MODEL

Our project model divides mainly into 2 parts: a signal model that couples the visual signal of the beetector with the number of engineered phages present; and a kinetic model that quantifies the sortase enzymatic reaction. The sessions below detail both of these models.

Signal Model

The first part of our signal model attempts to determine the number of phages that bind to a spore and, depending on this number, the intensity of the chromoprotein signal - that ultimately will vary with the level of infection. The model also helps us identify the ideal phage candidates that either have a higher binding affinity or multiple binding sites on the spore. The candidate phages for our ideal spore binding must be closest to phage titer of optimal packing to provide a greater number of amajLime per spore ratio, to increase the visual identification of the bound spores on paper chromatography.

Phage-spore packing

The model assumes that our minimum signal would be produced in the case where one reporter phage binds to one spore, and the maximum signal would be produced where the spatially maximal packing of reporter phages takes place on the spore. For the maximal packing case, we calculated the number of phages of dimensions 880 nm in length and 6.6 nm in height that could fit into a spore of dimensions 5.2 μm by 2.5 μm as a hexagonal arrangement.

From the spatial packing of M13 pVIII capsid protein by sortase enzyme, there are 91 ± 20 EGFP conjugated to M13 phage [1]. This is based on the protein of similar size to amajLime filling up space to prevent further sortase modifications on the N-terminus of pVIII. Based on similar size and dimensions of amajLime and EGFP, it is assumed that amajLime will fill up space of similar size. The spore is assumed to be an ellipsoid of 34.7μm² and the M13 phage with the conjugated amajLime is of surface area 6.65E-04μm² facing the spore in a vertical arrangement. Based on the optimal packing of ellipses by spheres [2], 39,000 reporter phages can maximally bind to the spore. The minimal packing of the reporter phage on the spore is determined to be one reporter phage per spore. As we do not have any information about the density of the phage receptor or ligand on the spore surface, we solved our signal model for this hypothetical range. A parasitic N. ceranae outbreak would show at a minimum of 1 million spores per bee in a sample collected from a hive, this is the count recommended before treatment is applied in North America.[3]

Figure 1: A crystal structure of amajLime(2A46) was oriented such that its C-terminus was adjacent to the N-terminus of pVIII. By analyzing this image, it was determined that one amajLime molecule blocked the N-termini of the six pVIII proteins from further sortase modification. The image shows similar bonding to the EGFP-M13 sortase modification portrayed in [1].

The expected phage-spore binding is based on the results of biopanning enrichment. The phage titer of the acid elution provides the average number of ligands for the spore binding. Individual selection of each candidate will increase or decrease the phage titer obtained in acid elution. It is assumed that all phages that are bound to spore will have a binding affinity and multiple binding sites on the spore wall. The phage titer from the third round of biopanning also gives us the overall average for the phage binding sites on the spore wall, based on the phage population and the spore titer. During biopanning, the phages were not conjugated with the reporter protein, so spatial constraints are accounted for in the range of minimal to maximal spore binding to determine the amajLime molecule to spore ratio.

Figure 2: The Binding Capacity for individual phages and the reporter protein conjugated to the phage are plotted on a Logarithmic scale.

Signal intensity

The purpose of this model is to provide the beekeeper with instructions on what they should be looking at when the application of our product takes place, as well as to establish the concentration of amajLime-tagged phages that our Beetector solution needs to have for a successful diagnosis of a diseased bee.

The process of diagnosing an N. ceranae infection with the Beetector starts by sampling bees and crushing them in 1 mL of water per bee. The crushed bee solution is mixed with a non-ionic detergent, Tween 20, as this provides good band resolution on paper in the neutral solution used for the paper-based chromatography. A paper strip is dipped into the bee solution and the Beetector solution containing the amajLime-tagged phages. The engineered phages would bind to the spores. Unbound phages would elute faster on paper compared to bound phages. After a certain amount of time (based on biopanning incubation it is expected to be around 10 minutes, pending further testing), the beekeeper observes the signal and compares it with a range of intensities provided in our kit instructions. If no amajLime-spore band is observed below the amajLime band then the hive does not face an, N. ceranae outbreak. If the amajLime-spore band is detected below the AmajLime-band, the beekeeper matches the observed colour proportion with the range given in the instructions to determine the level of infection and take further action.

Based on the results from Lab:

Figure 3: The amount of amajLime molecules are determined to distinguish between different dilution factors of the Beetector signal solution.

Based on the visual signal proportions, the guidelines for distinguishing between amajLime bands is observed from the quantity data. The lowest signal dilution possible to distinguish a visual band on paper will be a dilution factor of 16 as any lower quantity will be too low to be differentiated from the band colour intensity.

Figure 4: Visual Signal correlation with the number of infected bee count(1 million spores per bee) in test sample.

The amajLime-tagged phage must be closer to maximal binding to reduce the number of infected bees needed to obtain a visual signal. The candidate phage ligands must bind greater than 15 phages per spore ratio or less than 39,000 phage per spore ratio. This can be increased by creating a more enriched biopanning sample and selecting for phages that provide a phage titer higher than 15. These phages would have either higher binding affinity to spores in order to bind a larger proportion of spores in solution or greater binding sites on the spore wall. An alternative is to use a reporter protein that is smaller in size and more visible to compensate for greater signalling than 1,400 amajLime per spore as shown in Figure 1. The volume for the Beetector solution chromatography could also be decreased by 100-fold to reduce the bee input needed for an infected signal. The risk of reducing the Beetector solution volume is decreasing the signal visibility. Below 64-fold dilution, the band is very low to distinguish.

Kinetics Model

Our second model is more straightforward, in which we characterize the kinetics of the enzymatic reaction used on the components of the Beetector.

From binding proteins to recycling enzymes as shown by Figure 7, this model includes 3 main reactions in which we assemble our genetically modified phage. All rate constants were obtained from literature and they represent the kinetics of a Sortase-A mediated ligation of enhanced Green Fluorescence Protein (EGFP) and the M13 phage [4]. These rate parameters were used because the structure of our chromoprotein, amajLime, is similar to the EGFP used in that study; we also used the same mutant of Sortase-A (SrtA) characterized by this paper.

Multiple simulations were performed and analyzed using the SimBiology module in MATLAB, and their respective reports are presented below.

First Reaction

The production of our Beetector begins by binding the sortase enzyme SrtA to our sortase-tagged chromoprotein, amajLime. This takes place in a reversible reaction with a rate constant of 1.0 mM [5].

Figure 5: Binding of Sortase-A to amajLime diagram.

Second Reaction

The next step in the process is the conjugation of our SrtA-tagged amajLime with our M13 bacteriophage. This is also a reversible reaction with a rate constant of 2.779 mM [6] .

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Figure 6: Phage Complex (amajLime-Sortase-A-M13) formation diagram.

Third Reaction

The last step in the assembly of the engineered phage would be the unbinding of SrtA from our phage complex. This is a non-reversible reaction with a rate constant of 3.7 s^-1 [7]. After this, we are left with our final product, the Beetector, consisting of multiple amajLime chromoproteins attached to our phage and free SrtA enzyme in the reaction mixture.

Figure 7: Unbinding of Sortase-A diagram.

With all of the parameters in place, we proceeded to investigate a couple of different scenarios, the first one being a simultaneous conjugation in which everything happens at the same time in the same space. The following simulation was obtained, and a SimBiology report of this case can be found here here.

Figure 8: Simultaneous Conjugation of M13 Complex diagram.

Figure 9: Simulation of first scenario, simultaneous conjugation in the same space. M13 phage and amajLime get consumed over time. Sortase-A gets used until maximum production of amajLime-Sortase-A-M13 complex is achieved, then it starts to unbind and return to its initial concentration.

For the second scenario, we tried simulating the case in which every reaction happens in a different space. This case needs optimization as everything is happening simultaneously. Diffusion parameters across cells or delays in supply may be added to this scenario to improve the quality of this simulation. The following data is obtained and a full report for this case is found here here.

Figure 10: Simultaneous conjugation of M13 Complex in separate compartments.

Figure 11: M13-amajLime-sortase modification reaction performed sequentially, amajLime-sortase reaction reaches equilibrium and then sortase tagged M13 is introduced for reaction to be complete and sortase enzyme to be released from the complex

In contrast to the first simulation, we note that under identical initial amounts, the Sortase-A-tagged amajLime is not completely used, and this is also represented on the Sortase-A curve as the enzyme is not able to return to its initial condition. However, faster production of our final product (M13-amajLime) is observed as the amajLime-M13 is produced within 4 seconds compared to Figure 8 with the product release occurring past 7 seconds.

References

[1 ]G.T. Hess, et al., “M13 Bacteriophage Display Framework That Allows Sortase-Mediated Modification of Surface-Accessible Phage Proteins”, Bioconjugate Chemistry, vol. 23, no. 7, July 2012, [Online serial]. Available: https://pubs.acs.org/doi/10.1021/bc300130z. [Accessed Sept. 9, 2019].

[2] E.W. Weisstein, “Decimal expansion of (24*sqrt(2) - 6*sqrt(3) - 2*Pi)*Pi/72.”, The On-Line Encyclopedia of Integer Sequences,. Oxford University Press, [online document], 2007. Available: The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A093824 . [Accessed: Aug. 25, 2007].

[3] College of Agricultural and Life Sciences, “Nosema Disease: Information for Identification & Control in New York”, College of Agricultural and Life Sciences, April 2015, [Online] Available: https://pollinator.cals.cornell.edu/sites/pollinator.cals.cornell.edu/files/shared/documents/Nosema.pdf [Accessed Aug 20, 2019].

[4],[5],[6],[7] L. Chen, et al., “Improved variants of SrtA for site-specific conjugation on antibodies and proteins with high efficiency”, Scientific Reports, vol. 6, no. 31899, August 2016. [Online serial]. Available: https://www.nature.com/articles/srep31899. [Accessed Oct. 20, 2019].

Team:UAlberta/Model