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Mathematical and theoretical biology

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Mathematical and theoretical biology, or biomathematics, is a part of biology that uses math and ideas to study how living things grow, behave, and are built. It is different from experimental biology, which tests ideas by doing experiments.[1] Sometimes, it is called mathematical biology to show it uses more math, or theoretical biology to show it uses more ideas.[2] Theoretical biology makes general rules for biology, while mathematical biology uses math to study living things, even though people sometimes mix up the words.[3][4]

Mathematical biology makes math models to explain how living things work. It uses math tools to help with research. By using math to describe living things, scientists can make predictions and learn things they might not see in real experiments. This needs accurate math models.

Since living things are very complex, theoretical biology uses many kinds of math and has helped create new ways to solve problems.[5]

History

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Early history

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Math has been used in biology since the 13th century. Fibonacci used the famous Fibonacci numbers to explain how rabbits multiply. In the 1700s, Daniel Bernoulli used math to study how smallpox affects people. In 1789, Thomas Malthus wrote about how human populations grow quickly, using the idea of exponential growth. In 1836, Pierre François Verhulst made the logistic growth model.

In 1879, Fritz Müller used math to explain how some animals copy each other to survive better, called Müllerian mimicry. This was the first time math was used to show how strong natural selection is, except for Malthus's earlier idea that populations grow fast (he called it "geometric"), but food and resources grow slowly. This idea influenced Charles Darwin.[6]

The phrase "theoretical biology" was first used by Johannes Reinke in 1901 and later by Jakob von Uexküll in 1920. A key book in this field is On Growth and Form by D'Arcy Thompson in 1917.[7] Other early thinkers include Ronald Fisher, Hans Leo Przibram, Vito Volterra, Nicolas Rashevsky, and Conrad Hal Waddington.[8]

Recent growth

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Interest in this field has grown quickly since the 1960s. Some reasons why this happened are:

  • Big sets of data from the genomics revolution are hard to understand without using special math tools.
  • New math ideas like chaos theory help explain complex, non-linear systems in biology.[9]
  • More powerful computers make it easier to do large calculations and simulations that were impossible before.
  • More people are using computer-based experiments (in silico) because real experiments with humans and animals can be risky, unreliable, or have ethical problems.

Areas of research

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There are many areas of research in mathematical and theoretical biology. Projects from different universities and thousands of scientists working in this field are listed. Many examples involve very complicated systems. These systems can only be explained by using a mix of math, logic, physics, chemistry, molecular science, and computer models.[10][11][12][13][14]

Abstract relational biology

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Abstract relational biology (ARB) studies general models of complex living systems without focusing on shapes or body parts. One simple model is the Metabolic-Replication system (M,R-system) created by Robert Rosen in 1957–1958 to explain how cells and organisms are organized.

Other ideas in this field include autopoiesis by Maturana and Varela, Kauffman’s Work-Constraints cycles, and more recently, closure of constraints.[15]

Algebraic biology

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Algebraic biology, also called symbolic systems biology, uses algebra and symbolic math to solve biology problems. It is mostly used for studying genes, proteins, and molecular structures.[16][17][18]

Complex systems biology

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This field grew after 1970 to explain more complicated life processes. It connects molecular set theory, relational biology, and algebraic biology.

Computer models and automata theory

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Many studies by 1986 are grouped in this topic, including:[19][20][21]

  • Computer models for biology and medicine
  • Models of blood flow in arteries
  • Neuron models
  • Biochemical networks and oscillations
  • Quantum computers and automata in biology[22]
  • Cancer models
  • Neural networks and genetic networks
  • Mathematical categories in biology
  • Metabolic-replication systems
  • Cellular automata
  • Self-reproducing systems
  • Chaotic systems in living things
  • Theories about organisms[23][24]

Modeling cell and molecular biology

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This area is growing fast because of molecular biology’s importance.

Examples of biological models

  • Mechanics of biological tissues[25]
  • Enzyme activity and speed
  • Cancer models and simulations[26]
  • Movement of groups of cells[27]
  • Scar tissue formation[28]
  • Inside-the-cell activity[29][30]
  • Cell cycles
  • Cell death (apoptosis)[31]

Modeling body systems

  • Blood vessel disease[32]
  • Multi-scale heart models
  • Electrical properties of muscles

Computational neuroscience

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Also called theoretical or mathematical neuroscience, this uses math to study the nervous system.[33]

Evolutionary Biology

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Math has been important in studying evolution and ecology.[34]

Population genetics

This uses math to study changes in genes. It looks at:

  • New gene versions (mutations)
  • New gene combinations (recombination)
  • How often genes appear (allele frequencies)

Ronald Fisher helped develop this field with new ideas in statistics. Phylogenetics is a branch that studies family trees of living things based on traits.

Population dynamics

This studies how populations grow or shrink. It started with Thomas Malthus’s model in 1798. The Lotka–Volterra equations for predators and prey are also well-known. This field connects with mathematical epidemiology, which models how diseases spread.

Evolutionary game theory

John Maynard Smith and George Price created this field. It focuses on how inherited traits affect survival without worrying about genes. Adaptive dynamics is a related field that uses math to study how traits change over time.

Mathematical biophysics

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In the early days, mathematical biology focused on mathematical biophysics. This uses math to study how physical systems in living things work. It creates specific models of how parts of living systems behave.

Deterministic processes (dynamical systems)

These processes follow fixed rules where the same starting point always leads to the same result. Different paths never cross.

  • Difference equations/Maps: Time moves in steps, but states change smoothly.
  • Ordinary differential equations: Time flows smoothly, states change smoothly, no space changes.
  • Partial differential equations: Time and state change smoothly, space also changes.
  • Cellular automata: Time and states move in steps with fixed rules.

Stochastic processes (random Systems)

These processes use random rules. The outcome depends on probabilities, not fixed paths.

  • Non-Markovian processes: Time flows smoothly, and past events affect future states.
  • Jump Markov processes: Time flows smoothly, past events do not affect future states. Events happen after random waiting times.
  • Continuous Markov processes: Time and state flow smoothly, events follow random paths.

Spatial Modelling

This studies how things spread or form patterns in space. Alan Turing’s 1952 paper on how shapes and patterns form in living things is a famous example.

Examples:

  • Wound healing waves
  • Swarming behavior
  • Morphogenesis theory
  • Biological patterns[35]
  • Turing patterns[36]

Mathematical Methods

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A biological system is turned into equations. Solving these equations shows how the system behaves over time or when balanced. The model and equations assume certain rules about the system.

Molecular Set Theory (MST)

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This theory uses math to describe how molecules interact in chemical reactions. Anthony Bartholomay developed it. MST helps in biology and medicine to study biochemical changes and disease processes.

Organizational Biology

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This field studies how parts of living things depend on each other. It uses circular systems where actions repeat or feed back into themselves.

Example:

  • Abstract relational biology (ARB) uses general models to study complex systems, not focusing on shape or structure. Robert Rosen’s Metabolic-Replication (M,R) systems, from 1957–1958, explain how cells and organisms organize themselves.[37]

References

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Citations

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  1. ↑ "What is mathematical biology | Centre for Mathematical Biology | University of Bath". 2018-09-23. Archived from the original on 23 September 2018. Retrieved 2025-01-12.
  2. ↑ "Careers in Theoretical Biology". 2019-09-14. Archived from the original on 14 September 2019. Retrieved 2025-01-12.
  3. ↑ Longo, Giuseppe; Soto, Ana M. (2016-10-01). "Why do we need theories?". Progress in Biophysics and Molecular Biology. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches. 122 (1): 4–10. doi:10.1016/j.pbiomolbio.2016.06.005. ISSN 0079-6107. PMC 5501401. PMID 27390105.
  4. ↑ Montévil, Maël; Speroni, Lucia; Sonnenschein, Carlos; Soto, Ana M. (2016-10-01). "Modeling mammary organogenesis from biological first principles: Cells and their physical constraints". Progress in Biophysics and Molecular Biology. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches. 122 (1): 58–69. doi:10.1016/j.pbiomolbio.2016.08.004. ISSN 0079-6107. PMC 5563449. PMID 27544910.
  5. ↑ Robeva, Raina; Davies, Robin; Hodge, Terrell; Enyedi, Alexander (2010-09-13). "Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics". CBE—Life Sciences Education. 9 (3): 227–240. doi:10.1187/cbe.10-03-0019. PMC 2931670. PMID 20810955.
  6. ↑ Mallet, James (2001-07-31). "Mimicry: An interface between psychology and evolution". Proceedings of the National Academy of Sciences. 98 (16): 8928–8930. Bibcode:2001PNAS...98.8928M. doi:10.1073/pnas.171326298. PMC 55348. PMID 11481461.
  7. ↑ Stewart, Ian (1998). Life's other secret: the new mathematics of the living world. New York: John Wiley. ISBN 978-0-471-15845-5.
  8. ↑ Keller, Evelyn Fox (2009). Making Sense of Life: Explaining Biological Development with Models, Metaphors, and Machines. Evelyn Fox Keller. Cambridge: Harvard University Press. ISBN 978-0-674-01250-9.
  9. ↑ Reed, Michael C. (2015-11-01). "Mathematical Biology is Good for Mathematics" (PDF). Notices of the American Mathematical Society. 62 (10): 1172–1176. doi:10.1090/noti1288. ISSN 0002-9920.
  10. ↑ Baianu IC, Brown R, Georgescu G, Glazebrook JF (2006). "Complex Non-linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks". Axiomathes. 16 (1–2): 65–122. doi:10.1007/s10516-005-3973-8. S2CID 9907900.
  11. ↑ Baianu IC (2004). "Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models" (PDF). Archived from the original on 2007-07-13. Retrieved 2011-08-07.
  12. ↑ Baianu I, Prisecaru V (April 2012). "Complex Systems Analysis of Arrested Neural Cell Differentiation during Development and Analogous Cell Cycling Models in Carcinogenesis". Nature Precedings. doi:10.1038/npre.2012.7101.1.
  13. ↑ "Research in Mathematical Biology". Maths.gla.ac.uk. Retrieved 2008-09-10.
  14. ↑ Jungck JR (May 1997). "Ten equations that changed biology: mathematics in problem-solving biology curricula" (PDF). Bioscene. 23 (1): 11–36. Archived from the original (PDF) on 2009-03-26.
  15. ↑ Montévil, Maël; Mossio, Matteo (2015-05-07). "Biological organisation as closure of constraints". Journal of Theoretical Biology. 372: 179–191. Bibcode:2015JThBi.372..179M. doi:10.1016/j.jtbi.2015.02.029. ISSN 0022-5193. PMID 25752259.
  16. ↑ Baianu IC (1987). "Computer Models and Automata Theory in Biology and Medicine". In Witten M (ed.). Mathematical Models in Medicine. Vol. 7. New York: Pergamon Press. pp. 1513–1577.
  17. ↑ Barnett MP (2006). "Symbolic calculation in the life sciences: trends and prospects" (PDF). In Anai H, Horimoto K (eds.). Algebraic Biology 2005. Computer Algebra in Biology. Tokyo: Universal Academy Press. Archived from the original (PDF) on 2006-06-16.
  18. ↑ Preziosi L (2003). Cancer Modelling and Simulation (PDF). Chapman Hall/CRC Press. ISBN 1-58488-361-8. Archived from the original (PDF) on March 10, 2012.
  19. ↑ Witten M, ed. (1986). "Computer Models and Automata Theory in Biology and Medicine" (PDF). Mathematical Modeling : Mathematical Models in Medicine. Vol. 7. New York: Pergamon Press. pp. 1513–1577.
  20. ↑ Lin HC (2004). "Computer Simulations and the Question of Computability of Biological Systems" (PDF).
  21. ↑ Computer Models and Automata Theory in Biology and Medicine. 1986.
  22. ↑ Baianu, Professor Ion (1983-09-01). Landahl, H. D. (ed.). "NATURAL TRANSFORMATION MODELS IN MOLECULAR BIOLOGY". Proceedings of SIAM & Society for Mathematical Biology Meeting. AMS (SIAM): 230–232.
  23. ↑ Weisstein, Eric W. "Dual Tessellation". mathworld.wolfram.com. Retrieved 2025-01-12.
  24. ↑ Preston Jr., Kendall (1985). Modern Cellular Automata: Theory and Applications. Advanced Applications in Pattern Recognition Ser. M. J. B. Duff. New York, NY: Springer. ISBN 978-0-306-41737-5.
  25. ↑ Wang, Yafei; Brodin, Erik; Nishii, Kenichiro; Frieboes, Hermann B.; Mumenthaler, Shannon M.; Sparks, Jessica L.; Macklin, Paul (2021-01-18). "Impact of tumor-parenchyma biomechanics on liver metastatic progression: a multi-model approach". Scientific Reports. 11 (1): 1710. Bibcode:2021NatSR..11.1710W. doi:10.1038/s41598-020-78780-7. ISSN 2045-2322. PMC 7813881. PMID 33462259.
  26. ↑ Oprisan, Sorinel A.; Oprisan, Ana (2006-03-01). "A Computational Model of Oncogenesis using the Systemic Approach". Axiomathes. 16 (1): 155–163. doi:10.1007/s10516-005-4943-x. ISSN 1572-8390.
  27. ↑ "Jonathan Sherratt's Research Interests". www.ma.hw.ac.uk. Retrieved 2025-01-12.
  28. ↑ "Jonathan Sherratt's Research: Scar Formation". www.ma.hw.ac.uk. Retrieved 2025-01-12.
  29. ↑ Kuznetsov, A. V.; Avramenko, A. A. (2009-04-01). "A macroscopic model of traffic jams in axons". Mathematical Biosciences. 218 (2): 142–152. doi:10.1016/j.mbs.2009.01.005. ISSN 0025-5564. PMID 19563741.
  30. ↑ Wolkenhauer, O.; Ullah, M.; Kolch, W.; Cho, Kwang-Hyun (2004-09-01). "Modeling and simulation of intracellular dynamics: choosing an appropriate framework". IEEE Transactions on NanoBioscience. 3 (3): 200–207. doi:10.1109/TNB.2004.833694. ISSN 1558-2639. PMID 15473072.
  31. ↑ Fussenegger, Martin; Bailey, James E.; Varner, Jeffrey (2000-07-01). "A mathematical model of caspase function in apoptosis". Nature Biotechnology. 18 (7): 768–774. doi:10.1038/77589. ISSN 1546-1696. PMID 10888847.
  32. ↑ Computational intelligence methods for bioinformatics and biostatistics. New York, NY: Springer Berlin Heidelberg. 2017. ISBN 978-3-319-67833-7.
  33. ↑ Trappenberg, Thomas P. (2007). Fundamentals of computational neuroscience (Reprinted ed.). Oxford: Oxford Univ. Press. ISBN 978-0-19-851582-1.
  34. ↑ Semple, Charles; Steel, M. A. (2003). Phylogenetics. Oxford lecture series in mathematics and its applications. Oxford ; New York: Oxford University Press. ISBN 978-0-19-850942-4.
  35. ↑ Hurlbert, Stuart H. (1990). "Spatial Distribution of the Montane Unicorn". Oikos. 58 (3): 257–271. Bibcode:1990Oikos..58..257H. doi:10.2307/3545216. ISSN 0030-1299. JSTOR 3545216.
  36. ↑ Copeland, B. Jack (2017). The Turing guide. Oxford: Oxford University Press. ISBN 978-0-19-874782-6.
  37. ↑ Rosen, Robert (1991). Life itself: a comprehensive inquiry into the nature, origin, and fabrication of life. Complexity in ecological systems series. New York: Columbia Univ. Press. ISBN 978-0-231-07565-7.
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