Insulin is a peptide hormone that stimulates uptake of glucose by cells to reduce blood sugar levels and is commonly used for patients suffering with both type 1 and type 2 diabetes to replenish low levels of naturally produced insulin in the body [11] . Insulin is a prominent therapeutic protein. However, synthetic insulin has been shown to aggregate at sites of repeated injection as well as during the production, transportation, and storage process [12] , highlighting a need to engineer a mechanism that can reduce insulin aggregation. For our study, we used a crystallized human insulin model [13] (PDB ID: 3I40) (Appendix A).

Human insulin binds to insulin receptors present in the membranes of liver and muscle cells. We modeled the receptor-ligand interaction using the first three domains of an insulin receptor expressed through C. griseus [14] (PDB ID: 2HR7).

Erythropoietin is a glycoprotein hormone that stimulates production of red blood cells (erythropoiesis). Recombinant human erythropoietin (rhEPO) is often administered to patients suffering from anemia or those with low hematocrit due to infection or kidney disease [15] . A previous study found that aggregation of rhEPO results in conformational changes that greatly affect its activity and function [16] . By exploring the effects of PTMs on harmful aggregation of rhEPO, we hope to increase its efficacy. We used a human EPO model expressed through E. coli in our study [17] (PDB ID: 1BUY) (Appendix A).

EPO and rhEPO bind to receptors located on erythroid progenitors in the bone marrow [15] . We modeled this receptor using the extracellular domain of the human EPO receptor [18] (PDB ID: 1ERN).

Human growth hormone is a peptide hormone that first received approval by the FDA to treat GH deficiency in children in 1985 [19] . Previous studies conducted by Fradkin et al. found that aggregation of growth hormone existing in commercial formulas of the drug stimulated increased levels of immunogenicity in mouse models [20] . To avoid potential side effects of this increased immune response, such as decreased therapeutic half-life and anaphylaxis in the patient [21] , methods to reduce aggregation must be found. We used a wild-type HGH model in our study [22] (PDB ID: 1HGU) (Appendix A).

To model the receptor-ligand interaction between HGH and its receptor, we used chain B of an HGH-receptor complex [23] (PDB ID: 1A22), which corresponded to just the receptor domain of the modeled complex.

Protein phosphorylation refers to the reversible attachment of a phosphate group by a protein kinase (attachment) or phosphorylase (detachment) to a side chain of an amino acid. The side chains of serine, threonine, and tyrosine are usually the most commonly phosphorylated. Phosphorylation has the ability to alter the function, structure, and stability of a protein [24] , making it an intriguing and ideal PTM to analyze in this study.

Protein acetylation is a type of acylation, which is the addition of an acyl group to a protein’s amino acid residues by acyltransferases (attachment) and deacylases (detachment) [25] . Acetylation commonly occurs at serine and lysine residues, and has been implicated in the regulation of protein stability, localization, activity, and affinity for DNA-binding [25] . Because of the ubiquitous functions of acetylation, many metabolic enzymes are tightly regulated with the addition or removal of acetyl groups [26] . The many abilities of acetylation also make it a valid PTM to analyze in this study.

Protein methylation, catalyzed by methyltransferase (attachment) and demethylase (detachment), is the addition of a methyl group to amino acid residues—especially the lysines and arginines—of a protein [27] . Methylation can affect the function and conformation of molecules, making it ideal for modulating cell signaling and DNA repair pathways [27] . The implications of methylation on the characteristics of proteins prompted us to analyze it in this study.

We extracted PDB (protein data bank) format files of proteins for insulin, EPO, and HGH from the protein database RCSB.org, an open-source software containing numerous different protein structures and their relevant conformations [28] [29] . We identified the most recently uploaded files for and filtered searches for proteins classified as hormones originating from Homo sapiens, allowing us to get the most accurate representation of hormones in vivo. In addition, the common receptors for each of these proteins present in the human body were also obtained from the database. In cases where RCSB contained protein-receptor complexes, only the specific chain corresponding to the receptor itself was used.

We utilized an open-source server called Vienna-PTM 2.0 to apply post-translational modifications to the protein files we downloaded from RCSB.org. Vienna-PTM allows users to apply modifications to certain amino acids in uploaded PDB files [30] - [32] . Users can obtain new PDB files and force field parameters for a modified protein that can be used in molecular dynamics simulations such as GROMACS.

For our study, we selected two amino acids that are commonly modified for each group, and modified all instances of those amino acids in each protein to the most neutrally-charged relevant modification in Vienna-PTM to gauge the full effect of these PTMs. The amino acids modified in each group were chosen based on previously conducted studies (see section 2.2) and are detailed in Table 1 .

After modifying all specified amino acids, we exported and downloaded a new PDB file with the modified protein for analysis. This process was done for all experimentally modified insulin, EPO, and HGH files (See Appendix A).

We set out to characterize the affinity for these proteins to aggregate with each other by calculating the DS and BFE between two identical copies of the protein in question. To calculate these values, we used a web server called HawkDock, which specializes in structural prediction and analysis of protein-protein complexes using ATTRACT for global macromolecular docking and HawkRank for scoring [33] - [36] . It takes in one PDB file as a receptor and one PDB file as a ligand. It then outputs the ten most probable complexes, ranked by their docking scores. Docking scores are commonly used by scoring functions of algorithms to represent ligand binding affinities. A more negative DS generally correlates to stronger intermolecular interaction [37] . In the context of our study, we seek to maximize DS and bring it closer to zero, since we are aiming for weaker interactions between proteins and therefore less aggregation.

For our project, we inputted two PDB files of the same, identical protein we were analyzing during that trial into both the receptor and ligand fields of HawkDock. This way, we were able to simulate real interactions between identical proteins administered into the bloodstream through a single injection. The average of the ten complexes was taken in order to equally represent the variability in bonding that may occur in the natural world.

We then ran an MM/GBSA analysis, which is available on HawkDock, on the modeled complexes to obtain BFE values for each corresponding model in kJ/mol [38] - [40] . Binding free energy is a more well-established and practical measure of binding affinity; it quantifies the free energy difference between the bound and unbound states of a complex, often in kJ/mol of complex [41] . Similar to DS, more negative values mean the bound state of a complex is preferred; therefore, we aim to maximize BFE. Additionally, MM/GBSA analysis is a commonly used method to calculate ligand binding affinities computationally, since it does not require large amounts of calculations and boasts remarkable accuracy compared to experimental data [42] .

In order to minimize loss of function for these proteins from the PTMs, we analyzed the changes in modified proteins’ affinities for their native receptors.

We uploaded the respective receptor in the receptor field of HawkDock and the normal or modified protein we were analyzing in the ligand field, and recorded the top model. The top model, which is the most probable, is most likely to be the true receptor-ligand configuration in vivo. This assumption was also supported by the fact that the top model had a much lower docking score and binding energy than subsequent, lower-ranked models. Using this methodology, we were able to add an additional factor to our study’s consideration by weighing the change in biological function when making recommendations for most optimal PTMs.

Our overall method design consolidates all of these data points and is summarized in Figure 2 .

Since it is difficult to compare improvements in overall protein function (taking into account both aggregation and receptor affinity) without a numerical formula, we will mainly summarize improvements in the reduction of aggregation and comment on significant differences in receptor affinity (functional continuity). For a comparative, numerical analysis between modifications taking both aggregation and reception into account, refer to Section 3.3.

The average DS and BFE values for the top 10 models outputted by HawkDock for aggregation are summarized in Table 2 and Figure 3(a) & Figure 3(c) , and the DS and BFE values for the top (most optimal) model outputted by HawkDock for receptor-ligand interactions are summarized in Table 3 and Figure 3 (b) & Figure 3(d) .

More positive is more beneficial.

More negative is more beneficial.

To prove statistical significance between the normal (control) group and the experimental (phosphorylated, acetylated, methylated) groups for each protein, we used a single factor ANOVA test conducted through Google Sheets and the “XLMiner Analysis ToolPak” extension.

A single factor ANOVA (analysis of variance) test is a generalization of the two-sample t-test and calculates how much of the variance or discrepancy between data can be attributed to random error or the factor effect [43] . It outputs an F-statistic and P-value. In our study, we used the P-value at a significance level of 0.05 to indicate significance. A lower P-value, in this case p < 0.05, means the variance is likely not due to random chance and instead due to our treatment. Although we can see at a clear glance that the mean DS and BFEs are higher or lower between experimental groups, we applied this test to prove our findings were significant and identify which PTMs on which proteins had significant effects on their aggregation affinity. We also ran an overall ANOVA test with all four groups for each protein to show that the three PTMs we explored caused significant results in general. Since all overall P-values were significant at ɑ = 0.05, we rejected our null hypothesis. We did not run an ANOVA test on the receptor affinities since we only had one data point for each group. The P-values on the ANOVA analyses of aggregation affinity data between modified and unmodified proteins are shown in Table 4 and Table 5 .

*Statistically significant (p < 0.05).

*Statistically significant (p < 0.05).

In order to quantitatively and specifically measure the improvement or deterioration of each PTM treatment, we created a protein viability index equation by modifying a sigmoid function using the base aggregation and reception values from the normal protein, and the aggregation and reception values from the modified protein:

$\frac{2}{1+e^{-\left(0.25\cdot \frac{Valu{e}_{agg}-Bas{e}_{agg}}{\left|Bas{e}_{agg}\right|}+0.75\cdot \frac{Valu{e}_{rec}-Bas{e}_{rec}}{\left|Bas{e}_{rec}\right|}\right)}}-1$ (1)

The process taken to create the index was to first have a baseline value of comparison for each protein, which was the normal DS and BFE in aggregation and reception for an unmodified protein (DS and BFE are calculated in separate programs, using their specific base values for each protein). Using the base values for a protein, the post-translationally modified protein’s average DS or BFE values were inputted for aggregation and reception for that same protein. By taking the difference between the experimental value and the unmodified base value and dividing by the unmodified base value, the experimental value is normalized for that specific protein which allows other transformations to be put upon the data. A weight of 0.25 was used for the aggregation aspect, and 0.75 was used for the reception aspect. This 1:3 ratio between reception values and aggregation values represents the greater importance of the reception values, exhibiting how the original function of a protein staying constant is not something that can be changed while increasing protein longevity. Changes in the function of the protein as a whole causes longevity to become irrelevant in this setting and therefore must be prioritized in the index. The ratio chosen can vary depending on the difference in importance, however this is something that can be more studied using wet-lab experiments to study how PTMs can change the function of a protein more accurately.

After the normalized and weighted values were calculated, they were put into a sigmoid-shaped logistic function that was modified to output values from −1 to 1 in Python (see Appendix B). The logistic function transforms the given data around a midpoint, where the instantaneous rate of change is a vertical line, to a specified range. It is commonly used in data analysis. Here, the point of inflection was 0, representing no change, and the limits of the left and right sides of the function approached −1 and 1, respectively. A positive output represented an increase in the overall performance of the modified protein given the weighting of aggregation and reception, and a negative output represented a decrease in the performance of the modified protein. The normal insulin, EPO, and HGH proteins can automatically be assigned a value of 0 on the index as they are the baseline models. The values provided by this index is used as a quantitative measure to determine the change in the DS and BFE caused by the PTM and allows us to objectively and numerically compare the differences in differently modified proteins (see Appendix B for index source code). The index values are shown in Table 6 , Table 7 , and Figure 4 .

Note: Proteins structures shown below come from the RSCB.org database ( Figures A1(a)-(c) ) and the HawkDock software visual output ( Figures A1(d)-(e) ).

Note: A different file was used for each protein and for DS/BFE which had the baseline values unique to each. The file shown below is hghindex.py, specifically calculating BFE.