Stability and order preservation in chemical reaction networks

Networks of chemical reactions abound in nature. Despite internal and environmental fluctuations these "chemical reaction networks" (CRNs) often reliably perform some task. In biology, understanding this robustness is key to understanding processes such as metabolism and signalling, and thus to understanding disease; in chemical and biochemical engineering it is central to the design and control of reactors, and to minimising inefficiency and energy losses.

While robustness is often found, instabilities can also be observed in CRNs. Sometimes instability can be catastrophic - chemical engineering disasters can often be traced to an unpredicted instability; but instability may also be useful - the role of calcium oscillations in biological signalling is a familiar example. The experimental and mathematical literature in this area provides numerous examples which illustrate the complex and sometimes counterintuitive behaviour of CRNs, and the need for more "global" theory to inform us on their behaviour.

Some insight can be gained by simulating computer models of CRNs. But the limitations of this approach are illustrated by the fact that there are some well-known - and surprisingly small - systems of chemical reactions for which it is unknown whether there exists some choice of reaction rates (kinetics) at which the system can display "switching" behaviour or oscillation. The reason is that simulation, while extremely useful, can only provide information for some fixed kinetics. On the other hand, practical difficulties in experimental measurement mean that precise kinetic data is often not available for a CRN. Global theory fills the gap, begin able to explain reaction network behaviours from network structure. Recently, work in this area has seen some difficult problems reduced to simple, rapid calculation.

The aim of this project is to develop an important area in the global theory of chemical reaction networks. A key feature of this work will be to make minimal assumptions about the rates of reactions, and thus to derive conclusions based largely on qualitative knowledge of the interactions. A powerful body of mathematical theory, the theory of monotone dynamical systems, will provide the main tool for the analysis of CRNs. This requires both development of new areas within the theory, and imaginative application of the what is already available. Theoretical work, development of algorithms from the theory, and application to experimentally studied systems, will all be important components of the work to be undertaken.

The first and foremost impact will be on areas of science and applied mathematics: this work should have significant short to medium term impact on research communities working at the interfaces between mathematics, biology, chemistry and engineering. A number of steps will be taken to maximise the impact across communities, including publication in journals for different audiences, and presentation of the work at a variety of conferences. A one day workshop on chemical reaction network theory will be organised, and advertised widely.

Long term economic impact will come via the impacts on the life-sciences and engineering. The importance of global "systems" approaches of the kind in this proposal has been highlighted by the research councils, for example the EPSRC grand challenge "Systems chemistry: exploring the chemical roots of biological organisation", and the BBSRC priority "systems approach to biological research". Work in this area has already proved its ability to explain, to predict, and to suggest testable hypotheses for experimental work. It is beginning to provide a framework for the understanding of modularity in biology, namely how complex network behaviours emerge from the interconnection of simpler, well-understood components. The proposed work opens up exciting possibilities in synthetic biology for the design of reaction networks with prescribed properties. A coherent body of theory on the behaviour of systems of chemical reactions also has the potential to inform on practical questions of reactor efficiency and design in engineering. The concrete possibilities for collaboration with bio/chemical engineers will be explored during the project.