Kronecker delta

From Wikipedia, the free encyclopedia

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

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:

$${\displaystyle {\delta }_{ij}=\{\begin{array}{ll}0& \text{if }i\ne j,\\ 1& \text{if }i=j.\end{array}}$$

or with use of Iverson brackets:

$${\displaystyle {\delta }_{ij}=[i=j]}$$

For example, ${\displaystyle {\delta }_{12}=0}$ because ${\displaystyle 1\ne 2}$, whereas ${\displaystyle {\delta }_{33}=1}$ because ${\displaystyle 3=3}$.

The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above.

In linear algebra, the ${\displaystyle n\times n}$ identity matrix ${\displaystyle \mathbf{I}}$ has entries equal to the Kronecker delta:

$${\displaystyle I_{ij}={\delta }_{ij}}$$

where ${\displaystyle i}$ and ${\displaystyle j}$ take the values ${\displaystyle 1,2,\cdots ,n}$, and the inner product of vectors can be written as

$${\displaystyle \mathbf{a}\cdot \mathbf{b}=\sum _{i,j=1}^n{a}_i{\delta }_{ij}{b}_j=\sum _{i=1}^n{a}_i{b}_i.}$$

Here the Euclidean vectors are defined as n-tuples: ${\displaystyle \mathbf{a}=(a_1,a_2,\dots ,a_n)}$ and ${\displaystyle \mathbf{b}=(b_1,b_2,...,b_n)}$ and the last step is obtained by using the values of the Kronecker delta to reduce the summation over ${\displaystyle j}$.

It is common for i and j to be restricted to a set of the form {1, 2, ..., n} or {0, 1, ..., n − 1}, but the Kronecker delta can be defined on an arbitrary set.

Properties

The following equations are satisfied:

$${\displaystyle \begin{array}{rl}\sum _j{\delta }_{ij}{a}_j& =a_i,\\ \sum _i{a}_i{\delta }_{ij}& =a_j,\\ \sum _k{\delta }_{ik}{\delta }_{kj}& ={\delta }_{ij}.\end{array}}$$

Therefore, the matrix δ can be considered as an identity matrix.

Another useful representation is the following form:

$${\displaystyle {\delta }_{nm}=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{k=1}^N{e}^{2\pi i\frac{k}{N}(n-m)}}$$

This can be derived using the formula for the geometric series.

Alternative notation

Using the Iverson bracket:

$${\displaystyle {\delta }_{ij}=[i=j].}$$

Often, a single-argument notation ${\displaystyle {\delta }_i}$ is used, which is equivalent to setting ${\displaystyle j=0}$:

$${\displaystyle {\delta }_i={\delta }_{i0}=\{\begin{array}{ll}0,& \text{if }i\ne 0\\ 1,& \text{if }i=0\end{array}}$$

In linear algebra, it can be thought of as a tensor, and is written ${\displaystyle {\delta }_j^i}$. Sometimes the Kronecker delta is called the substitution tensor.

Digital signal processing

Unit sample function

In the study of digital signal processing (DSP), the unit sample function ${\displaystyle \delta [n]}$ represents a special case of a 2-dimensional Kronecker delta function ${\displaystyle {\delta }_{ij}}$ where the Kronecker indices include the number zero, and where one of the indices is zero. In this case:

$${\displaystyle \delta [n]\equiv {\delta }_{n0}\equiv {\delta }_{0n}\text{where}-\mathrm{\infty }<n<\mathrm{\infty }}$$

Or more generally where:

$${\displaystyle \delta [n-k]\equiv \delta [k-n]\equiv {\delta }_{nk}\equiv {\delta }_{kn}\text{where}-\mathrm{\infty }<n<\mathrm{\infty },-\mathrm{\infty }<k<\mathrm{\infty }}$$

However, this is only a special case. In tensor calculus, it is more common to number basis vectors in a particular dimension starting with index 1, rather than index 0. In this case, the relation ${\displaystyle \delta [n]\equiv {\delta }_{n0}\equiv {\delta }_{0n}}$ does not exist, and in fact, the Kronecker delta function and the unit sample function are different functions that overlap in the specific case where the indices include the number 0, the number of indices is 2, and one of the indices has the value of zero.

While the discrete unit sample function and the Kronecker delta function use the same letter, they differ in the following ways. For the discrete unit sample function, it is more conventional to place a single integer index in square braces; in contrast the Kronecker delta can have any number of indexes. Further, the purpose of the discrete unit sample function is different from the Kronecker delta function. In DSP, the discrete unit sample function is typically used as an input function to a discrete system for discovering the system function of the system which will be produced as an output of the system. In contrast, the typical purpose of the Kronecker delta function is for filtering terms from an Einstein summation convention.

The discrete unit sample function is more simply defined as:

$${\displaystyle \delta [n]=\{\begin{array}{ll}1& n=0\\ 0& n\text{ is another integer}\end{array}}$$

In addition, the Dirac delta function is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as:

$${\displaystyle \delta (t)=\{\begin{array}{ll}\mathrm{\infty }& t=0\\ 0& t\text{ is another real}\end{array}}$$

Unlike the Kronecker delta function ${\displaystyle {\delta }_{ij}}$ and the unit sample function ${\displaystyle \delta [n]}$, the Dirac delta function ${\displaystyle \delta (t)}$ does not have an integer index, it has a single continuous non-integer value t.

To confuse matters more, the unit impulse function is sometimes used to refer to either the Dirac delta function ${\displaystyle \delta (t)}$, or the unit sample function ${\displaystyle \delta [n]}$.

Notable properties

The Kronecker delta has the so-called sifting property that for ${\displaystyle j\in \mathbb{Z}}$:

$${\displaystyle \sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i{\delta }_{ij}=a_j.}$$

and if the integers are viewed as a measure space, endowed with the counting measure, then this property coincides with the defining property of the Dirac delta function

$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x-y)f(x)dx=f(y),}$$

and in fact Dirac's delta was named after the Kronecker delta because of this analogous property.^[2] In signal processing it is usually the context (discrete or continuous time) that distinguishes the Kronecker and Dirac "functions". And by convention, ${\displaystyle \delta (t)}$ generally indicates continuous time (Dirac), whereas arguments like ${\displaystyle i}$, ${\displaystyle j}$, ${\displaystyle k}$, ${\displaystyle l}$, ${\displaystyle m}$, and ${\displaystyle n}$ are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: ${\displaystyle \delta [n]}$. The Kronecker delta is not the result of directly sampling the Dirac delta function.

The Kronecker delta forms the multiplicative identity element of an incidence algebra.

Relationship to the Dirac delta function

In probability theory and statistics, the Kronecker delta and Dirac delta function can both be used to represent a discrete distribution. If the support of a distribution consists of points ${\displaystyle \mathbf{x}=\{x_1,\cdots ,x_n\}}$, with corresponding probabilities ${\displaystyle p_1,\cdots ,p_n}$, then the probability mass function ${\displaystyle p(x)}$ of the distribution over ${\displaystyle \mathbf{x}}$ can be written, using the Kronecker delta, as

$${\displaystyle p(x)=\sum _{i=1}^n{p}_i{\delta }_{x{x}_i}.}$$

Equivalently, the probability density function ${\displaystyle f(x)}$ of the distribution can be written using the Dirac delta function as

$${\displaystyle f(x)=\sum _{i=1}^n{p}_i\delta (x-x_i).}$$

Under certain conditions, the Kronecker delta can arise from sampling a Dirac delta function. For example, if a Dirac delta impulse occurs exactly at a sampling point and is ideally lowpass-filtered (with cutoff at the critical frequency) per the Nyquist–Shannon sampling theorem, the resulting discrete-time signal will be a Kronecker delta function.

Generalizations

If it is considered as a type ${\displaystyle (1,1)}$ tensor, the Kronecker tensor can be written ${\displaystyle {\delta }_j^i}$ with a covariant index ${\displaystyle j}$ and contravariant index ${\displaystyle i}$:

$${\displaystyle {\delta }_j^i=\{\begin{array}{ll}0& (i\ne j),\\ 1& (i=j).\end{array}}$$

This tensor represents:

• The identity mapping (or identity matrix), considered as a linear mapping ${\displaystyle V\to V}$ or ${\displaystyle V^{\ast }\to V^{\ast }}$
• The trace or tensor contraction, considered as a mapping ${\displaystyle V^{\ast }\otimes V\to K}$
• The map ${\displaystyle K\to V^{\ast }\otimes V}$, representing scalar multiplication as a sum of outer products.

The generalized Kronecker delta or multi-index Kronecker delta of order ${\displaystyle 2p}$ is a type ${\displaystyle (p,p)}$ tensor that is completely antisymmetric in its ${\displaystyle p}$ upper indices, and also in its ${\displaystyle p}$ lower indices.

Two definitions that differ by a factor of ${\displaystyle p!}$ are in use. Below, the version is presented has nonzero components scaled to be ${\displaystyle \pm 1}$. The second version has nonzero components that are ${\displaystyle \pm 1/p!}$, with consequent changes scaling factors in formulae, such as the scaling factors of ${\displaystyle 1/p!}$ in § Properties of the generalized Kronecker delta below disappearing.

Definitions of the generalized Kronecker delta

In terms of the indices, the generalized Kronecker delta is defined as:

$${\displaystyle {\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\{\begin{array}{ll}+1& \text{if }{\nu }_1\dots {\nu }_p\text{ are distinct integers and are an even permutation of }{\mu }_1\dots {\mu }_p\\ -1& \text{if }{\nu }_1\dots {\nu }_p\text{ are distinct integers and are an odd permutation of }{\mu }_1\dots {\mu }_p\\ 0& \text{in all other cases}.\end{array}}$$

Let ${\displaystyle {\mathrm{S}}_p}$ be the symmetric group of degree ${\displaystyle p}$, then:

$${\displaystyle {\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\sum _{\sigma \in {\mathrm{S}}_p}\mathrm{sgn}(\sigma ){\delta }_{{\nu }_{\sigma (1)}}^{{\mu }_1}\cdots {\delta }_{{\nu }_{\sigma (p)}}^{{\mu }_p}=\sum _{\sigma \in {\mathrm{S}}_p}\mathrm{sgn}(\sigma ){\delta }_{{\nu }_1}^{{\mu }_{\sigma (1)}}\cdots {\delta }_{{\nu }_p}^{{\mu }_{\sigma (p)}}.}$$

Using anti-symmetrization:

$${\displaystyle {\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=p!{\delta }_{[{\nu }_1}^{{\mu }_1}\dots {\delta }_{{\nu }_p]}^{{\mu }_p}=p!{\delta }_{{\nu }_1}^{[{\mu }_1}\dots {\delta }_{{\nu }_p}^{{\mu }_p]}.}$$

In terms of a ${\displaystyle p\times p}$ determinant:

$${\displaystyle {\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\left|\begin{array}{ccc}{\delta }_{{\nu }_1}^{{\mu }_1}& \cdots & {\delta }_{{\nu }_p}^{{\mu }_1}\\ ⋮& \ddots & ⋮\\ {\delta }_{{\nu }_1}^{{\mu }_p}& \cdots & {\delta }_{{\nu }_p}^{{\mu }_p}\end{array}\right|.}$$

Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:

$${\displaystyle \begin{array}{rl}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}& =\sum _{k=1}^p(-1{)}^{p+k}{\delta }_{{\nu }_k}^{{\mu }_p}{\delta }_{{\nu }_1\dots {\check{\nu }}_k\dots {\nu }_p}^{{\mu }_1\dots {\mu }_k\dots {\check{\mu }}_p}\\ & ={\delta }_{{\nu }_p}^{{\mu }_p}{\delta }_{{\nu }_1\dots {\nu }_{p-1}}^{{\mu }_1\dots {\mu }_{p-1}}-\sum _{k=1}^{p-1}{\delta }_{{\nu }_k}^{{\mu }_p}{\delta }_{{\nu }_1\dots {\nu }_{k-1}{\nu }_p{\nu }_{k+1}\dots {\nu }_{p-1}}^{{\mu }_1\dots {\mu }_{k-1}{\mu }_k{\mu }_{k+1}\dots {\mu }_{p-1}},\end{array}}$$

where the caron, ${\displaystyle \stackrel{ˇ}{}}$, indicates an index that is omitted from the sequence.

When ${\displaystyle p=n}$ (the dimension of the vector space), in terms of the Levi-Civita symbol:

$${\displaystyle {\delta }_{{\nu }_1\dots {\nu }_n}^{{\mu }_1\dots {\mu }_n}={\epsilon }^{{\mu }_1\dots {\mu }_n}{\epsilon }_{{\nu }_1\dots {\nu }_n}.}$$

More generally, for ${\displaystyle m=n-p}$, using the Einstein summation convention:

$${\displaystyle {\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\frac{1}{m!}{\epsilon }^{{\kappa }_1\dots {\kappa }_m{\mu }_1\dots {\mu }_p}{\epsilon }_{{\kappa }_1\dots {\kappa }_m{\nu }_1\dots {\nu }_p}.}$$

Contractions of the generalized Kronecker delta

Kronecker Delta contractions depend on the dimension of the space. For example,

$${\displaystyle {\delta }_{{\mu }_1}^{{\nu }_1}{\delta }_{{\nu }_1{\nu }_2}^{{\mu }_1{\mu }_2}=(d-1){\delta }_{{\nu }_2}^{{\mu }_2},}$$

where d is the dimension of the space. From this relation the full contracted delta is obtained as

$${\displaystyle {\delta }_{{\mu }_1{\mu }_2}^{{\nu }_1{\nu }_2}{\delta }_{{\nu }_1{\nu }_2}^{{\mu }_1{\mu }_2}=2d(d-1).}$$

The generalization of the preceding formulas is

$${\displaystyle {\delta }_{{\mu }_1\dots {\mu }_n}^{{\nu }_1\dots {\nu }_n}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=n!\frac{(d-p+n)!}{(d-p)!}{\delta }_{{\nu }_{n+1}\dots {\nu }_p}^{{\mu }_{n+1}\dots {\mu }_p}.}$$

Properties of the generalized Kronecker delta

The generalized Kronecker delta may be used for anti-symmetrization:

$${\displaystyle \begin{array}{rl}\frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{a}^{{\nu }_1\dots {\nu }_p}& =a^{[{\mu }_1\dots {\mu }_p]},\\ \frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{a}_{{\mu }_1\dots {\mu }_p}& =a_{[{\nu }_1\dots {\nu }_p]}.\end{array}}$$

From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta:

$${\displaystyle \begin{array}{rl}\frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{a}^{[{\nu }_1\dots {\nu }_p]}& =a^{[{\mu }_1\dots {\mu }_p]},\\ \frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{a}_{[{\mu }_1\dots {\mu }_p]}& =a_{[{\nu }_1\dots {\nu }_p]},\\ \frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{\delta }_{{\kappa }_1\dots {\kappa }_p}^{{\nu }_1\dots {\nu }_p}& ={\delta }_{{\kappa }_1\dots {\kappa }_p}^{{\mu }_1\dots {\mu }_p},\end{array}}$$

which are the generalized version of formulae written in § Properties. The last formula is equivalent to the Cauchy–Binet formula.

Reducing the order via summation of the indices may be expressed by the identity

$${\displaystyle {\delta }_{{\nu }_1\dots {\nu }_s{\mu }_{s+1}\dots {\mu }_p}^{{\mu }_1\dots {\mu }_s{\mu }_{s+1}\dots {\mu }_p}=\frac{(n-s)!}{(n-p)!}{\delta }_{{\nu }_1\dots {\nu }_s}^{{\mu }_1\dots {\mu }_s}.}$$

Using both the summation rule for the case ${\displaystyle p=n}$ and the relation with the Levi-Civita symbol, the summation rule of the Levi-Civita symbol is derived:

$${\displaystyle {\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\frac{1}{(n-p)!}{\epsilon }^{{\mu }_1\dots {\mu }_p{\kappa }_{p+1}\dots {\kappa }_n}{\epsilon }_{{\nu }_1\dots {\nu }_p{\kappa }_{p+1}\dots {\kappa }_n}.}$$

The 4D version of the last relation appears in Penrose's spinor approach to general relativity that he later generalized, while he was developing Aitken's diagrams, to become part of the technique of Penrose graphical notation. Also, this relation is extensively used in S-duality theories, especially when written in the language of differential forms and Hodge duals.

Integral representations

For any integer ${\displaystyle n}$, using a standard residue calculation we can write an integral representation for the Kronecker delta as the integral below, where the contour of the integral goes counterclockwise around zero. This representation is also equivalent to a definite integral by a rotation in the complex plane.

$${\displaystyle {\delta }_{x,n}=\frac{1}{2\pi i}{\oint }_{|z|=1}{z}^{x-n-1}dz=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{i(x-n)\phi }d\phi }$$

The Kronecker comb

The Kronecker comb function with period ${\displaystyle N}$ is defined (using DSP notation) as:

$${\displaystyle {\mathrm{\Delta }}_N[n]=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\delta [n-kN],}$$

where ${\displaystyle N}$ and ${\displaystyle n}$ are integers. The Kronecker comb thus consists of an infinite series of unit impulses N units apart, and includes the unit impulse at zero. It may be considered to be the discrete analog of the Dirac comb.

Kronecker integral

The Kronecker delta is also called degree of mapping of one surface into another. Suppose a mapping takes place from surface S_uvw to S_xyz that are boundaries of regions, R_uvw and R_xyz which is simply connected with one-to-one correspondence. In this framework, if s and t are parameters for S_uvw, and S_uvw to S_uvw are each oriented by the outer normal n:

$${\displaystyle u=u(s,t),v=v(s,t),w=w(s,t),}$$

while the normal has the direction of

$${\displaystyle (u_s\mathbf{i}+v_s\mathbf{j}+w_s\mathbf{k})\times (u_t\mathbf{i}+v_t\mathbf{j}+w_t\mathbf{k}).}$$

Let x = x(u, v, w), y = y(u, v, w), z = z(u, v, w) be defined and smooth in a domain containing S_uvw, and let these equations define the mapping of S_uvw onto S_xyz. Then the degree δ of mapping is 1/4π times the solid angle of the image S of S_uvw with respect to the interior point of S_xyz, O. If O is the origin of the region, R_xyz, then the degree, δ is given by the integral:

$${\displaystyle \delta =\frac{1}{4\pi }{\iint }_{R_{st}}{\left(x^2+y^2+z^2\right)}^{-\frac{3}{2}}\left|\begin{array}{ccc}x& y& z\\ \frac{\mathrm{\partial }x}{\mathrm{\partial }s}& \frac{\mathrm{\partial }y}{\mathrm{\partial }s}& \frac{\mathrm{\partial }z}{\mathrm{\partial }s}\\ \frac{\mathrm{\partial }x}{\mathrm{\partial }t}& \frac{\mathrm{\partial }y}{\mathrm{\partial }t}& \frac{\mathrm{\partial }z}{\mathrm{\partial }t}\end{array}\right|dsdt.}$$

Transfer RNA

From Wikipedia, the free encyclopedia

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

The interaction of tRNA and mRNA in protein synthesis.

Transfer RNA (abbreviated tRNA and formerly referred to as sRNA, for soluble RNA) is an adaptor molecule composed of RNA, typically 76 to 90 nucleotides in length (in eukaryotes), that serves as the physical link between the mRNA and the amino acid sequence of proteins. Transfer RNA (tRNA) does this by carrying an amino acid to the protein synthesizing machinery of a cell called the ribosome. Complementation of a 3-nucleotide codon in a messenger RNA (mRNA) by a 3-nucleotide anticodon of the tRNA results in protein synthesis based on the mRNA code. As such, tRNAs are a necessary component of translation, the biological synthesis of new proteins in accordance with the genetic code.

Typically, tRNAs genes from Bacteria are shorter (mean = 77.6 bp) than tRNAs from Archaea (mean = 83.1 bp) and eukaryotes (mean = 84.7 bp). The mature tRNA follows an opposite pattern with tRNAs from Bacteria being usually longer (median = 77.6 nt) than tRNAs from Archaea (median = 76.8 nt), with eukaryotes exhibiting the shortest mature tRNAs (median = 74.5 nt).

Overview

While the specific nucleotide sequence of an mRNA specifies which amino acids are incorporated into the protein product of the gene from which the mRNA is transcribed, the role of tRNA is to specify which sequence from the genetic code corresponds to which amino acid. The mRNA encodes a protein as a series of contiguous codons, each of which is recognized by a particular tRNA. One end of the tRNA matches the genetic code in a three-nucleotide sequence called the anticodon. The anticodon forms three complementary base pairs with a codon in mRNA during protein biosynthesis.

On the other end of the tRNA is a covalent attachment to the amino acid that corresponds to the anticodon sequence. Each type of tRNA molecule can be attached to only one type of amino acid, so each organism has many types of tRNA. Because the genetic code contains multiple codons that specify the same amino acid, there are several tRNA molecules bearing different anticodons which carry the same amino acid.

The covalent attachment to the tRNA 3’ end is catalysed by enzymes called aminoacyl tRNA synthetases. During protein synthesis, tRNAs with attached amino acids are delivered to the ribosome by proteins called elongation factors, which aid in association of the tRNA with the ribosome, synthesis of the new polypeptide, and translocation (movement) of the ribosome along the mRNA. If the tRNA's anticodon matches the mRNA, another tRNA already bound to the ribosome transfers the growing polypeptide chain from its 3’ end to the amino acid attached to the 3’ end of the newly delivered tRNA, a reaction catalysed by the ribosome. A large number of the individual nucleotides in a tRNA molecule may be chemically modified, often by methylation or deamidation. These unusual bases sometimes affect the tRNA's interaction with ribosomes and sometimes occur in the anticodon to alter base-pairing properties.

Structure

Secondary cloverleaf structure of tRNA

Tertiary structure of tRNA. CCA tail in yellow, Acceptor stem in purple, Variable loop in orange, D arm in red, Anticodon arm in blue with Anticodon in black, T arm in green.

3D animated GIF showing the structure of phenylalanine-tRNA from yeast (PDB ID 1ehz). White lines indicate base pairing by hydrogen bonds. In the orientation shown, the acceptor stem is on top and the anticodon on the bottom

The structure of tRNA can be decomposed into its primary structure, its secondary structure (usually visualized as the cloverleaf structure), and its tertiary structure (all tRNAs have a similar L-shaped 3D structure that allows them to fit into the P and A sites of the ribosome). The cloverleaf structure becomes the 3D L-shaped structure through coaxial stacking of the helices, which is a common RNA tertiary structure motif. The lengths of each arm, as well as the loop 'diameter', in a tRNA molecule vary from species to species. The tRNA structure consists of the following:

• The acceptor stem is a 7- to 9-base pair (bp) stem made by the base pairing of the 5′-terminal nucleotide with the 3′-terminal nucleotide (which contains the CCA 3′-terminal group used to attach the amino acid). In general, such 3′-terminal tRNA-like structures are referred to as 'genomic tags'. The acceptor stem may contain non-Watson-Crick base pairs.
• The CCA tail is a cytosine-cytosine-adenine sequence at the 3′ end of the tRNA molecule. The amino acid loaded onto the tRNA by aminoacyl tRNA synthetases, to form aminoacyl-tRNA, is covalently bonded to the 3′-hydroxyl group on the CCA tail. This sequence is important for the recognition of tRNA by enzymes and critical in translation. In prokaryotes, the CCA sequence is transcribed in some tRNA sequences. In most prokaryotic tRNAs and eukaryotic tRNAs, the CCA sequence is added during processing and therefore does not appear in the tRNA gene.
• The D loop is a 4- to 6-bp stem ending in a loop that often contains dihydrouridine.
• The anticodon loop is a 5-bp stem whose loop contains the anticodon. The tRNA 5′-to-3′ primary structure contains the anticodon but in reverse order, since 3′-to-5′ directionality is required to read the mRNA from 5′-to-3′.
• The ΨU loop is named so because of the characteristic presence of the unusual base ΨU in the loop, where Ψ is pseudouridine, a modified uridine. The modified base is often found within the sequence 5' -TΨUCG-3'.
• The variable loop sits between the anticodon loop and the ΨU loop and, as its name implies, varies in size from 3 to 21 bases.

Anticodon

An anticodon is a unit of three nucleotides corresponding to the three bases of an mRNA codon. Each tRNA has a distinct anticodon triplet sequence that can form 3 complementary base pairs to one or more codons for an amino acid. Some anticodons pair with more than one codon due to wobble base pairing. Frequently, the first nucleotide of the anticodon is one not found on mRNA: inosine, which can hydrogen bond to more than one base in the corresponding codon position. In genetic code, it is common for a single amino acid to be specified by all four third-position possibilities, or at least by both pyrimidines and purines; for example, the amino acid glycine is coded for by the codon sequences GGU, GGC, GGA, and GGG. Other modified nucleotides may also appear at the first anticodon position—sometimes known as the "wobble position"—resulting in subtle changes to the genetic code, as for example in mitochondria. Per cell, 61 tRNA types are required to provide one-to-one correspondence between tRNA molecules and codons that specify amino acids, as there are 61 sense codons of the standard genetic code. However, many cells have under 61 types of tRNAs because the wobble base is capable of binding to several, though not necessarily all, of the codons that specify a particular amino acid. At least 31 tRNAs are required to translate, unambiguously, all 61 sense codons.

Aminoacylation

See also: Aminoacyl-tRNA

Aminoacylation is the process of adding an aminoacyl group to a compound. It covalently links an amino acid to the CCA 3′ end of a tRNA molecule. Each tRNA is aminoacylated (or charged) with a specific amino acid by an aminoacyl tRNA synthetase. There is normally a single aminoacyl tRNA synthetase for each amino acid, despite the fact that there can be more than one tRNA, and more than one anticodon for an amino acid. Recognition of the appropriate tRNA by the synthetases is not mediated solely by the anticodon, and the acceptor stem often plays a prominent role. Reaction:

1. amino acid + ATP → aminoacyl-AMP + PPi
2. aminoacyl-AMP + tRNA → aminoacyl-tRNA + AMP

Certain organisms can have one or more aminophosphate-tRNA synthetases missing. This leads to charging of the tRNA by a chemically related amino acid, and by use of an enzyme or enzymes, the tRNA is modified to be correctly charged. For example, Helicobacter pylori has glutaminyl tRNA synthetase missing. Thus, glutamate tRNA synthetase charges tRNA-glutamine(tRNA-Gln) with glutamate. An amidotransferase then converts the acid side chain of the glutamate to the amide, forming the correctly charged gln-tRNA-Gln.

Interference with aminoacylation may be useful as an approach to treating some diseases: cancerous cells may be relatively vulnerable to disturbed aminoacylation compared to healthy cells. The protein synthesis associated with cancer and viral biology is often very dependent on specific tRNA molecules. For instance, for liver cancer charging tRNA-Lys-CUU with lysine sustains liver cancer cell growth and metastasis, whereas healthy cells have a much lower dependence on this tRNA to support cellular physiology. Similarly, hepatitis E virus requires a tRNA landscape that substantially differs from that associated with uninfected cells. Hence, inhibition of aminoacylation of specific tRNA species is considered a promising novel avenue for the rational treatment of a plethora of diseases.

Binding to ribosome

The ribosome has three binding sites for tRNA molecules that span the space between the two ribosomal subunits: the A (aminoacyl), P (peptidyl), and E (exit) sites. In addition, the ribosome has two other sites for tRNA binding that are used during mRNA decoding or during the initiation of protein synthesis. These are the T site (named elongation factor Tu) and I site (initiation). By convention, the tRNA binding sites are denoted with the site on the small ribosomal subunit listed first and the site on the large ribosomal subunit listed second. For example, the A site is often written A/A, the P site, P/P, and the E site, E/E. The binding proteins like L27, L2, L14, L15, L16 at the A- and P- sites have been determined by affinity labeling by A. P. Czernilofsky et al. (Proc. Natl. Acad. Sci, USA, pp. 230–234, 1974).

Once translation initiation is complete, the first aminoacyl tRNA is located in the P/P site, ready for the elongation cycle described below. During translation elongation, tRNA first binds to the ribosome as part of a complex with elongation factor Tu (EF-Tu) or its eukaryotic (eEF-1) or archaeal counterpart. This initial tRNA binding site is called the A/T site. In the A/T site, the A-site half resides in the small ribosomal subunit where the mRNA decoding site is located. The mRNA decoding site is where the mRNA codon is read out during translation. The T-site half resides mainly on the large ribosomal subunit where EF-Tu or eEF-1 interacts with the ribosome. Once mRNA decoding is complete, the aminoacyl-tRNA is bound in the A/A site and is ready for the next peptide bond to be formed to its attached amino acid. The peptidyl-tRNA, which transfers the growing polypeptide to the aminoacyl-tRNA bound in the A/A site, is bound in the P/P site. Once the peptide bond is formed, the tRNA in the P/P site is acylated, or has a free 3’ end, and the tRNA in the A/A site dissociates the growing polypeptide chain. To allow for the next elongation cycle, the tRNAs then move through hybrid A/P and P/E binding sites, before completing the cycle and residing in the P/P and E/E sites. Once the A/A and P/P tRNAs have moved to the P/P and E/E sites, the mRNA has also moved over by one codon and the A/T site is vacant, ready for the next round of mRNA decoding. The tRNA bound in the E/E site then leaves the ribosome.

The P/I site is actually the first to bind to aminoacyl tRNA, which is delivered by an initiation factor called IF2 in bacteria. However, the existence of the P/I site in eukaryotic or archaeal ribosomes has not yet been confirmed. The P-site protein L27 has been determined by affinity labeling by E. Collatz and A. P. Czernilofsky (FEBS Lett., Vol. 63, pp. 283–286, 1976).

tRNA genes

Organisms vary in the number of tRNA genes in their genome. For example, the nematode worm C. elegans, a commonly used model organism in genetics studies, has 29,647 genes in its nuclear genome, of which 620 code for tRNA. The budding yeast Saccharomyces cerevisiae has 275 tRNA genes in its genome. The number of tRNA genes per genome can vary widely, with bacterial species from groups such as Fusobacteria and Tenericutes having around 30 genes per genome while complex eukaryotic genomes such as the zebrafish (Danio rerio) can bear more than 10 thousand tRNA genes.

In the human genome, which, according to January 2013 estimates, has about 20,848 protein coding genes  in total, there are 497 nuclear genes encoding cytoplasmic tRNA molecules, and 324 tRNA-derived pseudogenes—tRNA genes thought to be no longer functional (although pseudo tRNAs have been shown to be involved in antibiotic resistance in bacteria). As with all eukaryotes, there are 22 mitochondrial tRNA genes in humans. Mutations in some of these genes have been associated with severe diseases like the MELAS syndrome. Regions in nuclear chromosomes, very similar in sequence to mitochondrial tRNA genes, have also been identified (tRNA-lookalikes). These tRNA-lookalikes are also considered part of the nuclear mitochondrial DNA (genes transferred from the mitochondria to the nucleus). The phenomenon of multiple nuclear copies of mitochondrial tRNA (tRNA-lookalikes) has been observed in many higher organisms from human to the opossum suggesting the possibility that the lookalikes are functional.

Cytoplasmic tRNA genes can be grouped into 49 families according to their anticodon features. These genes are found on all chromosomes, except the 22 and Y chromosome. High clustering on 6p is observed (140 tRNA genes), as well on 1 chromosome.

The HGNC, in collaboration with the Genomic tRNA Database (GtRNAdb) and experts in the field, has approved unique names for human genes that encode tRNAs.

Evolution

The top half of tRNA (consisting of the T arm and the acceptor stem with 5′-terminal phosphate group and 3′-terminal CCA group) and the bottom half (consisting of the D arm and the anticodon arm) are independent units in structure as well as in function. The top half may have evolved first including the 3′-terminal genomic tag which originally may have marked tRNA-like molecules for replication in early RNA world. The bottom half may have evolved later as an expansion, e.g. as protein synthesis started in RNA world and turned it into a ribonucleoprotein world (RNP world). This proposed scenario is called genomic tag hypothesis. In fact, tRNA and tRNA-like aggregates have an important catalytic influence (i.e., as ribozymes) on replication still today. These roles may be regarded as 'molecular (or chemical) fossils' of RNA world.

Genomic tRNA content is a differentiating feature of genomes among biological domains of life: Archaea present the simplest situation in terms of genomic tRNA content with a uniform number of gene copies, Bacteria have an intermediate situation and Eukarya present the most complex situation. Eukarya present not only more tRNA gene content than the other two kingdoms but also a high variation in gene copy number among different isoacceptors, and this complexity seem to be due to duplications of tRNA genes and changes in anticodon specificity.

Evolution of the tRNA gene copy number across different species has been linked to the appearance of specific tRNA modification enzymes (uridine methyltransferases in Bacteria, and adenosine deaminases in Eukarya), which increase the decoding capacity of a given tRNA. As an example, tRNAAla encodes four different tRNA isoacceptors (AGC, UGC, GGC and CGC). In Eukarya, AGC isoacceptors are extremely enriched in gene copy number in comparison to the rest of isoacceptors, and this has been correlated with its A-to-I modification of its wobble base. This same trend has been shown for most amino acids of eukaryal species. Indeed, the effect of these two tRNA modifications is also seen in codon usage bias. Highly expressed genes seem to be enriched in codons that are exclusively using codons that will be decoded by these modified tRNAs, which suggests a possible role of these codons—and consequently of these tRNA modifications—in translation efficiency.

It is important to note that many species have lost specific tRNAs during evolution. For instance, both mammals and birds lack the same 14 out of the possible 64 tRNA genes, but other life forms contain these tRNAs. For translating codons for which an exactly pairing tRNA is missing, organisms resort to a strategy called wobbling, in which imperfectly matched tRNA/mRNA pairs still give rise to translation, although this strategy also increases to propensity for translation errors. The reasons why tRNA genes have been lost during evolution remains under debate but may relate improving resistance to viral infection. Because nucleotide triplets can present more combinations than there are amino acids and associated tRNAs, there is redundancy in the genetic code, and several different 3-nucleotide codons can express the same amino acid. This codon bias is what necessitates codon optimization.

tRNA-derived fragments

tRNA-derived fragments (or tRFs) are short molecules that emerge after cleavage of the mature tRNAs or the precursor transcript. Both cytoplasmic and mitochondrial tRNAs can produce fragments. There are at least four structural types of tRFs believed to originate from mature tRNAs, including the relatively long tRNA halves and short 5’-tRFs, 3’-tRFs and i-tRFs. The precursor tRNA can be cleaved to produce molecules from the 5’ leader or 3’ trail sequences. Cleavage enzymes include Angiogenin, Dicer, RNase Z and RNase P. Especially in the case of Angiogenin, the tRFs have a characteristically unusual cyclic phosphate at their 3’ end and a hydroxyl group at the 5’ end. tRFs appear to play a role in RNA interference, specifically in the suppression of retroviruses and retrotransposons that use tRNA as a primer for replication. Half-tRNAs cleaved by angiogenin are also known as tiRNAs. The biogenesis of smaller fragments, including those that function as piRNAs, are less understood.

tRFs have multiple dependencies and roles; such as exhibiting significant changes between sexes, among races and disease status. Functionally, they can be loaded on Ago and act through RNAi pathways, participate in the formation of stress granules, displace mRNAs from RNA-binding proteins or inhibit translation. At the system or the organismal level, the four types of tRFs have a diverse spectrum of activities. Functionally, tRFs are associated with viral infection, cancer, cell proliferation  and also with epigenetic transgenerational regulation of metabolism.

tRFs are not restricted to humans and have been shown to exist in multiple organisms.

Two online tools are available for those wishing to learn more about tRFs: the framework for the interactive exploration of mitochondrial and nuclear tRNA fragments (MINTbase) and the relational database of Transfer RNA related Fragments (tRFdb). MINTbase also provides a naming scheme for the naming of tRFs called tRF-license plates (or MINTcodes) that is genome independent; the scheme compresses an RNA sequence into a shorter string.

Engineered tRNAs

Artificial suppressor elongator tRNAs are used to incorporate unnatural amino acids at nonsense codons placed in the coding sequence of a gene. Engineered initiator tRNAs (tRNA^fMet2 with CUA anticodon encoded by metY gene) have been used to initiate translation at the amber stop codon UAG. This type of engineered tRNA is called a nonsense suppressor tRNA because it suppresses the translation stop signal that normally occurs at UAG codons. The amber initiator tRNA inserts methionine and glutamine at UAG codons preceded by a strong Shine-Dalgarno sequence. An investigation of the amber initiator tRNA showed that it was orthogonal to the regular AUG start codon showing no detectable off-target translation initiation events in a genomically recoded E. coli strain.

tRNA biogenesis

In eukaryotic cells, tRNAs are transcribed by RNA polymerase III as pre-tRNAs in the nucleus. RNA polymerase III recognizes two highly conserved downstream promoter sequences: the 5′ intragenic control region (5′-ICR, D-control region, or A box), and the 3′-ICR (T-control region or B box) inside tRNA genes. The first promoter begins at +8 of mature tRNAs and the second promoter is located 30–60 nucleotides downstream of the first promoter. The transcription terminates after a stretch of four or more thymidines.

Bulge-helix-bulge motif of a tRNA intron

Pre-tRNAs undergo extensive modifications inside the nucleus. Some pre-tRNAs contain introns that are spliced, or cut, to form the functional tRNA molecule; in bacteria these self-splice, whereas in eukaryotes and archaea they are removed by tRNA-splicing endonucleases. Eukaryotic pre-tRNA contains bulge-helix-bulge (BHB) structure motif that is important for recognition and precise splicing of tRNA intron by endonucleases. This motif position and structure are evolutionarily conserved. However, some organisms, such as unicellular algae have a non-canonical position of BHB-motif as well as 5′- and 3′-ends of the spliced intron sequence. The 5′ sequence is removed by RNase P, whereas the 3′ end is removed by the tRNase Z enzyme. A notable exception is in the archaeon Nanoarchaeum equitans, which does not possess an RNase P enzyme and has a promoter placed such that transcription starts at the 5′ end of the mature tRNA. The non-templated 3′ CCA tail is added by a nucleotidyl transferase. Before tRNAs are exported into the cytoplasm by Los1/Xpo-t, tRNAs are aminoacylated. The order of the processing events is not conserved. For example, in yeast, the splicing is not carried out in the nucleus but at the cytoplasmic side of mitochondrial membranes.

Nonetheless, In March 2021, researchers reported evidence suggesting that a preliminary form of transfer RNA could have been a replicator molecule in the very early development of life, or abiogenesis.

History

The existence of tRNA was first hypothesized by Francis Crick as the "adaptor hypothesis" based on the assumption that there must exist an adapter molecule capable of mediating the translation of the RNA alphabet into the protein alphabet. Paul C Zamecnik and Mahlon Hoagland discovered tRNA  Significant research on structure was conducted in the early 1960s by Alex Rich and Donald Caspar, two researchers in Boston, the Jacques Fresco group in Princeton University and a United Kingdom group at King's College London. In 1965, Robert W. Holley of Cornell University reported the primary structure and suggested three secondary structures. tRNA was first crystallized in Madison, Wisconsin, by Robert M. Bock. The cloverleaf structure was ascertained by several other studies in the following years and was finally confirmed using X-ray crystallography studies in 1974. Two independent groups, Kim Sung-Hou working under Alexander Rich and a British group headed by Aaron Klug, published the same crystallography findings within a year.

Lone pair

From Wikipedia, the free encyclopedia

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

Lone pairs (shown as pairs of dots) in the Lewis structure of hydroxide

In chemistry, a lone pair refers to a pair of valence electrons that are not shared with another atom in a covalent bond and is sometimes called an unshared pair or non-bonding pair. Lone pairs are found in the outermost electron shell of atoms. They can be identified by using a Lewis structure. Electron pairs are therefore considered lone pairs if two electrons are paired but are not used in chemical bonding. Thus, the number of electrons in lone pairs plus the number of electrons in bonds equals the number of valence electrons around an atom.

Lone pair is a concept used in valence shell electron pair repulsion theory (VSEPR theory) which explains the shapes of molecules. They are also referred to in the chemistry of Lewis acids and bases. However, not all non-bonding pairs of electrons are considered by chemists to be lone pairs. Examples are the transition metals where the non-bonding pairs do not influence molecular geometry and are said to be stereochemically inactive. In molecular orbital theory (fully delocalized canonical orbitals or localized in some form), the concept of a lone pair is less distinct, as the correspondence between an orbital and components of a Lewis structure is often not straightforward. Nevertheless, occupied non-bonding orbitals (or orbitals of mostly nonbonding character) are frequently identified as lone pairs.

Lone pairs in ammonia (A), water (B), and hydrogen chloride (C)

A single lone pair can be found with atoms in the nitrogen group, such as nitrogen in ammonia. Two lone pairs can be found with atoms in the chalcogen group, such as oxygen in water. The halogens can carry three lone pairs, such as in hydrogen chloride.

In VSEPR theory the electron pairs on the oxygen atom in water form the vertices of a tetrahedron with the lone pairs on two of the four vertices. The H–O–H bond angle is 104.5°, less than the 109° predicted for a tetrahedral angle, and this can be explained by a repulsive interaction between the lone pairs.

Various computational criteria for the presence of lone pairs have been proposed. While electron density ρ(r) itself generally does not provide useful guidance in this regard, the Laplacian of the electron density is revealing, and one criterion for the location of the lone pair is where L(r) = –∇²ρ(r) is a local maximum. The minima of the electrostatic potential V(r) is another proposed criterion. Yet another considers the electron localization function (ELF).

Angle changes

Tetrahedral structure of water

The pairs often exhibit a negative polar character with their high charge density and are located closer to the atomic nucleus on average compared to the bonding pair of electrons. The presence of a lone pair decreases the bond angle between the bonding pair of electrons, due to their high electric charge, which causes great repulsion between the electrons. They are also involved in the formation of a dative bond. For example, the creation of the hydronium (H₃O⁺) ion occurs when acids are dissolved in water and is due to the oxygen atom donating a lone pair to the hydrogen ion.

This can be seen more clearly when looked at it in two more common molecules. For example, in carbon dioxide (CO₂), the oxygen atoms are on opposite sides of the carbon atom (linear molecular geometry), whereas in water (H₂O) the angle between the hydrogen atoms is 104.5° (bent molecular geometry). The repulsive force of the oxygen atom's two lone pairs pushes the hydrogen atoms further apart, until the forces of all electrons on the hydrogen atom are in equilibrium. This is an illustration of the VSEPR theory.

Dipole moments

Lone pairs can contribute to a molecule's dipole moment. NH₃ has a dipole moment of 1.42 D. As the electronegativity of nitrogen (3.04) is greater than that of hydrogen (2.2) the result is that the N-H bonds are polar with a net negative charge on the nitrogen atom and a smaller net positive charge on the hydrogen atoms. There is also a dipole associated with the lone pair and this reinforces the contribution made by the polar covalent N-H bonds to ammonia's dipole moment. In contrast to NH₃, NF₃ has a much lower dipole moment of 0.234 D. Fluorine is more electronegative than nitrogen and the polarity of the N-F bonds is opposite to that of the N-H bonds in ammonia, so that the dipole due to the lone pair opposes the N-F bond dipoles, resulting in a low molecular dipole moment.

Stereogenic lone pairs

	⇌
Inversion of a generic organic amine molecule at nitrogen

A lone pair can contribute to the existence of chirality in a molecule, when three other groups attached to an atom all differ. The effect is seen in certain amines, phosphines, sulfonium and oxonium ions, sulfoxides, and even carbanions.

The resolution of enantiomers where the stereogenic center is an amine is usually precluded because the energy barrier for nitrogen inversion at the stereo center is low, which allow the two stereoisomers to rapidly interconvert at room temperature. As a result, such chiral amines cannot be resolved, unless the amine's groups are constrained in a cyclic structure (such as in Tröger's base).

Unusual lone pairs

A stereochemically active lone pair is also expected for divalent lead and tin ions due to their formal electronic configuration of ns². In the solid state this results in the distorted metal coordination observed in the tetragonal litharge structure adopted by both PbO and SnO. The formation of these heavy metal ns² lone pairs which was previously attributed to intra-atomic hybridization of the metal s and p states has recently been shown to have a strong anion dependence. This dependence on the electronic states of the anion can explain why some divalent lead and tin materials such as PbS and SnTe show no stereochemical evidence of the lone pair and adopt the symmetric rocksalt crystal structure.

In molecular systems the lone pair can also result in a distortion in the coordination of ligands around the metal ion. The lone-pair effect of lead can be observed in supramolecular complexes of lead(II) nitrate, and in 2007 a study linked the lone pair to lead poisoning. Lead ions can replace the native metal ions in several key enzymes, such as zinc cations in the ALAD enzyme, which is also known as porphobilinogen synthase, and is important in the synthesis of heme, a key component of the oxygen-carrying molecule hemoglobin. This inhibition of heme synthesis appears to be the molecular basis of lead poisoning (also called "saturnism" or "plumbism").

Computational experiments reveal that although the coordination number does not change upon substitution in calcium-binding proteins, the introduction of lead distorts the way the ligands organize themselves to accommodate such an emerging lone pair: consequently, these proteins are perturbed. This lone-pair effect becomes dramatic for zinc-binding proteins, such as the above-mentioned porphobilinogen synthase, as the natural substrate cannot bind anymore – in those cases the protein is inhibited.

In Group 14 elements (the carbon group), lone pairs can manifest themselves by shortening or lengthening single bond (bond order 1) lengths, as well as in the effective order of triple bonds as well. The familiar alkynes have a carbon-carbon triple bond (bond order 3) and a linear geometry of 180° bond angles (figure A in reference). However, further down in the group (silicon, germanium, and tin), formal triple bonds have an effective bond order 2 with one lone pair (figure B) and trans-bent geometries. In lead, the effective bond order is reduced even further to a single bond, with two lone pairs for each lead atom (figure C). In the organogermanium compound (Scheme 1 in the reference), the effective bond order is also 1, with complexation of the acidic isonitrile (or isocyanide) C-N groups, based on interaction with germanium's empty 4p orbital.

Different descriptions for multiple lone pairs

Further information: Sigma-pi and equivalent-orbital models

The symmetry-adapted and hybridized lone pairs of H₂O

In elementary chemistry courses, the lone pairs of water are described as "rabbit ears": two equivalent electron pairs of approximately sp³ hybridization, while the HOH bond angle is 104.5°, slightly smaller than the ideal tetrahedral angle of arccos(–1/3) ≈ 109.47°. The smaller bond angle is rationalized by VSEPR theory by ascribing a larger space requirement for the two identical lone pairs compared to the two bonding pairs. In more advanced courses, an alternative explanation for this phenomenon considers the greater stability of orbitals with excess s character using the theory of isovalent hybridization, in which bonds and lone pairs can be constructed with sp^x hybrids wherein nonintegral values of x are allowed, so long as the total amount of s and p character is conserved (one s and three p orbitals in the case of second-row p-block elements).

To determine the hybridization of oxygen orbitals used to form the bonding pairs and lone pairs of water in this picture, we use the formula 1 + x cos θ = 0, which relates bond angle θ with the hybridization index x. According to this formula, the O–H bonds are considered to be constructed from O bonding orbitals of ~sp^4.0 hybridization (~80% p character, ~20% s character), which leaves behind O lone pairs orbitals of ~sp^2.3 hybridization (~70% p character, ~30% s character). These deviations from idealized sp³ hybridization (75% p character, 25% s character) for tetrahedral geometry are consistent with Bent's rule: lone pairs localize more electron density closer to the central atom compared to bonding pairs; hence, the use of orbitals with excess s character to form lone pairs (and, consequently, those with excess p character to form bonding pairs) is energetically favorable.

However, theoreticians often prefer an alternative description of water that separates the lone pairs of water according to symmetry with respect to the molecular plane. In this model, there are two energetically and geometrically distinct lone pairs of water possessing different symmetry: one (σ) in-plane and symmetric with respect to the molecular plane and the other (π) perpendicular and anti-symmetric with respect to the molecular plane. The σ-symmetry lone pair (σ(out)) is formed from a hybrid orbital that mixes 2s and 2p character, while the π-symmetry lone pair (p) is of exclusive 2p orbital parentage. The s character rich O σ(out) lone pair orbital (also notated n_O^(σ)) is an ~sp^0.7 hybrid (~40% p character, 60% s character), while the p lone pair orbital (also notated n_O^(π)) consists of 100% p character.

Both models are of value and represent the same total electron density, with the orbitals related by a unitary transformation. In this case, we can construct the two equivalent lone pair hybrid orbitals h and h' by taking linear combinations h = c₁σ(out) + c₂p and h' = c₁σ(out) – c₂p for an appropriate choice of coefficients c₁ and c₂. For chemical and physical properties of water that depend on the overall electron distribution of the molecule, the use of h and h' is just as valid as the use of σ(out) and p. In some cases, such a view is intuitively useful. For example, the stereoelectronic requirement for the anomeric effect can be rationalized using equivalent lone pairs, since it is the overall donation of electron density into the antibonding orbital that matters. An alternative treatment using σ/π separated lone pairs is also valid, but it requires striking a balance between maximizing n_O^(π)-σ* overlap (maximum at 90° dihedral angle) and n_O^(σ)-σ* overlap (maximum at 0° dihedral angle), a compromise that leads to the conclusion that a gauche conformation (60° dihedral angle) is most favorable, the same conclusion that the equivalent lone pairs model rationalizes in a much more straightforward manner. Similarly, the hydrogen bonds of water form along the directions of the "rabbit ears" lone pairs, as a reflection of the increased availability of electrons in these regions. This view is supported computationally. However, because only the symmetry-adapted canonical orbitals have physically meaningful energies, phenomena that have to do with the energies of individual orbitals, such as photochemical reactivity or photoelectron spectroscopy, are most readily explained using σ and π lone pairs that respect the molecular symmetry.

Because of the popularity of VSEPR theory, the treatment of the water lone pairs as equivalent is prevalent in introductory chemistry courses, and many practicing chemists continue to regard it as a useful model. A similar situation arises when describing the two lone pairs on the carbonyl oxygen atom of a ketone. However, the question of whether it is conceptually useful to derive equivalent orbitals from symmetry-adapted ones, from the standpoint of bonding theory and pedagogy, is still a controversial one, with recent (2014 and 2015) articles opposing and supporting the practice.

Implicit theories of intelligence

From Wikipedia, the free encyclopedia

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

In social and developmental psychology, an individual's implicit theory of intelligence refers to his or her fundamental underlying beliefs regarding whether or not intelligence or abilities can change, developed by Carol Dweck and colleagues.

History

Ellen Leggett developed implicit theories of intelligence in 1985. Her paper "Children's entity and incremental theories of intelligence: Relationships to achievement behavior" was presented at the 1985 meeting of the Eastern Psychological Association in Boston. As a result, Dweck and her collaborators began studying how individuals unknowingly (or implicitly) assess their own intelligence and abilities through interaction and interpretation of their environment. It was assumed that these assessments ultimately influenced the individual's goals, motivations, behaviors, and self-esteem. The researchers began by looking at students who were highly motivated to achieve, and students who were not, though the levels of self-achievement were not clarified. They noticed that the highly motivated students thrived in the face of challenge while the other students quit or withdrew from their work, but critically, a student's raw intelligence did not predict whether a student was highly motivated or not. Rather, they discovered that these two groups of students held different beliefs (or implicit theories) about intelligence, categorized as entity or incremental theories, which affected their classroom performance.

Entity theory vs. incremental theory

Carol Dweck identified two different mindsets regarding intelligence beliefs. The entity theory of intelligence refers to an individual's belief that abilities are fixed traits. For entity theorists, if perceived ability to perform a task is high, the perceived possibility for mastery is also high. In turn, if perceived ability is low, there is little perceived possibility of mastery, often regarded as an outlook of "learned helplessness" (Park & Kim, 2015). However, the incremental theory of intelligence proposes that intelligence and ability are malleable traits which can be improved upon through effort and hard work. For incremental theorists, there is a perceived possibility of mastery even when initial ability to perform a task is low. Those who subscribe to this theory of intelligence "don't necessarily believe that anyone can become an Einstein or Mozart, but they do understand even Einstein and Mozart had to put in years of effort to become who they were". This possibility of mastery contributes in part to intrinsic motivation of individuals to perform a task, since there is perceived potential for success in the task.

Individuals may fall on some spectrum between the two types of theories, and their views may change given different situations and training. By observing an individual's motivation and behavior towards achievement, an individual's general mindset regarding intelligence is revealed. About 40% of the general population believe the entity theory, 40% believe the incremental theory, and 20% do not fit well into either category.

Performance level on a task is not always predetermined by an individual's mindset. Previous research on the subject has shown that when faced with failure on an initial task, those with an entity theory mindset will perform worse on subsequent tasks that measure the same ability than those with an incremental theory mindset (Park & Kim, 2015). However, a 2015 research study published in the Personality and Social Psychology Bulletin found that when the subsequent task measured a different ability, entity theorists performed better than incremental theorists. In certain situations, the incremental theorists studied were self-critical about the previous failure; these thoughts disrupted their performance on the subsequent task. Incremental theorists' reactions to failure are traditionally seen as an "adaptive response", meaning they link the failure to insufficient effort and therefore search for ways to improve their performance. If there is no opportunity for improvement on the task, such as in the research study, thoughts of doubt about the failure affect future performance (Park & Kim, 2015).

For the individuals who believed in an entity theory of intelligence, there were no such feelings of doubt when performing the second task because they perceived the task as not measuring the ability that they lacked on the initial task. After the first failure, self-critical thoughts are less likely to linger in their minds while performing the second task; the study points out that "entity theorists will not necessarily feel helpless because the second task does not measure the ability they think they lack" (Park & Kim 2015). Therefore, in this study, entity theorists performed better on the subsequent task than did incremental theorists if the measured ability was different.

Motivation toward achievement

Different types of goals

An individual's motivation towards achievement is shaped by their implicit theory of intelligence (and their related implicit theories about domain-specific aptitudes) and its associated goals. J.G. Nicholls proposed two different types of goals related to achievement. Task involvement goals involve individuals aiming to improve their own abilities. Ego involvement goals involve individuals wanting to better themselves compared to others. Dweck modified Nicholls' ideas by proposing performance goals and mastery goals. Performance goals are associated with entity theory and lead individuals to perform actions in order to appear capable and avoid negative judgments about their skills. Mastery goals are associated with incremental theory and lead individuals to engage and work in order to gain expertise in new things.

Response to challenge

Individuals who believe they have the ability to grow and add to their knowledge gladly accept challenges that can aid in growth towards mastery. Individuals who believe their abilities are fixed will also accept and persist through challenges as long as they feel they will succeed and their abilities will not be questioned. However, when these individuals lack confidence in their abilities, they will avoid, procrastinate, or possibly cheat in challenging situations that might make them appear incompetent. These behaviors can lead to a sense of learned helplessness and stymied intellectual growth.

Attribution of failure

Attribution of failure and coping with that failure are highly related to mindset. Individuals who subscribe to an incremental view will attribute a failure to not yet having learned something, looking at something from the incorrect perspective, or not working hard enough. All of these problems can be corrected through effort, leading incrementalist individuals to continually seek any situation that will intellectually better themselves. Also they are more likely to engage in remedial action to correct mistakes if necessary. Those with fixed intelligence views attribute failure to their own lack of ability.

Self-regulated learning

Individuals with an incremental mindset will take feedback and channel that into determination to try new strategies for solving a given problem, a large part of self-regulated learning (or learning to effectively guide your own studies). As a result, incrementalist individuals are more effective at self-regulated learning, ultimately leading them to be more productive at developing plans for learning and making connections between topics which promotes deeper processing of information.

Self-esteem

Incrementalist individuals generally have positive and stable self-esteem and do not question their intelligence in the face of failure, instead remaining eager and curious. Individuals with entity beliefs mostly attribute failure or having to exert effort to a lack of ability. Therefore, if they do not succeed at some task, they are unlikely to seek similar tasks or will quit trying. They believe that putting in effort will undermine their competence because if they were smart enough to begin with, they would not need to put in effort. These individuals will limit themselves to situations where they believe they will succeed and may limit themselves in the face of negative feedback, which they will likely interpret as a personal attack on their ability. These individuals' self-esteem as well as their enjoyment of a task may suffer when they encounter failure and the associated feelings of helplessness. Many children who see failure as a reflection of their intelligence will even lie about their scores to strangers to preserve their self-esteem and competence, since they connect their judgments of self to their performance. Students who see the value of effort do not show such a tendency. The majority of what are considered "best students" are often concerned with failure. Students who achieve a great deal of academic success early on might be most likely to believe their intelligence is fixed because they so frequently have been praised regarding their intelligence. They may have faced fewer opportunities for setbacks and do not have much experience persisting through errors. Longitudinal research shows that individuals who endorse entity beliefs experience decreasing self-esteem throughout their college years, while individuals who endorse incremental beliefs experience an increase.

Development

Implicit theories of intelligence develop at an early age and are subtly influenced by parents and educators and the type of praise they give for successful work. Typically it has been assumed that any sort of praise will have a positive impact on a child's self-confidence and achievement. However, different types of praise can lead to the development of different views on intelligence. Young children who hear praise that values high intelligence as a measure of success, such as "You must be smart at these problems," may link failure with a lack of intelligence and are more susceptible to developing an entity mindset. Often children are given high praise for their intelligence after relatively easy success, which sets them up to develop counterproductive behaviors in dealing with academic setbacks, rather than fostering confidence and the enjoyment of learning. Praise for intelligence connects performance with ability, rather than effort, leading these individuals to develop "performance" goals to prove competence. However students who receive praise valuing hard work as a measure of success, such as "You must have worked hard at these problems," more often pursue mastery goals that underlie an incremental mindset.

Subtle differences in speech to children that promote non-generic praise (i.e. "You did a good job drawing") versus generic praise (i.e. "You are a good drawer"), lead children to respond to later criticism in a way that demonstrates an incremental mindset.

Shifting from entity to incremental mindset to improve achievement

Understanding differences between those who believe in entity theory versus incremental theory allows educators to predict how students will persevere in a classroom. Then, educators can change behaviors that may contribute to academic shortcomings for those with entity tendencies and low confidence in their abilities. While these implicit beliefs regarding where intelligence comes from are relatively stable across time and permeate all aspects of behavior, it is possible to change peoples' perspectives on their abilities for a given task with the right priming. Dweck's 2006 book Mindset: How You Can Fill Your Potential focuses on teaching individuals how they can encourage thinking with a growth mindset for a happier and more successful existence.

Elementary-aged students

Given the opportunity for fifth graders to choose their own learning tasks, when primed towards gaining skills, they will pick challenging tasks. When they are primed towards assessment, they will pick tasks that they think they will be successful at to show off their abilities. Thus, they will forgo new learning if it means the possibility of making mistakes. If the situation is framed in a manner that emphasizes learning and process rather than success, mindset can be altered.

Middle school-aged children

Transitioning between elementary school and middle school is a time when many students with an entity theory of intelligence begin to experience their first taste of academic difficulty. Transitioning students with low abilities can be oriented to a growth mentality when taught that their brains are like muscles that get stronger through hard work and effort. This lesson can result in a marked improvement in grades compared to students with similar abilities and resources available to them who do not receive this information on the brain.

College-aged students

One consequence for individuals experiencing the stereotype threat (or worrying about conforming to a negative stereotype associated with a member of one's group) is that they will also experience an entity mindset. College students are able to overcome this negative impact after participating in an incremental thinking intervention, afterwards reporting higher levels of happiness according to the theory.

Predictive power of knowing an individual's theory

Success in school and on tests

An individual's implicit theory of intelligence can predict future success, particularly navigating life transitions that are often associated with challenging situations, such as moving from elementary to middle school. Students followed throughout their middle school careers showed that those who possessed growth mindset tendencies made better grades and had a more positive view on the role of effort than students who possessed fixed mindset tendencies with similar abilities, two years following the initial survey. Those with theoretical entity beliefs worry more about tests even in situations where they have experienced some success, spend less time practicing before tests, and thus have shown reduced performance on IQ tests relative to others in their environment. If the situation is framed in a manner that emphasizes learning and process rather than success, mindset can be altered. Individuals with fixed mindsets may engage in less practice in order to allow themselves an excuse besides low ability for potentially poor performance in order to preserve their egos. Students who have learning goals (associated with incremental beliefs) are more internally motivated and successful in the face of a challenging college course. After the first test in a course, those who possess learning goals are likely to improve their grades on the next test whereas those with performance goals did not.

Negotiation skills

Incrementalist individuals tend to have stronger negotiating skills, believing that with effort a better deal can be reached. This finding may have implications for more favorable working conditions for those with incrementalist beliefs.

"Easily learned = easily remembered" heuristic

According to theory, individuals who believe their intelligence can grow think about information in their world differently even outside of academic challenges, seen by use of a different heuristic when making judgments of learning (JOLs), or estimates of learning. Those with entity views are generally guided by the principle "easily learned" means "easily remembered," which means that when learning information, these individuals will make low JOLs when a task is difficult. Those with incremental views did not follow this "easily learned" means "easily remembered" principle and gave higher judgments to more difficult tasks, perhaps believing that if more effort is put into the learning because it is harder, those items will be better remembered.

Other behaviors governed by implicit theories

Research into implicit theories of intelligence has led to additional discoveries expanding the idea of entity vs incremental mindset to other areas outside of just intelligence. Views about intelligence are just a single manifestation of a more general entity or incremental mindset which reveals a great deal about a person's view of the world and self. Generally those with entity views will see all characteristics, in addition to intelligence, as innate and static while those with incremental views see characteristics as malleable. Entity beliefs lead to more stereotyping, greater rigidity in prejudiced beliefs, and difficulties during conflict resolution. Incremental views are connected to more open beliefs and amenability during conflict resolutions. In intimate relationships, those who possess an incremental mindset tend to believe that people can change and exhibit more forgiveness than those with entity mindsets. Similarly, those with entity beliefs are likely to endorse the fundamental attribution error more than those with incremental views, who tend to focus much more on the situation than internal characteristics of an individual.