# Kinetic law

###### Calculation and Derivation

### Kinetic law of Modeling

Mathematic model helps us to predict the reaction result before experiment. First of all, we have to define which kinetic law can be applied into which metabolic pathway and chemical equation. There are three kinds of kinetic we chose in this model, mass action, Henri-Michaelis-Menten, and Ping-Pong-Bi-Bi.

- Mass action
- Henri-Michaelis-Menten
- Ping-Pong-Bi-Bi

In chemistry, the law of mass action is the proposition that the rate of a chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. Mass action kinetics describes this behavior as an equation where the velocity or rate of a chemical reaction is directly proportional to the concentration of the reactants.

$${aA + bB \xrightarrow{Kf} cC + dD}$$

$${aA + bB \xrightarrow{Kr} cC + dD}$$

$${Kf = {[C]^c[D]^d \over [A]^a[B]^b}}$$

$${Kr = {[A]^a[B]^b \over [C]^c[D]^d}}$$

Where K_{f} is forward rate of equilibrium, K_{r} is reverse rate of equilibrium

In 1913, Michaelis and Menten studied the invertase system and found some behavioral patterns of enzyme-matrix reaction. When the enzyme is catalyzed, the matrix will be combined with the enzyme to generate the product. However, the binding of the enzyme to the matrix is reversible. The concentration of ES will be constant when the reaction is steady state. They also proposed the mathematical description of the behavior of the enzyme. They can express the relationship between the reaction rate and the enzyme mathematically. Hence, the reaction rate is in a hyperbolic relationship with the concentration of substrate at a fixed amount of enzyme. That is the Michaelis-Menten kinetic law.

A. The Henri-Michaelis-Menten kinetic law is expressed as follows:

$${E + S \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} ES \xrightarrow{k_2} E + P}$$

The Enzyme (E) forms a Complex (ES) with the Substrate (S) and modifies it to the Product (P). The Product is then released and the Enzyme is free to perform another modification. This process is described by the ODE system.

The system cannot be solved analytically. Michaelis and Menten considered a quasi equilibrium between E and the ES-Complex to simplify the system.

$${{d[S] \over dt} = -k_1[E] \cdot [S]+ k_{-1}[ES]}$$

$${{d[ES]\over dt} = k_1[E] \cdot [S] - (k_{-1} + k_2)[ES]}$$

$${{d[E] \over dt} = k_1[E] \cdot [S] - (k_{-1} + k_2)[ES]}$$

$${{d[P] \over dt} = k_2[ES]}$$

After derivation $${v = {-ds \over dt} = {d[P] \over dt}}$$

where V_{m} represents the maximum rate achieved by the system,
and the Km is the substrate concentration at which the reaction rate is half of V_{m}.

The constants v_{max} and Km can be looked up at the public data base BRENDA.
The rate konstant k_{2} describes how many substrate molecules are transformed into product molecules per second and active site,
it is thus called turnover number.

$${Vmax = k_2 \cdot [E_{total}] = k_{cat} \cdot [E_{total}]}$$

B. A more realistic description of a enzymatic reactions than pure Michaelis Menten Kinetics is given by considering the product forming reaction step as reversible.

$${E + S \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} ES \xrightarrow{k_2} E + P}$$

The Enzyme (E) forms a Complex (ES) with the Substrate (S) and modifies it to the Product (P). In the reversible Michaelis-Menten reaction the Product can bind the enzyme again and react back to the substrate. This process is described by the ODE system.

$${{d[S] \over dt} = -k_1[E] \cdot [S] + k_{-1}[ES]}$$

$${{d[ES] \over dt} = k_1[E] \cdot [S] + k_{-2}[E] \cdot [P] - (k_{-1} + k_2)[ES]}$$

$${{d[E] \over dt} = k_1[E] \cdot [S] - k_{-2}[E] \cdot [P] - (k_{-1} + k_2)[ES]}$$

$${v = {d[P] \over dt} = k_2[ES] - k_{-2}[E]\cdot [P] = v_f - v_b}$$

After derivation $${v = {-ds \over dt} = {d[P] \over dt}}$$

Finally we use V_{fmax} = k_{2}⋅E_{total} and V_{bmax} = k_{−1}⋅E_{total} to get the common form for the reversible Michaelis Menten equation

$${v = {v^{max}_f S/K_{m,1} - v^{max}_b P/K_{m,2} \over (1 + S/K_{m,1} + P/K_{m,1})}}$$

Where V_{f} is the forward rate and the Vb is backwared rate while $${K_{m,1} = {K_{-1} + K_2 \over K_1}}$$ or represents as K_{S},
and $${K_{m,2} = {K_{-1} + K_2 \over K_{-2}}}$$ or represent as K_{P}.

For Ping-Pong-Bi-Bi, it is a double displacement reactions. Hence, there is more than one matrix in the reaction, and there may be several products for multi-matrix reactions in most of enzyme reactions. For example, S1 + S2 → P1 + P2. Binding of A and B are the different enzyme forms, E and F respectively. The double matrix reaction can still be applied to the Henri-Michaelis-Menten formula, but the Km of the two matrices should be determined separately. In addition, the concentration of S2 in the reaction should be saturated when S1 is measured.

### Method of our Modeling

We choose the simbiology in matlab to develop our model. SimBiology provides an app and programmatic tools to model, simulate, and analyze dynamic systems which can easily applied on systems biology.

### Pathway of our project

#### Part A : (CO_{2} uptake model)

CO_{2} diffuse into cell can use Fick’s law.

$${[CO_2]_{air} \rightarrow [CO_2]_{uptake}}$$

Assume *E. coli* was a sphere with 10^{-6} radius while the depth of membrane was 4*10^{-9}

$${J = {-D{dφ \over dx}}}$$

J is the "diffusion flux," of which the dimension is amount of substance per unit area per unit time. D is the diffusion coefficient or diffusivity

After CO_{2} diffused into cell, it will have a reversible reaction.

$${CO_2 + H_2O \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} H^+ + HCO_3^-}$$

That’s what CA catalyze the reversible interconversion of CO_{2} and HCO_{3}^{-}.
Although some CAs prefer the direction of CO_{2} hydration,
the carboxysomal CAs in cyanobacteria and some chemoautotrophic bacteria favor the direction of HCO_{3}^{-} dehydration.
The result of model and our experiments showed that CA improve amount of CO2 uptake and CO2 utilization.

Fig 1. CO_{2} uptake model diagram of simbiology

#### Part B : (Xylose – PRK – Rubisco model)

We clone our genes into *E. coli* to make it fix the carbon dioxide.
The genes include PRK, Rubisco. On the other hand,
we also want to produce the higher value product in our project.

Equation | Enzyme | Kinetic law |
---|---|---|

$${Xylose \rightarrow Xylulose}$$ $${Xylulose + ATP \rightarrow X5P}$$ $${X5P \rightarrow Ru5P}$$ | xylose isomerase | Henri-Michaelis-Menten |

$${Ru5P \xrightarrow{PRK} + RuBP}$$ | Phosphoribulokinase | Henri-Michaelis-Menten |

$${RuBP + CO_2 \xrightarrow{Rubisco} + 3PG}$$ | Ribulose-1,5-bisphosphate carboxylase/ oxygenase | Ping-Pong-Bi-Bi |

Fig 2. CO_{2}-utilization bypass model diagram of simbiology

#### Part C : (Original metabolism model)

It’s a original pathway which xylose is consumed by *E. coli*. Basically,
*E. coli* will comsume the xylose and then produce the temporary production, pyruvate.

Equation | Enzyme | Kinetic law |
---|---|---|

$${X5P + R5P \rightarrow S7P + GAP}$$ | transketolase | Henri-Michaelis-Menten |

$${S7P + GAP \rightarrow F6P + E4P}$$ | transaldolase | Henri-Michaelis-Menten |

$${E4P + X5P \rightarrow F6P + GAP}$$ | transaldolase | Henri-Michaelis-Menten |

$${FBP \rightarrow GAP}$$ | fructose-bisphosphate aldolase | Henri-Michaelis-Menten |

$${DHAP \rightarrow GAP}$$ | triose-phosphate isomerase | Reversible Henri-Michaelis-Menten |

$${F6P + ATP \rightarrow FBP + ADP}$$ | 6-phosphofructokinase | Henri-Michaelis-Menten |

$${F6P \rightarrow GAP}$$ | glucose-6-phosphate isomerase | Henri-Michaelis-Menten |

$${G6P + NADP^+ \rightarrow 6PG + NAPDH}$$ | 6-phosphogluconolactonase | Ping-Pong-Bi-Bi |

$${6PG + NADP^+ \rightarrow Ru5P + NAPDH}$$ | phosphogluconate dehydrogenase (NADP+-dependent, decarboxylating) | Henri-Michaelis-Menten |

$${GAP + NADP^+ + ADP \rightarrow 3PGA + NAPDH + ATP}$$ | glyceraldehyde-3-phosphate dehydrogenase (NADP+) | Henri-Michaelis-Menten |

phosphoglycerate kinase | ||

$${3PGA \rightarrow PEP}$$ | phosphoglycerate mutase | Henri-Michaelis-Menten |

phosphopyruvate hydratase | ||

$${PEP \rightarrow PYR}$$ | pyruvate kinase | Henri-Michaelis-Menten |

Fig 3. Original metabolism model diagram of simbiology

Fig 4. Xylose and pyruvate result of original pathway

Fig 5. X5P result of original pathway

Fig 6. Ru5P result of original pathway

Fig 7. 3PGA result of original pathway

Fig 8. R5P result of original pathway

Fig 9. GAP result of original pathway

Fig 10. S7P result of original pathway

Fig 11. F6P result of original pathway

Fig 12. E4P result of original pathway

Fig 13. FBP result of original pathway

Fig 14. DHAP result of original pathway

Fig 15. G6P result of original pathway

Fig 16. 6PG result of original pathway

Fig 17. PEP result of original pathway

#### Part D : (TCA cycle model)

In Tricarboxylic acid cycle (TCA cycle),
we can get final production in this cycle. The two carbons of Acetyl CoA are decomposed into carbon dioxide to produce energy,
form ATP or GTP, and reduce NADH and FADH_{2},
and then provide the energy needed for the reaction in the organism.

Equation | Enzyme | Kinetic law |
---|---|---|

$${PEP + HCO_3^- + NAD^+ \rightarrow OAA + NADH}$$ | PEP carboxylase | Reversible Henri-Michaelis-Menten |

$${AcCoA \rightarrow CIT}$$ | citrate synthase | Reversible Henri-Michaelis-Menten |

$${cIT \rightarrow ICIT}$$ | aconitate hydratase | Reversible Henri-Michaelis-Menten |

$${ICIT + NADP^+ \rightarrow AKG + NADPH + CO_2}$$ | isocitrate dehydrogenase | Reversible Henri-Michaelis-Menten |

$${AKG + NAD^+ \rightarrow SUCCoA + NADH + CO_2}$$ | dihydrolipoyllysine-residue succinyltransferase | Reversible Henri-Michaelis-Menten |

$${SUCCoA + ADP \rightarrow SUCC + ATP}$$ | succinate-CoA ligase (ADP-forming) | Reversible Henri-Michaelis-Menten |

$${SUCC + FAD \rightarrow FUM + FADH_2}$$ | succinate dehydrogenase | Reversible Henri-Michaelis-Menten |

$${FUM \rightarrow MAL}$$ | fumarate hydratase | Reversible Henri-Michaelis-Menten |

$${MAL + NAD^+ \rightarrow OAA + NADH}$$ | malate dehydrogenase | Reversible Henri-Michaelis-Menten |

$${OAA \rightarrow CIT}$$ | citrate synthase | Reversible Henri-Michaelis-Menten |

$${PYR \rightarrow NADH + AcCoA + CO_2}$$ | pyruvate dehydrogenase complex | Henri-Michaelis-Menten |

$${ICIT \rightarrow GOX}$$ | isocitrate lyase | Reversible Henri-Michaelis-Menten |

$${GOX \rightarrow MAL}$$ | malate synthase | Reversible Henri-Michaelis-Menten |

$${ICIT \rightarrow SUCC + GOX}$$ | isocitrate lyase | Reversible Henri-Michaelis-Menten |

Fig 18. TCA cycle model diagram of simbiology

### Result analysis

We built the whole xylose metabolism in our engineered *E. coli*,
which help us calculate the rate of xylose will go into CO_{2} utilization bypass.
The result of this model shows that CO_{2} uptake affect the production of pyruvate.
We can confirm that our engineered *E. coli* successfully uptake CO_{2} while the increase rate of pyruvate production followed the experiment data.
To see more analysis of CO_{2} utilization on analysis page.

Fig 19. TCA cycle model diagram of simbiology

Fig 20. whole xylose metabolism diagram of simbiology

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