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.

1. Mass action

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 Kf is forward rate of equilibrium, Kr is reverse rate of equilibrium

2. Henri-Michaelis-Menten

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.

$${{dS \over dt} = -k_1E \cdot S + k_{-1}ES}$$

$${{dES \over dt} = k_1E \cdot S - (k_{-1} + k_2)ES}$$

$${{dE \over dt} = k_1E \cdot S - (k_{-1} + k_2)ES}$$

$${{dP \over dt} = k_2ES}$$

After derivation $${v = {-ds \over dt} = {dP \over dt}}$$

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

The constants vmax and Km can be looked up at the public data base BRENDA. The rate konstant k2 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.

$${{dS \over dt} = -k_1E \cdot S + k_{-1}ES}$$

$${{dES \over dt} = k_1E \cdot S + k_{-2}E \cdot P - (k_{-1} + k_2)ES}$$

$${{dE \over dt} = k_1E \cdot S - k_{-2}E \cdot P - (k_{-1} + k_2)ES}$$

$${v = {dP \over dt} = k_2ES - k_{-2}E \cdot P = v_f - v_b}$$

After derivation $${v = {-ds \over dt} = {dP \over dt}}$$

Finally we use Vfmax=k2⋅Etotal and V_bmax=k₋₁⋅Etotal 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 Vf is the forward rate and the Vb is backwared rate while $${K_{m,1} = {K_{-1} + K_2 \over K_1}}$$ or represents as KS, and $${K_{m,2} = {K_{-1} + K_2 \over K_{-2}}}$$ or represent as KP.

3. Ping-Pong-Bi-Bi

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：CO2 uptake model

Part B：Xylose – PRK – Rubisco model

Part C：Original metabolism model

Part D：TCA cycle model

Part A : (CO₂ uptake model)

CO₂ 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 CO2 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 CO₂ and HCO₃^-. Although some CAs prefer the direction of CO₂ hydration, the carboxysomal CAs in cyanobacteria and some chemoautotrophic bacteria favor the direction of HCO₃⁻ dehydration. The result of model and our experiments showed that CA improve amount of CO2 uptake and CO2 utilization.

Fig. 1 CO₂ 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 CO2-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

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 FADH2, 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. 4 TCA cycle model diagram of simbiology

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$${d[CO_2] \over dt} = {-V_{max} \times [CO_2] \over Km + [CO_2]}$$

$${GAP + E^+ \rightleftharpoons C \xrightarrow{k_f} DHAP}$$

$${a \underset{k}{\stackrel{k_f}{\rightleftharpoons}} c}$$