—Calculation of Gel Points

According to polymer science, we know that a condensation polymerization comprises 2 kinds of reactions - linear polymerization and three-dimensional polymerization. The difference between the two reaction types is that the former can lead to the formation of covalently linked one-dimensional polymers, while the latter can lead to the formation of high-dimensional linkage hydrogel. To increase the contact area and the strength of the polymer, we decided to use monomers containing more than 3 functional groups, i.e., the multi-branched monomer Af (f>=3). We attempted to anticipate the gel point specific for the three-dimensional polymerization (TDP) with reaction extent p through Flory Theory and Carothers Theory.

In Flory’s description, the sol becomes a gel when its reaction extent reaches the gel point Pc. For a more direct route from initial experimental parameter configuration to its prediction, the linkage between the initial solution state and final reaction extent (when it reaches stability) must be established. A simple way to do this, with enough qualitative credibility, is to use the concentrations of A and B, together with the dissociation constant Kd (1).

The gel point also limits the obtainable states by confining Pf. Any gelation is not acceptable for the reasons described earlier. In Flory’s theory, the formula used to calculate the gel point is well described. We have therefore derived for our own case (2):

Assuming that the total number of a and b is unchanged, the differences of the propensities towards different kinds of configurations can easily be compared via their respective gel points. Hereby a tetrad SpyTag-SUP and double SpyTag reaction is a different configuration from a triple SpyTag-SUP and triple SpyTag reaction. The corresponding comparison chart is shown in Fig. 1.

— differences between 3A-SUP and 3A-mSA

It has been witnessed that changing the protein attached to the 3A part can notably affect the polymerization reaction. The result could be easily understood based on the corresponding SDS-PAGE picture (Fig. 2).

The figure illustrates the reaction process from 10-120 min after mixing of the reactants 3A and 3B. Both of these pre-experiments were done at 25 centigrade, pH=7.4 and an initial concentration of 1mg/ml for both 3A and 3B. The difference was that in the first group each 3A module was attached to an SUP module while in the second group mSA was substituted for SUP. The red vector in Fig. 3 indicates the passage of time and also a sharp decrease of the relevant band. Briefly, the decreasing band on the left represents 3B (55.4 kDa) and the one on the right represents 3A_mSA (24.2 kDa).

Such differences could be attributed to the effect of steric hindrance. Based on such a presumption, the terms and would change accordingly. Additionally, one of Flory’s assumptions must be reconsidered. That assumption is that all individual representatives of a single type of functional group in the system have the same probability of reacting. We thus developed a new method to deal with the probability terms , which can be expressed simply as “if one of the functional groups on a 3A or 3B monomer has already reacted, then a residual functional group has a lower/higher probability of reacting”. If the lower/higher probability has the format of min (1, Pa_d · Pa), the formula will change into (3):

Hereby a Pa_d larger than 1 indicates a recruiting effect of the first reacted “AB” of 3A on the rest of its functional groups. Conversely, a Pa_d lower than 1 indicates that the first “AB” bond hinders further reaction of functional groups in its immediate vicinity (the reaction of the rest of functional groups on its parent 3A monomer). This new approach has been programmed into the web-calculator “MWCal”. A monomer ratio experiment shows the validity of the formula via weight distribution analysis (Fig. 3).

Based on the experiment shown in Fig. 3, for the monomer functionalized with the SUP module, this ratio ranged from 1:1 to 1:3, and for mSA the ratio ranged from 1:1 to 3:1. The total initial mass concentrations (3A+3B) in the two experiments were set to 1mg/ml. The SDS-PAGE experiments were analyzed via densitometry, whereby each OD value roughly represents the protein mass located in a particular band. Hence we can get a coarse molecular weight distribution line graph with the help of gel analysis tools. Since there is no quantitative access to the mass concentrations from the light absorbance data, the shape and area of each peak (relative intensity) needs to be taken into account.

The bar charts below the line graphs were generated using the software “MWCal”. The blue bar charts have the same vertical values as the yellow bar charts below them, but with logarithmic coordinates, which helps when comparing them with the experimental results. The calculations in Figs. 28(A) and (B) share the following parameters: fa=3, fb=3 and Kd=4.64e-5. The molar concentrations and monomer weights correspond to the experimental facts. The Pa_d and Pb_d values were set to 0.9 and 1.38 for SUP, and 1.5 and 0.8 for mSA, respectively, which optimized the similarity between the relative intensities of the experimental and theoretical peaks, since said values were fitted based on this similarity. It is essential to bear in mind that proper choice of values for these two parameters can produce good similarity over a wide range (1:1 to 1:3) of cases, with 4 to 5 peaks in each case. This indicates the effectiveness of eq. (5).

The experiment shown in Fig. 28(C) is the same as the one in 28(B), while the calculation uses Pa_d=1, Pb_d=1. In other words, Fig. 6 shows the calculation without any correction for the Pa_d and Pb_d terms, from which we can see the limitation of the original eq. (4) (see below).

The result of this fitting work can be rough due to the low accuracy of the line graph. One can nevertheless qualitatively estimate the recruiting or hindering effect of the attached protein from the values of the optimized Pa_d and Pb_d terms. For example, in the SUP experiment, Pa_d=0.9 and Pb_d=1.38 indicate that the SUP protein reduces the likelihood of reaction for the second bond on its 3A monomer, and conversely makes it more likely that any 3B directly linked to this 3A monomer will react with another 3A.

Based on the definition of orthogonality for biological parts, the design of the SpyTag - SpyCatcher pair, which are the structural molecules, should ideally make this hindrance/recruiting effect negligible for any attached protein. In other words, whether it is SUP or mSA, the value Pa_d and Pb_d should remain unchanged. Otherwise, the resulting weight distribution will be uncontrollable, together with any related mechanical traits such as viscosity. We thus planned to optimize the design using Pa_d and Pb_d as index values.

It is easy to imagine that in cases where most of the polymer molecules in a colloidal sol are small in size, the system would require more biotin units for recovery than when the molecules are big on average. However, making the Reaper molecules too big may turn the polymer into a gel, which would make the relative surface area collapse to nil and with it the likelihood of effective SUP - uranyl collisions. Hence, finding a proper molecular size is important for the Reaper hydrogel’s adequate functioning. We consequently also focused on finding an optimal molecular weight distribution as a major research objective.

Since the target product should behave as a sol and not a gel, we found that the strategy introduced in Flory’s Principles of Polymer Chemistry, Chapter IX, is well supported and capable of predicting the molecular weight distribution for nonlinear polymers below the gel point, or in other words, when they constitute a sol. We have derived our own equation based on Flory’s strategy, which omits any intramolecular reactions and treats any functional group with the same reaction probability as its counterpart. The equation is as follows (4):

wherebyrepresent two kinds of monomers in the solution and represent the functional groups of and . The termsare the total numbers of and in a selected type of molecule, e. g. (2,4) molecules contain two and four -type functional groups. The terms describe the numbers of functional groups in monomersand, respectively, i.e. the monomers’ functionality (e.g. the functionality of triple SpyTag-SUP is 3). and are reaction extents defined as follows (5):

It is essential to note that the reaction extent Pf in the formula depends on the empirical result. The percentage given above just depicts all the possible distributions that should be actually implemented by adjusting the reagent concentrations to reach a particular Pf.

The corresponding SDS-PAGE experiments used to evaluate the molecular distribution can underscore the validity of this theory by measuring both the Pf values and the molecular weight distributions. The corresponding theoretical chart is shown in Fig. 5.