John Holland and colleagues have demonstrated experimentally this by using several mutants to increase the mutation rate of the vesicular stomatitis virus. As a result, the virus becomes non-infectious. Developmental biology is another field in biology that has benefited from mathematics. How can the process leading from one cell to a complex embryo be explained from basic principles? This was a total mystery reflecting a complete divorce between the 19th century physics focusing on systems at thermodynamic equilibrium, and so showing temporal evolutions toward disorder, and the dynamics toward increasing order and complexity characteristic of life.
The great mathematician Alan Turing provided the first theoretical approximation to solve this apparent paradox. Turing is famous for at least two other contributions, namely inventing the Turing machine, that is, the precursor of modern computers, and breaking the Enigma code used by the German Nazis to encrypt communications during the Second World War. After these two previous contributions, Turing had the brilliant idea in of writing down a mathematical model describing the dynamics of two chemical species called morphogens.
First, one morphogen called activator produces itself at a rate proportional to its abundance. This is a type of multiplicative, nonlinear process very common in biology: The more activator, the faster is produced. Second, the activator also produces a second morphogen called inhibitor, which in turn inhibits the former. Third, both activator and inhibitor diffuse through space, although the inhibitor does it at a faster rate. In summary, the system is described by local production and long-range inhibition. Turing showed that starting from a uniform spatial distribution of both morphogens, some random fluctuations will be amplified.
At the end, there will be patches with a high concentration of activator surrounded by empty areas. We have gone from a homogeneous distribution to a heterogeneous one. Imagine that this heterogeneous distribution of activator determines the formation of a head on one extreme, where the concentration of activator is beyond the average, and a tail on the other side, where the activator's concentration is below the average.
Symmetry has been broken through a bifurcation of the homogeneous solution. Structure has appeared. The size of the spatial domain in which these morphogens diffuse determines how many such bifurcations can be accommodated. Elegant extensions of the Turing model have been proposed to explain multiple examples of pattern formation in development, such as the pigmentation in the coat of some mammals. The great developmental biologist Pere Alberch, in collaboration with George Oster, James Murray, and others, used this type of mathematical formulation in combination with beautiful experiments in which the size of an amphibian extremity could be manipulated through mutagens.
Alberch and colleagues were able to show that evolution takes place through minor changes of a conserved developmental program[ 1 , 3 ]. Self-organization plays a very important role that cannot be anticipated by focusing exclusively on genes. These self-organizing spatial patterns show discrete bifurcations following a well-defined sequence, and so natural selection has only a limited set of possibilities to choose from. It seems through this brief description that the flow of ideas has always gone from mathematics to biology. Although this is the case in the majority of examples, there are also some cases in which the influence is the other way around.
That is, biology has also made a contribution to mathematics. This mathematical theory challenged a solid assumption arising from the Newtonian paradigm. Certainly, Newtonian mechanics had represented a triumph of science. Newton's laws were able to describe the dynamics of celestial bodies and, more spectacularly, to make powerful predictions.
Given an individual condition, let's say the position and speed of a comet right now, and the deterministic law of gravity, one can predict the position of such a comet or years from now. Or in the past, because for that matter the system is reversible, one can move the tape recorder either backward or forward. Newton started a new way to deal with nature in which deviations from what should be expected could be used to make specific claims. An example was the prediction of the planet Neptune on the basis of the modification of the gravitational field of the other planets.
Newton's contribution was so important, that Alexander Pope proposed the following epitaph for Newton, who died in And all was light. In the previous scheme, however, there is a small caveat: We cannot know perfectly an initial condition. The atmospheric temperature at a place and a time, for example, is a number with infinite decimal points. What we do is to round this number. Let's take five decimal points. This extremely small mistake remains small in systems, such as the ones studied by Newton.
Thus, if the arrival of the comet after years is predicted with an error of five decimal points, nobody would claim the theory is not good. In this type of systems, small errors remain small through time. This is because these are linear systems, that is, systems in which variables add one to another. However, lots of interesting systems, such as the weather and biological systems are nonlinear. In this case, variables do not add but multiply each other or are raised to an exponent. The consequence of this is that errors now will not remain small.
They will grow exponentially through time. The paradigmatic example is weather prediction. It is not that scientists do not understand the dynamics of fluids. They do so as well as they understand Newton's laws. A very small mistake after, let's say 10 days, will become so large that prediction just does not work for long temporal windows.
It is no longer true that knowledge implies prediction. The temporal dynamics of systems, such as weather show random-like behavior similar to what expect for a stochastic system, but they are generated by totally deterministic systems described by a small number of variables. This behavior was named deterministic chaos.
There were several contributions to the mathematics of deterministic chaos. Some were certainly done by mathematicians, but others by climatologists, such as Edward Lorenz, and by the theoretical ecologists Robert May. Lorenz was working on a simplified model of weather and found that after running a simulation a second time he obtained a different output.
Since his system of three differential equations was absolutely deterministic, he thought he had made a mistake. Finally he realized there was a tiny difference in the two initial conditions.
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He had encountered deterministic chaos and its strong dependence on initial conditions. Robert May, trained as a physicist, shifted to population biology and become one of the leading theoreticians in ecology and epidemiology. May co-discovered chaos by studying a simple, deterministic model of an ecological population.
Mathematical and theoretical biology
It was a difference equation or logistic map. 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 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.
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: 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. 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 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.
In particular the S and M checkpoints are regulated by means of special bifurcations called a Hopf bifurcation and an infinite period bifurcation. From Wikipedia, the free encyclopedia. This section does not cite any sources. Please help improve this section by adding citations to reliable sources.
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Mathematical and theoretical biology - Wikipedia
Biological applications of bifurcation theory Biostatistics Entropy and life Ewens's sampling formula Journal of Theoretical Biology Mathematical modelling of infectious disease Metabolic network modelling Molecular modelling Morphometrics Population genetics Statistical genetics Theoretical ecology Turing pattern. Retrieved 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.
Progress in Biophysics and Molecular Biology. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches. Cells and their physical constraints". Fall The American Society for Cell Biology. An interface between psychology and evolution". Transformations of Neuronal, Genetic and Neoplastic Networks". Archived from the original on CS1 maint: Ten Equations that Changed Biology: Journal of Theoretical Biology. Witten ed. Anai, K. Horimoto, Universal Academy Press, Tokyo, Preziosi, Cancer Modelling and Simulation. Mathematical Modeling: Mathematical Models in Medicine.
New York: Pergamon Press. Bethesda, MD. Accurately measuring the precise lengths and thicknesses of the beaks and averaging those differences within each population is very valuable raw data for drawing conclusions about how the species have differentiated over time. A sub-field of biological science is the field of biostatistics , a field in which statistics are used to describe and explain life sciences.
The purpose of statistical analyses is to find correlations , or interdependent relationships, between variables and to compare variables against each other. For example, a marine biologist might want to know if there is a positive statistical correlation between the presence of cone snails, a common predator in marine habitats, and tulip snails, who are often preyed upon by larger snails.
She may set up an experiment in which tulip snails are present in all testing areas, but cone snails are added to only a few of those areas. Over time, she records the number of tulip snails remaining in all areas, and finds that there is a difference between areas with cone snails and areas without cone snails.
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But is that difference significant enough to draw conclusions about snail behavior? She then runs a series of statistical tests to see if the difference is, indeed, statistically significant. In this example, she is using a single variable, the number of tulip snails, in a comparative test. There are many different types of statistical tests that can be run, depending on the number of variables and the type of data sought. Want to know another great thing about math? It can be used anywhere, anytime.
Sometimes there are limits on how biologists can perform an experiment or test a hypothesis, but mathematical models can be implemented instead to illustrate a probable outcome. There are certainly questions of accuracy when this occurs, and so any and all mathematical calculations and modeling involves establishing error margins , which are essentially quantifiable doubt about the accuracy of the results. The smaller the error margin, the more confidence the researchers have in their results.
In this chart taken from a study of the chambered nautilus, error margins are shown as the I-shaped ranges at the top of each data bar.
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Margins are useful in establishing confidence in the mathematical results. For example, environmental scientists are currently trying to estimate the degree of decline in fish populations. Of course, they cannot go out and count every fish in the ocean to do this. Nor can they know precisely how many baby fish hatched this year. They have likely used estimations from a sample size and created a model showing rates of population change from which they establish a hypothesis about how fish populations are changing. The hypothesis can be tested using predictive models that show predicted future numbers of the fish within an established error margin.
The field of mathematical biology examines the mathematical representations of biological systems and is a crucial aspect of better understanding the natural world. This includes the formulation of mathematical models , which can be used to predict or describe natural occurrences in a wide variety of useful ways. Mathematics are also used in biology for basic, raw data gathering that's useful in tracking changes over time. Biostatistics uses statistical analyses to form conclusions about biological phenomena, such as drawing comparisons or correlations between biological variables.
Using mathematics in biology can also be extremely useful in the process of testing hypotheses, particularly where a direct experiment cannot be conducted and predictive models are applied. Math is a very powerful tool to biology and medicine, from which we have learned so much about our natural world. To unlock this lesson you must be a Study. Create your account. Already a member? Log In.
The Role of Mathematics in Biology
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To learn more, visit our Earning Credit Page. Not sure what college you want to attend yet? The videos on Study. Sign Up. Explore over 4, video courses. Find a degree that fits your goals. The Role of Mathematics in Biology Mathematics is applied in all major fields of science, including biology. Here, we investigate the different applications of mathematics in biology and explore some examples. Try it risk-free for 30 days. An error occurred trying to load this video. Try refreshing the page, or contact customer support.
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