Dry Lab Model

To better back up the solution’s effectiveness and get a grasp on the quantitative data, the Team’s dry lab group proposed a bacterial conjugation model set to accomplish this task using mathematical pathways. It was aimed to provide valuable assessments on the reaction rate, plasmid spread, and other additional information regarding our proposed systems of BLIP-I and BLIP-II.

As we did not have any prior experience with the employment of computer-based models for microbial simulations. We turned to the dry lab people of NYCU-Formosa for pointers, where they generously suggested some processes that can be modeled using a depiction (Figure 1 is a remake).

With the first option (1. plasmid in) being the transformation of plasmids into the donor bacteria—transformation—where population dynamics of the plasmid, bacteria, and successfully transformed donors can be tested . The second (2. “penetration,” regards how plasmids enter the cells) being the one we picked to focus on—conjugation—the process of donors transmitting its plasmid to recipients. The third (3. enzyme) being the gene-of-interest’s expression, in this case BLIP, which describes the plasmid’s interaction with the original cell. The fourth (4. enzyme mec being “enzyme mechanism”) illustrates the enzyme effect, being the final stage of the system that answers the question of how effective the inhibition of the β-lactamases are.

After discussing and analyzing similar projects from previous Team’s (such as UFlorida 2021 [1], which heavy inspiration had been taken from), we’ve decided on modeling the process of conjugation.

Relating it back to the model, it would simulate bacterial conjugation as a means to track the process of plasmid spread between cells. By tracking the populations of donor, recipient, and transconjugant populations over time, we can compare the efficiency of different plasmids and predict the amount of donor bacteria that should be controlled to achieve a sufficient distribution of our BLIP genes. Using this framework and the formula we developed, the conjugation rate can be calculated, which quantifies the efficiency in which plasmids are shared. To get a better idea of what the process would be like, Figure 2 can be used to better visualize.

We used difference equations for the making of the model, as a few factors needed to be considered. However, initially, the dry lab group only tweaked the model from a previous team working on a similar topic—SDU-Denmark 2019 [2]—with respective variables and parameters that fit with the BLIP systems. The equations used were actual ordinary differential equations, which did not align well with the results we wanted. This situation was due to the initial lack of thought put into thinking about how cohesive it was to the overall project.

After having realized this, we turned to the mathematical expert suggested by our Primary Investigator, Mr. Peter Sun, who led us to the opportunity of using difference equations to relieve the complexity of the model and free the team from redundant work, contributing to an overall efficient outcome. Below is how the conjugation process goes with the appropriate parameters and variables on the sketched diagram.

We then revised our systems of equations to more effectively solve for the conjugation rate of BLIP-I and BLIP-II respectively. The equations below were also designed to complement the wet lab’s conjugation data to simulate the model in the most realistic scenario.

In addition, we have modeled conjugation dynamics under two scenarios: one in which donors were growing and one in which donor numbers remained constant. The purpose of comparing the two models (growing and constant) was to capture the range of possible biological outcomes. By testing both cases, we can evaluate how strongly donor growth influences conjugation efficiency and transconjugant accumulation over time. The non-growth model isolates the effect of conjugation alone, while the growth model shows the combined effect of conjugation and cell proliferation. Furthermore, tables with columns labeled "Hours," "Donors," "Recipients," and "Transconjugants" had been drafted for the wet lab group to collect the necessary data. Both scenarios have an expected incubating time of 24 hours. Below is one of the tables that outlines the circumstances of donor growth.

At last, the wet lab experiments could not progress to the conjugation stage around mid-September; numerous challenges and time constraints faced us as a new Team. So we relied more on computational simulations with most common conjugation input values to explore and predict potential outcomes. Doing this acknowledges experimental limitations while still providing meaningful predictions about how conjugation would behave under different assumptions. For better representation of what’s gained from the data, we plotted graphs by equations with the values input.

We chose donor and recipient populations of 1.0×10⁶ and 1.0×10⁷ cells/mL respectively, reflecting the common 1:10 ratio used in conjugation experiments [3].

Below are the graphs of growth—what the model represents, for both BLIP-I and BLIP-II under growing and ungrowing conditions. With the yellow function as donors, blue as recipients, and red representing transconjugants.

With results from the equations, we observed that the cells that consider growth of donors and recipients exhibit an exponential relationship with time as it progresses. This is due to the cells duplicating continuously across the entire time interval. In contrast, in a scenario where donors and recipients are set to without growth, the population trends become drastically different: the number of donors remains constant, acting as the plasmid reservoir, the recipient population gradually declines, and transconjugants rise in a logistic growth [5] curve. This highlights the inverse proportional relationship between transconjugants and recipients by isolating the effect of the conjugation process; an increase in transconjugants corresponds to a proportional decrease in recipients due to the continuous gene transfer.

However, interpretation of these relationships is limited as the wet lab experiment couldn’t progress to the process of performing BLIP expression assays, which renders us unaware of Transconjugant estimates for the system to successfully inhibit β-lactamase. Therefore, an alternative method had to be developed to simulate the gene’s expression and estimate the Transconjugant count, allowing for a more meaningful interpretation of the conjugation rate model. Finally, we chose to use COPASI. We can compensate for the absence of later experiments by again utilizing computational models. Key observations from these simulations are presented as Model 2 in the following section.

The results of the concentrations of substances in both BLIP expression models simulated in COPASI are presented in the graphs. Where cell concentration was measured in moles per liter and time was expressed in seconds. This modeling of BLIP expression emphasizes the activity of the type of BLIP (either BLIP-I or BLIP-II, shown in purple function as BLIP_1 or BLIP_2), what its impeding (β-lactamase, shown in green function as BLA), the enzyme inhibitor complex between BLA-BLIP interaction (shown in cyan function as Complex), and finally yet most importantly, the antibiotic of choice (ampicillin, shown in red function as Amp). Before the graphs, here are the rationales for the derived values of parameters and variables for the functions.

BLA mRNA Degradation:
The values used were based on published experimental data from PubMed (PMID: 12119387), which measured half-lives of mRNA in E. coli. The paper reports that about 80% of mRNAs have half-lives between 3 and 8 minutes [5]. We took the midpoint, being 5.5 minutes, as an approximate average half-life. Moreover, we employed the standard first-order decay equation, which had us calculate the corresponding mRNA degradation rate constant: about 0.126 min-1 (or 0.0021 s-1).

BLA Transcription:
We wanted to estimate how strong the promoter BBa_J23103[6] is in driving gene expression. First, we analyzed the published data from Baltimore BioCrew of 2019 [7], which showed that BBa_J23103 has a relative expression level of 0.73 compared to the standard promoter BBa_J23100, which was set to 1 [7]. To convert this into a more quantitative measure, we used the concept of Relative Promoter Units (RPU), which reflects how many RNA polymerases initiate transcription per second at a promoter.

The reference promoter BBa_J23101 has 1 RPU, corresponding to about 0.03 polymerases per second (PoPS), and BBa_J23103 was reported to have an RPU range of 0.77–0.96 [8]. So, we averaged it to be 0.87 RPU-Multiplying it by the PoPS of the reference promoter gives us an estimated transcription rate of 0.0261 polymerases per second for BBa_J23103. In other words, RNA polymerase starts transcription at this promoter roughly once every 38 seconds on average. This shows that BBa_J23103 is slightly weaker than BBa_J23101, but it still drives consistent gene expression.

To develop a model that simulates the interaction between BLIP and BLA, we modeled the following reactions:

BLA Translation:
To model β-lactamase (BLA) translation, we first estimated the rate at which TEM-1 (the BLA enzyme) is synthesized by E. coli ribosomes. TEM-1 consists of 286 amino acids, and ribosomes in E. coli typically elongate peptide chains at approximately 12–21 amino acids per second [9]. Using an average elongation rate of 18 amino acids per second, we calculated that it would take roughly 15.9 seconds to translate a full TEM-1 protein, corresponding to a production rate of 0.063 proteins per second. Converting this to minutes, approximately 3.8 proteins per minute, which matches our model parameter kₜₗ(bla) = 3.8/min. This calculation allowed us to set a biologically realistic translation rate for BLA in our model.

BLIP mRNA Degradation:
"A wide range of stabilities was observed for individual mRNAs of E. coli, although approximately 80% of all mRNAs had half-lives between 3 and 8 min." [5] Assuming that BLA’s mRNA and BLIP’s mRNA degrade at a similar rate:

BLIP Transcription:
Assumed to be the same as BLA Transcription.

BLIP Translation:
BLIP-1 is 186 amino acids in length. According to BioNumbers (ID 100059) [9], in E. coli ribosomes elongate peptide chains at about 12–21 amino acids per second (≈ 17 amino acids/s under typical conditions). Using an average rate of 18 aa/s, it would take ~ 186 ÷ 18 = 10.33 seconds for one ribosome to make a BLIP-1 protein. That means each ribosome produces ~ 1 ÷ 10.33 = 0.097 proteins per second under ideal conditions.

Amp Hydrolysis:
Function of Henri-Michaelis-Menten (irreversible) given by a ScienceDirect publication [10], shows:

How Henri-Michaelis-Menten applies to Amp hydrolysis:
- Substrate: Ampicillin ([Amp])
- Enzyme: β-lactamase
- Product: Waste

When there is only a little Amp, the reaction happens more slowly and is roughly proportional to the Amp concentration. When there is a lot of Amp, the enzyme becomes full and works at its maximum speed. Using this equation in our model lets us predict how fast Amp is degraded, which helps us understand how β-lactamase affects antibiotic activity in the system. Amp → Waste is represented using Michaelis-Menten kinetics to describe how efficiently β-lactamase can remove Amp over time. This helps simulate the decrease of Amp in the system and allows you to predict how much β-lactamase activity is needed to overcome antibiotic resistance.

Binding:

In our kinetic model, we explicitly simulate the binding interaction between BLIP (the inhibitor) and β-lactamase (BLA, the enzyme) to form a BLIP–BLA complex. To parameterize this interaction, we adopted binding kinetics values from the literature [11]. The association rate constant (kₒₙ) is given as 19 × 10⁵ M⁻¹ s⁻¹, and the dissociation rate constant (kₒff) is 29 × 10⁻⁷ s⁻¹. These constants define how quickly BLIP and BLA bind to each other and how fast the complex dissociates. These values were input into our COPASI model so that the binding step reflects realistic affinities: a relatively high kₒₙ (fast association) and a very low kₒff (slow dissociation) yield a stable inhibitor-enzyme complex. With those in place, our model can simulate dynamic changes in free BLIP, free BLA, and the complex over time under different BLIP-variant scenarios.

Analyzing the BLIP expression results modeled in COPASI, it can be clearly seen that the inhibition of β-lactamase exhibits different trends between BLIP-I and BLIP-II. With the former having the enzyme degraded consistently, however, at a relatively slow pace, resulting in the antibiotic getting hindered relatively quickly compared to BLIP-II, reaching below 1E-5 moles per liter at 1E-30 seconds. Turning the focus to BLIP-II dynamics, the relationship between BLIP and β-lactamase is rather the opposite than the former; BLIP concentration increases in very short time interval, resulting in β-lactamase being quickly impeded before it continues to degrade any more antibiotics, in turn, allowing the ampicillin to survive over 0.01 seconds before reaching below 1E-7 moles per liter in amount. With a difference of roughly 0.01 (0.01 - 1E-30) seconds of higher concentration (1E-7 mol/L) before it exits the graph, it’s indeed an extremely minor difference. Despite this, a minimal amount of variance can still prove a contrast; hence, BLIP-II is considered to degrade β-lactamase more efficiently than BLIP-I. However, this model has several limitations and is based on assumptions made to simplify it, including the absence of plasmid replication and simplified enzyme kinetics, which makes it less reflective of actual biological conditions. Therefore, although the simulations indicate that BLIP-II may be more effective, these findings should be interpreted with caution and confirmed through experimental validation.

Combining the results of both models, we concluded that transconjugants and recipients should have an inverse proportional relationship as conjugation proceeds, and that BLIP-II degrades β-lactamase more efficiently than BLIP-I. The former’s simple mathematical formula can easily be adopted by future Teams for modeling conjugative plasmids, RP4-based systems, or bacterial communication networks. The latter’s BLIP expression model clearly depicts BLIP concentration and β-lactamase activity, and can offer methods for drawing predictions from limited data. Overall, the model contributes a quantitative framework for evaluating protein-based inhibition strategies against antibiotic resistance enzymes, and future Teams will be able to adapt and expand from the proposed model.

1. Team: UFlorida/Model - 2021.igem.org. 2021.igem.org/Team:UFlorida/Model.
2. Team: SDU-Denmark/Model - 2019.igem.org. 2019.igem.org/Team:SDU-Denmark/Model.
3. Iannelli, F.; Santoro, F.; Fox, V.; Pozzi, G. A mating procedure for genetic transfer of integrative and conjugative elements (ICEs) of streptococci and enterococci. Methods and Protocols 2021, 4 (3), 59. https://doi.org/10.3390/mps4030059.
4. Logistic Growth Model.https://sites.math.duke.edu/education/ccp/materials/diffeq/logistic/logi1.html.
5. Bernstein, J. A.; Khodursky, A. B.; Lin, P.-H.; Lin-Chao, S.; Cohen, S. N. Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays. Proceedings of the National Academy of Sciences 2002, 99 (15), 9697–9702. https://doi.org/10.1073/pnas.112318199.
6. Part:BBA J23103 - parts.igem.org. https://parts.igem.org/Part:BBa_J23103.
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8. Kelly, J. R.; Rubin, A. J.; Davis, J. H.; Ajo-Franklin, C. M.; Cumbers, J.; Czar, M. J.; De Mora, K.; Glieberman, A. L.; Monie, D. D.; Endy, D. Measuring the activity of BioBrick promoters using an in vivo reference standard. Journal of Biological Engineering 2009, 3 (1), 4. https://doi.org/10.1186/1754-1611-3-4.
9. Rate of translation by ribosome - Bacteria Escherichia coli - BNID 100059. https://bionumbers.hms.harvard.edu/bionumber.aspx?id=100059&ver=26.
10. Naidja, A.; Huang, P. Significance of the Henri–Michaelis–Menten theory in abiotic catalysis: catechol oxidation by δ-MnO2. ScienceDirect 2001. https://www.sciencedirect.com/science/article/abs/pii/S0039602802013754.
11. Fryszczyn, B. G.; Adamski, C. J.; Brown, N. G.; Rice, K.; Huang, W.; Palzkill, T. Role of β‐lactamase residues in a common interface for binding the structurally unrelated inhibitory proteins BLIP and BLIP‐II. Protein Science 2014, 23 (9), 1235–1246. https://doi.org/10.1002/pro.2505.