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{{Even polygon db|Even polygon stat table|p10}}
A '''random coil''' is a [[polymer]] [[Chemical structure|conformation]] where the [[monomer]] subunits are oriented [[randomness|randomly]] while still being [[chemical bond|bonded]] to [[graph (mathematics)|adjacent]] units. It is not one specific [[shape]], but a [[statistics|statistical]] distribution of shapes for all the chains in a [[statistical population|population]] of [[macromolecule]]s. The conformation's name is derived from the idea that, in the absence of specific, stabilizing interactions, a polymer backbone will "sample" all possible conformations randomly. Many linear, [[Branching (polymer chemistry)|unbranched]] [[homopolymer]]s — in solution, or above their [[glass transition temperature|melting temperature]]s — assume ([[approximation|approximate]]) random coils. Even [[copolymer]]s with [[monomers]] of unequal [[length]] will distribute in random coils if the subunits lack any specific interactions. The parts of branched polymers may also assume random coils.
[[File:Gonbad-e Qabus.JPG|thumb|[[Gonbad-e Qabus (tower)|Gonbad-e Qabus]], the tallest pure brick tower in the world, is built on a decagonal plan.]]
In [[geometry]], a '''decagon''' is any [[polygon]] with ten sides and ten [[angle]]s. A [[regular polygon|regular]] decagon has all sides of equal length and each internal angle equal to 144°. Its [[Schläfli symbol]] is {10}.


Below their melting temperatures, most [[thermoplastic]] polymers ([[polyethylene]], [[nylon]], etc.) have [[amorphous solid|amorphous]] regions in which the chains approximate random coils, alternating with regions that are [[crystal]]line. The amorphous regions contribute [[elasticity (physics)|elasticity]] and the crystalline regions contribute strength and [[Stiffness|rigidity]].
==Regular decagon==
The [[area]] of a regular decagon is: (with ''t'' = edge length)
:<math>A = \frac{5}{2}t^2 \cot \frac{\pi}{10} = \frac{5t^2}{2} \sqrt{5+2\sqrt{5}} \simeq 7.694208843 t^2.</math>


More complex polymers such as [[protein]]s, with various interacting chemical groups attached to their backbones, [[Molecular self-assembly|self-assemble]] into well-defined structures. But segments of proteins, and [[peptide|polypeptides]] that lack [[secondary structure]], are often assumed to exhibit a random-coil conformation in which the only fixed relationship is the joining of adjacent [[amino acid]] [[residue (chemistry)|residue]]s by a [[peptide bond]]. This is not actually the case, since the [[statistical ensemble (mathematical physics)|ensemble]] will be [[energy]] weighted due to interactions between amino acid [[Side chain|side-chains]], with lower-energy conformations being present more frequently. In addition, even arbitrary sequences of amino acids tend to exhibit some [[hydrogen bond]]ing and secondary structure. For this reason, the term "statistical coil" is occasionally preferred. The [[conformational entropy]] associated with the random-coil state significantly contributes to its energetic stabilization and accounts for much of the energy barrier to [[protein folding]].  
An alternative formula is <math>\scriptstyle A\,=\,2.5dt</math> where ''d'' is the distance between parallel sides, or the height when the decagon stands on one side as base.<br>
By simple trigonometry <math>\scriptstyle d\,=\,2t(\cos{54^\circ}\,+\,\cos{18^\circ})</math>.


A random-coil conformation can be detected using spectroscopic techniques. The arrangement of the planar amide bonds results in a distinctive signal in [[circular dichroism]]. The [[chemical shift]] of amino acids in a random-coil conformation is well known in [[Protein NMR|nuclear magnetic resonance]] (NMR). Deviations from these signatures often indicates the presence of some secondary structure, rather than complete random coil. Furthermore, there are signals in multidimensional NMR experiments that indicate that stable, non-local amino acid interactions are absent for polypeptides in a random-coil conformation. Likewise, in the images produced by [[X-ray crystallography|crystallography]] experiments, segments of random coil result simply in a reduction in "electron density" or contrast. A randomly coiled state for any polypeptide chain can be attained by [[denaturation (biochemistry)|denaturing]] the system. However, there is evidence that proteins are never truly random coils, even when denatured (Shortle & Ackerman).
==Sides==
The side of a regular decagon inscribed in a unit circle is <math>\tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\phi}</math>, where ''&#x03D5;'' is the [[golden ratio]], <math>\tfrac{1+\sqrt{5}}{2}</math>.


==Random walk model: The Gaussian chain==
===Construction===
{{main|Ideal chain}}
A regular decagon is [[constructible polygon|constructible]] using [[compass and straightedge]]:
[[Image:Ideal chain random walk.png|thumb|200px|Short [[ideal chain|random chain]]]]


There are an enormous number of different [[Ludwig Boltzmann|ways]] in which a chain can be curled around in a relatively compact shape, like an unraveling ball of twine with lots of open [[space]], and comparatively few ways it can be more or less stretched out. So, if each conformation has an equal [[probability]] or [[statistics|statistical]] weight, chains are much more likely to be ball-like than they are to be extended — a purely [[entropy|entropic]] effect. In an [[statistical ensemble (mathematical physics)|ensemble]] of chains, most of them will, therefore, be loosely [[sphere|balled up]]. This is the kind of shape any one of them will have most of the time.
[[File:Regular Decagon Inscribed in a Circle.gif|Construction of a regular decagon]]


Consider a linear polymer to be a freely-jointed chain with ''N'' subunits, each of length <math>\scriptstyle\ell</math>, that occupy [[0 (number)|zero]] [[volume]], so that no part of the chain excludes another from any location. One can regard the segments of each such chain in an ensemble as performing a [[random walk]] (or "random flight") in three [[dimension]]s, limited only by the constraint that each segment must be joined to its neighbors. This is the ''[[ideal chain]]'' [[mathematical model]]. It is clear that the maximum, fully extended length ''L'' of the chain is <math>\scriptstyle N\,\times\,\ell</math>. If we assume that each possible chain conformation has an equal statistical weight, it can be [[ideal chain|shown]] that the probability ''P''(''r'') of a polymer chain in the [[statistical population|population]] to have distance ''r'' between the ends will obey a characteristic [[Probability distribution|distribution]] described by the formula
An alternative (but similar) method is as follows:
#Construct a pentagon in a circle by one of the methods shown in [[Pentagon#Construction_of_a_regular_pentagon|constructing a pentagon]].
#Extend a line from each vertex of the pentagon through the center of the [[circle]] to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon.
#The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.


: <math>P(r) = \frac{4 \pi r^2}{(2/3\; \pi \langle r^2\rangle)^{3/2}} \;e^{-\,\frac{3r^2}{2\langle r^2\rangle}}</math>
==Related figures==
There is one regular [[star polygon]], the [[decagram (geometry)|decagram]] {10/3}, using the same points, but connecting every third points. There are also two compounds: {10/4} is reduced to 2{5/2} as two [[pentagram]]s, and {10/2} is reduced to 2{5} as two [[pentagon]]s.


The ''average'' ([[root mean square]]) end-to-end distance for the chain, <math>\scriptstyle \sqrt{\langle r^2\rangle}</math>, turns out to be <math>\scriptstyle\ell</math> times the square root of&nbsp;''N'' &mdash; in other words, the average distance scales with ''N''<sup>0.5</sup>.
{| class=wikitable width=360
|- align=center
|[[File:Truncated pentagon.png|120px]]<BR>A [[truncation (geometry)|truncated]] regular pentagon
|[[File:Decagram_10_3.png|120px]]<br>{10/3}<BR>[[Decagram (geometry)|Decagram]]
|[[Image:Decagram 10 2.png|120px]]<br>{10/2} or 2{5}
|[[Image:Decagram 10 4.png|120px]]<br>{10/4} or 2{5/2}
|}


Note that although this model is termed a "Gaussian chain", the distribution function is not a [[normal distribution|gaussian (normal) distribution]].  The end-to-end distance probability distribution function of a Gaussian chain is non-zero only for ''r''&nbsp;>&nbsp;0.
===Petrie polygons===
<ref>In fact, the Gaussian chain's distribution function is also unphysical for real chains, because it has a non-zero probability for lengths that are larger than the extended chain.  This comes from the fact that, in strict terms, the formula is only valid for the limiting case of an infinite long chain.  However, it is not problematic since the probabilities are very small.</ref>
The regular decagon is the [[Petrie polygon]] for many higher dimensional polytopes, shown in these skew [[orthogonal projection]]s in various [[Coxeter plane]]s:


==Real polymers==
{| class=wikitable width=450
 
|- align=center valign=top
A real polymer is not freely-jointed. A -C-C- single [[chemical bond|bond]] has a fixed [[alkane#Molecular geometry|tetrahedral]] angle of 109.5 degrees. The value of ''L'' is well-defined for, say, a fully extended [[polyethylene]] or [[nylon]], but it is less than ''N''&nbsp;x&nbsp;''l'' because of the zig-zag backbone. There is, however, free rotation about many chain bonds. The model above can be enhanced. A longer, "effective" unit length can be defined such that the chain can be regarded as freely-jointed, along with a smaller ''N'', such that the constraint ''L''&nbsp;=&nbsp;''N''&nbsp;x&nbsp;''l'' is still obeyed. It, too, gives a Gaussian distribution. However, specific cases can also be precisely calculated. The average end-to-end distance for ''freely-rotating'' (not freely-jointed) polymethylene (polyethylene with each -C-C- considered as a subunit) is ''l'' times the square root of 2''N'', an increase by a factor of about 1.4. Unlike the zero volume assumed in a random walk calculation, all real polymers' segments occupy space because of the [[van der Waals radius|van der Waals radii]] of their atoms, including [[steric effects|bulky substituent groups]] that interfere with [[molecular geometry|bond rotations]]. This can also be taken into account in calculations. All such effects increase the mean end-to-end distance.
!valign=center|A<sub>9</sub>
 
|[[File:9-simplex_t0.svg|100px]]<br>[[9-simplex]]
Because their polymerization is [[stochastic]]ally driven, chain lengths in any real population of [[chemical synthesis|synthetic]] polymers will obey a statistical distribution. In that case, we should take ''N'' to be an average value. Also, many polymers have random branching.
|[[File:9-simplex_t1.svg|100px]]<br>[[Rectified 9-simplex]]
 
|[[File:9-simplex_t2.svg|100px]]<br>[[Birectified 9-simplex]]
Even with corrections for local constraints, the random walk model ignores steric interference between chains, and between distal parts of the same chain. A chain often cannot move from a given conformation to a closely related one by a small displacement because one part of it would have to pass through another part, or through a neighbor. We may still hope that the ideal-chain, random-coil model will be at least a qualitative indication of the shapes and [[dimension]]s of real polymers in [[solution]], and in the amorphous state, as long as there are only weak [[intermolecular force|physicochemical interactions]] between the monomers. This model, and the [[Flory-Huggins Solution Theory]],<ref>Flory, P.J. (1953) ''Principles of Polymer Chemistry'', Cornell Univ. Press, ISBN 0-8014-0134-8</ref><ref>Flory, P.J. (1969) ''Statistical Mechanics of Chain Molecules'', Wiley, ISBN 0-470-26495-0; reissued 1989, ISBN 1-56990-019-1</ref> for which [[Paul Flory]] received the [[Nobel Prize in Chemistry]] in 1974, ostensibly apply only to [[ideal solution|ideal, dilute solutions]]. But there is reason to believe (e.g., [[neutron diffraction]] studies) that [[steric effects|excluded volume effects]] may cancel out, so that, under certain conditions, chain dimensions in amorphous polymers have approximately the ideal, calculated size <ref>"Conformations, Solutions, and Molecular Weight" from "Polymer Science & Technology"  courtesy of Prentice Hall Professional publications [http://www.informit.com/content/images/chap3_0130181684/elementLinks/chap3_0130181684.pdf]</ref>
|[[File:9-simplex_t3.svg|100px]]<br>[[Trirectified 9-simplex]]
When separate chains interact cooperatively, as in forming crystalline regions in [[solid]] thermoplastics, a different mathematical approach must be used.
|[[File:9-simplex_t4.svg|100px]]<br>[[Quadrirectified 9-simplex]]
 
|- align=center valign=top
Stiffer polymers such as [[alpha helix|helical]] polypeptides, [[Kevlar]], and double-stranded [[DNA]] can be treated by the [[worm-like chain]] model.
!valign=center|BC<sub>5</sub>
|[[File:5-cube_t4.svg|100px]]<br>[[5-orthoplex]]
|[[File:5-cube_t3.svg|100px]]<br>[[Rectified 5-orthoplex]]
|[[File:5-cube_t2.svg|100px]]<br>[[Birectified 5-cube]]
|[[File:5-cube_t1.svg|100px]]<br>[[Rectified 5-cube]]
|[[File:5-cube_t0.svg|100px]]<br>[[5-cube]]
|- align=center valign=top
!valign=center|D<sub>6</sub>
|[[File:6-cube_t5_B5.svg|100px]]<br>[[5-orthoplex|t<sub>1</sub>(4<sub>31</sub>)]]
|[[File:6-cube_t4_B5.svg|100px]]<br>[[Rectified 5-orthoplex|t<sub>3</sub>(1<sub>31</sub>)]]
|[[File:6-cube_t3_B5.svg|100px]]<br>[[Birectified 5-orthoplex|t<sub>2</sub>(1<sub>31</sub>)]]
|[[File:6-demicube_t1_D6.svg|100px]]<br>[[Rectified 6-demicube|t<sub>1</sub>(1<sub>31</sub>)]]
|[[File:6-demicube_t0_D6.svg|100px]]<br>[[6-demicube]]<br>(1<sub>31</sub>)
|- align=center valign=top
!valign=center|H<sub>3</sub>
|[[File:Dodecahedron petrie.png|100px]]<br>[[Dodecahedron]]
|[[File:Icosahedron petrie.png|100px]]<br>[[Icosahedron]]
|[[File:Dodecahedron t1 H3.png|100px]]<br>[[Icosidodecahedron]]
|}


==See also==
==See also==
*[[protein folding]]
*[[decagonal number]]
*[[native state]]
*[[Gambrel]]
*[[molten globule]]
*[[Golden ratio]]
*[[probability theory]]


==References==
==External links==
{{Reflist}}
*{{MathWorld |urlname=Decagon |title=Decagon}}
*[http://www.mathopenref.com/decagon.html Definition and properties of a decagon] With interactive animation


==External links==
{{Polygons}}
*[http://arjournals.annualreviews.org/doi/pdf/10.1146/annurev.pc.25.100174.001143 polymer statistical mechanics]
*[http://www.iop.org/EJ/abstract/0305-4470/20/12/040/ A topological problem in polymer physics: configurational and mechanical properties of a random walk enclosing a constant are]
*[http://www.sciencemag.org/cgi/content/abstract/293/5529/487?view=abstract D. Shortle and M. Ackerman, Persistence of native-like topology in a denatured protein in 8 M urea, Science 293 (2001), pp. 487–489]
*[http://phptr.com/content/images/chap3_0130181684/elementLinks/chap3_0130181684.pdf Sample chapter "Conformations, Solutions, and Molecular Weight" from "Polymer Science & Technology"  courtesy of Prentice Hall Professional publications]


{{DEFAULTSORT:Random Coil}}
[[Category:Polygons]]
[[Category:Polymer physics]]
[[Category:Physical chemistry]]

Revision as of 10:41, 11 August 2014

My name is Madelaine and I am studying Industrial and Labor Relations and Continuing Education and Summer Sessions at Kleindorf Ii / Austria.

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Gonbad-e Qabus, the tallest pure brick tower in the world, is built on a decagonal plan.

In geometry, a decagon is any polygon with ten sides and ten angles. A regular decagon has all sides of equal length and each internal angle equal to 144°. Its Schläfli symbol is {10}.

Regular decagon

The area of a regular decagon is: (with t = edge length)

A=52t2cot⁡π10=5t225+25≃7.694208843t2.

An alternative formula is A=2.5dt where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry d=2t(cos⁡54∘+cos⁡18∘).

Sides

The side of a regular decagon inscribed in a unit circle is −1+52=1ϕ, where ϕ is the golden ratio, 1+52.

Construction

A regular decagon is constructible using compass and straightedge:

Construction of a regular decagon

An alternative (but similar) method is as follows:

  1. Construct a pentagon in a circle by one of the methods shown in constructing a pentagon.
  2. Extend a line from each vertex of the pentagon through the center of the circle to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon.
  3. The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.

There is one regular star polygon, the decagram {10/3}, using the same points, but connecting every third points. There are also two compounds: {10/4} is reduced to 2{5/2} as two pentagrams, and {10/2} is reduced to 2{5} as two pentagons.


A truncated regular pentagon
{10/3}
Decagram
{10/2} or 2{5}
{10/4} or 2{5/2}

Petrie polygons

The regular decagon is the Petrie polygon for many higher dimensional polytopes, shown in these skew orthogonal projections in various Coxeter planes:

A9
9-simplex 		File:9-simplex t1.svg
Rectified 9-simplex 		File:9-simplex t2.svg
Birectified 9-simplex 		File:9-simplex t3.svg
Trirectified 9-simplex 		File:9-simplex t4.svg
Quadrirectified 9-simplex
BC5 File:5-cube t4.svg
5-orthoplex 		File:5-cube t3.svg
Rectified 5-orthoplex 		File:5-cube t2.svg
Birectified 5-cube 		File:5-cube t1.svg
Rectified 5-cube 		File:5-cube t0.svg
5-cube
D6 File:6-cube t5 B5.svg
t1(431) 		File:6-cube t4 B5.svg
t3(131) 		File:6-cube t3 B5.svg
t2(131) 		File:6-demicube t1 D6.svg
t1(131) 		File:6-demicube t0 D6.svg
6-demicube
(131)
H3 File:Dodecahedron petrie.png
Dodecahedron 		File:Icosahedron petrie.png
Icosahedron 		File:Dodecahedron t1 H3.png
Icosidodecahedron

See also



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  • Definition and properties of a decagon With interactive animation

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