December  2016, 21(10): 3391-3405. doi: 10.3934/dcdsb.2016103

Computational methods for asynchronous basins

1. 

Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, 724 SW Harrison Street, Portland, OR 97201, United States

Received  December 2015 Revised  March 2016 Published  November 2016

For a Boolean network we consider asynchronous updates and define the exclusive asynchronous basin of attraction for any steady state or cyclic attractor. An algorithm based on commutative algebra is presented to compute the exclusive basin. Finally its use for targeting desirable attractors by selective intervention on network nodes is illustrated with two examples, one cell signalling network and one sensor network measuring human mobility.
Citation: Ian H. Dinwoodie. Computational methods for asynchronous basins. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3391-3405. doi: 10.3934/dcdsb.2016103
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[1]

J. Abbott and A. M. Bigatti, CoCoALib: A C++ library for doing Computations in Commutative Algebra, 2014., Available from: , ().

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D. Austin, R. M. Cross, T. Hayes and J. Kaye, Regularity and Predictability of Human Mobility in Personal Space,, PLoS One, 9 (2014). doi: 10.1371/journal.pone.0090256.

[3]

D. Austin and I. H. Dinwoodie, Monomials and Basin Cylinders for Network Dynamics,, SIAM J. Appl. Dyn. Syst., 14 (2015), 25. doi: 10.1137/140975929.

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M. Chaves, R. Albert and E. D. Sontag, Robustness and fragility of Boolean models for genetic regulatory networks,, Jour. Theoret. Biol., 235 (2005), 431. doi: 10.1016/j.jtbi.2005.01.023.

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D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, $2^{nd}$ edition,, Springer, (1997).

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W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 - A Computer Algebra System for Polynomial Computations, 2015., Available from: , ().

[9]

G. V. De Ferrari and N. C. Inestrosa, Wnt signaling function in Alzheimer's disease,, Brain Res. Rev., 33 (2000), 1.

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I. H. Dinwoodie, Conditional tests on basins of attraction with finite fields,, Methodol. Comput. Appl. Probab., 16 (2014), 161. doi: 10.1007/s11009-012-9304-9.

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I. H. Dinwoodie, Vanishing configurations in network dynamics with asynchronous updates,, Proc. Amer. Math. Soc., 142 (2014), 2991. doi: 10.1090/S0002-9939-2014-12044-2.

[12]

I. H. Dinwoodie, Polynomials for classification trees and applications,, Stat. Methods Appt., 19 (2009), 171. doi: 10.1007/s10260-009-0123-2.

[13]

I. H. Dinwoodie and K. Pandya, Exact tests for singular network data,, Ann. Inst. Statist. Math. 67 (2015), 67 (2015), 687. doi: 10.1007/s10463-014-0472-y.

[14]

D. R. Grayson and M. E. Stillman, Macaulay2, A Software System for Research in Algebraic Geometry, 2014., Available from: , ().

[15]

T. Handorf and E. Klipp, Modeling mechanistic biological networks: An advanced Boolean approach,, Bioinformatics, 28 (2012), 557.

[16]

T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, $2^{nd}$ edition,, Springer, (2009). doi: 10.1007/978-0-387-21606-5.

[17]

T. L. Hayes, T. Riley, M. Pavel and J. A. Kaye, Estimation of rest-activity patterns using motion sensors,, Conf. Proc. IEEE Eng. Med. Biol. Soc., 2010 (2010), 2147. doi: 10.1109/IEMBS.2010.5628022.

[18]

M. Hermes, G. Eichoff and O. Garaschuk, Intracellular calcium signalling in Alzheimer's disease,, J. Cell. Mol. Med., 14 (2009), 30. doi: 10.1111/j.1582-4934.2009.00976.x.

[19]

F. Hinkelmann, M. Brandon, B. Guang, R. McNeill, G. Blekherman, A. Veliz-Cuba and R. Laubenbacher, ADAM: Analysis of discrete models of biological systems using computer algebra,, BMC Bioinformatics, 12 (2011). doi: 10.1186/1471-2105-12-295.

[20]

F. Hinkelmann, D. Murrugarra, A. S. Jarrah and R. Laubenbacher, A mathematical framework for agent based models of complex biological networks,, Bull. Math. Biol. 73 (2011), 73 (2011), 1583. doi: 10.1007/s11538-010-9582-8.

[21]

J. A. Kaye, S. A. Maxwell, N. Mattek, T. L. Hayes, H. Dodge, M. Pavel, H. B. Jimison, K. Wild, L. Boise and T. A. Zitzelberger, Intelligent systems for assessing aging changes: Home-based, unobtrusive, and continuous assessment of ageing,, J. Gerontol. B: Psychol. Sci. and Soc. Sci., 66B (2011). doi: 10.1093/geronb/gbq095.

[22]

S. Klamt, J. Saez-Rodriquez, J. A. Lindquist, L. Simeoni and E. D. Gilles, A methodology for the structural and functional analysis of signalling and regulatory networks,, BMC Bioinformatics, 7 (2006), 1471.

[23]

M. Kreuzer and L. Robbiano, Computational Commutative Algebra I,, Springer, (2000). doi: 10.1007/978-3-540-70628-1.

[24]

R. Laubenbacher and B. Sturmfels, Computer Algebra in Systems Biology,, Amer. Math. Monthly, 116 (2009), 882. doi: 10.4169/000298909X477005.

[25]

R. K. Layek, A. Datta and E. R. Dougherty, From biological pathways to regulatory networks,, Molecular BioSystems, 7 (2011), 843. doi: 10.1109/CDC.2010.5716936.

[26]

R. K. Layek, A. Datta, M. Bittner and E. R. Dougherty, Cancer therapy design based on pathway logic,, Bioinformatics, 27 (2011), 548. doi: 10.1093/bioinformatics/btq703.

[27]

A. Liaw and M. Wiener, Classification and Regression by randomForest,, R News, 2 (2002), 18.

[28]

T. Lu, L. Aron, J. Zullo, Y. Pan, H. Kim, Y. Chen, T.-H. Yang, H.-M. Kim, D. Drake, X. S. Liu, D. A. Bennett, M. P. Colaiácovo and B. A. Yankner, REST and stress resistance in ageing and Alzheimer's disease,, Nature, 507 (2014), 448. doi: 10.1038/nature13163.

[29]

M. K. Morris, J. Saez-Rodriguez, P. K. Sorger and D. A. Lauffenburger, Logic-based models for the analysis of cell signaling networks,, Biochemistry, 49 (2010), 3216. doi: 10.1021/bi902202q.

[30]

D. Murrugarra, A. Veliz-Cuba, B. Aguilar, S. Arat and R. Laubenbacher, Modeling stochasticity and variability in gene regulatory networks,, EURASIP J. Bioinform. and Syst. Biol., 2012 (2012). doi: 10.1186/1687-4153-2012-5.

[31]

C. Müssel, M. Hopfensitz and H. A. Kestler, BoolNet - an R package for generation, reconstruction and analysis of Boolean networks,, Bioinformatics, 26 (2010), 1378.

[32]

J. Petersen, D. Austin, J. Kaye, M. Pavel and T. Hayes, Unobtrusive in-home detection of time spent out-of-home with applications to loneliness and physical activity,, IEEE J. Biomed. Health Inform., 18 (2014), 1590. doi: 10.1109/JBHI.2013.2294276.

[33]

R. C. Petersen, Mild cognitive impairment as a diagnostic entity,, J. Intern. Med., 256 (2004), 183. doi: 10.1111/j.1365-2796.2004.01388.x.

[34]

G. Pistone, E. Riccomagno and H. Wynn, Algebraic Statistics: Computational Commutative Algebra in Statistics,, Chapman and Hall, (2001).

[35]

R. Poltz and M. Naumann, Dynamics of p53 and NF-$\kappa$B regulation in response to DNA damage and identification of target proteins suitable for therapeutic intervention,, BMC Syst. Biol., 6 (2012).

[36]

A. Saadatpour, R-S. Wang, A. Liao, X. Liu, T. P. Loughran, I. Albert and R. Albert, Dynamical and structural analysis of a T cell survival network identifies novel candidate therapeutic targets for large granula lymphocyte leukemia,, PLoS Comp. Biol. 7 (2011), 7 (2011).

[37]

A. Saadatpour, I. Albert and R. Albert, Attractor analysis of asynchronous Boolean models of signal transduction networks,, J. Theor. Biol., 266 (2010), 641. doi: 10.1016/j.jtbi.2010.07.022.

[38]

R. Schlatter, K. Schmich, I. A. Vizcarra, P. Scheurich, T. Sauter, C. Borner, M. Ederer, I. Merfort and O. Sawodny, ON/OFF and Beyond - A Boolean Model of Apoptosis,, PLoS Comput. Biol., 5 (2009). doi: 10.1371/journal.pcbi.1000595.

[39]

I. Shmulevich, E. R. Dougherty, S. Kim and W. Zhang, Probabilistic Boolean Networks: A rule-based uncertainty model for gene regulatory networks,, Bioinformatics, 18 (2002), 261. doi: 10.1093/bioinformatics/18.2.261.

[40]

B. Stigler, Polynomial dynamical systems in systems biology,, AMS 2006 Proceedings of Symposia in Applied Mathematics, 64 (2007), 53. doi: 10.1090/psapm/064/2359649.

[41]

T. Therneau, B. Atkinson and B. Ripley, Rpart: Recursive Partitioning and Regression Trees, 2015., Available from: , ().

[42]

R. Thomas, Boolean formalization of genetic control circuits,, J. Theoret. Biol., 42 (1973), 563.

[43]

A. Veliz-Cuba, An algebraic approach to reverse engineering finite dynamical systems arising from biology,, SIAM Jour. Appl. Dyn. Systems, 11 (2012), 31. doi: 10.1137/110828794.

[44]

A. Wuensche, Complex and Chaotic Dynamics, Basins of Attraction, and Memory in Discrete Networks,, Acta Physica Polonica B, 3 (2010), 463.

[45]

R. Zhang, M. V. Shah, J. Yang, S. B. Nyland, X. Liu, J. Yun, R. Albert and T. P. Loughran, Network model of survival signaling in large granular lymphocyte leukemia,, Proc. Natl. Acad. Sci. USA, 105 (2008), 16308.

show all references

References:
[1]

J. Abbott and A. M. Bigatti, CoCoALib: A C++ library for doing Computations in Commutative Algebra, 2014., Available from: , ().

[2]

D. Austin, R. M. Cross, T. Hayes and J. Kaye, Regularity and Predictability of Human Mobility in Personal Space,, PLoS One, 9 (2014). doi: 10.1371/journal.pone.0090256.

[3]

D. Austin and I. H. Dinwoodie, Monomials and Basin Cylinders for Network Dynamics,, SIAM J. Appl. Dyn. Syst., 14 (2015), 25. doi: 10.1137/140975929.

[4]

D. E. Bredesen, Reversal of cognitive decline: A novel therapeutic program,, Aging, 6 (2014), 707. doi: 10.18632/aging.100690.

[5]

T. Buracchio, H. Dodge, D. Howieson, D. Wasserman and J. Kaye, The trajectory of gait speed preceding MCI,, Arch. Neurol., 67 (2010), 980.

[6]

M. Chaves, R. Albert and E. D. Sontag, Robustness and fragility of Boolean models for genetic regulatory networks,, Jour. Theoret. Biol., 235 (2005), 431. doi: 10.1016/j.jtbi.2005.01.023.

[7]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, $2^{nd}$ edition,, Springer, (1997).

[8]

W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 - A Computer Algebra System for Polynomial Computations, 2015., Available from: , ().

[9]

G. V. De Ferrari and N. C. Inestrosa, Wnt signaling function in Alzheimer's disease,, Brain Res. Rev., 33 (2000), 1.

[10]

I. H. Dinwoodie, Conditional tests on basins of attraction with finite fields,, Methodol. Comput. Appl. Probab., 16 (2014), 161. doi: 10.1007/s11009-012-9304-9.

[11]

I. H. Dinwoodie, Vanishing configurations in network dynamics with asynchronous updates,, Proc. Amer. Math. Soc., 142 (2014), 2991. doi: 10.1090/S0002-9939-2014-12044-2.

[12]

I. H. Dinwoodie, Polynomials for classification trees and applications,, Stat. Methods Appt., 19 (2009), 171. doi: 10.1007/s10260-009-0123-2.

[13]

I. H. Dinwoodie and K. Pandya, Exact tests for singular network data,, Ann. Inst. Statist. Math. 67 (2015), 67 (2015), 687. doi: 10.1007/s10463-014-0472-y.

[14]

D. R. Grayson and M. E. Stillman, Macaulay2, A Software System for Research in Algebraic Geometry, 2014., Available from: , ().

[15]

T. Handorf and E. Klipp, Modeling mechanistic biological networks: An advanced Boolean approach,, Bioinformatics, 28 (2012), 557.

[16]

T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, $2^{nd}$ edition,, Springer, (2009). doi: 10.1007/978-0-387-21606-5.

[17]

T. L. Hayes, T. Riley, M. Pavel and J. A. Kaye, Estimation of rest-activity patterns using motion sensors,, Conf. Proc. IEEE Eng. Med. Biol. Soc., 2010 (2010), 2147. doi: 10.1109/IEMBS.2010.5628022.

[18]

M. Hermes, G. Eichoff and O. Garaschuk, Intracellular calcium signalling in Alzheimer's disease,, J. Cell. Mol. Med., 14 (2009), 30. doi: 10.1111/j.1582-4934.2009.00976.x.

[19]

F. Hinkelmann, M. Brandon, B. Guang, R. McNeill, G. Blekherman, A. Veliz-Cuba and R. Laubenbacher, ADAM: Analysis of discrete models of biological systems using computer algebra,, BMC Bioinformatics, 12 (2011). doi: 10.1186/1471-2105-12-295.

[20]

F. Hinkelmann, D. Murrugarra, A. S. Jarrah and R. Laubenbacher, A mathematical framework for agent based models of complex biological networks,, Bull. Math. Biol. 73 (2011), 73 (2011), 1583. doi: 10.1007/s11538-010-9582-8.

[21]

J. A. Kaye, S. A. Maxwell, N. Mattek, T. L. Hayes, H. Dodge, M. Pavel, H. B. Jimison, K. Wild, L. Boise and T. A. Zitzelberger, Intelligent systems for assessing aging changes: Home-based, unobtrusive, and continuous assessment of ageing,, J. Gerontol. B: Psychol. Sci. and Soc. Sci., 66B (2011). doi: 10.1093/geronb/gbq095.

[22]

S. Klamt, J. Saez-Rodriquez, J. A. Lindquist, L. Simeoni and E. D. Gilles, A methodology for the structural and functional analysis of signalling and regulatory networks,, BMC Bioinformatics, 7 (2006), 1471.

[23]

M. Kreuzer and L. Robbiano, Computational Commutative Algebra I,, Springer, (2000). doi: 10.1007/978-3-540-70628-1.

[24]

R. Laubenbacher and B. Sturmfels, Computer Algebra in Systems Biology,, Amer. Math. Monthly, 116 (2009), 882. doi: 10.4169/000298909X477005.

[25]

R. K. Layek, A. Datta and E. R. Dougherty, From biological pathways to regulatory networks,, Molecular BioSystems, 7 (2011), 843. doi: 10.1109/CDC.2010.5716936.

[26]

R. K. Layek, A. Datta, M. Bittner and E. R. Dougherty, Cancer therapy design based on pathway logic,, Bioinformatics, 27 (2011), 548. doi: 10.1093/bioinformatics/btq703.

[27]

A. Liaw and M. Wiener, Classification and Regression by randomForest,, R News, 2 (2002), 18.

[28]

T. Lu, L. Aron, J. Zullo, Y. Pan, H. Kim, Y. Chen, T.-H. Yang, H.-M. Kim, D. Drake, X. S. Liu, D. A. Bennett, M. P. Colaiácovo and B. A. Yankner, REST and stress resistance in ageing and Alzheimer's disease,, Nature, 507 (2014), 448. doi: 10.1038/nature13163.

[29]

M. K. Morris, J. Saez-Rodriguez, P. K. Sorger and D. A. Lauffenburger, Logic-based models for the analysis of cell signaling networks,, Biochemistry, 49 (2010), 3216. doi: 10.1021/bi902202q.

[30]

D. Murrugarra, A. Veliz-Cuba, B. Aguilar, S. Arat and R. Laubenbacher, Modeling stochasticity and variability in gene regulatory networks,, EURASIP J. Bioinform. and Syst. Biol., 2012 (2012). doi: 10.1186/1687-4153-2012-5.

[31]

C. Müssel, M. Hopfensitz and H. A. Kestler, BoolNet - an R package for generation, reconstruction and analysis of Boolean networks,, Bioinformatics, 26 (2010), 1378.

[32]

J. Petersen, D. Austin, J. Kaye, M. Pavel and T. Hayes, Unobtrusive in-home detection of time spent out-of-home with applications to loneliness and physical activity,, IEEE J. Biomed. Health Inform., 18 (2014), 1590. doi: 10.1109/JBHI.2013.2294276.

[33]

R. C. Petersen, Mild cognitive impairment as a diagnostic entity,, J. Intern. Med., 256 (2004), 183. doi: 10.1111/j.1365-2796.2004.01388.x.

[34]

G. Pistone, E. Riccomagno and H. Wynn, Algebraic Statistics: Computational Commutative Algebra in Statistics,, Chapman and Hall, (2001).

[35]

R. Poltz and M. Naumann, Dynamics of p53 and NF-$\kappa$B regulation in response to DNA damage and identification of target proteins suitable for therapeutic intervention,, BMC Syst. Biol., 6 (2012).

[36]

A. Saadatpour, R-S. Wang, A. Liao, X. Liu, T. P. Loughran, I. Albert and R. Albert, Dynamical and structural analysis of a T cell survival network identifies novel candidate therapeutic targets for large granula lymphocyte leukemia,, PLoS Comp. Biol. 7 (2011), 7 (2011).

[37]

A. Saadatpour, I. Albert and R. Albert, Attractor analysis of asynchronous Boolean models of signal transduction networks,, J. Theor. Biol., 266 (2010), 641. doi: 10.1016/j.jtbi.2010.07.022.

[38]

R. Schlatter, K. Schmich, I. A. Vizcarra, P. Scheurich, T. Sauter, C. Borner, M. Ederer, I. Merfort and O. Sawodny, ON/OFF and Beyond - A Boolean Model of Apoptosis,, PLoS Comput. Biol., 5 (2009). doi: 10.1371/journal.pcbi.1000595.

[39]

I. Shmulevich, E. R. Dougherty, S. Kim and W. Zhang, Probabilistic Boolean Networks: A rule-based uncertainty model for gene regulatory networks,, Bioinformatics, 18 (2002), 261. doi: 10.1093/bioinformatics/18.2.261.

[40]

B. Stigler, Polynomial dynamical systems in systems biology,, AMS 2006 Proceedings of Symposia in Applied Mathematics, 64 (2007), 53. doi: 10.1090/psapm/064/2359649.

[41]

T. Therneau, B. Atkinson and B. Ripley, Rpart: Recursive Partitioning and Regression Trees, 2015., Available from: , ().

[42]

R. Thomas, Boolean formalization of genetic control circuits,, J. Theoret. Biol., 42 (1973), 563.

[43]

A. Veliz-Cuba, An algebraic approach to reverse engineering finite dynamical systems arising from biology,, SIAM Jour. Appl. Dyn. Systems, 11 (2012), 31. doi: 10.1137/110828794.

[44]

A. Wuensche, Complex and Chaotic Dynamics, Basins of Attraction, and Memory in Discrete Networks,, Acta Physica Polonica B, 3 (2010), 463.

[45]

R. Zhang, M. V. Shah, J. Yang, S. B. Nyland, X. Liu, J. Yun, R. Albert and T. P. Loughran, Network model of survival signaling in large granular lymphocyte leukemia,, Proc. Natl. Acad. Sci. USA, 105 (2008), 16308.

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