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Can somebody explain how is the following differential equation found? It is about the blood alcohol content.

See this article (pages 3 and 4).

They say the concentration of blood alcohol satisfies (in the drinking phase): $$\dfrac{dC}{dt}=u-\dfrac{\alpha C}{C+\beta}\quad; \quad C(0)=0,$$ where $\alpha, \beta$ are the liver and the kidney parameters. For the Recovery Phase we have: $$\dfrac{dC}{dt}=-\dfrac{\alpha C}{C+\beta}\quad; \quad C=C_{max}.$$

I can't see how they came to these equations. The only thing they say about the origin of this model is that it comes from a continuity argument: the amount of alcohol that is introduced in the body, equals the amount of the alcohol that leave the body.

I wonder if there is a model which takes into account all parameters in the above article plus the amount of food in the stomach.

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    $\begingroup$ Are you asking about how the Michaelis-Menten equation is derived or why it was deemed appropriate to model blood alchohol levels? The former is off-topic to BioSE since the Michaelis-Menten equation is used in many fields of science, and its derivation belongs to mathematics (maybe suitable for MathSE). $\endgroup$ – fileunderwater Jun 23 '15 at 21:54
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    $\begingroup$ @fileunderwater really? I always assumed the MM equation was closely related to biochemistry (enzyme-substrate interactions). I was unaware of any other application of the MM equation ? $\endgroup$ – Rover Eye Jun 23 '15 at 23:02
  • $\begingroup$ Given that the two individuals it is named after were noted biochemists Michaelis-Menten, it is difficult to imagine any other SE that would be more relevant than Biology. $\endgroup$ – mdperry Jun 23 '15 at 23:05
  • $\begingroup$ Based on the appearance of your source I would caution you about taking it too seriously. I does not seem to be published in a peer-reviewed journal and is just a manuscript (for example the first reference is to another unpublished URL and the second reference does not contain enough information to allow one to find it). $\endgroup$ – mdperry Jun 23 '15 at 23:43
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    $\begingroup$ Microbial growth, Prey-predator ecology, and enzyme-substrate kinetics in protein chemistry, yet you posit that it is off-topic for Biology SE? A better reason might be that there is no mention of any of these versions of the equation in the source our OP cites. $\endgroup$ – mdperry Jun 23 '15 at 23:53
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These kind of equations (the Michaelis-Menten [MM] like term) denote saturation kinetics. The basic mechanistic assumption behind saturation kinetics is this:

A rate (of lets say product formation) is dependent on the concentration of a molecule such that the rate increases linearly with increase in the concentration of the molecule.

  • Example 1: Substrate in an enzyme catalysed reaction
  • Example 2: Transcription factor in a transcriptional activation

However at a certain point increasing the concentration of the molecule will have no effect on the rate of reaction. For the above examples this point will arrive when (respectively):

  • Active sites in all the enzyme molecules are bound and there is no more space to accommodate a new substrate molecule.
  • All the binding sites in the DNA are occupied by the existing TF molecules and no more can bind.

In other words, the reaction kinetics changes from first order to zeroth order.

We can provide mechanistic explanations for some cases but in many we do not know how exactly the saturation happens. In modelling we just use a function that will mimic this kind of behaviour.

Why functions of the form $\dfrac{X}{K+X}$ ?

In deterministic modelling using differential equations, it would be preferred to use a continuous function (instead of using a piecewise linear function) because:

  • Numerical integration is better in case of continuous functions
  • In biochemical systems, operations are always continuous. There is no digital logic.

The MM function can actually be derived using an equilibrium or a quasi-steady state approximation (See the wikipedia page for the explanation).

In the MM function, when $X \gg K$ then the function is almost 1. This multiplied by the maximum rate constant will give you the max catalytic rate. At $X=0$ the rate will also be equal to zero. Note that in this model the catalytic rate will not indefinitely increase with $X$.

There are several functions that have values between 0 and 1 for increasing concentration of the substrate. Hill function and logistic function are common examples. However MM is a simpler model. You can always revise your modelling approach when you have better information of the biological system.

Now back to your question:

In this model $C$ denotes alcohol concentration $\alpha$ denotes maximum clearance rate and $\beta$ denotes half saturation concentration (concentration of alcohol at which the rate of clearance is half the maximum rate). Intuitively you can think about the mechanism like this: rate of clearance of alcohol depends on the level of alcohol i.e. if you have a lot of alcohol the amount of it removed per unit time will be higher. Think of it as water that is pumped through a pipe; more water in the tank will result in higher flow rate. However when the kidney (or pipe) is overloaded or "clogged" with an already excess of alcohol then clearance rate would remain the same (the maximum that the kidneys can excrete).

the amount of alcohol that is introduced in the body, equals the amount of the alcohol that leave the body

That is called mass balance which can be summarized as follows:

Accumulation = Input + Production - (Output + Degradation)

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  • $\begingroup$ Note that MM kinetics is outdated, currently the unified enzymatic activity modulation equation is the best approach. biology.stackexchange.com/a/23561/3703 $\endgroup$ – inf3rno Jun 24 '15 at 15:44
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    $\begingroup$ @inf3rno No it is not outdated. No model is outdated unless it is proven universally incorrect. Your choice of model depends on what you are trying to ask. You can always plug in more parameters to make it precise but the question is what level of precision do you want. $\endgroup$ – WYSIWYG Jun 24 '15 at 15:49
  • $\begingroup$ These equations are working, not doubt, but they are the bad approach. Ofc. we can talk about what "outdated" means in your dictionary and in mine, but I think that would be off-topic here. $\endgroup$ – inf3rno Jun 24 '15 at 16:03
  • $\begingroup$ @inf3rno I have had a huge debate about this elsewhere. A bad approach is, IMO, one where one does not select the method that is suitable for an approach, rationally. Quantum mechanics answers questions that classical mechanics does not. Moreover quantum laws are applicable for macroscopic systems as well- that does not mean that you should study the mileage of your car using quantum mechanics. $\endgroup$ – WYSIWYG Jun 24 '15 at 16:09
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    $\begingroup$ @inf3rno The post that you link to, seems unclear in this context. Also, I am not sure what you are trying to indicate here. Either the link is insufficient or the argument is not valid in this case. There are no inhibitors here. Why should you consider them? $\endgroup$ – WYSIWYG Jun 24 '15 at 16:42

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