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| <a id="suppl"></a> | | <a id="suppl"></a> |
| <div class="texttitle">Supplementary</div> | | <div class="texttitle">Supplementary</div> |
− |
| + | <div > |
| <p>The strategy shown below is based on the gelation theory by Flory<sup>[ “Monomer Size Distribution Obtained by Condensing A-R-Bf-1 Monomers”. Chapter IX. Flory, Paul J. Principle of Polymer Chemistry. 1953.]<sup>. | | <p>The strategy shown below is based on the gelation theory by Flory<sup>[ “Monomer Size Distribution Obtained by Condensing A-R-Bf-1 Monomers”. Chapter IX. Flory, Paul J. Principle of Polymer Chemistry. 1953.]<sup>. |
| </p> | | </p> |
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| </p> | | </p> |
| | | |
− |
| + | <figure> |
− | <p>For the convenience of enumeration only, the cA ‘a’ groups and cB ‘b’ groups are considered distinguishable. It is because there is no intracellular reaction that the number of ‘ab’ pairs in the molecule is cA+cB-1. It is easy to prove that in the molecule there is cA*fa-cA-cB+1 unreacted ‘a’ and cB*fb-cA-cB+1 unreacted ‘b’. If we choose a free ‘a’ as the root of the entire molecule (or a free ‘b’, if there is no free ‘a’), ‘ab’ pairs can be classified into two types: from ‘a’ to ‘b’ and from ‘b’ to ‘a’. The following picture can help in understanding their difference. For the record, their numbers are cB and cA-1 respectively.</p> | + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/d/dc/T--Peking--images_md_fig16.png" target="_blank"><img style="width: 70% ;" src=" https://static.igem.org/mediawiki/2016/d/dc/T--Peking--images_md_fig16.png" alt=""/></a></p> |
| + | <figcaption style="text-align:left;"> |
| + | Fig. 1. An example of a molecule with no loop (or intracellular reaction). cA and cB are 4 and 3 for this example. |
| + | </figcaption> |
| + | </figure> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/8/89/Figs2.png" target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/8/89/Figs2.png" alt=""/></a></p> |
| + | <figcaption style="text-align:left;"> |
| + | Fig. 2. The molecule is now treated as tree. |
| + | </figcaption> |
| + | </figure> |
| + | <p> |
| + | For the convenience of enumeration only, the cA <b>a</b> groups and cB <b>b</b> groups are considered distinguishable. It is because there is no intracellular reaction that the number of <b>ab</b> pairs in the molecule is <b>cA+cB-1</b>. It is easy to prove that in the molecule there is <b>cA•f-cA-cB+1</b> unreacted <b>a</b> and <b>cB•f<sub>b</sub>-cA-cB+1</b> unreacted <b>b</b>. If we choose a free <b>a</b> as the root of the entire molecule (or a free <b>b</b>, if there is no free <b>a</b>), <b>ab</b> pairs can be classified into two types: from <b>a</b> to <b>b</b> and from <b>b</b> to <b>a</b>. The following picture can help in understanding their difference. For the record, their numbers are cB and <b>cA-1</b> respectively.</p> |
| | | |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/6/6f/T--Peking--images_md_fig17.png" target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/6/6f/T--Peking--images_md_fig17.png" alt=""/></a></p> |
| + | <figcaption style="text-align:left;"> |
| + | Fig. 2. The molecule is now treated as tree. |
| + | </figcaption> |
| + | </figure> |
| | | |
− | <p>Selecting a random ‘a’ from the system, it will have a chance of ‘Pa’ to be bonded by ‘b’. This ‘Pa’ can be directly derived from the reaction degree Pf, which will be described later. Here for each free ‘a’, the chance it is on a cA & cB configured molecule equals the probability that the particular sequence of cA-1 ‘b’ have reacted and the remaining cB*fb-cB-cA+1 ‘b’ have not, while cB ‘a’ have reacted and the remaining cA*fa-cA-cB have not (the root not included). This probability | + | <p>Selecting a random <b>a</b> from the system, it will have a chance of P<sub>a</sub> to be bonded by <b>b</b>. This P<sub>a</sub> can be directly derived from the reaction degree P<sub>f</sub>, which will be described later. Here for each free <b>a</b>, the chance it is on a cA & cB configured molecule equals the probability that the particular sequence of <b>cA-1</b> <b>b</b> have reacted and the remaining <b>cB•f<sub>b</sub>-cB-cA+1</b> <b>b</b> have not, while cB <b>a</b> have reacted and the remaining <b>cA•f<sub>a</sub>-cA-cB</b> have not (the root not included). This probability |
| + | </p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/f/fc/T--Peking--image_md_ThyEdit_O7.png" target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/f/fc/T--Peking--image_md_ThyEdit_O7.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | <p>is the same for each configuration with cA <b>A</b> and cB <b>B</b>. Hence the probability that any unreacted <b>a</b> group is on a cA & cB configured molecule of any structural configuration is |
| </p> | | </p> |
− | <p>is the same for each configuration with cA ‘A’ and cB ‘B’. Hence the probability that any unreacted ‘a’ group is on a cA & cB configured molecule of any structural configuration is | + | |
− | </p> | + | <figure> |
− | <p>where is the total number of configurations. This probability has the following physical picture: | + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/f/ff/T--Peking--image_md_ThyEdit_O8.png" target="_blank"><img style="width: 80% ;" src="https://static.igem.org/mediawiki/2016/f/ff/T--Peking--image_md_ThyEdit_O8.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | <p>where ω<sub>cA,cB</sub> is the total number of configurations. This probability has the following physical picture: |
| </p> | | </p> |
− | <p>To get the number of cA&cB molecules, it needs to be multiplied with | + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/7/74/T--Peking--images_md_fig20.png" target="_blank"><img style="width: 75% ;" src="https://static.igem.org/mediawiki/2016/7/74/T--Peking--images_md_fig20.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | <p>To get the number of cA & cB molecules, it needs to be multiplied with |
| </p> | | </p> |
− | <p>which equals. is the number of A monomers. | + | |
− | </p> | + | <figure> |
− | <p>In order to evaluate, all the ‘a’ and ‘b’ groups in the molecule has been assumed distinguishable. First of all, we need to select those ‘a’ and ‘b’ who form the ‘ab’ or ‘ba’ pairs. Considering each monomer should has at least one of its functional group bonded, the number of combinations is | + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/0/08/T--Peking--images_md_fig21.png" target="_blank"><img style="width: 60% ;" src="https://static.igem.org/mediawiki/2016/0/08/T--Peking--images_md_fig21.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | <p>which equals <a href="https://static.igem.org/mediawiki/2016/5/5b/T--Peking--images_md_fig22.png" target="_blank"><img src="https://static.igem.org/mediawiki/2016/5/5b/T--Peking--images_md_fig22.png" alt="" style="width: 30% ;"/></a>, N<sub>A</sub> is the number of <b>A</b> monomers.</p> |
| + | |
| + | <p>In order to evaluate ω<sub>cA,cB</sub>, all the <b>a</b> and <b>b</b> groups in the molecule has been assumed distinguishable. First of all, we need to select those <b>a</b> and <b>b</b> who form the <b>ab</b> or <b>ba</b> pairs. Considering each monomer should has at least one of its functional group bonded, the number of combinations is |
| </p> | | </p> |
| | | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/c/c7/T--Peking--image_md_ThyEdit_O9.png" target="_blank"><img style="width: 30% ;" src="https://static.igem.org/mediawiki/2016/c/c7/T--Peking--image_md_ThyEdit_O9.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| | | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="2016.igem.org/File:T--Peking--image_md_ThyEdit_O10.png" target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/6/69/T--Peking--images_md_fig24.png" |
| + | alt=""/></a></p> |
| + | <figcaption style="text-align:left;"> |
| + | Fig. 4. The functional groups on a monomer are classified into parent-connecting ones and branchers. |
| + | </figcaption> |
| + | </figure> |
| | | |
| + | <p>The definition of parent is the same as that one of tree, which means the monomer connects a neighbor that is the nearest one to the root. Select cB <b>a</b> from a<sub>1</sub> to a<sub>n</sub> shown in the picture. That cB is the number of <b>ab</b> and <b>ba</b> pairs (cA+cB-1) substrates the number of <b>a</b> connected to parents (cA-1), which equals the number of reacted <b>a</b> branchers. The same goes for <b>b</b>. Notice that all the parent-connecting <b>b</b> should be paired with branching <b>a</b> and all the parent connecting <b>a</b> should be paired with branching <b>b</b>. The number of combinations here is |
| + | <a href="2016.igem.org/File:T--Peking--image_md_ThyEdit_O10.png" target="_blank"><img style="width: 70% ;" src="2016.igem.org/File:T--Peking--image_md_ThyEdit_O10.png" |
| + | alt=""/></a> |
| | | |
− | <p>The definition of parent is the same as that one of tree, which means the monomer’s neighbor nearest to the root. Select cB ‘a’ from a1 to an shown in the picture. That is the number of ‘ab’ and ‘ba’ pairs substrates the number of ‘a’ connected to parents, which equals the number of reacted ‘a’ branchers. The same goes for ‘b’. Notice that all the parent connecting ‘b’ should be paired with branching ‘a’ and all the parent connecting ‘a’ should be paired with branching ‘b’. The number of combinations here is
| + | </p> |
− | </p> | + | <p>The worry of not that loops may occur in such combinations can be eliminated if the expression given above is understood like this: |
− | <p>The worry of not that loops may occur in such combinations can be eliminated if the expression given above is understood like this: | + | |
| </p> | | </p> |
| | | |
− |
| + | <figure> |
− | <p>In the first step, all the conditions of branching ‘a’ to parent-connecting ‘b’ are enumerated, that is. Then, before going to a more complex molecule, we simplify the confirmed structures into many monomers. This structure is quite similar to Flory’s description in “Monomer Size Distribution Obtained by Condensing A-R-Bf-1 Monomers” Chapter IX, Principle of Polymer Chemistry. Any molecule consists of monomers always have one free ‘a’. In our model, this free ‘a’ is the root. Since the root is not an option for any brancher ‘b’ to bond, it is clear that no loops will form in the next step of enumeration attaching parent-connecting ‘a’ to brancher ‘b’ , which connects those monomers into one molecule. | + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/5/5a/T--Peking--images_md_fig25.png" target="_blank"><img style="width: 85% ;" src="https://static.igem.org/mediawiki/2016/5/5a/T--Peking--images_md_fig25.png" alt=""/></a></p> |
| + | <figcaption style="text-align:left;"> |
| + | Fig. 5. No loop will occur. |
| + | </figcaption> |
| + | </figure> |
| + | |
| + | <p>In the first step, all the conditions of branching <b>a</b> to parent-connecting <b>b</b> are enumerated, that is cB!. Then, before going to a more complex molecule, we simplify the confirmed structures into many a-b<sub>x</sub> monomers. This structure is quite similar to Flory’s description in “Monomer Size Distribution Obtained by Condensing A-R-B<sub>f-1</sub>Monomers” Chapter IX, Principle of Polymer Chemistry<sup>1</sup>. Any molecule consists of only a-b<sub>x</sub> monomers always have one free <b>a</b>. In our model, this free <b>a</b> is the root. Since the root is not an option for any brancher <b>b</b> to bond, it is clear that no loops will form in the next step of enumeration of attaching parent-connecting <b>a</b> to brancher <b>b</b>, which connects those a-b<sub>x</sub> monomers into one molecule. |
| </p> | | </p> |
− | <p>Until now, the number all the configurations of distinguished cA ‘A’ and cB ‘B’ (their functional groups also distinguished) condensing into a molecule has been derived: | + | <p>Until now, the number all the configurations of distinguished cA <b>A</b> and cB <b>B</b> (their functional groups also distinguished) condensing into a molecule has been derived: |
| </p> | | </p> |
− | <p>The effect of distinguishing A and B should be compensated by dividing, while the fact that when counting each functional groups on for example an ‘A’ monomer, the binomial distribution itself requires the first discussed functional group be different from the second one, makes it reasonable to distinguish functional groups in a monomer. Finally, the total number of cA&cB configurations starts from a root ‘a’ is | + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/d/da/T--Peking--image_md_ThyEdit_O11.png |
| + | " target="_blank"><img style="width: 90% ;" src="https://static.igem.org/mediawiki/2016/d/da/T--Peking--image_md_ThyEdit_O11.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | <p>The effect of distinguishing <b>A</b> and <b>B</b> should be compensated by dividing cA!•cB!, while the fact that when counting each functional groups on for example an <b>A</b> monomer, the binomial distribution itself requires the first discussed functional group be different from the second one, makes it reasonable to distinguish functional groups in a monomer. Finally, the total number of cA & cB configurations starts from a root <b>a</b> is |
| + | </p> |
| + | <figure> |
| + | <p style="text-align:center;"> |
| + | <a href="https://static.igem.org/mediawiki/2016/0/07/T--Peking--images_md_fig29.png" target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/0/07/T--Peking--images_md_fig29.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | |
| + | <p>The number of cA & cB configured molecule is</p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/6/68/T--Peking--image_md_ThyEdit_O12.png |
| + | " target="_blank"><img style="width: 90% ;" src="https://static.igem.org/mediawiki/2016/6/68/T--Peking--image_md_ThyEdit_O12.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | |
| + | <p>Because P<sub>a</sub>f<sub>A</sub>N<sub>A</sub> equals the number of total <b>ab</b> pairs in the system, thus it equals P<sub>b</sub>f<sub>B</sub>N<sub>B</sub>, the equation given above has symmetric form, which means that for molecules who have no free <b>a</b>, eq. (4) can be derived by using a free <b>b</b> as the root. |
| </p> | | </p> |
− | <p>Because equals the number of total ‘ab’ pairs in the system, thus equals, the equation given above has symmetric form, which means that for molecules who have no free ’a’, eq (4) can be derived by using a free ‘b’ as the root.
| + | <p>Next, from the definition of the reaction degree P<sub>f</sub>, P<sub>a</sub> and P<sub>b</sub>: |
− | </p>
| + | |
− | <p>Next, from the definition of the reaction degree Pf, Pa and Pb: | + | |
| </p> | | </p> |
− | <p>where x is the total number of ‘ab’ pairs in the system. Pa and Pb can be expressed by Pf. Finally, the following equation shown in the main body is got: | + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/e/e3/T--Peking--images_md_fig30.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/e/e3/T--Peking--images_md_fig30.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | <p>The following equations are obtained:</p> |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/9/9a/T--Peking--images_md_fig31.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/9/9a/T--Peking--images_md_fig31.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | <p>where x is the total number of <b>ab</b> pairs in the system. P<sub>a</sub> and P<sub>b</sub> can be expressed using P<sub>f</sub>. Finally, the following equation shown in the main body is got: |
| </p> | | </p> |
| | | |
| + | <figure> |
| + | <p style="text-align:center;"><a href=" https://static.igem.org/mediawiki/2016/4/41/T--Peking--images_md_fig32.png |
| + | " target="_blank"><img style="width: 100% ;" src=" https://static.igem.org/mediawiki/2016/4/41/T--Peking--images_md_fig32.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | <h1>Our Innovation: The SADP model</h1> |
| + | <p> |
| + | Under the principle of equal reactivity, the probability sequence described above </p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/9/93/T--Peking--image_md_ThyEdit_O13.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/9/93/T--Peking--image_md_ThyEdit_O13.png" alt=""/></a></p> |
| + | |
| + | </figure> |
| + | |
| + | <p>represents the each step of enumeration shown on the right of the following picture:</p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/1/19/T--Peking--images_md_fig34.png" target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/1/19/T--Peking--images_md_fig34.png" alt=""/></a></p> |
| + | <figcaption style="text-align:left;"> |
| + | Fig. 6. The difference in enumeration process between the original model and the SADP model. |
| + | </figcaption> |
| + | </figure> |
| + | |
| + | |
| + | <p>Our model assumes that the second and above reactions on a monomer have a different reaction probability from the first reaction, to wit the P<sub>b</sub>•P<sub>b_d</sub>. shown on the right of Fig. 6. For convenience, we call this new method SADP model (Second and Above have Different Probability). </p> |
| + | |
| + | <p>There is no change to the definitions of P<sub>f</sub>, P<sub>b</sub> and P<sub>a</sub>, which means the average expect of the reacted functional groups in the system. The same goes for that expect on an arbitrary monomer. The difference is in the numerical distribution of monomers with zero, one, two, three…of its functional groups having reacted. To be specific, the distribution changes from binomial to non-binomial: </p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/0/08/T--Peking--images_md_fig35.png" target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/0/08/T--Peking--images_md_fig35.png" alt=""/></a></p> |
| + | <figcaption style="text-align:left;"> |
| + | Fig. 7. P<sub>b1</sub> is the reaction probability in the condition that none of the monomers’ functional groups have been confirmed having reacted. P<sub>b2</sub> is the reaction probability in the other conditions, which equals P<sub>b_d</sub>•P<sub>b</sub>. |
| + | </figcaption> |
| + | </figure> |
| + | |
| + | <p>Though the actual numerical distribution without equal reactivity should not be limited to, or just isn’t in the form given in Fig. 7, such inaccurate distribution makes it possible for us to uniform the magnitude of the second, third, fourth…reaction probabilities, to wit P<sub>b_d</sub>•P<sub>b</sub>. This uniformity is preferred for the mathematical convenience, since other forms of reaction probability has been found quite insufferable when trying to enumerate all the cases. </p> |
| + | |
| + | <p>From Fig. 6, the probability sequence is</p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/2/24/T--Peking--image_md_ThyEdit_O14.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/2/24/T--Peking--image_md_ThyEdit_O14.png" alt=""/></a></p> |
| + | </figure> |
| + | |
| + | <p> |
| + | That is the case when the change (correction) only happen to <sub>b</sub> groups. In the previous proof of (7), a free <b>a</b> was used as the root. Under that condition, when considering the influence that P<sub>a_d</sub> (<b>a</b>’s correction) would have on the probability sequence, it would involve an annoying item of the root-monomer’s branchers. After one or two steps of thinking, one will find that the probability sequence is even dependent on the structural configuration. Here we suggest a shortcut which skips this burdensome procedure. |
| + | </p> |
| + | |
| + | <p>First, the final expression of Number of cA & cB molecule with P<sub>a_d</sub>=1 is certainly</p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/8/82/T--Peking--images_md_fig37.png |
| + | " target="_blank"><img style="width: 100% ;" src="https://static.igem.org/mediawiki/2016/8/82/T--Peking--images_md_fig37.png" alt=""/></a></p> |
| + | </figure> |
| + | <p>Deem P<sub>a</sub>, P<sub>b</sub>, P<sub>a_d</sub> and P<sub>b_d</sub> as the input parameters of those functions. Because the following part</p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/8/85/T--Peking--images_md_fig38.png |
| + | " target="_blank"><img style="width: 100% ;" src="https://static.igem.org/mediawiki/2016/8/85/T--Peking--images_md_fig38.png" alt=""/></a></p> |
| + | </figure> |
| + | |
| + | <p>is independent to P<sub>a</sub>, P<sub>b</sub>, P<sub>a_d</sub> and P<sub>b_d</sub>, the formula(<b>a</b> as a root) can be rewritten as:</p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/1/10/T--Peking--images_md_fig39.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/1/10/T--Peking--images_md_fig39.png" alt=""/></a></p> |
| + | </figure> |
| + | |
| + | <p>where <a href="https://static.igem.org/mediawiki/2016/6/66/T--Peking--image_md_ThyEdit_O15.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/6/66/T--Peking--image_md_ThyEdit_O15.png" alt=""/></a> |
| + | , F<sub>a</sub>(P<sub>a_d</sub>,P<sub>a</sub>) is the burdensome probability sequence of <b>a</b> functional groups’ enumeration, which is unknown. When P<sub>a_d</sub>=1, |
| + | <a href="https://static.igem.org/mediawiki/2016/6/64/T--Peking--image_md_ThyEdit_O16.png |
| + | " target="_blank"><img style="width: 65% ;" src="https://static.igem.org/mediawiki/2016/6/64/T--Peking--image_md_ThyEdit_O16.png" alt=""/></a>. This item of <b>a</b>, though may be related to the configuration, should be irrelevant to P<sub>b</sub> and P<sub>b_d</sub>. Because that all the bonding actions behind <b>b</b>’s probabilities in the sequence are independent incidents to <b>a</b> functional groups’ bonding actions, we can separate the variables as shown in (9). </p> |
| + | |
| + | <p> |
| + | Because of the symmetry between <b>a</b> and <b>b</b>, the conclusions from (8) to (9) also work if we choose a random free <b>b</b> as the root. Thus we have (10) and (11). |
| + | </p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/6/6b/T--Peking--images_md_fig42.png |
| + | " target="_blank"><img style="width: 100% ;" src="https://static.igem.org/mediawiki/2016/6/6b/T--Peking--images_md_fig42.png" alt=""/></a></p> |
| + | </figure> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/a/ac/T--Peking--images_md_fig43.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/a/ac/T--Peking--images_md_fig43.png" alt=""/></a></p> |
| + | </figure> |
| + | |
| + | <p>Since the value of the left side of (11) should be equal to that of (9), the following relationship always exist:</p> |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/d/d0/T--Peking--images_md_fig44.png |
| + | " target="_blank"><img style="width: 80% ;" src="https://static.igem.org/mediawiki/2016/d/d0/T--Peking--images_md_fig44.png" alt=""/></a></p> |
| + | </figure> |
| + | |
| + | <p>Considering that P<sub>a</sub>, P<sub>b</sub>, P<sub>a_d</sub> and P<sub>b_d</sub>are independent to each other, the relationships in (13) and (14) are essential for (12).</p> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/e/e0/T--Peking--images_md_fig45.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/e/e0/T--Peking--images_md_fig45.png" alt=""/></a></p> |
| + | </figure> |
| + | |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/5/5f/T--Peking--images_md_fig46.png |
| + | " target="_blank"><img style="width: 70% ;" src="https://static.igem.org/mediawiki/2016/5/5f/T--Peking--images_md_fig46.png" alt=""/></a></p> |
| + | </figure> |
| + | |
| + | <p> |
| + | Because |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/e/e8/T--Peking--image_md_ThyEdit_O17.png |
| + | " target="_blank"><img style="width: 60% ;" src="https://static.igem.org/mediawiki/2016/e/e8/T--Peking--image_md_ThyEdit_O17.png" alt=""/></a></p> |
| + | </figure> |
| + | , which equals G<sub>b</sub>(1,P<sub>a</sub>), the constant C in (13) and (14) equals 1. Substitute (14) or (13) to (11) or (9), finally we get the number of cA & cB configured molecules in the sol with correction factors. |
| + | </p> |
| + | <figure> |
| + | <p style="text-align:center;"><a href="https://static.igem.org/mediawiki/2016/6/63/T--Peking--images_md_fig48.png |
| + | " target="_blank"><img style="width: 100% ;" src="https://static.igem.org/mediawiki/2016/6/63/T--Peking--images_md_fig48.png" alt=""/></a></p> |
| + | </figure> |
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