# Isolated Enzymatic Kinetic Model

We start constructing our models by analyzing the localized genetic circuits. The in-depth study of concentration profiles, which is important with respect to the project as it predicts the theoretical yield of desired product (tagatose) in our system, is visualized through an enzymatic kinetic model.

The basics of our kinetic model is built upon two most fundamental rules: the law of Mass Action and the Michaelis-Menton model. Because these two laws combined are the most popular modelling paradigm for enzymatic kinetics, only preliminary explanations will be shown on this document. For the law of Mass Action, it is based on the assumption that the rate of a reaction is directly proportional to the product of the activities or concentrations of the reactants, thereby predicting the solutions in dynamic equilibrium of metabolites.

##### Figure 1. Three examples of reaction mechanisms and their respective rate laws

On the other hand, the Michaelis-Menten kinetics model is most often used to model enzyme kinetics. It is based on the idea that an enzyme reaction starts with the binding interaction of the enzyme and the substrate to form a complex; afterwards, it releases a product and regenerates the original enzyme.

##### Figure 2. Reaction mechanism for the Michaelis-Menton kinetics

Applying the law of mass action, the above model can be derived into a set of ODE’s to be solved simultaneously.

##### Figure 3. Incorporating the Michaelis-Menton kinetics to the Law of Mass Action

This system is relatively easy to solve; however, when a system involves hundreds of such ODE’s, we need to utilize packages specifically designed for computing concentration profiles.

The **M**ass-**A**ction Stoichiometric Simulation (MASS) toolbox is a modeling software package that focuses on the construction and analysis of kinetic models of biochemical reactions systems [1]. This Mathematica-based package is chosen as the carrier of our enzyme kinetic model, as it both has flexible means to manipulate the reactions and parameters and efficiently solves a set of ODE’s via linear algebra. However, the currently built model that is compatible with this tool box only includes model organisms such as E. coli, human RBC, CHO cell, etc. Given the perplexity of constructing a complete metabolic pathway of PCC 7942, only an **isolated model** specifically involving the conversion from sucrose to tagatose was constructed.

##### Figure 4. The MASS Toolbox combines all the dynamic mass balances for all concentrations x in a biochemical reaction network and present it by a matrix equation, where S is the stoichiometric matrix, v is the vector of reaction fluxes {vj}, and x is the vector of concentrations {xi}. This equation will be the “master” equation that will be used to describe network dynamics states.

##### Figure 5. Our proposed pathway involves 4 enzymes, 6 grand enzymatic reactions which are analyzed by a Michaelis-Menten approach and divided into 21 sub-reactions, 8 metabolites, 17 enzyme-substrate complexes, 21 pre-set parameters (Keq), and 14 fitted parameters (k).

Because enzyme kinetic models give concentration profiles, we also need to consider other parameters. The equilibrium constants (Keq) of each binding/catalyzing reaction are found in different articles. Note that sometimes this constant is portrayed as the activity of the enzyme with a certain substrate. To ensure the success of the model construction, all the rate constants are either fitted or arbitrarily set.

After combining all the catalytic branches and updating all the equilibrium constants, the rate constants of activation and catalysis are fitted, and those of inhibition interaction are set arbitrarily high. Since myo-inositol does not play the role of a reactant, its concentration is set arbitrarily. Also, sucrose and water concentration are set arbitrarily high, assuming an endless source of such substrates.

##### Figure 6. The catalytic branches (sub-reactions) of the three modules.

##### Table 1. Table of equilibrium constants found in literature

##### Table 2. Table of rate constants fitted by ourselves

After configuring the model, we can finally begin analyzing the result. First, we wanted to visualize the system by displaying the complete (excluding the enzymes and enzyme-substrate complexes) concentration profile of the two-cycled system. We can see that several reactants get readily consumed and formed to tagatose. On the other hand, since sucrose is in another order of magnitude, it does not experience significant changes in the graph.

After we have confirmed that the model is integrated as expected, we started to tweak the initial concentrations of the enzyme- and substrate-of-interest: GLA and GOL. GLA involves with both cycles of reactions and is essential for the production of galactose, the precursor of tagatose. GOL is the only reactant that is not naturally present in cyanobacteria, therefore its concentration needs to be manipulated to see how it affects the overall system. The following four figures represent the concentration profiles of tagatose and galactose under different GLA and GOL initial concentrations. **The results match the intuition, indicating the amount of enzyme added to the system affects the efficiency, while that of the substrate will increase the overall yield**.

At the end of the day, we want to see how by knocking out the STS gene module (thereby knocking out an entire reaction loop) will affect the overall tagatose yield and production efficiency. Therefore, in our model, we eliminated the gene STS and all the reactions/metabolites relating to that, creating a one-cycled system co-existing with our original two-cycled model. Now that we have these two parallel models, we plot the tagatose concentrations of both systems together given the same conditions. GLA and GOL are again being manipulated, reasons given above. The graphs imply that while the model containing an extra reaction cycle leads to significantly faster reaction, the final yield of tagatose is not that greater than that with only one reaction cycle. **We have concluded that it is justifiable for us to construct the prototype cyanobacterial strain with only one reaction cycle involving the addition of four genes.** Since it will be much easier to incorporate one less gene (STS is ~3kbp) in a wet-lab setting, it increases the success rate significantly while not compromising the product yield to a great extent.

Next, we perform the perturbation analysis to understand the sensitivity of our circuit with respect to different environmental changes. In order to do so, the model needs to be first set to steady states. After that, either the concentration of a certain metabolite or the parameter (usually a rate constant) is altered, and concentration profile is observed and analyzed. The figure below (Figure.4) demonstrates that the system is set to the steady state (all the concentrations remain constant as expected).

We want to see how sensitive our circuit will react to a change of sucrose concentration (via salt induction) at steady state. According to the law of Mass Action, more reactant should push the reactions to the right side, given a constant Keq. However, as we manipulate the sucrose concentration (the following four figures represents the concentration profile after an increase/decrease of sucrose initial concentration), even though the pattern of change of tagatose concentration follows intuition, the change of concentration magnitude is not obvious - the concentration remains relatively the same, suggested by the y-axis. **Therefore, we conclude that the system is relatively insensitive to change of sucrose environment, and there is no need to alter the sucrose concentration in order to produce more tagatose so long as it can be treated as an infinite source.**

#### Reference

[1] N. Jamshidi and B. Ø. Palsson, *Biophysical Journal*, 2010, 98, 175-185.

[2] T. Peterbauer, L. Mach, J. Mucha and A. Richter, *Planta*, 2002, 215, 839-846.

[3] T. Peterbauer and A. Richter, *Plant Physiology*, 1998, 117, 165-172.

[4] W. Tanner and O. Kandler, *European Journal of Biochemistry*, 1968, 4, 233-239.

[5] J. G. Ferreira, A. P. Reis, V. M. Guimarães, D. L. Falkoski, L. D. S. Fialho and S. T. D. Rezende, *Applied Biochemistry and Biotechnology*, 2011, 164, 1111-1125.