In our project, the mix between SpyTag003 and SpyCatcher003 would perform a reaction called condensation reaction in which the monomers,
SpyTag003 and SpyCatcher003 ,form larger mol-ecules through step-growth polymerization. The condensation reaction is also the reaction in
which monomers that have more than two functional groups react form high molecular com-pounds through reactions.

Gelation is a certain reaction extent polymerization would perform. During the process of gela-tion, the viscosity of the polymers increases. However, condensation reaction is normally reversible, the cross-linking network formed during the process of gelation make gelation impossible to be reversible. The mix of SpyTag003 and SpyCatcher003 would lead to the formation of a cross-linking network and gelation such that the viscosity and strength of the products increases significantly. Since increment of viscosity and strength is one of the goals we are looking for, the primary consideration in this model is to find the most appropriate ratio between SpyTag003 and SpyCatcher003 such that the gelation can be attained most quickly. Previously, 2016 PKU has succeeded in finding the relationship between the fraction of amount of SpyTag003 and SpyCatcher003 and the threshold for gelation to occur, which is the Gel point. Therefore, we choose to adopt the model in 2016 PKU. Recently, an article Approaching infinite affinity through engineering of peptide–protein interaction published on PANS has provided us with an optimized version of SpyTag003 and SpyCatcher003, which is the SpyTag003 and Spy-Catcher003 that we use in our program with significantly increased reaction rate. However, 2016 PKU used SpyTag001 and SpyCatcher001 in their model. Therefore, there are a few parameters from PKU 2016 that need to be recalibrated so that it fits better in our experiment. When the multifunctional monomer polymerizes to a certain degree, the cross-linking between them begins, and the viscosity increases suddenly, which is the phenomenon called gelation. The extent of reaction at this time is called the gel point which is defined as the critical extent of re-action at the moment when the gel starts to appear. Since the gel is insoluble in any solvent, it can be considered as the cross-linking network among linear large molecules of which the mo-lecular weight is infinite.

In our model, we would apply a theory provided by Paul J. Flory (1910-1985), Walter H. Stockmayer (1914-2004) to predict the gel point and thus
achieve our purpose of obtaining the appropriate ratio of between SpyTag003 and SpyCatcher003.

To begin with, Flory-Stockmayer Theory is based on a primary assumption that the polymerizing units unite in such a manner that the probability
that any particular functional group has undergone reaction is an independent probability. That is, all functional groups on a branch unit are
equally reactive. Besides, we would also ignore intramolecular reactions.

However, results of Kienle and his coworkers on glycerol-dibasic acid polymers and of Bradley on drying oil resins show that the decrease in the number of molecules is slightly less than the number of inter-unit linkages formed. Consequently, a small part of the reaction must be intramolecular [Flory,1941]. Considering our previous neglect of this reaction, it is to be expected on this account that the predicted gel point would be slightly lower than actually needed to create a gel.

Firstly, we introduce the concept of branching index α, which is the probability of the linkage between branching units. For system involving 2 monomers with same functionality, when the functionality of branching unit is f, the critical branching index to form the gel can be expressed by

For a common 2-2-3 System, ie., A-A, B-B and A_f, where f is the functionality of branching and f=3, let P_A, P_B be the degree of reaction of
group A and B respectively, \rho as the fraction of the number of groups A in the branching unit (A_f) in the total number of A in the mixture,
and (1-\rho) is the fraction of the number of groups A-A in the mixture of the total number of A, then

The group B and the branching unit A, the reaction probability is pB

The probability that group B reacts with the unbranched unit A-A is Pb (1-P).

Therefore, the total probability of forming the chain segment between the two branching points is the product of the reaction probabilities of
each step.

According to (1) and (2), we can find out the relationship between f and P, when PA=PB=Pf=Pc, we have,

Following the previous reasoning, we could derive the formula of our own case:

Where fA and fB are the functionality of SpyTag003 and SpyCatcher003, NA and NB are the The amount of monomers containing functional groups of SpyTag003 and SpyCatcher003

The reaction ends when the solution reach its equilibrium. Therefore, we write the relationship between the initial solution state and the final reaction extent, expressed by the equilibrium equation with equilibrium constant K:

Noted that K is the equilibrium constant for both intramolecular and intermolecular reaction here, where the former will lead to the formation of a loop and decrease the degree of cross linking. By our assumption, we ignore the intermolecular reaction then K is the equilibrium con-stant for exactly the intermolecular reaction.

Now define Pf as the reactive extent of all functional groups, then we have

Where N is the amount of functional groups consumed in the reaction, x is the initial amount of all functional groups in the reaction. A and B each means SpyTag003 and SpyCatcher003.

It is feasible to assume that the concentration of small molecule (H2O) is constant during the reaction, then the K and the dissociation constant Kd can be described as,

Then we can derive the relationship between P_F and N_A, N_B as the following:

The initial amount of all functional groups in the reaction, x, can be expressed using the follow-ing equations:

Thus, to achieve our goal of form a gel, we would need Pf>Pc Namely,

In given total mass of the SpyTag003 and SpyCatcher003, gel can form in certain range of ratio between SpyTag003 and SpyCatcher003. As the condition of the formation of the gel listed above, the critical ratio between SpyTag003 and SpyCatcher003 can be found.

Here is the graph of C(Na,Nb):

By solving C(Na,Nb)=0, the two functions of critical ratio are attained.

Then we need to find a mapping relation between the lower bound of the ratio of SpyTag003 and SpyCatcher003 that can form the gel and the mass total amount of SpyTag003 and SpyCatch-er003, which can be expressed as:

g1(Na+Nb)=Na/(Na+Nb)=(1-Nb)/(Na+Nb)

Then, a function need to be found to express Nb into the form of Na+Nb. Using the functions of lower critical ratio, it is derived:

By calculating the reverse function of it, Nb can be expressed by (Na+Nb):

Therefore, we could derive g(Na+Nb), the lower bound of the ratio of SpyTag003 and SpyCatch-er003 that can form the gel in relation to the total amount of SpyTag003 and SpyCatcher003, where x is the total amount of SpyTag003 and SpyCatcher003:

Doing the same calculation, we could find the upper bound of the ratio of SpyTag003 and Spy-Catcher003 that can form the gel in relation to the total amount of SpyTag003 and SpyCatch-er003, where x is the total amount of SpyTag003 and SpyCatcher003:

By plotting the graph, we found that both the lower critical bound and upper critical bound of the ratio of SpyTag003 and SpyCatcher003 that can form the gel are around 0.8. Therefore, we can conclude that the best ratio between SpyTag003 and SpyCatcher003 is 0.8.