Imagine that when most of the molecules in our colloidal sol are small in size, the system would take more biotin units for collection than when the molecules are big. Nevertheless, making each reaper molecule too big may finally turn the network into gel, which makes the relative superficial area collapse to nil and so does the probability of effective SUP - Uranyl collision. Hence, finding a proper molecular size is important for the reaper protein’s well-functioning. Thus, we also set the reaper protein’s molecular weight distribution as our major research object.

We found that the strategy introduced in Flory’s Principles of Polymer Chemistry, Chapter IX¹, is well proved and capable of predicting the molecular weight distribution for nonlinear polymers before its gel point, or in other words, when the reaction is at the state of Sol. Here are some brief descriptions of Flory’s method.

For some definitions, a polymer is a large molecule, or macromolecule, composed of many repeated subunits. Polymerization is a process of reacting monomer molecules together in a chemical reaction to form polymer chains or three-dimensional networks. And the word “functionality” means the amount of functional groups in a monomer molecule. Our Spytag & Spycatcher network belongs to the category of three-dimensional networks, which requires at least one kind of the reactant possess a functionality more than two (known as the polyfunctional unit).

In Flory’s description, the presence of polyfunctional units nearly always presents the possibility of forming chemical structures of macroscopic dimensions, to which the term infinite network is appropriately applied¹. To estimate the critical point that the infinite network takes its form, the concept of branching coefficient α was used. α was defined in the case when Flory used following reactants:

Condensation exclusively occurs between A and B. α is the probability that an A group selected at random from one of the trifunctional units is connected to a chain the far end of which connects to another trifunctional unit¹. In this trifunctional condition, α must be bigger than 1/2 for the system to form an infinite network. In an f-functional condition, the critical point of α is 1/ (f-1). That is because the far end must have an expectation of the number of potential branching functional groups larger than 1 to make the expectation of total branching points infinite. This critical value of α is called “Gel Point”.

Flory’s description has two assumptions which are necessary for the validity of his model. The first one assumes that there is no reaction between functional groups that belong to the same molecule. In other words, whatever the polymer’s size, no loop can be found in it. The second one is called the principle of equal reactivity, which assumes that every functional groups share the same probability to have reacted.

Flory suggested an example of a system with only A-B₂ as reactant. A large molecule containing x A-B₂ can be called an x-mer. Enumeration of the probability of each particular x-mer’s existence can be done with the branching coefficient α, or in a more universal definition the reaction probability P. Based on that enumeration, he derived the molecular weight distribution. For more details, please read his book Principle of Polymer Chemistry¹.

In our work, the condition changes into the reaction between two kinds of monomers with arbitrary numbers of a or b functional groups (one only has a, the other only has b). For the convenience of description only, A and B now refer to those two kinds of monomers and a, b refer to functional groups:

Let cA, cB be the counts of A and B in a selected type of molecule. We call this kind of molecule cA-cB-mer. For example, the molecule shown below can be called a 2-3-mer.

There isn’ t one type of branching unit in this case (there are two). The reaction probability are defined as the probability that a randomly selected functional group inside the system has reacted. For a and b, those probabilities are P_a and P_b. Under the principle of equal reactivity, P_a and P_b are obtained in the following way:

Such variables are also called reaction degree, indicating the progress of the reaction. P_f is the general reaction degree. For the record, all of those reaction degrees are treated as inputs in our model.

We have derived our own formula of the number of particular cA-cB-mers based on Flory’s strategy. Detailed deduction of this formula can be found in Supplementary Part. This final equation is shown as follows:

With the number of cA-cB-mers, the total weight of that kind of molecule can be got by multiplying this number with its molecular weight. Next, we can calculate the molecular weight distribution of the entire system by counting each cA-cB-mers.

Some examples of the outcomes based on eq. (2) are shown below.

For another important concept, the gel point, we also derived its formula of our own case. (Detailed process of deduction can be found in The Other Part Of our Modeling)

Note that eq. (2) is valid only when the reaction degree P_f is smaller than P_c.

In a pre-experiment, it has been witnessed that changing the protein attached to the 3A part can notably affect the reaction. The result could be easily understood with the following SDS-PAGE picture:

Such difference could be attributed to some effect similar with steric hindrance, which leads to bias upon the same monomer’s functional groups’ reaction probability. Based on such presumption, the concept of P_a and P_b should no longer be universally used. To be specific, one of Flory’s assumption in his book that all of a type of functional groups in the system have the same probability of having reacted, must be reconsidered. Here we brought up a new method to deal with the probability P_a, P _b, which can be expressed simply as “If it is confirmed that one of the functional groups on a 3A or 3B has reacted, then a rest functional group has a lower/higher probability of having reacted”. We assume the lower/higher probability has the formation of min (1, P_{a_d}·P_a) and our formula changes into:

The idea behind this formula can’t be directly understood. Detailed reasons that why we chose this form for correction are discussed in the supplementary files.

It is important to note that in this model, the definitions of P_a and P _b are the same as the definitions in eq. (1). However, the “first” reaction probability on a 3A or 3B won’t be P_a or P_b. It would be higher/lower than P_a to compensate the effect of lower/higher “P_{a_d}·P_a”, since the total expectation of the number of reacted groups on a 3A (or 3B) should be unchanged (to wit, P_a·f_a). The difference is in the fraction of monomers with certain number of its groups having reacted, not the total numbers of reacted functional groups. Though this method isn’t the universal formation of monomer distribution for monomers with functionality more than two, this form has been found quite friendly with formula deduction.

Apparently, a P_{a_d} higher than 1 indicates recruiting effect of the first reacted a on a A to this monomer’s rest functional groups, a P_{a_d} lower than 1 indicates that the first reacted a hinders its siblings’ reaction (the reaction of the rest functional groups on this A). This new approach has been programmed into a web-calculator “SoP”, which will be described later.

We have done some further experiments to testify our new approach:

In Fig. (6), the column charts bellow the line graphs were generated by our software “SoP”. The blue column charts have the same vertical value with the yellow column charts below them, but with logarithmic coordinates which help in comparing with the experimental results. The calculations in (A) and (B) share the following parameters: f_a=3, f_b=3, K_d=4.64e-5. The molar concentrations and monomer weights were set according to the experimental fact (use 1mg/ml total mass concentration and M_3A-SUP=21.4kDa, M_3B=55.4kDa and the mass ratio 1:1, 1:2…in Fig. (6) to calculate). P_{a_d} and P_{b_d} were set to “P_{a_d}=0.9, P_{b_d}=1.38” for SUP and “P_{a_d}=1.5, P_{b_d}=0.8” for mSA, which optimize the similarity of the experimental and theoretical peaks’ relative intensity (Actually those values are fitted based on the similarity). It is essential to notice that this two parameter’s proper being can lead to the similarity of a wide range (1:1 to 1:3) of cases, and each case 4 to 5 peaks. That shows using the form of eq. (5) is quite effective.

In Fig. 6(C), the experiment is the same one of Fig. 6(B), while the calculation uses “P_{a_d} =1, P_{b_d} =1”. In other words, Fig. 6(C) shows the calculation without “P_{a_d} & P_{b_d}” correction, from which we can see the limitation of the original form eq. (2). The discrepancy between theoretical columns and experimental lines increases as the correction removed.

The result of this fitting work could be rough since the low accuracy of the experimental line graphs, but from the values of the optimized P_{a_d} and P_{b_d}, qualitatively one can tell the recruiting or hindering effect of the attached protein. E.g. in the SUP’s experiment, “P_{a_d}=0.9, P_{b_d}=1.38” indicates that the SUP protein makes the second bond on its 3A less possible to happen, and bestows any functional groups on the 3B directly linked to this 3A more chance to react with another a.