Colloidal Gold, Mathematically Considered

Mathematical Consideration of Colloidal Nano Gold

Colloidal nano gold (gold nanoparticles, AuNPs) is being studied in medicine and biology because of its potential anti-inflammatory, antioxidant and cell-stimulating properties. The scientific effects can be described mathematically using various models.

1. Optical properties: plasmon resonance

Gold nanoparticles exhibit surface plasmon resonance (SPR) , which is important for their bioactive effect. The absorption and scattering properties can be described by the Mie theory :

$$\sigma_{ext} = \sigma_{abs} + \sigma_{sca}$$

with:

• $\sigma_{ext}$ = extinction cross section (total light interaction),
• $\sigma_{abs}$ = absorption cross section,
• $\sigma_{sca}$ = scattering cross section.

The plasmon resonance frequency $\omega_p$ the nanoparticles is:

$$\omega_p^2 = \frac{Ne^2}{\varepsilon_0 m_e}$$

with:

• $N$ = electron density,
• $e$ = elementary charge,
• $\varepsilon_0$ = electric field constant,
• $m_e$ = electron mass.

Meaning:

• SPR leads to the generation of reactive oxygen species (ROS), which can have antioxidant or oxidative effects.
• This resonance enables the use of colloidal gold for targeted medical applications (e.g. in cancer therapy).

---

2. Interaction with biological cells

Gold nanoparticles interact with cell membranes through electrostatic forces. The uptake into the cell can be described by the Langmuir kinetics :

$$\frac{d\theta}{dt} = k_a C (1 - \theta) - k_d \theta$$

with:

• $\theta$ = degree of coverage of the cell membrane by nanoparticles,
• $C$ = concentration of nanoparticles,
• $k_a$ = adsorption rate,
• $k_d$ = desorption rate.

Solution of this equation:

$$\theta(t) = \frac{k_a C}{k_a C + k_d} (1 - e^{-(k_a C + k_d)t})$$

Meaning:

• Shows how quickly and in what quantity colloidal gold is absorbed by cells.
• The absorption rate depends on the particle size and charge.

---

3. Influence on cell reactions: Oxidative stress

Gold nanoparticles can act as an antioxidant by neutralizing reactive oxygen species (ROS) or generating them through plasmon effects. The ROS concentration $[ROS]$ in the cell can be modeled by a differential equation:

$$\frac{d[ROS]}{dt} = P - k_{elim} [ROS]$$

with:

• $P$ = ROS production rate (depending on gold-nanoparticle interaction),
• $k_{elim}$ = Elimination rate by cellular antioxidants.

Solution:

$$[ROS](t) = \frac{P}{k_{elim}}$$ \left(1 - e^{-k_{elim} t} \right)

Meaning:

• At low $P$ -Rate, colloidal gold can act as an antioxidant.
• At high $P$ rate, it can induce oxidative stress, which can trigger cell damage or apoptosis.

---

4. Heat effects: Photothermal therapy

Gold nanoparticles can be excited by infrared light, causing them to generate heat. The temperature increase $\Delta T$ due to absorbed light energy $Q$ follows:

$$\Delta T = \frac{Q}{\rho c V}$$

with:

• $\rho$ = density of the medium,
• $c$ = heat capacity,
• $V$ = volume of the treated area.

Meaning:

• This is used in cancer therapy to specifically destroy tumor cells.
• The higher the particle concentration, the greater the photothermal effect.

Summary:

• The plasmon resonance of colloidal gold plays a central role in optical and therapeutic applications.
• Uptake into cells can be modeled using kinetic equations.
• The antioxidant or oxidative effect can be described using ROS dynamics.
• Photothermal effects can be predicted by energy absorption models.

Colloidal gold is therefore a promising material with complex biological interactions that can be described precisely mathematically.