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53-451: Mathematical biology aims at the mathematical representation and modeling of biological processes , using techniques and tools of applied mathematics . 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 . Because of

106-624: A mathematical model on a computer , the model being designed to represent the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determined by comparing their results to the real-world outcomes they aim to predict. Computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics ( computational physics ), astrophysics , climatology , chemistry , biology and manufacturing , as well as human systems in economics , psychology , social science , health care and engineering . Simulation of

159-519: A boost due to the growing importance of molecular biology . Modelling physiological systems Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is the theoretical study of the nervous system. 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,

212-511: A desert-battle simulation of one force invading another involved the modeling of 66,239 tanks, trucks and other vehicles on simulated terrain around Kuwait , using multiple supercomputers in the DoD High Performance Computer Modernization Program. Other examples include a 1-billion-atom model of material deformation; a 2.64-million-atom model of the complex protein-producing organelle of all living organisms,

265-400: A different answer for each execution. Although this might seem obvious, this is a special point of attention in stochastic simulations , where random numbers should actually be semi-random numbers. An exception to reproducibility are human-in-the-loop simulations such as flight simulations and computer games . Here a human is part of the simulation and thus influences the outcome in a way that

318-495: 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. 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 . One classic work in this area is Alan Turing 's paper on morphogenesis entitled The Chemical Basis of Morphogenesis , published in 1952 in

371-470: A map that uses numeric coordinates and numeric timestamps of events. Similarly, CGI computer simulations of CAT scans can simulate how a tumor might shrink or change during an extended period of medical treatment, presenting the passage of time as a spinning view of the visible human head, as the tumor changes. Other applications of CGI computer simulations are being developed to graphically display large amounts of data, in motion, as changes occur during

424-493: A simulation run. Generic examples of types of computer simulations in science, which are derived from an underlying mathematical description: Specific examples of computer simulations include: Notable, and sometimes controversial, computer simulations used in science include: Donella Meadows ' World3 used in the Limits to Growth , James Lovelock's Daisyworld and Thomas Ray's Tierra . In social sciences, computer simulation

477-646: A simulation". Computer simulation developed hand-in-hand with the rapid growth of the computer, following its first large-scale deployment during the Manhattan Project in World War II to model the process of nuclear detonation . It was a simulation of 12 hard spheres using a Monte Carlo algorithm . Computer simulation is often used as an adjunct to, or substitute for, modeling systems for which simple closed form analytic solutions are not possible. There are many types of computer simulations; their common feature

530-420: A state in which the system is in equilibrium. Such models are often used in simulating physical systems, as a simpler modeling case before dynamic simulation is attempted. Formerly, the output data from a computer simulation was sometimes presented in a table or a matrix showing how data were affected by numerous changes in the simulation parameters . The use of the matrix format was related to traditional use of

583-631: A system is represented as the running of the system's model. It can be used to explore and gain new insights into new technology and to estimate the performance of systems too complex for analytical solutions . Computer simulations are realized by running computer programs that can be either small, running almost instantly on small devices, or large-scale programs that run for hours or days on network-based groups of computers. The scale of events being simulated by computer simulations has far exceeded anything possible (or perhaps even imaginable) using traditional paper-and-pencil mathematical modeling. In 1997,

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636-421: 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

689-419: A wide variety of practical contexts, such as: The reliability and the trust people put in computer simulations depends on the validity of the simulation model , therefore verification and validation are of crucial importance in the development of computer simulations. Another important aspect of computer simulations is that of reproducibility of the results, meaning that a simulation model should not provide

742-651: 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

795-439: Is an integral component of the five angles of analysis fostered by the data percolation methodology, which also includes qualitative and quantitative methods, reviews of the literature (including scholarly), and interviews with experts, and which forms an extension of data triangulation. Of course, similar to any other scientific method, replication is an important part of computational modeling Computer simulations are used in

848-618: 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

901-480: Is considered to be On Growth and Form (1917) by D'Arcy Thompson , and other early pioneers include Ronald Fisher , Hans Leo Przibram , Vito Volterra , Nicolas Rashevsky and Conrad Hal Waddington . Interest in the field has grown rapidly from the 1960s onwards. Some reasons for this include: 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

954-414: Is hard, if not impossible, to reproduce exactly. Vehicle manufacturers make use of computer simulation to test safety features in new designs. By building a copy of the car in a physics simulation environment, they can save the hundreds of thousands of dollars that would otherwise be required to build and test a unique prototype. Engineers can step through the simulation milliseconds at a time to determine

1007-450: Is much harder is knowing what the accuracy (compared to measurement resolution and precision ) of the values are. Often they are expressed as "error bars", a minimum and maximum deviation from the value range within which the true value (is expected to) lie. Because digital computer mathematics is not perfect, rounding and truncation errors multiply this error, so it is useful to perform an "error analysis" to confirm that values output by

1060-503: Is the attempt to generate a sample of representative scenarios for a model in which a complete enumeration of all possible states of the model would be prohibitive or impossible. The external data requirements of simulations and models vary widely. For some, the input might be just a few numbers (for example, simulation of a waveform of AC electricity on a wire), while others might require terabytes of information (such as weather and climate models). Input sources also vary widely: Lastly,

1113-480: Is very important to perform a sensitivity analysis to ensure that the accuracy of the results is properly understood. For example, the probabilistic risk analysis of factors determining the success of an oilfield exploration program involves combining samples from a variety of statistical distributions using the Monte Carlo method . If, for instance, one of the key parameters (e.g., the net ratio of oil-bearing strata)

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1166-551: The Philosophical Transactions of the Royal Society . 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

1219-541: The ribosome , in 2005; a complete simulation of the life cycle of Mycoplasma genitalium in 2012; and the Blue Brain project at EPFL (Switzerland), begun in May 2005 to create the first computer simulation of the entire human brain, right down to the molecular level. Because of the computational cost of simulation, computer experiments are used to perform inference such as uncertainty quantification . A model consists of

1272-538: 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 ,

1325-678: The buildup of queues in the simulation of humans evacuating a building. Furthermore, simulation results are often aggregated into static images using various ways of scientific visualization . In debugging, simulating a program execution under test (rather than executing natively) can detect far more errors than the hardware itself can detect and, at the same time, log useful debugging information such as instruction trace, memory alterations and instruction counts. This technique can also detect buffer overflow and similar "hard to detect" errors as well as produce performance information and tuning data. Although sometimes ignored in computer simulations, it

1378-412: 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

1431-420: The complexity of the living systems , theoretical biology employs several fields of mathematics, and has contributed to the development of new techniques. Mathematics has been used in biology as early as the 13th 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

1484-400: 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

1537-492: 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. The term "theoretical biology" was first used as a monograph title by Johannes Reinke in 1901, and soon after by Jakob von Uexküll  in 1920. One founding text

1590-415: The equations used to capture the behavior of a system. By contrast, computer simulation is the actual running of the program that perform algorithms which solve those equations, often in an approximate manner. Simulation, therefore, is the process of running a model. Thus one would not "build a simulation"; instead, one would "build a model (or a simulator)", and then either "run the model" or equivalently "run

1643-428: The exact stresses being put upon each section of the prototype. Computer graphics can be used to display the results of a computer simulation. Animations can be used to experience a simulation in real-time, e.g., in training simulations . In some cases animations may also be useful in faster than real-time or even slower than real-time modes. For example, faster than real-time animations can be useful in visualizing

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1696-443: 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

1749-413: The field of adaptive dynamics . 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. A fixed mapping between an initial state and

1802-645: 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

1855-520: 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 (ARB)

1908-433: The human population. Thomas Malthus ' 1789 essay on the growth of the human population was based on the concept of exponential growth. Pierre François Verhulst formulated the logistic growth model in 1836. 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

1961-416: 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) is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of

2014-399: 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

2067-452: The matrix concept in mathematical models . However, psychologists and others noted that humans could quickly perceive trends by looking at graphs or even moving-images or motion-pictures generated from the data, as displayed by computer-generated-imagery (CGI) animation. Although observers could not necessarily read out numbers or quote math formulas, from observing a moving weather chart they might be able to predict events (and "see that rain

2120-403: 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

2173-609: The notion of closure of constraints. 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 . 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. A monograph on this topic summarizes an extensive amount of published research in this area up to 1986, including subsections in

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2226-405: The persistence and transformation of life forms. Regulation of biological processes occurs when any process is modulated in its frequency, rate or extent. Biological processes are regulated by many means; examples include the control of gene expression , protein modification or interaction with a protein or substrate molecule . Computer modeling Computer simulation is the running of

2279-622: 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 . Biological process Biological processes are those processes that are necessary for an organism to live and that shape its capacities for interacting with its environment. Biological processes are made of many chemical reactions or other events that are involved in

2332-527: 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

2385-490: 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. 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 have produced several models of

2438-436: The simulation will still be usefully accurate. Models used for computer simulations can be classified according to several independent pairs of attributes, including: Another way of categorizing models is to look at the underlying data structures. For time-stepped simulations, there are two main classes: For steady-state simulations, equations define the relationships between elements of the modeled system and attempt to find

2491-443: 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

2544-446: The time at which data is available varies: Because of this variety, and because diverse simulation systems have many common elements, there are a large number of specialized simulation languages . The best-known may be Simula . There are now many others. Systems that accept data from external sources must be very careful in knowing what they are receiving. While it is easy for computers to read in values from text or binary files, what

2597-440: 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

2650-473: 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 (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

2703-481: The values of the variables), the progression of the system is calculated by solving the equations at each time-frame in small increments. 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

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2756-402: Was headed their way") much faster than by scanning tables of rain-cloud coordinates . Such intense graphical displays, which transcended the world of numbers and formulae, sometimes also led to output that lacked a coordinate grid or omitted timestamps, as if straying too far from numeric data displays. Today, weather forecasting models tend to balance the view of moving rain/snow clouds against

2809-644: Was introduced by Anthony Bartholomay , and its applications were developed in mathematical biology and especially in mathematical medicine. 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. Theoretical approaches to biological organization aim to understand

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