Nonlinear Dynamics at the Free University Berlin

Summer 2021

Seminar on Chemical and Metabolic Networks

Prof. Dr. Bernold Fiedler

Dr. Isabelle Schneider

Dr. Nicola Vassena

Schedule, Summer 2021

Wednesdays, 10:15-12:00


With minimal prerequisites, we will explore the dynamics of reaction networks of ordinary differential equations. Starting with Feinberg's notation, for high-school chemistry, we connect mathematical results, such as the existence of positive steady states, with real-life applications, such as citric acid cycle, cell-differentiation, and sleep-wake 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.


1 How come many chemical systems possess a unique stable steady-state?

  • Examples of chemical reaction networks and the corresponding ODEs
  • Mass action kinetics
  • The Deficiency-Zero Theorem
  • References: [1]

2 Only one steady-state?

2.1 Concordant chemical reaction networks
  • Definition and properties of concordant networks
  • The Species-Reaction Graph: when is a network concordant?
  • References: [2], [3]
2.2 Beyond mass action: fully-open systems
  • P-matrices and Strongly Signed Determined (SSD) matrices
  • Stoichiometric matrix SSD ⇒ only one steady-state
  • The Species-Reaction Graph, revisited.
  • References: [4], [5]

3 3 Sensitivity of metabolic networks: how do steady-states respond to perturbations?

3.1 Monomolecular networks
  • Monomolecular networks as directed graphs
  • Flux-response 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
  • Monomolecular sign analysis
  • References: [9], [10]

  • 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 o -diagonal delays
  • Characterization of asymptotic stability for any choice of delays
  • An application to biochemical networks
  • References: [16] , [17]


  • [1]
  • [2] G. Shinar and M. Feinberg, Concordant chemical reaction networks, Mathematical Bio-sciences 240 (2013) 92-113
  • [3] G. Shinar and M. Feinberg, Concordant chemical reaction networks and the Species-Reaction Graph, Mathematical Biosciences 241 (2013) 1-23
  • [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, 1523-1547.
  • [5] M. Banaji, G. Craciun, Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems, (2008) Advances in Applied Mathematics, volume 44, number 2, 168-184
  • [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, 3519-3537
  • [7] N. Vassena, H. Matano Monomolecular reaction networks: Flux-influenced sets and balloons, Math. Meth. in the App. Sc., (2017), Volume 40, Number 18, 7722-77366
  • [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, 1344-1376
  • [9] N. Vassena Sensitivity of monomolecular reaction networks: signed flux-response 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, 7621-7644
  • [12] G. Shinar, U. Alon and M. Feinberg, Sensitivity and robustness in chemical reaction networks, SIAM J. Appl. Math. 69 4 (2009) 977-998
  • [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, 39-48
  • [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/s10884-013-9311-8
  • [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) 130-146
  • [16] J. Hofbauer, J.W.H. So Diagonal dominance and harmless o -diagonal delays Proc. of the American Math. Society. (2000), Volume 128, Number 9, Pages 2675-2682 S 0002-9939(00)05564-7
  • [17] M. Mincheva, M.R. Roussel Graph-theoretic methods for the analysis of chemical and bio-chemical networks. II. Oscillations in networks with delays J. Math. Biol. (2007) 55:87-104 DOI 10.1007/s00285-007-0098-2
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