Mathematical and theoretical biology

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Yellow chamomile head showing the Fibonacci numbers in spirals consisting of 21 (blue) and 13 (aqua). Such arrangements have been noticed since the Middle Ages and can be used to make mathematical models of a wide variety of plants. FibonacciChamomile.PNG
Yellow chamomile head showing the Fibonacci numbers in spirals consisting of 21 (blue) and 13 (aqua). Such arrangements have been noticed since the Middle Ages and can be used to make mathematical models of a wide variety of plants.

Mathematical and theoretical biology is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to prove and validate the scientific theories. [1] The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. [2] Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study biological systems, even though the two terms are sometimes interchanged. [3] [4]

Biology is the natural science that studies life and living organisms, including their physical structure, chemical processes, molecular interactions, physiological mechanisms, development and evolution. Despite the complexity of the science, there are certain unifying concepts that consolidate it into a single, coherent field. Biology recognizes the cell as the basic unit of life, genes as the basic unit of heredity, and evolution as the engine that propels the creation and extinction of species. Living organisms are open systems that survive by transforming energy and decreasing their local entropy to maintain a stable and vital condition defined as homeostasis.

Organism Any individual living physical entity

In biology, an organism is any individual entity that propagates the properties of life. It is a synonym for "life form".

Experimental biology is the set of approaches in the field of biology concerned with the conduction of experiments to investigate and understand biological phenomena. The term is opposed to theoretical biology which is concerned with the mathematical modelling and abstractions of the biological systems. Due to the complexity of the investigated systems, biology is primarily an experimental science. However, as a consequence of the modern increase in computational power, it is now becoming more feasible to find approximate solutions and validate mathematical models of complex living organisms.


Mathematical biology aims at the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics and it can be useful in both theoretical and practical research. Describing systems in a quantitative manner means their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter. This requires precise mathematical models.

Biological process are the processes vital for a living organism to live, and that shape its capacities for interacting with its environment. Biological processes are made up of many chemical reactions or other events that are involved in the persistence and transformation of life forms. Metabolism and homeostasis are examples.

Applied mathematics Application of mathematical methods to other fields

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

Applied science is the application of existing scientific knowledge to practical applications, like technology or inventions.

Because of the complexity of the living systems, theoretical biology employs several fields of mathematics, [5] and has contributed to the development of new techniques.


Early history

Mathematics has been used in biology as early as the 12th century, when Fibonacci used the famous Fibonacci series to describe a growing population of rabbits. In the 18th century Daniel Bernoulli applied mathematics to describe the effect of smallpox on the human population. Thomas Malthus' 1789 essay on the growth of the human population was based on the concept of exponential growth. Pierre Francois Verhulst formulated the logistic growth model in 1836.

Fibonacci Italian mathematician and writer

Fibonacci was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".

Daniel Bernoulli Swiss mathematician

Daniel Bernoulli FRS was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. His name is commemorated in the Bernoulli's principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the airplane wing.

Fritz Müller described the evolutionary benefits of what is now called Müllerian mimicry in 1879, in an account notable for being the first use of a mathematical argument in evolutionary ecology to show how powerful the effect of natural selection would be, unless one includes Malthus's discussion of the effects of population growth that influenced Charles Darwin: Malthus argued that growth would be exponential (he uses the word "geometric") while resources (the environment's carrying capacity) could only grow arithmetically. [6]

Fritz Müller German-born Brazilian biologist (1822–1897)

Johann Friedrich Theodor Müller, better known as Fritz Müller, and also as Müller-Desterro, was a German biologist who emigrated to southern Brazil, where he lived in and near the German community of Blumenau, Santa Catarina. There he studied the natural history of the Atlantic forest south of São Paulo, and was an early advocate of Darwinism. He lived in Brazil for the rest of his life. Müllerian mimicry is named after him.

Müllerian mimicry natural phenomenon in which two or more distasteful species, that may or may not be closely related and share one or more common predators, have come to mimic each others warning signals

Müllerian mimicry is a natural phenomenon in which two or more unprofitable species, that may or may not be closely related and share one or more common predators, have come to mimic each other's honest warning signals, to their mutual benefit, since predators can learn to avoid all of them with fewer experiences. It is named after the German naturalist Fritz Müller, who first proposed the concept in 1878, supporting his theory with the first mathematical model of frequency-dependent selection, one of the first such models anywhere in biology.

Evolutionary ecology Study of how interactions among species and between species and their environment affect species through selection and adaptation

Evolutionary ecology lies at the intersection of ecology and evolutionary biology. It approaches the study of ecology in a way that explicitly considers the evolutionary histories of species and the interactions between them. Conversely, it can be seen as an approach to the study of evolution that incorporates an understanding of the interactions between the species under consideration. The main subfields of evolutionary ecology are life history evolution, sociobiology, the evolution of inter specific relations and the evolution of biodiversity and of communities.

The term "theoretical biology" was first used by Johannes Reinke in 1901. One founding text is considered to be On Growth and Form (1917) by D'Arcy Thompson, [7] and other early pioneers include Ronald Fisher, Hans Leo Przibram, Nicolas Rashevsky and Vito Volterra. [8]

Johannes Reinke German botanist

Johannes Reinke was a German botanist and philosopher who was a native of Ziethen, Lauenburg. He is remembered for his research of benthic marine algae.

<i>On Growth and Form</i> book by DArcy Thompson

On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.

Ronald Fisher British statistician, evolutionary biologist, geneticist, and eugenicist

Sir Ronald Aylmer Fisher was a British statistician and geneticist. For his work in statistics, he has been described as "a genius who almost single-handedly created the foundations for modern statistical science" and "the single most important figure in 20th century statistics". In genetics, his work used mathematics to combine Mendelian genetics and natural selection; this contributed to the revival of Darwinism in the early 20th-century revision of the theory of evolution known as the modern synthesis. For his contributions to biology, Fisher has been called "the greatest of Darwin’s successors".

Recent growth

Interest in the field has grown rapidly from the 1960s onwards. Some reasons for this include:

Areas of research

Several areas of specialized research in mathematical and theoretical biology [9] [10] [11] [12] [13] 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, and supercomplex mechanisms, as it is being increasingly recognised that the result of such interactions may only be 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. [14]

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. [15] [16] [17]

Complex systems biology

An elaboration of systems biology to understanding 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, [18] [19] [20] 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, [21] cancer modelling, [22] neural nets, genetic networks, abstract categories in relational biology, [23] metabolic-replication systems, category theory [24] applications in biology and medicine, [25] automata theory, cellular automata, [26] tessellation models [27] [28] and complete self-reproduction, chaotic systems in organisms, relational biology and organismic theories. [15] [29]

Modeling cell and molecular biology

This area has received a boost due to the growing importance of molecular biology. [12]

Modelling physiological systems

Computational neuroscience

Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is the theoretical study of the nervous system. [40] [41]

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 [42] 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.

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.

Spatial modelling

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.

Mathematical methods

A model of a biological system is converted into a system of equations, although the word 'model' is often used synonymously with the system of corresponding equations. The solution of the equations, by either analytical or numerical means, describes 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 (MST) 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. [49] In a more general sense, MST 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. [49] [50]

Organizational biology

Theoretical approaches to biological organization aim to understand the interdependence between the parts of organisms. They emphasize the circularities that these interdependences lead to. Theoretical biologists developed several concepts to formalize this idea.

For example, abstract relational biology (ARB) [51] 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. [52]

Model example: the cell cycle

The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation leads to cancers. It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results. Two research groups [53] [54] have produced several models of the cell cycle simulating several organisms. They have recently produced a generic eukaryotic cell cycle model that can represent a particular eukaryote depending on the values of the parameters, demonstrating that the idiosyncrasies of the individual cell cycles are due to different protein concentrations and affinities, while the underlying mechanisms are conserved (Csikasz-Nagy et al., 2006).

By means of a system of ordinary differential equations these models show the change in time (dynamical system) of the protein inside a single typical cell; this type of model is called a deterministic process (whereas a model describing a statistical distribution of protein concentrations in a population of cells is called a stochastic process).

To obtain these equations an iterative series of steps must be done: first the several models and observations are combined to form a consensus diagram and the appropriate kinetic laws are chosen to write the differential equations, such as rate kinetics for stoichiometric reactions, Michaelis-Menten kinetics for enzyme substrate reactions and Goldbeter–Koshland kinetics for ultrasensitive transcription factors, afterwards the parameters of the equations (rate constants, enzyme efficiency coefficients and Michaelis constants) must be fitted to match observations; when they cannot be fitted the kinetic equation is revised and when that is not possible the wiring diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such as protein half-life and cell size.

To fit the parameters, the differential equations must be studied. This can be done either by simulation or by analysis. In a simulation, given a starting vector (list of the values of the variables), the progression of the system is calculated by solving the equations at each time-frame in small increments.

Cell cycle bifurcation diagram.jpg

In analysis, the properties of the equations are used to investigate the behavior of the system depending on the values of the parameters and variables. A system of differential equations can be represented as a vector field, where each vector described the change (in concentration of two or more protein) determining where and how fast the trajectory (simulation) is heading. Vector fields can have several special points: a stable point, called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an unstable point, either a source or a saddle point, which repels (forcing the concentrations to change away from a certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the concentrations oscillate).

A better representation, which handles the large number of variables and parameters, is a bifurcation diagram using bifurcation theory. The presence of these special steady-state points at certain values of a parameter (e.g. mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the cell cycle has phases (partially corresponding to G1 and G2) in which mass, via a stable point, controls cyclin levels, and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a bifurcation event (Cell cycle checkpoint), the system cannot go back to the previous levels since at the current mass the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a Hopf bifurcation and an infinite period bifurcation.[ citation needed ]

Societies and institutes

See also


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A biologist is a scientist who has specialized knowledge in the field of biology, the scientific study of life. Biologists involved in fundamental research attempt to explore and further explain the underlying mechanisms that govern the functioning of living matter. Biologists involved in applied research attempt to develop or improve more specific processes and understanding, in fields such as medicine and industry.

René Thomas (biologist) scientist, professor

René Thomas (14 May 1928 - 9 January 2017 was a Belgian scientist. His research included DNA biochemistry and biophysics, genetics, mathematical biology, and finally dynamical systems. He devoted his life to the deciphering of key logical principles at the basis of the behaviour of biological systems, and more generally to the generation of complex dynamical behaviour. He was professor and laboratory head at the Université Libre de Bruxelles, and taught and inspired several generations of researchers.

Cellular model

Creating a cellular model has been a particularly challenging task of systems biology and mathematical biology. It involves developing efficient algorithms, data structures, visualization and communication tools to orchestrate the integration of large quantities of biological data with the goal of computer modeling.

Natural computing, also called natural computation, is a terminology introduced to encompass three classes of methods: 1) those that take inspiration from nature for the development of novel problem-solving techniques; 2) those that are based on the use of computers to synthesize natural phenomena; and 3) those that employ natural materials to compute. The main fields of research that compose these three branches are artificial neural networks, evolutionary algorithms, swarm intelligence, artificial immune systems, fractal geometry, artificial life, DNA computing, and quantum computing, among others.

Complex systems biology branch or subfield of mathematical and theoretical biology

Complex systems biology (CSB) is a branch or subfield of mathematical and theoretical biology concerned with complexity of both structure and function in biological organisms, as well as the emergence and evolution of organisms and species, with emphasis being placed on the complex interactions of, and within, bionetworks, and on the fundamental relations and relational patterns that are essential to life. CSB is thus a field of theoretical sciences aimed at discovering and modeling the relational patterns essential to life that has only a partial overlap with complex systems theory, and also with the systems approach to biology called systems biology; this is because the latter is restricted primarily to simplified models of biological organization and organisms, as well as to only a general consideration of philosophical or semantic questions related to complexity in biology. Moreover, a wide range of abstract theoretical complex systems are studied as a field of applied mathematics, with or without relevance to biology, chemistry or physics.

Virtual Cell (VCell) is an open-source software platform for modeling and simulation of living organisms, primarily cells. It has been designed to be a tool for a wide range of scientists, from experimental cell biologists to theoretical biophysicists.

The Institute of Mathematical Problems of Biology RAS is a research institute specializing in computational biology and bioinformatics. The objective of the institute is elaboration of mathematical and computational methods for biological research, as well as implementation of these methods directly addressing the problems of computational biology.

Cell-based models are mathematical models that represent biological cells as a discrete entities. They are used in the field of computational biology for simulating the biomechanics of multicellular structures such as tissues. Their main advantage is the easy integration of cell level processes such as cell division, intracellular processes and single-cell variability within a cell population.


Theoretical biology

Further reading