# Biomod/2013/NanoUANL/Reactor

## What is a reactor?

### Introduction

The CCMV capsid was considered as a continuous stirred-tank reactor with accumulation of the product.

For an enzymatic reaction of the type:

$E + S \leftrightarrow ES \rightarrow E^0 + P$

with a reaction rate of:

$\frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES]$

We established the following in our system:

• Uniform distribution throughout the reactor
• K-1 >> K1 and K2
• One enzyme per reactor/VLP
• Tortuosity approaches zero during diffusion

Mass balance was presented as such:

INFLOW= OUTFLOW- DISAPPEARANCE BY REACTION + ACCUMULATION

where

Inflow= F0

Outflow= F0(1-XS)

Disappearance = V(-rS

Accumulation = $\tfrac{d[P]}{dt}$

F0 = F0(1-XS) - V(-rS) + $\tfrac{d[P]}{dt}$

The intake and outflow flux were determined by diffusion , considering a spherical container.

For the simplification of the diffusion phenomenon we considered:

• Constant temperature
• Constant pressure
• Species B stays in a stationary state (it does not diffuse in A)
• The container (VLP) has a spherical shape

A mass balance, taking into account a spherical envelope leads to:

$\frac{d}{dr}(r^2N_{Ar})=0$

where NAr represents molar flux. For NBr we obtain:

$\frac{d}{dr}(r^2\frac{cD_{AB}}{1-x_A}\frac{dx_A}{dr})=0$

At a constant temperature the product (cDAB) is equally constant and xA=1-xB, the equation can be integrated into the following expression:

$F_A=4\pi r_1^2N_{Ar}|_{r=r1}= \frac{4\pi cD_{AB}}{1/r_1=1/r_2} \ln\frac{x_{B2}}{x_{B1}}$

where x are the fractions, c is the concentration and r are the respective radii.

This equation defines the nanoreactor inflow; a similar analysis yields the reactor outflow.

The ionic silver diffusion coefficient in function to the solution is described by Nerst's equation (1888)1:

$D_{AB}°= \frac{RT}{F^2} \frac{\lambda^o_+\lambda^o_-}{\lambda^o_++\lambda^o_-} \frac{|Z_-|+|Z_+|}{|Z_+Z_-|}$

where

F = Faraday's constant DAB°=Diffusion coefficient at infinite dilution λ+°=Cationic conductivity at infinite dilution λ-°=Anionic conductivity at infinite dilution Z+=Cation valence Z-=Anionic valence T=Absolute temperature