# Stability and order preservation in chemical reaction networks

### Abstract

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.

### Planned Impact

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.

## People |
## ORCID iD |

Murad Banaji (Principal Investigator) |

*A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems.*in Journal of mathematical biology

*A graph-theoretic condition for irreducibility of a set of cone preserving matrices*in Linear Algebra and its Applications

*CoNtRol: an open source framework for the analysis of chemical reaction networks.*in Bioinformatics (Oxford, England)

*Global convergence in systems of differential equations arising from chemical reaction networks*in Journal of Differential Equations

*Local and Global Stability of Equilibria for a Class of Chemical Reaction Networks*in SIAM Journal on Applied Dynamical Systems

*Some Results on Injectivity and Multistationarity in Chemical Reaction Networks*in SIAM Journal on Applied Dynamical Systems

*Some results on injectivity and multistationarity in chemical reaction networks*in arXiv (accepted for publication in SIADS)

*Some results on the structure and spectra of matrix-products*in Linear Algebra and its Applications

Description | With my research associate and collaborators, I have developed tools to identify families of chemical reaction networks (CRNs - these might occur, for example, as subnetworks in living cells) with behaviour restricted by the property of being "monotone". These tools are both theoretical (we have proved that certain families necessarily are monotone); and algorithmic - we have written programs which are able to search for networks (upto certain size limitations) which are monotone. We were able to characterise those families which were monotone in certain ways (e.g. preserved "simplicial orders"), but were not successful in finding a complete characterisation in terms of network structure in general: however our attempts in this direction have convinced us that this is a difficult problem, and that there may never be a simple computable condition both necessary and sufficient to identify a network as monotone. In addition to the work on identifying monotone chemical reaction networks, we have explored the implications of monotonicity combined with other commonly occurring features of chemical systems. This mathematical work has allowed us to prove that systems with certain combinations of features necessarily have very simple behaviour which does not depend on initial data or kinetic parameters. Networks with such simple behaviour provide the natural "building blocks" of more complex networks and provide insight into how, for example, biological networks can be controlled to encourage or discourage some particular behaviour. My collaborators and I have also tackled a number of related questions on chemical networks, complementary to questions about monotonicity. With co-workers at Imperial, I did delicate and mathematically nontrivial work on necessary features for a chemical oscillator; and I have also been consolidating earlier strands of work identifying reaction networks which permit/ forbid multiple equilibria. Recently, with a co-worker at Kent, I have been developing theorems which can be broadly described as "thermodynamical" which explain a number of earlier results on CRNs as a consequence of a single feature, nonexpansivity with respect to an appropriate norm. Finally, the applications-oriented work has also inspired some elegant new mathematical theory (mainly in combinatorics, and in analysis) which has a life of its own beyond the applications. I have found, for example, interesting new results on qualitative matrices and graphs, and on the number and stability of equilibria in classes of dynamical systems which need not arise from chemistry. This work, which has been presented at conferences aimed mainly at pure mathematicians, has publicised the fact that chemical reaction networks are an application which generates interesting and nontrivial mathematical questions, potentially opening the door for the influx of more mathematicians into the area. |

Exploitation Route | Our work is already well-regarded and referred to in academic communities. Several of our theoretical results quite naturally lend themselves to development and generalisation, and we expect this to happen during the coming two-three years. In some cases we have theorems which have proved hard to cast as algorithms (namely where checking that a given CRN satisfies conditions of the theorem is algorithmically nontrivial), and we expect to see work on this occurring on the future. Regarding non-academic communities, we believe that publicising the algorithmic side of our work is the most important route towards impact. To this end we published a short paper in Bioinformatics aimed at experimental/ systems biologists (rather than mathematicians), describing this algorithmic work. We expect to see analysis via CoNtRol (our web-based platform for the autamated analysis of CRNs) figuring as a precursor to experimental studies, and also to see its batch-processing capabilities generating new hypotheses about CRNs. |

Sectors |
Chemicals,Healthcare,Pharmaceuticals and Medical Biotechnology |

Description | The findings have not so far been used outside of research communities. However, this area of research has potential for considerable impact. Some of our papers and software produced during the EPSRC grant have had significant impact on research in this area, with the potential for impact beyond research communities. Most important amongst the outputs is the open source, web-based, tool we have produced (CoNtRol) allowing a non-expert user to get a rigorous, detailed, structure-based analysis of an arbitrary reaction network (up to size limitations). This opens up the possibility that experimental scientists/ engineers will begin to use this tool as a precursor to numerical work or experimentation. This structural analysis potentially allows more interesting questions to be asked, and can lead to more targeted numerical/ experimental work. While I am not aware of direct societal or economic impacts so far, the choice to focus considerable effort on building a powerful, user-friendly, cross-platform piece of software, greatly enhances the possibility of long-term impact. |

Description | LMS workshop grant (towards costs of 3-day CRNT workshop at Portsmouth) |

Amount | £3,200 (GBP) |

Funding ID | WS-1213-05 |

Organisation | London Mathematical Society |

Sector | Learned Society |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

Start | 06/2014 |

End | 06/2014 |

Description | Collaboration with University of Kent (Nonexpansivity and monotonicity results applied to prove convergence in CRNs) |

Organisation | University of Kent |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

Sector | Academic/University |

PI Contribution | Collaboration between Portsmouth group (Murad Banaji and Pete Donnell) and Bas Lemmens (University of Kent) exploring the relationship between nonexpansivity and monotonicity applied to CRNs. This began with informal discussions at the POSTA conference (2012), and was developed via a seminar given by Murad at Kent, followed by two full day meetings between all three of us, held in London (Thursday 26th Semptember 2013), and one in Portsmouth (11th Feb 2014). The first draft of a new paper resulted. |

Collaborator Contribution | Our partner in Kent, Bas Lemmens, comes with extensive technical expertise in functional analysis and the mathematics of monotone dynamical systems. Using this expertise, we were able to give several existing results in CRNT new and simpler proofs, and to arrive at some new theorems, potentially applicable to chemical systems. |

Impact | We proved several results on nonexpansivity in CRNs which are currently in preprint form, in the process of being finalised for submission. A joint publication with Bas Lemmens (University of Kent) will result. |

Start Year | 2012 |

Description | Collaboration with West Virginia University (General themes in chemical reaction network theory) |

Organisation | West Virginia University |

Country | United States of America |

Sector | Academic/University |

PI Contribution | This collaboration with Casian Pantea, formerly at Imperial, now at West Virginia University, began before the EPSRC grant, but developed considerably during the EPSRC grant. It resulted in the development of CoNtRol (open source CRN analysis software), the reaction networks wiki, and several joint publications. |

Collaborator Contribution | Apart from close collaboration on a number of papers and computational projects, Casian Pantea was able to invite Pete Donnell (the research associate working on the EPSRC grant) for an extended visit to WVU, primarily aimed at further development of CoNtRol. WVU met the funding for this visit which resulted in the addition of various modules to CoNtRol. |

Impact | Outputs are: - CoNtRol (open source CRN analysis software); - reaction-networks.net (wiki), - "Combinatorial approaches to Hopf bifurcations in systems of interacting elements" (publication, Communications in Mathematical Sciences); - "Some results on injectivity and multistationarity in chemical reaction networks" (publication, preprint submitted to SIAM Journal on Applied Dynamical Systems); - "CoNtRol: an open source framework for the analysis of chemical reaction networks" (publication, bioinformatics) |

Start Year | 2011 |

Title | CoNtRol: Chemical Reaction Network analysis tool |

Description | CoNtRol is an open source, web-based (and hence platform independent) framework for analysis of chemical reaction networks (CRNs), designed to be both extensible and simple to use. It currently includes a Java-based tool to visualise DSR graphs, and implements a number of necessary and/or sufficient structural tests for multiple equilibria, stable periodic orbits, convergence to equilibria, persistence, etc. and is coded in C, Java, Octave and PHP. As the front-end is web-based, the complexity of the underlying platform is hidden from the user. It also provides a visualisation tool which can produce the so-called DSR graph of a reaction system, and implements some graph theoretic analysis. |

Type Of Technology | Software |

Year Produced | 2014 |

Impact | The tool, which is under continuous development, has become quite widely used within the chemical reaction network theory (CRNT) community. Development has also been opened up to the wider community and modules have subsequently been added by researchers not originally part of the development team. The goal is both to allow new results which naturally take algorithmic forms to be made widely available, and to provide an open source alternative to some of the closed source CRN analysis software currently available. We expect long term impact to be significant. |

URL | http://reaction-networks.net/control/ |

Title | Mathematics of Reaction Networks (wiki) |

Description | This is a collaboratively developed wiki on chemical reaction network theory (CRNT) which came about as a consequence of discussions at SIAM Life Sciences 2012. The goal is to provide an ever-expanding source of basic reference material on CRNT. The main technical development of the wiki, and initial material was provided by Pete Donnell as part of the EPSRC grant. Subsequently a number of experts in CRNT have contributed modules. The wiki also links to, and provides documentation for, the CRN analysis system, CoNtRol. |

Type Of Technology | Webtool/Application |

Year Produced | 2013 |

Impact | Anecdotally, the wiki has become widely used in the CRNT community. This was visible recently at a CRNT workshop in Portsmouth, where we found that a variety of academics from the UK, Europe and North America all regularly use it as a reference tool when writing papers in this area. One expected long term result is a harmonisation of language and basic assumptions across CRNT research communities with somewhat different frameworks and histories, allowing greater scope for collaboration and communication between these communities. |

URL | http://reaction-networks.net/wiki/Mathematics_of_Reaction_Networks |

Description | Convergence to equilibria in order-preserving systems with an increasing linear first integral |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Other academic audiences (collaborators, peers etc.) |

Results and Impact | Seminar at Portsmouth. The goal was to present more mathematical aspects of this research to a local audience of mathematicians |

Year(s) Of Engagement Activity | 2012 |

Description | Identifying monotone and strongly monotone dynamical systems |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Other academic audiences (collaborators, peers etc.) |

Results and Impact | Talk given at Kent Mathematics Colloquium (Jan 2013). Seminar outlining the questions of identifying monotone dynamical systems, with examples drawn from chemical reaction networks. This talk played an important part in the development of the collaboration with Bas Lemmens at the University of Kent. |

Year(s) Of Engagement Activity | 2012 |

Description | Logarithmic norms, compound matrices, and applications to chemical reaction networks |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | Network seminar at the University of Wisconsin, Madison |

Year(s) Of Engagement Activity | 2015 |

URL | https://www.math.wisc.edu/wiki/index.php/Networks_Seminar |

Description | Monotonicity of Chemical Reaction Networks with Respect to Non-Simplicial Cones |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other academic audiences (collaborators, peers etc.) |

Results and Impact | Talk given by Pete Donnell in a minisymposium at SIAM Life Sciences conference 2014. This minisymposium provided a gathering place for the CRNT community, and also for others, both mathematicians and life scientists, with an interest in CRNT. The session was well attended and resulted in lively discussion. Increased interest in the intersection between Monotone Dynamical Systems and Chemical Reaction Network Theory (this intersection is at the heart of the EPSRC grant). |

Year(s) Of Engagement Activity | 2014 |

Description | Monotonicity, nonexpansivity and chemical reaction networks |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Other academic audiences (collaborators, peers etc.) |

Results and Impact | This was a technical talk given by Pete Donnell (the research associate on the EPSRC grant) to the mathematics department at West Virginia University where he was visiting Casian Pantea. The talk cemented what is likely to be a long-term collaboration with the department of mathematics at WVU. |

Year(s) Of Engagement Activity | 2014 |

URL | http://waw2011.math.wvu.edu/node/596 |

Description | Nonexpansivity in chemical reaction networks |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | Applied and Computational Mathematics Seminar in the department of mathematics at the University of Wisconsin, Madison |

Year(s) Of Engagement Activity | 2015 |

URL | https://today.wisc.edu/events/view/83558 |

Description | Positivity and conditions for Hopf bifurcation and oscillation in biological networks (POSTA, Rome) |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | The goal was to report recent results on computationally checkable conditions for oscillation in biological networks. The work built on work done during my EPSRC award including the study of monotone chemical reaction networks. Abstract: Oscillation is known to occur in a great variety of biological contexts, from the natural rhythms of "body clocks" and ovulation, biochemical oscillations in cellular signalling, the cyclic nature of various diseases, cyclic behaviour in Lotka-Volterra-type models of interacting populations, and so forth. A key question is: which underlying networks, be they biochemical, ecological, etc. permit (stable) oscillation? Can we tell from the network topology whether oscillation should be observed? Several necessary conditions for oscillation in network dynamical systems are available, and these partial results often involve (the absence of) "positivity" in one sense or another. The focus in this talk is on oscillation in chemical reaction networks (CRNs) which are at the root of much biological oscillation, but can also be interpreted to include apparently different model classes, such as population models in ecology and gene-interaction networks. One approach which provides conclusions about oscillation involves the use of monotone dynamical systems. If we can identify that a dynamical system (modelled via ODEs) is a monotone system, in the sense that some partial order on its trajectories is preserved by the evolution, this provides a sufficient condition to rule out stable oscillation of the system. The complicating factor is that the partial order in question may not be a "standard" order, and indeed CRNs have often been observed to preserve non-standard partial orders. Nevertheless, some computational and theoretical tools are available for identifying CRNs as monotone systems and thus ruling out stable oscillation. A complementary class of approaches involves examining certain polynomials, generally occurring as minors (or combinations of minors) of various symbolic matrices associated with a dynamical system. To be more specific, these polynomials arise from powers or compounds of the Jacobian matrix of the system. Showing positivity of these polynomials is sufficient to rule out Hopf bifurcation, and hence the most likely pathway to oscillation. Sometimes, we may even infer positivity of the polynomials from various combinatorial conditions on the network, leading to a nice interplay between graph-theoretic and algebraic approaches. I will describe some theoretical and computational results on Hopf bifurcation and oscillation in dynamical systems with network structure, including some results from a large study of small CRNs. |

Year(s) Of Engagement Activity | 2016 |

URL | http://www.posta2016.org/ |

Description | Some combinatorial and algebraic problems arising from the study of chemical reaction networks |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | Invited talk at workshop "Dynamics in networks with special properties" at the Mathematical Biosciences Institute (University of Ohio), bringing together mathematicians and biologists. |

Year(s) Of Engagement Activity | 2016 |

URL | https://mbi.osu.edu/event/?id=896#schedule |

Description | Some mathematical problems arising in the study of chemical reaction networks |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other academic audiences (collaborators, peers etc.) |

Results and Impact | Invited talk given at the Tata Institute of Fundamental Research (TIFR), Mumbai, India, in August 2013. This was the first talk I have given on mathematical approaches to chemical reaction networks outside of the UK or US. There is a small group of academics at TIFR (mainly computer scientists) working in this area, and this talk reached an audience who are not normally present at the conferences I attend. |

Year(s) Of Engagement Activity | 2013 |

URL | http://www.tcs.tifr.res.in/events/some-mathematical-problems-arising-study-chemical-reaction-network... |

Description | Some results on nonexpansive dynamical systems with applications to chemical reaction networks |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | National |

Primary Audience | Professional Practitioners |

Results and Impact | Applied Mathematics seminar at the University of Leicester |

Year(s) Of Engagement Activity | 2015 |