# Mathematical Analysis of Biological Interaction Networks (17w5099)

Arriving in Banff, Alberta Sunday, June 4 and departing Friday June 9, 2017

## Organizers

Casian Pantea (West Virginia University)

Anne Shiu (Texas A&M University)

Lea Popovic (Concordia University)

David Anderson (University of Wisconsin-Madison)

Matthew Johnston (San Jose State University)

## Objectives

The goal of this workshop is to bring together researchers with expertise in a variety of mathematical areas to address major open questions of biological interaction modeling. The confirmed and expected participants include researchers working in dynamical systems and control theory, probability and stochastic processes, computational algebraic geometry, optimization and computation, and systems biology. The focus of the workshop will be on discussion and collaboration among these areas of expertise to solve important problems of common interest.

The workshop will focus on the following problems:

(1) Clarifying the relationship between continuous-state differential equation models and discrete-state stochastic models. Although the stochastic model is known, in an appropriate scaling limit, to converge to the corresponding deterministic solution, the result holds on compact time intervals only [20]. In fact, the two frameworks often produce vastly different dynamical predictions on long time-frames. In particular, the discrete models may predict an "extinction event'' not present in the deterministic model. Recent work has identified a broad classes of biochemical reaction mechanisms for which this disparity necessarily occurs [2,26]. This workshop will further develop these results with particular focus on recently established connections with Petri Net Theory [3,6,24].

(2) Applications of network-based methods to algebraic systems biology. Since the early 2000s, there has been significant interest in the algebraic structure of the steady states of mass-action systems, which is the most common modeling framework in systems biology. Network-based approaches have been found to be a useful alternative to computational approaches such as Groebner bases, which are often computationally intractable for realistic biochemical systems. Much work has focused recently on the topics of multiple equilibria and switch-like behavior, and led to (and benefited from): (i) in-depth investigations of systems with toric steady states [8,25]; (ii) results on the existence of steady-state invariants, with application to in-vivo mechanism validation [13,14,16,19,21]; and (iii) development of algebraic conditions for injectivity of vector fields arising from networks [4,12,22]. One focus of the workshop is on the connection between network structure and algebraic ideals in systems biology.

(3) Computational implementation of results. Significant work has been conducted in recent years in building algorithms and computational packages which can be used to implement the latest results in biochemical reaction modeling. Some notable examples are: (i) CoNtRol, an open-source browser-based program which has modules for verifying a network's capacity for multistationarity, steady-state stability, and persistence [11,17]; (ii) Bertini, a numerical solver for roots of systems of polynomial equations using continuation methods [5,15]; and (iii) CRNreals, which uses optimization methods to identify dynamically equivalent network representations of reaction networks [18,27]. As the understanding of biochemical reaction models grows, however, there is increased need for implementation which is comprehensive, up-to-date, and accessible to the wider biochemical modeling community. Therefore, discussion of directions for algorithm development and implementation will be a focus of the workshop.

(4) Further discussion and unification of dynamical approaches. In recent years, several novel approaches to studying the dynamical properties of chemical reaction systems have been developed, including: (i) a reformulation of monotone systems theory into reaction-coordinates [9,10]; (ii) the introduction of piecewise linear in rates (PWLR) Lyapunov functions [1]; (iii) the development of toric differential inclusion approaches to persistence and global stability [7]; and (iv) the formalization of generalized mass action systems theory [23]. A focus of the workshop will be to explore connections among these distinct approaches, and explore new application areas within biochemical modeling.

The format will feature limited overview lectures from leading experts in the relevant fields in the mornings, with significant time reserved for discussion and break-out sessions in the afternoons. The core research community has found this model to be very conducive to establishing long-lasting collaborations and new approaches to classical problems. Similar formats have been adapted at recent related workshops in 2012 (Castle Dagstuhl, Germany), 2013 (American Institute of Mathematics, USA), 2014 (University of Portsmouth, UK), and 2015 (University of Copenhagen, Denmark).

The workshop will actively promote participation from young researchers and women. We note that three of the organizers are pre-tenure and two are women. We anticipate broad international representation, and participation from a number of postdoctoral researchers and graduate students.

References:

[1] M.A. Al-Radhawi and D. Angeli. New approach to the stability of chemical reaction networks: Piecewise linear in rates Lyapunov functions. 2015. DOI 10.1109/TAC.2015.2427691.

[2] D.F. Anderson, G. Enciso, and M.D. Johnston. Stochastic analysis of chemical reaction networks with absolute concentration robustness. J. R. Soc. Interface, 11(93):20130943, 2014.

[3] D. Angeli, P. de Leenheer, and E. Sontag. A petri net approach to the study of persistence in chemical reaction networks. Math. Biosci., 210(2):598--618, 2007.

[4] M. Banaji and C. Pantea. Some results on injectivity and multistationarity in chemical reaction networks, submitted, available on arXiv, 2015.

[5] D.J. Bates, J.D. Hauenstein, A.J. Sommese, and C.W. Wampler. Bertini: Software for Numerical Algebraic Geometry. Available at bertini.nd.edu with permanent doi: dx.doi.org/10.7274/R0H41PB5

[6] R. Brijder. Dominant and T-Invariants for Petri Nets and Chemical Reaction Networks. Lecture Notes in Comput. Sci., 9211:1--15, 2015.

[7] G. Craciun. Toric differential inclusions and a proof of the global attractor conjecture, submitted, available on arXiv, 2015.

[8] G. Craciun, A. Dickenstein, A. Shiu, and B. Sturmfels. Toric dynamical systems. J. Symbolic Comput., 44(11):1551--1565, 2009.

[9] P. de Leenheer, D. Angeli, and E. Sontag. Monotone chemical reaction networks. J. Math. Chem., 41:295--314, 2007.

[10] P. Donnell and M. Banaji. Local and global stability of equilibria for a class of chemical reaction networks. SIAM J. Appl. Dyn. Syst., 12(2):899--920, 2013.

[11] P. Donnell, M. Banaji, A. Marginean, and C. Pantea. CoNtRol: an open source framework for the analysis of chemical reaction networks. Bioinformatics, 30(11):1633--1634,2014.

[12] E. Feliu. Injectivity, multiple zeros, and multistationarity in reaction networks. Proc. R. Soc. A., 471:2173, 2015.

[13] E. Feliu and C. Wiuf. Enzyme sharing as a cause of multistationarity in signaling systems. J. Roy. Soc. Int., 9(71):1224--1232, 2012.

[14] E. Gross, B. Davis, K.L. Ho, D.J. Bates, H.A. Harrington. Numerical algebraic geometry for model selection. Submitted. arXiv:1507.04331.

[15] E. Gross, H.A. Harrington, Z. Rosen, B. Sturmfels. Algebraic systems biology: a case study for the Wnt pathway. Submitted. arXiv:1502.03188.

[16] J. Gunawardena. Distributivity and processivity in multisite phosphorylation can be distinguished through steady-state invariants. Biophys. J., 93:3828--3834, 2007.

[17] M.D. Johnston, C. Pantea, and P. Donnell. A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems. J. Math. Biol. 2015. DOI: 10.1007/s00285-015-0892-1

[18] M.D. Johnston, D. Siegel, and G. Szederkenyi. A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks. J. Math. Chem., 50(1):274--288, 2012.

[19] R.L. Karp, M. Perez Millan, T. Dasgupta, A. Dickenstein, and J. Gunawardena. Complex linear invariants of biochemical networks. J. Theor. Biol., 311:130--138, 2012.

[20] T.G. Kurtz. The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys., 57:2976--2978, 1972.

[21] A. Manrai and J. Gunawardena. The geometry of multisite phosphorylation. Biophys. J., 95:5533--5543, 2009.

[22] S. Muller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu, and A. Dickenstein. Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. Found. Comput. Math., 2015.

[23] S. Muller and G. Regensburger. Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces. SIAM J. Appl. Math., 72(6):1926--1947, 2012.

[24] L. Pauleve, G. Craciun, and H. Koeppl. Dynamical properties of discrete reaction networks. J. Math. Biol., 69(1):55--72, 2014.

[25] M. Perez Millan, A. Dickenstein, A. Shiu, and C. Conradi. Chemical reaction systems with toric steady states. Bull. Math. Biol., 74(5):1027--1065, 2012.

[26] G. Shinar and M. Feinberg. Structural sources of robustness in biochemical reaction networks. Science, 327(5971):1389--1391, 2010.

[27] G. Szederkenyi, J.R. Banga, and A.A. Alonso. CRNreals: a toolbox for distinguishability and identifiability analysis of biochemical reaction networks. Bioinformatics, 28(11):1549-50, 2012.

The workshop will focus on the following problems:

(1) Clarifying the relationship between continuous-state differential equation models and discrete-state stochastic models. Although the stochastic model is known, in an appropriate scaling limit, to converge to the corresponding deterministic solution, the result holds on compact time intervals only [20]. In fact, the two frameworks often produce vastly different dynamical predictions on long time-frames. In particular, the discrete models may predict an "extinction event'' not present in the deterministic model. Recent work has identified a broad classes of biochemical reaction mechanisms for which this disparity necessarily occurs [2,26]. This workshop will further develop these results with particular focus on recently established connections with Petri Net Theory [3,6,24].

(2) Applications of network-based methods to algebraic systems biology. Since the early 2000s, there has been significant interest in the algebraic structure of the steady states of mass-action systems, which is the most common modeling framework in systems biology. Network-based approaches have been found to be a useful alternative to computational approaches such as Groebner bases, which are often computationally intractable for realistic biochemical systems. Much work has focused recently on the topics of multiple equilibria and switch-like behavior, and led to (and benefited from): (i) in-depth investigations of systems with toric steady states [8,25]; (ii) results on the existence of steady-state invariants, with application to in-vivo mechanism validation [13,14,16,19,21]; and (iii) development of algebraic conditions for injectivity of vector fields arising from networks [4,12,22]. One focus of the workshop is on the connection between network structure and algebraic ideals in systems biology.

(3) Computational implementation of results. Significant work has been conducted in recent years in building algorithms and computational packages which can be used to implement the latest results in biochemical reaction modeling. Some notable examples are: (i) CoNtRol, an open-source browser-based program which has modules for verifying a network's capacity for multistationarity, steady-state stability, and persistence [11,17]; (ii) Bertini, a numerical solver for roots of systems of polynomial equations using continuation methods [5,15]; and (iii) CRNreals, which uses optimization methods to identify dynamically equivalent network representations of reaction networks [18,27]. As the understanding of biochemical reaction models grows, however, there is increased need for implementation which is comprehensive, up-to-date, and accessible to the wider biochemical modeling community. Therefore, discussion of directions for algorithm development and implementation will be a focus of the workshop.

(4) Further discussion and unification of dynamical approaches. In recent years, several novel approaches to studying the dynamical properties of chemical reaction systems have been developed, including: (i) a reformulation of monotone systems theory into reaction-coordinates [9,10]; (ii) the introduction of piecewise linear in rates (PWLR) Lyapunov functions [1]; (iii) the development of toric differential inclusion approaches to persistence and global stability [7]; and (iv) the formalization of generalized mass action systems theory [23]. A focus of the workshop will be to explore connections among these distinct approaches, and explore new application areas within biochemical modeling.

The format will feature limited overview lectures from leading experts in the relevant fields in the mornings, with significant time reserved for discussion and break-out sessions in the afternoons. The core research community has found this model to be very conducive to establishing long-lasting collaborations and new approaches to classical problems. Similar formats have been adapted at recent related workshops in 2012 (Castle Dagstuhl, Germany), 2013 (American Institute of Mathematics, USA), 2014 (University of Portsmouth, UK), and 2015 (University of Copenhagen, Denmark).

The workshop will actively promote participation from young researchers and women. We note that three of the organizers are pre-tenure and two are women. We anticipate broad international representation, and participation from a number of postdoctoral researchers and graduate students.

References:

[1] M.A. Al-Radhawi and D. Angeli. New approach to the stability of chemical reaction networks: Piecewise linear in rates Lyapunov functions. 2015. DOI 10.1109/TAC.2015.2427691.

[2] D.F. Anderson, G. Enciso, and M.D. Johnston. Stochastic analysis of chemical reaction networks with absolute concentration robustness. J. R. Soc. Interface, 11(93):20130943, 2014.

[3] D. Angeli, P. de Leenheer, and E. Sontag. A petri net approach to the study of persistence in chemical reaction networks. Math. Biosci., 210(2):598--618, 2007.

[4] M. Banaji and C. Pantea. Some results on injectivity and multistationarity in chemical reaction networks, submitted, available on arXiv, 2015.

[5] D.J. Bates, J.D. Hauenstein, A.J. Sommese, and C.W. Wampler. Bertini: Software for Numerical Algebraic Geometry. Available at bertini.nd.edu with permanent doi: dx.doi.org/10.7274/R0H41PB5

[6] R. Brijder. Dominant and T-Invariants for Petri Nets and Chemical Reaction Networks. Lecture Notes in Comput. Sci., 9211:1--15, 2015.

[7] G. Craciun. Toric differential inclusions and a proof of the global attractor conjecture, submitted, available on arXiv, 2015.

[8] G. Craciun, A. Dickenstein, A. Shiu, and B. Sturmfels. Toric dynamical systems. J. Symbolic Comput., 44(11):1551--1565, 2009.

[9] P. de Leenheer, D. Angeli, and E. Sontag. Monotone chemical reaction networks. J. Math. Chem., 41:295--314, 2007.

[10] P. Donnell and M. Banaji. Local and global stability of equilibria for a class of chemical reaction networks. SIAM J. Appl. Dyn. Syst., 12(2):899--920, 2013.

[11] P. Donnell, M. Banaji, A. Marginean, and C. Pantea. CoNtRol: an open source framework for the analysis of chemical reaction networks. Bioinformatics, 30(11):1633--1634,2014.

[12] E. Feliu. Injectivity, multiple zeros, and multistationarity in reaction networks. Proc. R. Soc. A., 471:2173, 2015.

[13] E. Feliu and C. Wiuf. Enzyme sharing as a cause of multistationarity in signaling systems. J. Roy. Soc. Int., 9(71):1224--1232, 2012.

[14] E. Gross, B. Davis, K.L. Ho, D.J. Bates, H.A. Harrington. Numerical algebraic geometry for model selection. Submitted. arXiv:1507.04331.

[15] E. Gross, H.A. Harrington, Z. Rosen, B. Sturmfels. Algebraic systems biology: a case study for the Wnt pathway. Submitted. arXiv:1502.03188.

[16] J. Gunawardena. Distributivity and processivity in multisite phosphorylation can be distinguished through steady-state invariants. Biophys. J., 93:3828--3834, 2007.

[17] M.D. Johnston, C. Pantea, and P. Donnell. A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems. J. Math. Biol. 2015. DOI: 10.1007/s00285-015-0892-1

[18] M.D. Johnston, D. Siegel, and G. Szederkenyi. A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks. J. Math. Chem., 50(1):274--288, 2012.

[19] R.L. Karp, M. Perez Millan, T. Dasgupta, A. Dickenstein, and J. Gunawardena. Complex linear invariants of biochemical networks. J. Theor. Biol., 311:130--138, 2012.

[20] T.G. Kurtz. The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys., 57:2976--2978, 1972.

[21] A. Manrai and J. Gunawardena. The geometry of multisite phosphorylation. Biophys. J., 95:5533--5543, 2009.

[22] S. Muller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu, and A. Dickenstein. Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. Found. Comput. Math., 2015.

[23] S. Muller and G. Regensburger. Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces. SIAM J. Appl. Math., 72(6):1926--1947, 2012.

[24] L. Pauleve, G. Craciun, and H. Koeppl. Dynamical properties of discrete reaction networks. J. Math. Biol., 69(1):55--72, 2014.

[25] M. Perez Millan, A. Dickenstein, A. Shiu, and C. Conradi. Chemical reaction systems with toric steady states. Bull. Math. Biol., 74(5):1027--1065, 2012.

[26] G. Shinar and M. Feinberg. Structural sources of robustness in biochemical reaction networks. Science, 327(5971):1389--1391, 2010.

[27] G. Szederkenyi, J.R. Banga, and A.A. Alonso. CRNreals: a toolbox for distinguishability and identifiability analysis of biochemical reaction networks. Bioinformatics, 28(11):1549-50, 2012.