SequenceIn mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position.
Limit of a sequenceAs the positive integer becomes larger and larger, the value becomes arbitrarily close to . We say that "the limit of the sequence equals ." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
DNADeoxyribonucleic acid (diːˈɒksᵻˌraɪboʊnjuːˌkliːᵻk,_-ˌkleɪ-; DNA) is a polymer composed of two polynucleotide chains that coil around each other to form a double helix. The polymer carries genetic instructions for the development, functioning, growth and reproduction of all known organisms and many viruses. DNA and ribonucleic acid (RNA) are nucleic acids. Alongside proteins, lipids and complex carbohydrates (polysaccharides), nucleic acids are one of the four major types of macromolecules that are essential for all known forms of life.
Sequence analysisIn bioinformatics, sequence analysis is the process of subjecting a DNA, RNA or peptide sequence to any of a wide range of analytical methods to understand its features, function, structure, or evolution. Methodologies used include sequence alignment, searches against biological databases, and others. Since the development of methods of high-throughput production of gene and protein sequences, the rate of addition of new sequences to the databases increased very rapidly.
Sequence alignmentIn bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. Aligned sequences of nucleotide or amino acid residues are typically represented as rows within a matrix. Gaps are inserted between the residues so that identical or similar characters are aligned in successive columns.
Cauchy sequenceIn mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: the consecutive terms become arbitrarily close to each other – their differences tend to zero as the index n grows.
Scientific methodThe scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific method for additional detail.) It involves careful observation, applying rigorous skepticism about what is observed, given that cognitive assumptions can distort how one interprets the observation.
Sequence spaceIn functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication.
Sequence homologySequence homology is the biological homology between DNA, RNA, or protein sequences, defined in terms of shared ancestry in the evolutionary history of life. Two segments of DNA can have shared ancestry because of three phenomena: either a speciation event (orthologs), or a duplication event (paralogs), or else a horizontal (or lateral) gene transfer event (xenologs). Homology among DNA, RNA, or proteins is typically inferred from their nucleotide or amino acid sequence similarity.
Chain ruleIn calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation, or, equivalently, The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y.
Nucleic acid sequenceA nucleic acid sequence is a succession of bases within the nucleotides forming alleles within a DNA (using GACT) or RNA (GACU) molecule. This succession is denoted by a series of a set of five different letters that indicate the order of the nucleotides. By convention, sequences are usually presented from the 5' end to the 3' end. For DNA, with its double helix, there are two possible directions for the notated sequence; of these two, the sense strand is used.
LengthLength is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in.
Nucleic acid double helixIn molecular biology, the term double helix refers to the structure formed by double-stranded molecules of nucleic acids such as DNA. The double helical structure of a nucleic acid complex arises as a consequence of its secondary structure, and is a fundamental component in determining its tertiary structure. The term entered popular culture with the publication in 1968 of The Double Helix: A Personal Account of the Discovery of the Structure of DNA by James Watson.
Spectral sequenceIn homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology.
Multiple sequence alignmentMultiple sequence alignment (MSA) may refer to the process or the result of sequence alignment of three or more biological sequences, generally protein, DNA, or RNA. In many cases, the input set of query sequences are assumed to have an evolutionary relationship by which they share a linkage and are descended from a common ancestor. From the resulting MSA, sequence homology can be inferred and phylogenetic analysis can be conducted to assess the sequences' shared evolutionary origins.
Quotient ruleIn calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is It is provable in many ways by using other derivative rules. Given , let , then using the quotient rule: The quotient rule can be used to find the derivative of as follows: Reciprocal rule The reciprocal rule is a special case of the quotient rule in which the numerator .
EnergyIn physics, energy () is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat and light. Energy is a conserved quantity—the law of conservation of energy states that energy can be converted in form, but not created or destroyed. The unit of measurement for energy in the International System of Units (SI) is the joule (J).
Computational anatomyComputational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures. The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, machine learning, computational mechanics, computational science, biological imaging, neuroscience, physics, probability, and statistics; it also has strong connections with fluid mechanics and geometric mechanics.
Computational neuroscienceComputational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to understand the principles that govern the development, structure, physiology and cognitive abilities of the nervous system. Computational neuroscience employs computational simulations to validate and solve mathematical models, and so can be seen as a sub-field of theoretical neuroscience; however, the two fields are often synonymous.
Bond energyIn chemistry, bond energy (BE), also called the mean bond enthalpy or average bond enthalpy is a measure of bond strength in a chemical bond. IUPAC defines bond energy as the average value of the gas-phase bond-dissociation energy (usually at a temperature of 298.15 K) for all bonds of the same type within the same chemical species. The bond dissociation energy (enthalpy) is also referred to as bond disruption energy, bond energy, bond strength, or binding energy (abbreviation: BDE, BE, or D).
