Geometric, Topological, and Statistical Dynamics in Soft Matter and Mathematical Biology

Lead Research Organisation: University of Cambridge
Department Name: Applied Maths and Theoretical Physics

Abstract

This is an intradisciplinary proposal to study both long-standing and new problems within Fluid and Continuum Mechanics and Mathematical Biology, and which, due to their dynamical or statistical nature, need to be addressed with new tools from Geometry, Topology and Statistics. Uniquely for a proposal in Mathematics, the research will be carried out with strong support from experimental work.

Topological ideas have played an essential role in science ever since Kelvin's Vortex theory of atoms. Though short-lived, in part because its knotted vortices are mostly unstable (one of the root causes of turbulence) the theory nevertheless served to initiate Tait's monumental work on topological properties of knots. Since then, despite recognition of the great importance of dynamic reconfigurations of topology in fields ranging from fluid dynamics to developmental biology, progress has been limited, in part due to the nonlinear character of the processes involved and lack of suitable model systems. The proposed research divides into three sub-projects.

1. Statistics of physical fibre bundles. In collaboration with scientists at Unilever Research (Port Sunlight), we shall
continue a line of research initiated several years ago on the physics of fibre arrays such as hair. Begun originally to
understand how to improve such products as shampoos and conditioners, the research uses ideas from statistical physics and fluid mechanics to construct a theory for the elastic properties of physical fibre bundles. We will extend this work to address fundamental mathematical issues related to the entanglement of physical bundles, including the statistics of contacts and crossing within bundles, the relationship between true knottedness, physical knottedness, and entanglements, and the dynamics of bundles with quenched random intrinsic curvature.

2. Topological rearrangements and singularities of soap films. From magnetohydrodynamics to proteins there are fundamental open questions regarding how knotted configurations relax to energetic minima under dissipation, subject to topological constraints. The experimental challenges in visualization and control of these processes motivate the search for laboratory realizations in which the dynamics occurs reproducibly and on demand. We have identified such an example: interconversions of minimal surfaces (represented by soap films) triggered by slow boundary deformation.
In spite of the large amount of existing work on minimal surfaces, little, if anything, has been studied regarding the dynamics of interconversion between them. We aim to develop a classification scheme for these processes and predictive PDEs describing the dynamical evolution and location of singularities for each universality class.

3. Topological inversion of embryonic algae. When Lewis Wolpert said [i]t is not birth, marriage, or death, but gastrulation which is truly the most important time in your life, he was referring to the process by which an animal embryo changes its topology from simply connected to non-simply connected by developing an inward passage that eventually becomes its gastric system. While animal embryos are notoriously difficult to study systematically because of difficulties to control and visualize, it has recently become clear that a process that occurs in multicellular algae, and which is analogous to gastrulation, provides a much simpler but faithful alternative that is amenable to careful experimentation. This process of embryonic inversion operates in an entire class of multicellular algae, with systematic variations determined by the organism size. In spite of the experimental and empirical information currently available, there is as yet no mathematical description of how such an object can turn itself inside out. Our goal is to utilize the principles of Continuum Mechanics and Differential Geometry to understand this fascinating phenomenon in Mathematical Biology

Planned Impact

The three sub-projects within this proposal will have different types of potential impact, as summarized below.

The work on statistics, entanglements, and dynamics of fibre array bundles grew out of industrial interest (Unilever) associated with very basic questions about the properties of hair. Our work thus far has had direct impact within Unilever through the development of imaging and analysis methods to quantify the properties of hair, as needed to assess personal care products. We expect continued impact on this industry as we seek to address problems involving friction within bundles and their movement. The longstanding work with Unilever is also relevant to our work on Plateau border dynamics, which is very important to the wetting of fibres, a problem of central importance to the action of surfactants on hair. Continued collaboration with Unilever scientists will assure close alignment with their research goals. As well, the wealth of materials that consist of fibre arrays is enormous, from felt to insulation, and we anticipate that development of mathematical models for their gross properties, based on the particular details of the microstructure, will be of broad interest. The intense worldwide interest generated by our first paper on this subject serves to indicate how the research area can be used as a case study for the wider public in the application of mathematics to real-world problems.

The work on dynamical transformations that take one minimal surface to another is based on a comprehensive approach, involving theory, numerics and experiments. Here too there has already been impact on the wider public, through the BBC Four program "POP! The Science of Bubbles" that aired in 2013. We are in discussion with the producer of that program about possible future programs on the science of everyday phenomena. With these studies we have shown again how mathematics (and laboratory experiments) can be used to address universal questions on a broad range of length scales. Our work is regularly used in the classroom to demonstrate singularities and bifurcations, and we will work to create a standardized module for physics education that illustrates these points. Although the mathematical details of the work in this proposal may be technical and abstract, the problem that we shall investigate is so easily demonstrated with little equipment, and has such a natural beauty, that it is a perfect vehicle to introduce abstract mathematical concepts to a lay audience. In particular, the fact that a simplified version of the experiment can be done in one's kitchen means that children will naturally be drawn to the subject. There are many opportunities to present this research to the general public, for DAMTP and the University of Cambridge have in place programs specifically for this. These include the annual Cambridge Summer Science Festival, lectures for school-age children through the Millenium Mathematics Project, and annual Open Houses in DAMTP.

Our work on embryonic inversion will have obvious impact on developmental biology, and the PI has forged links with scientists and laboratories worldwide interested in these problems. As developmental processes analogous to inversion are so ubiquitous, we envision that the theoretical and computational advances made in this sub-project will be of use to biomedical researchers studying development. We will conduct public outreach activities through various learned societies, as we have done in the past (including the Royal Society Summer Science Exhibition), and also through an upcoming Royal Society Discussion meeting on Bio-Inspiration in 2015. But we also envision interesting avenues for the design of new materials with tunable shapes through the incorporation of shape-changing elements. Development of a theoretical framework for the optimality of such shape changes may lead to interesting developments in materials science.

Publications

10 25 50
publication icon
Azeem M (2020) Optimal Design of Multilayer Fog Collectors. in ACS applied materials & interfaces

publication icon
De Canio G (2017) Spontaneous oscillations of elastic filaments induced by molecular motors. in Journal of the Royal Society, Interface

publication icon
Goldstein R (2016) Batchelor Prize Lecture Fluid dynamics at the scale of the cell in Journal of Fluid Mechanics

publication icon
Goldstein RE (2015) Green Algae as Model Organisms for Biological Fluid Dynamics. in Annual review of fluid mechanics

publication icon
Goldstein RE (2018) Are theoretical results 'Results'? in eLife

publication icon
Haas P (2019) Shape-shifting polyhedral droplets in Physical Review Research

publication icon
Haas P (2020) Subpopulations and stability in microbial communities in Physical Review Research

publication icon
Haas PA (2017) Theory of Shape-Shifting Droplets. in Physical review letters

publication icon
Haas PA (2015) Elasticity and glocality: initiation of embryonic inversion in Volvox. in Journal of the Royal Society, Interface

publication icon
Ishikawa T (2020) Stability of dancing Volvox in Journal of Fluid Mechanics

publication icon
Kirkegaard J (2018) The role of tumbling frequency and persistence in optimal run-and-tumble chemotaxis in IMA Journal of Applied Mathematics

publication icon
Kirkegaard JB (2016) Aerotaxis in the closest relatives of animals. in eLife

publication icon
Lushi E (2017) Scattering of biflagellate microswimmers from surfaces. in Physical review. E

 
Description During the course of this grant we have made significant progress in generating new knowledge and opening up new research directions in four areas.

The first involves studies of a particular process of tissue folding driven by changes in the shapes of constituent cells as a model for generic events of this type in a wide range
of biological contexts. Focusing on "embryonic inversion" in green algae, in which daughter organisms literally turn themselves inside out through a sequence of cell shape changes, we have generated the first high-resolution three-dimensional data on the shape evolution and developed a quantitative elastic theory to explain the phenomenon. This opens up the study of tissue folding in general to quantitative approaches.

The second area of progress concerns the dynamics of topological rearrangements in fluid mechanics such as the reconnections of magnetic field lines emanating from the surface of the sun. By focusing on the transitions that occur when soap films supported by wire frames are rendered unstable by deforming those frames, we have been able to break new ground in the mathematical analysis of such problems, particularly by generating laboratory experiments to test the various new concepts. This has included the development of new computational approaches to the partial differential equations that describe these processes, and mappings between them and well-known phenomena on condensed matter physics.

A third significant body of results concerns the phenomenon of cytoplasmic streaming in biological cells. This is the persistent circulatory motion of the fluid contents of large eukaryotic cells driven the action of certain "motor proteins" along filaments. Such motion is particularly prominent during certain stages in embryonic development of model organisms, but has remained unexplained on any quantitative level. Through a combination of analytical and computational studies we have now provided such a quantitative theory that reveals the existence of a novel "swirling instability" that arises from the coupling between fluid motion and the deformability of the filaments. In the course of this work we contributed to the development of new type of continuum theory for the kinds of dense fiber arrays that occur in a variety of biological and industrial processes.

A fourth area of our research involved the phenomenon of bioluminescence - spontaneous light emission - generated by marine microorganisms. This is has been of
interest to mankind for millennia, as it lights up nighttime breaking waves and water sliced by a ship's prow. While the internal biochemical mechanism of light generation has
been well understood for many years, the mechanism by which fluid motion triggers cells to give off light has remained unclear. Our combined experimental and theoretical
work has now opened up new research questions by providing a novel, quantitative method of studying this process at the single cell level that revealed some very clear systematics in the response of cells to external stimuli.
Exploitation Route I expect our research outcomes to continue to influence academic researchers by having answered a number of longstanding scientific questions and opening up new research areas in biological physics and mathematics. These include the subjects of shape-shifting droplets, tissue folding, interconversions of minimal surfaces, cytoplasmic streaming, and bioluminescence. We anticipate this activity will also lead to workshops and/or conferences on the national UK scene. In terms of societal impact, we look forward to outreach activities sponsored by professional societies and scientific media organizations, similar to those already undertaken. Finally, a new collaboration with Procter and Gamble on the physical properties of hair is likely to lead to direct industrial impact.
Sectors Education,Environment,Healthcare

URL https://www.damtp.cam.ac.uk/user/gold
 
Description Our general findings on biological physics and soft matter were highlighted in the BBC4 program "The River - A Year in the Life of the Tay" that aired in the summer of 2019. This program followed the physical course of the Tay from its origins to the sea, and described the many fascinating organisms that live in the water. Research from my laboratory was shown to explain aspects of multicellularity, and I had the opportunity on camera to explain our scientific questions and methodologies. This program overall did much to bring to a general audience current questions in evolutionary biology and biological physics.
First Year Of Impact 2019
Sector Education,Environment
Impact Types Societal

 
Description Marine Microbiology Initiative
Amount $1,018,750 (USD)
Funding ID 7523 
Organisation Gordon and Betty Moore Foundation 
Sector Charity/Non Profit
Country United States
Start 05/2018 
End 11/2020
 
Description Physics of Water Harvesting by Fog Collectors 
Organisation Adolfo Ibáñez University
Country Chile 
Sector Academic/University 
PI Contribution Dr Adriana Pesci and I contributed to initial experimental and theoretical analyses of the impacts of fog droplets on long slender fibres as occur in harp-style fog collectors used in areas of the world with little rainfall.
Collaborator Contribution Our partners (primarly Prof Jacques Dumais and his postdoc and student) built a bespoke experimental apparatus to test various designs of water collectors and further developed the theory to help understand how to maximize the efficiency of such collectors.
Impact Our primary output is a set of design principles for fog collectors that pinpoints multilayer harp collectors with a particular wire spacing and number of layers. These criteria should have a considerable impact on water harvesting in economically deprived areas of the world. The major publication resulting from this work appeared in 2020.
Start Year 2017
 
Description Byre2017 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Public/other audiences
Results and Impact Public Lecture, Byre Theatre, University of St. Andrews, Scotland, 29 August, 2017
Year(s) Of Engagement Activity 2017
 
Description CUSC 2019 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact Lecture, Cambridge University Scientific Society, Cambridge, 3 December, 2019
Year(s) Of Engagement Activity 2019
URL http://scisoc.com/event/lessons-from-the-old-literature-prof-ray-goldstein/
 
Description Cambridge Physics Centre 2020 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Schools
Results and Impact Outreach lecture to a broad audience of school children from the region. Provided an opportunity to discuss unusual research motivations in the context of a subject that has had broad public interest over the years.
Year(s) Of Engagement Activity 2020
URL http://outreach.phy.cam.ac.uk/programme/cpc/cpcposters/feb2020
 
Description FM105 interview 
Form Of Engagement Activity A broadcast e.g. TV/radio/film/podcast (other than news/press)
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Public/other audiences
Results and Impact A radio interview on FM105 Cambridgeshire, focusing on my research and musical interests.
Year(s) Of Engagement Activity 2018
URL https://www.mixcloud.com/duncan_odom/ray-goldstein-interview-in-2018-on-cambridge-105-science/
 
Description INIArt2017 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact Lecture, Newton Institute Workshop on "Form & Deformation in Art, Toys and Games, Cambridge, 1 December, 2017
Year(s) Of Engagement Activity 2017
URL http://www.turing-gateway.cam.ac.uk/event/ofbw36
 
Description King's College 2020 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact Physical Sciences Seminar, King's College, Cambridge 29 January, 2020
Year(s) Of Engagement Activity 2020
 
Description Lecture at Gresham College 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Public/other audiences
Results and Impact This was one of the public lectures sponsored by Gresham College, London, as part of their lecture series in Mathematics. It was entitled "Leonardo, Rapunzel, and the Mathematics of Hair".
Year(s) Of Engagement Activity 2016
 
Description Trinity Math Society 2019 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact Lecture, Trinity Mathematical Society, Cambridge, 11 November, 2019
Year(s) Of Engagement Activity 2019
URL https://tms.soc.srcf.net/wp-content/uploads/2019/10/termcard2019Michaelmas.pdf