Mathematical expressions of the curves of measurements in biology.
The goal of this article is to present a value without dimension characteristic of each enzyme, or a trace which makes it possible to identify each particular function. The means suggested to obtain this effect is the suppression of our usual base of time. In other words, to use the variation of measurement as time bases, which brings back to express the equations under only one dimension in the place of two dimensions.
This kind of development implies some guiding principles which are :
a) Time (called variation) appears only between two steady balance.
b) The shape of the curve will depend on the number of factors, the way in which the factors interact, and on the system (naturally stable, or naturally unstable).
c) Measuring accuracy, because often with starting it appears a particular function which delays the development. This function is decreasing and disappears rather quickly.
In example of application, we find a development of the article appeared in BIOKIN : Program DYNAFIT for the analysis of enzyme kinetic data: application to HIV proteinase."
Kuzmic, P. (1996) Anal. Biochem. 237, 260-73.
The first curves FIG. 1, indicate to us the value of the signal according to time and on various concentrations.
We determine with the reading of the curve the two states of balances.
The state of initial balance is not visible on this figure and the state of final balance will depend on the concentration and will be thus different for each curves.
The measured values are :
Curve A being the witness.
Curve A.
final balance = 11,6
Curve B.
final balance = 9,1
Curve C.
final balance = 8,8
Curve D.
final balance = 7,17
Curve E.
final balance = 7
We can say that the variation appears between 0,01 and final balance ; and by seeing the shape of the curve, we can say that the factor dominating is form : signal = équilibre initial + équilibre final (1-exp(-t/jo)).
By supposing that only one factor intervenes, we can say that the value of OJ will be given to 63% of the variation. But this case, is not very probable on enzymatic reactions. Moreover we can say that the damping ratio of the beginning, or the factor which delays starting, appears on curves of highher degrees of accuracy. The principal encountered problem is to determine how the loop of reaction intervenes. This is two functions in series which are followed or is this a function integrated in another function, in other words isn't jo in fact, a new function ?
By using the footbridge of time or more exactly of times ( the time of planets towards the real time, that of the variation), the mathematical expression becomes :
For curve A : signal = signal max. ( 1 - exp ( -t/jo))
from where signal = 11,6 ( 1 - exp ( - t/ jo ))
with jo = 32,5
If we observe with more precision the first two points of the curve, we appercevons ourselves that jo is in fact the result of a function who is not other than the representation of a loop of reaction in the first loop of reaction, already represented. In other words jo is written :
jo = 32,5 ( 1 - exp ( -t / jo' ))
from where jo = 32,5 ( 1 - exp ( -t / 17,6 ))
It should be noted that this second loop is apparent only on the first two points of the curve A.
from where the signal = 11,6 { 1 - exp [ - t/ 32,5 ( 1 - exp ( -t / 17,6 ))]}
For curve B : signal = signal max. ( 1 - exp ( -t/jo))
from where signal = 9,1 ( 1 - exp ( - t/ jo ))
with jo = 42 ( 1 - exp ( - t / 53))
d'où le signal = 9,1 { 1 - exp [ - t / 42 ( 1 - exp ( - t / 53))]}
Pour la courbe C : signal = signal max. ( 1 - exp ( -t/jo))
d'où signal = 8,8 ( 1 - exp ( - t/ jo ))
avec jo = 42 ( 1 - exp ( - t / 53))
from where the signal = 8,8 { 1 - exp [ - t / 42 ( 1 - exp ( - t / 53))]}
For curve D :
For curve E :
While returning in the usual temporal reference mark for the curve B, this expression enables us to write that with
t = 0,1 signal = signal(measured) =
t = 25 signal = 7,2 signal(measured) = 7,2
t = 50 signal = 7,8 signal(measured) = 7,8
t = 75 signal = 8,2 signal(measured) = 8,2
t = 100 signal = 8,5 signal(measured) = 8,5
t = 125 signal = 8,7 signal(measured) = 8,7
t = 150 signal = 8,9 signal(measured) = 8,9
t = 175 signal = 9 signal(measured) = 9
We can observe that the model suggested makes it possible to follow the curve B exactly measured.
While returning in the usual temporal reference mark for the curve C,
the formula : signal = 8,8 { 1 - exp [ - t / 42 ( 1 - exp ( - t / 53))]}, us allows to write that with
t = 0,1 signal= signal(measured) =
t = 25 signal= 7 signal(measured) = 7
t = 50 signal= 7,5 signal(measured) = 7,5
t = 75 signal= 8 signal(measured) = 8
t = 100 signal= 8,3 signal(measured) = 8,3
t = 125 signal= 8,5 signal(measured) = 8,5
t = 150 signal= 8,6 signal(measured) = 8,6
t = 175 signal= 8,7 signal(measured) = 8,7
We can observe that the model suggested makes it possible to follow the curve C exactly measured, and that we find the same value for jo, some is the concentration.
While returning in the usual temporal reference mark for the curve D, us allows to write that with
t = 0,1 signal= signal(measured) =
t = 25 signal= signal(measured) =
t = 50 signal= signal(measured) =
t = 75 signal= signal(measured) =
t = 100 signal= signal(measured) =
t = 125 signal= signal(measured) =
t = 150 signal= signal(measured) =
t = 175 signal= signal(measured) =
While returning in the usual temporal reference mark for the curve E, us allows to write that with
t = 0,1 signal= signal(measured) =
t = 25 signal= signal(measured) =
t = 50 signal= signal(measured) =
t = 75 signal= signal(measured) =
t = 100 signal= signal(measured) =
t = 125 signal= signal(measured) =
t = 150 signal= signal(measured) =
t = 175 signal= signal(measured) =
While returning in the usual temporal reference mark for the curve A, us allows to write that with
t = 0,1 signal= signal(measured) =
t = 25 signal = 7,4 signal(measured) = 7,4
t = 50 signal = 9,3 signal(measured) = 9,3
t = 75 signal = 10,5 signal(measured) = 10,5
t = 100 signal = 11 signal(measured) = 11
t = 125 signal = 11,3 signal(measured) = 11,3
t = 150 signal = 11,5 signal(measured) = 11,5
t = 175 signal = 11,55 signal(measured) = 11,55
We can observe that the model suggested makes it possible to follow exactly measured curve A. This curve being the pilot curve.
