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.