Stochasticity and anomalous transport phenomena in complex systems

Lead Research Organisation: University of Manchester
Department Name: Physics and Astronomy

Abstract

Transport and diffusion are phenomenon which are ubiquitous throughout nature, being intimately connected with the second law of thermodynamics. Transport is particularly important in biological cell regulatory processes. Due to the complicated interactions of the many participants involved in such complex systems, the diffusion of the constituents under study may not always be of the classical Brownian type. When this is the case, the transport is classed as anomalous. Finding the right underpinning mathematics and statistical physics, and connecting it with experimental evidence for anomalous diffusion is largely an open question. This is the case in particular when noise is present in the system. The constituents of systems which exhibit diffusion are discrete by nature and their complicated interactions are modelled probabilistically. As a consequence, observed quantities tend to fluctuate about an average, both temporally and between realisations of the system. However, such intrinsic fluctuations are often ignored in existing studies of anomalous transport.

The main aims of the PhD project are: (i) to study the effects of noise in subdhiffusive and non-Markovian systems, (ii) to develop new mathematics to describe these fluctuations, and (iii) to use these tools to characterise transport observed experimentally in the production of cellulose by by plant cells.

Some EPSRC areas of study which this project encompasses are Complexity science, Computational and theoretical chemistry, Mathematical biology, Non-linear systems, Statistics and applied probability.

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509565/1 01/10/2016 30/09/2021
1788412 Studentship EP/N509565/1 08/09/2016 31/03/2020 Joseph William Baron
 
Description In the 1950s, Alan Turing presented a simple mathematical model, which described the reaction and diffusion of two chemicals. He used this model to demonstrate how period patterns could be formed from the interaction of a slowly diffusing "activator" (which tends to produce more of itself and the other chemical) and a quickly diffusing "inhibitor" (the opposite of an activator). One issue with this theory is the large difference between the diffusion rates of the activator and inhibitor required for patterns to form -- this is quite difficult to realise in experimental systems. If Turing's mechanism is to be a viable model for spontaneous pattern formation in nature, this issue must be resolved. As a result of work done for this award, we found that the combination of anomalous diffusion and chemical fluctuations (two solutions previously proposed for the aforementioned issue) can reduce the difference between the diffusion coefficients required for pattern formation to a greater extent than either effect on its own. We also derived for the first time equations describing the random fluctuations in a chemical reaction system with chemicals that diffuse in an anomalous manner. This was done using sophisticated "path integral" mathematical techniques, which could have broad application to other problems involving a combination of memory effects and noisy fluctuations.

In another publication, we used similar mathematical techniques to quantify the fluctuations in protein and mRNA numbers in a model of a gene regulatory network. The production of mRNA a protein are inherently random process, meaning that the levels in the cell fluctuate over time. We looked at a specific gene regulatory network, which functions as a cellular clock in the so-called "somite segmentation process" -- this is process in which block of cells (somites) which are precursors to vertebrae form in the embryo. Here, the levels of protein and mRNA are required oscillate in a regular, periodic manner within cells in the embryo to so that the somites may form in a regimented manner. Via an intercell signalling mechanism known as delta-notch signalling, cells synchronise with their neighbours. Using a simple mathematical model, we showed that taking into account noisy fluctuations was vital -- such noise can give rise to cycles for sets of model parameters where a system without noise would not. Secondly, we showed that the delta-notch signalling mechanism plays a key role in not only synchronising noisy cycles between cells, but also in improving their coherence and in raising their amplitude. That is, delta-notch signalling gives rise to all-round better quality cellular clock.

Finally, in a manuscript that is soon to be submitted to a journal for publication, we examine the stability of model ecosystems. In the 1970s, Robert May proposed a simple statistical model of an ecosystem. Using this model he argued that, counter to intuition, food webs with larger numbers of species or greater connectivity between species ought to be more unstable. This was counter to the intuition of many ecologists at the time and in contrast to the great diversity seen in many natural ecosystems. May's work sparked the so-called "stability-diversity" debate, which continues to this day. Since May's work, people have added more details to his quite austere and simplified model to analyse the effect on stability of the ecosystem. In our work, we included the effects of diffusion in the model and, inspired by Turing's mechanism for pattern formation (discussed above) we showed that diffusion (or dispersal in the ecology jargon) can be a destabilising influence on the ecosystem. This is in contrast some other recent work which in fact claimed the opposite, but failed to take into account a Turing-like mechanism. We also discuss how the complex interaction between many species can affect the nature of the Turing-instability and find that more complexity in the interactions between species can reduce the difference between coefficients required for the Turing instability to occur.
Exploitation Route The work on noise in systems with anomalous diffusion can also be applied to any system with the combination of memory effects and stochasticity. For example, models of socio-technical systems like the voter model can be modified to include aging dynamics. Including demographic noise in such a model is a non-trivial exercise -- our work provides a way to do this.

The work on noise in the somite segmentaion clock can be taken forward in at least two possible ways: (1) Researchers building models of the somite segmentation clock should be aware that the a deterministic model (which doesn't include noise), may not capture the oscillatory nature of the protein dynamics, even if the model is complete and the reaction rates are all roughly correct. The cycles in gene expression may be "noise-induced". Our message is that care should be taken to include the effects of noise in models of the segmentation clock. (2) We predict that delta-notch signalling not only plays a role in synchronising cells but also plays a role in improving the quality of the cycles in an individual cell. By encumbering the signalling between cells (as has been done previously with so-called knock-in gene manipulation), this prediction can be tested. This will improve our understanding of the role of the delta-notch signalling mechanism in the somite segmentation process.

The work on ecosystems also improves understanding on several fronts. We not only improve on established random-matrix calculation techniques in this work but also further understanding of the Turing-mechanism and the stability of ecosystems. Our random matrix approach can be generalised for other food-web structures and adapted for other application outside of ecology, such as neuronal networks. Our study also highlights the need for careful consideration about how one goes about modelling the dispersal process in models of ecosystems and that it is not always an effect that can be neglected.
Sectors Chemicals,Environment,Pharmaceuticals and Medical Biotechnology

 
Description Didsbury high school biology week 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Schools
Results and Impact As part of Didsbury highschool's inaugural biology week, I was asked to come and speak to year 7 pupils about the work I did in my PhD as a cross-disciplinary scientist. I spoke braodly about Physics and Biology and the cross-over between the two.
Year(s) Of Engagement Activity 2019
 
Description Fluctuations, tipping points and emergence in eco-evolutionary dynamics -- Leeds 2019 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave at a talk at an international conference at the University of Leeds based on the work done in my PhD. This led to productive discussion/questions and potential future collaboration
Year(s) Of Engagement Activity 2019,2020
URL https://conferences.leeds.ac.uk/fluctuations/
 
Description IWEE Leeds 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a talk at an international workshop IWEE -- Leeds on my work. This generated interesting discussions and generated possible future collaborations and contacts.
Year(s) Of Engagement Activity 2019
URL http://extendedevolutionarysynthesis.com/iwee-leeds-19/
 
Description Internal seminar -- GRIND series -- Biology department -- Univrsity of Manchester 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I spoke to a group of experimental Biologists working at the University of Manchester for 1 hour about my theoretical work on gene regulatory networks. The talk sparked an interesting discussion between myself and the audience. I stayed and talked with members of the group afterwards about various projects they are working on and about applying the ideas that I mentioned in my talk.
Year(s) Of Engagement Activity 2019
 
Description Maths-Life seminar 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I gave a talk in the University of Manchester's Maths-Life sciences seminary series about the work I have done in my PhD. This was attended by people from various scientific backgrounds from the Univeristy.
Year(s) Of Engagement Activity 2019
 
Description PDS in the pub 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Postgraduate students
Results and Impact I gave a talk about my work to other holders of the PDS award at the University of Manchester. These were PhD students from across the Universty, covering all research disciplines. My talk sparked lively conversation and interesting questions.
Year(s) Of Engagement Activity 2019
 
Description PhDiscussion University of Manchester 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact PhD students from various departments at the University were invited to give 30 minute presentations on their work and surrounding topics aimed at Undergraduate students (and non-specialist PhD students). The aim of this event was give Undergraduates a taste for Postgraduate research and the various topics that they could study.

The title of my talk:
Statistical mechanics and complex systems: from Boltzmann to Turing and beyond

Abstract:
The pioneering work of Boltzmann, along with Maxwell and Gibbs, was the first successful attempt to explain macroscopic thermodynamics in terms of the uncountable interactions of constituent microscopic particles. This work, championed most notably by Einstein in his papers on Brownian motion, was expanded throughout the 20th century, allowing us to understand not only equilibrium phenomena in thermodynamic systems, but also fluctuations about equilibrium and even processes far from equilibrium. Such was the success and universality of the statistical theory, that the mathematical techniques used to describe non-equilibrium statistical mechanics have since been applied to phenomena as diverse as predator-prey interaction in ecology and pattern formation in chemical and biological systems. The use of statistical theories to provide insight into complex macroscopic phenomena remains a fruitful and active area of research today. This talk will give a brief overview of this history and present some examples of current work in statistical mechanics and complex systems.
Year(s) Of Engagement Activity 2018
 
Description Solstice conference -- Dresden 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a talk on my work at an international conference in Dresden, Germany on Discrete models of complex systems. My talk generated interesting and useful discussion and potential collaborations.
Year(s) Of Engagement Activity 2019
URL https://solstice2019.loria.fr/
 
Description Talk at IFISC institute -- UIB -- Mallorca, Spain 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave an hour-long talk about my research to other researchers in the field at IFISC in Palma de Mallorca, Spain. This talk led to collaboration on an on-going project.
Year(s) Of Engagement Activity 2019