Model | SPC - iGEM 2025

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Traditional Michaelis-Menten Model

PETase, as an enzyme, is able to shorten the time needed for breaking down PET, which would otherwise take hundreds of years.

Given that the model is used to fasten the rate to break down PET using PETase, it is useful to find out the factors that affect the rate of reaction, aside from temperature and pH as room temperature water is used. By finding an equation to depict the relation between reaction rate and concentration, we can also find out whether there is a maximum reaction rate if their relation is not linear.

Michaelis-Menten Equation

The Michaelis-Menten equation used in enzyme kinetics, mainly used to find the rate of reaction of converting one substrate to one product, the equation is given:

v = \frac{V_{\max} [S]}{K_{m} + [S]}

Annotations:
$[S]$ : Substrate concentration.
$v$ : Reaction rate (rate of product formation), typically reported in mM/s.
$K_{m}$ : The Michaelis constant — the substrate concentration at which $v = \frac{V_{\max}}{2}$ .
$V_{\max}$ : The maximum reaction velocity. It is given by $V_{\max} = k_{cat} [E]$ .
$k_{cat}$ : Turnover number — the number of substrate molecules converted to product per enzyme active site per second.
$[E]$ : Enzyme concentration.

It is used to find out the homogeneous reaction rate of an enzyme converting substrate to product, while homogeneous refers to that substrate is soluble in water. However, PET has a heterogeneous nature in contrast with the MM model, that is being insoluble in water. In addition, the enzyme can only access the bonds on the surface of PET, so the reaction is limited to the surface area of PET, not the total PET concentration. As such a modified MM model is required.

Modified Michaelis-Menten model

A general formula for the enzyme-substrate reaction (E = enzyme, S = substrate, P = product) is:

E + S ⇌_{k_{- 1}}^{k_{1}} ES \overset{k_{2}}{\to} EP ⇌_{}^{k_{p}} E + P

The equilibrium between free and substrate-adsorbed enzyme is defined by the adsorption constant $K_{a}$ :

K_{a} = \frac{k_{1}}{k - 1}

Applying a Langmuir adsorption model to account for enzyme adhesion to the PET surface gives the surface coverage $θ$ :

θ = \frac{K_{a} [E]}{1 + K_{a} [E]}

where $θ$ is the fraction of surface ester bonds on PET that form an enzyme–substrate complex.

To find the initial reaction rate:

v = k_{2} [E S]

Here $[E S]$ is the concentration of the enzyme–substrate complex. Since the ES concentration is related to the area of substrate bound to enzyme, we can write:

v = k_{2} A

where $A$ is the surface-area concentration of the substrate (proportional to the amount of PET surface available to the enzyme).

Combining the Langmuir adsorption model with the above model gives:

v = \frac{k_{2} A K_{a} [E]}{1 + K_{a} [E]}

Further Modifications

PET does not have a fixed area, so a standard area will need to be used, which may cause some percentage errors. This could be fixed by making the PET used have similar areas, but to do that would likely not be worth the effort.

Analytical Methods

Now that the products are formed from the reaction, as they are all soluble in water, we need to find a method that will allow us to determine the amount of products actually produced by the reaction. We can make use of the light absorbing properties of the soluble products and use light with their corresponding wavelength, that being around 240 nm, to find out their concentration.

Absorbance Determination

The increase in absorbance of the reaction mixtures in the ultraviolet region of the light spectrum (at 240 nm) indicates the release of soluble TPA or its esters (BHET and MHET) from an insoluble PET substrate. These compounds share an identical strong absorbance peak around 240–244 nm with an identical extinction coefficient (ε_{240 nm} 13800 M^-1 cm^-1) as all three compounds contain the same number of carbonyl groups. Measurements can be performed by withdrawing aliquots of the reaction mixture at specific time intervals and transferring them into quartz cuvettes or UV-transparent plate wells (12–96 wells).^[1]

Beer-Lambert Law

The Beer-Lambert Law describes the relationship between absorbance of light and concentration of the absorbing substance, it is usually expressed as:

A = ε c l

Annotations:
$A$ : Absorbance (no units). It can be calculated by
$A = - \log (\frac{I_{i}}{I_{t}})$ or measured directly with a spectrophotometer.
$I_{i}$ : Intensity of incident light.
$I_{t}$ : Intensity of transmitted light.
$ε$ : Molar absorption coefficient (given as $13800 M^{- 1} {cm}^{- 1}$ ).
$l$ : Path length (usually $1 cm$ ).
$c$ : Concentration of absorbing species (e.g., TPA, MHET, BHET); unit: $M$ .

Rearragement

By rearranging the variables, we can find the concentration of products when UV light at 240 nm is used:

c_{products} = \frac{A_{240 nm}}{ε_{240 nm} \cdot l}

(where $A_{240 nm}$ is the absorbance at 240 nm)

The concentration of the products released by PETase can then be used to find the rate of PET hydrolysis.

Range of wavelengths

The substances which we want to detect after the reaction is completed all have a strong affinity for absorbing UV light at around 240 nm. Soluble TPA, MHET, BHET all have an absorbance value at around 240-244 nm (within UV light wavelength) as they have the same number of carbonyl groups. If a single wavelength is used (e.g. 240nm), there is no confirmation on what the absorbing compound is. As such, a wavelength range of 220 to 300 nm is recommended. In addition to that, there will be other substances that would also absorb UV light, causing interference.

Interfering substances:

1. PETase (absorbance value around 260-280 nm)

2. DMSO (dimethyl sulfoxide) (absorbance value around 268 nm)

3. Contaminants

By measuring a range of wavelengths, there could be multiple peaks due to other substances, causing the 240 nm peak to be inflated. Hence, these factors will have to be excluded to improve accuracy and validity.

Further Modifications

The concentrations of MHET and BHET may not be consistent as they are used to form other products, as such, there may be some differences between the calculated and actual values.

Modelling flow rate with beads

Before modelling, we assume that:

Assumptions:

1. The water flow is steady and incompressible

2. The density and dynamic viscosity of water is constant

3. The power of the pump is constant

4. The temperature and pH do not affect the function of the pump

5. The calcium alginate beads are of 1 cm diameter and are perfect spheres

Reasons for modelling

To maximise the flow rate but also to maximise the number of calcium alginate beads in chamber.

Table of Variables

Parameters
Symbol	Description	Value / Units
Q	Flow rate	m³ s^-1
P_p	Power of water pump	20 W
A₁	Cross-sectional area (top funnel, 15 cm dia.)	0.017671458 m²
A₂	Cross-sectional area (neck, 5 cm dia.)	0.001963495408 m²
A₃	Cross-sectional area (chamber, 25 cm dia.)	0.039760782 m²
ρ	Density of seawater	1025 kg m^-3
μ	Dynamic viscosity of seawater	0.001003 Pa s
d	Diameter of calcium alginate bead	0.01 m
L	Length filled by beads in chamber	0.06 m
Φ	Bead packing fraction (V_b/V_tube)	(unitless)
V_tube	Volume of tube (A3 × length)	0.06 × 0.039760782 = 0.0023856 m³

Equation for modelling with Bernoulli's principle, Darcy's Law:

P_{p} = \frac{1}{2} ρ^{2} Q^{3} (\frac{1}{A^{2}} - \frac{1}{A^{1}}) + \frac{180 μ L ρ φ d^{2}}{{(1 - φ)}^{3} d^{2} A 3} Q^{2}

Bead volume and numeric substitution

Bead volume $V_{b}$ and the number of beads $n$ :

V_{b} = \frac{4}{3} π {(\frac{d}{2})}^{3} n

n: number of calcium alginate beads

After substituting numbers and simplifying the equation:

20 = 1.34578214 \cdot 10^{11} Q^{3} + 0.1345226 {(1 - 2.194835681 \cdot 10^{- 4} \cdot n)}^{3} n^{2} Q^{2}

Graph

Figure 1. Flow rate against number of beads

Annotations:

Y-axis: value Q (flow rate m³s^-1)

X-axis: value n (number of beads)

Having 2000 calcium alginate beads would cause the flow rate to drop from 0.0005297 to 0.0005225 (m³s^-1), approximately decrease for 7.2 mL s^-1

References

[1] Pirillo, V., Pollegioni, L., & Molla, G. (2021). Analytical methods for the investigation of enzyme‐catalyzed degradation of polyethylene terephthalate. FEBS Journal, 288(16), 4730–4745. https://doi.org/10.1111/febs.15850