Mathematical and theoretical biology

**Mathematical and theoretical biolog**y, 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. Sometimes, it is called mathematical biology to show it uses more math, or theoretical biology to show it uses more ideas. Theoretical biology makes general rules for biology, while mathematical biology uses math to study living things, even though people sometimes mix up the words.

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.

### Early history

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 slowl<nowiki/>y. This idea influenced Charles Darwin.

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. Other early thinkers include Ronald Fisher, Hans Leo Przibram, Vito Volterra, Nicolas Rashevsky, and Conrad Hal Waddington.

### Recent growth

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.
- 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

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.

### Abstract relational biology

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.

### Algebraic biology

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.

### Complex systems biology

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

Many studies by 1986 are grouped in this topic, including:

- Computer models for biology and medicine
- Models of blood flow in arteries
- Neuron models
- Biochemical networks and oscillations
- Quantum computers and automata in biology
- 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

### Modeling cell and molecular biology

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

**Examples of biological models**

- Mechanics of biological tissues
- Enzyme activity and speed
- Cancer models and simulations
- Movement of groups of cells
- Scar tissue formation
- Inside-the-cell activity
- Cell cycles
- Cell death (apoptosis)

**Modeling body systems**

- Blood vessel disease
- Multi-scale heart models
- Electrical properties of muscles

### Computational neuroscience

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

### Evolutionary Biology

Math has been important in studying evolution and ecology.

**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

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
- Turing patterns

### Mathematical Methods

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)

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

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.