Background knowledge
1. Michaelis-Menten kinetics
Michaelis–Menten kinetics is one of the best-known models of enzyme kinetics. It is named after German biochemist Leonor Michaelis and Canadian physician Maud Menten. The model takes the form of an equation describing the rate of enzymatic reactions, by relating reaction rate to , the concentration of a substrate S. Its formula is given by
Here, represents the maximum rate achieved by the system, at maximum (saturating) substrate concentrations. The Michaelis constant is the substrate concentration at which the reaction rate is half of .[11]Biochemical reactions involving a single substrate are often assumed to follow Michaelis–Menten kinetics, without regard to the model's underlying assumptions.
The mathematical model of the reaction [10] involves an enzyme E binding to a substrate S to form a complex ES, which in turn is converted into a product P and the enzyme. This may be represented schematically as
where , , and denote the rate constants,[12]and the double arrows between S and ES represent the fact that enzyme-substrate binding is a reversible process.
Under certain assumptions – such as the enzyme concentration being much less than the substrate concentration – the rate of product formation is given by
The reaction rate increases with increasing substrate concentration , asymptotically approaching its maximum rate , attained when all enzyme is bound to substrate. It also follows that , where is the enzyme concentration. , the turnover number, is the maximum number of substrate molecules converted to product per enzyme molecule per second.
The Michaelis constant is the substrate concentration at which the reaction rate is at half-maximum,[10] and is an inverse measure of the substrate's affinity for the enzyme—as a small indicates high affinity, meaning that the rate will approach more quickly.[13] The value of is dependent on both the enzyme and the substrate, as well as conditions such as temperature and pH.
2. Hill Equation
In biochemistry, the binding of a ligand to a macromolecule is often enhanced if there are already other ligands present on the same macromolecule (this is known as cooperative binding). The Hill coefficient provides a way to quantify this effect.
It describes the fraction of the macromolecule saturated by ligand as a function of the ligand concentration; it is used in determining the degree of cooperativeness of the ligand binding to the enzyme or receptor. It was originally formulated by Archibald Hill in 1910 to describe the sigmoidal O2 binding curve of hemoglobin.[1]
A coefficient of 1 indicates completely independent binding, regardless of how many additional ligands are already bound. Numbers greater than one indicate positive cooperativity, while numbers less than one indicate negative cooperativity. The Hill coefficient of oxygen binding to hemoglobin is 2.3-3.0.
Hill equation :
- free (unbound) ligand concentration.
- Apparent dissociation constant derived from the law of mass action (equilibrium constant for dissociation).
- ligand concentration producing half occupation (ligand concentration occupying half of the binding sites). This is also the microscopic dissociation constant. In recent literature, this constant is sometimes referred to as
.
- Hill coefficient, describing cooperativity (or possibly other biochemical properties, depending on the context in which the Hill equation is being used).
Taking the reciprocal of both sides, rearranging, inverting again, and then taking the logarithm on both sides of the equation leads to an alternative formulation of the Hill equation:
When appropriate, the value of the Hill coefficient describes the cooperativity of ligand binding in the following way :
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- Positively cooperative binding: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases.
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- Negatively cooperative binding: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases.
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- Noncooperative binding: The affinity of the enzyme for a ligand molecule is not dependent on whether or not other ligand molecules are already bound. In this case, the Hill equation (as a relationship between the concentration of a compound adsorbing to binding sites and the fractional occupancy of the binding sites) is equivalent to the Langmuir equation.
molecule is not dependent on whether or not other ligand molecules are already bound. In this case, the Hill equation (as a relationship between the concentration of a compound adsorbing to binding sites and the fractional occupancy of the binding sites) is equivalent to the Langmuir equation.
3. Deterministic modeling
Deterministic modeling consists of a set of ordinary differential equations (ODEs) describing the relationship between concentration and time. It based on the chemistry principle of mass balance of the system.We compare three kinds of model proposed from different source.Following are the three model we use in the simulation :
Model1 :
In model 1,we follow the model proposed in DREAM[1](Dialogue for Reverse Engineering Assessments and Methods,a competition of computational biology)
Model2 :
In model 2, we use the model proposed in [7] to simulate our circuit design. You will get more details in “Our simulation progress part”.
Model 3 :
In model 3, we use the model proposed in [9] to simulate our circuit design. You will get more details in “Our simulation progress part”.
*Comparison of three model :
4. Stochastic modeling
The core concept of stochastic modeling is that the reaction in bio-system is not continuous process, and the concentration profile isn’t uniform, so using deterministic modeling is not reasonable. In stochastic modeling, we will add a noise term,which is a random generated number,to the deterministic model.Then, we will run the simulation in the same manner as deterministic model many times. In the final stage, we will average all simulation and get the result.
5. Boolean Network [6]
In Boolean network , each gene, each input, and each output is represented by a node, and only can be only of two states (on and off). The relationship between nodes is represented by arrows forming a directed graph, and time is viewed as a series of discrete steps. At each step, the new state of each node is a Boolean function of prior state determined by the arrows pointing towards it.We would like to elaborate this part with our model operation for profound description.
6. Some Biochemical Facts are Associated with Our Model
FadR forms dimer in the absence of bound DNA and long chain acyl-CoA thioester,and fadR binds to DNA in the same manner[4]. On the other hand, two molecules acyl-CoA thioester would bind with fadR dimer causing conformational change ,and make the fadR separate with DNA [5].By the way, fadR dimer also regulate the gene expression of fadR and fadL.[8]
Conclusion:
If anyone ask to refine our project into one single sentence, it would be: Pursuit of beauty. We used the perfectness of skin as the indicator of beauty, therefore decided the directions of our project: skin whitening, corneum decomposition and producing fragrance. We want to reveal the nature fairness hiding behind the grease.
Factors influencing skin problems can be many, but we choose fatty acid to be point of penetration. It’s the foundation for follow-up studies. Throughout the researches on fatty acid metabolism, we found the way to achieve our goal: le everyone go with their own glow. On top of that, we successfully built up a regulation pathway controlled by fadR system, which may have great potential in further applications.
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