5 International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9

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: nizamuddin4research@gmail.com, 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].

2. Functions for Rate of Enzymatic Reaction: The Rate of Enzymatic reaction (V) is affected by

concentration of substrate (S), Temperature (T),

concentration of enzyme (E). Increasing their values are increase the rate of Enzymatic reaction (V). We take their

value in interval and defined the functions:

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].

4. Interpolate the rate of Reaction with Temperature:

Let V = f(T) be a function defined by “n” points (V1, T1), (V2, T2), ………… (Vn, Tn ). Where “V” is the rate of

reaction and “T” is the Temperature of reaction. And other

factors are to be constant.

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International Journal of Research in Biochemistry and Biophysics

Universal Research Publications. All rights reserved

6 International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9

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]

If we use this relationship

[where “u” is variable],

Then

T-T1 = hu

T-T2 = T-(T1+h) [same interval]

T-T2 = T-T1-h

T-T2= hu – h [Because T-T1 = hu]

T-T2= h (u-1)

.

.

.

.

.

T-Tn= h [u-(n-1)

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 T1,T2

T3,……..Tn are equally spaced with interval “h”:-

[Where is backward difference operator]

If we use this relationship:

Then

T-Tn = hz

T-Tn-1 = T-(Tn-h)

T-Tn-1 = T-Tn+ h

T-Tn-1 = hz + h

T-Tn-1 = h(z+ 1)

.

.

.

.

T-T1 = h [z+(n-1)]

Substituting these value (T-Tn), (T-Tn-1)… (T-T1) in the

equation (2), We got:

7 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 [Because S-S1 = hw]

S-S2 = h (w-1)

.

.

.

.

S-Sn = h [w-(n-1)]

Substituting these value (S-S1), (S-S2),……………,(S-Sn) In the equation (3), We got:

Simplifying, we got:

8 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)

.

.

.

S-S1 = h [p+(n-1)]

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 E1, E2,

E3,……..En are equally spaced with interval “h”:-

If we use this relationship:

9 International Journal of Research in Biochemistry and Biophysics 2012; 2(2): 5-9

Then

E-E1 = hq

E-E2 = E-(E1+h) [same interval]

E-E2 = E-E1-h

E-E2 = hq – h [Because E-E1 = hq]

E-E2 = h (q-1)

.

.

.

.

E-En = h [q-(n-1)]

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 E1, 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)

.

.

.

E-E1 = h [Q+(n-1)]

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].

10 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.

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

sincere thank due to Soumya V G. My thank is also due to

Munavvar Ali.

References:

1. “Vaidehi bhagat”, “computer oriented numerical

methods”, First Edition(2004-05), kamal prakashan, Indore-2

2. “S.R. Gupta”, “A text book of computer oriented

numerical methods”, First Edition(2002), Nakoda

publisher & printers, Indore-2

3. “B.D. singh”, “Biotechnology”, second Edition

(2007), kalyani publishers,new delhi

4. “A.C. Deb”, “Fundamental of biochemistry”, Eight

edition(2002), new central book agency, kolkata-9

5. “Ram Nivas singh”, “Biology”, yugbodh

publication, Raipur

6. student.ccbcmd.edu

Source of support: Nil; Conflict of interest: None declared