DNA supercoil

In a "relaxed" double-helical segment of B-DNA, the two strands twist around the helical axis once every 10.4–10.5 base pairs of sequence. Adding or subtracting twists, as some enzymes do, imposes strain. If a DNA segment under twist strain is closed into a circle by joining its two ends, and then allowed to move freely, it takes on different shape, such as a figure-eight. This shape is referred to as a supercoil. (The noun form "supercoil" is often used when describing DNA topology.)

The DNA of most organisms is usually negatively supercoiled. It becomes temporarily positively supercoiled when it is being replicated or transcribed. These processes are inhibited (regulated) if it is not promptly relaxed. The simplest shape of a supercoil is a figure eight; a circular DNA strand assumes this shape to accommodate more or few helical twists. The two lobes of the figure eight will appear rotated either clockwise or counterclockwise with respect to one another, depending on whether the helix is over- or underwound. For each additional helical twist being accommodated, the lobes will show one more rotation about their axis.^[2]

Lobal contortions of a circular DNA, such as the rotation of the figure-eight lobes above, are referred to as writhe. The above example illustrates that twist and writhe are interconvertible. Supercoiling can be represented mathematically by the sum of twist and writhe. The twist is the number of helical turns in the DNA and the writhe is the number of times the double helix crosses over on itself (these are the supercoils). Extra helical twists are positive and lead to positive supercoiling, while subtractive twisting causes negative supercoiling. Many topoisomerase enzymes sense supercoiling and either generate or dissipate it as they change DNA topology.

In part because chromosomes may be very large, segments in the middle may act as if their ends are anchored. As a result, they may be unable to distribute excess twist to the rest of the chromosome or to absorb twist to recover from underwinding—the segments may become supercoiled, in other words. In response to supercoiling, they will assume an amount of writhe, just as if their ends were joined.

Supercoiled DNA forms two structures; a plectoneme or a toroid, or a combination of both. A negatively supercoiled DNA molecule will produce either a one-start left-handed helix, the toroid, or a two-start right-handed helix with terminal loops, the plectoneme. Plectonemes are typically more common in nature, and this is the shape most bacterial plasmids will take. For larger molecules it is common for hybrid structures to form – a loop on a toroid can extend into a plectoneme. If all the loops on a toroid extend then it becomes a branch point in the plectonemic structure. DNA supercoiling is important for DNA packaging within all cells, and seems to also play a role in gene expression.^[3][4]

Discover more about Overview related topics

Nucleic acid double helix

Nucleic acid double helix

In 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.

Base pair

Base pair

A base pair (bp) is a fundamental unit of double-stranded nucleic acids consisting of two nucleobases bound to each other by hydrogen bonds. They form the building blocks of the DNA double helix and contribute to the folded structure of both DNA and RNA. Dictated by specific hydrogen bonding patterns, "Watson–Crick" base pairs allow the DNA helix to maintain a regular helical structure that is subtly dependent on its nucleotide sequence. The complementary nature of this based-paired structure provides a redundant copy of the genetic information encoded within each strand of DNA. The regular structure and data redundancy provided by the DNA double helix make DNA well suited to the storage of genetic information, while base-pairing between DNA and incoming nucleotides provides the mechanism through which DNA polymerase replicates DNA and RNA polymerase transcribes DNA into RNA. Many DNA-binding proteins can recognize specific base-pairing patterns that identify particular regulatory regions of genes.

Enzyme

Enzyme

Enzymes are proteins that act as biological catalysts by accelerating chemical reactions. The molecules upon which enzymes may act are called substrates, and the enzyme converts the substrates into different molecules known as products. Almost all metabolic processes in the cell need enzyme catalysis in order to occur at rates fast enough to sustain life. Metabolic pathways depend upon enzymes to catalyze individual steps. The study of enzymes is called enzymology and the field of pseudoenzyme analysis recognizes that during evolution, some enzymes have lost the ability to carry out biological catalysis, which is often reflected in their amino acid sequences and unusual 'pseudocatalytic' properties.

Writhe

Writhe

In knot theory, there are several competing notions of the quantity writhe, or . In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve in such a way that does not change its topology, one may still change its writhe.

Topoisomerase

Topoisomerase

DNA topoisomerases are enzymes that catalyze changes in the topological state of DNA, interconverting relaxed and supercoiled forms, linked (catenated) and unlinked species, and knotted and unknotted DNA. Topological issues in DNA arise due to the intertwined nature of its double-helical structure, which, for example, can lead to overwinding of the DNA duplex during DNA replication and transcription. If left unchanged, this torsion would eventually stop the DNA or RNA polymerases involved in these processes from continuing along the DNA helix. A second topological challenge results from the linking or tangling of DNA during replication. Left unresolved, links between replicated DNA will impede cell division. The DNA topoisomerases prevent and correct these types of topological problems. They do this by binding to DNA and cutting the sugar-phosphate backbone of either one or both of the DNA strands. This transient break allows the DNA to be untangled or unwound, and, at the end of these processes, the DNA backbone is resealed. Since the overall chemical composition and connectivity of the DNA do not change, the DNA substrate and product are chemical isomers, differing only in their topology.

Chromosome

Chromosome

A chromosome is a long DNA molecule with part or all of the genetic material of an organism. In most chromosomes the very long thin DNA fibers are coated with packaging proteins; in eukaryotic cells the most important of these proteins are the histones. These proteins, aided by chaperone proteins, bind to and condense the DNA molecule to maintain its integrity. These chromosomes display a complex three-dimensional structure, which plays a significant role in transcriptional regulation.

Plasmid

Plasmid

A plasmid is a small, extrachromosomal DNA molecule within a cell that is physically separated from chromosomal DNA and can replicate independently. They are most commonly found as small circular, double-stranded DNA molecules in bacteria; however, plasmids are sometimes present in archaea and eukaryotic organisms. In nature, plasmids often carry genes that benefit the survival of the organism and confer selective advantage such as antibiotic resistance. While chromosomes are large and contain all the essential genetic information for living under normal conditions, plasmids are usually very small and contain only additional genes that may be useful in certain situations or conditions. Artificial plasmids are widely used as vectors in molecular cloning, serving to drive the replication of recombinant DNA sequences within host organisms. In the laboratory, plasmids may be introduced into a cell via transformation. Synthetic plasmids are available for procurement over the internet.

Based on the properties of intercalating molecules, i.e. fluorescing upon binding to DNA and unwinding of DNA base-pairs, in 2016, a single-molecule technique has been introduced to directly visualize individual plectonemes along supercoiled DNA^[5] which would further allow to study the interactions of DNA processing proteins with supercoiled DNA. In that study, Sytox Orange (an intercalating dye) was used to induce supercoiling on surface tethered DNA molecules.

Using this assay, it was found that the DNA sequence encodes for the position of plectonemic supercoils.^[6] Furthermore, DNA supercoils were found to be enriched at the transcription start sites in prokaryotes.

Discover more about Intercalation-induced supercoiling of DNA related topics

Intercalation (biochemistry)

Intercalation (biochemistry)

In biochemistry, intercalation is the insertion of molecules between the planar bases of deoxyribonucleic acid (DNA). This process is used as a method for analyzing DNA and it is also the basis of certain kinds of poisoning.

Fluorescence

Fluorescence

Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, and therefore a lower photon energy, than the absorbed radiation. A perceptible example of fluorescence occurs when the absorbed radiation is in the ultraviolet region of the electromagnetic spectrum, while the emitted light is in the visible region; this gives the fluorescent substance a distinct color that can only be seen when the substance has been exposed to UV light. Fluorescent materials cease to glow nearly immediately when the radiation source stops, unlike phosphorescent materials, which continue to emit light for some time after.

Single-molecule experiment

Single-molecule experiment

A single-molecule experiment is an experiment that investigates the properties of individual molecules. Single-molecule studies may be contrasted with measurements on an ensemble or bulk collection of molecules, where the individual behavior of molecules cannot be distinguished, and only average characteristics can be measured. Since many measurement techniques in biology, chemistry, and physics are not sensitive enough to observe single molecules, single-molecule fluorescence techniques caused a lot of excitement, since these supplied many new details on the measured processes that were not accessible in the past. Indeed, since the 1990s, many techniques for probing individual molecules have been developed.

Assay

Assay

An assay is an investigative (analytic) procedure in laboratory medicine, mining, pharmacology, environmental biology and molecular biology for qualitatively assessing or quantitatively measuring the presence, amount, or functional activity of a target entity. The measured entity is often called the analyte, the measurand, or the target of the assay. The analyte can be a drug, biochemical substance, chemical element or compound, or cell in an organism or organic sample. An assay usually aims to measure an analyte's intensive property and express it in the relevant measurement unit.

Prokaryote

Prokaryote

A prokaryote is a single-celled organism that lacks a nucleus and other membrane-bound organelles. The word prokaryote comes from the Greek πρό and κάρυον. In the two-empire system arising from the work of Édouard Chatton, prokaryotes were classified within the empire Prokaryota. But in the three-domain system, based upon molecular analysis, prokaryotes are divided into two domains: Bacteria and Archaea. Organisms with nuclei are placed in a third domain, Eukaryota. In biological evolution, prokaryotes are deemed to have arisen before eukaryotes.

Genome packaging

DNA supercoiling is important for DNA packaging within all cells. Because the length of DNA can be thousands of times that of a cell, packaging this genetic material into the cell or nucleus (in eukaryotes) is a difficult feat. Supercoiling of DNA reduces the space and allows for DNA to be packaged. In prokaryotes, plectonemic supercoils are predominant, because of the circular chromosome and relatively small amount of genetic material. In eukaryotes, DNA supercoiling exists on many levels of both plectonemic and solenoidal supercoils, with the solenoidal supercoiling proving most effective in compacting the DNA. Solenoidal supercoiling is achieved with histones to form a 10 nm fiber. This fiber is further coiled into a 30 nm fiber, and further coiled upon itself numerous times more.

DNA packaging is greatly increased during mitosis when duplicated sister DNAs are segregated into daughter cells. It has been shown that condensin, a large protein complex that plays a central role in mitotic chromosome assembly, induces positive supercoils in an ATP hydrolysis-dependent manner in vitro.^[7][8] Supercoiling could also play an important role during interphase in the formation and maintenance of topologically associating domains (TADs).^[9]

Supercoiling is also required for DNA/RNA synthesis. Because DNA must be unwound for DNA/RNA polymerase action, supercoils will result. The region ahead of the polymerase complex will be unwound; this stress is compensated with positive supercoils ahead of the complex. Behind the complex, DNA is rewound and there will be compensatory negative supercoils. Topoisomerases such as DNA gyrase (Type II Topoisomerase) play a role in relieving some of the stress during DNA/RNA synthesis.^[10]

Gene expression

Specialized proteins can unzip small segments of the DNA molecule when it is replicated or transcribed into RNA. But work published in 2015 illustrates how DNA opens on its own.^[11][12]

Simply twisting DNA can expose internal bases to the outside, without the aid of any proteins. Also, transcription itself contorts DNA in living human cells, tightening some parts of the coil and loosening it in others. That stress triggers changes in shape, most notably opening up the helix to be read. Unfortunately, these interactions are very difficult to study because biological molecules morph shapes so easily. In 2008 it was noted that transcription twists DNA, leaving a trail of undercoiled (or negatively supercoiled) DNA in its wake. Moreover, they discovered that the DNA sequence itself affects how the molecule responds to supercoiling.^[3][4]

For example, the researchers identified a specific sequence of DNA that regulates transcription speed; as the amount of supercoil rises and falls, it slows or speeds the pace at which molecular machinery reads DNA.^[3] It is hypothesized that these structural changes might trigger stress elsewhere along its length, which in turn might provide trigger points for replication or gene expression.^[3][4] This implies that it is a very dynamic process in which both DNA and proteins each influences how the other acts and reacts.^[3]

Gene Expression during cold shock

Almost half of the genes of the bacterium E. coli that are repressed during cold shock are similarly repressed when Gyrase is blocked by the antibiotic Novobiocin.^[13] Moreover, during cold shocks, the density of nucleoids increases, and the protein gyrase and the nucleoid become colocalized (which is consistent with a reduction in DNA relaxation). This is evidence that the reduction of negative supercoiling of the DNA is one of the main mechanisms responsible for the blocking of transcription of half of the genes that conduct the cold shock transcriptional response program of bacteria. Based on this, a stochastic model of this process has been proposed. This model is illustrated in the figure, where reactions 1 represent transcription and its locking due to supercoiling. Meanwhile, reactions 2 to 4 model, respectively, translation, and RNA and protein degradation.^[13]

Illustration of how cold shock affects the supercoiling state of the DNA, by blocking the activity of Gyrase. The signs ‘ − ’ and ‘+’ represent negative and positive supercoiling, respectively. Created with BioRender.com. Also shown is a stochastic model of gene expression during cold shock as a function of the global DNA supercoiling state. The transition from ON to OFF of the promoter (P) causes the locking of transcription (i.e. RNA production). When ON, the promoter can produce RNA, from which proteins can be produced. RNA and proteins are always subject to degradation or dilution due to cell division.

Discover more about Functions related topics

Cell nucleus

Cell nucleus

The cell nucleus is a membrane-bound organelle found in eukaryotic cells. Eukaryotic cells usually have a single nucleus, but a few cell types, such as mammalian red blood cells, have no nuclei, and a few others including osteoclasts have many. The main structures making up the nucleus are the nuclear envelope, a double membrane that encloses the entire organelle and isolates its contents from the cellular cytoplasm; and the nuclear matrix, a network within the nucleus that adds mechanical support.

Eukaryote

Eukaryote

Eukaryota, whose members are known as eukaryotes, is a diverse domain of organisms whose cells have a nucleus. All animals, plants, fungi, and many unicellular organisms, are eukaryotes. They belong to the group of organisms Eukaryota or Eukarya, which is one of the three domains of life. Bacteria and Archaea make up the other two domains.

Mitosis

Mitosis

In cell biology, mitosis is a part of the cell cycle in which replicated chromosomes are separated into two new nuclei. Cell division by mitosis gives rise to genetically identical cells in which the total number of chromosomes is maintained. Therefore, mitosis is also known as equational division. In general, mitosis is preceded by S phase of interphase and is often followed by telophase and cytokinesis; which divides the cytoplasm, organelles and cell membrane of one cell into two new cells containing roughly equal shares of these cellular components. The different stages of mitosis altogether define the mitotic (M) phase of an animal cell cycle—the division of the mother cell into two daughter cells genetically identical to each other.

Condensin

Condensin

Condensins are large protein complexes that play a central role in chromosome assembly and segregation during mitosis and meiosis. Their subunits were originally identified as major components of mitotic chromosomes assembled in Xenopus egg extracts.

Adenosine triphosphate

Adenosine triphosphate

Adenosine triphosphate (ATP) is an organic compound that provides energy to drive many processes in living cells, such as muscle contraction, nerve impulse propagation, condensate dissolution, and chemical synthesis. Found in all known forms of life, ATP is often referred to as the "molecular unit of currency" of intracellular energy transfer. When consumed in metabolic processes, it converts either to adenosine diphosphate (ADP) or to adenosine monophosphate (AMP). Other processes regenerate ATP. The human body recycles its own body weight equivalent in ATP each day. It is also a precursor to DNA and RNA, and is used as a coenzyme.

Interphase

Interphase

Interphase is the portion of the cell cycle that is not accompanied by visible changes under the microscope, and includes the G1, S and G2 phases. During interphase, the cell grows (G1), replicates its DNA (S) and prepares for mitosis (G2). A cell in interphase is not simply quiescent. The term quiescent would be misleading since a cell in interphase is very busy synthesizing proteins, copying DNA into RNA, engulfing extracellular material, processing signals, to name just a few activities. The cell is quiescent only in the sense of cell division. Interphase is the phase of the cell cycle in which a typical cell spends most of its life. Interphase is the 'daily living' or metabolic phase of the cell, in which the cell obtains nutrients and metabolizes them, grows, replicates its DNA in preparation for mitosis, and conducts other "normal" cell functions.

Topologically associating domain

Topologically associating domain

A topologically associating domain (TAD) is a self-interacting genomic region, meaning that DNA sequences within a TAD physically interact with each other more frequently than with sequences outside the TAD. The median size of a TAD in mouse cells is 880 kb, and they have similar sizes in non-mammalian species. Boundaries at both side of these domains are conserved between different mammalian cell types and even across species and are highly enriched with CCCTC-binding factor (CTCF) and cohesin. In addition, some types of genes appear near TAD boundaries more often than would be expected by chance.

Transcription (biology)

Transcription (biology)

Transcription is the process of copying a segment of DNA into RNA. The segments of DNA transcribed into RNA molecules that can encode proteins are said to produce messenger RNA (mRNA). Other segments of DNA are copied into RNA molecules called non-coding RNAs (ncRNAs). mRNA comprises only 1–3% of total RNA samples. Less than 2% of the human genome can be transcribed into mRNA, while at least 80% of mammalian genomic DNA can be actively transcribed, with the majority of this 80% considered to be ncRNA.

Polymerase

Polymerase

A polymerase is an enzyme that synthesizes long chains of polymers or nucleic acids. DNA polymerase and RNA polymerase are used to assemble DNA and RNA molecules, respectively, by copying a DNA template strand using base-pairing interactions or RNA by half ladder replication.

DNA gyrase

DNA gyrase

DNA gyrase, or simply gyrase, is an enzyme within the class of topoisomerase and is a subclass of Type II topoisomerases that reduces topological strain in an ATP dependent manner while double-stranded DNA is being unwound by elongating RNA-polymerase or by helicase in front of the progressing replication fork. The enzyme causes negative supercoiling of the DNA or relaxes positive supercoils. It does so by looping the template so as to form a crossing, then cutting one of the double helices and passing the other through it before releasing the break, changing the linking number by two in each enzymatic step. This process occurs in bacteria, whose single circular DNA is cut by DNA gyrase and the two ends are then twisted around each other to form supercoils. Gyrase is also found in eukaryotic plastids: it has been found in the apicoplast of the malarial parasite Plasmodium falciparum and in chloroplasts of several plants. Bacterial DNA gyrase is the target of many antibiotics, including nalidixic acid, novobiocin, albicidin, and ciprofloxacin.

RNA

RNA

Ribonucleic acid (RNA) is a polymeric molecule essential in various biological roles in coding, decoding, regulation and expression of genes. RNA and deoxyribonucleic acid (DNA) are nucleic acids. Along with lipids, proteins, and carbohydrates, nucleic acids constitute one of the four major macromolecules essential for all known forms of life. Like DNA, RNA is assembled as a chain of nucleotides, but unlike DNA, RNA is found in nature as a single strand folded onto itself, rather than a paired double strand. Cellular organisms use messenger RNA (mRNA) to convey genetic information that directs synthesis of specific proteins. Many viruses encode their genetic information using an RNA genome.

Drawing showing the difference between a circular DNA chromosome (a plasmid) with a secondary helical twist only, and one containing an additional tertiary superhelical twist superimposed on the secondary helical winding.

In nature, circular DNA is always isolated as a higher-order helix-upon-a-helix, known as a superhelix. In discussions of this subject, the Watson–Crick twist is referred to as a "secondary" winding, and the superhelices as a "tertiary" winding. The sketch at right indicates a "relaxed", or "open circular" Watson–Crick double-helix, and, next to it, a right-handed superhelix. The "relaxed" structure on the left is not found unless the chromosome is nicked; the superhelix is the form usually found in nature.

For purposes of mathematical computations, a right-handed superhelix is defined as having a "negative" number of superhelical turns, and a left-handed superhelix is defined as having a "positive" number of superhelical turns. In the drawing (shown at the right), both the secondary (i.e., "Watson–Crick") winding and the tertiary (i.e., "superhelical") winding are right-handed, hence the supertwists are negative (–3 in this example).

The superhelicity is presumed to be a result of underwinding, meaning that there is a deficiency in the number of secondary Watson–Crick twists. Such a chromosome will be strained, just as a macroscopic metal spring is strained when it is either overwound or unwound. In DNA which is thusly strained, supertwists will appear.

DNA supercoiling can be described numerically by changes in the linking number Lk. The linking number is the most descriptive property of supercoiled DNA. Lk_o, the number of turns in the relaxed (B type) DNA plasmid/molecule, is determined by dividing the total base pairs of the molecule by the relaxed bp/turn which, depending on reference is 10.4;^[14] 10.5;^[15][16] 10.6.^[17]

${\displaystyle Lk_{o}=bp/10.4}$

Lk is the number of crosses a single strand makes across the other, often visualized as the number of Watson–Crick twists found in a circular chromosome in a (usually imaginary) planar projection. This number is physically "locked in" at the moment of covalent closure of the chromosome, and cannot be altered without strand breakage.

The topology of the DNA is described by the equation below in which the linking number is equivalent to the sum of Tw, which is the number of twists or turns of the double helix, and Wr, which is the number of coils or "writhes." If there is a closed DNA molecule, the sum of Tw and Wr, or the linking number, does not change. However, there may be complementary changes in Tw and Wr without changing their sum:

${\displaystyle Lk=Tw+Wr}$

Tw, called "twist," is the number of Watson–Crick twists in the chromosome when it is not constrained to lie in a plane. We have already seen that native DNA is usually found to be superhelical. If one goes around the superhelically twisted chromosome, counting secondary Watson–Crick twists, that number will be different from the number counted when the chromosome is constrained to lie flat. In general, the number of secondary twists in the native, supertwisted chromosome is expected to be the "normal" Watson–Crick winding number, meaning a single 10-base-pair helical twist for every 34 Å of DNA length.

Wr, called "writhe," is the number of superhelical twists. Since biological circular DNA is usually underwound, Lk will generally be less than Tw, which means that Wr will typically be negative.

If DNA is underwound, it will be under strain, exactly as a metal spring is strained when forcefully unwound, and that the appearance of supertwists will allow the chromosome to relieve its strain by taking on negative supertwists, which correct the secondary underwinding in accordance with the topology equation above.

The topology equation shows that there is a one-to-one relationship between changes in Tw and Wr. For example, if a secondary "Watson–Crick" twist is removed, then a right-handed supertwist must have been removed simultaneously (or, if the chromosome is relaxed, with no supertwists, then a left-handed supertwist must be added).

The change in the linking number, ΔLk, is the actual number of turns in the plasmid/molecule, Lk, minus the number of turns in the relaxed plasmid/molecule Lk_o:

${\displaystyle \Delta Lk=Lk-Lk_{o}}$

If the DNA is negatively supercoiled, ${\displaystyle \Delta Lk$. The negative supercoiling implies that the DNA is underwound.

A standard expression independent of the molecule size is the "specific linking difference" or "superhelical density" denoted σ, which represents the number of turns added or removed relative to the total number of turns in the relaxed molecule/plasmid, indicating the level of supercoiling.

${\displaystyle \sigma =\Delta {Lk/Lk_{o}}}$

The Gibbs free energy associated with the coiling is given by the equation below^[18]

${\displaystyle {\Delta G/N=10RT\sigma ^{2}}}$

The difference in Gibbs free energy between the supercoiled circular DNA and uncoiled circular DNA with N > 2000 bp is approximated by:

${\displaystyle \Delta G/N=700\,{\text{Kcal}}/bp*(\Delta Lk/N)}$

or, 16 cal/bp.

Since the linking number L of supercoiled DNA is the number of times the two strands are intertwined (and both strands remain covalently intact), L cannot change. The reference state (or parameter) L₀ of a circular DNA duplex is its relaxed state. In this state, its writhe W = 0. Since L = T + W, in a relaxed state T = L. Thus, if we have a 400 bp relaxed circular DNA duplex, L ~ 40 (assuming ~10 bp per turn in B-DNA). Then T ~ 40.

• Positively supercoiling:

  T = 0, W = 0, then L = 0

  T = +3, W = 0, then L = +3

  T = +2, W = +1, then L = +3

• Negatively supercoiling:

  T = 0, W = 0, then L = 0

  T = -3, W = 0, then L = -3

  T = -2, W = -1, then L = -3

Negative supercoils favor local unwinding of the DNA, allowing processes such as transcription, DNA replication, and recombination. Negative supercoiling is also thought to favour the transition between B-DNA and Z-DNA, and moderate the interactions of DNA binding proteins involved in gene regulation.^[19]

Discover more about Mathematical description related topics

Linking number

Linking number

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.

Base pair

Base pair

A base pair (bp) is a fundamental unit of double-stranded nucleic acids consisting of two nucleobases bound to each other by hydrogen bonds. They form the building blocks of the DNA double helix and contribute to the folded structure of both DNA and RNA. Dictated by specific hydrogen bonding patterns, "Watson–Crick" base pairs allow the DNA helix to maintain a regular helical structure that is subtly dependent on its nucleotide sequence. The complementary nature of this based-paired structure provides a redundant copy of the genetic information encoded within each strand of DNA. The regular structure and data redundancy provided by the DNA double helix make DNA well suited to the storage of genetic information, while base-pairing between DNA and incoming nucleotides provides the mechanism through which DNA polymerase replicates DNA and RNA polymerase transcribes DNA into RNA. Many DNA-binding proteins can recognize specific base-pairing patterns that identify particular regulatory regions of genes.

Gibbs free energy

Gibbs free energy

In thermodynamics, the Gibbs free energy is a thermodynamic potential that can be used to calculate the maximum amount of non-volume expansion work that may be performed by a thermodynamically closed system at constant temperature and pressure. It also provides a necessary condition for processes such as chemical reactions that may occur under these conditions. The Gibbs free energy is expressed as where p is pressure, T is the temperature, U is the internal energy, V is volume, H is the enthalpy, and S is the entropy.

DNA replication

DNA replication

In molecular biology, DNA replication is the biological process of producing two identical replicas of DNA from one original DNA molecule. DNA replication occurs in all living organisms acting as the most essential part of biological inheritance. This is essential for cell division during growth and repair of damaged tissues, while it also ensures that each of the new cells receives its own copy of the DNA. The cell possesses the distinctive property of division, which makes replication of DNA essential.

Genetic recombination

Genetic recombination

Genetic recombination is the exchange of genetic material between different organisms which leads to production of offspring with combinations of traits that differ from those found in either parent. In eukaryotes, genetic recombination during meiosis can lead to a novel set of genetic information that can be further passed on from parents to offspring. Most recombination occurs naturally and can be classified into two types: (1) interchromosomal recombination, occurring through independent assortment of alleles whose loci are on different but homologous chromosomes ; & (2) intrachromosomal recombination, occurring through crossing over.

Z-DNA

Z-DNA

Z-DNA is one of the many possible double helical structures of DNA. It is a left-handed double helical structure in which the helix winds to the left in a zigzag pattern, instead of to the right, like the more common B-DNA form. Z-DNA is thought to be one of three biologically active double-helical structures along with A-DNA and B-DNA.

Some stochastic models have been proposed to account for the effects of positive supercoiling buildup (PSB) in gene expression dynamics (e.g. in bacterial gene expression), differing in, e.g., the level of detail. In general, the detail increases when adding processes affected by and affecting supercoiling. As this addition occurs, the complexity of the model increases.

For example, in ^[20] two models of different complexity are proposed. In the most detailed one, events were modeled at the nucleotide level, while in the other the events were modeled at the promoter region alone, and thus required much less events to be accounted for.

Stochastic, prokaryotic model of the dynamics of RNA production and transcription locking at the promoter region, due to PSB.

Examples of stochastic models that focus on the effects of PSB on a promoter’s activity can be found in: ^[21][22]. In general, such models include a promoter, Pro, which is the region of DNA controlling transcription and, thus, whose activity/locking is affected by PSB. Also included are RNA molecules (the product of transcription), RNA polymerases (RNAP) which control transcription, and Gyrases (G) which regulate PSB. Finally, there needs to be a means to quantify PSB on the DNA (i.e. the promoter) at any given moment. This can be done by having some component in the system that is produced over time (e.g., during transcription events) to represent positive supercoils, and that is removed by the action of Gyrases. The amount of this component can then be set to affect the rate of transcription.

Figure showing the various conformational changes which are observed in circular DNA at different pH. At a pH of about 12 (alkaline), there is a dip in the sedimentation coefficient, followed by a relentless increase up to a pH of about 13, at which pH the structure converts into the mysterious "Form IV".

The topological properties of circular DNA are complex. In standard texts, these properties are invariably explained in terms of a helical model for DNA, but in 2008 it was noted that each topoisomer, negative or positive, adopts a unique and surprisingly wide distribution of three-dimensional conformations.^[4]

When the sedimentation coefficient, s, of circular DNA is ascertained over a large range of pH, the following curves are seen. Three curves are shown here, representing three species of DNA. From top-to-bottom they are: "Form IV" (green), "Form I" (blue) and "Form II" (red).

"Form I" (blue curve) is the traditional nomenclature used for the native form of duplex circular DNA, as recovered from viruses and intracellular plasmids. Form I is covalently closed, and any plectonemic winding which may be present is therefore locked in. If one or more nicks are introduced to Form I, free rotation of one strand with respect to the other becomes possible, and Form II (red curve) is seen.

Form IV (green curve) is the product of alkali denaturation of Form I. Its structure is unknown, except that it is persistently duplex, and extremely dense.

Between pH 7 and pH 11.5, the sedimentation coefficient s, for Form I, is constant. Then it dips, and at a pH just below 12, reaches a minimum. With further increases in pH, s then returns to its former value. It doesn’t stop there, however, but continues to increase relentlessly. By pH 13, the value of s has risen to nearly 50, two to three times its value at pH 7, indicating an extremely compact structure.

If the pH is then lowered, the s value is not restored. Instead, one sees the upper, green curve. The DNA, now in the state known as Form IV, remains extremely dense, even if the pH is restored to the original physiologic range. As stated previously, the structure of Form IV is almost entirely unknown, and there is no currently accepted explanation for its extraordinary density. About all that is known about the tertiary structure is that it is duplex, but has no hydrogen bonding between bases.

These behaviors of Forms I and IV are considered to be due to the peculiar properties of duplex DNA which has been covalently closed into a double-stranded circle. If the covalent integrity is disrupted by even a single nick in one of the strands, all such topological behavior ceases, and one sees the lower Form II curve (Δ). For Form II, alterations in pH have very little effect on s. Its physical properties are, in general, identical to those of linear DNA. At pH 13, the strands of Form II simply separate, just as the strands of linear DNA do. The separated single strands have slightly different s values, but display no significant changes in s with further increases in pH.

A complete explanation for these data is beyond the scope of this article. In brief, the alterations in s come about because of changes in the superhelicity of circular DNA. These changes in superhelicity are schematically illustrated by four little drawings which have been strategically superimposed upon the figure above.

Briefly, the alterations of s seen in the pH titration curve above are widely thought to be due to changes in the superhelical winding of DNA under conditions of increasing pH. Up to pH 11.5, the purported "underwinding" produces a right-handed ("negative") supertwist. But as the pH increases, and the secondary helical structure begins to denature and unwind, the chromosome (if we may speak anthropomorphically) no longer "wants" to have the full Watson–Crick winding, but rather "wants", increasingly, to be "underwound". Since there is less and less strain to be relieved by superhelical winding, the superhelices therefore progressively disappear as the pH increases. At a pH just below 12, all incentive for superhelicity has expired, and the chromosome will appear as a relaxed, open circle.

At higher pH still, the chromosome, which is now denaturing in earnest, tends to unwind entirely, which it cannot do so (because L_k is covalently locked in). Under these conditions, what was once treated as "underwinding" has actually now become "overwinding". Once again there is strain, and once again it is (in part at least) relieved by superhelicity, but this time in the opposite direction (i.e., left-handed or "positive"). Each left-handed tertiary supertwist removes a single, now undesirable right-handed Watson–Crick secondary twist.

The titration ends at pH 13, where Form IV appears.