# Mathematics of Reaction Networks

**Mathematical modeling of chemical reaction networks** consists of a variety of methods for approaching questions about the dynamical behaviour of chemical reactions arising in real world applications.

This wiki is intended to serve the dual purpose of being an accessible primer for students and researchers new to the area of mathematical modeling of chemical reactions, and a summary of the current state of the discipline for those who are active in the field.

## Resources for research

The following resources are intended to assist people who are familiar with, and actively involved in, research in modeling of reaction networks. See the active research topics pages for discussion of current research initiatives.

### List of active research topics

- Applications to biochemical settings
- Injectivity and multiple equilibria
- Foundations of Chemical Reaction Network Theory
- Model reduction and the QSSA
- Monotone systems theory applied to reaction networks
- Oscillations in chemical reaction networks
- Persistence and the Global Attractor Conjecture
- Stochastic modelling of biochemical reaction networks

### List of upcoming events

#### 2017

- May 25–27, Mathematics in (bio)Chemical Kinetics and Engineering (MaCKiE) (Budapest, Hungary)
- June 4–9, Mathematical Analysis of Biological Interaction Networks (BIRS, Banff, Alberta, Canada)

### List of previous events

#### 2012

- August 7–10, SIAM Conference on the Life Sciences (San Diego, California, USA)
- November 12–16, Symbolic Methods for Chemical Reaction Networks (Castle Dagstuhl, Germany)
- November 26–27, Workshop November 2012 (Imperial College, London, UK)

#### 2013

- March 25–29, Mathematical problems arising from biochemical reaction networks (American Institute of Mathematics, Palo Alto, California)
- August 1–4, SIAM Conference on Applied Algebraic Geometry (Colorado State University, Fort Collins, Colorado)

#### 2014

- June 23–25, Combinatorial and algebraic approaches to chemical reaction networks, (University of Portsmouth, Portsmouth, UK)
- August 4–7, Mini symposium at the SIAM Conference on Life Sciences, (Charlotte, North Carolina, USA)

#### 2015

- July 1–3, Workshop on Mathematical Trends in Reaction Network Theory (University of Copenhagen, Copenhagen, Denmark)
- August 3–7, SIAM Conference on Applied Algebraic Geometry (National Institute for Mathematical Sciences, Daejeon, South Korea)
- October 2–4, Central Fall Sectional Meeting (Loyola University, Chicago, Illinois)
- October 5–7, Mathematics of Reaction Networks Workshop (University of Wisconsin-Madison, Madison, Wisconsin)

#### 2016

- January 25–29, Dynamics of Biologicaly-Inspired Networks, (Mathematical Biosciences Institute, Columbus, Ohio, USA)
- March 11–13, Workshop on the Global Attractor Conjecture, (San Jose State University, San Jose, California, USA)
- July 11–14, SIAM Conference on Life Sciences, (Boston, Massachusetts, USA)

### List of people

- Muhammad Ali Al-Radhawi
- Zahra Aminzare
- David Anderson
- David Angeli
- Murad Banaji
- Julio Banga
- Balázs Boros
- Daniele Cappelletti
- Gheorghe Craciun
- Carsten Conradi
- Patrick De Leenheer
- Alicia Dickenstein
- Mirela Domijan
- Pete Donnell
- Martin Feinberg
- Elisenda Feliu
- Dietrich Flockerzi
- Attila Gabor
- Gilles Gnacadja
- Manoj Gopalkrishnan
- Alexander Gorban
- Jeremy Gunawardena
- Juliette Hell
- Bill Helton
- Bayu Jayawardhana
- Matthew Johnston
- Badal Joshi
- Igor Klep
- Bence Mélykúti
- Nicolette Meshkat
- Ezra Miller
- Maya Mincheva
- Mark Muldoon
- Stefan Müller
- Jost Neigenfind
- Zoran Nikoloski
- Irene Otero-Muras
- Casian Pantea
- Mercedes Perez Millan
- Shodham Rao
- Georg Regensburger
- Alan Rendall
- Anne Shiu
- David Siegel
- Guy Shinar
- Eduardo Sontag
- Gabor Szederkenyi
- Janos Toth
- Carsten Wiuf
- Pencho Yordanov

### List of literature resources

### List of software packages

- Martin Feinberg's Chemical Reaction Network Toolbox
- ERNEST
- CRNreals
- CoNtRol ( documentation)
- Mathematica CRN Jacobian matrix files by Igor Klep et al.
- GraTeLPy

## Resources for education

This section is intended as a primer for those who are curious about the mathematics underlying the study of reaction networks.

### Chemical reaction network theory

Chemical reaction network theory is a framework for modeling the evolution of chemical concentrations resulting from simultaneously occurring chemical reactions. A key feature of the theory is the relationship between the graphical structure of the reaction network and the resulting dynamics. A strong emphasis, consequently, is placed on results which hold regardless of the parameter values of the network, i.e. results which depend on the network structure alone.

The foundations of chemical reaction network theory were laid down in a series of seminal papers by Fritz Horn, Roy Jackson and Martin Feinberg in the early 1970's ^{[1]}^{[2]}^{[3]}. In these papers, the authors were primarily focused on developing conditions sufficient for uniqueness and stability of equilibrium concentrations, but their foundation has since between adapted to questions of multistability, injectivity, monotonicity, persistence, equivalence of mass-action systems, model reduction, oscillations, and applications. The models have also been adapted to the stochastic framework, reaction-diffusion equations, and kinetic schemes other than mass-action (e.g. Michaelis-Menten, Hill kinetics, etc.).

See also:

- Boundedness
- Biochemistry
- Chemical reaction networks
- Chemical reaction network theory
- Complex balanced systems
- Deficiency theory
- Detailed balanced systems
- Dynamical equivalence
- Generalized mass action systems
- Global attractor conjecture
- Hill kinetics
- Injectivity
- Linear conjugacy
- Linkage class
- Lumping
- Mass action systems
- Michaelis-Menten kinetics
- Model reduction
- Monotonicity
- Multistability
- Persistence
- Reaction graph
- Reversibility
- Reaction-diffusion models
- Stochastic models
- Toric dynamical systems
- Toric steady states
- Weak reversibility

## References

- ↑ R. Horn, R. Jackson, General mass action kinetics,
*Arch. Ration. Mech. Anal.*47 (1972) 81-116 - ↑ F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics,
*Arch. Ration. Mech. Anal.*, 49 (1972) 172-186 - ↑ M. Feinberg, Complex balancing in general kinetic systems,
*Arch. Ration. Mech. Anal.*, 49 (1972) 187-194

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