International Journal of Recent Academic Research (ISSN: 2582-158X) Vol. 01, Issue 09, pp.532-542, December, 2019
Available online at http://www.journalijrar.com
RESEARCH ARTICLE
EULER’S LINE FOR ENZYME KINETICS
1, *Vitthalrao Bhimasha Khyade, 2Avram Hershko and 3Seema Karna Dongare
1Department of Zoology, Shardabai Pawar Mahila Mahavidyalaya, Shardanagar Tal., Baramati Dist., Pune – 413115, India 2Unit of Biochemistry, The B. Rappaport Faculty of Medicine, and the Rappaport Institute for Research in the Medical Sciences, Technion-Israel Institute of Technology, Haifa 31096, Israel 3P.G. Student, Department of Microbiology, Maharashtra Education Society's, Abasaheb Garware College, Karve Road, Pune – 411004, India
ARTICLE INFO ABSTRACT
Article History: A graph of the double-reciprocal equation is also called a Line weaver-Burk, reciprocal of velocity of enzyme
Received 10th September 2019, reaction (1÷v) against reciprocal of substrate concentration [1÷S]. Lineweaver-Burk graphs are particularly useful Received in revised form for analyzing the changes in enzyme kinetics in the presence of inhibitors, competitive, non-competitive, or a 28th October 2019, mixture of the two. One more attempt is carried out for establishment of Euler’s line through the use of Line Accepted 04th November 2019, weaver-Burk plot. Line weaver-Burk plot (double reciprocal plot) is with positive value of (Km÷Vmax) as a Published online th slope. Euler Line for Enzyme Kinetics is with negative value of (Km÷Vmax) as a slope. The intercept on y-axis 30 December 2019. for Line weaver-Burk plot (double reciprocal plot) for Enzyme Kinetics correspond to: (1 ÷ Vmax). The intercept on y-axis for Euler Line for Enzyme Kinetics correspond to: [(Km +2) ÷ Vmax)]. Lineweaver-Burk plot (double reciprocal plot) and Euler Line for Enzyme Kinetics are intersecting at the point, x – co-ordinate of which correspond to: (1÷2) and y- co-ordinate of which correspond to: [(Km+2) ÷ Vmax]. The centroid for enzyme kinetics is always located between the orthocenter and the circumcenter of enzyme kinetics. The distance from the centroid to the orthocenter is always twice the distance from the centroid to the circumcenter of enzyme kinetics. Attempt may open a new avenue for three dimensional enzyme structure of and mechanism of enzyme involved reactions. *Corresponding Author: Vitthalrao Bhimasha Khyade Key Words: Centroid, Orthocenter, Circumcenter, Euler’s line.
Copyright © 2019, Vitthalrao Bhimasha Khyade. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The rate of the forward reaction from E + S to ES may be INTRODUCTION termed k1, and the reverse reaction as k-1. Likewise, for the reaction from the ES complex to E and P, the forward reaction Enzymes are protein molecules that manipulate other rate is k2, and the reverse is k-2. Therefore, the ES complex molecules the enzymes' substrates. These target molecules may dissolve back into the enzyme and substrate, or move bind to an enzyme's active site and are transformed into forward to form product. At initial reaction time, when t ≈ 0, products through a series of steps known as the enzymatic little product formation occurs, therefore the backward reaction mechanism. Enzyme kinetics is the study of the chemical rate of k-2 may be neglected. The new reaction becomes: reactions that are catalyzed by enzymes. In enzyme kinetics, the reaction rate is measured and the effects of varying the E + S ↔ ES → E + P conditions of the reaction are investigated. Studying an enzyme's kinetics in this way can reveal the catalytic Assuming steady state, the following rate equations may be mechanism of this enzyme, its role in metabolism, how its written as: activity is controlled, and how a drug or an agonist might inhibit the enzyme (Kraut, Carroll and Herschlag (2003). The Rate of formation of ES = k1[E][S] Michaelis-Menten equation arises from the general equation Rate of breakdown of ES = (k-1 + k2) [ES] and set equal to for an enzymatic reaction: each other (Note that the brackets represent concentrations).
E + S ↔ ES ↔ E + P, Therefore:
k1[E][S] = (k-1 + k2) [ES] Where E is the enzyme, Rearranging terms, S is the substrate, [E][S]/[ES] = (k-1 + k2)/k1 ES is the enzyme-substrate complex, and P is the product. The fraction [E][S]/[ES] has been coined Km, or the Michaelis constant. According to Michaelis-Menten's kinetics equations, Thus, the enzyme combines with the substrate in order to form at low concentrations of substrate, [S], the concentration is the ES complex, which in turn converts to product while almost negligible in the denominator as KM >> [S], so the preserving the enzyme. equation is essentially 533 International Journal of Recent Academic Research, Vol. 01, Issue 09, pp.532-542, December, 2019
V0 = Vmax [S]/KM A graph of the double-reciprocal equation is also called a Lineweaver-Burk, 1/Vo vs 1/[S]. The y-intercept is 1/Vmax; Which resembles a first order reaction. the x-intercept is -1/KM; and the slope is KM/Vmax. Lineweaver-Burk graphs are particularly useful for analyzing At High substrate concentrations, [S] >> KM, and thus the term how enzyme kinematics change in the presence of inhibitors, [S]/([S] + KM) becomes essentially one and the initial velocity competitive, non-competitive, or a mixture of the two. There approached Vmax, which resembles zero order reaction. are four reversible inhibitors: competitive, uncompetitive, non- competitive and mixed inhibitors. They can be plotted on The Michaelis-Menten equation is: double reciprocal plot. Competitive inhibitors are molecules that look like substrates and they bind to active site and slow down the reactions. Therefore, competitive inhibitors increase Km value (decrease affinity, less chance the substrates can go to active site), and Vmax stays the same. On double reciprocal
plot, competitive inhibitor shifts the x-axis (1/[s]) to the right Michaelis-Menten Equation towards zero compared to the slope with no inhibitor present.
Uncompetitive inhibitors can bind close to the active site but In this equation: don't occupy the active site. As a result, uncompetitive
inhibitors lower Km (increase affinity) and lower Vmax. On V is the initial velocity of the reaction. 0 double reciprocal plot, x-axis (1/[s]) is shifted to the left and up V is the maximal rate of the reaction. max on the y-axis (1/V) compared to the slope with no inhibitor. [Substrate] is the concentration of the substrate. Non-competitive inhibitors are not bind to the active site but somewhere on that enzyme which changes its activity. It has K is the Michaelis-Menten constant which shows the m the same Km but lower Vmax to those with no inhibitors. On concentration of the substrate when the reaction velocity is the double reciprocal plot, the slope goes higher on y-axis equal to one half of the maximal velocity for the reaction. It (1/V) than the one with no inhibitor. Km value is numerically can also be thought of as a measure of how well a substrate equal to the substrate concentration at which the half of the complexes with a given enzyme, otherwise known as its enzyme molecules are associated with substrate. km value is an binding affinity. An equation with a low K value indicates a m index of the affinity of enzyme for its particular substrate. large binding affinity, as the reaction will approach V more max The velocity (v) of biochemical reaction catalyzed by the rapidly. An equation with a high K indicates that the enzyme m enzyme vary according to the status of factors like: does not bind as efficiently with the substrate, and V will max concentration of the substrate [S]; hydrogen ion concentration; only be reached if the substrate concentration is high enough to temperature; concentration of the respective enzyme; activators saturate the enzyme. As the concentration of substrates and inhibitors. There is no linear response of velocity (v) of increases at constant enzyme concentration, the active sites on biocatalyzed reaction to the concentration of the substrate [s]. the protein will be occupied as the reaction is proceeding. This may be due to saturable nature of enzyme catalyzed When all the active sites have been occupied, the reaction is biochemical reactions. If the initial velocity (v) or rate of the complete, which means that the enzyme is at its maximum enzyme catalyzed biochemical reaction is expressed in terms capacity and increasing the concentration of substrate will not of substrate-concentration of [S], it appears to increase. That is increase the rate of turnover. Here is an analogy which helps to to say, initial velocity (v) of the enzyme catalyzed biochemical understand this concept easier. reaction get increase according to the increase in the
concentration of substrate [S]. This tendency of increase in Vmax is equal to the product of the catalyst rate constant (kcat) initial velocity (v) of the enzyme catalyzed biochemical and the concentration of the enzyme. The Michaelis-Menten reaction according to the increase in the concentration of equation can then be rewritten as V= Kcat [Enzyme] [S] / (Km substrate [S] is observed up to certain level of the + [S]). Kcat is equal to K2, and it measures the number of concentration of substrate [S]. At this substrate concentration substrate molecules "turned over" by enzyme per second. The [S], the enzyme exhibit saturation and exert the initial velocity unit of Kcat is in 1/sec. The reciprocal of Kcat is then the time (v) of the biocatalyzed reaction to achieve maximum velocity required by an enzyme to "turn over" a substrate molecule. The (V ). higher the Kcat is, the more substrates get turned over in one max second. MATERIALS AND METHODS Km is the concentration of substrates when the reaction reaches half of Vmax. A small Km indicates high affinity since The medians, altitudes and bisectors are the requirements for it means the reaction can reach half of Vmax in a small number establishment of Euler’s line for a triangle. The intersection of of substrate concentration. This small Km will approach Vmax the three medians yields the point of “Centroid” for a triangle. more quickly than high Km value. The intersection of the three altitudes yields the point of “Orthocenter” for a triangle. The intersection of the three When Kcat/ Km, it gives us a measure of enzyme efficiency bissectors yields the point of “Circumcenter” for a triangle. with a unit of 1/(Molarity*second)= L/ (mol*s). The enzyme Material and methods for the present attempt is divided into efficiency can be increased as Kcat has high turnover and a the steps, which include: (A) Establishment of a Right angled small number of Km. Taking the reciprocal of both side of the Triangle Through the Linewever-Burk Plot (line Y.1); (B) Michaelis-Menten equation gives: Establishment of Geometrical Centroid of Right Angled Triangle; (C) Establishment of Geometrical Orthocenter of = + Right Angled Triangle; (D) Establishment of Geometrical 534 International Journal of Recent Academic Research, Vol. 01, Issue 09, pp. 532-542, December, 2019
Circumcenter of Right Angled Triangle and (E) Establishment y1= = + of Euler’s line for a right angled triangle.
This Lineweaver–Burk plot deserve wide applicability. It is (A). Establishment of a Right angled Triangle Through the useful for the determination of K , the most significant factor Linewever-Burk Plot (line Y.1): m in enzyme kinetics. The intercept on y – axis of “Lineweaver–
Burk-Plot” is the reciprocal of V or (1/ V ). And intercept Hans Lineweaver and Dean Burk (1934) suggested the double max max on X – axis of “Lineweaver–Burk-Plot” is the reciprocal of - reciprocal plot for presenting the information in the form of K or (−1/K ). Reciprocals of both, [S] and (v) are utilized in readings or the data on the concentration of substrate [S] and m m the Lineweaver-Burk plot. That is to say, this plot is pertaining rate or velocity (v) of the biocatalyzed reaction. In enzyme and . Therefore, “Lineweaver–Burk-Plot” is also termed as kinetics, double reciprocal plot suggested by Hans Lineweaver and Dean Burk is well esteemed graphical presentation of the a double reciprocal graph. This attempt through the data on concentration of the substrate [S] and velocity (v) of “Lineweaver-Burk-Plot”, is giving quick, concept or idea of the biocatalyzed reaction recognized as, the “Lineweaver–Burk the biochemical reaction. It also allow to understand the plot”. This plot deserve wide applicability. The most mechanism of activation of enzyme and inhibition of enzyme. significant application of Lineweaver–Burk plot lies in the Researchers including authors of present attempt designating determination of concentration of substrate [S] which is the double reciprocal plot as a Nobel Plot. Most of researchers responsible for achievement of the half the maximum rate or entertaining the enzyme kinetics through this double reciprocal the velocity (Vmax ÷ 2) of the biochemical reaction catalyzed plot are non-mathematical academicians. Present attempt is by the enzymes. The “Km” or Michaelis constant is the trying it’s best to minimize the errors in understanding the concentration of substrate [S] responsible for yield of the concepts in enzyme kinetics through modification in the reaction rate, which is corresponding to exactly half the rate or “Lineweaver-Burk-Plot”. Each and every method is with velocity of maximal (Vmax ÷ 2) for enzyme involved positive and negative points of advantages. According to biochemical reaction. For practical purposes, this “Km” or Hayakawa, et al (2006), there is distortion of error structure Constant of Michaelis is the reading pertaining [S] that allows through this double reciprocal plot of “Lineweaver-Burk-Plot”. velocity to achieve half with reference to maximum rate or It is therefore, method of graphical presentation of velocity (Vmax). The affinity of enzymes for their substrate “Lineweaver–Burk-Plot” (double-reciprocal-plot) appears to vary. Generally, the enzyme with a higher Km value has little attempt to minimize errors. This may yield easier method of bit lower affinity for its substrate. According to Keith J. calculation of constants or parameters of enzyme kinetics. On Laidler (1997), enzymes with lower affinity for their substrate, this line of improvement of method of calculation of constants requires a greater volume of substrate or substrate or parameters of enzyme kinetics, much more work is already concentration for the purpose to achieve maximum rate or exist. According to Hayakawa, et al (2006), methods of velocity of enzyme involved biochemical reactions. The wide improvement in the calculation of constants or parameters of range of applicability is the distinguishing feature of enzyme kinetics are under the title, “non-linear regression or Lineweaver–Burk plot. In the past, there was no computer alternative linear forms of equations”. And they include: the facilities as today. In such a critical situation, the parameters of plot of “Hans-Woolf”; the plot of “Eadie-Hofste”; such and the enzyme kinetics, Km and Vmax served a lot through this others (Greco and Hakala, 1979). Lineweaver-Burk plot for fortified concept of enzyme kinetics. In this Lineweaver-Burk plot, reading the inverse of maximum Dick (2011) explained type of inhibition of enzyme activity or velocity of biocatalyzed reaction (1÷ Vmax) take the position stoping the working of enzymes. Of course, this discussion is of y-intercept (Fig.1). The negative value of inverse of Km based exclusively on “Lineweaver–Burk-Plot” (reciprocals ob (1÷Km) take the position of x-intercept. The quick visual both the axes) is able to group or classify the inhibitors of impression of the inverse form of substrate concentration and actions of enzymes. Accordingly, the inhibitors of enzyme can rate or velocity of reaction is one more advantage of basically be grouped into the types like: The “Competitive Lineweaver-Burk plot. And this feature help for understanding Inhibitors”; “Non-competitive inhibitors” and “uncompetitive the concept of enzyme inhibition. Accordingly, mathematical inhibitors”. The inhibitors of enzyme of “Competitive” class equation suggested by Lineweaver and Dean Burk (1934) can deserve one and the same point of intersection on the Y-axis. It be written as: = + clearly means, inhibitors of enzyme of “Competitive” class are not affecting on maximal rate or velocity of reaction
(competitive inhibitors provide protection the maximum velocity Vmax . They keep this maximum velocity Vmax non- affected). But, slopes of equations are not same. Slopes are different slopes. The inhibitors of enzyme of “Non- competitive” class deserve one and the same point of intersection on the X-axis. It clearly means, inhibitors of enzyme of “Non-competitive” class are not affecting on the Km, the [S] for half the maximal rate or velocity of reaction (Km is remains unaffected by non-competitive inhibitors. The inverse of Km doesn't change). The non-competitive inhibition produces plots with the same x-intercept (−1/Km) as uninhibited enzyme (Km is unaffected) but different slopes and y-intercepts. Uncompetitive inhibition causes different intercepts on both the y- and x-axes (Berg, et al, 2002). John E.
Fig. 1. Regular form of Linrweaver-Burk Plot (Double Reciprocal Plot) Dowd and Douglas Briggs (1965) reviewed the literature on “Estimates of Michaelis – Menten kinetic constants through 535 International Journal of Recent Academic Research, Vol. 01, Issue 09, pp.532-542, December, 2019 the use of different linear transformation” and listed some any triangle that is not equilateral. It is a central line of the problems with Lineweaver–Burk plot (double reciprocal plot). triangle, and it passes through several important points Accordingly, Lineweaver–Burk plot (double reciprocal graph) determined from the triangle, including the orthocenter, the is appearing in most of the new as well older books of circumcenter, the centroid, the Exeter point and the center of biochemistry. It seems in having prone to error. There may be the nine-point circle of the triangle. The centroid is also called mistake in understanding the expected for researchers. The as geometric center of a plane figure. It is the arithmetic mean readings of inverse of “v” are on Y – axis. The readings of position of all the points in the figure. Altshiller-Court, Nathan inverse of “[S]” are on X – axis. The lower values of both the (1925) informally defined the centroid as the point at which a readings (inverse of “v” and inverse of “[S]”) are occupying cutout of the shape could be perfectly balanced on the tip of a higher (signifiacant) position in graph. And… and… higher pin. The barycenter is a synonym used for centroid. The values of both the readings (inverse of “v” and inverse of centroid of a triangle is the point of intersection of it’s three “[S]”) are occupying lower (non-significant) position in graph. medians. It is a point of concurrency of the triangle. It is the Both the conditions may be interpreted wrongly. point, where all the three medians intersect. It is often described as the center of gravity of triangle. The geometrical properties of centroid of a triangle include: Intersection of the three medians exert to form the centroid; The centroid is one of the points of concurrency of a triangle; The centroid is always located inside the triangle; The centroid divides each median in a ratio of 2:1. The centroid will always be 2/3 of the way along any given median. According to Johnson (1929) and Wells (1991), the geometric centroid (center of mass) of the of a triangle is the point formed through the intersections of the three medians of the triangle. The point of centroid, is therefore also called as the median point. It has equivalent triangle center functions. Orthocenter of triangle is the point of intersection of the altitudes. Each leg in a right triangle forms altitude. In a right-angled triangle, the orthocenter lies at the vertex containing the right angle. Altitude of a triangle is a line Fig. 2. Right angled triangle through the use of “Regular form of segment through a vertex and perpendicular to a line Linrweaver-Burk Plot (Double Reciprocal Plot) (Y.1) containing the base (the side opposite the vertex). This line
In a Regular form of Linrweaver-Burk Plot (Double Reciprocal containing the opposite side is called as the extended base of altitude. The circumcenter of a triangle is the point of Plot) [ y1= (Km÷Vmax) x + (1÷Vmax)]; when the value of x is one, the y value correspond to [(Km +1)÷Vmax]. The y – intersections of the three perpendicular bisectors of the sides of intercept of the regular form of Linrweaver-Burk Plot (Double that particular triangle intersects. In other words, the point of Reciprocal Plot) correspond to (1÷Vmax)]. In figure 2; the y – concurrency of the bisector of the sides of a triangle is called intercept of the regular form of Linrweaver-Burk Plot (Double the circumcenter. Every triangle has exactly three medians, one Reciprocal Plot) is designated as point: “A”. The co-ordinates from each vertex, and they all intersect each other at the of the point “A” are zero and reciprocal of maximum velocity triangle's centroid. For getting the point of centroid in right of biochemical reaction involving the interplay of the enzymes. angled triangle (∆ ABC) (figure – 2) in present attempt, it is It can be written as: [A (0, 1÷Vmax)]. The line segment necessary to establish the three medians, one from each vertex. perpendicular to y – axis and passing through as point: “A” up to the point: “B” is considered as one of the side of triangle. At (B.1). Establishment of the line Y.2 (For one of the median the point “B”, x equals to one and y equals to the reciprocal of of (∆ ABC) (Fig.3): maximum velocity of biochemical reaction involving the Let us first consider the line y2. The slope and intercept on y- interplay of the enzymes (figure 2). Therefore the co-ordinates axis of this y line are considered as: and of the point “B” can be written as: [A (1, 1÷Vmax). As stated 2 earlier, when the value of x is one, the y value of Regular form respectively. Therefore, the mathematical equation in “Slope- of Linrweaver-Burk Plot (Double Reciprocal Plot) (y1) intercept” form (in the form of typical y= m x +c) of this line correspond to: [(Km +1)÷Vmax]. The point (with x – value y3 can be written as: equals to one) in regular form of Linrweaver-Burk Plot (Double Reciprocal Plot) (y1) claiming the y value is Y = x + 2 designated as “C”. The co-ordinates of the point “C” can be written as: C [ 1, (Km +1) ÷Vmax]. Joining the point “A” to This equation may also be writte as: the point “B”; the point “B” to the point “C” and the point “C” to “A” yields right angled triangle (figure – 2). A right-angled Y = x + + 2 triangle is a triangle in which measure of one angle is ninety degree (or a right angle). The above equation contain the term: (Km÷Vmax) for two times. (B). Geometrical Centroid of Right Angled Triangle: Let us take out one of the term: (Km÷Vmax) as a common The relation between the sides and angles of a right angled factor and let us simplyfy the above equation. triangle is the basis for trigonometry. The side opposite the right angle is called the hypotenuse. The sides adjacent to the Y = [ + ] + 2 right angle are called legs. In geometry, the Euler line, named Y = [ ] + after Leonhard Euler. The Euler line is a line determined from 2 536 International Journal of Recent Academic Research, Vol. 01, Issue 09, pp. 532-542, December, 2019
If we replace the “x” by (1÷ S) If we replace the “x” by (1÷ S) and substitute the Km = [S (Vmax –v) ÷ v]; the mathematical equation for the line y2 is going to transform into: Y2 = [ ] + ( ) = + Let us now substitute the Km = [S (Vmax –v) ÷ v]; the mathematical equation for the line y2 is going to transform into: Simplification of this equation is going to yields into
( ) ( ) = + Y2 = [ ] + . ( )( ) = = + = Simplification will yields into
( )( ) It definitely means for plotting the y3; it is necessary to = + consider X = (1÷ S) and
( )( ) == Y =
It definitely means for plotting the y2; it is necessary to Through replacing the values of Vmax; V and S, it is possible consider X = (1÷ S) and to calculate the respective values of Y. This is going to serve the purpose of plotting this new line (y3) along with y1 and y2 ( )( ) Y = This new line (y3) is passing from the point “A” (one of the vertex of ∆ ABC) and attain half the measurement of the Through replacing the values of Vmax; V and S, it is possible segment “BC” at point “F”. The line (y3) is dividing the to calculate the respective values of Y. This is going to serve segment BC into two equal parts. Therefore, the segment “AF” the purpose of plotting this new line (y2) along with y1. This is designated as median of ∆ ABC drawn from the vertex “A” new line (y2) is passing from the point “B” (one of the vertex on the side segment “BC”. of ∆ ABC) and intersecting the line y1 at the point “D” and dividing the segment AC into two equal parts. Therefore, the segment BD is designated as median of ∆ ABC drawn from the vertex “B” on the side segment “AC”.
Fig. 4. Line Y.2 and Y.3 as the two medians in Right angled triangle (∆ ABC)
(B.3). Establishment of the line Y.4 (For one of the median of (∆ ABC) (Fig.5):
Let us now consider the line y4 (Fig.5). The slope and intercept ( )
on y- axis of this y4 line are considered as: and Fig. 3. Line Y.2 as one of the median in Right angled triangle (∆ ABC) respectively. Therefore, the mathematical equation in “Slope- (B.2). Establishment of the line Y.3 (For one of the median intercept” form (in the form of typical y= m x +c) of this line of (∆ ABC) (Fig.4): y4 can be written as: