Team:Exeter/Model
Introducing Our Models
Due to restricted lab time, modelling has been a hugely important part of our project this summer. Our modelling efforts have been focussed on the diffusion of gases within hydrogels and modelling the mechanical properties of hydrogels. We have also done some statistical modelling on UK and global carbon dioxide emissions in the hope of gaining insight and inferences into the scale on which climate change is affecting our planet.
We would also like to thank University College London for performing Flux Balance Analysis on our Bacillus subtilis terminal chassis. This has provided us with great data to help better our project in terms of its use in the real world as well as future steps that we could look to take with our project!
Modelling Carbon Dioxide Diffusion into a Hydrogel
Overview
The aim of this model is to simulate how carbon dioxide diffuses into a hydrogel. The colour change is due to carbon dioxide acidifying in the hydrogel (much like how oceans acidify). The reason why this simulation was made was to understand to what depth carbon dioxide thoroughly diffuses into a hydrogel of a given viscosity and therefore informing us of what is the upper limit of depth we would have to consider when printing using the bacteria/hydrogel complex such that the bacteria are exposed to as much carbon dioxide as possible.
Introduction
Diffusion of gases and the investigation of the viscosity of hydrogels are themselves very active areas of research, but only make up part of our project. We still tasked ourselves with the building of such a model in order to inform ourselves, from a theoretical as well as an experimental stand point, on which hydrogel is optimal for use in our printer. The essence of scientific modelling is to simplify the model as much as possible until the observed behaviour matches with the predicted behaviour of the model. The key assumptions of the model include:
- 1. Carbon dioxide diffusion still follows Brownian motion in the hydrogel.
- 2. The hydrogel is homogenous and has uniform viscosity.
With the foresight that the project direction will vary a lot, the modelling team built the model to be as generalised as possible. As such, it can be easily adapted to various experimental scenarios.
The greatest areas of success of the model are:
- 1. Its generalisation - the model boundary conditions for the fixed space can easily be adjusted for length and for whether they allow flow. Hence, the model can show the diffusion of carbon dioxide into mediums of varying viscosity and parameters.
- 2. User feedback – with the careful adjustment of space and time steps, as well as using less error-prone numerical methods, the model is able to show how the diffusion of carbon dioxide occurs in real time.
The scientific paper shows how greatly the model has been adapted over the course of the project. The main sections of the model are described below along with the paper itself.
Section 1
This section aims to introduce the reader to the model being used which is the diffusion equation in 1D.
Section 2
Here the first numerical approach is considered; explicit finite differences method. The section primarily focusses on the mathematical discretisation of the model in 1D, the stability analysis of the numerical method and the boundary conditions being used. This section documents the initial model-building stage.
Section 3
The boundary conditions are fundamental to the behaviour of the model. This section focusses on taking a more sophisticated approach to them and applying them to the code.
Section 4
In this section, the reader is first introduced to how the real-time timer has been implemented into the code as well as a deliberation of how to improve the aesthetics of the graphic output.
Section 5
This section expands on Section 4 where the presentation of the model is improved further in order to create a colour grid.
Section 6
By the time Section 6 was reached, the model had started being used in the laboratory. It was quickly discovered that there are several problems with the model when put in practical use, namely; limitations of stability and that the model was limited to 1D.
At this point, the modelling team began work on building an identical model but using a different and far more stable numerical method - implicit finite differences method - thereby extending the model beyond 1D to 2D.
This section documents how the combination of different boundary conditions can influence the output of the code. Furthermore, model functionality was further expanded by incorporating theory of fluid dynamics into it.
Section 7
Section 7 is aimed to be a synopsis for the reader of what they have read so far and how each part has contributed and steered the modelling work.
Section 8
In this section, the 1D diffusion model is improved to the extent that it is unconditionally stable. No longer does the user need to use only certain combinations of time-steps, space-steps and diffusion constants in order for the code to run. It can now work with any combination.
Section 9
At this point in the paper, the final model output is displayed after all considerations for the hydrogel experiments have been made.
Section 10
This section explains where the parameters, which have been used in the model displayed in section 9, have been deduced from and the experimental and simulated inferences have been made to draw conclusions for the project.
Section 11
In the concluding section, a brief overview of the the success of the model and experiments has been made and a final inference for the engineering of the project has been drawn with mathematical justification.
If you have trouble viewing the PDF please click here!
Modelling Carbon Dioxide emissions from the UK and Globally
Overview
This modelling section is comprised of four different data sets of carbon dioxide emissions each analysed in a similar way whereby a linear model using the data is constructed. The suitability of the imposed models relative to the data is discussed and model extrapolations are used to make predictions for the future of carbon emissions. The rates from the FBA (Flux Balance Analysis) have been used to understand the magnitude of the carbon problem.
Introduction
In this paper, the modelling team has worked to produce an accompanying paper going into greater detail to show how and where the statistical inferences have been drawn from for Human Practices.
Statistical analysis has been carried out on source data for net carbon emissions from the UK and Globally. The inferences from data help to quantify for the reader how the project is good for the world by looking at carbon emissions during recent history.
If you have trouble viewing the PDF please click here!
Modelling the Mechanical Properties of Hydrogel
The aim of the project is to create a new way of producing calcium carbonate using genetically engineered bacteria that take in carbon dioxide from the environment. The calcium will need to be embedded in a hydrogel which has a certain shape. This will produce an object which is of the same shape as the hydrogel.
The modelling team therefore decided to try to model how the hydrogel would change shape in response to the environmental stresses. In the printer, these stresses would force the hydrogel into the shape of choice.
Hydrogels exhibit a property known as viscoelasticity, which means that they show both elastic behaviour, such as in a solid, and viscous behaviour, such as in fluids.
Section 1 - Theory of Mechanical Behaviour
Hydrogels are either rubber elastic or viscoelastic, depending on whether they're swollen. If the hydrogel isn't swollen it shows rubber elastic properties, however the more swollen it is, it shows more viscoelastic behaviour.
The hydrogel can be modelled as a series of cross linked polymers. Because there are more permutations of states of hydrogels when they are folded, the act of unfolding them creates a state of lower entropy. From the second law of thermodynamics which states that the entropy of an isolated system must always increase, the hydrogel experiences an elastic force that wants to fold it again in order to maximise the entropy.
The result of this is that when a hydrogel is stretched to a significant fraction of its original length the stress forces ceases to increase, unlike with a simple elastic material. The shear modulus G is also no longer a constant and is actually a function of the strain itself. This is true for hydrogels even when they're in a swollen state.
Section 2 - Viscoelasticity
When hydrogels are highly swollen, they exhibit a property called viscoelasticity. In simple terms, this is when a material behaves both elastically, such as in a solid, or viscously, such as in a liquid. This can be modelled physically using what's known as the "spring and dashpot" model. The spring represents the elastic component and the dashpot represents the viscous component. How these components are combined can produce different models of viscoelasticity.
Section 3 - Maxwell Model
This consists of a spring and dashpot connected in series. In this model, if the polymer is put under constant strain, the stresses gradually relax. However, the viscous component grows with time as long as the stress is applied.
Section 4 - Kelvin-Voigt Model
This consists of a spring and dashpot in parallel. This model represents a material undergoing constant a reversable viscoelastic strain. Under constant stress, the material is found to deform at a decreasing rate.
Section 5 - Modelling Viscoelasticity
The modelling team were able to translate these models into differential equations, which were then solved numerically in MATLAB. The solver was a fourth order Runge-Kutta method, and used the process of discretisation to produce a plot of the solution of the differential equation. This was done for various different experimental conditions of the material, including the material being held at a constant stress or a constant strain. The result was a graph which showed how the dependent variable (either stress or strain) changed over time.
Below are two graphs comparing the different models. They are both of how the strain responds to the material when it is held at a constant stress.
Section 6 - Comparison between Models
From the analysis, the Kelvin Voigt model more accurately predicts what would happen to the strain in real life. Rather than constantly increasing forever (according to the Maxwell model), it asymptotically approaches a constant value which corresponds to elastic behaviour. In reality, a viscoelastic material can be modelled by using a combination of both Maxwell and Kelvin-Voigt components.
Section 7 - Relevance to the Project
Our aim is to prototype a 3D bioprinter, and a large part of it is to know how quickly a print can be made. The timescale at which a 3D shape can be printed depends highly on its mechanical properties such as viscosity and elastic moduli, which affects both the rate at which different shapes can be formed. It is the viscoelastic property of the hydrogel which makes it an ideal choice of material for use in a bioprinter.
If you have trouble viewing the PDF please click here!
Flux Balance Analysis
Flux Balance Analysis (FBA) is a widely used approach to analyze the flow of metabolites through a metabolic network. In our case, this made it possible to predict the rate of production of our terminal Bacillus subtilis (B.subtilis) chassis and the production rate of biotechnologically important metabolites such as our carbonate ions [1]. Flux Balance analysis is particularly useful for maximising productivity through metabolic engineering strategies e.g. gene deletion.
The FBA performed on our behalf by UCL gave us a critical insight into several elements of the biology of B.subtilis. A 2D line plot produced by this analysis informed us that a lactose uptake rate of around 4 mmol/gDCW/h gave us the greatest growth rate for the least amount of lactose. At lactose uptake rates greater than 4mmol/gDCW/h, there was no further increase in growth. A 3D simulation performed as an extension of this analysis revealed that the minimum carbonate production (at an uptake of -1mmol/gDCW/h for both CO2 and urea) was 42.85mmol/gDCW/h. At a more realistic uptake of -20mmol/gDCW/h for both CO2 and urea, the B.subtilis was producing 80.85mmol/gDCW/h. These simulations were incredibly useful and revealed a clear, expected trend; carbonate production directly positively correlates with urea and CO2 uptake.
In addition to this, analysing potential metabolic engineering strategies through a single gene deletion simulation revealed that the rate of carbonate production could be increased by knocking out the non-essential BSU21920 gene, which encodes a UDP-glucose diacylglyceroltransferase. Taking ‘CalcifEXE’ technology further would require taking insights such as this into consideration for future laboratory research and genetic circuit design.
