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107-468: 5925 19645 ENSG00000139687 ENSMUSG00000022105 P06400 P13405 NM_000321 NM_009029 NP_000312 NP_000312.2 NP_033055 The retinoblastoma protein (protein name abbreviated Rb or pRb ; gene name abbreviated Rb , RB or RB1 ) is a tumor suppressor protein that is dysfunctional in several major cancers . One function of pRb is to prevent excessive cell growth by inhibiting cell cycle progression until

214-443: A 0 x + c = c + ∑ i = 0 n a i x i + 1 i + 1 {\displaystyle {\frac {a_{n}x^{n+1}}{n+1}}+{\frac {a_{n-1}x^{n}}{n}}+\dots +{\frac {a_{2}x^{3}}{3}}+{\frac {a_{1}x^{2}}{2}}+a_{0}x+c=c+\sum _{i=0}^{n}{\frac {a_{i}x^{i+1}}{i+1}}} where c is an arbitrary constant. For example, antiderivatives of x + 1 have

321-532: A 2 x 2 + a 1 x + a 0 {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}} that evaluates to f ( x ) {\displaystyle f(x)} for all x in the domain of f (here, n is a non-negative integer and a 0 , a 1 , a 2 , ..., a n are constant coefficients). Generally, unless otherwise specified, polynomial functions have complex coefficients, arguments, and values. In particular,

428-410: A 2 x 2 + a 1 x + a 0 , {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0},} where a 0 , … , a n {\displaystyle a_{0},\ldots ,a_{n}} are constants that are called the coefficients of the polynomial, and x {\displaystyle x}

535-420: A 3 ) x + a 2 ) x + a 1 ) x + a 0 . {\displaystyle (((((a_{n}x+a_{n-1})x+a_{n-2})x+\dotsb +a_{3})x+a_{2})x+a_{1})x+a_{0}.} A polynomial function in one real variable can be represented by a graph . A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value ). If

642-433: A k x k {\displaystyle \sum _{k=0}^{n}a_{k}x^{k}} That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms . Each term consists of the product of a number – called the coefficient of the term  – and a finite number of indeterminates, raised to non-negative integer powers. The exponent on an indeterminate in

749-402: A n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0 = 0. {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}=0.} For example, 3 x 2 + 4 x − 5 = 0 {\displaystyle 3x^{2}+4x-5=0}

856-417: A constant term and a constant polynomial . The degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below). For example: − 5 x 2 y {\displaystyle -5x^{2}y} is a term. The coefficient is −5 , the indeterminates are x and y ,

963-425: A denotes a number, a variable, another polynomial, or, more generally, any expression, then P ( a ) denotes, by convention, the result of substituting a for x in P . Thus, the polynomial P defines the function a ↦ P ( a ) , {\displaystyle a\mapsto P(a),} which is the polynomial function associated to P . Frequently, when using this notation, one supposes that

1070-499: A is a number. However, one may use it over any domain where addition and multiplication are defined (that is, any ring ). In particular, if a is a polynomial then P ( a ) is also a polynomial. More specifically, when a is the indeterminate x , then the image of x by this function is the polynomial P itself (substituting x for x does not change anything). In other words, P ( x ) = P , {\displaystyle P(x)=P,} which justifies formally

1177-414: A polynomial is a mathematical expression consisting of indeterminates (also called variables ) and coefficients , that involves only the operations of addition , subtraction , multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x is x − 4 x + 7 . An example with three indeterminates

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1284-422: A ( x )/ b ( x ) results in two polynomials, a quotient q ( x ) and a remainder r ( x ) , such that a = b q + r and degree( r ) < degree( b ) . The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division . When the denominator b ( x ) is monic and linear, that is, b ( x ) = x − c for some constant c , then

1391-553: A bifurcation point: Presence of un-phosphorylated pRb drives cell cycle exit and maintains senescence. At the end of mitosis, PP1 dephosphorylates hyper-phosphorylated pRb directly to its un-phosphorylated state. Furthermore, when cycling C2C12 myoblast cells differentiated (by being placed into a differentiation medium), only un-phosphorylated pRb was present. Additionally, these cells had a markedly decreased growth rate and concentration of DNA replication factors (suggesting G0 arrest). This function of un-phosphorylated pRb gives rise to

1498-581: A cell is ready to divide. When the cell is ready to divide, pRb is phosphorylated , inactivating it, and the cell cycle is allowed to progress. It is also a recruiter of several chromatin remodeling enzymes such as methylases and acetylases . pRb belongs to the pocket protein family , whose members have a pocket for the functional binding of other proteins. Should an oncogenic protein, such as those produced by cells infected by high-risk types of human papillomavirus , bind and inactivate pRb, this can lead to cancer. The RB gene may have been responsible for

1605-514: A cell passes the restriction point, Cyclin E - Cdk 2 hyper-phosphorylates all mono-phosphorylated isoforms. While the exact mechanism is unknown, one hypothesis is that binding to the C-terminus tail opens the pocket subunit, allowing access to all phosphorylation sites. This process is hysteretic and irreversible, and it is thought accumulation of mono-phosphorylated pRb induces the process. The bistable, switch like behavior of pRb can thus be modeled as

1712-408: A cell sustain only one mutation in the other RB gene, all pRb in that cell would be ineffective at inhibiting cell cycle progression, allowing cells to divide uncontrollably and eventually become cancerous. Furthermore, as one allele is already mutated in all other somatic cells, the future incidence of cancers in these individuals is observed with linear kinetics. The working allele need not undergo

1819-452: A consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial. There may be several meanings of "solving an equation" . One may want to express the solutions as explicit numbers; for example, the unique solution of 2 x − 1 = 0 is 1/2 . This is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express

1926-430: A general formula in radicals. However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree. The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the complex solutions are counted with their multiplicity . This fact is called the fundamental theorem of algebra . A root of

2033-545: A hypothesis for the lack of cell cycle control in cancerous cells: Deregulation of Cyclin D - Cdk 4/6 phosphorylates un-phosphorylated pRb in senescent cells to mono-phosphorylated pRb, causing them to enter G1. The mechanism of the switch for Cyclin E activation is not known, but one hypothesis is that it is a metabolic sensor. Mono-phosphorylated pRb induces an increase in metabolism, so the accumulation of mono-phosphorylated pRb in previously G0 cells then causes hyper-phosphorylation and mitotic entry. Since any un-phosphorylated pRb

2140-419: A mutation per se, as loss of heterozygosity (LOH) is frequently observed in such tumours. However, in the sporadic form, both alleles would need to sustain a mutation before the cell can become cancerous. This explains why sufferers of sporadic retinoblastoma are not at increased risk of cancers later in life, as both alleles are functional in all their other cells. Future cancer incidence in sporadic pRb cases

2247-430: A nonzero univariate polynomial P is a value a of x such that P ( a ) = 0 . In other words, a root of P is a solution of the polynomial equation P ( x ) = 0 or a zero of the polynomial function defined by P . In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered. A number a is a root of a polynomial P if and only if

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2354-411: A polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also restricted to the reals, the resulting function is a real function that maps reals to reals. For example, the function f , defined by f ( x ) = x 3 − x , {\displaystyle f(x)=x^{3}-x,}

2461-541: A recruiter that allows for the binding of proteins that alter chromatin structure onto the site E2F-regulated promoters. Access to these E2F-regulated promoters by transcriptional factors is blocked by the formation of nucleosomes and their further packing into chromatin. Nucleosome formation is regulated by post-translational modifications to histone tails. Acetylation leads to the disruption of nucleosome structure. Proteins called histone acetyltransferases (HATs) are responsible for acetylating histones and thus facilitating

2568-554: A region that is independent to its E2F-binding site. pRb recruitment of histone deacetylases leads to the repression of genes at E2F-regulated promoters due to nucleosome formation. Some genes activated during the G1/S transition such as cyclin E are repressed by HDAC during early to mid-G1 phase. This suggests that HDAC-assisted repression of cell cycle progression genes is crucial for the ability of pRb to arrest cells in G1. To further add to this point,

2675-401: A sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial. The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called rational fractions , rational expressions , or rational functions , depending on context. This is analogous to

2782-427: A term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any term with nonzero coefficient. Because x = x , the degree of an indeterminate without a written exponent is one. A term with no indeterminates and a polynomial with no indeterminates are called, respectively,

2889-433: A tumor suppressor and cell cycle regulator developed through research investigating mechanisms of interactions with E2F family member proteins. Yet, more data generated from biochemical experiments and clinical trials reveal other functions of pRb within the cell unrelated (or indirectly related) to tumor suppression. In proliferating cells, certain pRb conformations (when RxL motif if bound by protein phosphatase 1 or when it

2996-726: Is x + 2 xyz − yz + 1 . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations , which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions , which appear in settings ranging from basic chemistry and physics to economics and social science ; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties , which are central concepts in algebra and algebraic geometry . The word polynomial joins two diverse roots :

3103-403: Is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the function defined by the polynomial, then x represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably. A polynomial P in the indeterminate x is commonly denoted either as P or as P ( x ). Formally, the name of

3210-415: Is a polynomial equation. When considering equations, the indeterminates (variables) of polynomials are also called unknowns , and the solutions are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a polynomial identity like ( x + y )( x − y ) = x − y , where both expressions represent

3317-412: Is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in f ( x , y ) = 2 x 3 + 4 x 2 y + x y 5 + y 2 − 7. {\displaystyle f(x,y)=2x^{3}+4x^{2}y+xy^{5}+y^{2}-7.} According to

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3424-423: Is a polynomial with real coefficients. When it is used to define a function , the domain is not so restricted. However, a real polynomial function is a function from the reals to the reals that is defined by a real polynomial. Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients. A polynomial in one indeterminate

3531-522: Is able to bind to the activation domain of the activator E2Fs which blocks their activity, repressing transcription of the genes controlled by that E2F-promoter. The preinitiation complex (PIC) assembles in a stepwise fashion on the promoter of genes to initiate transcription. The TFIID binds to the TATA box in order to begin the assembly of the TFIIA , recruiting other transcription factors and components needed in

3638-636: Is acetylated or methylated) are resistant to CDK phosphorylation and retain other function throughout cell cycle progression, suggesting not all pRb in the cell are devoted to guarding the G1/S transition. Tumor suppressor Too Many Requests If you report this error to the Wikimedia System Administrators, please include the details below. Request from 172.68.168.226 via cp1108 cp1108, Varnish XID 829459475 Upstream caches: cp1108 int Error: 429, Too Many Requests at Fri, 29 Nov 2024 08:42:04 GMT Polynomial In mathematics ,

3745-654: Is also common to say simply "polynomials in x , y , and z ", listing the indeterminates allowed. Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law ) and combining of like terms. For example, if P = 3 x 2 − 2 x + 5 x y − 2 {\displaystyle P=3x^{2}-2x+5xy-2} and Q = − 3 x 2 + 3 x + 4 y 2 + 8 {\displaystyle Q=-3x^{2}+3x+4y^{2}+8} then

3852-425: Is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots . The graph of the zero polynomial, f ( x ) = 0 , is the x -axis. In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n . The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree

3959-462: Is available) cellular energy for anabolism . In vivo, it is still not entirely clear how and which cell types cancer initiation occurs with solely loss of pRb, but it is clear that the pRb pathway is altered in large number of human cancers.[110] In mice, loss of pRb is sufficient to initiate tumors of the pituitary and thyroid glands, and mechanisms of initiation for these hyperplasia are currently being investigated. The classic view of pRb's role as

4066-428: Is called a univariate polynomial , a polynomial in more than one indeterminate is called a multivariate polynomial . A polynomial with two indeterminates is called a bivariate polynomial . These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance, when working with univariate polynomials, one does not exclude constant polynomials (which may result from

4173-466: Is called the multiplicity of a as a root of P . The number of roots of a nonzero polynomial P , counted with their respective multiplicities, cannot exceed the degree of P , and equals this degree if all complex roots are considered (this is a consequence of the fundamental theorem of algebra ). The coefficients of a polynomial and its roots are related by Vieta's formulas . Some polynomials, such as x + 1 , do not have any roots among

4280-535: Is crucial for recruitment of p130/p107 to the MuvB core and thus DREAM assembly. Consequences of loss of pRb function is dependent on cell type and cell cycle status, as pRb's tumor suppressive role changes depending on the state and current identity of the cell. In G0 quiescent stem cells, pRb is proposed to maintain G0 arrest although the mechanism remains largely unknown. Loss of pRb leads to exit from quiescence and an increase in

4387-498: Is immediately phosphorylated, the cell is then unable to exit the cell cycle, resulting in continuous division. DNA damage to G0 cells activates Cyclin D - Cdk 4/6, resulting in mono-phosphorylation of un-phosphorylated pRb. Then, active mono-phosphorylated pRb causes repression of E2F-targeted genes specifically. Therefore, mono-phosphorylated pRb is thought to play an active role in DNA damage response, so that E2F gene repression occurs until

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4494-502: Is now known that pRb is un-phosphorylated in G0 cells and mono-phosphorylated in early G1 cells, prior to hyper-phosphorylation after the restriction point in late G1. When a cell enters G1, Cyclin D- Cdk4/6 phosphorylates pRb at a single phosphorylation site. No progressive phosphorylation occurs because when HFF cells were exposed to sustained cyclin D- Cdk4/6 activity (and even deregulated activity) in early G1, only mono-phosphorylated pRb

4601-483: Is observed with polynomial kinetics, not exactly quadratic as expected because the first mutation must arise through normal mechanisms, and then can be duplicated by LOH to result in a tumour progenitor . RB1 orthologs have also been identified in most mammals for which complete genome data are available. RB / E2F -family proteins repress transcription . pRb is a multifunctional protein with many binding and phosphorylation sites. Although its common function

4708-441: Is reversible. Following induced knockout of pRb, cells treated with cisplatin , a DNA-damaging agent, were able to continue proliferating, without cell cycle arrest, suggesting pRb plays an important role in triggering chronic S-phase arrest in response to genotoxic stress. One such example of E2F-regulated genes repressed by pRb are cyclin E and cyclin A . Both of these cyclins are able to bind to Cdk2 and facilitate entry into

4815-656: Is seen as binding and repressing E2F targets, pRb is likely a multifunctional protein as it binds to at least 100 other proteins. pRb has three major structural components: a carboxy-terminus, a "pocket" subunit, and an amino-terminus. Within each domain, there are a variety of protein binding sites, as well as a total of 15 possible phosphorylation sites. Generally, phosphorylation causes interdomain locking, which changes pRb's conformation and prevents binding to target proteins. Different sites may be phosphorylated at different times, giving rise to many possible conformations and likely many functions/activity levels. pRb restricts

4922-434: Is the indeterminate. The word "indeterminate" means that x {\displaystyle x} represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a function , called a polynomial function . This can be expressed more concisely by using summation notation : ∑ k = 0 n

5029-509: Is the polynomial n a n x n − 1 + ( n − 1 ) a n − 1 x n − 2 + ⋯ + 2 a 2 x + a 1 = ∑ i = 1 n i a i x i − 1 . {\displaystyle na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\dots +2a_{2}x+a_{1}=\sum _{i=1}^{n}ia_{i}x^{i-1}.} Similarly,

5136-446: Is too complicated to be useful, the unique way of solving it is to compute numerical approximations of the solutions. There are many methods for that; some are restricted to polynomials and others may apply to any continuous function . The most efficient algorithms allow solving easily (on a computer ) polynomial equations of degree higher than 1,000 (see Root-finding algorithm ). For polynomials with more than one indeterminate,

5243-464: Is undefined. For example, x y + 7 x y − 3 x is homogeneous of degree 5. For more details, see Homogeneous polynomial . The commutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x ", with the term of largest degree first, or in "ascending powers of x ". The polynomial 3 x − 5 x + 4

5350-432: Is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers , the irreducible factors are linear. Over the real numbers , they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree. For example,

5457-454: Is written in descending powers of x . The first term has coefficient 3 , indeterminate x , and exponent 2 . In the second term, the coefficient is −5 . The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two. Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using

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5564-506: The DREAM complex composed of DP, E2F4/5, RB-like (p130/p107) And MuvB (Lin9:Lin37:Lin52:RbAbP4:Lin54). The DREAM complex is assembled in Go/G1 and maintains quiescence by assembling at the promoters of > 800 cell-cycle genes and mediating transcriptional repression. Assembly of DREAM requires DYRK1A (Ser/Thr kinase) dependant phosphorylation of the MuvB core component, Lin52 at Serine28. This mechanism

5671-430: The distributive law , into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0. Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial , a two-term polynomial is called a binomial , and a three-term polynomial is called a trinomial . A real polynomial

5778-415: The linear polynomial x − a divides P , that is if there is another polynomial Q such that P = ( x − a ) Q . It may happen that a power (greater than 1 ) of x − a divides P ; in this case, a is a multiple root of P , and otherwise a is a simple root of P . If P is a nonzero polynomial, there is a highest power m such that ( x − a ) divides P , which

5885-404: The polynomial remainder theorem asserts that the remainder of the division of a ( x ) by b ( x ) is the evaluation a ( c ) . In this case, the quotient may be computed by Ruffini's rule , a special case of synthetic division. All polynomials with coefficients in a unique factorization domain (for example, the integers or a field ) also have a factored form in which the polynomial

5992-399: The quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), although much more complicated, are known for equations of degree three and four (see cubic equation and quartic equation ). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, Niels Henrik Abel proved

6099-423: The real numbers . If, however, the set of accepted solutions is expanded to the complex numbers , every non-constant polynomial has at least one root; this is the fundamental theorem of algebra . By successively dividing out factors x − a , one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as

6206-560: The E2F family and thereby inhibit their function. When pRb is chronically activated, it leads to the downregulation of the necessary DNA replication factors. Within 72–96 hours of active pRb induction in A2-4 cells, the target DNA replication factor proteins—MCMs, RPA34, DBF4 , RFCp37, and RFCp140—all showed decreased levels. Along with decreased levels, there was a simultaneous and expected inhibition of DNA replication in these cells. This process, however,

6313-526: The Greek poly , meaning "many", and the Latin nomen , or "name". It was derived from the term binomial by replacing the Latin root bi- with the Greek poly- . That is, it means a sum of many terms (many monomials ). The word polynomial was first used in the 17th century. The x occurring in a polynomial is commonly called a variable or an indeterminate . When the polynomial is considered as an expression, x

6420-544: The HDAC-pRb complex is shown to be disrupted by cyclin D/Cdk4 which levels increase and peak during the late G1 phase. Senescence in cells is a state in which cells are metabolically active but are no longer able to replicate. pRb is an important regulator of senescence in cells and since this prevents proliferation, senescence is an important antitumor mechanism. pRb may occupy E2F-regulated promoters during senescence. For example, pRb

6527-571: The PIC. Data suggests that pRb is able to repress transcription by both pRb being recruited to the promoter as well as having a target present in TFIID. The presence of pRb may change the conformation of the TFIIA/IID complex into a less active version with a decreased binding affinity. pRb can also directly interfere with their association as proteins, preventing TFIIA/IID from forming an active complex. pRb acts as

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6634-485: The S phase of the cell cycle. Through the repression of expression of cyclin E and cyclin A, pRb is able to inhibit the G1/S transition . There are at least three distinct mechanisms in which pRb can repress transcription of E2F-regulated promoters . Though these mechanisms are known, it is unclear which are the most important for the control of the cell cycle. E2Fs are a family of proteins whose binding sites are often found in

6741-481: The association of transcription factors on DNA promoters. Deacetylation, on the other hand, leads to nucleosome formation and thus makes it more difficult for transcription factors to sit on promoters. Histone deacetylases (HDACs) are the proteins responsible for facilitating nucleosome formation and are therefore associated with transcriptional repressors proteins. pRb interacts with the histone deacetylases HDAC1 and HDAC3 . pRb binds to HDAC1 in its pocket domain in

6848-405: The cell cycle and acting as a growth suppressor. The pRb-E2F/DP complex also attracts a histone deacetylase (HDAC) protein to the chromatin , reducing transcription of S phase promoting factors, further suppressing DNA synthesis. pRb has the ability to reversibly inhibit DNA replication through transcriptional repression of DNA replication factors. pRb is able to bind to transcription factors in

6955-510: The cell cycle by activating cyclin-dependent kinases, and a molecule called proliferating cell nuclear antigen, or PCNA , which speeds DNA replication and repair by helping to attach polymerase to DNA. Since the 1990s, pRb was known to be inactivated via phosphorylation. Until, the prevailing model was that Cyclin D- Cdk 4/6 progressively phosphorylated it from its unphosphorylated to its hyperphosphorylated state (14+ phosphorylations). However, it

7062-485: The cell's ability to replicate DNA by preventing its progression from the G1 ( first gap phase ) to S ( synthesis phase ) phase of the cell division cycle. pRb binds and inhibits E2 promoter-binding–protein-dimerization partner (E2F-DP) dimers, which are transcription factors of the E2F family that push the cell into S phase. By keeping E2F-DP inactivated, RB1 maintains the cell in the G1 phase, preventing progression through

7169-401: The combinations of values for the variables for which the polynomial function takes the value zero are generally called zeros instead of "roots". The study of the sets of zeros of polynomials is the object of algebraic geometry . For a set of polynomial equations with several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number

7276-401: The complex numbers. The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems . Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions. The derivative of

7383-519: The damage is fixed and the cell can pass the restriction point. As a side note, the discovery that damages causes Cyclin D - Cdk 4/6 activation even in G0 cells should be kept in mind when patients are treated with both DNA damaging chemotherapy and Cyclin D - Cdk 4/6 inhibitors. During the M-to-G1 transition, pRb is then progressively dephosphorylated by PP1 , returning to its growth-suppressive hypophosphorylated state. pRb family proteins are components of

7490-457: The definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression ( 1 − x 2 ) 2 , {\displaystyle \left({\sqrt {1-x^{2}}}\right)^{2},} which takes the same values as the polynomial 1 − x 2 {\displaystyle 1-x^{2}} on

7597-427: The degree is higher than one, the graph does not have any asymptote . It has two parabolic branches with vertical direction (one branch for positive x and one for negative x ). Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior. A polynomial equation , also called an algebraic equation , is an equation of the form a n x n +

7704-1000: The degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3 . Forming a sum of several terms produces a polynomial. For example, the following is a polynomial: 3 x 2 ⏟ t e r m 1 − 5 x ⏟ t e r m 2 + 4 ⏟ t e r m 3 . {\displaystyle \underbrace {_{\,}3x^{2}} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {1} \end{smallmatrix}}\underbrace {-_{\,}5x} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {2} \end{smallmatrix}}\underbrace {+_{\,}4} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {3} \end{smallmatrix}}.} It consists of three terms:

7811-459: The degrees may be applied to the polynomial or to its terms. For example, the term 2 x in x + 2 x + 1 is a linear term in a quadratic polynomial. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial . Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). The zero polynomial

7918-448: The differentiating potential of cycling cells, increases chromosomal instability, prevents induction of cellular senescence, promotes angiogenesis, and increases metastatic potential. Although most cancers rely on glycolysis for energy production ( Warburg effect ), cancers due to pRb loss tend to upregulate oxidative phosphorylation . The increased oxidative phosphorylation can increase stemness , metastasis , and (when enough oxygen

8025-573: The evolution of multicellularity in several lineages of life including animals. In humans, the protein is encoded by the RB1 gene located on chromosome 13 —more specifically, 13q14.1-q14.2 . If both alleles of this gene are mutated in a retinal cell, the protein is inactivated and the cells grow uncontrollably, resulting in development of retinoblastoma cancer, hence the "RB" in the name 'pRb'. Thus most pRb knock-outs occur in retinal tissue when UV radiation-induced mutation inactivates all healthy copies of

8132-927: The example, the product of polynomials is always a polynomial. Given a polynomial f {\displaystyle f} of a single variable and another polynomial g of any number of variables, the composition f ∘ g {\displaystyle f\circ g} is obtained by substituting each copy of the variable of the first polynomial by the second polynomial. For example, if f ( x ) = x 2 + 2 x {\displaystyle f(x)=x^{2}+2x} and g ( x ) = 3 x + 2 {\displaystyle g(x)=3x+2} then ( f ∘ g ) ( x ) = f ( g ( x ) ) = ( 3 x + 2 ) 2 + 2 ( 3 x + 2 ) . {\displaystyle (f\circ g)(x)=f(g(x))=(3x+2)^{2}+2(3x+2).} A composition may be expanded to

8239-515: The existence of two notations for the same polynomial. A polynomial expression is an expression that can be built from constants and symbols called variables or indeterminates by means of addition , multiplication and exponentiation to a non-negative integer power. The constants are generally numbers , but may be any expression that do not involve the indeterminates, and represent mathematical objects that can be added and multiplied. Two polynomial expressions are considered as defining

8346-400: The fact that mutated pRb could be inherited and lent support for the two-hit hypothesis . This states that only one working allele of a tumour suppressor gene is necessary for its function (the mutated gene is recessive ), and so both need to be mutated before the cancer phenotype will appear. In the familial form, a mutated allele is inherited along with a normal allele. In this case, should

8453-455: The fact that the ratio of two integers is a rational number , not necessarily an integer. For example, the fraction 1/( x + 1) is not a polynomial, and it cannot be written as a finite sum of powers of the variable x . For polynomials in one variable, there is a notion of Euclidean division of polynomials , generalizing the Euclidean division of integers. This notion of the division

8560-635: The factored form of 5 x 3 − 5 {\displaystyle 5x^{3}-5} is 5 ( x − 1 ) ( x 2 + x + 1 ) {\displaystyle 5(x-1)\left(x^{2}+x+1\right)} over the integers and the reals, and 5 ( x − 1 ) ( x + 1 + i 3 2 ) ( x + 1 − i 3 2 ) {\displaystyle 5(x-1)\left(x+{\frac {1+i{\sqrt {3}}}{2}}\right)\left(x+{\frac {1-i{\sqrt {3}}}{2}}\right)} over

8667-541: The first is degree two, the second is degree one, and the third is degree zero. Polynomials of small degree have been given specific names. A polynomial of degree zero is a constant polynomial , or simply a constant . Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials . For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for

8774-418: The form ⁠ 1 / 3 ⁠ x + x + c . For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number p , or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient ka k understood to mean the sum of k copies of a k . For example, over

8881-442: The gene, but pRb knock-out has also been documented in certain skin cancers in patients from New Zealand where the amount of UV radiation is significantly higher. Two forms of retinoblastoma were noticed: a bilateral, familial form and a unilateral, sporadic form. Sufferers of the former were over six times more likely to develop other types of cancer later in life, compared to individuals with sporadic retinoblastoma. This highlighted

8988-404: The general antiderivative (or indefinite integral) of P {\displaystyle P} is a n x n + 1 n + 1 + a n − 1 x n n + ⋯ + a 2 x 3 3 + a 1 x 2 2 +

9095-432: The indeterminate x ". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name(s) of the indeterminate(s) do not appear at each occurrence of the polynomial. The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials. If

9202-427: The indicated multiplications and additions. For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner's method , which consists of rewriting the polynomial as ( ( ( ( ( a n x + a n − 1 ) x + a n − 2 ) x + ⋯ +

9309-454: The integers modulo p , the derivative of the polynomial x + x is the polynomial 1 . A polynomial function is a function that can be defined by evaluating a polynomial. More precisely, a function f of one argument from a given domain is a polynomial function if there exists a polynomial a n x n + a n − 1 x n − 1 + ⋯ +

9416-451: The interval [ − 1 , 1 ] {\displaystyle [-1,1]} , and thus both expressions define the same polynomial function on this interval. Every polynomial function is continuous , smooth , and entire . The evaluation of a polynomial is the computation of the corresponding polynomial function; that is, the evaluation consists of substituting a numerical value to each indeterminate and carrying out

9523-1689: The multiplication in each term produces P Q = 4 x 2 + 10 x y + 2 x 2 y + 2 x + 6 x y + 15 y 2 + 3 x y 2 + 3 y + 10 x + 25 y + 5 x y + 5. {\displaystyle {\begin{array}{rccrcrcrcr}PQ&=&&4x^{2}&+&10xy&+&2x^{2}y&+&2x\\&&+&6xy&+&15y^{2}&+&3xy^{2}&+&3y\\&&+&10x&+&25y&+&5xy&+&5.\end{array}}} Combining similar terms yields P Q = 4 x 2 + ( 10 x y + 6 x y + 5 x y ) + 2 x 2 y + ( 2 x + 10 x ) + 15 y 2 + 3 x y 2 + ( 3 y + 25 y ) + 5 {\displaystyle {\begin{array}{rcccrcrcrcr}PQ&=&&4x^{2}&+&(10xy+6xy+5xy)&+&2x^{2}y&+&(2x+10x)\\&&+&15y^{2}&+&3xy^{2}&+&(3y+25y)&+&5\end{array}}} which can be simplified to P Q = 4 x 2 + 21 x y + 2 x 2 y + 12 x + 15 y 2 + 3 x y 2 + 28 y + 5. {\displaystyle PQ=4x^{2}+21xy+2x^{2}y+12x+15y^{2}+3xy^{2}+28y+5.} As in

9630-663: The number of cells without loss of cell renewal capacity. In cycling progenitor cells, pRb plays a role at the G1, S, and G2 checkpoints and promotes differentiation. In differentiated cells, which make up the majority of cells in the body and are assumed to be in irreversible G0, pRb maintains both arrest and differentiation. Loss of pRb therefore exhibits multiple different responses within different cells that ultimately all could result in cancer phenotypes. For cancer initiation, loss of pRb may induce cell cycle re-entry in both quiescent and post-mitotic differentiated cells through dedifferentiation. In cancer progression, loss of pRb decreases

9737-476: The polynomial P = a n x n + a n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0 = ∑ i = 0 n a i x i {\displaystyle P=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i}} with respect to x

9844-400: The polynomial is P , not P ( x ), but the use of the functional notation P ( x ) dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let P ( x ) be a polynomial" is a shorthand for "let P be a polynomial in

9951-547: The promoter regions of genes for cell proliferation or progression of the cell cycle. E2F1 to E2F5 are known to associate with proteins in the pRb-family of proteins while E2F6 and E2F7 are independent of pRb. Broadly, the E2Fs are split into activator E2Fs and repressor E2Fs though their role is more flexible than that on occasion. The activator E2Fs are E2F1, E2F2 and E2F3 while the repressor E2Fs are E2F4 , E2F5 and E2F6. Activator E2Fs along with E2F4 bind exclusively to pRb. pRb

10058-2459: The result is another polynomial. Subtraction of polynomials is similar. Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, if P = 2 x + 3 y + 5 Q = 2 x + 5 y + x y + 1 {\displaystyle {\begin{aligned}\color {Red}P&\color {Red}{=2x+3y+5}\\\color {Blue}Q&\color {Blue}{=2x+5y+xy+1}\end{aligned}}} then P Q = ( 2 x ⋅ 2 x ) + ( 2 x ⋅ 5 y ) + ( 2 x ⋅ x y ) + ( 2 x ⋅ 1 ) + ( 3 y ⋅ 2 x ) + ( 3 y ⋅ 5 y ) + ( 3 y ⋅ x y ) + ( 3 y ⋅ 1 ) + ( 5 ⋅ 2 x ) + ( 5 ⋅ 5 y ) + ( 5 ⋅ x y ) + ( 5 ⋅ 1 ) {\displaystyle {\begin{array}{rccrcrcrcr}{\color {Red}{P}}{\color {Blue}{Q}}&{=}&&({\color {Red}{2x}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{1}})\\&&+&({\color {Red}{3y}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{1}})\\&&+&({\color {Red}{5}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{5}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{5}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{5}}\cdot {\color {Blue}{1}})\end{array}}} Carrying out

10165-468: The same polynomial if they may be transformed, one to the other, by applying the usual properties of commutativity , associativity and distributivity of addition and multiplication. For example ( x − 1 ) ( x − 2 ) {\displaystyle (x-1)(x-2)} and x 2 − 3 x + 2 {\displaystyle x^{2}-3x+2} are two polynomial expressions that represent

10272-477: The same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. In elementary algebra , methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations . For higher degrees, the Abel–Ruffini theorem asserts that there can not exist

10379-470: The same polynomial; so, one has the equality ( x − 1 ) ( x − 2 ) = x 2 − 3 x + 2 {\displaystyle (x-1)(x-2)=x^{2}-3x+2} . A polynomial in a single indeterminate x can always be written (or rewritten) in the form a n x n + a n − 1 x n − 1 + ⋯ +

10486-423: The solutions as algebraic expressions ; for example, the golden ratio ( 1 + 5 ) / 2 {\displaystyle (1+{\sqrt {5}})/2} is the unique positive solution of x 2 − x − 1 = 0. {\displaystyle x^{2}-x-1=0.} In the ancient times, they succeeded only for degrees one and two. For quadratic equations ,

10593-414: The start of Galois theory and group theory , two important branches of modern algebra . Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation ). When there is no algebraic expression for the roots, and when such an algebraic expression exists but

10700-436: The striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see Abel–Ruffini theorem ). In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it. This result marked

10807-465: The subtraction of non-constant polynomials), although strictly speaking, constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as bivariate , trivariate , and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It

10914-825: The sum P + Q = 3 x 2 − 2 x + 5 x y − 2 − 3 x 2 + 3 x + 4 y 2 + 8 {\displaystyle P+Q=3x^{2}-2x+5xy-2-3x^{2}+3x+4y^{2}+8} can be reordered and regrouped as P + Q = ( 3 x 2 − 3 x 2 ) + ( − 2 x + 3 x ) + 5 x y + 4 y 2 + ( 8 − 2 ) {\displaystyle P+Q=(3x^{2}-3x^{2})+(-2x+3x)+5xy+4y^{2}+(8-2)} and then simplified to P + Q = x + 5 x y + 4 y 2 + 6. {\displaystyle P+Q=x+5xy+4y^{2}+6.} When polynomials are added together,

11021-454: The transcription of cell cycle progression genes leads to the S phase arrest that prevents replication of damaged DNA. When it is time for a cell to enter S phase, complexes of cyclin-dependent kinases (CDK) and cyclins phosphorylate pRb, allowing E2F-DP to dissociate from pRb and become active. When E2F is free it activates factors like cyclins (e.g. cyclin E and cyclin A), which push the cell through

11128-433: Was detected on the cyclin A and PCNA promoters in senescent cells. Cells respond to stress in the form of DNA damage, activated oncogenes, or sub-par growing conditions, and can enter a senescence-like state called "premature senescence". This allows the cell to prevent further replication during periods of damaged DNA or general unfavorable conditions. DNA damage in a cell can induce pRb activation. pRb's role in repressing

11235-432: Was detected. Furthermore, triple knockout, p16 addition, and Cdk 4/6 inhibitor addition experiments confirmed that Cyclin D- Cdk 4/6 is the sole phosphorylator of pRb. Throughout early G1, mono-phosphorylated pRb exists as 14 different isoforms (the 15th phosphorylation site is not conserved in primates in which the experiments were performed). Together, these isoforms represent the "hypo-phosphorylated" active pRb state that

11342-472: Was recently shown that pRb only exists in three states: un-phosphorylated, mono-phosphorylated, and hyper-phosphorylated. Each has a unique cellular function. Before the development of 2D IEF , only hyper-phosphorylated pRb was distinguishable from all other forms, i.e. un-phosphorylated pRb resembled mono-phosphorylated pRb on immunoblots. As pRb was either in its active "hypo-phosphorylated" state or inactive "hyperphosphorylated" state. However, with 2D IEF, it

11449-578: Was thought to exist. Each isoform has distinct preferences to associate with different exogenous expressed E2Fs. A recent report showed that mono-phosphorylation controls pRb's association with other proteins and generates functional distinct forms of pRb. All different mono-phosphorylated pRb isoforms inhibit E2F transcriptional program and are able to arrest cells in G1-phase. Importantly, different mono-phosphorylated forms of pRb have distinct transcriptional outputs that are extended beyond E2F regulation. After

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