Multiscale modelling and analysis of mechanical properties of plant cells and tissues

Lead Research Organisation: University of Dundee
Department Name: Mathematics

Abstract

One of the major challenges facing mankind is to provide enough food for the expanding world population. This problem is compounded by extreme wet-dry weather cycles induced by rapid climate change, which alters both soil structure and nutrient availability leading to yield reduction in staple and cash crops. These factors combine to present a significant problem for agriculture in both developed and developing countries. Hence, there is an immediate need to develop a new range of crops that can maintain or indeed increase yields in the face of worsening conditions and reduction in the availability and use of fertilisers and pesticides. In order to manipulate and improve plant responses to environmental changes and external mechanical forces we need to evaluate the most important physical and biochemical factors responsible for plant cell biomechanics and for the growth limitation of plant tissues.

The mechanical properties and growth of plant tissues are strongly determined by the structure of the cell wall (the main structural feature of plant cells) and the adhesion between the cells. The high complexity of the microstructure and biochemical processes in the cell wall requires mathematical modelling at the scale of its structural elements to help to close some gaps in the experimentally obtained understanding of the plant mechanics and biochemistry. New mathematical microscopic models for biomechanics of the plant cell wall and tissue will be developed in this project. A microscopic model on the scale of cell wall microfibrils will allow us to consider non-homogeneous distributions of cell wall structural elements and the biochemical interactions between them, as well as changes in the microstructure in response to internal and external stimuli.

As there are thousands of microfibrils in a cell wall and of cells in a plant tissue, effective numerical simulations of the complex microscopic models on the time and length scales of practical interest are not possible and asymptotic analysis techniques need to be applied. The techniques of periodic and locally-periodic homogenisation will be generalised to address non-periodic microstructures of plant cell walls and tissues. By applying asymptotical analysis, the macroscopic properties of plant tissues will be defined from the microscopic description of biochemical and mechanical processes. This multiscale approach and analysis of the macroscopic model will enable us to predict the influence of microscopic interactions on the macroscopic mechanical behaviour.

The new modelling and analytical approaches to be developed in this project will help us to better understand the biomechanics of plant cells and the influence of external mechanical forces on bioche- mical processes inside plant cells. The analytical and numerical results of the mathematical models combined with data from biological experiments will help us to identify new approaches to select, breed and genetically engineer improved cultivars. A better understanding of plant cell biomechanics will enable experimentalists to manipulate plant cell wall properties which in turn will lead to an improvement in the efficiency of wood, paper and biofuel production.

Planned Impact

The proposed fundamental mathematical research, devoted to the development of new modelling approaches and multiscale analysis techniques, will find its path toward impact on society and economy through collaboration with experimentalists in biology and engineering. The biologically-focussed modelling and analysis will impact the agricultural sector, whilst development of new multiscale analysis techniques will lead to improvement in engineering material performance with beneficial consequences for environment and commercial profitability.

Direct collaboration with experimental groups at The James Hutton Institute working with plant breeders, and interactions with research groups in the College of Life Sciences, Universities of Dundee, studying lignin biosynthesis in the context of biofuel production, will facilitate and ensure the pathway to impact of the proposed research on the society and economy. Both collaborating institutions have direct connections with the industrial and commercial sectors. Discussions with experimentalists in concrete research in the Civil Engineering Department, University of Dundee, and establishing new collaborations with research groups and companies in biomimetics will link the multiscale analysis techniques with the design of new engineering materials. Presentation of the results obtained from the multiscale modelling and analysis to experimental and industrial communities at both national and international conferences and workshops will support the transfer of knowledge and expertise.

There are several ways in which this research will directly contribute to industry-relevant problems:

1. Mathematical modelling of the mechanical properties of plant cells and tissues will give new insights into the control mechanisms of the root growth process and will help experimental biologists to specify the beneficial root traits to select crops ideal for problematic soils and to enhance agricultural efficiency in regions with degraded, hard and unproductive soil.

2. A better understanding of pectin biochemistry will help to manipulate the quality of pectin in plants and will contribute to more efficient production of pectin as a food ingredient.

3. The new knowledge obtained from the mathematical modelling of plant cell walls and changes in the microscopic properties in response to external and internal stimuli will provide new ideas for the possible manipulation of cell wall structural and mechanical properties and impact the efficiency and quality of pulps, paper, wood, and biofuel production.

4. Studies of the influence of the hierarchical microstructure of plant tissues on their mechanical performance will help to improve the development and production of innovative composite materials.

5. The multiscale analysis techniques that will be developed here, when applied to models describing sound and temperature propagation through concrete walls could provide a quantitative relationship between the microscopic structure and propagation impedance and allow the thermal and acoustic isolation of building materials to be optimised. This will have positive consequences for environment and quality of life.

In the context of food production for the growing world population the impact of the proposed work has also an essential international aspect and is especially important for developing countries.

The topic of the proposed research is of wide interest and its dissemination to the public will raise awareness of the innovation and impact of applied mathematical research.

Publications

10 25 50
 
Description The main results of the research supported by this award are the multiscale modelling and analysis of mechanical properties of plant cell walls and tissues as well as the generalisation of multiscale analysis techniques.
The mechanical properties and growth of plants are strongly determined by plant cell walls, that have a complex microscopic structure composed of thousands of microfibrils embedded in the cell wall matrix of proteins and polymers (hemicellulose and pectins). Thus, to analyse the mechanical properties and growth of plant tissues, it is important to consider microscopic modelling approach and to describe microstructure, chemical processes, and mechanical deformations of cell walls and tissues on the scale of microfibrils. Multiscale analysis techniques are then applied to derive macroscopic properties and behaviour of plant cell walls and tissues from the microscopic description of the physical and/or biological processes. In our novel microscopic models for plant cell wall and tissue biomechanics we consider the interactions between chemical processes (calcium-pectin chemistry) in cell walls and elastic/viscoelastic deformations of plant cell walls and tissues on the scale of cell wall microfibrils. Applying the homogenization techniques we derived macroscopic models describing the effective (macroscopic) mechanical properties of plant cell walls and tissues. Numerical simulations of the macroscopic models are used to analyse how the changes in biochemical processes influence the mechanical properties of plant cell walls and tissues and how mechanical forces affect chemical reactions inside the cell walls. For numerical simulations we used Finite Element Methods implemented in FEniCS and DUNE. The large difference in elastic properties of cell wall matrix and microfibrils required a rigorous derivation and careful numerical calculations of macroscopic elastic properties of cell walls.
It was observed that the patterns in the distribution of load-bearing (calcium-pectin) cross-links in the cell wall matrix depend on the microstructure and heterogeneous distribution of mechanical forces (stresses) in plant cell walls and tissues. Using the mathematical model we also analysed the impact of microfibrils orientation in upper and lower parts of cell walls of plant roots (difficult to visualise and analyse experimentally) on mechanical properties of plant tissues. Our analysis showed that the microfibrils orientation in upper and lower parts of root cell walls is important in the presence of tissue tension.
This research was conducted in the collaboration with the James Hutton Institute, UK and Centre for Organismal Studies, University of Heidelberg, Germany. Our microscopic modelling of plant cell wall biomechanics attracted interest from molecular biologists, resulting in an invitation to write a book chapter on mathematical modelling of calcium-pectin chemistry in "Plant Biomechanics", A. Geitmann, J. Gril, Editors, Springer 2018.
The multiscale analysis methods (the method of two-scale convergence on surfaces of locally periodic microstructures and the locally periodic unfolding method), developed here, allow us to consider a wide range of non-periodic microstructures. Using those new methods we analysed a nonlinear model describing biological processes (e.g. intercellular signalling processes) in perforated domains (e.g. extracellular space) with non-periodical distribution of perforations (e.g. cells).
The research results are published in twelve research articles and one book chapter. The research of this project was also presented to general public on the Maths Events at Dundee Science Festival, Women in Science Festival and D'Arcy Thompson Exhibition at Dundee Science Centre.
Exploitation Route The generalised homogenization methods (the method of two-scale convergence on surfaces of locally periodic microstructures and the locally periodic unfolding method) developed in this project will be / are used by researchers from the mathematics, physics and engineering communities to analyse the influence of the microstructure on the macroscopic behaviour of biological and engineering materials and to compute the macroscopic properties of composite materials with non-periodic microstructures. A better understanding of relations between the microstructure and macroscopic behaviour of a material or a tissue will provide new ideas for the engineering companies to design improved composite materials.
The derivation, multiscale analysis and numerical simulations of our mathematical models for plant cell wall and tissue biomechanics on the scale of cell wall microfibrils attracted interest from researchers in plant biology. Our multiscale models will be used to analyse experimental results on plant tissue biomechanics and are used to derive further models for mechanical properties of biological tissues. Our microscopic description of mechanical properties of plant tissues is used to developed microscopic cell-based simulations for plant root growth.
Multiscale mathematical models for plant tissue biomechanics allow us to analyse the interactions between external forces and biochemical processes in plant cell walls. This will be used to analyse the experiments on plant root growth impedance to better understand the mechanisms underlying the root growth impedance caused by hard soil. The gained understanding will help plant breeders to identify novel approached to select and breed improved cultivars, which then, through the agricultural sector, will result in production of high-yield crops. Mathematical modelling and analysis of mechanical properties of plant tissues can also help to select plants best suited for soil reinforcement, important for the prevention of slope failures.
Sectors Agriculture, Food and Drink,Construction,Environment,Manufacturing, including Industrial Biotechology

URL https://orcid.org/0000-0003-4091-5080
 
Description EMS Research Support Fund
Amount £600 (GBP)
Organisation Edinburgh Mathematical Society 
Sector Academic/University
Country United Kingdom
Start 04/2016 
End 06/2016
 
Description EMS travel grant
Amount £400 (GBP)
Organisation Edinburgh Mathematical Society 
Sector Academic/University
Country United Kingdom
Start 05/2015 
End 06/2015
 
Description NRP Early Career Researcher Exchanges
Amount £7,500 (GBP)
Organisation Northern Research Partnership 
Sector Academic/University
Country United Kingdom
Start 12/2014 
End 07/2015
 
Description Small Grant
Amount £600 (GBP)
Organisation Institute of Mathematics and its Applications 
Sector Academic/University
Country United Kingdom
Start 06/2016 
End 07/2016
 
Description Stochastic Multiscale Modelling and Analysis of Cell Signalling Processes
Amount £70,000 (GBP)
Funding ID 2003343 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 10/2015 
End 03/2019
 
Title Locally periodic unfolding method 
Description The locally periodic unfolding method is a generalisation of the periodic unfolding method, widely used for homogenisation of partial differential equations with periodically oscillating coefficients or posed in media with periodic microstructures, to locally-periodic situations (composite materials with non-periodic microstructures). The locally periodic unfolding method enables a multiscale analysis of nonlinear problems posed in domains with non-periodic microstructures. This allows us to analyse models for biological and physical processes defined in domains with non-periodic microstructures and derive effective macroscopic properties for composite materials and biological tissues with complex non-periodic microstructures. 
Type Of Material Improvements to research infrastructure 
Year Produced 2014 
Provided To Others? Yes  
Impact The development of new homogenization methods has an academic impact on the further development of multiscale analysis. The new multiscale analysis method developed in this research project is of subsequent use to researchers from the mathematics, physics and engineering communities studying the influence of microscopic structures on the macroscopic behaviour of biological or engineering materials, especially considering non-periodic microstructures, e.g. plywood-like structure of biomaterials or locally-periodic microstructure of concrete. The knowledge of effective macroscopic properties of composite materials is important for e.g. civil engineering and biomimetics. A better understanding of relations between microstructure and macroscopic behaviour will provide new ideas for the engineering companies to design improved composite materials. 
URL http://epubs.siam.org/doi/pdf/10.1137/140978405
 
Title Multiscale mathematical models for plant biomechanics 
Description Novel mathematical models for plant cell wall and tissue biomechanics are derived and analysed. In the microscopic model for cell wall biomechanics the interactions between chemical processes in a plant cell wall and its mechanical properties are defined on the scale of cell wall microfibrils. Using homogenisation techniques the macroscopic model for cell wall biomechanics is derived. In the microscopic model for plant tissue biomechanics we considered the chemical processes and mechanical deformations on the scale of plant cells. Then using the homogenization techniques we derived the macroscopic model for plant tissue biomechanics. This multiscale approach and analysis of the macroscopic models allow us to define the influence of microscopic interactions and heterogeneity of cell wall matrix and plant tissue on the patterns in the dynamics of chemical processes and on the macroscopic mechanical deformations of plant cell walls and tissues. Numerical codes for simulations of the macroscopic mathematical models for plant cell wall biomechanics were written using the free softwares FEniCs and DUNE (Distributed and Unified Numerics Environment). 
Type Of Material Improvements to research infrastructure 
Year Produced 2015 
Provided To Others? Yes  
Impact The analysis of the mathematical models and comparison of numerical solutions of the mathematical models with experimental results will allow us to better understand the mechanisms controlling plant growth and development at the cellular level and give new insight into how plants respond to changes in environmental stresses. The gained understanding will help plant breeders to identify novel approached to select and breed improved cultivars, which then, through the agricultural sector, will result in production of high-yield crops. 
URL http://www.esaim-m2an.org/articles/m2an/pdf/2016/02/m2an140104.pdf
 
Title Two-scale convergence on the surfaces of locally periodic microstructures and locally periodic boundary unfolding method 
Description The methods of locally periodic two-scale convergence on oscillating surfaces and the locally periodic boundary unfolding operator allow us to analyse differential equations defined on boundaries of non-periodic microstructures and consider nonhomogeneous Neumann conditions on the boundaries of perforations, distributed non periodically, e.g. processes on the cell membranes in tissues with non-periodic distribution of cells. There new homogenisation methods allows us to define rigorously macroscopic properties and behaviour of composite materials with non-periodic microstructures from the microscopic description of physical and/or biological processes. 
Type Of Material Improvements to research infrastructure 
Year Produced 2014 
Provided To Others? Yes  
Impact The development of new homogenization methods has an academic impact on the further development of multiscale analysis. The new multiscale analysis methods developed in this research project are of subsequent use to researchers from the mathematics, physics and engineering communities studying the influence of microscopic structures on the macroscopic behaviour of biological or engineering materials, especially considering non-periodic microstructures, e.g. plywood-like structure of biomaterials or locally-periodic microstructure of concrete. The knowledge of effective macroscopic properties of composite materials is important for e.g. civil engineering and biomimetics. A better understanding of relations between microstructure and macroscopic behaviour will provide new ideas for the engineering companies to design improved composite materials. 
URL http://epubs.siam.org/doi/pdf/10.1137/140978405
 
Description Collaboration with COS, University of Heidelberg 
Organisation Heidelberg University
Country Germany 
Sector Academic/University 
PI Contribution The discussions with the collaboration partner from the Centre for Organismal Studies (COS), University of Heidelberg, during the development of the mathematical model for plant cell wall biomechanics induced new interesting biological equations, which will be tested experimentally. I also visited the research group of the collaboration partner at COS, University of Heidelberg, to discuss the experimental results on the dynamics and distribution of pectins in plant cell walls and tissues.
Collaborator Contribution Many discussions with the collaboration partner, his excellent expertise in molecular biology of plants, and the experimental results, obtained by the group of the collaboration partner, were important for the development of our new mathematical models for plant cell wall and tissue biomechanics. The collaboration partner also visited the University of Dundee to discuss biological questions arising during the development of our mathematical models and to give a seminar talk on his research.
Impact This is a multi-disciplinary collaboration between a biologist and a mathematician. The main outcomes of this collaboration at the moment are the new mathematical models for plant cell wall and tissue biomechanics, numerical implementation of the model equations, and the analysis of the interplay between mechanical forces and distribution of calcium-pectin cross-links using numerical simulations of the model equations. Together with a PhD student we study the coupling between the dynamics of calcium-pectin chemistry in plant cell walls and intercellular signalling processes. This coupling between mechanical properties of cell walls and intercellular signalling pathways is a part of cell wall signalling processes, which are studied experimentally in the group of the collaborator from COS, University of Heidelberg.
Start Year 2014
 
Description Collaboration with JHI (Dr. Glyn Bengough) 
Organisation James Hutton Institute
Country United Kingdom 
Sector Charity/Non Profit 
PI Contribution The multiscale mathematical models for plant cell wall and tissue biomechanics developed in this research project is used to analyse the impact of external forces on mechanical properties of plant tissues and will be applied to analyse the impact of external forces on mechanical properties and growth of plant roots, which is one of the main research interests of the experimental collaborator from the James Hutton Institute.
Collaborator Contribution The excellent experimental expertise of the collaboration partner in mechanical properties of plant tissues supported the development of mathematical models for plant cell wall and tissue biomechanics.
Impact This is a multi-disciplinary collaboration between a mathematician and a biologist/engineer. The development of novel mathematical models for plant cell wall and tissue biomechanics and numerical implementations of the model equations are the main outcomes of this collaboration up-to-date. The next step in this collaboration is to use the mathematical models and experimental results to analyse the impact of external forces on the mechanical properties and growth of plant tissues.
Start Year 2014
 
Description Multiscale modelling of plant tissue biomechanics 
Organisation UiT The Arctic University of Norway
Country Norway 
Sector Academic/University 
PI Contribution The multiscale modelling and analysis of plant tissue biomechanics are the main aims of this research collaboration. To work on this research problem the collaborator from the UiT - The Arctic University of Norway, campus Narvik, visited the University of Dundee for two weeks in January/February 2015. To continue to work on the problem and to finalise the article I was visiting the The Arctic University of Norway in Narvik in April and August 2015. These visits were supported by my Northern Research Partnership Early Career Research Exchanges grant.
Collaborator Contribution In the context of this research collaboration I visited The Arctic University of Norway, campus Narvik, twice and during these visits we finalised our paper on the multiscale modelling and analysis of plant tissue biomechanics when considering periodic distribution of cells in a plant tissue (published in 2017). We also started the second manuscript on the multiscale analysis of a biomechanical model for plant tissues with randomly distributed cells.
Impact Together with the collaborator from The Arctic University of Norway we published in 2017 an article on the multiscale analysis of the biomechanical model for a plant tissue. In this work we considered periodic distribution of cells in a plant tissue. The manuscript on the homogenization of the biomechanical model for plant tissues with randomly distributed cells is in the final stage of preparation.
Start Year 2015
 
Description "Meet the Mathematicians" Event at the Dundee Science Centre as part of the D'Arcy Thompson Exhibition 2015 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Public/other audiences
Results and Impact "Meet the Mathematicians" Event was a part of the D'Arcy Thompson Exhibition 2015. The D'Arcy Thompson Exhibition 2015 was organised by the D'Arcy Thompson Museum and Division of Mathematics, University of Dundee, as part of celebrating D'Arcy Thompson 150th anniversary. Numerical simulations, the app for conforming and non-conforming maps (transformations) which were applied by D'Arcy Thompson on plants and animals to show similarities in their morphologies, many different mathematical games were used to present the main ideas of the research of D'Arcy Thompson and of the research in Mathematical Biology at the Division of Mathematics, University of Dundee.
Many visitors of the event and many questions indicated the interest of the general public in the research in Mathematical Biology.
Banners and a movie describing the research in Mathematical Biology at the Division of Mathematics, University of Dundee, were displayed through the whole time of the exhibition (21 Aug 2015 - 25 Oct 2015). One of the banners and a part of the movie represented also the main ideas and results of my research on the multiscale modelling and analysis of plants.
Year(s) Of Engagement Activity 2015
URL http://www.dundee.ac.uk/museum/exhibitions/zoology/equations/
 
Description Life and Light in Numbers, Maths Event at Dundee Science Festival 2014 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? Yes
Geographic Reach Regional
Primary Audience Public/other audiences
Results and Impact Maths Event, Life and Light in Numbers, was an event for general public at the Dundee Science Centre as part of Dundee Science Festival 2014. We were awarded the main auditorium at the Dundee Science Centre and in total 16 staff manned 10 separate display stations. Our visitors (kids and adults) had the opportunity to learn how mathematics is used to better understand the world around us. They were fascinated to hear that mathematical research is used to improve not only the cancer treatment, but also crop and biofuel production. Hands-on activities, films, and numerical simulations were used to present our research to the general public. The event had positive impact on the general public's interest in mathematical research.

Feedback has highlighted the popularity and effectiveness of this approach of using hands-on exhibits, physical and virtual, that introduce mathematical concepts in a fun and interactive way and provide a starting point to explore mathematical ideas.

Our Maths Event at Dundee Science Festival increased the interest of general public in studying Mathematical and in mathematical research. This event also provided our visitors the possibility to see how much mathematics is in the world around us and how mathematical methods can help to solve some real world problems. An additional possible impact will be an increasing interest in our forthcoming Maths Event at Dundee Science Festival 2014 on 16th November.
Year(s) Of Engagement Activity 2014
URL http://www.maths.dundee.ac.uk/sciencefestival/
 
Description Maths Even at Women in Science Festival 2015, Dundee Science Centre 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Public/other audiences
Results and Impact Maths Event at the "Meet the Expert" Day at the Dundee Science Centre as part of Women in Science Festival 2015 was an event for general public. Our visitors (kids and adults) had the opportunity to learn how mathematics is used to better understand the world around us. Physical demonstrations were used to engage the general public and to introduce mathematical concepts in a fun and interactive way. Each hands-on exhibit presented an interesting phenomenon related to our research. Computer simulations were used to communicate the importance of computing in modern mathematical/scientific work as well as to present our research results; discussion of mathematical concepts were aiming to inspire audiences with the beauty and power of mathematics. They were fascinated to hear how mathematical research is used to better understand development and growth of plants. This event had positive impact on the general public's interest in mathematical research in general and in mathematical modelling and analysis of plants in particular.
Year(s) Of Engagement Activity 2015
 
Description Maths Week Scotland 2017 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Public/other audiences
Results and Impact Hand-on activities and displays popularising Mathematics and demonstrating research in Mathematics in general and in Mathematical Biology in particular. Different types of expositions and maths games were interesting to children and adults.
Year(s) Of Engagement Activity 2017
URL https://blogs.gov.scot/making-maths-count/2017/09/06/maths-week-scotland-whats-on/
 
Description Nuffield Research project for school students 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? Yes
Geographic Reach Regional
Primary Audience Schools
Results and Impact In the context of Nuffield Research Placements programme a school student was working with me for four weeks during the summer on a research project in mathematical biology. Together with the student we developed a simple model for chemical reactions describing changes in chemical configurations of pectins in plant cell walls. During this activity the school student learned some modelling and computation techniques. The numerical simulations of the mathematical model provided some interesting information about the dynamics of processes related to the changes in chemical configurations of pectins. This information can be used to improve further our simple mathematical model. This activity had positive impact on the interest of school students in research and in studying at a university and, hence, on the increase of a skilled scientific and engineering workforce.

It activity had a positive impact on the interest of school students in studying mathematics as well as in mathematical research related to mathematical modelling of biological systems, especially in mathematical modelling of plants. This activity also increased interest in mathematical research in the public.
Year(s) Of Engagement Activity 2014
URL http://www.nuffieldfoundation.org/nuffield-research-placements
 
Description Nuffield Research project for school students 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Schools
Results and Impact In the context of Nuffield Research Placements programme, together with two school students and one of my colleagues we were working on a five weeks research project and studied the properties of mathematical and physical percolation in different systems. Students developed an algorithm for mathematical description of percolation, evaluated results of numerical simulations, investigated critical phenomena and performed experiments of percolation in porous media, e.g. soil. This activity contributed positively to the school students interest in science and mathematical research.
Year(s) Of Engagement Activity 2015
URL http://www.nuffieldfoundation.org/nuffield-research-placements