Model

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The aim of our work is to model the diffusion of the enzyme through the skin and to determine the velocity of the enzyme reaction.

Enzyme diffusion through the skin
To construct the model, we'll make the following simplifying assumptions:

1. Homogeneous Skin Tissue: The skin is considered as a homogeneous medium without distinct layers.
2. Constant Diffusion Coefficient: The diffusion coefficient D of the active ingredient in the skin is constant.
3. First-Order Elimination: The elimination (metabolism or clearance) of the active ingredient from the skin follows first-order kinetics.
4. Semi-Infinite Medium: The skin is treated as a semi-infinite medium, meaning it extends infinitely in depth compared to the penetration depth of the active ingredient.
5. Constant Surface Concentration: The concentration of the active ingredient at the ointment-skin interface remains constant over time after application.

The primary mechanism for the penetration of the active ingredient into the skin is diffusion, which is described by the second law of Fick:

Where:

• C(x,t) is the concentration of the active ingredient at depth x and time t.
• D is the diffusion coefficient (cm²/s).
• k is the first-order elimination rate constant (1/s).
• x is the depth into the skin (cm).
• t is the time elapsed since application (s).

Solution for the differential equation:

Where:

• erfc(z) is the complementary error function, defined as erfc(z)=1−erf(z).
• The term exp(−kt) accounts for the first-order elimination of the active ingredient.
• D_DAO=58 μm^2/s
• D_HNMT=70 μm^2/s
• C_0=0.5 μg/ml

To apply this model practically, parameters need to be estimated experimentally:

• Diffusion Coefficient (D): Can be measured using techniques like the Franz diffusion cell method.
• Elimination Rate Constant (k): Determined through pharmacokinetic studies involving the clearance of the active ingredient from the skin.
• Surface Concentration (C0): Based on the formulation of the ointment and the initial loading of the active ingredient.

Plotting for t with fixed x:

DAO

HNMT

Plotting for x with fixed t:

DAO

HNMT

Velocity of enzyme reaction – with fixed enzyme concentration
Two different enzymes were used: DAO and HNMT
The velocity of the enzyme reaction can be described with the following equation:

Where:

• [S] is the concentration of the substrate under the skin (nM)
• [E] is the concentration of the enzyme under the skin (nM)
• q is a constant describing the enzyme, q=k_cat/K_M
• DAO: K_M=2.8 μM k_cat=2.32/s
• HNMT: K_M=5.47 μM k_cat=1.27/s
• q_DAO=0.83 /μMs
• q_HNMT=0.23 /μMs

Solution to the differential equation:

Plotting [S](t) for DAO enzyme:

Plotting [S](t) for HNMT enzyme:

Velocity of enzyme reaction – with [E](t)=C(t)
The equation is the same as the previous one, but instead of taking [E] as a fixed value, now [E] is a function of time.
Where:

• [E]=C(x, t)
• x is fixed

Solution to the differential equation for [S](t):

Plotting for [S](t):

DAO

HNMT

Note: For the plottings we have taken k as a unit (1) due to lack of data.

Sources:

• k_cat and K_M values of DAO:
  Bradley O Elmore, John A Bollinger, David M Dooley (2002) Human kidney diamine oxidase: heterologous expression, purification, and characterization https://pubmed.ncbi.nlm.nih.gov/12072962/
• V_max for HNMT:
  https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4581600/
• KM for HNMT:
  Abolfazl Heidari , Chanakan Tongsook , Reza Najafipour et al. (2015)
  Mutations in the histamine N-methyltransferase gene, HNMT, are associated with nonsyndromic autosomal recessive intellectual disability https://pubmed.ncbi.nlm.nih.gov/26206890/
• Approximation of D_DAO and DHNMT:
  Yifei Zhang, Henry Hess (2019) Enhanced Diffusion of Catalytically Active Enzymes https://pubs.acs.org/doi/10.1021/acscentsci.9b00228#