Bioinformatics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Bioinformatics 

Early bioinformatics—computational alignment of experimentally determined sequences of a class of related proteins; see § Sequence analysis for further information.

 

Map of the human X chromosome (from the National Center for Biotechnology Information website)

Bioinformatics (/ˌbaɪ.oʊˌɪnfərˈmætɪks/) is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combines biology, computer science, information engineering, mathematics and statistics to analyze and interpret the biological data. Bioinformatics has been used for in silico analyses of biological queries using mathematical and statistical techniques.

Bioinformatics includes biological studies that use computer programming as part of their methodology, as well as a specific analysis "pipelines" that are repeatedly used, particularly in the field of genomics. Common uses of bioinformatics include the identification of candidates genes and single nucleotide polymorphisms (SNPs). Often, such identification is made with the aim of better understanding the genetic basis of disease, unique adaptations, desirable properties (esp. in agricultural species), or differences between populations. In a less formal way, bioinformatics also tries to understand the organizational principles within nucleic acid and protein sequences, called proteomics.

Overview

Bioinformatics has become an important part of many areas of biology. In experimental molecular biology, bioinformatics techniques such as image and signal processing allow extraction of useful results from large amounts of raw data. In the field of genetics, it aids in sequencing and annotating genomes and their observed mutations. It plays a role in the text mining of biological literature and the development of biological and gene ontologies to organize and query biological data. It also plays a role in the analysis of gene and protein expression and regulation. Bioinformatics tools aid in comparing, analyzing and interpreting genetic and genomic data and more generally in the understanding of evolutionary aspects of molecular biology. At a more integrative level, it helps analyze and catalogue the biological pathways and networks that are an important part of systems biology. In structural biology, it aids in the simulation and modeling of DNA, RNA, proteins as well as biomolecular interactions.

History

Historically, the term bioinformatics did not mean what it means today. Paulien Hogeweg and Ben Hesper coined it in 1970 to refer to the study of information processes in biotic systems. This definition placed bioinformatics as a field parallel to biochemistry (the study of chemical processes in biological systems).

Sequences

Sequences of genetic material are frequently used in bioinformatics and are easier to manage using computers than manually.

Computers became essential in molecular biology when protein sequences became available after Frederick Sanger determined the sequence of insulin in the early 1950s. Comparing multiple sequences manually turned out to be impractical. A pioneer in the field was Margaret Oakley Dayhoff. She compiled one of the first protein sequence databases, initially published as books and pioneered methods of sequence alignment and molecular evolution. Another early contributor to bioinformatics was Elvin A. Kabat, who pioneered biological sequence analysis in 1970 with his comprehensive volumes of antibody sequences released with Tai Te Wu between 1980 and 1991. In the 1970s, new techniques for sequencing DNA were applied to bacteriophage MS2 and øX174, and the extended nucleotide sequences were then parsed with informational and statistical algorithms. These studies illustrated that well known features, such as the coding segments and the triplet code, are revealed in straightforward statistical analyses and were thus proof of the concept that bioinformatics would be insightful.

Goals

To study how normal cellular activities are altered in different disease states, the biological data must be combined to form a comprehensive picture of these activities. Therefore, the field of bioinformatics has evolved such that the most pressing task now involves the analysis and interpretation of various types of data. This also includes nucleotide and amino acid sequences, protein domains, and protein structures. The actual process of analyzing and interpreting data is referred to as computational biology. Important sub-disciplines within bioinformatics and computational biology include:

• Development and implementation of computer programs that enable efficient access to, management and use of, various types of information.
• Development of new algorithms (mathematical formulas) and statistical measures that assess relationships among members of large data sets. For example, there are methods to locate a gene within a sequence, to predict protein structure and/or function, and to cluster protein sequences into families of related sequences.

The primary goal of bioinformatics is to increase the understanding of biological processes. What sets it apart from other approaches, however, is its focus on developing and applying computationally intensive techniques to achieve this goal. Examples include: pattern recognition, data mining, machine learning algorithms, and visualization. Major research efforts in the field include sequence alignment, gene finding, genome assembly, drug design, drug discovery, protein structure alignment, protein structure prediction, prediction of gene expression and protein–protein interactions, genome-wide association studies, the modeling of evolution and cell division/mitosis.

Bioinformatics now entails the creation and advancement of databases, algorithms, computational and statistical techniques, and theory to solve formal and practical problems arising from the management and analysis of biological data.

Over the past few decades, rapid developments in genomic and other molecular research technologies and developments in information technologies have combined to produce a tremendous amount of information related to molecular biology. Bioinformatics is the name given to these mathematical and computing approaches used to glean understanding of biological processes.

Common activities in bioinformatics include mapping and analyzing DNA and protein sequences, aligning DNA and protein sequences to compare them, and creating and viewing 3-D models of protein structures.

Relation to other fields

Bioinformatics is a science field that is similar to but distinct from biological computation, while it is often considered synonymous to computational biology. Biological computation uses bioengineering and biology to build biological computers, whereas bioinformatics uses computation to better understand biology. Bioinformatics and computational biology involve the analysis of biological data, particularly DNA, RNA, and protein sequences. The field of bioinformatics experienced explosive growth starting in the mid-1990s, driven largely by the Human Genome Project and by rapid advances in DNA sequencing technology.

Analyzing biological data to produce meaningful information involves writing and running software programs that use algorithms from graph theory, artificial intelligence, soft computing, data mining, image processing, and computer simulation. The algorithms in turn depend on theoretical foundations such as discrete mathematics, control theory, system theory, information theory, and statistics.

Sequence analysis

Since the Phage Φ-X174 was sequenced in 1977, the DNA sequences of thousands of organisms have been decoded and stored in databases. This sequence information is analyzed to determine genes that encode proteins, RNA genes, regulatory sequences, structural motifs, and repetitive sequences. A comparison of genes within a species or between different species can show similarities between protein functions, or relations between species (the use of molecular systematics to construct phylogenetic trees). With the growing amount of data, it long ago became impractical to analyze DNA sequences manually. Computer programs such as BLAST are used routinely to search sequences—as of 2008, from more than 260,000 organisms, containing over 190 billion nucleotides.

DNA sequencing

Before sequences can be analyzed they have to be obtained from the data storage bank example the Genbank. DNA sequencing is still a non-trivial problem as the raw data may be noisy or afflicted by weak signals. Algorithms have been developed for base calling for the various experimental approaches to DNA sequencing.

Sequence assembly

Most DNA sequencing techniques produce short fragments of sequence that need to be assembled to obtain complete gene or genome sequences. The so-called shotgun sequencing technique (which was used, for example, by The Institute for Genomic Research (TIGR) to sequence the first bacterial genome, Haemophilus influenzae) generates the sequences of many thousands of small DNA fragments (ranging from 35 to 900 nucleotides long, depending on the sequencing technology). The ends of these fragments overlap and, when aligned properly by a genome assembly program, can be used to reconstruct the complete genome. Shotgun sequencing yields sequence data quickly, but the task of assembling the fragments can be quite complicated for larger genomes. For a genome as large as the human genome, it may take many days of CPU time on large-memory, multiprocessor computers to assemble the fragments, and the resulting assembly usually contains numerous gaps that must be filled in later. Shotgun sequencing is the method of choice for virtually all genomes sequenced today, and genome assembly algorithms are a critical area of bioinformatics research.

Genome annotation

In the context of genomics, annotation is the process of marking the genes and other biological features in a DNA sequence. This process needs to be automated because most genomes are too large to annotate by hand, not to mention the desire to annotate as many genomes as possible, as the rate of sequencing has ceased to pose a bottleneck. Annotation is made possible by the fact that genes have recognisable start and stop regions, although the exact sequence found in these regions can vary between genes.

The first description of a comprehensive genome annotation system was published in 1995 by the team at The Institute for Genomic Research that performed the first complete sequencing and analysis of the genome of a free-living organism, the bacterium Haemophilus influenzae. Owen White designed and built a software system to identify the genes encoding all proteins, transfer RNAs, ribosomal RNAs (and other sites) and to make initial functional assignments. Most current genome annotation systems work similarly, but the programs available for analysis of genomic DNA, such as the GeneMark program trained and used to find protein-coding genes in Haemophilus influenzae, are constantly changing and improving.

Following the goals that the Human Genome Project left to achieve after its closure in 2003, a new project developed by the National Human Genome Research Institute in the U.S appeared. The so-called ENCODE project is a collaborative data collection of the functional elements of the human genome that uses next-generation DNA-sequencing technologies and genomic tiling arrays, technologies able to automatically generate large amounts of data at a dramatically reduced per-base cost but with the same accuracy (base call error) and fidelity (assembly error).

Computational evolutionary biology

Evolutionary biology is the study of the origin and descent of species, as well as their change over time. Informatics has assisted evolutionary biologists by enabling researchers to:

• trace the evolution of a large number of organisms by measuring changes in their DNA, rather than through physical taxonomy or physiological observations alone,
• compare entire genomes, which permits the study of more complex evolutionary events, such as gene duplication, horizontal gene transfer, and the prediction of factors important in bacterial speciation,
• build complex computational population genetics models to predict the outcome of the system over time
• track and share information on an increasingly large number of species and organisms

Future work endeavours to reconstruct the now more complex tree of life.

The area of research within computer science that uses genetic algorithms is sometimes confused with computational evolutionary biology, but the two areas are not necessarily related.

Comparative genomics

The core of comparative genome analysis is the establishment of the correspondence between genes (orthology analysis) or other genomic features in different organisms. It is these intergenomic maps that make it possible to trace the evolutionary processes responsible for the divergence of two genomes. A multitude of evolutionary events acting at various organizational levels shape genome evolution. At the lowest level, point mutations affect individual nucleotides. At a higher level, large chromosomal segments undergo duplication, lateral transfer, inversion, transposition, deletion and insertion. Ultimately, whole genomes are involved in processes of hybridization, polyploidization and endosymbiosis, often leading to rapid speciation. The complexity of genome evolution poses many exciting challenges to developers of mathematical models and algorithms, who have recourse to a spectrum of algorithmic, statistical and mathematical techniques, ranging from exact, heuristics, fixed parameter and approximation algorithms for problems based on parsimony models to Markov chain Monte Carlo algorithms for Bayesian analysis of problems based on probabilistic models.

Many of these studies are based on the detection of sequence homology to assign sequences to protein families.

Pan genomics

Pan genomics is a concept introduced in 2005 by Tettelin and Medini which eventually took root in bioinformatics. Pan genome is the complete gene repertoire of a particular taxonomic group: although initially applied to closely related strains of a species, it can be applied to a larger context like genus, phylum etc. It is divided in two parts- The Core genome: Set of genes common to all the genomes under study (These are often housekeeping genes vital for survival) and The Dispensable/Flexible Genome: Set of genes not present in all but one or some genomes under study. A bioinformatics tool BPGA can be used to characterize the Pan Genome of bacterial species.

Genetics of disease

With the advent of next-generation sequencing we are obtaining enough sequence data to map the genes of complex diseases infertility, breast cancer or Alzheimer's disease. Genome-wide association studies are a useful approach to pinpoint the mutations responsible for such complex diseases. Through these studies, thousands of DNA variants have been identified that are associated with similar diseases and traits. Furthermore, the possibility for genes to be used at prognosis, diagnosis or treatment is one of the most essential applications. Many studies are discussing both the promising ways to choose the genes to be used and the problems and pitfalls of using genes to predict disease presence or prognosis.

Analysis of mutations in cancer

In cancer, the genomes of affected cells are rearranged in complex or even unpredictable ways. Massive sequencing efforts are used to identify previously unknown point mutations in a variety of genes in cancer. Bioinformaticians continue to produce specialized automated systems to manage the sheer volume of sequence data produced, and they create new algorithms and software to compare the sequencing results to the growing collection of human genome sequences and germline polymorphisms. New physical detection technologies are employed, such as oligonucleotide microarrays to identify chromosomal gains and losses (called comparative genomic hybridization), and single-nucleotide polymorphism arrays to detect known point mutations. These detection methods simultaneously measure several hundred thousand sites throughout the genome, and when used in high-throughput to measure thousands of samples, generate terabytes of data per experiment. Again the massive amounts and new types of data generate new opportunities for bioinformaticians. The data is often found to contain considerable variability, or noise, and thus Hidden Markov model and change-point analysis methods are being developed to infer real copy number changes.

Two important principles can be used in the analysis of cancer genomes bioinformatically pertaining to the identification of mutations in the exome. First, cancer is a disease of accumulated somatic mutations in genes. Second cancer contains driver mutations which need to be distinguished from passengers.

With the breakthroughs that this next-generation sequencing technology is providing to the field of Bioinformatics, cancer genomics could drastically change. These new methods and software allow bioinformaticians to sequence many cancer genomes quickly and affordably. This could create a more flexible process for classifying types of cancer by analysis of cancer driven mutations in the genome. Furthermore, tracking of patients while the disease progresses may be possible in the future with the sequence of cancer samples.

Another type of data that requires novel informatics development is the analysis of lesions found to be recurrent among many tumors.

Gene and protein expression

Analysis of gene expression

The expression of many genes can be determined by measuring mRNA levels with multiple techniques including microarrays, expressed cDNA sequence tag (EST) sequencing, serial analysis of gene expression (SAGE) tag sequencing, massively parallel signature sequencing (MPSS), RNA-Seq, also known as "Whole Transcriptome Shotgun Sequencing" (WTSS), or various applications of multiplexed in-situ hybridization. All of these techniques are extremely noise-prone and/or subject to bias in the biological measurement, and a major research area in computational biology involves developing statistical tools to separate signal from noise in high-throughput gene expression studies. Such studies are often used to determine the genes implicated in a disorder: one might compare microarray data from cancerous epithelial cells to data from non-cancerous cells to determine the transcripts that are up-regulated and down-regulated in a particular population of cancer cells.

Analysis of protein expression

Protein microarrays and high throughput (HT) mass spectrometry (MS) can provide a snapshot of the proteins present in a biological sample. Bioinformatics is very much involved in making sense of protein microarray and HT MS data; the former approach faces similar problems as with microarrays targeted at mRNA, the latter involves the problem of matching large amounts of mass data against predicted masses from protein sequence databases, and the complicated statistical analysis of samples where multiple, but incomplete peptides from each protein are detected. Cellular protein localization in a tissue context can be achieved through affinity proteomics displayed as spatial data based on immunohistochemistry and tissue microarrays.

Analysis of regulation

Gene regulation is the complex orchestration of events by which a signal, potentially an extracellular signal such as a hormone, eventually leads to an increase or decrease in the activity of one or more proteins. Bioinformatics techniques have been applied to explore various steps in this process.

For example, gene expression can be regulated by nearby elements in the genome. Promoter analysis involves the identification and study of sequence motifs in the DNA surrounding the coding region of a gene. These motifs influence the extent to which that region is transcribed into mRNA. Enhancer elements far away from the promoter can also regulate gene expression, through three-dimensional looping interactions. These interactions can be determined by bioinformatic analysis of chromosome conformation capture experiments.

Expression data can be used to infer gene regulation: one might compare microarray data from a wide variety of states of an organism to form hypotheses about the genes involved in each state. In a single-cell organism, one might compare stages of the cell cycle, along with various stress conditions (heat shock, starvation, etc.). One can then apply clustering algorithms to that expression data to determine which genes are co-expressed. For example, the upstream regions (promoters) of co-expressed genes can be searched for over-represented regulatory elements. Examples of clustering algorithms applied in gene clustering are k-means clustering, self-organizing maps (SOMs), hierarchical clustering, and consensus clustering methods.

Analysis of cellular organization

Several approaches have been developed to analyze the location of organelles, genes, proteins, and other components within cells. This is relevant as the location of these components affects the events within a cell and thus helps us to predict the behavior of biological systems. A gene ontology category, cellular component, has been devised to capture subcellular localization in many biological databases.

Microscopy and image analysis

Microscopic pictures allow us to locate both organelles as well as molecules. It may also help us to distinguish between normal and abnormal cells, e.g. in cancer.

Protein localization

The localization of proteins helps us to evaluate the role of a protein. For instance, if a protein is found in the nucleus it may be involved in gene regulation or splicing. By contrast, if a protein is found in mitochondria, it may be involved in respiration or other metabolic processes. Protein localization is thus an important component of protein function prediction. There are well developed protein subcellular localization prediction resources available, including protein subcellular location databases, and prediction tools.

Nuclear organization of chromatin

Data from high-throughput chromosome conformation capture experiments, such as Hi-C (experiment) and ChIA-PET, can provide information on the spatial proximity of DNA loci. Analysis of these experiments can determine the three-dimensional structure and nuclear organization of chromatin. Bioinformatic challenges in this field include partitioning the genome into domains, such as Topologically Associating Domains (TADs), that are organised together in three-dimensional space.

Structural bioinformatics

3-dimensional protein structures such as this one are common subjects in bioinformatic analyses.

Protein structure prediction is another important application of bioinformatics. The amino acid sequence of a protein, the so-called primary structure, can be easily determined from the sequence on the gene that codes for it. In the vast majority of cases, this primary structure uniquely determines a structure in its native environment. (Of course, there are exceptions, such as the bovine spongiform encephalopathy (mad cow disease) prion.) Knowledge of this structure is vital in understanding the function of the protein. Structural information is usually classified as one of secondary, tertiary and quaternary structure. A viable general solution to such predictions remains an open problem. Most efforts have so far been directed towards heuristics that work most of the time.

One of the key ideas in bioinformatics is the notion of homology. In the genomic branch of bioinformatics, homology is used to predict the function of a gene: if the sequence of gene A, whose function is known, is homologous to the sequence of gene B, whose function is unknown, one could infer that B may share A's function. In the structural branch of bioinformatics, homology is used to determine which parts of a protein are important in structure formation and interaction with other proteins. In a technique called homology modeling, this information is used to predict the structure of a protein once the structure of a homologous protein is known. This currently remains the only way to predict protein structures reliably.

One example of this is hemoglobin in humans and the hemoglobin in legumes (leghemoglobin), which are distant relatives from the same protein superfamily. Both serve the same purpose of transporting oxygen in the organism. Although both of these proteins have completely different amino acid sequences, their protein structures are virtually identical, which reflects their near identical purposes and shared ancestor.

Other techniques for predicting protein structure include protein threading and de novo (from scratch) physics-based modeling.

Another aspect of structural bioinformatics include the use of protein structures for Virtual Screening models such as Quantitative Structure-Activity Relationship models and proteochemometric models (PCM). Furthermore, a protein's crystal structure can be used in simulation of for example ligand-binding studies and in silico mutagenesis studies.

Network and systems biology

Network analysis seeks to understand the relationships within biological networks such as metabolic or protein–protein interaction networks. Although biological networks can be constructed from a single type of molecule or entity (such as genes), network biology often attempts to integrate many different data types, such as proteins, small molecules, gene expression data, and others, which are all connected physically, functionally, or both.

Systems biology involves the use of computer simulations of cellular subsystems (such as the networks of metabolites and enzymes that comprise metabolism, signal transduction pathways and gene regulatory networks) to both analyze and visualize the complex connections of these cellular processes. Artificial life or virtual evolution attempts to understand evolutionary processes via the computer simulation of simple (artificial) life forms.

Molecular interaction networks

Interactions between proteins are frequently visualized and analyzed using networks. This network is made up of protein–protein interactions from Treponema pallidum, the causative agent of syphilis and other diseases.

Tens of thousands of three-dimensional protein structures have been determined by X-ray crystallography and protein nuclear magnetic resonance spectroscopy (protein NMR) and a central question in structural bioinformatics is whether it is practical to predict possible protein–protein interactions only based on these 3D shapes, without performing protein–protein interaction experiments. A variety of methods have been developed to tackle the protein–protein docking problem, though it seems that there is still much work to be done in this field.

Other interactions encountered in the field include Protein–ligand (including drug) and protein–peptide. Molecular dynamic simulation of movement of atoms about rotatable bonds is the fundamental principle behind computational algorithms, termed docking algorithms, for studying molecular interactions.

Others

Literature analysis

The growth in the number of published literature makes it virtually impossible to read every paper, resulting in disjointed sub-fields of research. Literature analysis aims to employ computational and statistical linguistics to mine this growing library of text resources. For example:

• Abbreviation recognition – identify the long-form and abbreviation of biological terms
• Named entity recognition – recognizing biological terms such as gene names
• Protein–protein interaction – identify which proteins interact with which proteins from text

The area of research draws from statistics and computational linguistics.

High-throughput image analysis

Computational technologies are used to accelerate or fully automate the processing, quantification and analysis of large amounts of high-information-content biomedical imagery. Modern image analysis systems augment an observer's ability to make measurements from a large or complex set of images, by improving accuracy, objectivity, or speed. A fully developed analysis system may completely replace the observer. Although these systems are not unique to biomedical imagery, biomedical imaging is becoming more important for both diagnostics and research. Some examples are:

• high-throughput and high-fidelity quantification and sub-cellular localization (high-content screening, cytohistopathology, Bioimage informatics)
• morphometrics
• clinical image analysis and visualization
• determining the real-time air-flow patterns in breathing lungs of living animals
• quantifying occlusion size in real-time imagery from the development of and recovery during arterial injury
• making behavioral observations from extended video recordings of laboratory animals
• infrared measurements for metabolic activity determination
• inferring clone overlaps in DNA mapping, e.g. the Sulston score

High-throughput single cell data analysis

Computational techniques are used to analyse high-throughput, low-measurement single cell data, such as that obtained from flow cytometry. These methods typically involve finding populations of cells that are relevant to a particular disease state or experimental condition.

Biodiversity informatics

Biodiversity informatics deals with the collection and analysis of biodiversity data, such as taxonomic databases, or microbiome data. Examples of such analyses include phylogenetics, niche modelling, species richness mapping, DNA barcoding, or species identification tools.

Ontologies and data integration

Biological ontologies are directed acyclic graphs of controlled vocabularies. They are designed to capture biological concepts and descriptions in a way that can be easily categorised and analysed with computers. When categorised in this way, it is possible to gain added value from holistic and integrated analysis.

The OBO Foundry was an effort to standardise certain ontologies. One of the most widespread is the Gene ontology which describes gene function. There are also ontologies which describe phenotypes.

Databases

Databases are essential for bioinformatics research and applications. Many databases exist, covering various information types: for example, DNA and protein sequences, molecular structures, phenotypes and biodiversity. Databases may contain empirical data (obtained directly from experiments), predicted data (obtained from analysis), or, most commonly, both. They may be specific to a particular organism, pathway or molecule of interest. Alternatively, they can incorporate data compiled from multiple other databases. These databases vary in their format, access mechanism, and whether they are public or not.

Some of the most commonly used databases are listed below. For a more comprehensive list, please check the link at the beginning of the subsection.

• Used in biological sequence analysis: Genbank, UniProt
• Used in structure analysis: Protein Data Bank (PDB)
• Used in finding Protein Families and Motif Finding: InterPro, Pfam
• Used for Next Generation Sequencing: Sequence Read Archive
• Used in Network Analysis: Metabolic Pathway Databases (KEGG, BioCyc), Interaction Analysis Databases, Functional Networks
• Used in design of synthetic genetic circuits: GenoCAD

Software and tools

Software tools for bioinformatics range from simple command-line tools, to more complex graphical programs and standalone web-services available from various bioinformatics companies or public institutions.

Open-source bioinformatics software

Many free and open-source software tools have existed and continued to grow since the 1980s. The combination of a continued need for new algorithms for the analysis of emerging types of biological readouts, the potential for innovative in silico experiments, and freely available open code bases have helped to create opportunities for all research groups to contribute to both bioinformatics and the range of open-source software available, regardless of their funding arrangements. The open source tools often act as incubators of ideas, or community-supported plug-ins in commercial applications. They may also provide de facto standards and shared object models for assisting with the challenge of bioinformation integration.

The range of open-source software packages includes titles such as Bioconductor, BioPerl, Biopython, BioJava, BioJS, BioRuby, Bioclipse, EMBOSS, .NET Bio, Orange with its bioinformatics add-on, Apache Taverna, UGENE and GenoCAD. To maintain this tradition and create further opportunities, the non-profit Open Bioinformatics Foundation have supported the annual Bioinformatics Open Source Conference (BOSC) since 2000.

An alternative method to build public bioinformatics databases is to use the MediaWiki engine with the WikiOpener extension. This system allows the database to be accessed and updated by all experts in the field.

Web services in bioinformatics

SOAP- and REST-based interfaces have been developed for a wide variety of bioinformatics applications allowing an application running on one computer in one part of the world to use algorithms, data and computing resources on servers in other parts of the world. The main advantages derive from the fact that end users do not have to deal with software and database maintenance overheads.

Basic bioinformatics services are classified by the EBI into three categories: SSS (Sequence Search Services), MSA (Multiple Sequence Alignment), and BSA (Biological Sequence Analysis). The availability of these service-oriented bioinformatics resources demonstrate the applicability of web-based bioinformatics solutions, and range from a collection of standalone tools with a common data format under a single, standalone or web-based interface, to integrative, distributed and extensible bioinformatics workflow management systems.

Bioinformatics workflow management systems

A bioinformatics workflow management system is a specialized form of a workflow management system designed specifically to compose and execute a series of computational or data manipulation steps, or a workflow, in a Bioinformatics application. Such systems are designed to

• provide an easy-to-use environment for individual application scientists themselves to create their own workflows,
• provide interactive tools for the scientists enabling them to execute their workflows and view their results in real-time,
• simplify the process of sharing and reusing workflows between the scientists, and
• enable scientists to track the provenance of the workflow execution results and the workflow creation steps.

Some of the platforms giving this service: Galaxy, Kepler, Taverna, UGENE, Anduril, HIVE.

BioCompute and BioCompute Objects

In 2014, the US Food and Drug Administration sponsored a conference held at the National Institutes of Health Bethesda Campus to discuss reproducibility in bioinformatics. Over the next three years, a consortium of stakeholders met regularly to discuss what would become BioCompute paradigm. These stakeholders included representatives from government, industry, and academic entities. Session leaders represented numerous branches of the FDA and NIH Institutes and Centers, non-profit entities including the Human Variome Project and the European Federation for Medical Informatics, and research institutions including Stanford, the New York Genome Center, and the George Washington University.

It was decided that the BioCompute paradigm would be in the form of digital 'lab notebooks' which allow for the reproducibility, replication, review, and reuse, of bioinformatics protocols. This was proposed to enable greater continuity within a research group over the course of normal personnel flux while furthering the exchange of ideas between groups. The US FDA funded this work so that information on pipelines would be more transparent and accessible to their regulatory staff.

In 2016, the group reconvened at the NIH in Bethesda and discussed the potential for a BioCompute Object, an instance of the BioCompute paradigm. This work was copied as both a "standard trial use" document and a preprint paper uploaded to bioRxiv. The BioCompute object allows for the JSON-ized record to be shared among employees, collaborators, and regulators.

Education platforms

Software platforms designed to teach bioinformatics concepts and methods include Rosalind and online courses offered through the Swiss Institute of Bioinformatics Training Portal. The Canadian Bioinformatics Workshops provides videos and slides from training workshops on their website under a Creative Commons license. The 4273π project or 4273pi project also offers open source educational materials for free. The course runs on low cost Raspberry Pi computers and has been used to teach adults and school pupils. 4273π is actively developed by a consortium of academics and research staff who have run research level bioinformatics using Raspberry Pi computers and the 4273π operating system.

MOOC platforms also provide online certifications in bioinformatics and related disciplines, including Coursera's Bioinformatics Specialization (UC San Diego) and Genomic Data Science Specialization (Johns Hopkins) as well as EdX's Data Analysis for Life Sciences XSeries (Harvard). University of Southern California offers a Masters In Translational Bioinformatics focusing on biomedical applications.

Conferences

There are several large conferences that are concerned with bioinformatics. Some of the most notable examples are Intelligent Systems for Molecular Biology (ISMB), European Conference on Computational Biology (ECCB), and Research in Computational Molecular Biology (RECOMB).

Mathematical and theoretical biology

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology 

 

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 or, Biomathematics, 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. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. 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.

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 the complexity of the living systems, theoretical biology employs several fields of mathematics, 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 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 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 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 by Johannes Reinke in 1901. One founding text is considered to be On Growth and Form (1917) by D'Arcy Thompson, and other early pioneers include Ronald Fisher, Hans Leo Przibram, Nicolas Rashevsky and Vito Volterra.

Recent growth

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
• 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 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.

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 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, 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.

• Mechanics of biological tissues
• 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.

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
• Swarming behaviour
• A mechanochemical theory of morphogenesis
• Biological pattern formation
• Spatial distribution modeling using plot samples
• Turing patterns

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. 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.

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) 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

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 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.

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.

Societies and institutes

• National Institute for Mathematical and Biological Synthesis
• Society for Mathematical Biology
• ESMTB: European Society for Mathematical and Theoretical Biology
• The Israeli Society for Theoretical and Mathematical Biology
• Société Francophone de Biologie Théorique
• International Society for Biosemiotic Studies
• School of Computational and Integrative Sciences, Jawaharlal Nehru University

 

Population dynamics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Population_dynamics 

Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.

History

Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 220 years, although over the last century the scope of mathematical biology has greatly expanded.

The beginning of population dynamics is widely regarded as the work of Malthus, formulated as the Malthusian growth model. According to Malthus, assuming that the conditions (the environment) remain constant (ceteris paribus), a population will grow (or decline) exponentially. This principle provided the basis for the subsequent predictive theories, such as the demographic studies such as the work of Benjamin GompertzPierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.

A more general model formulation was proposed by F. J. Richards in 1959, further expanded by Simon Hopkins, in which the models of Gompertz, Verhulst and also Ludwig von Bertalanffy are covered as special cases of the general formulation. The Lotka–Volterra predator-prey equations are another famous example, as well as the alternative Arditi–Ginzburg equations.

Logistic function

Simplified population models usually start with four key variables (four demographic processes) including death, birth, immigration, and emigration. Mathematical models used to calculate changes in population demographics and evolution hold the assumption ('null hypothesis') of no external influence. Models can be more mathematically complex where "...several competing hypotheses are simultaneously confronted with the data." For example, in a closed system where immigration and emigration does not take place, the rate of change in the number of individuals in a population can be described as:

${\displaystyle {\frac {dN}{dT}}=B-D=bN-dN=(b-d)N=rN,}$

where N is the total number of individuals in the specific experimental population being studied, B is the number of births and D is the number of deaths per individual in a particular experiment or model. The algebraic symbols b, d and r stand for the rates of birth, death, and the rate of change per individual in the general population, the intrinsic rate of increase. This formula can be read as the rate of change in the population (dN/dT) is equal to births minus deaths (B - D).

Using these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the logistic equation:

${\displaystyle {\frac {dN}{dT}}=aN\left(1-{\frac {N}{K}}\right),}$

where N is the biomass density, a is the maximum per-capita rate of change, and K is the carrying capacity of the population. The formula can be read as follows: the rate of change in the population (dN/dT) is equal to growth (aN) that is limited by carrying capacity (1-N/K). From these basic mathematical principles the discipline of population ecology expands into a field of investigation that queries the demographics of real populations and tests these results against the statistical models. The field of population ecology often uses data on life history and matrix algebra to develop projection matrices on fecundity and survivorship. This information is used for managing wildlife stocks and setting harvest quotas.

Intrinsic rate of increase

The rate at which a population increases in size if there are no density-dependent forces regulating the population is known as the intrinsic rate of increase. It is

${\displaystyle {\dfrac {dN}{dt}}{\dfrac {1}{N}}=r}$

where the derivative ${\displaystyle dN/dt}$ is the rate of increase of the population, N is the population size, and r is the intrinsic rate of increase. Thus r is the maximum theoretical rate of increase of a population per individual – that is, the maximum population growth rate. The concept is commonly used in insect population ecology or management to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth.

Epidemiology

Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analysed, and provide important results that may be applied to health policy decisions.

Geometric populations

Operophtera brumata populations are geometric.

The mathematical formula below can used to model geometric populations. Geometric populations grow in discreet reproductive periods between intervals of abstinence, as opposed to populations which grow without designated periods for reproduction. Say that N denotes the number of individuals in each generation of a population that will reproduce.

${\displaystyle N^{t}+1=N^{t}+B^{t}+I^{t}-D^{t}-E^{t}}$

Where: N^t is the population size in generation t, and N^t+1 is the population size in the generation directly after N^t; B^t is the sum of births in the population between generations t and t+1 (i.e. the birth rate); I^t is the sum of immigrants added to the population between generations; D^t is the sum of deaths between generations (death rate); and E^t is the sum of emigrants moving out of the population between generations.

When there is no migration to or from the population, ${\displaystyle N^{t}+1=N^{t}+B^{t}-D^{t}}$

Assuming in this case that the birth and death rates are constants, then the birth rate minus the death rate equals R, the geometric rate of increase.

N_t+1 = N_t + RN_t

N_t+1 = (N_t + RN_t)

Take the term N_t out of the brackets again.

N_t+1 = (1 + R)N_t

1 + R = λ, where λ is the finite rate of increase.

N_t+1 = λN_t

At t+1	Nt+1 = λNt
At t+2	Nt+2 = λNt+1 = λλNt = λ2Nt
At t+3	Nt+3 = λNt+2 = λλNt+1 = λλλNt = λ3Nt

Therefore:

N_t = λ^tN₀

Term	Definition
λt	Finite rate of increase raised to the power of the number of generations (e.g. for t+2 [two generations] → λ2 , for t+1 [one generation] → λ1 = λ, and for t [before any generations - at time zero] → λ0 = 1

Doubling time

G. stearothermophilus has a shorter doubling time (td) than E. coli and N. meningitidis. Growth rates of 2 bacterial species will differ by unexpected orders of magnitude if the doubling times of the 2 species differ by even as little as 10 minutes. In eukaryotes such as animals, fungi, plants, and protists, doubling times are much longer than in bacteria. This reduces the growth rates of eukaryotes in comparison to Bacteria. G. stearothermophilus, E. coli, and N. meningitidis have 20 minute, 30 minute, and 40 minute^[23] doubling times under optimal conditions respectively. If bacterial populations could grow indefinitely (which they do not) then the number of bacteria in each species would approach infinity (∞). However, the percentage of G. stearothermophilus bacteria out of all the bacteria would approach 100% whilst the percentage of E. coli and N. meningitidis combined out of all the bacteria would approach 0%. This graph is a simulation of this hypothetical scenario. In reality, bacterial populations do not grow indefinitely in size and the 3 species require different optimal conditions to bring their doubling times to minima.

Time in minutes	% that is G. stearothermophilus
30	44.4%
60	53.3%
90	64.9%
120	72.7%
→∞	100%

Time in minutes	% that is E. coli
30	29.6%
60	26.7%
90	21.6%
120	18.2%
→∞	0.00%

Time in minutes	% that is N. meningitidis
30	25.9%
60	20.0%
90	13.5%
120	9.10%
→∞	0.00%

Disclaimer: Bacterial populations are logistic instead of geometric. Nevertheless, doubling times are applicable to both types of populations.

The doubling time (t_d) of a population is the time required for the population to grow to twice its size. We can calculate the doubling time of a geometric population using the equation: N_t = λ^tN₀ by exploiting our knowledge of the fact that the population (N) is twice its size (2N) after the doubling time.

N_td = λ^t_d × N₀

2N₀ = λ^t_d × N₀

λ^t_d = 2N₀ / N₀

λ^t_d = 2

The doubling time can be found by taking logarithms. For instance:

t_d × log₂(λ) = log₂(2)

log₂(2) = 1

t_d × log₂(λ) = 1

t_d = 1 / log₂(λ)

Or:

t_d × ln(λ) = ln(2)

t_d = ln(2) / ln(λ)

t_d = 0.693... / ln(λ)

Therefore:

t_d = 1 / log₂(λ) = 0.693... / ln(λ)

Half-life of geometric populations

The half-life of a population is the time taken for the population to decline to half its size. We can calculate the half-life of a geometric population using the equation: N_t = λ^tN₀ by exploiting our knowledge of the fact that the population (N) is half its size (0.5N) after a half-life.

N_t1/2 = λ^t_1/2 × N₀

0.5N₀ = λ^t_1/2 × N₀

Term	Definition
t1/2	Half-life.

λ^t_1/2 = 0.5N₀ / N₀

λ^t_1/2 = 0.5

The half-life can be calculated by taking logarithms (see above).

t_1/2 = 1 / log_0.5(λ) = ln(0.5) / ln(λ)

Geometric (R) growth constant

R = b - d

N_t+1 = N_t + RN_t

N_t+1 - N_t = RN_t

N_t+1 - N_t = ΔN

Term	Definition
ΔN	Change in population size between two generations (between generation t+1 and t).

ΔN = RN_t

ΔN/N_t = R

Finite (λ) growth constant

1 + R = λ

N_t+1 = λN_t

λ = N_t+1 / N_t

Mathematical relationship between geometric and logistic populations

In geometric populations, R and λ represent growth constants. In logistic populations however, the intrinsic growth rate, also known as intrinsic rate of increase (r) is the relevant growth constant. Since generations of reproduction in a geometric population do not overlap (e.g. reproduce once a year) but do in an exponential population, geometric and exponential populations are usually considered to be mutually exclusive. However, both sets of constants share the mathematical relationship below.

The growth equation for exponential populations is

N_t = N₀e^rt

Term	Definition
e	Euler's number - A universal constant often applicable in logistic equations.
r	intrinsic growth rate

Assumption: N_t (of a geometric population) = N_t (of a logistic population).

Therefore:

N₀e^rt = N₀λ^t

N₀ cancels on both sides.

N₀e^rt / N₀ = λ^t

e^rt = λ^t

Take the natural logarithms of the equation. Using natural logarithms instead of base 10 or base 2 logarithms simplifies the final equation as ln(e) = 1.

rt × ln(e) = t × ln(λ)

Term	Definition
ln	natural logarithm - in other words ln(y) = loge(y) = x = the power (x) that e needs to be raised to (ex) to give the answer y. In this case, e1 = e therefore ln(e) = 1.

rt × 1 = t × ln(λ)

rt = t × ln(λ)

t cancels on both sides.

rt / t = ln(λ)

The results:

r = ln(λ)

and

e^r = λ

Evolutionary game theory

Evolutionary game theory was first developed by Ronald Fisher in his 1930 article The Genetic Theory of Natural Selection. In 1973 John Maynard Smith formalised a central concept, the evolutionarily stable strategy.

Population dynamics have been used in several control theory applications. Evolutionary game theory can be used in different industrial or other contexts. Industrially, it is mostly used in multiple-input-multiple-output (MIMO) systems, although it can be adapted for use in single-input-single-output (SISO) systems. Some other examples of applications are military campaigns, water distribution, dispatch of distributed generators, lab experiments, transport problems, communication problems, among others.

Trivia

The computer game SimCity, Sim Earth and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics.