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International Journal of Research in Biochemistry and Biophysics
Universal Research Publications. All rights reserved
ISSN 2249-8524
Original Article
Interpolate the Rate of Enzymatic Reaction: Temperature, Substrate Concentration
and Enzyme Concentration based Formulas using Newton’s Method
NIZAM UDDIN*
*Post Graduate Student, M. B. Khalsa College, Indore (M.P.), India.
Email: [email protected], Contact Number: +919630775404
Received 27 March 2012; accepted 17 April 2012
Abstract
This research paper is based on Interpolate the rate of Enzymatic Reaction. The Rate of Enzymatic reaction is affected by
concentration of substrate, Temperature, concentration of enzyme and other factors. Take other factors are constant. I take the
values of substrate concentration, Temperature, Enzyme concentration in interval and defined the functions. Increasing their
values are increased the rate of Enzymatic reaction. All functions are followed the limit “n” which is optimum limit. Apply
Newton’s method on the functions. If the point lies in the upper half then used Newton’s forward interpolation formula. If the
point lies in the lower half then we used Newton’s backward interpolation formula. And when the interval is not equally spaced
then used Newton’s divide difference interpolation formula.
©2011 Universal Research Publications. All rights reserved
Key words: Newton’s method, concentration of substrate, Temperature, concentration of enzyme, Interpolation.
1. Introduction:
Interpolation is the technique of obtaining the value
of a function for any intermediate value of independent
variable [1]. Mathematically, we can say that given a set of
value of the function f(x) for certain value of the independent
variable “x” , the method of finding the value f(x) for any
given value of “x” is known as Interpolation thus the value of
f(x) determined is known as interpolated[2]. Newton’s
method of interpolation are Newton’s forward interpolation
formula, Newton’s backward
interpolation formula,
Newton’s divide difference interpolation formula[1][2].The
rate of Enzymatic reaction is affected by concentration of
substrate, Temperature, concentration of enzyme and other
factors[3][4].
V = f(T)
V = f(S)
V = f(E)
Above functions are be “n” interval and other factors are be
constant in each function.
3. Effect of Temperature:
The rise in Temperature accelerates an Enzyme
reaction but at the same time causes inactivation of the
protein. At certain Temperature known as the optimum
Temperature the activity is maximum [4][5].
2. Functions for Rate of Enzymatic Reaction:
The Rate of Enzymatic reaction (V) is affected by 4. Interpolate the rate of Reaction with Temperature:
Let V = f(T) be a function defined by “n” points (V1,
concentration of substrate (S), Temperature (T),
T
),
(V
,
T2), ………… (Vn, Tn ). Where “V” is the rate of
1
2
concentration of enzyme (E). Increasing their values are
increase the rate of Enzymatic reaction (V). We take their reaction and “T” is the Temperature of reaction. And other
factors are to be constant.
value in interval and defined the functions:
International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9
5
4.1 formula (i):
If the point lies in the upper half then we used Newton’s forward interpolation formula [1][2]. When T1, T2,
T3………..Tn are equally spaced with interval “h”:-
[Where ∆ is forward difference operator]
T-T2 = T-(T1+h)
If we use this relationship
T-T2 = T-T1-h
T-T2= hu – h
[same interval]
[Because T-T1 = hu]
T-T2= h (u-1)
.
.
.
.
.
T-Tn= h [u-(n-1)
[where “u” is variable],
Then
T-T1 = hu
Substituting these values (T-T1), (T-T2),……………(T-Tn) In the equation (1), We got:
Simplifying, we got:
4.2 Formula (ii):
If the point lies in the lower half then we used Newton’s backward Interpolation formula [1][2]. When T 1,T2
T3,……..Tn are equally spaced with interval “h”:-
[Where
is backward difference operator]
If we use this relationship:
T-Tn-1
T-Tn-1 = hz + h
T-Tn-1
Then
T-T1
T-Tn
= hz
T-Tn-1
= T-(Tn-h)
6
= T-Tn+ h
= h(z+ 1)
.
.
.
.
= h [z+(n-1)]
Substituting these value (T-Tn), (T-Tn-1)… (T-T1) in the
equation (2), We got:
International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9
Simplifying, we got:
4.3 Formula (iii):
If T1, T2 T3…Tn are not be equally spaced. We use Newton’s divide difference interpolation formula[1][2].
[Where
is divide difference operator]
5. Effect of Concentration of Substrate:
At the constant enzyme concentration and other factor, the concentration of substrate is the limiting factor, as the
substrate concentration increases, the Enzyme reaction rate increases. However, at very high substrate concentration, the
Enzyme becomes saturated with substrate and a higher concentration of substrate does not increase the reaction rate [6].
6 Interpolate the rate of Reaction with concentration of substrate:
Let V = f(S) be a function defined by “n” points (V1,S1), (V2,S2), ………… (Vn, Sn). Where “V” is the rate of reaction
and “S” is the concentration of substrate. And other factors are to be constant.
6.1 formula (i):
If the point lies in the upper half then we used Newton’s forward interpolation formula [1][2]. When S1, S2, S3…Sn are
equally spaced with interval “h”:-
If we use this relationship:
Then
S-S1
= hw
S-S2
= S-(S1+h)
[same interval]
S-S2
= S-S1-h
S-S2
= hw – h
S-S2
= h (w-1)
.
.
.
.
= h [w-(n-1)]
S-Sn
[Because S-S1 = hw]
Substituting these value (S-S1), (S-S2),……………,(S-Sn) In the equation (3), We got:
Simplifying, we got:
7
International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9
6.2 Formula (ii):
If the point lies in the lower half then we used Newton’s backward Interpolation formula [1][2]. When S1, S2
,Sn……….Sn are equally spaced with interval “h”:
[Where
is backward difference operator]
If we use this relationship:
Then
S-Sn
= hp
S-Sn-1
= S-(Sn-h)
S-Sn-1
= S-Sn+ h
S-Sn-1
= hp + h
S-Sn-1
= h(p+ 1)
.
.
.
= h [p+(n-1)]
S-S1
Substituting these value (S-Sn), (S-Sn-1)…(S-S1), in the equation (4), We got:
Simplifying, we got:
6.3 Formula (iii):
If S1, S2, Sn……....Sn are not be equally spaced. We use Newton’s divide difference interpolation formula[1][2].
[Where
is divide difference operator]
7. Effect of concentration of Enzyme:
Assuming a sufficient concentration of substrate is available, increasing Enzyme concentration will increase the
Enzymatic reaction rate [6]
8. Interpolate the rate of Reaction with concentration of Enzyme:
Let V = f(E) be a function defined by “n” points (V1,E1), (V2,E2), ………… (Vn,En). Where “V” is the rate of reaction
and “E” is the concentration of Enzyme. And other factors are to be constant.
8.1 Formula (i): If the point lies in the upper half then we used Newton’s forward interpolation formula [1][2]. When E 1, E2,
E3,……..En are equally spaced with interval “h”:-
If we use this relationship:
8
International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9
Then
E-E1
= hq
E-E2
= E-(E1+h)
E-E2
= E-E1-h
[same interval]
E-E2
= hq – h
E-E2
= h (q-1)
.
.
.
.
= h [q-(n-1)]
E-En
[Because E-E1 = hq]
Substituting these value (E-E1), (E-E2),……………,(E-En) In the equation (3), We got:
Simplifying, we got:
8.2 Formula (ii):
If the point lies in the lower half then we used Newton’s backward Interpolation formula [1][2]. When E 1, E2
,En……….En are equally spaced with interval “h”:-
[Where
is backward difference operator]
If we use this relationship:
Then
E-En
= hQ
E-En-1
= E-(En-h)
E-En-1
= E-En+ h
E-En-1
= hQ + h
E-En-1
= h(Q+ 1)
.
.
.
= h [Q+(n-1)]
E-E1
Substituting these value (E-En), (E-En-1)… (E-E1), in the equation (6), We got:
Simplifying, we got:
8.3 Formula (iii):
If E1, E2,En……....En are not be equally spaced. We use Newton’s divide difference interpolation formula[1][2].
9
International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9
[Where
is divide difference operator]
9.Concluding Remarks:
My concluding remarks sated thus; when the value
of Temperature (T) is increased in interval then the value of
the Rate of Enzymatic Reaction (V) is increased in interval.
Where “n” is optimum limit of Temperature (T).
When the value of concentration of substrate (S) is increased
in interval then the value of the Rate of Enzymatic Reaction
(V) is increased in interval. Where “n” is optimum limit of
substrate concentration (S).
When the value of Enzyme concentration (E) is increased in
interval then the value of the Rate of Enzymatic Reaction (V)
is increased in interval. Where “n” is optimum limit of
Enzyme concentration (E).
10. Conclusion: We Interpolate the value of Enzymatic
reaction rate using formulas in the interval. These formulas
are to be applied on given conditions. Means point lies in
upper half or lower half or unequally interval spaced.
sincere thank due to Soumya V G. My thank is also due to
Munavvar Ali.
References:
1.
2.
3.
4.
5.
6.
“Vaidehi bhagat”, “computer oriented numerical
methods”, First Edition(2004-05), kamal prakashan,
Indore-2
“S.R. Gupta”, “A text book of computer oriented
numerical methods”, First Edition(2002), Nakoda
publisher & printers, Indore-2
“B.D. singh”, “Biotechnology”, second Edition
(2007), kalyani publishers,new delhi
“A.C. Deb”, “Fundamental of biochemistry”, Eight
edition(2002), new central book agency, kolkata-9
“Ram Nivas singh”, “Biology”, yugbodh
publication, Raipur
student.ccbcmd.edu
11. Acknowledgments: I would like to gratefully and
sincerely thank Sayyad Maksud Ali, Mansur Ali, Abdul Ali
and Isahaq Uddin Sheikh for continuous their support. My
Source of support: Nil; Conflict of interest: None declared
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International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9