In our project, we utilized the latest version of SpyTag and SpyCatcher: SpyTag003 and SpyCatcher003 for protein crosslinking^{1}. Particularly, one goal for us to achieve is the gel formation mediated by the interconnection between SpyCatcher003 and SpyTag003. However, to achieve gel formation, the ratio of SpyCatcher003 and SpyTag003 should be optimized. Therefore, from the perspective of engineering, we built our mathematic model to characterize SpyTag003 and SpyCatcher003 by calculating the optimal ratio based on the concept of gelation^{2}.

Firstly，we can describe the relationship of gel point Pc and the amount of functionality in SpyTag003
and SpyCatcher003 with the following equation:

In which and are the functionality of SpyTag003 and SpyCatcher003, and are the number of monomers
containing functional groups in SpyTag003 and SpyCatcher003.

*Figure 1. The graph colored as yellow represent C(Na,Nb).*

By Solving for the critical condition where the function equals 0, the two
functions of critical ratio
are attained.

We can therefore derive the function of the maximum and minimum ratio for gelation to occur of the
fraction of Na and Nb：

Where x=Na+Nb，and g1 and g2 are the functions describing the two critical conditions.

By plotting the graph, we find out 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.

*Figure 2. The physical map of our hardware — RDR*

See more information in our model webpage.

ModelReference

1. Keeble, A. H. et al. Approaching infinite affinity through engineering of peptide-protein interaction. Proc Natl Acad Sci U S A, doi:10.1073/pnas.1909653116 (2019).

2. Flory, P. J. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation1. Journal of the American Chemical Society 63, 3083-3090, doi:10.1021/ja01856a061 (1941).