Derivation of simple integrated rate equations

Introduction

     As with many physicochemical principles, the rules governing rate of reaction were first established empirically, and later subject to extensive theoretical analysis, leading to our present understanding of the controlling factors. The empirical observations centered around establishment of reaction conditions under which the rate could be measured as a function of concentration of reactants. It was found that the rate was related to concentration in a predictable way, leading to the concept of the rate constant, and equations for simple reactions that fell into several classes:

1. 1st-order reactions
• Reactions in which the rate varies with concentration of a single species, and the change in concentration is exponential, so that a plot of ln(concentration) v. time is linear. The stoichiometric coefficient is 1. Radioactive decay is an example usually cited, but many electron transfer processes, and most enzyme mechanisms contain intermediate reactions that are first order.
2. 2nd-order reactions of Class I
• Reactions in which the rate varies with concentration of a single species, but the stoichiometric coefficient is 2. The rate varies with the reciprocal of the concentration, so that a plot of 1/(concentration) v. time is linear.
3. 2nd-order reactions of Class II
• Reactions in which the rate varies with the concentration of two substrates, each of which has a stoichiometric coefficient of 1. A plot of ln(conc.A/conc.B) v. time is linear.
4. Higher-order reactions
• Reactions in which more than two species are involved, or one species reacts with stoichiometric coefficient >2. Many different classes, of which the simplest involves a single species with stoichiometric coefficient greater than 2. Although many biochemical reactions might appear to be higher-order, in general, the enzyme mechanism functions through binding of multiple substrates to a single protein, and the empirical order is simpler, and often appears to be zero-order (see below).
5. Zero-order reactions
   • Many reactions in biochemistry appear to occur at a rate independent of substrate concentration. These are said to show zero-order behavior. In general, this is because the reaction is enzyme-catalyzed, and the rate is determined by the concentration of enzyme, as long as the substrate is in excess, so that the rate is saturated. Enzyme mechanisms will be discussed later in the course.

     Before deriving of the equations describing these processes, a brief discussion of order of reaction is necessary.

Reaction equation, stoichiometric coefficients, and order of reaction

     Reactions are described by reaction equations of the sort already discussed at length in earlier lectures. The general properties of the reaction equation are:

• The chemical or electrochemical species involved, identified by an appropriate symbol (usually the chemical formula, but often in biochemical discussions, a name, conventional abbreviation, or some other symbol), are divided into reactants and products, separated by an operator, usually shown as the equilibrium arrow (), indicating that the reaction is reversible. Since in principle all reactions are microscopically reversible, this is generally appropriate, even when the equilibrium constant is very large.
• By convention, the reaction is usually written in the direction appropriate to the context. Most often this is the direction favored by the equilibrium constant (K_eq > 1), but not always. For example, when considering a series of reaction such as those of glycolysis, where some of the reactions have positive DG^o' values, it is appropriate to write the reaction in the direction with an unfavorable K_eq. In general, reactants decrease, and products increase in concentration during the course of the reaction, which proceeds from left to right.
• The stoichiometry of the reaction is indicated by placing stoichiometric coefficients before the chemical species. These are usually integer numbers, but in some bioenergetic processes non-interger stoichiometries are known to pertain.
• The species included in the reaction equation are those that undergo changes in concentration as a result of the reaction. Other components may be involved, for example, catalysts, the solvent, etc., but if their concentration does not change, they are not included in the reaction equation (though their involvement might be indicated in some way).

For example:
glucose + ATP  glucose-6-P + ADP
glucose + 2ADP + 2 Pi  2lactate + 2ATP In general:
aA + bB  pP
     The stoichiometric coefficients are important in the context of this kinetic discussion, because they determine the order of reaction. This term describes how the rate of the reaction depends on the concentrations of the species involved. In general, the rate of a reaction, v, is described by an equation such as the following:
v = k[A]^a[B]^b[P]^p
where k is the rate constant, A and B are reactants, and P is the product, with stoichiometric coefficients a, b, p, respectively. Then the overall order of reaction is given by the sum of the stoichiometric coefficients:

order of reaction = a + b + p + .....n where n indicates the possible involvement of other species not included in the general mechanism above.
     However, use of the term "overall order of reaction" is a little muddled, because it is generally recognized as an empirical term, so that the value found by experiment might not correspond to that expected by applying the above general rule. Most frequently, for simple chemical processes, the order of reaction found experimentally turns out to be equal to the sum of the stoichiometric coefficients of the reactants. This is generally the case for reactions with a large K_eq, starting from the condition of [P] = 0, since [P] will then be negligible during the time of measurement, and the reverse reaction will not be significant. Then, for the general reaction above:
v = k[A]^a[B]^b
If a and b are both 1, then the overall order of reaction will be 2nd-order (1 + 1 = 2).
For most biochemical processes, enzyme catalysis and the saturation effects resulting from this, determine that the steady-state reaction does not obey the simple rules. However, if the pre-steady-state kinetics are measured, in which the enzyme is considered as a reactant, then the simple rate laws pertain. We will examine this case separately in a later lecture.
     While the overall order of reaction is described as above, a second term is also often used, - the order of reaction with respect to a particular species. For example, in a reaction involving 2A, the reaction is said to be 2^nd.-order in A. The order is given simply by the stoichiometric ratio. From this it can be seen that measurement of the order of reaction can provide a value for the coefficient if this is otherwise unknown. A useful protocol for determining the order of reaction with respect to a particular component is to measure the concentration dependence of rate when all other reactants are in great excess. Under these circumstances, their concentrations will not vary significantly during the reaction, and the rate law revealed by experiment will give the order of reaction with respect to the tested component:
v = k[A]^a_excess[B]^bv = k'[B]^b
k' = k[A]^a_excess

Relation between rate constants and the equilibrium constant

An important relation between the forward and reverse rate constants for a process, the equilibrium constant, and DG^o' is covered elsewhere.

Derivation of the rate equations

• 1st-order processes

  • Later in the course, we will deal with the reactions of photosynthesis, in which absorption of light in a photochemical reaction center leads to separation of charge, and stabilization through electron transfer. Much of our current knowledge of electron transfer reaction mechanism comes from study of these processes, because the rates can be measured by ultra-fast spectrophotmetric methods (ps or fs spectrophotometry) after initiation with a short (<1 ps) flash from a laser. Because the protein structures are known at high resolution, the reaction kinetics can be studied in this context, and used to test theories of electron transfer. Because the rapid electron transfer reactions are intramolecular, they provide examples of 1st-order processes. A reaction that can be measured with relatively simple apparatus is the back-reaction by which the photochemical reaction relaxes to the dark state if forward electron transfer is prevented: PA  P^*A  P⁺A^-  PA
    Here, P is the primary electron donor to the photochemical reaction, and A is the first stable acceptor. P^* is the excited state formed on absortion of a photon, and P⁺A^- the stable product of the photochemical reaction. The last step represents the back-reaction. Because the formation of the P⁺A^- state takes < 1 ns, and the back-reaction has a half-time of ~100 ms, the kinetics of the back-reaction are well separated from the forward reaction, and can be readily measured.
    P⁺A^-  PArate of reaction = v = -d[P⁺A^-]/dt = k₁[P⁺A^-] Another case of general interest is found in all enzyme catalyzed reactions. In simple Michaelis-Menton kinetics, the reaction procedes through two steps, - formation of the enzyme-substrate (ES-) complex, and followed by the breakdown of the ES-complex to products:
    E + S  ES  E + P The formation of the ES-complex is a second-order process, while the breakdown to products is 1st-order.For the general case , we will consider a 1st-order reaction:
    
    A  B
    rate of reaction = v = -d[A]/dt = d[B]/dt = k₁[A] where k₁ is the 1st-order rate constant for the forward reaction, [A] is the reactant concentration, and [B] is the product concentration. The rate of the reaction (or its velocity v) is given either by the rate of disappearance of [A] or appearance of [B]. It is convenient to keep the terms to a minimum, so we use disappearance of [A] in our treatment. We re-arrange this equation to bring terms with [A] to one side, and t to the other: Then, using the standard integral, we integrate both sides:
    In order to find a value for the constant C, we note that the equation must apply for all concentrations, including that at t=0. If we denote the concentration of A at zero time as [A]_o, and substitute into the last equation above, we get ln[A]_o = C There are several useful forms of the equation we get by back-substitution, - a general logathmic form representing the progress of the reaction from zero time, or a more versatile form, in which [A]₁ and [A]₂ are concentrations at times t₁ and t₂ during the reaction. An equivalent expression for the first of these is the exponential form of the general equation.Finally, a useful term allowing comparison of rates for different processes is the the half-time for the process, t_½, - the time for half completion of the reaction. A similar term is also commonly used in the context of radioactive decay, to specify the half-life of a radioactive species. We can derive an equation for t_½ by substitution into the above, as follows:
    ln ½ = -k₁t_½t_½ = 0.6931 / k₁ Note that the concentration of [A] does not appear in this equation, - the half-time of the reaction is independent of the starting concentration.
    

• 2nd-order processes

  • Class IWhile reactions involving a single species with a stoichiometric coefficient of 2 are quite common in chemistry, they are relatively rare in biochemistry. An interesting example from earlier in the course is the mechanism of action of gramicidin. When this ionophore forms a pore across a biological membrane, two monomers (M) interact in the membrane to form the active dimeric form (D). The frequency of formation of active dimers can be assayed by measuring the current across a black lipid membrane, - the current jump when a single channel forms can be readily detected. It is found that the probability of formation of channels is proportional to [M]²
    
    M + M  D
    rate of reaction = frequency of formation of channels = v = -d[M] / dt = k₂[M]² Derivation of the formal equations describing this sort of reaction follows a similar approach to that for 1st-order reactions. We describe a general reaction:
    2A  P
    rate of reaction = v = -d[A]/dt = d[P]/dt = k₂[A]² We chose a set of parameters to minimize terms:
    We re-arrange to group like terms We integrate both sides of the equation We find an appropriate term for the constant of integration, C We solve for t_½ Note that the integrated rate equation shows that a plot of 1 / [A] against time will give a straight line for a 2nd-order, Class I reaction, with an intercept at 1 / [A]₀. Note also that a concentration term for [A] appears in the equation for t_½, so the half-time depends on initial concentration.
    
  • Class IIAs noted above, for most enzyme catalyzed reactions, the formation of the enzyme-substrate complex (ES-complex) involves a collisional reaction between substrate and enzyme. This can be considered as a 2nd-order reaction of Class II, in which the substrate and enzyme are reactants, and the ES-complex is the product.
    E + S  ES
    For most enzyme reactions, the substrate is at a concentration greatly in excess of the enzyme, and the reaction follows pseudo-first-order kinetics. An interesting case is that of ubihydroquinone (quinol, or QH₂) oxidation by the bc₁ complex, studied in a photosynthetic bacterium, Rb. sphaeroides. In this case, the substrate (QH₂) can be generated in the membrane by flash-activation of the photochemical reaction center at a concentration similar to that of the enzyme. Photochemical activation also generates the oxidizing substrate, cyt c₂⁺, so the reaction can be started up rapidly. At this low concentration of the QH₂ substrate, the rate of reaction is limited by the rate of formation of the ES-complex, so that the reaction above can be measured by watching the appearance of the product, monitored through reduction of heme b_H in the presence of antimycin (see the bc₁ complex pages for a discussion of mechanism).Again, derivation of the formal equations describing this sort of reaction follows a similar approach to that for 2nd-order reactions of Class I. However, because the reactants can have initial concentrations that are different, the formalism differs from that for Class I reactions. As noted below, in the special case that the initial concentrations of the two reactants is the same, this formalism fails, but in that case the equations derived for Class I reactions can be applied.
    First, we describe a general reaction:
    A + B  P
    rate of reaction = v = -d[A]/dt = d[P]/dt = k₂[A][B] We would like to chose a set of parameters to minimize terms, but are now stuck with the participation of the second reactant, and therefore have to include an appropriate concentration term.
    v = k₂[A][B] We can simplify the treatment somewhat by recognizing that, as the reaction proceeds, the loss of reactants (and the increase in product) will be stoichiometrically linked. Setting the loss of reactants (or appearance of product) = x, we get We re-arrange to group like terms The integration of this equation is not trivial, but we can look it up in integration tables, and find a solution. On substitution back for x, we get: Note that the integrated rate equation shows that a plot of ln [A]/[B] against time will give a straight line for a 2nd-order, Class II reaction. Note also that the treatment fails if the initial concentrations of the two substrates are the same, - the logarithmic term becomes zero. In this case, the reaction can be treated by the same formalism as for Class I reactions, or alternatively, the initial concentrations can be handle if the values are very slightly different.