Several areas of specialized research in mathematical and theoretical biology as well as external links to related projects in various universities are concisely presented in the following subsections, including also a large number of appropriate validating references from a list of several thousands of published authors contributing to this field. Many of the included examples are characterised by highly complex, nonlinear mechanisms, as it is being increasingly recognised that such examples may be best understood through a combination of mathematical, logical, physical/chemical, molecular and computational models.
Abstract relational biology Abstract relational biology (ARB) is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of the simplest models in ARB are the Metabolic-Replication, or (M,R)--systems introduced by Robert Rosen in 1957–1958 as abstract, relational models of cellular and organismal organization. Other approaches include the notion of
autopoiesis developed by
Maturana and
Varela,
Kauffman's Work-Constraints cycles, and more recently the notion of closure of constraints.
Algebraic biology Algebraic biology (also known as symbolic
systems biology) applies the algebraic methods of
symbolic computation to the study of biological problems, especially in
genomics,
proteomics, analysis of
molecular structures and study of
genes.
Complex systems biology An elaboration of systems biology to understand the more complex life processes was developed since 1970 in connection with molecular set theory, relational biology and algebraic biology.
Computer models and automata theory A monograph on this topic summarizes an extensive amount of published research in this area up to 1986, including subsections in the following areas:
computer modeling in biology and medicine, arterial system models,
neuron models, biochemical and
oscillation networks, quantum automata,
quantum computers in
molecular biology and
genetics, cancer modelling,
neural nets,
genetic networks, abstract categories in relational biology, metabolic-replication systems,
category theory applications in biology and medicine,
automata theory,
cellular automata,
tessellation models and complete self-reproduction,
chaotic systems in
organisms, relational biology and organismic theories.
Modeling cell and molecular biology This area has received a boost due to the growing importance of
molecular biology. • Theoretical enzymology and
enzyme kinetics •
Cancer modelling and simulation • Modelling the movement of interacting cell populations • Mathematical modelling of scar tissue formation • Mathematical modelling of intracellular dynamics • Mathematical modelling of the cell cycle • Mathematical modelling of apoptosis
Modelling physiological systems • Modelling of
arterial disease • Multi-scale modelling of the
heart • Modelling electrical properties of muscle interactions, as in
bidomain and
monodomain models
Computational neuroscience Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is the theoretical study of the nervous system.
Evolutionary biology Ecology and
evolutionary biology have traditionally been the dominant fields of mathematical biology. Evolutionary biology has been the subject of extensive mathematical theorizing. The traditional approach in this area, which includes complications from genetics, is
population genetics. Most population geneticists consider the appearance of new
alleles by
mutation, the appearance of new
genotypes by
recombination, and changes in the frequencies of existing alleles and genotypes at a small number of
gene loci. When
infinitesimal effects at a large number of gene loci are considered, together with the assumption of
linkage equilibrium or
quasi-linkage equilibrium, one derives
quantitative genetics.
Ronald Fisher made fundamental advances in statistics, such as
analysis of variance, via his work on quantitative genetics. Another important branch of population genetics that led to the extensive development of
coalescent theory is
phylogenetics. Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics. Traditional population genetic models deal with alleles and genotypes, and are frequently
stochastic. Many population genetics models assume that population sizes are constant. Variable population sizes, often in the absence of genetic variation, are treated by the field of
population dynamics. Work in this area dates back to the 19th century, and even as far as 1798 when
Thomas Malthus formulated the first principle of population dynamics, which later became known as the
Malthusian growth model. The
Lotka–Volterra predator-prey equations are another famous example. Population dynamics overlap with another active area of research in mathematical biology:
mathematical epidemiology, the study of infectious disease affecting populations. Various models of the spread of
infections have been proposed and analyzed, and provide important results that may be applied to health policy decisions. In
evolutionary game theory, developed first by
John Maynard Smith and
George R. Price, selection acts directly on inherited phenotypes, without genetic complications. This approach has been mathematically refined to produce the field of
adaptive dynamics.
Mathematical biophysics The earlier stages of mathematical biology were dominated by mathematical
biophysics, described as the application of mathematics in biophysics, often involving specific physical/mathematical models of biosystems and their components or compartments. The following is a list of mathematical descriptions and their assumptions.
Deterministic processes (dynamical systems) A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in time, a deterministic process always generates the same trajectory, and no two trajectories cross in state space. •
Difference equations/Maps – discrete time, continuous state space. •
Ordinary differential equations – continuous time, continuous state space, no spatial derivatives.
See also: Numerical ordinary differential equations. •
Partial differential equations – continuous time, continuous state space, spatial derivatives.
See also: Numerical partial differential equations. •
Logical deterministic cellular automata – discrete time, discrete state space.
See also: Cellular automaton.
Stochastic processes (random dynamical systems) A random mapping between an initial state and a final state, making the state of the system a
random variable with a corresponding
probability distribution. • Non-Markovian processes –
generalized master equation – continuous time with memory of past events, discrete state space, waiting times of events (or transitions between states) discretely occur. • Jump
Markov process –
master equation – continuous time with no memory of past events, discrete state space, waiting times between events discretely occur and are exponentially distributed.
See also: Monte Carlo method for numerical simulation methods, specifically
dynamic Monte Carlo method and
Gillespie algorithm. • Continuous
Markov process –
stochastic differential equations or a
Fokker–Planck equation – continuous time, continuous state space, events occur continuously according to a random
Wiener process.
Spatial modelling and dynamical systems One classic work in this area is
Alan Turing's paper on
morphogenesis entitled
The Chemical Basis of Morphogenesis, published in 1952 in the
Philosophical Transactions of the Royal Society. • Travelling waves in a wound-healing assay •
Swarming behaviour • A mechanochemical theory of
morphogenesis •
Biological pattern formation • Spatial distribution modeling using plot samples •
Turing patterns
Geometric organisation and spatial patterning Many biological systems exhibit recurring
geometric and
spatial patterns, and the analysis of these forms is an established area of biomathematics. Mathematical models and dynamical systems are used to describe how such patterns arise, how they
scale with size or number, and how they relate to underlying biological processes and constraints. File:Dnaconformations.png|Different
conformations of
DNA illustrating molecular‑scale geometric organisation. File:Mariposa 88 Diaethria clymena (cropped).png|Wing pattern of
Diaethria clymena with a distinctive “88” marking, often modelled using
reaction–diffusion mechanisms described by
Alan Turing. File:Bienen auf Honigwabe (cropped).jpg|Worker honeybees (
Apis mellifera) build
hexagonal wax combs that exemplify geometric optimisation . Geometric organization appears across multiple levels of
biological organization . At the molecular scale, mathematical approaches are used to study the geometry of
protein folding,
DNA packing,
membrane structures, and the shapes of
biomolecules, which can often be described using concepts from
molecular geometry,
VSEPR theory, and
stereochemistry . At this level, regular
polyhedral and
symmetric forms, such as the capsids of many
icosahedral or
helical viruses , provide classic examples of mathematically constrained biological structures. At the
organismal level, well‑known examples include
phyllotaxis in plants, where leaves and florets form
spiral arrangements often related to
golden‑angle packing;
animal coat patterns such as spots and stripes; and branching structures such as the
vascular system,
bronchial tree, neuronal arborisation, and
tree canopies. In
marine and other organisms,
shells and skeletons can exhibit helical,
logarithmic spiral,
lattice‑like, or
radially symmetric forms that can be described with geometric and growth models. At
ecological and
landscape scales, spatial
vegetation patterns,
coral growth forms, and other large‑scale structures can also be analysed using geometric and dynamical models. Examples include banded and spotted vegetation in
semi‑arid ecosystems, patchy distributions of organisms arising from
spatial interactions, and branching or
reef‑like structures in marine environments. Several mathematical frameworks are used to study these phenomena. Reaction–diffusion models describe the emergence of spatial patterns such as stripes, spots, and spirals in developing tissues.
Fractal and fractal‑based models are used to analyse branching networks and
self‑similar structures in organisms and ecosystems. Models of
Phyllotaxis explain the appearance of spiral arrangements and regular
packing in plant growth.
Mathematical methods A mathematical model of a biological system consists of a system of mathematical equations or relationships which describes various properties of a system, their relationship, and their evolution over time. The solution of these equations, by either analytical or numerical means, predicts how the biological system behaves either over time or at
equilibrium. There are many different types of equations and the type of behavior that can occur is dependent on both the model and the equations used. The model often makes assumptions about the system. The equations may also make assumptions about the nature of what may occur.
Molecular set theory Molecular set theory is a mathematical formulation of the wide-sense
chemical kinetics of biomolecular reactions in terms of sets of molecules and their chemical transformations represented by set-theoretical mappings between molecular sets. It was introduced by
Anthony Bartholomay, and its applications were developed in mathematical biology and especially in mathematical medicine. In a more general sense, Molecular set theory is the theory of molecular categories defined as categories of molecular sets and their chemical transformations represented as set-theoretical mappings of molecular sets. The theory has also contributed to biostatistics and the formulation of clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to Physiology, Clinical Biochemistry and Medicine. is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of the simplest models in ARB are the Metabolic-Replication, or
(M,R)--systems introduced by
Robert Rosen in 1957–1958 as abstract, relational models of cellular and organismal organization. == Model example: the cell cycle ==