Attractor Stability in Finite Asynchronous Biological System Models

We present mathematical techniques for exhaustive studies of long-term dynamics of asynchronous biological system models. Specifically, we extend the notion of κ -equivalence developed for graph dynamical systems to support systematic analysis of all possible attractor configurations that can be gen...

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Veröffentlicht in:Bulletin of mathematical biology 2019-05, Vol.81 (5), p.1442-1460
Hauptverfasser: Mortveit, Henning S., Pederson, Ryan D.
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description We present mathematical techniques for exhaustive studies of long-term dynamics of asynchronous biological system models. Specifically, we extend the notion of κ -equivalence developed for graph dynamical systems to support systematic analysis of all possible attractor configurations that can be generated when varying the asynchronous update order (Macauley and Mortveit in Nonlinearity 22(2):421, 2009 ). We extend earlier work by Veliz-Cuba and Stigler (J Comput Biol 18(6):783–794, 2011 ), Goles et al. (Bull Math Biol 75(6):939–966, 2013 ), and others by comparing long-term dynamics up to topological conjugation : rather than comparing the exact states and their transitions on attractors, we only compare the attractor structures. In general, obtaining this information is computationally intractable. Here, we adapt and apply combinatorial theory for dynamical systems from Macauley and Mortveit (Proc Am Math Soc 136(12):4157–4165, 2008 . https://doi.org/10.1090/S0002-9939-09-09884-0 ; 2009 ; Electron J Comb 18:197, 2011a ; Discret Contin Dyn Syst 4(6):1533–1541, 2011b . https://doi.org/10.3934/dcdss.2011.4.1533 ; Theor Comput Sci 504:26–37, 2013 . https://doi.org/10.1016/j.tcs.2012.09.015 ; in: Isokawa T, Imai K, Matsui N, Peper F, Umeo H (eds) Cellular automata and discrete complex systems, 2014 . https://doi.org/10.1007/978-3-319-18812-6_6 ) to develop computational methods that greatly reduce this computational cost. We give a detailed algorithm and apply it to (i) the  lac operon model for  Escherichia coli proposed by Veliz-Cuba and Stigler ( 2011 ), and (ii) the regulatory network involved in the control of the cell cycle and cell differentiation in the  Caenorhabditis elegans vulva precursor cells proposed by Weinstein et al. (BMC Bioinform 16(1):1, 2015 ). In both cases, we uncover all possible limit cycle structures for these networks under sequential updates. Specifically, for the lac operon model, rather than examining all  10 ! > 3.6 · 10 6 sequential update orders, we demonstrate that it is sufficient to consider 344 representative update orders, and, more notably, that these 344 representatives give rise to 4 distinct attractor structures . A similar analysis performed for the C. elegans model demonstrates that it has precisely 125 distinct attractor structures. We conclude with observations on the variety and distribution of the models’ attractor structures and use the results to discuss their robustness.
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Specifically, we extend the notion of κ -equivalence developed for graph dynamical systems to support systematic analysis of all possible attractor configurations that can be generated when varying the asynchronous update order (Macauley and Mortveit in Nonlinearity 22(2):421, 2009 ). We extend earlier work by Veliz-Cuba and Stigler (J Comput Biol 18(6):783–794, 2011 ), Goles et al. (Bull Math Biol 75(6):939–966, 2013 ), and others by comparing long-term dynamics up to topological conjugation : rather than comparing the exact states and their transitions on attractors, we only compare the attractor structures. In general, obtaining this information is computationally intractable. Here, we adapt and apply combinatorial theory for dynamical systems from Macauley and Mortveit (Proc Am Math Soc 136(12):4157–4165, 2008 . https://doi.org/10.1090/S0002-9939-09-09884-0 ; 2009 ; Electron J Comb 18:197, 2011a ; Discret Contin Dyn Syst 4(6):1533–1541, 2011b . https://doi.org/10.3934/dcdss.2011.4.1533 ; Theor Comput Sci 504:26–37, 2013 . https://doi.org/10.1016/j.tcs.2012.09.015 ; in: Isokawa T, Imai K, Matsui N, Peper F, Umeo H (eds) Cellular automata and discrete complex systems, 2014 . https://doi.org/10.1007/978-3-319-18812-6_6 ) to develop computational methods that greatly reduce this computational cost. We give a detailed algorithm and apply it to (i) the  lac operon model for  Escherichia coli proposed by Veliz-Cuba and Stigler ( 2011 ), and (ii) the regulatory network involved in the control of the cell cycle and cell differentiation in the  Caenorhabditis elegans vulva precursor cells proposed by Weinstein et al. (BMC Bioinform 16(1):1, 2015 ). In both cases, we uncover all possible limit cycle structures for these networks under sequential updates. Specifically, for the lac operon model, rather than examining all  10 ! &gt; 3.6 · 10 6 sequential update orders, we demonstrate that it is sufficient to consider 344 representative update orders, and, more notably, that these 344 representatives give rise to 4 distinct attractor structures . A similar analysis performed for the C. elegans model demonstrates that it has precisely 125 distinct attractor structures. 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Specifically, we extend the notion of κ -equivalence developed for graph dynamical systems to support systematic analysis of all possible attractor configurations that can be generated when varying the asynchronous update order (Macauley and Mortveit in Nonlinearity 22(2):421, 2009 ). We extend earlier work by Veliz-Cuba and Stigler (J Comput Biol 18(6):783–794, 2011 ), Goles et al. (Bull Math Biol 75(6):939–966, 2013 ), and others by comparing long-term dynamics up to topological conjugation : rather than comparing the exact states and their transitions on attractors, we only compare the attractor structures. In general, obtaining this information is computationally intractable. Here, we adapt and apply combinatorial theory for dynamical systems from Macauley and Mortveit (Proc Am Math Soc 136(12):4157–4165, 2008 . https://doi.org/10.1090/S0002-9939-09-09884-0 ; 2009 ; Electron J Comb 18:197, 2011a ; Discret Contin Dyn Syst 4(6):1533–1541, 2011b . https://doi.org/10.3934/dcdss.2011.4.1533 ; Theor Comput Sci 504:26–37, 2013 . https://doi.org/10.1016/j.tcs.2012.09.015 ; in: Isokawa T, Imai K, Matsui N, Peper F, Umeo H (eds) Cellular automata and discrete complex systems, 2014 . https://doi.org/10.1007/978-3-319-18812-6_6 ) to develop computational methods that greatly reduce this computational cost. We give a detailed algorithm and apply it to (i) the  lac operon model for  Escherichia coli proposed by Veliz-Cuba and Stigler ( 2011 ), and (ii) the regulatory network involved in the control of the cell cycle and cell differentiation in the  Caenorhabditis elegans vulva precursor cells proposed by Weinstein et al. (BMC Bioinform 16(1):1, 2015 ). In both cases, we uncover all possible limit cycle structures for these networks under sequential updates. Specifically, for the lac operon model, rather than examining all  10 ! &gt; 3.6 · 10 6 sequential update orders, we demonstrate that it is sufficient to consider 344 representative update orders, and, more notably, that these 344 representatives give rise to 4 distinct attractor structures . A similar analysis performed for the C. elegans model demonstrates that it has precisely 125 distinct attractor structures. 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Specifically, we extend the notion of κ -equivalence developed for graph dynamical systems to support systematic analysis of all possible attractor configurations that can be generated when varying the asynchronous update order (Macauley and Mortveit in Nonlinearity 22(2):421, 2009 ). We extend earlier work by Veliz-Cuba and Stigler (J Comput Biol 18(6):783–794, 2011 ), Goles et al. (Bull Math Biol 75(6):939–966, 2013 ), and others by comparing long-term dynamics up to topological conjugation : rather than comparing the exact states and their transitions on attractors, we only compare the attractor structures. In general, obtaining this information is computationally intractable. Here, we adapt and apply combinatorial theory for dynamical systems from Macauley and Mortveit (Proc Am Math Soc 136(12):4157–4165, 2008 . https://doi.org/10.1090/S0002-9939-09-09884-0 ; 2009 ; Electron J Comb 18:197, 2011a ; Discret Contin Dyn Syst 4(6):1533–1541, 2011b . https://doi.org/10.3934/dcdss.2011.4.1533 ; Theor Comput Sci 504:26–37, 2013 . https://doi.org/10.1016/j.tcs.2012.09.015 ; in: Isokawa T, Imai K, Matsui N, Peper F, Umeo H (eds) Cellular automata and discrete complex systems, 2014 . https://doi.org/10.1007/978-3-319-18812-6_6 ) to develop computational methods that greatly reduce this computational cost. We give a detailed algorithm and apply it to (i) the  lac operon model for  Escherichia coli proposed by Veliz-Cuba and Stigler ( 2011 ), and (ii) the regulatory network involved in the control of the cell cycle and cell differentiation in the  Caenorhabditis elegans vulva precursor cells proposed by Weinstein et al. (BMC Bioinform 16(1):1, 2015 ). In both cases, we uncover all possible limit cycle structures for these networks under sequential updates. Specifically, for the lac operon model, rather than examining all  10 ! &gt; 3.6 · 10 6 sequential update orders, we demonstrate that it is sufficient to consider 344 representative update orders, and, more notably, that these 344 representatives give rise to 4 distinct attractor structures . A similar analysis performed for the C. elegans model demonstrates that it has precisely 125 distinct attractor structures. We conclude with observations on the variety and distribution of the models’ attractor structures and use the results to discuss their robustness.</abstract><cop>New York</cop><pub>Springer US</pub><pmid>30656504</pmid><doi>10.1007/s11538-018-00565-x</doi><tpages>19</tpages></addata></record>
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subjects Algorithms
Animals
Attractors (mathematics)
Automata theory
Biological models (mathematics)
Caenorhabditis elegans - cytology
Cell Biology
Cell cycle
Cell Cycle Checkpoints
Cell Differentiation
Cellular automata
Combinatorial analysis
Complex systems
Computer applications
Conjugation
Differentiation (biology)
Dynamical systems
E coli
Escherichia coli - genetics
Escherichia coli - metabolism
Lac Operon
Lactose operon
Life Sciences
Mathematical and Computational Biology
Mathematical Concepts
Mathematical models
Mathematics
Mathematics and Statistics
Models, Biological
Nonlinear systems
Robustness (mathematics)
Software
Systems Biology
Vulva
title Attractor Stability in Finite Asynchronous Biological System Models
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