Conserved sequence and Linear independence: Difference between pages

Latest revision as of 20:16, 12 March 2015

Linearly independent vectors in R³.

Linearly dependent vectors in a plane in R³.

Linear independence is a concept from linear algebra. It is used to talk about vector spaces. Each vector space has a null vector. This vector is expressed as a linear combination (a sum) of other vectors. A set of these vectors is called linearly independent if and only if all of them are needed to express this null vector. This is equivalent to saying that at least one of the vectors can be expressed as a linear combination of the others. If the vectors are not linearly independent, they are called linearly dependent.

As an example, take the three-dimensional Euclidean space:

[math]\displaystyle{ \begin{matrix} \mbox{independent}\qquad\\ \underbrace{ \overbrace{ \begin{bmatrix}0\\0\\1\end{bmatrix}, \begin{bmatrix}0\\2\\-2\end{bmatrix}, \begin{bmatrix}1\\-2\\1\end{bmatrix} }, \begin{bmatrix}4\\2\\3\end{bmatrix} }\\ \mbox{dependent}\\ \end{matrix} }[/math]

Here the first three vectors are linearly independent; but the fourth vector equals nine times the first plus five times the second plus four times the third, so the four vectors together are linearly dependent. Linear dependence is a property of the family, not of any particular vector; for example in this case we could just as well write the first vector as a linear combination of the last three.

[math]\displaystyle{ \bold{v}_1 = \left(-\frac{5}{9}\right) \bold{v}_2 + \left(-\frac{4}{9}\right) \bold{v}_3 + \frac{1}{9} \bold{v}_4 . }[/math]

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-[[Image:Conserved residues.svg|thumb|Residues conserved among various [[G protein coupled receptor]]s are highlighted in green.]]
+[[File:Vec-indep.png|thumb|right|Linearly independent vectors in '''R'''<sup>3</sup>.]]
+[[File:Vec-dep.png|thumb|right|Linearly dependent vectors in a plane in '''R'''<sup>3</sup>.]]
+'''Linear independence''' is a concept from [[linear algebra]]. It is used to talk about [[vector space]]s. Each vector space has a [[null vector]]. This vector is expressed as a [[linear combination]] (a sum) of other vectors. A [[set]] of these vectors is called ''linearly independent'' if and only if all of them are needed to express this null vector. This is equivalent to saying that at least one of the vectors can be expressed as a linear combination of the others. If the vectors are not linearly independent, they are called ''linearly dependent''.
-'''Conserved sequences''' are similar or identical [[DNA sequencing|sequences]] which occur in [[DNA]], and cause sequences in [[RNA]], [[proteins]] and [[carbohydrates]].
+As an example, take the three-dimensional [[Euclidean space]]:
+:<math>
+\begin{matrix}
+\mbox{independent}\qquad\\
+\underbrace{
+  \overbrace{
+    \begin{bmatrix}0\\0\\1\end{bmatrix},
+    \begin{bmatrix}0\\2\\-2\end{bmatrix},
+    \begin{bmatrix}1\\-2\\1\end{bmatrix}
+  },
+  \begin{bmatrix}4\\2\\3\end{bmatrix}
+}\\
+\mbox{dependent}\\
+\end{matrix}
+</math>
+Here the first three vectors are linearly independent; but the fourth vector equals nine times the first plus five times the second plus four times the third, so the four vectors together are linearly dependent. Linear dependence is a property of the family, not of any particular vector; for example in this case we could just as well write the first vector as a linear combination of the last three.
+:<math>\bold{v}_1 = \left(-\frac{5}{9}\right) \bold{v}_2 + \left(-\frac{4}{9}\right) \bold{v}_3 + \frac{1}{9} \bold{v}_4 . </math>
-These sequences occur across species. This shows that the sequences has been maintained in evolution despite [[speciation]]. The further back up the [[phylogenetic tree]] a particular conserved sequence occurs, the more highly conserved it is. Since sequence information is normally transmitted from parents to progeny by [[gene]]s, a conserved sequence implies that there is a '''conserved gene'''.
+{{math-stub}}
+[[Category:Linear algebra]]
-Conservation of a sequence happens when [[mutations]] in a highly conserved region lead to non-viable life forms, that is, a form which is eliminated through [[natural selection]]. In other words, the product of the gene is vital to life, and its function is destroyed by almost all changes (mutations) to the sequence.
-== Conserved nucleic acid sequences ==
-The basic theory, widely agreed, is that highly conserved DNA sequences must have functional value, though the role for many of these highly conserved non-coding DNA sequences is not known. One recent study that eliminated four highly conserved non-coding DNA sequences in mice yielded viable mice with no significant phenotypic differences; the authors described their findings as "unexpected".<ref name="pmid17803355">{{cite journal | author = Ahituv N. Zhu Y. & Visel A. ''et al'' 2007. Deletion of ultraconserved elements yields viable mice | journal = PLoS Biol. | volume = 5 | issue = 9 | pages = e234 | pmid = 17803355 | doi = 10.1371/journal.pbio.0050234 | pmc = 1964772}}</ref> So there is clearly something here which is not understood.
-Many regions of the DNA, including highly conserved DNA sequences, consist of repeated sequence elements. If only one of a set of a repeated sequences was removed, and the repetitions were not needed, then no difference would be seen in the mice. The paper did not report whether the eliminated sequences were repeated sequences.
-==Conserved protein sequences and structures==
-Highly conserved proteins are often required for cells to work or divide.  Conservation of protein sequences is shown by the presence of identical [[amino acid]] residues at analogous parts of proteins.  Conservation of protein structures is indicated by the presence of functionally equivalent, though not necessarily identical, amino acid residues and structures between analogous parts of proteins.
-Shown below is an [[amino acid]] sequence alignment between two human [[zinc finger]] proteins. Conserved amino acid sequences are marked by strings of '''<math>\mathrm{*}</math>''' on the third line of the sequence alignment. As can be seen from this alignment, these two [[proteins]] contain a number of conserved amino acid sequences (represented by identical letters aligned between the two sequences).
-[[Image:Zinc-finger-seq-alignment2.png|frame|center]]
-== Comparative genomics ==
-The research field which studies the [[evolution]] and function of multigene families is called '''comparative genomics'''.<ref>Klug, William S. ''et al'' 2010. ''Concepts of genetics''. 10th ed, Pearson, p600. ISBN 0-321-79578-4</ref>
-== References ==
-{{Reflist}}
-[[Category:Genetics]]

Conserved sequence and Linear independence: Difference between pages

Latest revision as of 20:16, 12 March 2015

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