Summer 2021
Seminar on Chemical and Metabolic Networks
Prof. Dr. Bernold Fiedler
Dr. Isabelle Schneider
Dr. Nicola Vassena
Schedule, Summer 2021
Wednesdays, 10:1512:00
 ONLINE COURSE!
Description
With minimal prerequisites, we will explore the dynamics of reaction networks of ordinary differential equations. Starting with Feinberg's notation, for highschool
chemistry, we connect mathematical results, such as the existence of positive steady states,
with reallife applications, such as citric acid cycle, celldifferentiation, and sleepwake
rhythms. The seminar is particularly suitable for students of the Berlin Mathematical
School, the Integrate Research Training Group of the CRC910, or others interested in
network techniques. Deeper knowledge of advanced mathematics, biology or chemistry
is neither a prerequisite, nor an obstacle.
References are understood as alternative sources and only small excerpts are required.
Topics
1 How come many chemical systems possess
a unique stable steadystate?
 Examples of chemical reaction networks and the corresponding ODEs
 Mass action kinetics
 The DeficiencyZero Theorem
 References: [1]
2 Only one steadystate?
2.1 Concordant chemical reaction networks
 Definition and properties of concordant networks
 The SpeciesReaction Graph: when is a network concordant?
 References: [2], [3]
2.2 Beyond mass action: fullyopen systems
 Pmatrices and Strongly Signed Determined (SSD) matrices
 Stoichiometric matrix SSD ⇒ only one steadystate
 The SpeciesReaction Graph, revisited.
 References: [4], [5]
3 3 Sensitivity of metabolic networks:
how do steadystates respond to perturbations?
3.1 Monomolecular networks
 Monomolecular networks as directed graphs
 Fluxresponse theorem: nonzero response
 Transitivity of influence
 References: [6], [7]
3.2 General multimolecular networks
 Child selections: single children have no influence
 Transitivity theorem in full glory
 References: [8]
3.3 Some elements of sign analysis
a)
 Monomolecular sign analysis
 References: [9], [10]
b)
 Good and bad children
 References: [11]
4 Dynamics and control at feedback vertex sets
4.1 From feedback vertex sets to determining nodes
 Definitions: feedback vertex sets, determining nodes
 Proof
 Examples
 References: [14] , [15]
4.2 From determining nodes to feedback vertex sets
 Proof
 One example: Cell differentiation, or signal transduction, or control of circadian rhythm
 References: [14] , [15]
5 Do delays matter?
 Harmless odiagonal delays
 Characterization of asymptotic stability for any choice of delays
 An application to biochemical networks
 References: [16] , [17]
References
 [1] http://www.crnt.osu.edu/LecturesOnReactionNetworks
 [2] G. Shinar and M. Feinberg, Concordant chemical reaction networks, Mathematical Biosciences 240 (2013) 92113
 [3] G. Shinar and M. Feinberg, Concordant chemical reaction networks and the SpeciesReaction Graph, Mathematical Biosciences 241 (2013) 123
 [4] M. Banaji, P. Donnell, S. Baigent P matrix properties, injectivity, and stability in chemical reaction systems, (2007) SIAM Journal on Applied Mathematics, volume 67, number 6,
15231547.
 [5] M. Banaji, G. Craciun, Graphtheoretic criteria for injectivity and unique equilibria in general chemical reaction systems, (2008) Advances in Applied Mathematics, volume 44, number 2, 168184
 [6] B. Fiedler and A. Mochizuki, Sensitivity of chemical reaction networks: a structural approach. 2. Regular monomolecular systems, (2015) Mathematical Methods in the Applied Sciences, Volume 38, Number 16, 35193537
 [7] N. Vassena, H. Matano Monomolecular reaction networks: Fluxinfluenced sets and balloons, Math. Meth. in the App. Sc., (2017), Volume 40, Number 18, 772277366
 [8] B. Brehm, B. Fiedler Sensitivity of chemical reaction networks: a structural approach. 3.Regular multimolecular systems, Mathematical Methods in the Applied Sciences (2018), volume 41, number 4, 13441376
 [9] N. Vassena Sensitivity of monomolecular reaction networks: signed fluxresponse to reaction rate perturbations, Proceedings of 2017 European Conference on Circuit Theory and Design (ECCTD).
 [10] N. Vassena Sensitivity of Metabolic Networks (2020) PhD Thesis, Free University Berlin.
 [11] N. Vassena Good and bad children in metabolic networks, Mathematical Biosciences and Engineering, (2020) Volume 17, Number 6, 76217644
 [12] G. Shinar, U. Alon and M. Feinberg, Sensitivity and robustness in chemical reaction networks, SIAM J. Appl. Math. 69 4 (2009) 977998
 [13] G. Shinar and M. Feinberg, Design principles for robust biochemical reaction networks: What works, what cannot work, and what might almost work, Mathematical Biosciences (2011) 231, 3948
 [14] B. Fiedler, A. Mochizuki, G. Kurosawa, D. Saito Dynamics and control at feedback vertex sets. I: Informative an determining nodes in regulatory networks, J. Dynamics Di. Equations (2013), DOI 10.1007/s1088401393118
 [15] A. Mochizuki, B. Fiedler, G. Kurosawa, D. Saito Dynamics and control at feedback vertex sets. II: A faithful monitor to determine the diversity of molecular activities in regulatory networks, J. Theor. Biology 335 (2013) 130146
 [16] J. Hofbauer, J.W.H. So Diagonal dominance and harmless odiagonal delays Proc. of the American Math. Society. (2000), Volume 128, Number 9, Pages 26752682 S 00029939(00)055647
 [17] M. Mincheva, M.R. Roussel Graphtheoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays J. Math. Biol. (2007) 55:87104 DOI 10.1007/s0028500700982
