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Discrete measure - Revision history
2024-03-28T19:42:00Z
Revision history for this page on the wiki
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https://en.formulasearchengine.com/index.php?title=Discrete_measure&diff=252452&oldid=prev
en>Brirush at 12:59, 24 August 2014
2014-08-24T12:59:13Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:59, 24 August 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In [[polymer science]], the ''</del>'<del style="font-weight: bold; text-decoration: none;">Lifson–Roig model''' is a [[helix-coil transition model]] applied to the [[alpha helix]]-[[random coil]] transition of [[polypeptide]]</del>s<del style="font-weight: bold; text-decoration: none;">;<ref name="Lifson">{{cite journal |author=Lifson S, Roig A |year=1961 |title=On the theory of helix-coil transition in polypeptides |journal=J Chem Phys |volume=34 |pages=1963–1974 |doi=10.1063/1.1731802 |issue=6}}</ref> it is a refinement of the [[Zimm-Bragg model]] that recognizes that a polypeptide [[alpha helix]] is only stabilized by a [[hydrogen bond]] only once three consecutive residues have adopted the helical conformation. To consider three consecutive residues each with two states (helix and coil), the Lifson–Roig model uses a 4x4 transfer matrix instead of the 2x2 transfer matrix of the Zimm-Bragg model, which considers only two consecutive residues. However, the simple nature of the coil state allows this to be reduced to a 3x3 matrix for most applications. </del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">They contact me Emilia. Minnesota is where he</ins>'s been <ins style="font-weight: bold; text-decoration: none;">residing </ins>for <ins style="font-weight: bold; text-decoration: none;">years</ins>. <ins style="font-weight: bold; text-decoration: none;">Managing people </ins>is <ins style="font-weight: bold; text-decoration: none;">what I do </ins>and the <ins style="font-weight: bold; text-decoration: none;">wage has been really fulfilling</ins>. <ins style="font-weight: bold; text-decoration: none;">Playing baseball </ins>is the <ins style="font-weight: bold; text-decoration: none;">hobby he will </ins>by <ins style="font-weight: bold; text-decoration: none;">no means quit doing</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Feel free to visit my web site</ins>: <ins style="font-weight: bold; text-decoration: none;">std testing </ins>at <ins style="font-weight: bold; text-decoration: none;">home </ins>([<ins style="font-weight: bold; text-decoration: none;">http</ins>://<ins style="font-weight: bold; text-decoration: none;">www</ins>.<ins style="font-weight: bold; text-decoration: none;">teenvalley</ins>.<ins style="font-weight: bold; text-decoration: none;">net</ins>/<ins style="font-weight: bold; text-decoration: none;">user</ins>/<ins style="font-weight: bold; text-decoration: none;">RLay please click </ins>the <ins style="font-weight: bold; text-decoration: none;">following webpage</ins>])</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The Zimm-Bragg and Lifson–Roig models are but the first two in a series of analogous transfer-matrix methods in polymer science that have also </del>been <del style="font-weight: bold; text-decoration: none;">applied to [[nucleic acid]]s and branched polymers. The transfer-matrix approach is especially elegant </del>for <del style="font-weight: bold; text-decoration: none;">homopolymers, since the statistical mechanics may be solved exactly using a simple [[eigenanalysis]]</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Parameterization==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The Lifson–Roig model </del>is <del style="font-weight: bold; text-decoration: none;">characterized by three parameters: the [[statistical weight]] for [[nucleation|nucleating]] a helix, the weight for propagating a helix </del>and the <del style="font-weight: bold; text-decoration: none;">weight for forming a hydrogen bond, which is granted only if three consecutive residues are in a helical state</del>. <del style="font-weight: bold; text-decoration: none;">Weights are assigned at each position in a polymer as a function of the conformation of the residue in that position and as a function of its two neighbors. A statistical weight of 1 </del>is <del style="font-weight: bold; text-decoration: none;">assigned to </del>the <del style="font-weight: bold; text-decoration: none;">"reference state" of a coil unit whose neighbors are both coils, and a "nucleation" unit is defined (somewhat arbitrarily) as two consecutive helical units neighbored </del>by <del style="font-weight: bold; text-decoration: none;">a coil. A major modification of the original Lifson–Roig model introduces "capping" parameters for the helical termini, in which the N- and C-terminal capping weights may vary independently</del>.<<del style="font-weight: bold; text-decoration: none;">ref name="Doig"</del>><del style="font-weight: bold; text-decoration: none;">{{cite journal |doi=10.1002/pro.5560040708 |author=Doig AJ, Baldwin RL |year=1995 |title=N- and C-capping preferences for all 20 amino acids in alpha-helical peptides |journal=Protein Sci |volume=4 |issue=7 |pages=1325–1336 |pmid=7670375 |pmc=2143170}}</del><<del style="font-weight: bold; text-decoration: none;">/ref</del>> <del style="font-weight: bold; text-decoration: none;">The correlation matrix for this modification can be represented as a matrix M, reflecting the statistical weights of the helix state ''h'' and coil state ''c''. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{| class="wikitable" style="text-align</del>:<del style="font-weight: bold; text-decoration: none;">center" align="right"</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|-</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">! ''M'' !! hh !! hc !! ch !! cc</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|-</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">! hh</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| w || v || 0 || 0</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">! hc</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| 0 || 0 || <math>\sqrt{nc}</math> || c</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|-</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">! ch</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| v || v || 0 || 0</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|-</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">! cc</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| 0 || 0 || n || 1</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The Lifson–Roig model may be solved by the [[transfer-matrix method]] using the transfer matrix '''M''' shown </del>at <del style="font-weight: bold; text-decoration: none;">the right, where ''w'' is the [[statistical weight]] for helix propagation, ''v'' for initiation, ''n'' for N-terminal capping, and ''c'' for C-terminal capping. </del>(<del style="font-weight: bold; text-decoration: none;">In the traditional model ''n'' and ''c'' are equal to 1.) The </del>[<del style="font-weight: bold; text-decoration: none;">[partition function (statistical mechanics)|partition function]] for the helix-coil transition equilibrium is </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del style="font-weight: bold; text-decoration: none;"><math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Z = V \left(\prod_{i=0}^{N+1} M(i)\right)\tilde{V}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><</del>/<del style="font-weight: bold; text-decoration: none;">math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">where ''V'' is the end [[coordinate vector|vector]] <math>V=[0 0 0 1]<</del>/<del style="font-weight: bold; text-decoration: none;">math>, arranged to ensure the coil state of the first and last residues in the polymer</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">This strategy for parameterizing helix-coil transitions was originally developed for [[alpha helix|alpha helices]], whose [[hydrogen bond]]s occur between residues ''i'' and ''i+4''; however, it is straightforward to extend the model to [[310 helix|3<sub>10</sub> helices]] and [[pi helix|pi helices]], with ''i+3'' and ''i+5'' hydrogen bonding patterns respectively</del>. <del style="font-weight: bold; text-decoration: none;">The complete alpha/3<sub>10</sub></del>/<del style="font-weight: bold; text-decoration: none;">pi transfer matrix includes weights for transitions between helix types as well as between helix and coil states. However, because 3<sub>10<</del>/<del style="font-weight: bold; text-decoration: none;">sub> helices are much more common in </del>the <del style="font-weight: bold; text-decoration: none;">[[tertiary structure</del>]<del style="font-weight: bold; text-decoration: none;">]s of proteins than pi helices, extension of the Lifson–Roig model to accommodate 3<sub>10</sub> helices - resulting in a 9x9 transfer matrix when capping is included - has found a greater range of application.<ref name="Rohl">{{cite journal |author=Rohl CA, Doig AJ |year=1996 |title=Models for the 3(10)-helix/coil, pi-helix/coil, and alpha-helix/3(10</del>)<del style="font-weight: bold; text-decoration: none;">-helix/coil transitions in isolated peptides |journal=Protein Sci |pmid=8844857 |volume=5 |issue=8 |pmc=2143481 |pages=1687–1696 |doi=10.1002/pro.5560050822}}</ref> Analogous extensions of the Zimm-Bragg model have been put forth but have not accommodated mixed helical conformations.<ref name="Sheinerman">{{cite journal |doi=10.1021/ja00145a022 |author=Sheinerman FB, Brooks CL |year=1995 |title=310 helices in peptides and proteins as studied by modified Zimm-Bragg theory |journal=J Am Chem Soc |volume=117 |issue=40 |pages=10098–10103}}</ref></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==References==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{Reflist}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Lifson-Roig model}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Polymer physics]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Protein structure]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Statistical mechanics]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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