Mathematical and theoretical biology
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Mathematical and theoretical biology, or biomathematics, is a branch of
Mathematical biology aims at the mathematical representation and modeling of
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.
History
Early history
Mathematics has been used in biology as early as the 13th century, when
The term "theoretical biology" was first used as a monograph title by
Recent growth
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Interest in the field has grown rapidly from the 1960s onwards. Some reasons for this include:
- The rapid growth of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical tools[9]
- Recent development of mathematical tools such as chaos theory to help understand complex, non-linear mechanisms in biology
- An increase in computing power, which facilitates calculations and simulations not previously possible
- An increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other complications involved in human and animal research
Areas of research
Several areas of specialized research in mathematical and theoretical biology[10][11][12][13][14] 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.[15]
Algebraic biology
Algebraic biology (also known as symbolic systems biology) applies the algebraic methods of
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,
Modeling cell and molecular biology
This area has received a boost due to the growing importance of molecular biology.[13]
- Mechanics of biological tissues[31][32]
- Theoretical enzymology and enzyme kinetics
- Cancer modelling and simulation[33][34]
- Modelling the movement of interacting cell populations[35]
- Mathematical modelling of scar tissue formation[36]
- Mathematical modelling of intracellular dynamics[37][38]
- Mathematical modelling of the cell cycle[39]
- Mathematical modelling of apoptosis[40]
Modelling physiological systems
- Modelling of arterial disease[41]
- Multi-scale modelling of the heart[42]
- 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.[43][44]
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[45] 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
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.
- Numerical ordinary differential equations.
- 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
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[46]
- Swarming behaviour[47]
- A mechanochemical theory of morphogenesis[48]
- Biological pattern formation[49]
- Spatial distribution modeling using plot samples[50]
- Turing patterns[51]
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
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.[52] 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.[52]
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)
Model example: the cell cycle
The eukaryotic cell cycle is very complex and has been the subject of intense study, 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 [55][56] 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
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
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
See also
- Biological applications of bifurcation theory
- Biophysics
- Biostatistics
- Entropy and life
- Ewens's sampling formula
- Journal of Theoretical Biology
- Logistic function
- Mathematical modelling of infectious disease
- Metabolic network modelling
- Molecular modelling
- Morphometrics
- Population genetics
- Spring school on theoretical biology
- Statistical genetics
- Theoretical ecology
- Turing pattern
Notes
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- ^ "There is a subtle difference between mathematical biologists and theoretical biologists. Mathematical biologists tend to be employed in mathematical departments and to be a bit more interested in math inspired by biology than in the biological problems themselves, and vice versa." Careers in theoretical biology Archived 2019-09-14 at the Wayback Machine
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- ^ 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.
- .
- ^ a b "Research in Mathematical Biology". Maths.gla.ac.uk. Retrieved 2008-09-10.
- ^ 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.
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- ^ a b 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.
- ^ 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.
- ISBN 1-58488-361-8. Archived from the original(PDF) on March 10, 2012.
- ^ 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.
- ^ Lin HC (2004). "Computer Simulations and the Question of Computability of Biological Systems" (PDF).
- ^ Computer Models and Automata Theory in Biology and Medicine. 1986.
- ^ "Natural Transformations Models in Molecular Biology". SIAM and Society of Mathematical Biology, National Meeting. N/A. Bethesda, MD: 230–232. 1983.
- ^ Baianu IC (2004). "Quantum Interactomics and Cancer Mechanisms" (PDF). Research Report Communicated to the Institute of Genomic Biology, University of Illinois at Urbana.
- ^ Kainen PC (2005). "Category Theory and Living Systems" (PDF). In Ehresmann A (ed.). Charles Ehresmann's Centennial Conference Proceedings. University of Amiens, France, October 7–9th, 2005. pp. 1–5.
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References
- Edelstein-Keshet L (2004). Mathematical Models in Biology. SIAM. ISBN 0-07-554950-6.
- Hoppensteadt F (1993) [1975]. Mathematical Theories of Populations: Demographics, Genetics and Epidemics (Reprinted ed.). Philadelphia: SIAM. ISBN 0-89871-017-0.
- Renshaw E (1991). Modelling Biological Populations in Space and Time. C.U.P. ISBN 0-521-44855-7.
- Rubinow SI (1975). Introduction to Mathematical Biology. John Wiley. ISBN 0-471-74446-8.
- Strogatz SH (2001). Nonlinear Dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering. Perseus. ISBN 0-7382-0453-6.
- "Biologist Salary | Payscale". Payscale.Com, 2021, Biologist Salary | PayScale. Accessed 3 May 2021.
- Theoretical biology
- Bonner JT (1988). The Evolution of Complexity by Means of Natural Selection. Princeton: Princeton University Press. ISBN 0-691-08493-9.
- Mangel M (2006). The Theoretical Biologist's Toolbox. Quantitative Methods for Ecology and Evolutionary Biology. Cambridge University Press. ISBN 0-521-53748-7.
Further reading
- Hoppensteadt F (September 1995). "Getting Started in Mathematical Biology" (PDF). Notices of the American Mathematical Society.
- May RM (February 2004). "Uses and abuses of mathematics in biology". Science. 303 (5659): 790–3. S2CID 24844494.
- Murray JD (1988). "How the leopard gets its spots?". Scientific American. 258 (3): 80–87. .
- Reed MC (March 2004). "Why Is Mathematical Biology So Hard?" (PDF). Notices of the American Mathematical Society.
- Kroc J, Balihar K, Matejovic M (2019). "Complex Systems and Their Use in Medicine: Concepts, Methods and Bio-Medical Applications". )