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Introduction: The topic kinetics in biological system is mainly concerned with the study of rate of change of kinetics of a reaction or a chemical process that occur in a living organism. In a biological system the reaction kinetics mainly depends upon the enzymes involved in that reaction. By the change in quantity of enzyme secreted or due to any change in chemical composition of the enzymes, the rate of chemical reaction can be affected. Enzymes are single or multiple-chain proteins that act as a biological catalyst with the ability to promote specific chemical reaction under the mild conditions that prevail in most living organism.
The substance upon which enzymes are acting is known as substrate. Enzymes bind their substrate at a specific binding site, generally called as a active or a catalytic site. The resulting enzyme-substrate complex promotes a chemical reaction, facilitated by specific amino acids residues in the catalytic site, resulting in the formation of the product. Different amino acid residues in the site may be involved in the binding of the substrate and hence promoting the reaction.
Enzyme assays are undertaken for a variety of reasons but the two most common reasons are:
To determine the amount of enzyme present in a particular preparation.
To gain an insight into the kinetic characteristics of the reaction and hence to determine a range of kinetic constants such as Km,Vmax and Kcat.
Initial rates: when an enzyme is mixed with an excess of substrate there is an initial short period of time during which intermediates leading to the formation of the product gradually build up. This is so called pre-steady state requires special techniques for study. After this pre-steady state, the reaction rate and the concentration of intermediates change relatively slow with time and so called steady-state kinetics exist. The tangent drawn through the origin to the curves of substrate concentration and product concentration versus time allow the initial rate vo to be calculated. This the maximum rate for the given concentration of enzyme and the substrate under the defined experimental conditions. Measurement of the initial rate of the enzyme catalyzed reaction is perquisite to a complete understanding of the mechanism by which the enzyme works, as well as the estimation of the activity of an enzyme in an biological sample. Its numerical value is influenced by many factors, including substrate and enzyme concentration, pH, temperature and the presence of activators or inhibitors.
Initial rates are sometimes determined experimentally on the basis of single measurement of amount of substrate consumed or product produced in a given time rather than the tangent method. This approach is valid over only the short period of time when the reaction is proceeding effectively at a constant rate.
Methods for steady-state studies: The various methods are
Visible and ultraviolet spectrophotometric method:
Many substrates or products absorbs light in the visible or ultraviolet region and the change in the absorbance during the reaction can be used as a basis for the enzyme assay. Hence the Beer-lamberts law should be obeyed. The number of units of enzymes
Enzyme steady-state kinetics:
Monosubstrate enzyme reactions: For many enzymes, the initial rate vo , varies hyperbolically with substrate concentration for a fixed concentration of enzyme. The mathematical equation expressing this hyperbolic relationship between initial rate and substrate concentration is known as Michealis-Menten equation:
Vo = vmax[s]/km + [s]
Where vmax is the limiting value of the initial rate when all the active sites are occupied, km is the Michaelis constant, and [s] is the substrate concentration. At low substrate concentration the occupancy of the active sites on the enzyme molecules is low and the reaction rate is directly related to the number of sites occupied. This approximation to the first order kinetics in that the rate is proportional to substrate concentration.
At high substrate concentrations effectively all the active sites are occupied and the reaction becomes independent of the substrate concentration and hence no more enzyme-substrate complex can be formed and zero-order or saturation kinetics are observed. Under these conditions the reaction rate is dependent upon the conversion of the enzyme-substrate complex products and the diffusion of the complex products and the diffusion of the products from the enzyme.
It van be noted from the equation that when vo = 0.5vmax , km = [s]. thus km is numerically equal to the substrate concentration at which the initial rate is one-half the maximum rate and has the units of molarity.
Enzyme-catalyzed reactions proceed via the formation of an enzyme-substrate complex in which the substrate (S) is non-covalently bonded to the active site of the enzyme (E). the formation of this complex for majority of enzymes is rapid and reversible and is characterized by the dissociation constant, Ks , of the complex:
E + S ES
Where K+1 and k -1 are the rate constants for the forward and reverse reactions. At equilibrium, the rates of the forward and reverse reactions are equal and the law of mass action can be applied to the reversible process:
K+1 [E][S] = K -1 [ES]
Ks = [E][S] / [ES] = k+1 / k -1 =1/ks
Where ks is the association(or affinity) constant. Therefore, when ks is numerically large, the equilibrium is in favor of unbound E and S, while if ks is numerically small, the equilibrium is in favor of the formation of ES. Thus ks is inversely proportional to the affinity of the enzyme for its substrate.
The conversion of ES to product can be most simply represented by the
ES ------------> E + P
Where k+2 is the first order rate constant of the reaction.
In some cases the conversion of ES to E and P may involve several stages and may not necessarily be essentially irreversible. The rate constant k+2 is generally smaller than both k+1 and k -1 and in some cases very much smaller. Therefore, the conversion of ES to products is the rate limiting step such that the concentration of ES is essentially constant but not necessarily the equilibrium concentration. Under these conditions the Michaelis constant, km , is given by:
Km = k+2 + k -1 / k+1 = ks +k+2/k+1
It is evident tha under these circumstances, km must be numerically larger than ks and only when k+2 is very small do km and ks approximately equal each other. The relationship between these two constants is further complicated by the fact that, for some enzyme reactions, two products are formed sequentially, each controlled by different rate constants:
E + S ES ïƒ p1 +EA ïƒ E + p2
Where p1 and p2 are products, and A is a metabolic product of S that is further metabolized to p2 .
In such circumstances it can be shown that:
Km = ks [ k+3 / (k+2 + k+3 ) ]
So that km is numerically smaller than ks . it is obvious therefore that care must be taken in the interpretation of the significance of km relative to ks . only when the complete reaction mechanism is known can the mathematical relationship between km and ks is fully appreciated.
Lineweaver-Burk equation: it is obtained by taking reciprocal of the Michealis-Menten equation. It is a linear transformation of the Michealis-Menten equation. Hence the equation becomes:
1/v0 = ( km/Vmax x 1/[s] )+ 1/vmax
Bisubstrate enzyme reactions:
Bisubstrate reactions are those catalyzed by the Transferaces, kinases and dehydrogenases, in which two substrates s1 and s2 are converted to two products p1 and p2 , and these are inherently more complicated than Monosubstrate reactions.
Effect of enzyme concentration: it can be shown that for Monosubstrate enzymatic reactions that they obey simple Michaelis-Menten kinetics:
Vo = k+2 [E][S] / km + s
And hence that
Vo = k+2 [E] / ( km / [s] +1)
Thus when the substrate concentration is very large, the equation reduces to
v0 = k+2 [E], i.e. the initial rate is directly proportional to enzyme concentration. This is the basis of experimental determination of enzyme activity in a particular biological sample.
Effect of temperature: the initial rate of the enzyme reaction varies with temperature according to the ARRHENIUS equation.
Rate = A e-Ea/ RT , where E is the activation energy and R is the gas constant and A is constant known as pre exponential factor, which is related to the frequency at which molecules of the enzyme and substrate collide in correct orientation to produce the enzyme-substrate complex.
Effect of pH: The state of ionization of amino acids in residues in the catalytic site of an enzyme is pH dependent. Since catalytic activity relies on specific state of ionization of these residues, enzyme activity is also pH dependent. As a consequence, plots of log Km and log Vmax against pH are either bell shaped(indicating two important ionisable amino acid residues in the active site), giving a narrow pH optimum, or a plateau( one important ionisable amino acid residues in the active site). In either case, the enzyme is generally studied at a pH at which its activity is maximal. By studying the variation of log Km and log vmax with pH, it is possible to identify the pKa values of key amino acid residues involved in the binding and catalytic processes.
Effect of enzyme inhibitor:
Irreversible inhibitor - An enzyme inhibitor binds to an enzyme in such a way as to reduce the ability of the enzyme to either bind substrate and/or convert it to product. Irreversible inhibitors such as organomercury compounds, cyanide, hydrogen sulphide etc, combine with the enzyme to form a covalent bond. The effect of their inhibition is reduced amount of enzyme available for reaction. Hence irreversible inhibitors reduce the rate of reaction. This inhibition can,t be removed by simple physical techniques.
Competitive reversible inhibition: Reversible inhibitors combine non-covalently with the enzyme and hence reduce the rate of the reaction. This inhibition can be removed by dialysis.
Substrate inhibition: A number of enzymes at a high substrate concentration display substrate inhibition characterized by a decrease in initial rate with increased substrate concentration.
Significance of kinetic studies: kinetic studies using a range of substrates and/or competitive inhibitors and the determination of the associated Km, Kcat, and Ki values allows correlation to be drawn between molecular structures and kinetic constants and hence deductions to be made about the structure of the active site. In the case of Bisubstrate reactions, information about the reaction mechanism and substrate binding sequence can be deduced. Further information about the structure of the active site can be gained by studying the influence of pH on the kinetic constants. The effect of pH on km (i.e. on binding of E to S) and on Vmax or Kcat (i.e. conversion of ES to products) is studied. Plots are then made on the variation of log Km with pH and of log Vmax or log Kcat with pH. The intersection of the tangents drawn to the curve gives an indication of the pKa values of ionisable groups involved in the active site. These are then compared with the pKa values of the ionisable groups known to be in the proteins. For e.g. pH sensitivity around the range 6-8 could reflect the importance of one or more imidazole side-chains of a histamines residue in the active site because of its known pKa in this range.