SynPETic

Model

Overview

Dry Lab members initially generated mathematical models and computer simulations based purely on theory and reading. After the Wet Lab had collected real data, it became possible to compare theoretical expectations with their findings.

Theoretical Models of Enzymatic PET Degradation

Every simulation reviewed here tracks the same chemistry — PET broken down by PETase, its intermediate cleaved by MHETase into terephthalic acid (TPA) and ethylene glycol (EG) — but each encodes a different mathematical assumption about how fast that happens. This section lays out the governing equations side by side for the five models retained after review.

1. Saturable (Michaelis–Menten) kinetics · Simulation X · baseline

The baseline model treats both enzymes as classic single-substrate catalysts. Reaction velocity saturates as substrate becomes abundant, because enzyme active sites are the limiting resource:

$$ v_{\text{PETase}} = \frac{V_{\max}^{P}\,S}{K_m^{P} + S}, \qquad v_{\text{MHETase}} = \frac{V_{\max}^{M}\,[\text{MHET}]}{K_m^{M} + [\text{MHET}]} $$ $$ V_{\max}^{P} = 2.5\,[\text{PETase}], \quad K_m^{P} = 150 \qquad\qquad V_{\max}^{M} = 4.0\,[\text{MHETase}], \quad K_m^{M} = 50 $$

Crystalline PET is harder to attack than amorphous PET, so velocity is split by phase and the crystalline fraction is damped exponentially by a crystallinity index $x = S_{\text{cryst}}/S$:

$$ v_{\text{amorph}} = v_{\text{PETase}}\cdot\frac{S_a}{S}, \qquad v_{\text{cryst}} = v_{\text{PETase}}\cdot\frac{S_c}{S}\cdot e^{-2.5x} $$

Temperature enters through an Arrhenius term centered on a 30°C optimum, with an extra penalty past 55°C:

$$ f(T) = \exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\text{opt}}}-\frac{1}{T}\right)\right] \times \begin{cases} 1 & T \le 55^{\circ}\text{C} \\ e^{-0.3(T-55)} & T > 55^{\circ}\text{C} \end{cases} $$

MHET cleavage splits into TPA and EG at a fixed, mass-conserving ratio (molar composition of MHET): $ \Delta\text{TPA} = 0.73\,v_{\text{MHETase}}, \; \Delta\text{EG} = 0.27\,v_{\text{MHETase}} $. Time is advanced in discrete ticks (an explicit Euler-style update), not continuous integration.

2. Cooperative (Hill) kinetics with a sharper thermal cliff · Simulation Z · variant

Here PETase activity is modeled as cooperative rather than simple Michaelis–Menten, using a Hill coefficient $n = 1.8$ — velocity rises more sigmoidally with substrate concentration:

$$ v_{\text{PETase}} = \frac{V_{\max}^{P}\,S^{\,n}}{\big(K_m^{P}\big)^{n} + S^{\,n}}, \qquad n = 1.8,\;\; V_{\max}^{P} = 3.5\,[\text{PETase}],\;\; K_m^{P} = 200 $$

MHETase kinetics stay Michaelis–Menten (identical to X). What changes most is thermal behavior: instead of X's gentle penalty above 55°C, Z applies a much steeper denaturation cliff starting ten degrees earlier:

$$ f(T) = \underbrace{\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\text{opt}}}-\frac{1}{T}\right)\right]}_{\text{Arrhenius term, same as X}} \times \begin{cases} 1 & T \le 45^{\circ}\text{C} \\ e^{-0.8(T-45)} & T > 45^{\circ}\text{C} \end{cases} $$

Crystallinity hindrance is also linearized rather than exponential: $ e^{-2.5x} \to 1 - 0.95x $. Stoichiometry reverts to the correct 0.73/0.27 TPA/EG split, so unlike Y, Z conserves mass.

3. Continuous-time ODE integration · alternate_simulation.html

Rather than stepping forward in discrete ticks, this model expresses the same PET→MHET→TPA pathway as a coupled system of ordinary differential equations and integrates it continuously with an explicit Runge–Kutta solver (scipy.integrate.solve_ivp, RK45):

$$ \frac{d[\text{PET}]}{dt} = -\,0.15\,k_1\,\frac{[\text{PET}]}{[\text{PET}]+20} $$ $$ \frac{d[\text{MHET}]}{dt} = 0.15\,k_1\,\frac{[\text{PET}]}{[\text{PET}]+20} \;-\; 0.25\,k_2\,\frac{[\text{MHET}]}{[\text{MHET}]+10} $$ $$ \frac{d[\text{TPA}]}{dt} = 0.25\,k_2\,\frac{[\text{MHET}]}{[\text{MHET}]+10} $$

Here $k_1, k_2$ (PETase/MHETase expression levels) are the free parameters a user drags on sliders; the interface also perturbs each by a small $\delta$ and re-integrates to show a live finite-difference sensitivity, $ \partial Y/\partial k_i \approx \big[Y(k_i+\delta)-Y(k_i)\big]/\delta $.

4. Gene-circuit-coupled degradation · petase_circuit.html

This is the only model where enzyme concentration is not a fixed input — it is itself the output of an upstream transcription/translation cascade, driven by promoter strength $p$, plasmid copy number $c$, and cell density $d$:

$$ \Delta\text{mRNA}_i = 0.002\,p\,c\,d \cdot w_i, \qquad w_1 = 1,\; w_2 = 0.8 $$ $$ [\text{PETase}] \mathrel{+}= 0.2\,\text{mRNA}_1, \qquad [\text{MHETase}] \mathrel{+}= 0.2\,\text{mRNA}_2, \qquad \text{mRNA}_i \mathrel{*}= 0.85 $$

PET breakdown itself is then zero-order in substrate and first-order only in enzyme concentration — there is no $K_m$ term at all, unlike every other model here:

$$ \frac{d[\text{PET}]}{dt} = -\,0.03\,[\text{PETase}], \qquad \frac{d[\text{MHET}]}{dt} = 0.03\,[\text{PETase}] - 0.04\,[\text{MHETase}] $$

5. Molar-mass stoichiometric pathway · intermediate.html (v7) & 2033_intermediate.html (v5)

The most detailed model in the set: it tracks an explicit intermediate (BHET) between PET and MHET, converts every flux through real molar masses, and feeds temperature and pH back into the reaction rates rather than treating them as fixed constants:

$$ f_1 = \frac{[\text{PET}]}{192.17}\cdot 0.16 \cdot \phi(\text{pH}) \cdot \theta(T), \qquad f_2, f_3 \;\text{analogous for BHET$\to$MHET, MHET$\to$TPA} $$ $$ \Delta[\text{PET}] = -f_1 M_{\text{PET}}, \quad \Delta[\text{BHET}] = (f_1-f_2)M_{\text{BHET}}, \quad \Delta[\text{MHET}] = (f_2-f_3)M_{\text{MHET}} $$ $$ \Delta[\text{TPA}] = f_3 M_{\text{TPA}}, \qquad \Delta[\text{EG}] = (f_2+f_3)M_{\text{EG}} $$

Molar masses (g/mol): PET 192.17, BHET 254.24, MHET 210.18, TPA 166.13, EG 62.07.

Two feedback loops make this model distinctive: reaction extent raises temperature ($\Delta T \mathrel{+}= 1.5f_1$, an exothermic-heat proxy), and acid byproduct lowers pH, which in turn slows $\phi(\text{pH})$ — a self-limiting reaction absent from every other model above. The v5 file (2033_intermediate.html) adds one further wrinkle: a crystallinity step-modifier, $ \text{crystMod} = 0.3 $ once $[\text{PET}] < 0.1\times\text{PET}_0$, representing a recalcitrant crystalline core that resists the final stage of digestion.

At a glance

Model	File	Time	Rate law	Mass-conserving?	Distinctive feature
X	`SimulationX_22May2026.html`	discrete	Michaelis–Menten	yes	baseline / reference case
Z	`SimulationZ_22May2026.html`	discrete	Hill, $n=1.8$	yes	cooperative binding + steep thermal cliff
ODE	`alternate_simulation.html`	continuous	saturable (RK45-integrated)	yes	built-in parameter sensitivity analysis
Circuit	`petase_circuit.html`	discrete	zero-order in substrate	yes	enzyme level is a dynamic output, not an input
Stoichiometric	`intermediate.html` (v7) `2033_intermediate.html` (v5)	discrete	molar-mass flux	yes	explicit BHET intermediate; pH/temperature feedback

In short: X and Z form a matched pair testing how one modeling choice (saturation law, thermal response, crystallinity penalty) changes outcomes on an otherwise identical scaffold — X assumes standard Michaelis–Menten kinetics, Z assumes cooperative (Hill) kinetics, reflecting the two candidate assumptions considered before the team had real experimental data to test between them. The ODE, circuit, and stoichiometric models each change the scaffold itself — continuous vs. discrete time, fixed vs. dynamically-produced enzyme, and lumped vs. explicit intermediates, respectively. Source simulations: SimulationX/Z (24 May 2026), alternate_simulation.html (13 May 2026), petase_circuit.html & intermediate.html/2033_intermediate.html (12 Apr 2026).

Simulation Y, an early linearized-kinetics variant, was removed from this comparison after review — it approximated saturating kinetics as first-order (valid only when substrate concentration is much smaller than K_m, which did not hold here) and its stoichiometric bookkeeping did not conserve total mass.