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Tuesday, September 30, 2025

Set theory

From Wikipedia, the free encyclopedia

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

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.

The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.

Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy, formal semantics, and evolutionary dynamics. Its foundational appeal, together with its paradoxes, and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

History

Early history

The basic notion of grouping objects has existed since at least the emergence of numbers, and the notion of treating sets as their own objects has existed since at least the Tree of Porphyry in 3rd-century AD. The simplicity and ubiquity of sets makes it hard to determine the origin of sets as now used in mathematics; however, Bernard Bolzano's Paradoxes of the Infinite (Paradoxien des Unendlichen, 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded on Galileo's paradox, and introduced one-to-one correspondence of infinite sets, for example between the intervals $[0, 5]$ and $[0, 12]$ by the relation $5 y = 12 x$ . However, he resisted saying these sets were equinumerous, and his work is generally considered to have been uninfluential in mathematics of his time.

Before mathematical set theory, basic concepts of infinity were considered to be in the domain of philosophy (see: Infinity (philosophy) and Infinity § History). Since the 5th century BC, beginning with Greek philosopher Zeno of Elea in the West (and early Indian mathematicians in the East), mathematicians had struggled with the concept of infinity. With the development of calculus in the late 17th century, philosophers began to generally distinguish between potential and actual infinity, wherein mathematics was only considered in the latter. Carl Friedrich Gauss famously stated: "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics."

Development of mathematical set theory was motivated by several mathematicians. Bernhard Riemann's lecture On the Hypotheses which lie at the Foundations of Geometry (1854) proposed new ideas about topology. His lectures also introduced the concept of basing mathematics in terms of sets or manifolds in the sense of a class (which he called Mannigfaltigkeit) now called point-set topology. The lecture was published by Richard Dedekind in 1868, along with Riemann's paper on trigonometric series (which presented the Riemann integral), The latter was a starting point a movement in real analysis for the study of “seriously” discontinuous functions. A young Georg Cantor entered into this area, which led him to the study of point-sets. Around 1871, influenced by Riemann, Dedekind began working with sets in his publications, which dealt very clearly and precisely with equivalence relations, partitions of sets, and homomorphisms. Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. However, he did not publish a formal explanation of his set theory until 1888.

Naive set theory

Set theory, as understood by modern mathematicians, is generally considered to be founded by a single paper in 1874 by Georg Cantor titled On a Property of the Collection of All Real Algebraic Numbers. In his paper, he developed the notion of cardinality, comparing the sizes of two sets by setting them in one-to-one correspondence. His "revolutionary discovery" was that the set of all real numbers is uncountable, that is, one cannot put all real numbers in a list. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument.

Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed a theory of transfinite numbers, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter $ℵ$ (ℵ, aleph) with a natural number subscript; for the ordinals he employed the Greek letter $ω$ (ω, omega).

Set theory was beginning to become an essential ingredient of the new “modern” approach to mathematics. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections (see: Controversy over Cantor's theory). Dedekind's algebraic style only began to find followers in the 1890s.

Despite the controversy, Cantor's set theory gained remarkable ground around the turn of the 20th century with the work of several notable mathematicians and philosophers. Richard Dedekind, around the same time, began working with sets in his publications, and famously constructing the real numbers using Dedekind cuts. He also worked with Giuseppe Peano in developing the Peano axioms, which formalized natural-number arithmetic, using set-theoretic ideas, which also introduced the epsilon symbol for set membership. Possibly most prominently, Gottlob Frege began to develop his Foundations of Arithmetic.

In his work, Frege tries to ground all mathematics in terms of logical axioms using Cantor's cardinality. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept horse in the barn. Frege attempted to explain our grasp of numbers through cardinality ('the number of...', or $N x : F x$ ), relying on Hume's principle.

However, Frege's work was short-lived, as it was found by Bertrand Russell that his axioms lead to a contradiction. Specifically, Frege's Basic Law V (now known as the axiom schema of unrestricted comprehension). According to Basic Law V, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. The contradiction, called Russell's paradox, is shown as follows:

Let R be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols:

Let R = {x ∣ x \notin x}, then R \in R ⟺ R \notin R

This came around a time of several paradoxes or counter-intuitive results. For example, that the parallel postulate cannot be proved, the existence of mathematical objects that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with Peano arithmetic. The result was a foundational crisis of mathematics.

Basic concepts and notation

Set theory begins with a fundamental binary relation between an object $o$ and a set $A$ . If $o$ is a member (or element) of $A$ , the notation $o \in A$ is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets.

A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set $A$ are also members of set $B$ , then $A$ is a subset of $B$ , denoted $A \subseteq B$ . For example, ${1, 2}$ is a subset of ${1, 2, 3}$ , and so is ${2}$ but ${1, 4}$ is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined, variously denoted $A \subset B$ , $A ⊊ B$ , or $A ⫋ B$ (note however that the notation $A \subset B$ is sometimes used synonymously with $A \subseteq B$ ; that is, allowing the possibility that $A$ and $B$ are equal). We call $A$ a proper subset of $B$ if and only if $A$ is a subset of $B$ , but $A$ is not equal to $B$ . Also, 1, 2, and 3 are members (elements) of the set ${1, 2, 3}$ , but are not subsets of it; and in turn, the subsets, such as ${1}$ , are not members of the set ${1, 2, 3}$ . More complicated relations can exist; for example, the set ${1}$ is both a member and a proper subset of the set ${1, {1}}$ .

Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The following is a partial list of them:

Union of the sets $A$ and $B$ , denoted $A \cup B$ , is the set of all objects that are a member of $A$ , or $B$ , or both. For example, the union of ${1, 2, 3}$ and ${2, 3, 4}$ is the set ${1, 2, 3, 4}$ .
Intersection of the sets $A$ and $B$ , denoted $A \cap B$ , is the set of all objects that are members of both $A$ and $B$ . For example, the intersection of ${1, 2, 3}$ and ${2, 3, 4}$ is the set ${2, 3}$ .
Set difference of $U$ and $A$ , denoted $U ∖ A$ , is the set of all members of $U$ that are not members of $A$ . The set difference ${1, 2, 3} ∖ {2, 3, 4}$ is ${1}$ , while conversely, the set difference ${2, 3, 4} ∖ {1, 2, 3}$ is ${4}$ . When $A$ is a subset of $U$ , the set difference $U ∖ A$ is also called the complement of $A$ in $U$ . In this case, if the choice of $U$ is clear from the context, the notation $A c$ is sometimes used instead of $U ∖ A$ , particularly if $U$ is a universal set as in the study of Venn diagrams.
Symmetric difference of sets $A$ and $B$ , denoted $A △ B$ or $A ⊖ B$ , is the set of all objects that are a member of exactly one of $A$ and $B$ (elements which are in one of the sets, but not in both). For instance, for the sets ${1, 2, 3}$ and ${2, 3, 4}$ , the symmetric difference set is ${1, 4}$ . It is the set difference of the union and the intersection, $(A \cup B) ∖ (A \cap B)$ or $(A ∖ B) \cup (B ∖ A)$ .
Cartesian product of $A$ and $B$ , denoted $A \times B$ , is the set whose members are all possible ordered pairs $(a, b)$ , where $a$ is a member of $A$ and $b$ is a member of $B$ . For example, the Cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.

Some basic sets of central importance are the set of natural numbers, the set of real numbers and the empty set – the unique set containing no elements. The empty set is also occasionally called the null set, though this name is ambiguous and can lead to several interpretations. The empty set can be denoted with empty braces " ${}$ " or the symbol " $\emptyset$ " or " $\emptyset$ ".

The power set of a set $A$ , denoted $P (A)$ , is the set whose members are all of the possible subsets of $A$ . For example, the power set of ${1, 2}$ is ${ {}, {1}, {2}, {1, 2} }$ . Notably, $P (A)$ contains both $A$ and the empty set.

Ontology

A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number $α$ , known as its rank. The rank of a pure set $X$ is defined to be the least ordinal that is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set containing only the empty set is assigned rank 1. For each ordinal $α$ , the set $V_{α}$ is defined to consist of all pure sets with rank less than $α$ . The entire von Neumann universe is denoted $V$ .

Formalized set theory

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.

The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:

Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Fragments of ZFC include:
- Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
- General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
- Kripke–Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
Sets and proper classes. These include Von Neumann–Bernays–Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse–Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC.

The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets and do not have any members. Zermelo set theory was originally defined over a domain consisting of both sets and urelements.

The New Foundations systems of NFU (allowing urelements) and NF (lacking them), associate with Willard Van Orman Quine, are not based on a cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, Thomas Forster has argued that it does reflect an iterative conception of set.

Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject.

An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977.

Applications

Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, vector spaces, and relational algebras can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.

Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic. For example, properties of the natural and real numbers can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes human-written, computer-verified derivations of more than 12,000 theorems starting from ZFC set theory, first-order logic and propositional logic.

Areas of study

Set theory is a major area of research in mathematics with many interrelated subfields:

Combinatorial set theory

Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdős–Rado theorem.

Descriptive set theory

Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.

The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.

A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.

Fuzzy set theory

In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.

Inner model theory

An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.

The study of inner models is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).

Large cardinals

A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo–Fraenkel set theory.

Determinacy

Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure.

Forcing

Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models.

Cardinal invariants

A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.

Set-theoretic topology

Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

Controversy

From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop's influential book Foundations of Constructive Analysis.

A different objection put forth by Henri Poincaré is that defining sets using the axiom schemas of specification and replacement, as well as the axiom of power set, introduces impredicativity, a type of circularity, into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]".

Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism. He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and finitism. Meta-mathematical statements – which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory – are not mathematics. Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in Remarks on the Foundations of Mathematics: Wittgenstein attempted to refute Gödel's incompleteness theorems after having only read the abstract. As reviewers Kreisel, Bernays, Dummett, and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as Crispin Wright begun to rehabilitate Wittgenstein's arguments.

Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory. Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces.

An active area of research is the univalent foundations and related to it homotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types. Principles such as the axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results.

Mathematical education

As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics of naive set theory early in mathematics education.

In the US in the 1960s, the New Math experiment aimed to teach basic set theory, among other abstract concepts, to primary school students but was met with much criticism. The math syllabus in European schools followed this trend and currently includes the subject at different levels in all grades. Venn diagrams are widely employed to explain basic set-theoretic relationships to primary school students (even though John Venn originally devised them as part of a procedure to assess the validity of inferences in term logic).

Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition) of sets (e.g. "months starting with the letter A"), which may be useful when learning computer programming, since Boolean logic is used in various programming languages. Likewise, sets and other collection-like objects, such as multisets and lists, are common datatypes in computer science and programming.

In addition to that, certain sets are commonly used in mathematical teaching (such as the sets $N$ of natural numbers, $Z$ of integers, $R$ of real numbers, etc.). These are commonly used when defining a mathematical function as a relation from one set (the domain) to another set (the range).

Protein metabolism

From Wikipedia, the free encyclopedia

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

Protein metabolism denotes the various biochemical processes responsible for the synthesis of proteins and amino acids (anabolism), and the breakdown of proteins by catabolism.

The steps of protein synthesis include transcription, translation, and post translational modifications. During transcription, RNA polymerase transcribes a coding region of the DNA in a cell producing a sequence of RNA, specifically messenger RNA (mRNA). This mRNA sequence contains codons: 3 nucleotide long segments that code for a specific amino acid. Ribosomes translate the codons to their respective amino acids. In humans, non-essential amino acids are synthesized from intermediates in major metabolic pathways such as the Citric Acid Cycle. Essential amino acids must be consumed and are made in other organisms. The amino acids are joined by peptide bonds making a polypeptide chain. This polypeptide chain then goes through post translational modifications and is sometimes joined with other polypeptide chains to form a fully functional protein.

Dietary proteins are first broken down to individual amino acids by various enzymes and hydrochloric acid present in the gastrointestinal tract. These amino acids are absorbed into the bloodstream to be transported to the liver and onward to the rest of the body. Absorbed amino acids are typically used to create functional proteins, but may also be used to create energy. They can also be converted into glucose. This glucose can then be converted to triglycerides and stored in fat cells.

Proteins can be broken down by enzymes known as peptidases or can break down as a result of denaturation. Proteins can denature in environmental conditions the protein is not made for.

Protein synthesis

Protein anabolism is the process by which proteins are formed from amino acids. It relies on five processes: amino acid synthesis, transcription, translation, post translational modifications, and protein folding. Proteins are made from amino acids. In humans, some amino acids can be synthesized using already existing intermediates. These amino acids are known as non-essential amino acids. Essential amino acids require intermediates not present in the human body. These intermediates must be ingested, mostly from eating other organisms.

Amino Acid Synthesis

Pathways that form each amino acid
Amino Acid	R-group^‡	Pathway*
Glycine	H−	Serine + THF → Glycine (hydroxymethyltransferase)
Alanine	CH₃−	Pyruvate → Alanine (aminotransferase)
Valine^§	(CH₃)₂−CH−	Hydroxyethyl-TPP + Pyruvate → α-acetolactate → Valine
Leucine^§	(CH₃)₂−CH−CH₂−	Hydroxyethyl-TPP + Pyruvate → α-ketobutyrate → Leucine
Isoleucine^§	CH₃−CH₂−CH(CH₃)−	Hydroxyethyl-TPP + Pyruvate → α-acetolactate → Isoleucine
Methionine^§	CH₃−S−(CH₂)₂−	Homocysteine → Methionine (methionine synthase)
Proline	−(CH₂)₃−	Glutamic Acid → Glutamate-5-semialdehyde → Proline (γ-glutamyl kinase)
Phenylalanine^§	Ph−CH₂−	Phosphoenolpyruvate → 2-keto-3-deoxy arabino heptulosonate-7-phosphate → Chorismate → Phenylalanine
Tryptophan^§	Ph−NH−CH=C−CH₂−	Phosphoenolpyruvate → 2-keto-3-deoxy arabino heptulosonate-7-phosphate → Chorismate → Tryptophan
Tyrosine	HO−Ph−CH₂−	Phenylalanine → Tyrosine (phenylalanine hydroxylase)
Serine	HO−CH₂−	3-phosphoglycerate → 3-phosphohydroxypyruvate (3-phosphoglycerate dehydrogenase) → 3-phosphoserine (aminotransferase) → Serine (phosphoserine phosphatase)
Threonine^§	CH₃−CH(OH)−	Aspartate → β-aspartate-semialdehyde → Homoserine → Threonine
Cysteine	HS−CH₂−	Serine → Cystathionine → α-ketobutyrate → Cysteine
Asparagine	H₂N−CO−CH₂−	Aspartic Acid → Asparagine (asparagine synthetase)
Glutamine	H₂N−CO−(CH₂)₂−	Glutamic Acid → Glutamine (glutamine synthetase)
Arginine	⁺H₂N=C(NH₂)−NH−(CH₂)₃−	Glutamate → Glutamate-5-semialdehyde (γ-glutamyl kinase) → Arginine
Histidine^§	NH−CH=N−CH=C−CH₂−	Glucose → Glucose-6-phosphate → Ribose-5-phosphate → Histidine
Lysine^§	⁺H₃N−(CH₂)₄−	Aspartate → β-aspartate-semialdehyde → Homoserine + lysine
Aspartic Acid	⁻OOC−CH₂−	Oxaloacetate → Aspartic Acid (aminotransferase)
Glutamic Acid	⁻OOC−(CH₂)₂−	α-ketoglutarate → Glutamic Acid (aminotransferase)
^‡Shown at physiological conditions. Complexes that are italicized are enzymes.* ^§Cannot be synthesized in humans.

Polypeptide synthesis

Transcription

In transcription, RNA polymerase reads a DNA strand and produces an mRNA strand that can be further translated. In order to initiate transcription, the DNA segment that is to be transcribed must be accessible (i.e. it cannot be tightly packed). Once the DNA segment is accessible, the RNA polymerase can begin to transcribe the coding DNA strand by pairing RNA nucleotides to the template DNA strand. During the initial transcription phase, the RNA polymerase searches for a promoter region on the DNA template strand. Once the RNA polymerase binds to this region, it begins to “read” the template DNA strand in the 3’ to 5’ direction. RNA polymerase attaches RNA bases complementary to the template DNA strand (Uracil will be used instead of Thymine). The new nucleotide bases are bonded to each other covalently. The new bases eventually dissociate from the DNA bases but stay linked to each other, forming a new mRNA strand. This mRNA strand is synthesized in the 5’ to 3’ direction. Once the RNA reaches a terminator sequence, it dissociates from the DNA template strand and terminates the mRNA sequence as well.

Transcription is regulated in the cell via transcription factors. Transcription factors are proteins that bind to regulatory sequences in the DNA strand such as promoter regions or operator regions. Proteins bound to these regions can either directly halt or allow RNA polymerase to read the DNA strand or can signal other proteins to halt or allow RNA polymerase reading.

Translation

Formation of a dipeptide via a peptide bond.

During translation, ribosomes convert a sequence of mRNA (messenger RNA) to an amino acid sequence. Each 3-base-pair-long segment of mRNA is a codon which corresponds to one amino acid or stop signal. Amino acids can have multiple codons that correspond to them. Ribosomes do not directly attach amino acids to mRNA codons. They must utilize tRNAs (transfer RNAs) as well. Transfer RNAs can bind to amino acids and contain an anticodon which can hydrogen bind to an mRNA codon. The process of bind an amino acid to a tRNA is known as tRNA charging. Here, the enzyme aminoacyl-tRNA-synthetase catalyzes two reactions. In the first one, it attaches an AMP molecule (cleaved from ATP) to the amino acid. The second reaction cleaves the aminoacyl-AMP producing the energy to join the amino acid to the tRNA molecule.

Ribosomes have two subunits, one large and one small. These subunits surround the mRNA strand. The larger subunit contains three binding sites: A (aminoacyl), P (peptidyl), and E (exit). After translational initiation (which is different in prokaryotes and eukaryotes), the ribosome enters the elongation period which follows a repetitive cycle. First a tRNA with the correct amino acid enters the A site. The ribosome transfers the peptide from the tRNA in the P site to the new amino acid on the tRNA in the A site. The tRNA from the P site will be shifted into the E site where it will be ejected. This continually occurs until the ribosome reaches a stop codon or receives a signal to stop. A peptide bond forms between the amino acid attached to the tRNA in the P site and the amino acid attached to a tRNA in the A site. The formation of a peptide bond requires an input of energy. The two reacting molecules are the alpha amino group of one amino acid and the alpha carboxyl group of the other amino acids. A by-product of this bond formation is the release of water (the amino group donates a proton while the carboxyl group donates a hydroxyl).

Translation can be downregulated by miRNAs (microRNAs). These RNA strands can cleave mRNA strands they are complementary to and will thus stop translation. Translation can also be regulated via helper proteins. For example, a protein called eukaryotic initiation factor-2 (eIF-2) can bind to the smaller subunit of the ribosome, starting translation. When elF-2 is phosphorylated, it cannot bind to the ribosome and translation is halted.

Post-translational Modifications

Once the peptide chain is synthesized, it still must be modified. Post-translational modifications can occur before protein folding or after. Common biological methods of modifying peptide chains after translation include methylation, phosphorylation, and disulfide bond formation. Methylation often occurs to arginine or lysine and involves adding a methyl group to a nitrogen (replacing a hydrogen). The R groups on these amino acids can be methylated multiple times as long as the bonds to nitrogen does not exceed 4. Methylation reduces the ability of these amino acids to form hydrogen bonds so arginine and lysine that are methylated have different properties than their standard counterparts. Phosphorylation often occurs to serine, threonine, and tyrosine and involves replacing a hydrogen on the alcohol group at the terminus of the R group with a phosphate group. This adds a negative charge on the R groups and will thus change how the amino acids behave in comparison to their standard counterparts. Disulfide bond formation is the creation of disulfide bridges (covalent bonds) between two cysteine amino acids in a chain which adds stability to the folded structure.

Protein folding

A polypeptide chain in the cell does not have to stay linear; it can become branched or fold in on itself. Polypeptide chains fold in a particular manner depending on the solution they are in. The fact that all amino acids contain R groups with different properties is the main reason proteins fold.

In a hydrophilic environment such as cytosol, the hydrophobic amino acids will concentrate at the core of the protein, while the hydrophilic amino acids will be on the exterior. This is entropically favorable since water molecules can move much more freely around hydrophilic amino acids than hydrophobic amino acids.
In a hydrophobic environment, the hydrophilic amino acids will concentrate at the core of the protein, while the hydrophobic amino acids will be on the exterior. Since the new interactions between the hydrophilic amino acids are stronger than hydrophobic-hydrophilic interactions, this is enthalpically favorable.

Once a polypeptide chain is fully folded, it is called a protein. Often many subunits will combine to make a fully functional protein although physiological proteins do exist that contain only one polypeptide chain. Proteins may also incorporate other molecules such as the heme group in hemoglobin, a protein responsible for carrying oxygen in the blood.

Protein breakdown

Protein catabolism is the process by which proteins are broken down to their amino acids. This is also called proteolysis and can be followed by further amino acid degradation.

Protein catabolism via enzymes

Proteases

Originally thought to only disrupt enzymatic reactions, proteases (also known as peptidases) actually help with catabolizing proteins through cleavage and creating new proteins that were not present before. Proteases also help to regulate metabolic pathways. One way they do this is to cleave enzymes in pathways that do not need to be running (i.e. gluconeogenesis when blood glucose concentrations are high). This helps to conserve as much energy as possible and to avoid futile cycles. Futile cycles occur when the catabolic and anabolic pathways are both in effect at the same time and rate for the same reaction. Since the intermediates being created are consumed, the body makes no net gain. Energy is lost through futile cycles. Proteases prevent this cycle from occurring by altering the rate of one of the pathways, or by cleaving a key enzyme, they can stop one of the pathways. Proteases are also nonspecific when binding to substrate, allowing for great amounts of diversity inside the cells and other proteins, as they can be cleaved much easier in an energy efficient manner.

Because many proteases are nonspecific, they are highly regulated in the cell. Without regulation, proteases will destroy many proteins which are essential to physiological processes. One way the body regulates proteases is through protease inhibitors. Protease inhibitors can be other proteins, small peptides, or molecules. There are two types of protease inhibitors: reversible and irreversible. Reversible protease inhibitors form non-covalent interactions with the protease limiting its functionality. They can be competitive inhibitors, uncompetitive inhibitors, and noncompetitive inhibitors. Competitive inhibitors compete with the peptide to bind to the protease active site. Uncompetitive inhibitors bind to the protease while the peptide is bound but do not let the protease cleave the peptide bond. Noncompetitive inhibitors can do both. Irreversible protease inhibitors covalently modify the active site of the protease so it cannot cleave peptides.

Exopeptidases

Exopeptidases are enzymes that can cleave the end of an amino acid side chain mostly through the addition of water.^[6] Exopeptidase enzymes exist in the small intestine. These enzymes have two classes: aminopeptidases are a brush border enzyme and carboxypeptidases which is from the pancreas. Aminopeptidases are enzymes that remove amino acids from the amino terminus of protein. They are present in all lifeforms and are crucial for survival since they do many cellular tasks in order to maintain stability. This form of peptidase is a zinc metalloenzyme and it is inhibited by the transition state analog. This analog is similar to the actual transition state, so it can make the enzyme bind to it instead of the actual transition state, thus preventing substrate binding and decreasing reaction rates.^[22] Carboxypeptidases cleave at the carboxyl end of the protein. While they can catabolize proteins, they are more often used in post-transcriptional modifications.

Endopeptidases

Endopeptidases are enzymes that add water to an internal peptide bond in a peptide chain and break that bond. Three common endopeptidases that come from the pancreas are pepsin, trypsin, and chymotrypsin. Chymotrypsin performs a hydrolysis reaction that cleaves after aromatic residues. The main amino acids involved are serine, histidine, and aspartic acid. They all play a role in cleaving the peptide bond. These three amino acids are known as the catalytic triad which means that these three must all be present in order to properly function. Trypsin cleaves after long positively charged residues and has a negatively charged binding pocket at the active site. Both are produced as zymogens, meaning they are initially found in their inactive state and after cleavage though a hydrolysis reaction, they becomes activated. Non-covalent interactions such as hydrogen bonding between the peptide backbone and the catalytic triad help increase reaction rates, allowing these peptidases to cleave many peptides efficiently.

Protein catabolism via environmental changes

pH

Cellular proteins are held in a relatively constant pH in order to prevent changes in the protonation state of amino acids. If the pH drops, some amino acids in the polypeptide chain can become protonated if the pka of their R groups is higher than the new pH. Protonation can change the charge these R groups have. If the pH raises, some amino acids in the chain can become deprotonated (if the pka of the R group is lower than the new pH). This also changes the R group charge. Since many amino acids interact with other amino acids based on electrostatic attraction, changing the charge can break these interactions. The loss of these interactions alters the proteins structure, but most importantly it alters the proteins function, which can be beneficial or detrimental. A significant change in pH may even disrupt many interactions the amino acids make and denature (unfold) the protein.

Temperature

As the temperature in the environment increases, molecules move faster. Hydrogen bonds and hydrophobic interactions are important stabilizing forces in proteins. If the temperature rises and molecules containing these interactions are moving too fast, the interactions become compromised or even break. At high temperatures, these interactions cannot form, and a functional protein is denatured. However, it relies on two factors; the type of protein used and the amount of heat applied. The amount of heat applied determines whether this change in protein is permanent or if it can be transformed back to its original form.

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