying the properties of genetic networks, the strong analogies between the activation and inhibition of gene expression and electric circuits suggest frameworks based on logical switching circuits. This focus issue provides a selection of papers reflecting current research directions in the quantitative analysis of genetic networks. The work extends from molecular models for the binding of proteins, to realistic detailed models of cellular metabolism. Between these extremes are simplified models in which genetic dynamics are modeled using classical methods of systems engineering, Boolean switching networks, differential equations that are continuous analogues of Boolean switching networks, and differential equations in which control is based on power law functions. The mathematical techniques are applied to study: (i) naturally occurring gene networks in living organisms including: cyanobacteria, Mycoplasma genitalium, fruit flies, immune cells in mammals; (ii) synthetic gene circuits in Escherichia coli and yeast; and (iii) electronic circuits modeling genetic networks using field-programmable gate arrays. Mathematical analyses will be essential for understanding naturally occurring genetic networks in diverse organisms and for providing a foundation for the improved development of synthetic genetic networks.
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Affiliation(s)
- Réka Albert
- Department of Physics, Penn State University, University Park, Pennsylvania 16802, USA
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