at the

American Institute of Mathematics, San Jose, California

organized by

Alicia Dickenstein, Jeremy Gunawardena, and Anne Shiu

Recent developments have suggested that mathematical analysis of such networks may be feasible, using methods from computational algebra, algebraic geometry and dynamical systems. For instance, one class of dynamical systems, coming from mass-action kinetics, gives rise to polynomial dynamical systems. Here, the biologically-relevant steady states are the positive real solutions to a system of polynomial equations and from computational algebra and algebraic geometry have been used to shed light on steady-state behaviour. Difficult questions still remain regarding the transient dynamics and generalisations to non mass-action systems.

The primary aim of this workshop is to bring together mathematicians as well as researchers who are closer to the experimental side of systems biology, in order to formulate precise open problems and to explore appropriate mathematical methods to tackle them. We hope to sustain an open dialogue between theory and experiment, so that they may continue to stimulate each other. Among the problems we may consider initially are the following.

- Decomposition of parameter spaces according to qualitative dynamical behaviour.
- Computation of steady state polynomial invariants.
- The Global Attractor Conjecture.
- Elimination theory and the QSSA (quasi-steady state approximation) approach.
- Extension of mass-action kinetics results to more general kinetics.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Numerical algebraic geometry for model selection

by Elizabeth Gross, Brent Davis, Kenneth L. Ho, Daniel J. Bates, and Heather A. Harrington

Lyapunov functions, stationary distributions, and non-equilibrium potential for chemical reaction networks

by David F. Anderson, Gheorghe Craciun, Manoj Gopalkrishnan, and Carsten Wiuf, *Bull. Math. Biol. 77 (2015), no. 9, 1744-1767 * MR3423060

Parameter-free methods distinguish Wnt pathway models and guide design of experiments

by Adam L. MacLean, Zvi Rosen, Helen M. Byrne, and Heather A. Harrington

ATP concentration c

by Jasmine Nirody and Padmini Rangamani

A global convergence result for processive multisite phosphorylation systems

by by Carsten Conradi and Anne Shiu, *Bull. Math. Biol. 77 (2015), no. 1, 126-155 * MR3303108

Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry

by by Stefan Muller, Elisenda Feliu, Georg Regensburger, Carsten Conradi, Anne Shiu, and Alicia Dickenstein, *Found. Comput. Math. 16 (2016), no. 1, 69-97 * MR3451424

Translated chemical reaction networks

by Matthew D. Johnston, *Bull. Math. Biol. 76 (2014), no. 5, 1081-1116 * MR3201336

Stochastic analysis of biochemical reaction networks with absolute concentration robustness

by David F. Anderson, German Enciso and Matthew Johnston, *J. R. Soc. Interface 11: 20130943, 2014 The Author(s) Published by the Royal Society*

Rate-independent computation in continuous chemical reaction networks

by H. L. Chen, D. Doty, and D. Soloveichik, *TCS'14-Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science, 313-325, ACM, New York, 2014.* MR3359486

Deterministic detailed balance in chemical reaction networks is sufficient but not necessary for stochastic detailed balance

by Badal Joshi, *Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 4, 1077–1105 * MR3315487