**EXAMPLE OF THE METHODAPPLIES IN AN ARTICLE OF BIOKIN**

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.

initial balance

= 0.01

final balance = 11,6

Curve B.

initial balance = 0.01

final balance = 9,1

Curve C.

initial balance = 0.01

final balance = 8,8

Curve D.

initial balance = 0.01

final balance = 7,17

Curve E.

initial balance = 0.01

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.