An introduction to mathematical modelling
How we understand and control the natural world with maths
Mathematical modelling has been thrown into the spotlight recently, as governments around the world turn to scientific advisors and mathematicians to guide their management of the COVID-19 pandemic. But what are mathematical models, and how do scientists go about writing natural phenomena in terms of maths in the first place?
Mathematical models vs. mathematical laws
Cast your mind back to school physics or chemistry and you’re likely to recall a certain amount of mathematics involved. Boyle’s law (relating pressure and volume of a gas), the law of gravity, Newton’s second law (relating force, mass, and acceleration) — all mathematical formalisms of laws underpinning the world around us.
These are mathematical models based on theories; if the maths doesn’t match reality, it indicates the underlying theory may not be complete. These theorems are significant and underpin many of the advancements in engineering that enable our modern lives. However, much of the natural world is messier, more random, and more complex than systems to which laws can be applied.
Mathematical modelling offers insight into how to predict, control, and optimise systems.
Instead, mathematical modelling aims to break down complex systems into their key dynamics and interactions. These system features (say, the R rate…) are set out in mathematical form, with each term of the model equation representing a particular feature (usually, though not always the case). Isolating each feature as part of the wider model enables us to experiment with the system computationally, to determine which dynamics are most significant, and explore what impact any changes may have on the system, beyond the limitations of testing in real-life.
These systems occur most frequently in biology, but the following discussion of the steps to build a successful mathematical model can be applied elsewhere in epidemiology, finance, weather forecasting, and managing systems such as agriculture and healthcare. Mathematical modelling offers unparalleled insight into how to predict, control, and optimise systems, that has become a fundamental part of how we interact with the natural world.
Step 1: Forming a conceptual model
The first step is to break a complex system down into the main features and interactions involved. Information about different parts of the system is drawn together from scientific literature, and specifically designed observation/experiments required to fill in any knowledge gaps, as we will see later. This puts together enough information for a conceptual model, sufficient to draw a diagram of what’s going on, but not enough information to make predictions about the system with any certainty.
Let’s look at an example of a morphogen, a general term given to chemicals in the body that act as cues for cells to switch on/off certain behaviours. Such chemicals are responsible for the patterning of animal coats, and signalling cells in the early embryo to form organs, vessels, and fingers in the right places.
If we want to model the amount of morphogen at a particular point and time, we would likely include the following features:
- production of the morphogen, by a specific cell type;
- diffusion of the morphogen (spreading out into the surrounding area, like perfume droplets in the air);
- uptake (absorption) of the morphogen by other cell types, and
- degradation of the morphogen (i.e., they have a shelf-life, usually short in the unideal storage conditions of the body).
Further dynamics, such as the proliferation (cell division) of the cells responsible for morphogen production and uptake, cell death, and cell migration could also be considered. Deciding which features are necessary to include require consideration of the time and spatial scale you are interested in for the purpose of your model.
The importance of scale
Biology in particular works across vast scales; from gene dynamics inside the nucleus, to morphogen production and other cell-cell interactions, to tissue-level dynamics of a whole organ or the mechanics of a joint in the skeleton. Dynamics working across these different time and spatial scales will hold varying importance in a particular model, and may not need to be explicitly considered depending on the level of detail required.
For example, production of a morphogen begins with DNA transcription in the cell’s nucleus, with a flurry of other activity responsible for the secretion of each morphogen molecule by the cell. However, if we are on a tissue level, in which thousands of these molecules can be found per square area, we do not need to model the discrete number of molecules, and can instead model the concentration (a continuous variable). Similarly, if our timescale spans minutes or hours, rather than seconds or milliseconds, we will not need to explicitly consider what is happening inside each cell. Internal cell processes happen on such a comparatively fast scale that only the end result of the internal production dynamics is needed — the overall rate of production, rather than rate of transcription, etc.
Step 2: Writing down the maths
To make informed predictions about a complex system, we are required to gather enough data to turn the conceptual model into a rigorous, tightly defined mathematical model that can be tested against further data collection. So how can something such as the movement of a cell, the flow of water, the spread of a disease, be written in terms of mathematics?
For this, we have the culmination of great minds turning their attention to the applications of mathematics to thank. A typical equation of a mathematical model today is, roughly speaking, formed of several terms that are each models in their own right.
Each of our morphogen model features are a mini-system that have previously been researched and modelled. Diffusion models are tested and confirmed by experiments using dye in water; the model of production of a chemical by a cell is a result of many experiments over the years, that has shown us the dependence of such a model on both the type of chemical, cell type, and potentially the environment around it; the uptake of chemicals by cells is a similarly involved process.
Let’s look at one element: production of the morphogen. First, we consider the key factors influencing production:
- Environmental factors such as oxygen level, and the presence of other morphogens.
- Since cells produce the morphogen, we expect an explicit dependence on the number of cells.
- Extreme numbers of cells (overcrowding or very few cells) may halt or boost production.
- Similarly, very high or low concentrations of morphogen may halt or boost production.
We can illustrate these scenarios, of which one or more may be true, by plotting what the rate of morphogen concentration would look like in each case.
- The first graph shows a linear increase in the rate of production with an increase in cell density. This equation is simply f(X) = X
where f(X) denotes rate of production, and X is the cell density. - The second graph shows a ‘saturating’ rate of production, as the production rate reaches a maximum level as the cell density increases. This equation is f(X)= X/(1+X).
- The third graph plots the rate of production compared to morphogen concentration already present, with production rate decreasing as the morphogen concentration increases. This equation is f(X)= 1/(1+X), where X in this case is the morphogen concentration.
Hence, in many cases, constructing a mathematical model is a case of piecing together the appropriate building blocks to form a mathematical representation of the system. To personalise the model to the system — the particular morphogen, and specific cell types involved, we require data.
Step 3: Making use of data
Data are used for two purposes in modelling: to increase the accuracy of the model by providing required parameters, and to validate the model, by providing a real-life comparison to the model outcomes.
Parameters
Parameters are the name given to the values required in the model: for the morphogen this may include the rate of diffusion, rate of production (by the cells), and rate of degradation.
These parameters make the model specific to the system being modelled, and are usually based on observation and experiment. Consider the equation below, where the two terms on the right-hand side can be used to model morphogen production and decay respectively. Here, X is the concentration of the morphogen, a is the rate of production, c is the rate of decay, and b is a saturating term that limits production at high concentration levels. Each morphogen and cell type will exhibit different rates of production/decay that inform the relevant values.
We can see the impact of the parameters by plotting this function for various values below. This function value corresponds to the rate of change of the morphogen concentration; to find the morphogen concentration at a particular time requires solving a differential equation.
Validation
Model validation involves comparing the model outcomes (such as morphogen concentration distribution) to equivalent measurements made of the real-life system. This usually requires several data points (that were not used to find the parameters for the model), to check the accuracy of the model’s predictions.*
It is at this point that a loop between model refinement and validation is formed, as model outcomes may motivate slight changes to parameter values, or inclusion of dynamics that were previously excluded. This can also involve further experiment to investigate other features, or test new model predictions, where possible.
Importantly, the model is only made as complex as is necessary; it must be accurate, but also simple enough to offer insights into what is going on in the system of interest. In the same way, we have a strong preference for only including model components that are directly related to physical properties of the system.
An unparalleled tool
Models such as those based on the example given have endless uses and possibilities in biology and other areas mentioned in the introduction. Identifying which morphogens are responsible for different aspects of embryologic development help identify how things go wrong, and aid understanding and prevention of birth defects or complication (e.g spina bifida). Famously, the interactions of two morphogens lead to what are known as Turing patterns, capable of replicating patterning on fish and other animal coats. Switch morphogens out for pharmaceutical drug delivery, with the term ‘production’ replaced with ‘drug release’, and we have the foundations of a model predicting the concentration and distribution of a drug delivered to a part of the body — such as chemotherapy in a tumour.
Mathematical models not only offer predictions, but benefit the advance of scientific knowledge in this way — and the more tools we have to unpick phenomena that on the surface appear complex, mysterious, and unpredictable, the more we can understand and better our world.
*You may notice that in the case of modelling an ongoing pandemic, the model is validated in real-time, as there are no such experiments that can be conducted to confirm the model’s predictions, and the number of variables are too great to compare well to previous pandemics. This is why feeding into the SAGE committee are up to 10 modelling groups, each using a model with slightly different assumptions, that collectively advise the government of possible outcomes and consequences of policy decisions. This is also why government ministers and the media, in my opinion, should make clear that the modelling is a two-way street: models and model outcomes must adapt to live policy making, and ongoing changes in government response lead to continuous updating of model predictions.
Further examples of mathematical modelling:
- Sugar, Mining, and Energy Conservation, The Conversation
- Delivery chemotherapy to prevent drug resistance in cancer cells, The Conversation
- Movement of vesicles in cells, The Conversation
- Modelling brain and skull growth, Oxford University
- Infectious disease modelling, NIH
- Mathematical model of communication, Medium
- Modelling population growth and Neanderthals, Medium
- Maths of weather prediction, The Guardian
