For the software given bellow, it was programmed based on eq (5):
Most of these parameters’ definitions can be found from the passage. The “Weight Constraint” sets the confine of the molecular weight coordinate for both displaying and calculating. The “Monomer Weight”, defined as its name, may affect the height and the horizontal position of a bar in the chart. If one wish to compare the generated chart with experimental data that uses mass concentration as unit, the molar concentration must be changed according to the “Monomer Weight” as well. Kd, Pf and Pc are automatically generated after inputting other parameters (Kd and Pf affect each other). Click “Generate Chart” to obtain the theoretical outcome from eq (5). Click “Use log” to change these bars’ horizontal position for comparing it with Adjusting Pa_d, Pb_d and Pf according to your experimental data to find a similar point. The optimized value of Pa_d and Pb_d would be the indicator of hindrance or recruiting.
The strategy shown below is based on the gelation theory by Flory[ “Monomer Size Distribution Obtained by Condensing A-R-Bf-1 Monomers”. Chapter IX. Flory, Paul J. Principle of Polymer Chemistry. 1953.].
Consider there are two kinds of monomers ‘A’ and ‘B’ in the system, one with fA functional groups ‘a’ and the other with fB functional groups ‘b’. With the principle of equal reactivity, we can say an arbitrary ‘a’ selected from the system have a probability “Pa” of having reacted. And the assumption that reactions between ‘a’ and ‘b’ groups on the same molecule are forbidden (no intracellular reaction) enable us to enumerate the probable shape of a certain molecule with cA number of A and cB number of B from a root to all the molecule’s branches.
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.
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
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
where is the total number of configurations. This probability has the following physical picture:
To get the number of cA&cB molecules, it needs to be multiplied with
which equals. is the number of A monomers.
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
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
The worry of not that loops may occur in such combinations can be eliminated if the expression given above is understood like this:
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.
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:
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
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.
Next, from the definition of the reaction degree Pf, Pa and Pb:
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:
