European Conference on Mathematical and Theoretical Biology (ECMTB 2026)

Biochemical reaction networks: systematic design, limit cycles and properties of Hilbert-like numbers

MS50-06

13 Jul 2026, 17:20

20m

11.02 - HS (University of Graz)

Minisymposium Talk Systems Biology and Biochemical Networks Dynamical Analysis of Biochemical Reaction Networks

Speaker

Radek Erban (University of Oxford)

Description

I will discuss two types of mathematical models of biochemical reaction networks: (i) deterministic models described by reaction-rate equations, i.e., ordinary differential equations (ODEs) for the concentrations of the involved biochemical species~\cite{ref1,ref2}, and (ii) stochastic models described by the Gillespie stochastic simulation algorithm, which provides more detailed information about the simulated system than ODEs~\cite{ref3,ref4,ref5}. I will present methods for the systematic design of relatively simple reaction systems with prescribed dynamical behaviours, including systems with multiple oscillating solutions (limit cycles). We will focus on chemical reaction systems with two species, which under mass-action kinetics are described by planar autonomous ODEs whose right-hand sides are polynomials. The Hilbert number $H(n)$ is defined as the maximum number of limit cycles of a planar autonomous system of ODEs whose right-hand sides are polynomials of degree at most $n$. I will discuss analogues of the Hilbert number $H(n)$ for several classes of chemical reaction systems, including systems with reactions up to the $n$-th order and systems with up to $n$-molecular chemical reactions. Lower bounds on the modified Hilbert numbers will be presented~\cite{ref1}.

Bibliography

@article{ref1,
title={Planar chemical reaction systems with algebraic and non-algebraic limit cycles},
author={Craciun, G. and Erban, R.},
journal={Journal of Mathematical Biology},
volume={90},
number={6},
pages={64},
year={2025}
}

@article{ref2,
title={Chemical systems with limit cycles},
author={Erban, R. and Kang, H.},
journal={Bulletin of Mathematical Biology},
volume={85},
number={8},
pages={76},
year={2023}
}

@article{ref3,
title={Identifiability of stochastically modelled reaction networks},
author={Enciso, G. and Erban, R. and Kim, J.},
journal={European Journal of Applied Mathematics},
volume={32},
number={5},
pages={865--887},
year={2021}
}

@article{ref4,
title={Noise control for molecular computing},
author={Plesa, T. and Zygalakis, K. and Anderson, D. and Erban, R.},
journal={Journal of the Royal Society Interface},
volume={15},
number={144},
year={2018}
}

@book{ref5,
title={Stochastic {M}odelling of {R}eaction-{D}iffusion {P}rocesses},
author={Erban, R. and Chapman, S. J.},
ISBN = {9781108498128},
doi = {10.1017/9781108628389},
year={2020},
publisher={Cambridge University Press}
}