By: Moriah Echlin
What is mathematical biology?
Mathematics and biology have a long history together. From evolution to molecular biology, there have been many contributions from mathematics to the investigation of life, and vice versa. While mathematics is often used as an analytical tool for understanding biological data, mathematical biologists are more concerned with using math to describe biological systems and phenomena in the form of mathematical models. For a simple example, a model of disease spread within a population could be described as a process of the interaction between healthy and sick individuals, accounting for the rate at which disease spreads. That is, the number of infected people next week is a product of the number of susceptible people that encountered infected people this week and the chance that the disease spread between them. In mathematical terms this would look like:
S(t+1) = r*S(t)*I(t)
Where S and I are susceptible and infected populations, r is the rate of infection, and t is the unit of time. In this way, mathematical biologists create models for systems such as the growth of bacteria in biofilms, the firing of neurons, the biomechanical movements of the human body, and the evolution of species. In general, mathematical models are used to ask questions about the underlying biology of the modeled system. Questions like “what mechanisms make a certain behavior possible?”, “how will the system behave in the future?”, or “what would happen if a specific change occurred in the system?”. In cancer biology, many of these questions revolve around how cancer develops, how it changes and gains resistance to drugs over time, and what possible treatment strategies are. If there are countless mathematical models across biology, there are more still within the discipline of cancer biology.
Gene regulatory networks and their models
The gene regulatory network, or GRN is the network of DNA, RNA and proteins that interact to form a master regulator for which genes are being expressed in a cell at any given time. Gene regulatory networks are a major determinant in the functional behavior of a cell and have been studied for their roles in cellular differentiation during development, cell response to their environment, and cellular malfunction in disease among other areas. In the lab, we work with mathematical models of GRNs in prostate cancer in order to better understand how changes in the GRN can enable cancerous cellular behaviors to develop. We can also use these models to investigate how cancer cells evolve over time or what targeted interventions could be used to ameliorate disease.
The art of model creation
One of the hardest parts of being a mathematical biologist is the creation of the model itself. No model will perfectly capture all the details of biology, so many aspects of the system have to be abstracted or simplified. The choice of which details to retain in a model as well as what type of mathematics to use are part of the art of the science. In our work, we take a very simplified approach and use Boolean network models of the GRN, which treat genes as either being ON or OFF. These models had their beginning in evolutionary biology in the late 1960s but have since expanded to other fields of biology as well as to outside disciplines. As models of the GRN, Boolean networks are appealing because they capture the qualitative behavior of the gene regulatory network while still being relatively simple to formulate and analyze.
From data to systems of equations
A model formulated directly from knowledge of the system is useful for investigating the consequences of either known or hypothesized mechanisms, for example the role of nutrient production in the competition between microbial species over shared resources. A model formulated using collected data to inform the model is useful for investigating systems for which many interactions are unknown, such as in our prostate cancer model, in which there are many unknowns within the gene regulatory network. Therefore, we rely on data collected from donated prostate tissue.
In particular, we utilize single-cell transcriptomic data to infer our gene regulatory network models. As single-cell technology has advanced, more and more data is being generated and made publicly available. This data provides a unique resource as it captures the range and heterogeneity of cellular behavior within a tissue which is vital to understand when trying to model the mechanisms underlying that behavior.
Where will we go from here?
Currently we are in the process of inferring our network models from single-cell data. After we have generated our Boolean network GRNs, we can use the relevant mathematical tools to analyze the cellular behaviors that the GRN proscribes. By creating models of prostate cancer at different stages of disease, we can then compare our analyses to identify what properties of the GRN relate to different cancerous behaviors. With more knowledge of the mechanisms driving cancerous behavior, we are one step closer in understanding this disease.
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