CHAPTER 6 - TRANSPORT AND KINETICS
D. MORE COMPLICATED ENZYMES
ONLINE STUDY GUIDE
BIOCHEMISTRY - DR. JAKUBOWSKI
Last Update: 04/11/12
http://employees.csbsju.edu/hjakubowski/classes/ch331/transkinetics/olco
mplicatedenzyme.html
Learning Goals/Objectives for Chapter 6D: After class and this reading, students will be able
to

draw Cleland chemical reaction diagrams showing enzyme, substate, and product
interactions for multisubstrate and multiproduct sequential and ping-pong
enzyme-catalyzed reactions;

draw double reciprocal 1/v vs 1/A plots at different fixed B concentrations for
sequential and ping-pong multisubstrate reactions;

draw v vs S graphs in the presence and absence of allosteric inhibitors and
activators for multi-subunit enzymes that display sigmoidal cooperative behavior
(K systems) conforming to the MWC model;

differentiate between K and V systems for allosterically regulated enzymes using
the MWC model and explain shifts in graphs of v vs S in the presence of
allosteric effectors
MULTI-SUBSTRATE ENZYMES
In reality, many enzymes have more than one substrate (A, B) and more than one product (P,
Q). For example, the enzyme alcohol dehydrogenase catalyzes the oxidation of ethanol with NAD (a
biological oxidizing agent) to form acetaldehyde and NADH. How do you do enzymes kinetics on
these more complicated systems? The answer is fairly straightforward. You keep one of the
substrates (B, for example) fixed, and vary the other substrate (A) and obtain a series of hyperbolic
plots of vo vs A at different fixed B concentrations. This would give a series of linear 1/v vs 1/A
double-reciprocal plots (Lineweaver-Burk plots) as well. The pattern of Lineweaver-Burk plots
depends on how the reactants and products interact with the enzyme.
Sequential Mechanism: In this mechanism, both substrates must bind to the enzyme before any
products are made and released. The substrates might bind to the enzyme in a random fashion (A
first then B or vice-versa) or in an ordered fashion (A first followed by B). An abbreviated
notation scheme developed by W.W. Cleland is shown below for the sequential random and
sequential ordered mechanisms. For both mechanisms, Lineweaver-Burk plots at varying A and
different fixed values of B give a series of intersecting lines. Derivative curves can be solved to
obtain appropriate kinetic constants.
Ping-Pong Mechanism: In this mechanism, one substrate binds first to the enzyme followed by
product P release. Typically, product P is a fragment of the original substrate A. The rest of the
substrate is covalently attached to the enzyme E , which is designated as E'. Now the second
reactant, B, binds and reacts with the E' and forms a covalent bond to the fragment of A still attached
to the enzyme, forming product Q. This is now released and the enzyme is restored to its initial
form, E. This represents a ping-pong mechanism
An abbreviated notation scheme is shown
below for ping-pong mechanisms. For this mechanism, Lineweaver-Burk plots at varying A and
different fixed values of B give a series of parallel lines. One example of a ping-pong enzyme is low
molecular weight protein tyrosine phosphatase. It reacts with the small substrate pinitrophenylphosphate (A) which binds to the enzyme covalently with the expulsion of the product P,
the p-nitrophenol leaving group. Water (B) then comes in and covalently attacks the enzyme,
forming an adduct with the covalently bound phosphate releasing it as inorganic phosphate. In this
particular example, however, you can't vary the water concentration and it would be impossible to
generate the parallel Lineweaver-Burk plots characteristic of ping-pong kinetics.
What are the meanings of the kinetic parameters, Km and Vm, for
multisubstrate/multiproduct mechanisms? Consider a random sequential bi-bi reaction
for a "simple" case in which the rapid equilibrium assumption defines the binding of
substrates A and B.
By inspection, there would appear to be two types of "effective" dissociation constants
for reactant A. One describes the binding of A to E (Kia) and the other the binding of A
to EB (Ka). Using mass balance for E and relationship that v0=kcat[EAB], the following
initial rate equation can be derived.
Note that Kib does not appear in the final equation. How can that be? The answer lies
in the fact that the final concentration of EAB can be derived from the path E to EA to
EAB or from the path E to EB to EAB. Assuming rapid equilibrium, KiaKb=KibKa. The
following equation can be derived from ping-pong bi-bi mechanism.
For simplicity, all of the enzyme kinetic equations have been derived assuming no
products are present.
Inhibitors in Multisubstrate Reactions:
Product Inhibition: Interpretation of kinetic experiments can be complicated by the fact
that the reactions can be reversed. Even if the catalytic conversions of the reverse
steps have too high an activation energy to actual proceed, the products, which
obviously have some structural resemblance to the reactants could inhibit the enzyme
as they could compete with reactants for binding to the enzyme. In contrast to
studying enzyme inhibition using varying concentrations of substrate at different fixed
concentrations of inhibitor, the concentration of products produced by an enzyme are
constantly increasing over the time course of the reaction. This suggests one
immediate reason that most kinetic parameters are determined by initial rate methods
in which the inhibitor-product has not yet build to a sufficient concentration to alter the
rate of conversion of substrate to product.
Product inhibition can occur in single
substrate reactions as well.
Dead End Inhibition: How do added inhibitors affect the double reciprocal plots of
multisubstrate reactions? Let's consider a special case of inhibitors call dead-end
inhibitors. These reversible inhibitors bind to a form of the enzyme and inhibits product
formation but does not participate in the reaction. It would be represented on a
Cleland diagram as a vertical line coming off the the horizontal line which represent
different enzymes forms (E, EA, EAB, E'Q, EP, EQ, etc) that lead to product
formation.
A quick inspection of Cleland diagrams lead to two simple rules that helps
in the interpretation of double reciprocal plots in the presence of different fixed and
nonsaturating concentrations of dead-end inhibitors (I) in multisubstrate reactions
(when one substrate S is varied):
1. Slope changes when:


I and S bind to the same form of the enzyme (for example E) OR
I binds to a form of E (on the horizontal line) which is connected to the form that
S binds, and I binds first.
2. Y Intercept changes when:

I and S bind to different forms of the enzyme (for example E) unless I binds first
and the binding of I and S are in rapid equilibrium.
These same rules apply for product inhibition. Consider the rapid equilibrium ordered
bibi reaction above when the concentration of the other substrate is around its Kxvalue:

Ia (an inhibitor that resembles A) and A both bind to E and EB, so the inhibition
is competitive as the slope changes but not the Y intercept

Ia and B both bind to the same enzyme form (E) so the slope should change, but
Ia also binds to EB (to which B can not bind) so the Y intercept would change,
which when combined would give either uncompetitive or noncompetitive.
ALLOSTERIC ENZYMES
Many enzymes do not demonstrate hyperbolic saturation kinetics, or typical Michaelis-Menten
kinetics. Graphs of initial velocity vs substrate demonstrate sigmoidal dependency of v on S, much
as we discussed with hemoglobin binding of dioxygen. Enzymes that display this non MichaelisMenten behavior have common characteristics. They :

are multi-subunit

bind other ligands at sites other than the active site (allosteric sites)

can be either activated or inhibited by allosteric ligands

exist in two major conformational states, R and T

often control key reactions in major pathways, which must be regulated.
Examples of these enzymes include glycogen phosphorylase (the enzyme which breaks down
intracellular glycogen reserves) and aspartate transcarbamyolase, which catalyzes the first step in
the synthesis of pyrimidine nucleotides. CTP is an allosteric inhibitor of this enzyme, which makes
physiological sense since high levels of this pyrimidine nucleotide should inhibit the first enzyme in
the synthesis of pyrimidines. ATP is a allosteric activator. This also makes sense since if high levels
of the purine nucleotide ATP are present, one would also want to balance the level of pyrimidine
nucleotides.
Figure: Aspartate transcarbamoylase: reactions
Figure: Aspartate transcarbamoylase: Non Michaelis-Menten Kinetics
Earlier we saw that cooperative binding equilibrium could be modeled with the Hill Equation, which
we introduced through the equation Y = Ln/(Kd + Ln) = Ln/(P50n + Ln) where n is the cooperativity or
Hill Coefficient. Likewise, for an enzyme which demonstrates cooperative (sigmoidal) initial rates
plots,
vo = VmSn/K0.50n + Ln)
When n=1, the equation reduces to the classical hyperbolic Michaelis Menten equation. For values
of n>1, sigmoidal plots are observed. We found a more easily understandable molecular
interpretation of cooperative binding of oxygen to hemoglobin using the MWC model (T and R
states). The MWC model has also been applied successfully to multi-subunit enzymes which
display cooperative, sigmoidal kinetics. In this model, allosteric inhibitors (which often don't
resemble the substrate) bind preferentially to the T state, leading to lower activity, while allosteric
activators bind preferentially to the R state, leading to greater activity. Activators shift the vo vs S
curve to the left while inhibitors shift it to the right (much like protons and carbon dioxide in
hemoglobin binding). These allosteric ligands induce their effects by shifting the To <=> Ro
equilibrium.
How do allosteric effectors change Vm and Km?
We have just studied how competitive, uncompetitive, and noncompetitive (or mixed) inhibitors
influence the apparent Km and Vm values for enzymes that display Michaelis-Menten kinetics. How
are Vm and Km influenced in allosteric enzymes? The example given above (ATCase), analogous
to effects observed in hemoglobin:oxgyen binding, influence the apparent Km, but not the
Vm. Remember that in the case of hemoglobin binding curves, the allosteric activators and
effectors we discussed shifted the sigmoidal binding curves to the left or right, but all reached a
plateau at the same fractional saturation value of 1. Allosteric enzyme systems that behave like this
are called K systems. Enzymes in which allosteric regulators change Vm, called V systems,
are also known. V systems display hyperbolic vo vs S curves in which activators display a
greater apparent Vm and inhibitors display a lower Vm without affecting the apparent Km. In these
systems, both the T and R forms have the same affinity for substrate (hence the same apparent
Km). This would be analogous to a situation in the MWC model where KR/KT =1 which also gave
hyperbolic, not sigmoidal Y vs  curves. This difference in V systems is that the R and T states
have different catalytic rate constants, kcat, for turnover of the bound substrate (hence different
apparent Vm values). In addition, the activator A and inhibitor I bind to the R and T forms with
different affinity, which again shifts the To <=> Ro equilibrium in the presence of the allosteric
effectors.
Cooperative binding of dioxygen to hemoglobin, regulated by allosteric effectors (protons and carbon
dioxide), was ideal for an oxygen transport system which must load and unload oxygen over a
narrow range of oxygen concentrations and allosteric effectors. Allosteric enzymes are usually
positioned at key metabolic steps which can be regulated to activate or inhibit whole pathways.
Web Links

Enzyme Kinetics - Allosteric Mechanisms (scroll to that part in the links)
Enzyme Regulation by Covalent Modification
Many enzymes are regulated not by allosteric ligands (activators and inhibitors), but by covalent
modification. Often the covalent modification involves phosphorylation (by enzymes called kinases
which transfer a phosphate from ATP to a Ser, Thr, or Tyr on the target enzyme) or
phosphatases (which remove the phosphates from phospho-Ser, Thr, or Tyr in the target
protein). In fact 1-2% of all genes in the human genome code for kinases and phosphatases.
Integration of Binding, Facilitated Diffusion and Kinetics
By now you should realize the similarities among the chemical and mathematical equations used to
describe binding, facilitated diffusion, and enzyme kinetics of macromolecule interactions with
ligands and substrates. In simple, noncooperative systems, it all starts with a macromolecule,
usually a protein, binding a ligand, which can be a small molecule or another macromolecule.
M + L <=> ML
In some cases, the only function of M is to bind and release ligand (as in the case of myoglobin). In
other cases, the ligand is transformed. In the simplest case involving transformation, the ligand is
moved across a membrane down a concentration gradient (facilitated diffusion), a purely physical
step.
R + Aout <=> RA --> R + Ain
In a yet more complicated case, the ligand can be transformed chemically (enzyme
kinetics) into product.
E + S <==> ES --> E + P
In this chapter we added interactions (multiple substrates, inhibitors) and saw that the
basic form of the binding, facilitated diffusion, and enzyme kinetic equations and graphs
for noncooperative system were extremely similar since the biological function always
depended in some fashion on the concentration of the macromolecule complex ( ML,
RA, ES, EI, ESI etc). The PowerPoint below reviews the similarity in the results of the
mathematical analyses and resulting graphs showing the concentration dependencies of
complex formation, facilitated diffusion, and enzyme-catalyzed chemical reactions.

PowerPoint: Summary of Graphical Descriptions of Noncooperative
Binding, Facilitated Diffusion, and Enzyme Kinetics
A feel for the dissociation constant
In all of the curves shown in the above PowerPoint, a ligand concentration can be found
at which the biological effect is half maximum (either activation or inhibition of a
biological function). Remember, only in certain conditions is that number equal to the
dissociation constant for ligand. This is clearly the case when the only interaction is a
1:1 binding of a macromolecule and ligand. If an inhibitor was unknowingly present
during the direct or indirect measurement of a binding reaction, the ligand
concentration at half-maximal binding would equal the apparent Kd, not the actual
dissociation constant. In the case of facilitated diffusion and enzyme-catalyzed
chemical transformation of a single substrate, the ligand concentration at half-maximal
binding is equal to the dissociation constant only in the rapid equilibrium assumption
holds. It clearly doesn't in the steady state assumption, and clearly not in more
complicated systems. Consider the case when a ligand binds a neurotransmitter
receptor and alters intracellular the calcium ion concentration through a complicated
signal transduction system. If the step leading to the releases of stored intracellular
calcium is several steps removed from the actual binding of ligand to a
neurotransmitter receptor, the likelihood that the ligand concentration at half-maximal
increase in intracellular calcium is equal to the Kd for ligand binding is small. If
however, the actual ligand dissociation constant for the receptor can be determined
(using radiolabeled ligand, for example), and it is equal to the ligand concentration for
half-maximal calcium increase, then it might be argued the binding is "rate or effect
determining". When inhibitors of biological function are used, IC50 values (the inhibitor
concentration at which the response is reduced to 50%) are usually reported.
From the Simple to the Complex
Enzymes don't work in isolation. Most are part of complex synthetic/degradative or
signal transduction systems in which their biological response (product formation or
degradation, pathway regulation, etc) are functions of ligand concentration and are
subject to signals further upstream or downstream in their resident pathways. How are
the dose responses for biological activity modulated by other parts of the pathway?
Although the equations we've derived don't directly predict the responses in lieu of
knowing the concentrations, rate constants, and dissociation constants for all the steps
in the pathway, an approach based on simple assumptions might illuminate the
possibilities. A recent paper by Bashor et al and an analysis of it by Pryciak describe
the input and output responses of signaling enzymes attached "in series" to a scaffold
protein and how different dose responses outputs can be engineered to meet the
requirements of the pathway.
Examples of different dose-response curves from
Bashor analysis are shown below. We've actually seen many of these response curves
for individual protein/enzymes or chemical reaction systems. Curve A represents a
typical response curve which could be produced by a process governed by a simple M +
L <=> ML equilibrium. Curves B and C are sigmoidal and could be produced by a
process governed by a multisubunit enzyme following the MWC model with different
values for the parameters L and c. Curve D is similar to the output of consecutive
irreversible reactions such as A --> B --> C, where the response would be similar to the
rise and fall B with time (not concentration, however). One could envision a
dose/response, however, that could produce a rise/fall as in curve D.
would reflect the rise of C in the same chemical reaction.
Likewise Curve E
To a first approximation,
one could image that the responses shown below are produced by steps that are ratelimiting in the overall pathway such that the overall response is governed by those
steps.
Obviously a complex mathematical systems analysis involving the solving of
matrices of differential equation for each step would be required for a more realistic
understanding of actual biological responses, but I hope you can see that an
understanding of real biological networks must start with a kinetic and thermodynamic
understanding of the "simple" steps that constitute larger system pathways.
Navigation

Biochemistry Online: An Approach Based on Chemical Logic - Table of
Contents

Biochemistry BCHM 321: Home Page
References
1. Pryciak, P. Customized Signaling Circuits. Science 319, pg 1489 (2008)
2. Bashor, C. Science 319, pg 1539 (2008)