- Model for the Tea Pack
- Parameters
- Basic Equations
- Simulation
- Operating life

Parameters

Parameter | Definition | Value |

$\rho_1$ | Concentration of caffeine | Variable |

$\rho_2$ | Concentration of 1,3,7-trimethyluric acid | Variable |

$\rho_d$ | Concentration of SAH | Variable |

$N$ | Amount of CkTcS in the tea pack | Variable |

$T$ | Temperature | Variable |

$K_\text{cat, 1}$ | Turnover number of Reaction. Step 1 |
$7.48$s^{-1 [1]} |

$K_\text{cat, 2}$ | Turnover number of Reaction. Step 2 |
$0.186$s^{-1 [2]} |

$K_\text{m, 1}$ | Michael-Menten constant of Reaction. Step 1 |
$10.2\mu$M |

$K_\text{m, 2}$ | Michael-Menten constant of Reaction. Step 2 |
$109\mu$M |

$\alpha_1$ | Temperature sensitivity (based on Arrhenius equations) of Reaction. Step 1 |
$3000$K |

$\alpha_2$ | Temperature sensitivity (based on Arrhenius equations) of Reaction. Step 2 |
$3000$K |

$E_\text{a, d}$ | Apparent activation energy of CkTcS denaturation | $34.0$kJ/mol |

$k_c$ | Temperature change speed | $0.001$s^{-1} |

$k_\text{ad}$ | dsorption rate of SAH | $0.01$mmol^{-1}s^{-1} |

$T_\text{m}$ | Melting temperature of CkTcS | $343$K |

$T_0$ | Room temperature | $298$K |

$\lambda$ | Constant related to denaturation rate of CkTcS | $3.46$mmol/s |

$\mu$ | Constant related to thermal sensitivity of CkTcS | $0.05$ |

$R$ | Gas constant | $8.31$J/(mol$\cdot$K) |

$c_1$ | Equivalent concentration of Cdh | $0.003mM$ |

$c_2$ | Equivalent concentration of CkTcS | $0.05mM$ |

Basic equations

We consider a simple version of the model, where molecules diffuse with an infinite speed; enzymes are embedded in a tea pack, and the distributions of both enzymes are uniform.

The two-step reaction is shown below:

*Reaction. Step 1* Caffeine + CoQ $\stackrel{\text{Cdh}}{\longrightarrow}$
1,3,7-trimethyluric acid + CoQH_{2}

*Reaction. Step 2* 1,3,7-trimethyluric acid + SAM
$\stackrel{\text{CkTcS}}{\longrightarrow}$ Theacrine + SAH

Neither of the mechanisms of enzymes are clear, but the kinetics are similar. For example, we consider A + B $\stackrel{\text{E}}{\longrightarrow}$ P + Q, if A is the zymolyte, then the kinetics can be written as \begin{equation} v=\dfrac{v_m[A][B]}{[A][B]+K_{m,B}[A]+K_{m,A}[B]+K_{?}} \end{equation}

If the reaction is an ordered reaction, then $K_{?}=K_{s,A}K_{m,B}$. If the reaction is a double-displacement reaction, then $K_{?}=0$. Here we use $K_{m,x}$ to describe the Michaelis-Menten constant of $x$, and $K_{s,x}$ the dissociation constant of $x$.

To discuss the concentrations of the components, parameters listed below are used.

Parameter | Definition |

$\rho_i$ | Concentration of component $i$ in standard temperature, where $i=$map([1,2,3,$a$,$b$,$c$,$d$] |

$v_{m,j},K_{?,j},E_{a,j}$ | Maximum speed, $K_{?}$, and activation energy of reaction $j$, where $j=$[1,2] |

$K_{m,i,j}$ | Michaelis-Menten constant of component $i$ in reaction $j$ |

Therefore, inside the tea pack, we have \begin{align} \dfrac{\mathrm{d}\rho_1}{\mathrm{d} t}&=-\dfrac{v_{m,1}\rho_1\rho_a}{\rho_1\rho_a+K_{m,a,1}\rho_1+K_{m,1,1}\rho_a+K_{?,1}}\\ \dfrac{\mathrm{d}\rho_2}{\mathrm{d} t}&=\dfrac{v_{m,1}\rho_1\rho_a}{\rho_1\rho_a+K_{m,a,1}\rho_1+K_{m,1,1}\rho_a+K_{?,1}}-\dfrac{v_{m,2}e^{-E_{a,2}/RT}\rho_2\rho_c}{\rho_2\rho_c+K_{m,c,2}\rho_2+K_{m,2,2}\rho_c+K_{?,2}}\\ \dfrac{\mathrm{d}\rho_3}{\mathrm{d} t}&=\dfrac{v_{m,2}\rho_2\rho_c}{\rho_2\rho_c+K_{m,c,2}\rho_2+K_{m,2,2}\rho_c+K_{?,2}}\\ \dfrac{\mathrm{d}\rho_a}{\mathrm{d} t}&=-\dfrac{v_{m,1}\rho_1\rho_a}{\rho_1\rho_a+K_{m,a,1}\rho_1+K_{m,1,1}\rho_a+K_{?,1}}\\ \dfrac{\mathrm{d}\rho_b}{\mathrm{d} t}&=\dfrac{v_{m,1}\rho_1\rho_a}{\rho_1\rho_a+K_{m,a,1}\rho_1+K_{m,1,1}\rho_a+K_{?,1}}\\ \dfrac{\mathrm{d}\rho_c}{\mathrm{d} t}&=-\dfrac{v_{m,2}\rho_2\rho_c}{\rho_2\rho_c+K_{m,c,2}\rho_2+K_{m,2,2}\rho_c+K_{?,2}}\\ \dfrac{\mathrm{d}\rho_d}{\mathrm{d} t}&=\dfrac{v_{m,2}\rho_2\rho_c}{\rho_2\rho_c+K_{m,c,2}\rho_2+K_{m,2,2}\rho_c+K_{?,2}} \end{align}

Additionally, as the equations above suggest, we need to consider the factor of temperature. For the simplest case, the temperature of the whole system changes linearly: \begin{equation} \dfrac{\mathrm{d}T}{\mathrm{d}t}=-k_c(T-T_0) \end{equation} where $T_0$ is the environment temperature. In addition, although we have taken measures to improve the thermal stability of enzymes using PROSS, heat denaturation is yet unavoidable. We assume that the process of enzyme inactivation can be regarded as an isomeraization reaction. Some of the hydrogen bonds in the protein are broken and then some other ones form, so the original structure is converted into another through a high energy intermediate state. Therefore, the relationship between reaction rate constant and temperature conforms to Arrhenius equations. In addition, as hydrogen bonds hardly break in the low temperature, we use a sigmoid function to fit such effect. In conclusion, we have \begin{equation} \dfrac{\mathrm{d}N}{\mathrm{d}t}=-\dfrac{\lambda Ne^{-E_{a,d}/RT}}{1+e^{-\mu(T-T_m)}} \end{equation}

Our model will then try to get solutions to these questions below. For the situations, we can assume that the concentrations and diffuse rates of CoQ and SAM are infinite, so Eqs.(2)-(8) can be simplified to \begin{align} \dfrac{\mathrm{d}\rho_1}{\mathrm{d} t}&=-\dfrac{K_{cat,1}c_1e^{-\alpha_1(1/T-1/T_m)}\rho_1}{K_{m,1}+\rho_1}\\ \dfrac{\mathrm{d}\rho_2}{\mathrm{d} t}&=\dfrac{K_{cat,1}c_1e^{-\alpha_1(1/T-1/T_m)}\rho_1}{K_{m,1}+\rho_1}-\dfrac{K_{cat,2}c_2e^{-\alpha_2(1/T-1/T_m)}\rho_2}{K_{m,2}+\rho_2}\\ \dfrac{\mathrm{d}\rho_d}{\mathrm{d} t}&=\dfrac{K_{cat,2}c_2e^{-\alpha_2(1/T-1/T_m)}\rho_2}{K_{m,2}+\rho_2}-k_{ad}\rho_d \end{align} where the last item indicates the adsorption effect to SAH, if concentration of the adsorption medium is a constant.

Simulation

To evaluate our design, we take a common example, where our goal is to eliminate all the caffeine in a cup of Starbucks Grande Caffe Americano, that is, 225mg caffeine in 473mL coffee, that is, $\rho_1(0)=2.45$mM$^{[3]}$. We simulate Eqs.(11)-(13) and the result is shown below, indicating that roughly the concentration of SAH drops to nearly zero after 1500 seconds.

One thing we need to determine is the initial amount of Cdh. When we change the concentration of Cdh, the SAH adsorption curves change as is shown below.

This indicates that when we add the concentration of Cdh from 0.001mM to 0.003mM, the speed of SAH adsorption will have an significant increase, but that won't happen when it increases from 0.003mM. Therefore, in our product, concentration of Cdh is set to 0.003mM so that the cost will be not so high and the performance will be good enough.

Operating Life

From Eq.(10), we find that the amount of enzymes will denature in the process of time. When our system can never satisfy users' demands, or it takes too much time to satisfy, then the tea pack cannot operate. For example, when the initial temperature is 343K(=$T_m$), from Eqs.(9)-(10), we have \begin{equation} \dfrac{\mathrm{d}N}{\mathrm{d}t}=-\dfrac{\lambda Ne^{-E_{a,d}/R(T_0+(T_m-T_0)e^{-k_ct})}}{1+e^{\mu(T_m-T_0)(1-e^-{k_ct})}} \end{equation} We simulate the equation above by setting initial value of $N$ to 1, and the result is shown below, meaning that about 0.5\% of CkTcS will denaturate after one operation.