The modelling and simulation team aims to describe the whole system with an equational system which characterizes the expression and secretion of all enzymes, the degradation time courses as well as the produced rose scent. In order to find the best enzyme composition for the maximal hair and lipid breakdown in a specific time period the 2nd modelling block is used to perform an optimization.
We use a simplified formula  from the book Biomolecular Feedback Systems(1) to describe the enzyme expression:
α : mRNA production rate of enzyme E
δ : mRNA degradation rate
κ : mRNA translation rate
γ : degradation rate
The dynamics of the enzymes production is dependant on the building rate, which depends on mRNA dynamics, and its degradation rate γ. The expression is followed by the enzyme secretion which can be depicted as follows (2):
rsecretion : secretion rate
[enzyme] : enzyme concentration
s : secretion efficency
t : time
The secretion depends on the secretion efficency, which is a value between zero and one (2). Secreted Enzymes can degrade Hair, which is modelled in Block 2.
As the kinetics of our lipases, esterases and keratinases are not well characterized in the literature, we use the simple Michaelis-Menten formula  to describe the enzymes’ kinetics.
Furthermore, there exists no knowledge on the interaction of these three types of enzymes that is why we assumed that lipases, esterases and keratinases do not interfere with each other. Human hair is built up of proteins, lipids, water, trace elements and pigments. The protein or keratin proportion depicts 80% by weight and lipids constitute 5% by weight(3). For simplification, we assumed that the lipids of hair can be found only on its outside, conjecting that the hair degradation occurs in a sequential manner: First, the lipases must start breaking down the lipids, before esterases can continue with the lipid degradation, finally keratinases break down keratin, which has to be free of any lipids otherwise no keratin degradation is possible. As we cannot know size of the blockage, consequently, we have no knowledge about the contained hair amount, we modelled with an excess of hair compared to the used enzyme amount. Furthermore, we assume that only undegraded hair can be found in the blockage at the beginning. The simulation is carried out for a time period of 12 h as the aim is to clear the blockage over night.
The amino acid phenylalanine, which is a by-product of the hair degradation with 1.3% by weight(3), is then used as substrate for the synthesis of 2-phenylethylacetate which has a rose like odor. The expression of the therefor needed enzymes is described by formula  and the Michaelis-Menten formula  can be used to depict their kinetics.
The model was first set up in Simulink, which is an easy clickable graphical surface for implementing simulations. Figure 2 shows the three building blocks for our model in Simulink. The functions in the second block follow the interaction graph:
The equations derived from the interaction graph with assumption of Michaelis-Menten kinetiks [3,4] are as follows:
At first hair with lipids (H_wl) is degraded by lipases to hair without lipids (H_wol), which is then degraded via esterases to esterase degraded hair (H_est). H_est is is further consumed by keratinases to keratinase degraded hair (H_ker).
The Simulink model files can be downloaded here. Each block contains functions and interactions with another. Over time there is enzyme expressed and secreted, which then degrades hair. A part of the degradation product then is used to produce a rose scent.
For people, who do not want to use Simulink, ther is also a sole matlab model of Block 2 available.
Due to the fact that there is almost no data available of our enzymes and systems in the literature we focused our work on the 2nd block. Nevertheless, all three blocks are functionally set up and can be used right away if the needed data is available. To verify our code, we use literature data of a keratinase from D. microsporus (4) for all three types of enzymes. The optimizer was set in two different ways: Either the amount of keratinase degraded product gets maximized or the time to obtain 10% keratinase degraded hair is minimized. We assume that the enzyme concentrations remain constant throughout the whole simulation process.
Results and discussion
As the substrate is available in excess Km in the fraction plays a minor role in the first degradation step. In the following degradation steps the substrate is limited. Therefore, Km becomes more important leading to saturation curve known from Michaelis-Menten kinetics. Kcat as well as Km are specific enzyme properties that is why we can only vary the enzyme concentration in our system. If the same enzyme values are assumed for all three of them, the maximal degradation is always reached with maximal possible enzyme concentrations. In contrast, if there is a bottle neck right in the beginning, meaning that there is little lipase and/ or lipase with low activity and comparatively a lot more esterases and keratinases available, the result is that the difference between the degradation rates can be neglected. In this case it makes sense to use less esterases and keratinases compared to lipases. Thereby, a higher efficiency of our biological drain cleaner is achieved whilst ensuring cost reduction at the same time. Because of the assumption of constant enzyme concentrations our results can be logically deducted from the rate equation. However, for dynamic enzyme concentrations meaning that the 1st modelling block is included in the simulation it can get very complex. Our optimizer is designed to deal with this task. In case of available data our mathematical model can be used right away as all three blocks are functionally set up.
- Biomolecular Feedback Systems (2014),Domitilla Del Vecchio and Richard M. Murray
- Towards the Identification of Type II Secretion Signals in a Nonacylated Variant of Pullulanase from Klebsiella oxytoca (2005), Olivera Francetić and Anthony P. Pugsley.
- Chemical and Physical Behavior of Human Hair (2012), Clarence R. Robbins
- Comparison of Keratinases of Paecilomyces marquandii and Doratomyces microsporus to Some Known Proteases (2005), Gradisar et al.