Modeling Modeling
Introduction

Overview

In synthetic biology, Ordinary Differential Equation (ODE)-based models are essential for the precise modeling of gene regulatory circuits. These models allow for the mathematical description of dynamic interactions between genes, proteins, and molecules, facilitating insights into how gene expression changes over time1. Using ODE-based models, precise control over gene expression is achieved, ensuring the engineering of tailored biological functions in synthetic biology2. For our project’s modeling, we have developed a comprehensive mathematical model tailored to the biological components of our project. The primary objective of our model is to enhance the design process in the wet lab by providing a structured, in silico representation that emphasizes modularity.

The System

The toxicity of oil mill wastewater (OMW) makes it a particularly destructive by-product capable of causing serious environmental complications. This is primarily associated with the high concentrations of phenolic compounds in OMW. Given these circumstances, the iGEM Thessaly wet lab team undertook the development of a system comprising two bacteria that collaborate in effectively degrading these phenolic compounds.
P. putida possesses the necessary mechanisms for the degradation of synthetic phenolic compounds, concurrently yielding a valuable by-product known as Polyhydroxyalkanoates (PHAs). The use of E. coli is equally decisive, as it has an assistive role in the degradation of phenolic compounds. This is achieved through a mechanism based on the regulation of laccase production. The mechanisms are based on a system of controlled protein expression by different regulatory factors. Utilizing activation and inhibition, the expression of each gene in the system can be regulated and controlled to produce the required output value. The integration of P. Putida and E. Coli inside a microbial consortium provides a promising answer for minimizing the environmental effect of OMW and additionally provides a very valuable product (PHAs). For further information regarding this bacterial synergy, visit our Design page.

Our Aim

Using differential equations as the fundamental mathematical approach, the focus of the modeling is to define the dynamics of the two-bacteria system with sufficient accuracy to obtain a better estimation of protein and enzyme production rates. Some molecules with high monitoring value are PcaU, FadR and LacI as they can regulate the expression of other genes as inhibitors or activators. The simulation should provide useful information for the clarification of the dependencies regulating the kinetics of molecule productions. It should also help determine the best possible promoter to achieve a faster expression of the gene. The main objective is to develop a model that approximates reality to a limited but acceptable degree. Our modeling work serves the following key purposes:

Requirements

Assumptions

The model developed does not fully correspond to reality as it is exceedingly complicated to represent multiple complex processes with a simple deterministic approach. In consideration of the project's optimal advancement, several assumptions have been defined.

  • In enzymatic reactions, the conversion into an enzyme-product complex and its dissociation into complex and product happens directly and without intermediate steps.
  • The binding of transcription factors to gene promoters is not modelled since the transcription reaction rate of the regulated genes also shows their association with the enzymes.
  • Enzymes that act as transcription factors are considered constant values. The modelers can change their values, but they remain fixed in each iteration.
  • The transcription rate is proportional to the gene copy number, and the translation rate is proportional to the mRNA number when the process isn’t controlled by an inhibitor or an activator.
  • There are no external disturbances that could affect the system. Environmental conditions are fixed.
  • The concentration of RNA polymerase is constant and sufficient throughout the duration of transcription.
  • The exit of mRNA from the nucleus is negligible.
  • The nutrients necessary for gene expression are sufficient for the process.
  • All the molecules are evenly distributed in the cell.
  • Binding and Unbinding processes are much faster than transcription and translation.
  • Protein degradation includes growth-associated dilution.

    • P. putida Model

    Model Overview



    Employing the MATLAB tool, SimBiology, we formulated a computational model of Pseudomonas putida composed of three distinct subsystems. These subsystems are:

  • The pcaU system. With the PCA intermediate and the regulatory protein PcaU, we can control the production of the enzymes PhaG and AlkK. PCA binds to the PcaU and forms a complex. The produced complex (PcaU-PCA) connects to the promoter of the phaG-alkK gene and acts as an activator of the gene’s expression.
  • The fadR system. In this system FadR binds with FFAs producing a complex (FadR-FFAs). Naturally, FadR connects with the promoter of the phaJ, repressing the expression of the gene.
  • The phaC1-phaC2 system. The gene pathway regulating the expression of PHAC1 and PHAC2 enzymes is the focus of analysis in this experimental system. The resulting polycistronic mRNA molecule is translated, producing the key enzymes PHAC1 and PHAC2. These enzymes help by catalyzing the final process of PHA biosynthesis.

    • Parameters & Equations

    To analyze the mechanisms of our model, it is essential to specify and define the associated parameters. Careful selection of parameters is critical, as it is the foundation on which further evaluations are based. This involves a thorough examination of scientific papers, MATLAB Projects and model architectures.







    In developing our model, we used a combination of ordinary differential equations (ODEs) to describe key processes such as translation, binding and transcription. This deterministic approach allows us to understand the complex interaction of molecules, enabling a better visualization of the regulation of gene expressions. While most of the equations are relatively simple, the repression of a gene’s expression from another transcriptional factor (enzyme) is considered and approached with the necessary equations.



    Model Simulation

    The plots generated by the model simulation provide a visual representation of the concentrations of each species in the subsystems. This visualization provides valuable insights into the dynamic behavior of the biochemical network. By communicating effectively with the wet lab team, we successfully conducted experiments to modify various parameters in the model, to meet specific requirements of the laboratory research. Main objective of the simulation was to uncover the regulatory mechanisms underlying the expression of the enzymes and the prior processes that controlled their concentration levels. The plots in figure 3 showed the concentration levels of the target enzymes that are mainly Phaj, PHAC1, PHAC2, AlkK and PhaG. We adjusted the parameters that controlled their expression such as the translation rates, binding and unbinding rates and the transcription rates. On example is the adjustments on the transcription rate of phaj mRNA where the repression equation was used.



    E. coli Model

    Model Overview





    Employing the MATLAB tool, SimBiology, we formulated a computational model of Escherichia Coli composed of three distinct subsystems. These subsystems are:

    1. The araC system. This system involves regulating the production of the enzyme laccase by utilizing the enzyme AraC while using Arabinose. AraC reacts with arabinose, and they form a complex which acts as an activator for the transcription of the lac gene.
    2. The TE system. This is a rather simple subsystem that can provide useful information about the transcription of the TE gene and its translation to produce Thioesterase.
    3. The FapR system. This system serves to regulate the expression of the enzyme carboxylase of acetyl-CoA (ACC), which is crucial for the catalytic conversion of acetyl-CoA to malonyl-CoA. This regulatory mechanism is accomplished through the regulated intervention of transcription factors, LacI and FapR, which act as repressors. Basically, the system incorporates a negative feedback loop, with FapR and LacI having inhibitory effects. The whole process depends on the presence of two critical molecules, Malonyl-CoA and isopropyl β-D-1-thiogalactopyranoside (IPTG). LacI forms a complex with IPTG, blocking its interaction with the promoter region of the ACC gene. Similarly, FapR forms a complex with Malonyl-CoA. Therefore, this prevents FapR from affecting the expression of the gene. The gene expression proceeds successfully. By adjusting the values of these molecules, we could extract useful information about how this oscillatory behavior affects enzyme production.



    Parameters & Equations

    To analyze the mechanisms of our model, it is essential to specify and define the associated parameters. Careful selection of parameters is critical, as it is the foundation on which further evaluations are based. This involves a thorough examination of scientific papers, MATLAB Projects and model architectures.







    When developing this model, we focused on understanding how oscillators and feedback loops work. For the FapR system to operate, a series of rules and abbreviations needs to be followed. The negative feedback loop is rather sensitive, and a small mistake can alter results. With this system that we proposed, our aim is to provide a solid foundation for one to build upon and understand the biological mechanism of E. coli.



    Model Simulation

    In our simulation, the plots strongly display the concentrations of each species in all systems, providing a complete picture of their dynamic interactions. Our system features a negative feedback loop, which is visually evident in the diagrams through discrete oscillations. These oscillations, characteristic of dynamical systems, reflect periodic variations in the concentrations of species over time. By changing the hill coefficient (n), oscillations became more frequent with its increase and less frequent with its decrease. Experimenting with other parameters helped when analyzing the plots and the results with the wet lab team.





    PHA Extraction Model

    Model Overview





    Employing the MATLAB tool, SimBiology, we formulated a computational model of the PHA recovery model that focuses on biosafety composed of one distinct subsystem. This model was used primarily as an experimental design rather than a tool to extract analytical data.

    1. The mekR system. This regulatory mechanism focuses on the controlled production of the Lys enzyme using a MekR enzyme in combination with the MEK substrate. The interaction between MekR and MEK leads to the formation of a complex, which then serves as a potent activator for the transcriptional process of the lys gene. This tuned regulatory cascade highlights a key mechanism in cellular homeostasis.



    Parameters & Equations

    The parameters used in the construction of this system's model are depicted below:





    The equations used in this model are similar to the equations of araC system of P. putida Model.

    Model Simulation

    This simlutation helped our team observe the Lys enzyme regulation. Here, the plots display the concentrations of each species in this system, providing a complete picture of their dynamic interactions.



    Conclusion

    Discussion

    Despite the limitations associated with the modeling process, the result is encouraging, as the resulting models provided useful information. The models demonstrated a good level of efficiency, providing powerful data that enriched our understanding of the project’s biological phenomena. By utilizing these data, we were able to provide valuable insights to our colleagues in the wet lab, aiding their experimental design. In particular, the implementation of the FapR negative feedback loop emerged as the most effective regulatory mechanism, showing effectiveness in understanding regulating system behavior. The results highlight the potential of our model as a fundamental structure for constructing more complex and integrated models for the microbial consortium. Eventually, our ambition is to reflect the complex reality of the oPHAelia microbial consortium within our simulations, trying to provide future iGEM teams with a strong basis in their attempts to work with similar complex systems.

    You can access our complete modeling documentation here:



    References

    1. Chandran, D., Copeland, W. B., Sleight, S. C., & Sauro, H. M. (2008). Mathematical modeling and synthetic biology. Drug discovery today. Disease models, 5(4), 299–309. https://doi.org/10.1016/j.ddmod.2009.07.002
    2. Zheng, Y., & Sriram, G. (2010). Mathematical modeling: bridging the gap between concept and realization in synthetic biology. Journal of biomedicine & biotechnology, 2010, 541609. https://doi.org/10.1155/2010/541609
    3. Liu, D., & Zhang, F. (2018). Metabolic Feedback Circuits Provide Rapid Control of Metabolite Dynamics. ACS Synthetic Biology, 7(2), 347–356. doi:10.1021/acssynbio.7b00342
    4. Selinger, Saxena, Cheung, Church, & Rosenow. Global RNA half-life analysis in Escherichia coli reveals positional patterns of transcript degradation. Retrieved October 11, 2023, from https://arep.med.harvard.edu/rna_decay/
    5. Sato, S., Kanazawa, H. & Tsuge, T. Expression and characterization of (R)-specific enoyl coenzyme A hydratases making a channeling route to polyhydroxyalkanoate biosynthesis in Pseudomonas putida . Appl Microbiol Biotechnol 90, 951–959 (2011). https://doi.org/10.1007/s00253-011-3150-5
    6. Trautwein, G., & Gerischer, U. (2001, February). Effects Exerted by Transcriptional Regulator PcaU from Acinetobacter sp. Strain ADP1. Journal of Bacteriology, 183(3), 873–881. https://doi.org/10.1128/jb.183.3.873-881.2001
    7. Zimmermann, T., Sorg, T., Siehler, S. Y., & Gerischer, U. (2009, April 15). Role of Acinetobacter baylyi Crc in Catabolite Repression of Enzymes for Aromatic Compound Catabolism. Journal of Bacteriology, 191(8), 2834–2842. https://doi.org/10.1128/jb.00817-08
    8. Wang, X., Lin, L., Dong, J., Ling, J., Wang, W., Wang, H., … Yu, X. (2018). Simultaneous improvements of cell growth and polyhydroxyalkanoate production in Pseudomonas from a lignin derivative for lignin consolidated bioprocessing. Applied and Environmental Microbiology, AEM.01469–18. doi:10.1128/aem.01469-18
    9. J. Rawlings, B. (1998). Biosynthesis of fatty acids and related metabolites. Natural Product Reports, 15(3), 275. doi:10.1039/a815275y
    10. Wang, Q., Tappel, R. C., Zhu, C., & Nomura, C. T. (2011). Development of a New Strategy for Production of Medium-Chain-Length Polyhydroxyalkanoates by Recombinant Escherichia coli via Inexpensive Non-Fatty Acid Feedstocks. Applied and Environmental Microbiology, 78(2), 519–527. doi:10.1128/aem.07020-11
    11. Thomas, B. S., Okamoto, K., Bankowski, M. J., & Seto, T. B. (2013). A Lethal Case of Pseudomonas putida Bacteremia Due to Soft Tissue Infection. Infectious diseases in clinical practice (Baltimore, Md.), 21(3), 147–213. https://doi.org/10.1097/IPC.0b013e318276956b
    12. Ballerstedt, H., Volkers, R. J., Mars, A. E., Hallsworth, J. E., dos Santos, V. A., Puchalka, J., van Duuren, J., Eggink, G., Timmis, K. N., de Bont, J. A., & Wery, J. (2007). Genomotyping of Pseudomonas putida strains using P. putida KT2440-based high-density DNA microarrays: implications for transcriptomics studies. Applied microbiology and biotechnology, 75(5), 1133–1142. https://doi.org/10.1007/s00253-007-0914-z
    13. Loeschcke, A., & Thies, S. (2015). Pseudomonas putida-a versatile host for the production of natural products. Applied microbiology and biotechnology, 99(15), 6197–6214. https://doi.org/10.1007/s00253-015-6745-4
    14. Bonner, W. M., & Bloch, K. (1972). Purification and properties of fatty acyl thioesterase I from Escherichia coli. Journal of Biological Chemistry, 247(10), 3123-3133.
    15. Yang, J., Wang, G., Ng, T. B., Lin, J., & Ye, X. (2016). Laccase Production and Differential Transcription of Laccase Genes in Cerrena sp. in Response to Metal Ions, Aromatic Compounds, and Nutrients. Frontiers in microbiology, 6, 1558. https://doi.org/10.3389/fmicb.2015.01558
    16. Janusz, G., Pawlik, A., Świderska-Burek, U., Polak, J., Sulej, J., Jarosz-Wilkołazka, A., & Paszczyński, A. (2020). Laccase Properties, Physiological Functions, and Evolution. International journal of molecular sciences, 21(3), 966. https://doi.org/10.3390/ijms21030966
    17. Stricker, J., Cookson, S., Bennett, M. et al. A fast, robust and tunable synthetic gene oscillator. Nature 456, 516–519 (2008). https://doi.org/10.1038/nature07389
    18. Daniel, R., Rubens, J., Sarpeshkar, R. et al. Synthetic analog computation in living cells. Nature 497, 619–623 (2013). https://doi.org/10.1038/nature12148
    19. Megerle, J. A., Fritz, G., Gerland, U., Jung, K., & Rädler, J. O. (2008). Timing and dynamics of single cell gene expression in the arabinose utilization system. Biophysical journal, 95(4), 2103–2115. https://doi.org/10.1529/biophysj.107.127191
    20. Shraddha, Shekher, R., Sehgal, S., Kamthania, M., & Kumar, A. (2011). Laccase: microbial sources, production, purification, and potential biotechnological applications. Enzyme research, 2011, 217861. https://doi.org/10.4061/2011/217861
    21. Uri, M. (n.d.). Half-life of AraC protein in vivo. Half-life of AraC protein in vivo - Bacteria Escherichia coli - BNID 105138. https://bionumbers.hms.harvard.edu/bionumber.aspx?id=105138&ver=1&trm=araC%2BmRNA&org=
    22. Leon Y Chan, Christopher F Mugler, Stephanie Heinrich, Pascal Vallotton, Karsten Weis (2018) Non-invasive measurement of mRNA decay reveals translation initiation as the major determinant of mRNA stability eLife, https://doi.org/10.7554/eLife.32536
    23. Schujman, G. E., Paoletti, L., Grossman, A. D., & de Mendoza, D. (2003). FapR, a Bacterial Transcription Factor Involved in Global Regulation of Membrane Lipid Biosynthesis. Developmental Cell, 4(5), 663–672. doi:10.1016/s1534-5807(03)00123-0
    24. Heath, R. J., White, S. W., & Rock, C. O. (2001). Lipid biosynthesis as a target for antibacterial agents. Progress in Lipid Research, 40(6), 467–497. doi:10.1016/s0163-7827(01)00012-1
    25. Parameters of the lactose repressor according to mathematical model. BioNumbers. Retrieved October 11, 2023, from https://bionumbers.hms.harvard.edu/bionumber.aspx?id=106217&ver=2&trm=LacI&org=
    26. Graf, N., Altenbuchner, J. Functional characterization and application of a tightly regulated MekR/P mekA expression system in Escherichia coli and Pseudomonas putida . Appl Microbiol Biotechnol 97, 8239–8251 (2013). https://doi.org/10.1007/s00253-013-5030-7
    27. Hoffmann, J., & Altenbuchner, J. (2015). Functional Characterization of the Mannitol Promoter of Pseudomonas fluorescens DSM 50106 and Its Application for a Mannitol-Inducible Expression System for Pseudomonas putida KT2440. PLOS ONE, 10(7), e0133248. doi:10.1371/journal.pone.0133248
    28. Inoue, I., & Rechsteiner, M. (1994). On the relationship between the metabolic and thermodynamic stabilities of T4 lysozymes. Measurements in Escherichia coli. The Journal of biological chemistry, 269(46), 29241–29246. https://doi.org/10.1016/S0021-9258(19)62036-3
    29. Bergamo, A., Gerdol, M., Pallavicini, A., Greco, S., Schepens, I., Hamelin, R., Armand, F., et al. (2019). Lysozyme-Induced Transcriptional Regulation of TNF-α Pathway Genes in Cells of the Monocyte Lineage. International Journal of Molecular Sciences, 20(21), 5502. MDPI AG. Retrieved from http://dx.doi.org/10.3390/ijms20215502
    30. Lin, J., & Amir, A. (2018, October 29). Homeostasis of protein and mRNA concentrations in growing cells. Nature Communications; Nature Portfolio. https://doi.org/10.1038/s41467-018-06714-zBergamo, A., Gerdol, M., Pallavicini, A., Greco, S., Schepens, I., Hamelin, R., Armand, F., et al. (2019). Lysozyme-Induced Transcriptional Regulation of TNF-α Pathway Genes in Cells of the Monocyte Lineage. International Journal of Molecular Sciences, 20(21), 5502. MDPI AG. Retrieved from http://dx.doi.org/10.3390/ijms20215502
    31. MIS for BCMP 309 (drug delivery and clearance example based on SimBiology Intro file by JW). Modeling with SimBiology. Quantitative Methods Boot Camp.
    32. Semsey, S., Jauffred, L., Csiszovszki, Z., Erdossy, J., Stéger, V., Hansen, S., & Krishna, S. (2013). The effect of LacI autoregulation on the performance of the lactose utilization system in Escherichia coli. Nucleic acids research, 41(13), 6381–6390. https://doi.org/10.1093/nar/gkt351