## Kinetics and Reaction Rates

Chemical or biochemical kinetics is the study of chemical or biochemical reactions with respect to reaction rates, effect of conditions reactions are subject to, rearrangement of molecules, formation of intermediates, and involvement of catalyst. The word kinetics is originated from the Greek kinesis, meaning movement. Thus, the kinetics of chemical or biochemical reactions are mainly concerned with rate of reaction and anything else affecting it.

In general, the reaction rate depends on the concentration of reactants. It may also depend on the concentrations of other species that do not appear in the stoichiometric equation. The dependence of reaction rate on concentrations of reactants can be expressed mathematically in terms of the reaction rate constant and the powers of concentrations of reactants. Recall Equation 1.6 for a general reaction form:

The rate of reaction can be expressed as the following (Equation 1.13):

r = kCaACbB

where k is the reaction rate constant, a and b are exponents that may or may not be equal to those coefficients appearing in Equation 1.6, and CA and CB are concentrations of reactants A and B. The summation of a and b is called reaction order—i.e., reaction order for the reaction shown in Equation 1.6 is (a+b). Generally, reactions are categorized as zero-order, first-order, second-order, or mixed-order (higher-order) reactions, based on the value of (a+b). The unit of k is (concentration)1 ~a~b (time) -1.

### Zero-order reactions

Zero-order reactions (order = 0) have a constant rate. This rate is independent of the concentration of the reactants. The rate law is r = k, with k having the units of (concentration)1 (time)-1, e.g., M/sec.

### First-order reactions

A first order reaction (order = 1) has a rate proportional to the concentration of one of the reactants. A common example of a first-order reaction is the phenomenon of radioactive decay. The rate law is r = kCA (or CB instead of CA), with k having the units of (time)-1, e.g., sec-1.

### Second-order reactions

A second-order reaction (order = 2) has a rate proportional to the concentration of the square of a single reactant or the product of the concentration of two reactants: rate = kCA2 (or substitute B for A or k multiplied by the concentration of A times the concentration of B), with the unit of the rate constant k = (concentration)-1 (time)-1, e.g., M-1sec-1.

### Mixed-order or higher-order reactions

Mixed-order reactions, such as some biochemical reactions, have a fractional order for their rate, e.g., rate = kCA1/3 The unit of the rate constant k is (concentration)23 (time)-1, e.g., M2/3/sec.

26 Food and Agricultural Wastewater Utilization and Treatment Catalytic reactions

Almost all biochemical reactions involve catalysts, enzymes that are specialized proteins synthesized by microorganisms. A catalyst is a substance (enzyme for biocatalyst) that increases the rate of reaction without undergoing permanent (bio)chemical change. The primary function of a catalyst is to lower the activation energy of a reaction so that the reaction can be easily carried out but does not affect the reaction equilibrium. In biochemical reactions, the enzyme is believed to possess certain active sites consisting of amino acid side chains or functional groups to which the specific functional groups of substrate molecules bind. Thus, the enzyme is reaction-specific. The active sites of the enzyme act as the donors or acceptors of electrons from the substrate molecules and speed up the reaction. It is assumed that the enzymatic reaction involves a series of step-by-step elementary reactions forming complexes with substrate molecules along the way and is described by Michaelis-Menten kinetics (Equation 1.14):

The terms k1, k^1, and k2 are rate constants for, respectively, the association of substrate and enzyme, the dissociation of unaltered substrate from the enzyme, and the dissociation of product (= altered substrate) from the enzyme. The overall rate of the reaction (rP) is limited by the step ES to E + P, and this will depend on two factors—the rate of that step (i.e., k2) and the concentration of enzyme that has substrate bound, i.e., CES (Equation 1.15):

rP = k2CES

At this point, we make two assumptions. The first is the availability of a vast excess of substrate, so that CS is markedly larger than CE. Secondly, it is assumed that the system is in pseudosteady state, i.e., that the ES complex is being formed and broken down at the same rate, so the overall CES is constant. The formation of ES will depend on the rate constant k1 and the availability of enzyme and substrate, i.e., CE and CS. The breakdown of CES can occur in two ways, either the conversion of substrate to product or the nonreactive dissociation of substrate from the complex. In both instances, the CES will be significant. Thus, at steady state we can write the following (Equation 1.16):

In the following term (Equation 1.17): (k-1 + k2)/ k1= Km

### Km is named the Michaelis-Menten constant.

The total amount of enzyme in the system must be the same throughout the experiment, but it may either be free (unbound) E or in complex with substrate, CES. If we term the total enzyme CE0, this relationship is expressed as the following (Equation 1.18):

in which CE0 represents initial enzyme concentration.

Inserting Equations 1.18 and 1.17 into Equation 1.16 and rearranging the resulting equation leads to the following (Equation 1.19):

So substituting this right side into Equation 1.15 in place of CES results in the following (Equation 1.20):

The maximum rate, which we can call rmax, would be achieved when all active sites of the enzyme molecules have saturated with substrate molecules. Under conditions when CS is much greater than CE, it is reasonable to assume that all CE will be in the form CES. Therefore, CE0 = CES. We may substitute the term rmax for r and CE0 for CES in Equation 1.15. This gives us the following (Equation 1.21):

rmax = k2CE0

So, we now have the following (Equation 1.22):

This equation is commonly referred to as the Michaelis-Menten equation.

The significance of Michaelis-Menten equation is that when rp is half of rmax, from Equation 1.22, we would have the following (Equation 1.23):

The Km of the enzyme is the substrate concentration at which the reaction occurs at half of the maximum rate and is therefore an indicator of the affinity that the enzyme has for a given substrate, and hence the stability of the enzyme-substrate complex. This interpretation may be better presented by plotting rp vs. CS, which is called the Michaelis plot.

It is obvious that at low CS, it is the availability of substrate that is the limiting factor. Therefore, as more substrate is added there is a rapid increase in the initial rate of the reaction—any substrate is rapidly gobbled up and converted to product. At the Km, 50% of active sites have the substrate occupied. At higher CS, a point is reached (at least theoretically) where all sites of the enzyme have substrate occupied. Adding more substrate will not increase the rate of the reaction—hence, the leveling-out observed in the Michaelis plot.

In order to use the Michaelis-Menten equation, we need to know the values of Km and rmax. The common approach is to linearize the MichaelisMenten equation by plotting 1/rp vs. 1/ CS (named the Lineweaver-Burk linearization), which results in a slope of the linearized line, Km/rmax and an intercept on the 1/rp axis, 1/rmax. Other linearization schemes of the Michaelis-Menten equation, such as Hanes and Hofstee schemes, would accomplish the same objective as the Lineweaver-Burk linearization.

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