Enzyme kinetics

The enzyme kinetics is a branch of biophysical chemistry. It describes how fast run enzyme-catalyzed chemical reactions. The enzyme kinetics is widely used in biology and medicine, as well as biological substrates ( reactants ) - including those that occur in humans - are examined. A main objective of the enzyme kinetics is the description of the concentration dependence of the response speed by suitable formulas, as well as the determination of the corresponding parameters for a specific protein (enzyme activity and catalytic efficiency). Since enzymes are used to speed up reactions and steer, the enzyme kinetic analysis to the understanding of enzyme function is essential.

• 4.1 Competitive
• 4.2 uncompetitive
• 4.3 Non -competitive
• 4.4 Mixed - competitive inhibition

Theory

The first to describe the relationship between substrate concentration [ S] and reaction rate of an enzyme v only after Sørensen 1909 the pH was the French physical chemist Victor Henri in 1902. However, the significance of the hydrogen ion concentration was for enzymatic reactions is not known at the time, had defined and implemented the buffering, the German Leonor Michaelis and his Canadian post- doctoral student Maud Menten able to confirm experimentally in 1913 the results of Henri's. The Henri- Michaelis -Menten equation was generalized in 1925 by GE Briggs and JBS Haldane ( Michaelis- Menten kinetics ).

Henri's key idea is to decompose the enzymatic reaction in two phases, the binding of the substrate to the enzyme and the reaction of the enzyme -substrate complex into enzyme and product. Was

The binding reaction can be described by the law of mass action, it is characterized by dissociation constant. It is the concentration of the substrate (in the absence of conversion to product) half of the enzyme molecules have substrate bound at the. It is inversely proportional to the affinity of the enzyme for the substrate.

The reaction of the enzyme -substrate complex to enzyme and product is determined by a rate constant of first order. The actual rate of the enzymatic reaction is given by. However, the decomposition reaction reduces the concentration of [ES ], it is therefore necessary to take these to calculate with. Then arises

This is the equation of a hyperbola, it approaches infinity for high substrate concentrations, when all enzyme molecules have substrate bound, a limit on the amount. This represents the total enzyme concentration. The concentration at which the reaction rate reached to just half of the maximum value is the Michaelis constant. For the determination of and from measurements of and are computer-based methods, such as non-linear regression analysis ( simplex or Levenberg -Marquardt method ). Graphical extrapolation ( linearizations ), such as the double -reciprocal Lineweaver and Burk should not be used for this, as they are too inaccurate. However, it is very well suited for the presentation of the results of enzyme kinetic experiments, because the human eye can detect deviations from a straight line more easily than that of a curve.

Direct linear plot

Enzyme-kinetic parameters can be easily and accurately directly from a Sättigungshyperbel shown in the figure derived ( " direct- linear plot " also called " Cornish - Bowden diagram" ). In this hyperbola enzymatic reaction rate is V ( y-axis) as a function of the substrate concentration [ S].

For the direct- linear plot one transfers the initial velocities of the enzymatic turnover directly into the v - [ S]- diagram. The [S ] values ​​are known before the experiment (discontinued substrate concentrations ); during the test series, the ordinate value for v is then nachzutragen ( the initial velocity ). Vmax from the maximum reaction rate can be half the maximum conversion rate of 0.5 derived vmax. Graphically, you can use it to determine the coordinate value for Km. The catalytic efficiency follows the way from the slope of the tangent at the origin: Vmax / Km; this results in kcat / Km.

The exception is the direct- linear plot greatly simplified: Averaging then the likely values ​​for the parameters Km and Vmax. For inspection of the scatter of the data points ( not the same as the standard deviation) can Outliers easily identified and so-called Median be read.

At this point it should be mentioned that all have ( and the following ) evaluation method not only for enzymes, but also for the binding processes of carriers or receptors validity. All of these methods (Hanes and Eadie - Hofstee plot for enzymes, Scatchard and Hill plots for Carrier) Historically originally developed by Woolf.

Linearization method

Were linearization procedure in the past very often used for quick graphical determination of the important kinetic parameters Km and Vmax. Although they are easy to remember and disseminated, but lead to a sometimes considerable distortion of the result by measurement error and error analysis for more or less suitable. Meanwhile, the determination of the Michaelis- Menten parameters by non-linear regression has become increasingly important, which leads to significantly more accurate results. Therefore, the linearization method should only be touched upon here.

Lineweaver- Burk plot

Hans Lineweaver (1907-2009) and Dean Burk (1904-1988) who in 1934 a double -reciprocal representation presented in the 1 / v is plotted as a function of 1 / [ S].

From the Michaelis- Menten equation

Is forming:

Therefore the graph crosses the x -axis at -1/Km, and the y -axis at 1/vmax, which can be read directly from the graphic plot. Although it is mostly used for data representation, but this method of analysis is unreliable. Small errors in v can be determined at low [S ] values ​​a large deviation in 1 / v, at large [S ] values ​​is these tend to neglect. The authors of the method have emphasized the uncertainty of large 1 / v values ​​and pointed out that this will generally be less weighted. Subsequent users have this mostly ignored. Wherever possible this should be replaced by computer method for determining enzyme kinetic parameters.

Eadie - Hofstee diagram

The Eadie - Hofstee diagram also Woolf - Eadie - Hofstee or Eadie Augustinsson - Augustinsson diagram occupies a central position. Here, v is plotted against (v / [ S] ).

A transformation of the Michaelis -Menten equation yields the following equation:

From the graph as the y -intercept and Km can be derived as vmax negative slope of the regression line.

The error grows with v / [ S]. Since v is received by two coordinates converge all deviations to the origin.

Scatchard plot

The Scatchard plot of

Similar to the previous (axes are reversed ) and is mostly used for the representation of binding measurements (instead of enzyme kinetic data). Scatchard and Eadie - Hofstee plots are considered the best tools for the diagnosis of cooperative phenomena. In the case of negative cooperativity or non-identical, isolated binding sites results in a concave curve with linear terminal branch. The slopes here correspond to the affinities ( Kd or Km) and the total number of binding sites ( active sites ) can be read from the intersection with the x- axis.

Hanes - Woolf plot ( Hanes ( Wilkinson ) graph)

The Hanes - Woolf chart is the best possible linear Auftragungsmöglichkeit. It goes back to Charles Samuel Hanes (1903-1990) and Barnet Woolf ( 1902-1983 ). Where [ S ] / v is plotted as a function of [S ], which yields the following equation for dissolving:

Error in [S ] / v is a far better approximation of the error in v. Because a genuine spread of the measurement points along the [ S ] axis, the result is in principle less distorted by individual outliers. Since, however, dependent and independent variable is also be mixed here a data optimization by linear regression does not make sense. The increase of the regression line is 1/vmax, the intersection with the y -axis corresponds to Km / Vmax, and with the x-axis - km.

Hill Chart

The Hill - diagram is a graphical method of Archibald Vivian Hill, which was originally used to determine the cooperative binding of oxygen to hemoglobin. The Hill plot is used primarily for the determination of a cooperative binding process for a protein ( enzyme). Be bound substrates S ( or ligands L) The graph is aware of Vmax ( or n ) operating systems. The application of

Is a straight line of slope 1 ( nH = 1 ) when the binding sites are independent. In cooperative systems, the slope at the origin point (the Hill coefficient nH ) can in theory the number of subunits ( n ) the same, but will remain practically underneath. If the individual Km - ( Kd ) values ​​not tangent to the 45 ° - foothills of the curve can be extrapolated, nH a surrogate measure of cooperativity is to be used; only for nH = 1 is equal to the zero crossing ln Km ( or ln Kd). For hemoglobin ( n = 4), high- protein co-operative is a classic, nH was determined to be 2.8 to 2.9.

The Hill equation:

With the proportion of occupied binding sites at which the pharmacophore of the ligand, as the concentration of unbound ligand, as the apparent dissociation constant of the mass action law, as the concentration of the ligand with half-maximal binding, as Hill coefficient.

A Hill coefficient of 1 indicates a two mutually independent binding ligand, a value greater than 1 indicates a positive cooperativity, less than 1 indicates negative cooperativity. Independence at a ( n = 1), the Hill equation to the Langmuir equation approximates.

If the reciprocal formed, changed, re- formed and the reciprocal of the logarithm, we obtain an alternative formulation of the Hill equation:

Inhibitors

Many drugs and poisons are inhibitors (inhibitors) of enzymes. Therefore, the elucidation of the mechanism of action is always approached a special meaning. The nomenclature of the Hemmtypen was by William Wallace Cleland (* 1930) found in 1963 on a systematic basis, unfortunately terms are used in many textbooks still used differently.

Here, however, it should be noted that classical limit analysis to reversible binding substances. Irreversible binding of a substance to an enzyme leads to inactivation, not to inhibition.

Derived from the Michaelis -Menten equation represents the general Inhibitionsgleichung as follows:

Then the ratio of the value ( dissociation constant of the complex EI ) and the value used ( dissociation constant of the complex EIS) for deriving the Inhibitionstyps:

Competitive

Inhibitor and substrate are mutually exclusive from the binding to the enzyme. However, this does not necessarily mean that the inhibitor binds to the same binding site as the substrate. Even if the binding of the substrate or inhibitor lead to conformational changes in the enzyme, which block the binding site for the other, the inhibition is competitive. If substrate and inhibitor, however, have the same binding site, the type of inhibition is necessary competitively.

In the competitive inhibition, the inhibitor may be displaced by the substrate of the enzyme, that does not change. However, a higher need for any desired speed, the apparent becomes higher with increasing so. Lineweaver - Burk plot, this leads to a family of straight lines which have a common intersection point on the y- axis ().

Uncompetitive

The inhibitor does not bind to the free enzyme, but to the ES complex. Higher concentrations of the substrate can therefore not displace the inhibitor from the enzyme, but lead to increased binding. Conversely, decreased binding of the inhibitor, the concentration of ES, according to the principle of Le Chatelier, therefore, additional ES from E and S must form: The apparent decreases, the affinity of the enzyme for the substrate increases with increasing. At the same time naturally decreases. In the Lineweaver- Burk plot, we find a family of parallel straight lines.

Non-competitive

The inhibitor can bind to both E and to ES. In the simplest case is in the process which means that the substrate does not change the binding affinity of the enzyme for the inhibitor, such as conformational. Then, of course, it also follows that the binding of the inhibitor does not alter the affinity of the enzyme for the substrate and. Because of the relationship between the binding of inhibitor and thus also does not change.

It can now show ( by substitution and elimination from the definitions of and ) that. If so, then follows and the apparent increases with. On the other hand, it follows and decreases with increasing apparent.

The non- competitive inhibition results in Lineweaver- Burk plot to a family of straight lines with the common point of intersection to the left of the y-axis, the point of intersection is located on the x- axis, if it is above the x-axis if necessary and the x if axis.

Mixed - competitive inhibition

The mechanism of this Hemmtyps ( which in practice is of little importance ) is similar to the non- competitive inhibition, however, the EIS complex has a catalytic activity. Also, the Lineweaver- Burk plot looks like in the non-competitive inhibition ( with all 3 options ). In the so-called secondary plot ( slope y-intercept or Lineweaver -Burk diagram as a function of ) but you can see in the case of non-competitive inhibition lines, in the case of mixed-competition but curves.