Biomathematics

Summary

Biomathematics is a field that applies mathematical techniques to analyze and model biological phenomena. Often a collaborative effort, mathematicians and biologists work together using mathematical tools such as algorithms and differential equations to understand and illustrate a specific biological function. Biomathematics is used in a wide variety of applications from medicine to agriculture. As new technologies lead to a rise in the amount of biological data available, biomathematics will become a discipline increasingly in demand to help analyze and effectively utilize the data.

Definition and Basic Principles

Biomathematics is a discipline that quantifies biological occurrences using mathematical tools. Biomathematics is related to and may be a part of other disciplines including bioinformatics, biophysics, bioengineering, and computational biology, as these disciplines include using mathematical tools in biology.

Biologists explain biological functions in many ways, often employing words or pictures. Biomathematics allows biologists to illustrate these functions using techniques such as algorithms and differential equations. Biological phenomena vary in scale and complexity, encompassing everything from molecules to ecosystems. Therefore, creating a model requires scientists to make some assumptions to simplify the process. Biomathematical models vary in length and complexity and several models may be tested.

The use of biomathematics is not limited to modeling a biological function. Other techniques include structuring and analyzing data. Scientists may use biomathematics to organize data or analyze data sets, and statistics are often considered an integral tool.

Background and History

As early as the 1600s, mathematics was used to explain biological phenomena, although the mathematical tools used date back even further. In 1628, British physician William Harvey used mathematics to prove that blood circulates in the body. His model changed the belief that there were two kinds of blood. In the mid-1800s, Gregor Mendel, an Augustinian monk, used mathematics to analyze the data he obtained from his experiments with pea plants. His experiments became the basis for genetics. In the early 1900s, British mathematician R. A. Fisher applied statistical methods to population biology, providing a better framework for studying the field. In 1947, theoretical physicist Nicolas Rashevsky argued that mathematical tools should be applied to biological processes and created a group dedicated to mathematical biology. Though some dismiss Rashevsky's work as too theoretical, many view him as one of the founders of mathematical biology. In the 1950s, the Hodgkin-Huxley equations were developed to describe a cellular function known as ion channels. These equations are still used. In the 1980s, the Smith-Waterman algorithm was created to aid scientists in comparing DNA sequences. While the algorithm was not particularly efficient, it paved the way for the BLAST (Basic Local Alignment Search Tool) software, which has allowed scientists to compare DNA sequences since 1990. Although mathematical tools have been applied to some biological problems during the second half of the twentieth century, the practice has not been all-inclusive. In the twenty-first century, there has been a renewed interest in biology becoming more quantitative, partly due to an increase in new data.

How It Works

Basic Mathematical Tools. Biomathematicians may use mathematical tools at different points during the investigation of a biological function. Mathematical tools may be used to organize, analyze, or generate data. Algorithms, which use symbols and procedures for solving problems, are employed in biomathematics in several ways. They may analyze data, as in sequence analysis. Sequence analysis uses specifically developed algorithms to detect similarity in pieces of DNA. Specifically developed algorithms are also used to predict the structure of different biological molecules, such as proteins. Algorithms have led to the development of more useful biological instruments such as specific types of microscopy. Statistics are another common way of analyzing biological data. Statistics are used to analyze data, and this data may help create an equation to describe a theory. Statistics were used to analyze the movement of single cells. The data taken from the analysis was then used to create partial equations describing cell movement.

Differential equations, which use variables to express changes over time, are another common biomathematics technique. There are two kinds of differential equations—linear and nonlinear. Nonlinear equations are commonly used in biomathematics. Differential equations and other tools have been used to model the functions of intercellular processes. Differential equations are utilized in several important systems used in biomathematics for modeling, including mean-field approaches. Other modeling systems include patch models, reaction-diffusion equations, stochastic models, and interacting particle systems. Each modeling system provides a different approach based on varying assumptions. Computers provide an easier way to apply and solve complex equations. Computer modeling of dynamic systems, such as the motion of proteins, is also a work in progress.

New methods and technology have increased the amount of data obtained from biological experimentation. The data gained through experimentation and analysis may be structured in different ways. Mathematics may be used to determine the structure. For example, phylogenetic trees (treelike graphs that illustrate how pieces of data relate to one another) use different mathematical tools, including matrices, to determine their structure. Phylogenetic trees also provide a model for how a particular piece of data evolved. Another way to organize data is a site graph, or hidden Markov model, which uses probability to illustrate relationships between the data.

Modeling a Biological Function. The scientist may be at different starting points when considering a mathematical model. They may start with data already analyzed or organized by a mathematical technique or previously described by a visual depiction or written theory. However, there are several considerations that scientists must take into account when creating a mathematical model. As biology covers a wide range of matter from molecules to ecosystems, when constructing a model the scale of phenomena must be considered. The time scale and complexity must be considered, as many biological systems are dynamic or interact with their environment. The scientist must make assumptions about the biological phenomena to reduce the parameters used in the model. The scientist may then define important variables and the relationships between them. Often, more than one model may be created and tested.

Applications and Products

The field of biomathematics applies to every area of biology. For example, biomathematics has been used to study population growth, evolution, genetic variation, and inheritance. Mathematical models have also been created for communities, modeling competition or predators, often using differential equations. Whether the scale is large or small, biomathematics allows scientists a greater understanding of biological phenomena.

Molecules and Cells. Biomathematics has been applied to various biological molecules, including DNA, ribonucleic acid (RNA), and proteins. Biomathematics may be used to help predict the structure of these molecules or help determine how certain molecules are related to one another. Scientists have used biomathematics to model how bacteria can obtain new, important traits by transferring genetic material between different strains. This information is important because bacteria may, through sharing genetic material, acquire a trait like resistance to an antibiotic. To model the sharing of a trait, scientists have combined two ways to structure data—the phylogenetic tree and the site graph. The phylogenetic tree illustrates how the types of bacteria are related. The site graph illustrates how pieces of genetic material interact. Then, scientists use a particular algorithm to determine the model’s parameters. Using such tools, scientists can predict which areas of genetic material are most likely to transfer between the bacterium.

Biomathematics has been used in cellular biology to model various cellular functions, including cellular division. The models can help scientists organize information and provide a deeper understanding of cellular functions. Cellular movement is one example of an application of biomathematics to cellular biology. Cellular movements can be seen as a set of steps. The scientists first considered certain cellular steps or functions, including how a cell senses a signal and how this signal is used within the cell to start movement. Scientists also considered the environment surrounding the cell, how the signal was provided, and the processes occurring within the cell to read the signal and start movement. The scientists could then build a mathematical function that considers these steps. Depending on the particular question, the scientists may focus on any of these steps. Therefore, more than one model may be used.

Organisms and Agriculture.  Biomathematics has been used to create mathematical models for different functions of organisms. One popular area has been organism movement, where models have been created for bacteria movement and insect flight. A more complete understanding of organisms through mathematical models supports new technologies in agriculture. Biomathematics may also be used to help protect harvests. For example, biomathematics has been used to model a type of algae bloom known as brown tide. In the late 1980s, the brown tide appeared near Long Island, New York, badly affecting the shellfish population by blocking sunlight and depleting oxygen. Four years later, the algae blooms receded. Both mathematicians and scientists collaborated to create a model of the brown tide to understand why it bloomed and whether it will bloom in the future. To create a model, the collaborators used differential equations. They focused on population density, which included factors such as temperature and nutrients. The collaborators had to consider many variables and remove the ones they did not consider important. For instance, they hypothesized that a drought followed by rain may have affected growth. They also considered fertilizers and pesticides that were used in the area. A better understanding of the brown tide may help protect the shellfish harvest in future years.

Medical Uses. Biomathematical models have been developed to illustrate various functions within the human body, including the heart, kidneys, and cardiac and neural tissue. Biomathematics is useful in modeling cancer, enabling scientists to learn more about the type of cancer, thereby allowing them to study the efficacy of different types of treatment. One project focused on modeling colon cancer on a genetic and molecular level. Not only did scientists gain information about the genetic mutations present during colon cancer, but they also developed a model that predicted when tumor cells would be sensitive to radiation, which is the most common way to treat colon cancer. Studies like this can be built on in future experimentations, the results of which may someday be used by doctors to create more effective cancer treatments.

Biomathematics has also been used to organize and analyze data from experiments concerning drug efficacy and gene expression in cancer cells. Using matrices, statistics, and algorithms, scientists understand if a particular drug is more likely to work based on the patient's cancer cell's gene expression. Biomathematics has also been integral in epidemiology, the field that studies diseases within a population. Biomathematics may be used to model various aspects of a disease such as human immunodeficiency virus (HIV), allowing for more comprehensive planning and treatment.

Careers and Course Work

Degree programs in biomathematics are gaining popularity in universities. Some schools have biomathematics departments, and others have biomathematics programs within the mathematics or biology departments. Undergraduate coursework for a bachelor’s degree in biomathematics encompasses mathematics, biology, and computer science, including calculus, chemistry, genetics, physics, software development, probability, statistics, organic chemistry, epidemiology, population biology, molecular biology, and physiology. A student may also choose to receive a B.S. in biology or mathematics. Students may seek additional opportunities outside their program. Ohio State's Mathematical Biosciences Institute offers summer programs for undergraduate and graduate students.

In the field of biomathematics, a doctorate is required for most careers. Doctoral programs in biomathematics include coursework in statistics, biology, probability, differential equations, linear algebra, cellular modeling, genetics modeling, computer programming, pharmacology, and clinical research methods. In addition, doctoral candidates often perform biomathematics research with support from departmental faculty. As with the undergraduate degrees, a student may pursue a doctorate in biology or mathematics.

With the influx of biological data from new technologies and tools, a degree in biomathematics is imperative. Those who receive a Ph.D. may choose to enter a postdoctoral program, like Ohio State's Mathematical Biosciences Institute, which offers postdoctoral fellowships, mentorships, and research opportunities. Other career paths include research in medicine, biology, and mathematics with universities and private research institutions; work with software development and computer modeling; teaching; or collaborating with other professionals and consulting in an industry like pharmaceuticals or bioengineering.

Social Context and Future Prospects

While mathematical tools have been applied to biology for some time, many scientists believe there is a need for increased quantitative analysis of biology. Some call for more emphasis on mathematics in high school and undergraduate biology classes. They believe this will advance biomathematics. As more universities develop biomathematics departments and degrees, more mathematics classes will be added to the curriculum. There is concern in the biomathematics field about the assumptions used to create simplified mathematical models. More complex and accurate models will likely be developed.

Important future applications for biomathematics will be in the bioengineering and medical industries. Developing mathematical models for complex biological phenomena will aid scientists in a deeper understanding that can lead to more effective treatments in areas such as tumor therapy. As new tools and methods continue to develop, biomathematics will be a field that expands to sort and analyze the large influx of data.

Bibliographies

Barros, Laécio Carvalho de, et al. A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics: Theory and Applications. 2nd ed., Springer, 2024.

Chaitlin, Gregory J. Proving Darwin: Making Biology Mathematical. Vintage, 2013.

Hochberg, Robert, and Kathleen Gabric. “A Provably Necessary Symbiosis.” The American Biology Teacher, vol. 72, no. 5, 2010, pp. 296-300. doi.org/10.1525/abt.2010.72.5.7.

Kondrashov, Dmitry A. Quantifying Life: A Symbiosis of Computation, Mathematics, and Biology. U of Chicago P, 2016.

Mondaini, Rubern. Trends in Biomathematics: Chaos and Control in Epidemics, Ecosystems, and Cells. Springer Nature, 2022.

Schnell, Santiago, Ramon Grima, and Philip Maini. “Multiscale Modeling in Biology.” American Scientist, vol. 95, no. 2, 2007, pp. 134-142. doi.org/10.1511/2007.64.1018.