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
References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[7]

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

[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: , ().   Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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).   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[42]

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

[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.  Google Scholar

[44]

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

[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.   Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[7]

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

[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: , ().   Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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).   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[42]

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

[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.  Google Scholar

[44]

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

[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.   Google Scholar

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