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Cover
Title Page
Preface
Acknowledgments
Part I: Ligand Binding to Single Binding Site Targets
1. 1 Theory of Ligand Binding to Monomeric Proteins
  1. 1.1 Importance of Ligand‐Binding Phenomena in Biology
  2. 1.2 Preliminary Requirements for Ligand‐Binding Study
  3. 1.3 Chemical Equilibrium and the Law of Mass Action
  4. 1.4 The Hyperbolic and Sigmoidal Representations of the Ligand‐Binding Isotherms
  5. 1.5 The Important Concept of X_1/2
  6. 1.6 Other Representations of the Ligand‐Binding Isotherm
  7. 1.7 Effect of Temperature: Thermodynamic Relationships
  8. 1.8 Replacement Reactions: Competitive Ligands
  9. 1.9 Heterotropic Linkage: Non‐Competitive Binding of Two Ligands
  10. 1.10 Allostery and Allosteric Phenomena in Monomeric Proteins
  11. 1.11 The Special Case of Cys Ligands (and Similar Reactions)
  12. 1.12 Other Special Cases
  13. References
2. 2 Ligand‐Binding Kinetics for Single‐Site Proteins
  1. 2.1 Basic Concepts of Chemical Kinetics: Irreversible Reactions
  2. 2.2 Reversible Reactions: Equilibrium and Kinetics
  3. 2.3 More Complex Kinetic Mechanisms
  4. 2.4 Reactions with Molecularity Higher Than Two
  5. 2.5 Classical Methods for the Study of Ligand‐Binding Kinetics
  6. 2.6 Photochemical Kinetic Methods
  7. 2.7 The Kinetics of Replacement Reactions
  8. References
  9. Appendix to Chapter 2 Appendix to Chapter 2: Principles of Data Analysis
3. 3 Practical Considerations and Commonly Encountered Problems
  1. 3.1 Design of the Experiment: The Free Ligand Concentration
  2. 3.2 The Signal and the Concentration of the Target
  3. 3.3 Test of the Reversibility of the Reaction
  4. 3.4 Frequent Abuses of the Concept of X_1/2
  5. 3.5 Two Common Problems: Protein Precipitation and Baseline Shifts
  6. 3.6 Low‐Affinity Ligands
  7. 3.7 High‐Affinity Ligands
  8. 3.8 Determination of Binding Stoichiometry
  9. 3.9 Ligands Occupying a Thermodynamic Phase Different from the Protein
  10. 3.10 Mixtures of Isoforms
  11. 3.11 Poor or Absent Signal
  12. References
Part II: Ligand Binding to Multiple Binding Site Proteins
1. 4 Proteins with Multiple Binding Sites
  1. 4.1 Multiple Binding Sites: Determination of the Binding Stoichiometry
  2. 4.2 The Binding Polynomial of a Homooligomeric Protein Made Up of Identical Subunits
  3. 4.3 Intramolecular Heterogeneity
  4. 4.4 Oligomeric Proteins with Interacting Binding Events: Homotropic Linkage
  5. 4.5 Cooperativity: Biochemistry and Physiology
  6. 4.6 Allostery and Symmetry: The Allosteric Model of Cooperativity
  7. 4.7 Two Alternative Concepts of Cooperativity
  8. 4.8 Ligand Replacement in Oligomeric Proteins
  9. 4.9 Heterotropic Linkage in Multimeric Proteins
  10. 4.10 Heterotropic Linkage and the Allosteric Model
  11. References
  12. Appendix 4.1 Statistical Distribution of the Ligand Among the Binding Sites: Statistical Factors
  13. Appendix 4.2 Symmetry of the X&c.macr; Versus Log([X]) Plot: The Concept of Xm
2. 5 Ligand‐Linked Association and Dissociation
  1. 5.1 Quaternary Constraint and Quaternary Enhancement
  2. 5.2 The Reversibly Dissociating Homodimer Devoid of Ligand‐Linked Association Equilibria
  3. 5.3 Ligand‐Linked Association‐Dissociation in the Non‐Cooperative Homodimer
  4. 5.4 Oligomers That Dissociate Into Monomers Upon Ligand Binding
  5. 5.5 Monomers That Self‐Associate to Homodimers Upon Ligation
  6. 5.6 Ligand‐Linked Association‐Dissociation in Cooperative Proteins
  7. 5.7 One Ligand Per Dimer: Ligand‐Binding Sites at Intersubunit Interfaces
  8. 5.8 Ligand‐Linked Association‐Dissociation in the Framework of the Allosteric Model
  9. 5.9 Practical Considerations
  10. References
3. 6 Kinetics of Ligand Binding to Proteins with Multiple Binding Sites
  1. 6.1 Stepwise Ligand Binding to Homooligomeric Proteins
  2. 6.2 Ligand Association to Heterooligomeric Proteins
  3. 6.3 Study of the Time Course of Ligand Dissociation
  4. 6.4 Practical Problems in the Study of Ligand‐Binding Kinetics with Oligomeric Proteins
  5. 6.5 Advanced Techniques for the Study of Ligation Intermediates
  6. 6.6 Integration of Equilibrium and Kinetic Data for Cooperative Systems
  7. 6.7 Ligand‐Binding Kinetics in the Framework of the Allosteric Model
  8. References
  9. Appendix 6.1 Kinetic Statistical Factors
4. 7 Hemoglobin and its Ligands
  1. 7.1 The Heme and Its Ligands
  2. 7.2 Reversible Ligand Binding and Cooperativity
  3. 7.3 The Structure of Hemoglobin
  4. 7.4 Ligation‐Dependent Structural Changes
  5. 7.5 Quaternary Constraint
  6. 7.6 Structural Aspects of Cooperativity: Allostery
  7. 7.7 Structure and Energy Degeneracy
  8. 7.8 Kinetics of Ligand Binding
  9. 7.9 Ligation Intermediates: Measurement and Structure
  10. 7.10 Ligand‐Linked Dissociation Into Dimers
  11. 7.11 Non‐Human Hemoglobins and Human Hemoglobin Mutants
  12. References
Part III: Enzymes
1. 8 Single‐Substrate Enzymes and their Inhibitors
  1. 8.1 Enzymes, Substrates, and Inhibitors: A Special Case of Ligand Binding
  2. 8.2 Importance of Initial Velocity Studies: Zero Order Kinetics
  3. 8.3 Linearizations of the Michaelis‐Menten Hyperbola
  4. 8.4 Enzymatic Catalysis of Reversible Reactions
  5. 8.5 The Study of Enzyme Inhibitors Under the Pseudo‐Equilibrium Approximation
  6. 8.6 Inhibitors that Bind to the Same Site as the Substrate (Pure Competitive Inhibitors)
  7. 8.7 Different Types of Heterotropic (Non‐Competitive) Inhibitors
  8. 8.8 Heterotropic Regulation of Enzyme Activity
  9. References
2. 9 Two‐Substrate Enzymes and their Inhibitors
  1. 9.1 Two Basic Catalytic Mechanisms for Two‐Substrate Enzymes
  2. 9.2 Steady‐State Parameters of Two‐Substrate Enzymes that Do Not Form a Ternary Complex
  3. 9.3 Competitive Inhibitors of Two‐Substrate Enzymes That Do Not Form a Ternary Complex
  4. 9.4 Steady‐State Parameters of Two‐Substrate Enzymes Forming a Ternary Complex
  5. 9.5 Competitive Inhibitors of Two‐Substrate Enzymes Forming a Ternary Complex
  6. References
3. 10 Beyond the Steady State
  1. 10.1 Structural and Catalytic Properties of Copper‐Containing Amine Oxidases
  2. 10.2 Experimentally Accessible Information on Copper‐Amine Oxidases
  3. 10.3 From Kinetic Constants to Steady‐State Parameters
  4. 10.4 The Method of King‐Altman to Derive Steady‐State Parameters
  5. References
4. 11 Slowly Binding and Irreversible Enzyme Inhibitors
  1. 11.1 Definitions and Classifications
  2. 11.2 Test of Reversibility of Binding
  3. 11.3 Slowly Equilibrating Competitive Inhibitors
  4. 11.4 Rapidly Binding Irreversible Inhibitors
  5. 11.5 Slowly Binding Irreversible Inhibitors
  6. 11.6 Mechanism‐Based Inhibitors
  7. References
Index
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List of Tables

Chapter 01
1. Table 1.1 Relationships between X_1/2 and K_d in monomeric single‐site proteins.
Chapter 03
1. Table 3.1 Common possible causes of broadened binding isotherms.
Chapter 04
1. Table 4.1 Possible causes of cooperativity.
Chapter 07
1. Table 7.1 Partition constants for gaseous ligands of ferrous human Hb.
2. Table 7.2 Carbon dioxide and oxygen content of human blood.
3. Table 7.3 Oxygen affinity of ferrous human hemoglobin. Experimental conditions as in Figure 7.2 (from Imai, 1982). O₂ affinities are given as P_1/2, the O₂ partial pressure required to half‐saturate Hb under the given experimental condition. As a term of comparison, one may consider that PO₂ is ∼ 150 mmHg in air, ∼100 mmHg in the arterial blood, and ∼ 40 mmHg in mixed venous blood (for a resting subject).
4. Table 7.4 Interatomic distances as markers of the ligand‐dependent structural changes in hemoglobin.
5. Table 7.5 Ionic interactions of the C‐terminal residues in unliganded Hb.
6. Table 7.6 Overall kinetic rate constants for the gaseous ligands of ferrous Hb and Mb.
7. Table 7.7 Kinetic processes observed in carbonmonoxy‐hemoglobin after a 10 ns laser pulse capable of inducing full photochemical breakdown of the complex (from Hofrichter et al., 1983).
8. Table 7.8 Cooperative free energies for the 10 ligation intermediates of tetrameric Hb, determined using three different mimics of the liganded heme.
9. Table 7.9 A selection of well‐characterized naturally occurring or site‐directed point mutations at the α1β2 interface (from Bellelli et al., 2006).
Chapter 08
1. Table 8.1 Different types of heterotropic inhibition.
Chapter 10
1. Table 10.1 Kinetic parameters of bovine serum amine oxidase.
2. Table 10.2 Steady‐state parameters of bovine serum amine oxidase.
Chapter 11
1. Table 11.1 Some important definitions.

List of Illustrations

Chapter 01
1. Figure 1.1 Two common representations of ligand‐binding isotherms. Panel A: the versus [X] plot; Panel B: same as A but [X] is logarithmically scaled. In both panels the points are calculated from eqn. 1.6. The two lines above and below the points on each panel are obtained by adding and subtracting a constant error of 0.01 units, and demonstrate that in these representations there is no distortion of the error.
2. Figure 1.2 Plot of increments versus [X] (on a log scale). Same data set as in Figure 1.1; because the errors added to calculated values are constant and independent of , in this plot the points and the lines exactly overlap. Notice that the values on the ordinate axis depend on the dimension of the intervals of log [X] explored and that to obtain a readable plot, these should be kept constant.
3. Figure 1.3 The Hill (A) and Scatchard (B) plots.
4. Figure 1.4 van't Hoff plot of the reaction of sperm whale myoglobin with oxygen. The ordinate reports the logarithm of the oxygen partial pressure required to half saturate the protein (P_1/2) that for a gaseous ligand is equivalent to log X_1/2.
5. Figure 1.5 A typical ligand replacement experiment. The dashed line represents the binding isotherm of ligand X in the absence of competing ligand Y; the three lines on the right represent the replacement of ligand Y by ligand X at constant fixed concentrations of ligand Y (from left to right: [Y]=10 K_d,Y, 25 K_d,Y, 50 K_d,Y). Notice the right shift of each curve, that is, the decrease of the apparent affinity for ligand X, as a response to successive increases of the concentration of ligand Y, and their shape invariance, anticipated in Section 1.4.
6. Figure 1.6 Reaction scheme for a protein binding two ligands at different binding sites. Notice that the ternary complex PXY is possible in this case, whereas it was impossible in the case of identical linkage.
7. Figure 1.7 A comparison of identical and heterotropic linkages.
8. Figure 1.8 Measurement of the affinity of human hemoglobin for glycerate 2,3 bis‐phosphate from its effect on the affinity for oxygen.
9. Figure 1.9 Reaction scheme of an allosteric monomeric protein binding two ligands. Under the limiting condition where each ligand has significant affinity for only one of the different conformational isomers of the protein, some intermediates may be neglected.
Chapter 02
1. Figure 2.1 Ligand‐binding time courses for the second order (continuous line) and pseudo‐first order (dashed line) conditions adjusted to have approximately the same t_1/2. (≈0.6 s) Time courses were calculated using eqns. 2.9 (k=800 M⁻¹s⁻¹, [P]₀ = negligible, [X]₀=1.5 mM; dashed line, marked as #1) and 2.8 (k=1000 M⁻¹s⁻¹, [P]₀ = 1 mM, [X]₀=1.5 mM; continuous line, marked as #2). Compared to the exponential, pseudo‐first order formation of the complex (dashed line), the second‐order reaction time course (continuous line) decelerates, reflecting the effect of decreasing ligand concentration. Inset: difference between time courses #2 and #1.
2. Figure 2.2 Panel A: time courses of reversible ligand association simulated using eqn. 2.11. Panel B plot of the end points of the time courses reported in Panel A as a function of ligand concentration. Data used in the simulation are as follows: k_a = 10⁵ M⁻¹s⁻¹; k_d = 1 s⁻¹; ligand concentrations 1, 3, 10, 30, 100 and 300 μM. Pseudo‐first order was assumed throughout (i.e., [P]<<0.1 μM). The equilibrium dissociation constant equals X_1/2 and corresponds to k_d/k_a = 10⁻⁵ M (Panel B). Notice that each time course is exponential (i.e., pseudo‐first order), but they become faster and faster as the ligand concentration increases. As a consequence, the time required to approach equilibrium is longer at low ligand concentrations.
3. Figure 2.3 Plot of the apparent rate constants from Figure 2.2A as a function of ligand concentration. The rate constants should be obtained from global fitting of experiments like that simulated in Figure 2.2A, rather than by fitting each time course to an exponential as the former procedure is statistically sounder. The plot is a straight line whose slope corresponds to k_a and whose intercept corresponds to k_d.
4. Figure 2.4 Simulated time courses for the reaction scheme . Conditions, all panels: total protein concentration 10 uM, k₁=10⁶ M⁻¹s⁻¹; k₂=1000 s⁻¹; k₃=20 s⁻¹; k₄=1 s⁻¹; K_d,1=10⁻³ M; K_d,2=0.05. Circles represent the concentration of unliganded protein (P); squares that of the intermediate complex P−X; triangles that of the end complex PX. Total ligand concentrations: 30 μM (Panel A); 100 μM (Panel B); 300 μM (Panel C); and 1 mM (Panel D).
5. Figure 2.5 Time courses of the approach of equilibrium (Panel A) and equilibrium ligand‐binding isotherm (Panel B) of the system simulated in Figure 2.4. Panel A: the time courses of formation of species PX at [X]=0.03, 0.1, 0.3, 1, and 10 mM were fitted to single exponentials; the apparent rate constants obtained were k=1.6, 2.8, 5.5, 11, and 19 s⁻¹. Only the initial part of the simulated time courses is represented in the figure in order to expand the differences. Panel B: the simulated end points of the time courses from Panel A (at 100 s, not shown in the panel) are plotted as a function of the free ligand concentration at the end of the time course. The line is calculated for a single site binding with K_d,app = K_d,1K_d,2 = 5x10⁻⁵ M and asymptote 9.5 μM. Inset: kinetic constants from panel A as a function of ligand concentration. In the absence of the intermediate this plot should be a straight line with slope k_a. and intercept k_d. (see Section 2.2 and Figure 2.3); in the present case the order of the reaction decreases from 2 to 1 as [X] increases.
6. Figure 2.6 The original stopped‐flow apparatus designed by Q.H. Gibson (Gibson and Milnes, 1964).
7. Figure 2.7 Rate of replacement of O₂ by CO in myoglobin (redrawn after Antonini and Brunori, 1971). In this series of experiments MbO₂ (equilibrated with several concentrations of O₂) was rapidly mixed with a solution of CO (at 0.5 mM constant concentration); the concentrations of both gases were largely in excess with respect of that of the protein, thus they may be assumed as constant throughout the reaction time course. As expected, the limiting value for the kinetic constant of the ligand replacement reaction extrapolated to zero concentration of O₂ equals the rate constant of O₂ dissociation. Data points are from Antonini and Brunori (1971); line was calculated from eqn. 2.19, using the following rate constants (also from Antonini and Brunori, 1971): k_a,O2 = 10⁴ atm⁻¹s⁻¹; k_d,O2 = 4 s⁻¹; k_a,CO = 3x10⁵ M⁻¹s⁻¹; k_d,CO = 0.04 s⁻¹.
Chapter 03
1. Figure 3.1 Binding isotherms of ferrous human hemoglobin and CO, a high‐affinity ligand. Panel A: at high hemoglobin concentration (6.4 to 6.9 uM on a heme basis, at pH=7 (open circles) and 9.2 (closed squares)) the binding isotherm is a stoichiometric titration and [CO]_free << [CO]_tot. The linear relationship between absorbance and binding is proven by the signal being a straight line, that breaks at the equivalence point [CO]_tot = [heme]_tot. Panel B: at low hemoglobin concentration (45 nM (closed squares) and 234 nM (open circles), both at pH=7) [CO]_free is large enough to be calculated with precision from , and a hyperbolic equilibrium curve is obtained. The concentration‐dependent difference in affinity is a consequence of ligand linked dissociation (see Chapter 5).
2. Figure 3.2 Panel A: Simulated time courses of irreversible ligand binding for a protein having infinite affinity and k_a = 10⁵ M⁻¹min⁻¹; ligand concentrations 0.1, 0.3, 1, 3 10, 30 μM. All time courses tend asymptotically to 1, with different rates. Panel B: Fractional saturation measured after 10 min. incubation (the points correspond to the full circles of Panel A). The true X_1/2 in this case is zero; however, given that the incubation time is insufficient to allow the reaction mixture to reach its equilibrium condition, a curve resembling equilibrium binding with non‐zero apparent X_1/2 is obtained (see text).
3. Figure 3.3 Comparison of the binding isotherms of two proteins that bind the same ligand with 100‐fold different affinity.
4. Figure 3.4 Effect of protein concentration on the shape of the ligand binding isotherms. Panel A: fractional saturation () as a function of total ligand concentration, for five different protein concentrations: [P]_tot = 0.1 K_d, 0.3 K_d, K_d, 3 K_d, and 10 K_d. Circles report the calculated values of for the highest protein concentration (10 K_d) plotted against [X] instead of [X]_tot; obviously in a simulation the experimental error is zero, and this allows us to demonstrate that increasing the protein concentration above K_d has no real effect on ligand affinity. Panel B: [X] versus [X]_tot as a function of [P]_tot. Note that when [P]_tot>K_d the free ligand concentration ([X] or [X]_free) is a small fraction of [X]_tot, hence its calculation is subjected to large errors (Section 3.2).
5. Figure 3.5 A tonometer sealed to a spectrophotometric cuvette, for the determination of the binding isotherms of gaseous ligands.
6. Figure 3.6 Reduced slope of the versus log([X]) plot for a mixture of isoforms (continuous line), as compared to a pure protein (dashed line). The binding curve for the two isoforms was simulated under the assumptions that they are present in equal amounts and that their K_ds are 0.3 and three times that of the simulated pure monomeric protein (dashed line).
Chapter 04
1. Figure 4.1 Fractional saturation of a non‐cooperative, non‐dissociating homodimeric protein.
2. Figure 4.2 Ligand‐binding isotherm of a non‐cooperative heterodimer with non‐equivalent binding sites. The continuous line P represents the fractional saturation of the heterodimer; M_a and M_b represent the corresponding curves for each of the constituent monomers (multiplied by 0.5 to take into account that the concentration of each monomer is half that of the heterodimer). The dashed line represents the fractional saturation of a non‐cooperative homodimer (eqn. 4.2) with the same X_1/2 for comparison; notice that it is steeper than that for the heterodimer P. In this simulation and .
3. Figure 4.3 Population of ligation intermediates in a heterodimer, as calculated from eqn. 4.14, using the same parameters as in Figure 4.2. Notice that one of the two liganded intermediates, M_aM_bX, is poorly populated, because it requires two unlikely conditions to be met, namely the higher affinity subunit must be in the unliganded state and the lower affinity subunit in the liganded state. The circles represent the sum of liganded species divided by the total number of subunits; they are superimposed on the curve of overall ligand saturation (curve P in Figure 4.2).
4. Figure 4.4 The Hill (panel A) and Scatchard plot (panel B) for data set P from Figure 4.2. In the Hill plot the heterodimer presents a slope <1, and its Scatchard plot is bent.
5. Figure 4.5 Ligand‐binding isotherms for a non‐interacting homodimer (ε = 1), and two interacting ones (with ε = 1/9 and ε = 9). It is evident that a ligand‐dependent intersubunit interaction with ε > 1 increases the steepness of ligand‐binding isotherm, whereas one with ε < 1 broadens the ligand‐binding isotherm. The three curves were simulated using the following parameter sets: K_d=1, ε = 1; K_d=3, ε = 1/9; K_d=⅓ ε = 9 (values of Kd in arbitrary units); all parameter sets have the same X_1/2=1 (in arbitrary units).
6. Figure 4.6 Pauling’s reaction scheme for a cooperative homotetramer presenting the tetrahedral functional geometry.
7. Figure 4.7 Population of unliganded, singly liganded and doubly liganded species in homodimeric proteins.
8. Figure 4.8 Structure‐ and ligand‐binding reactions of an allosteric symmetric homodimer. The allosteric homodimer (D) is postulated to exist in two quaternary conformations or structures, both symmetric, called ^TD and ^RD respectively. Each monomer either assumes the tertiary T or R structure (^TM and ^RM respectively), depending on the quaternary structure of the whole oligomer. The inter‐subunit interface is isologous, hence symmetric and is here represented with two structural features: a groove (indicated as G) and a protruding α helix (H), such that the helix of monomer 1 (H1) perfectly fits into the groove of monomer 2 (G2) and vice versa. As is characteristic of a isologous interface, the same contacts occur twice: G1‐H2 and G2‐H1. The structural features at the isologous interface are not superimposable in the two quaternary states (here represented as a greater G to H distance in the R‐structure than in the T structure). As a consequence the mixed, non‐symmetric state in which one subunit is tertiary T state and the other is tertiary R‐state would never be populated. The T and R states of the dimer freely interconvert in any ligation state, and their equilibrium is described by the allosteric constant L, whose value depends on the number of bound ligands (see text). Each state binds ligands with a characteristic intrinsic affinity constant (K_R or K_T).
9. Figure 4.9 Energy diagram for the two‐state concerted model, as compared to a sequential model.
10. Figure 4.10 Relationships between the fractional ligand saturation and the fraction of protein in the R state.
11. Figure 4.11 An induced‐fit structural model of a cooperative homodimer may admit the asymmetric ligation intermediate (here represented with an intermediate G to H distance; this structural feature would not be admissible within the two‐state framework depicted in Figure 4.9).
12. Figure 4.12 Binding of ligands Y and X to different binding sites of a homodimeric protein. The scheme assumes binding stoichiometries 2:1 for ligand X and 1:1 for ligand Y.
13. Figure 4.13 Positive homotropic cooperativity induced by the differential binding of the heterotropic effector. The protein is a non‐cooperative dimer (K_X,1=K_X,2 and ^YK_X,1=^YK_X,2) Parameters: ^YK_X = 5 K_X (this entails ^X2K_Y = 25 K_Y, see text); concentrations of the effector Y (from left to right, in multiples of K_Y) 0, 0.5, 1, 2, 5, 10, 20, 50, 100. Panel A: versus log([X]) curves. Panel B derivatives of the curves shown in panel A; notice that the maximum slope (indicative of cooperativity) peaks at intermediate concentrations of the effector, and tends toward that of a non‐cooperative isotherm at very high and very low concentrations of the effector.
14. Figure 4.14 Negative heterotropic linkage in a non‐cooperative homodimer.
15. Figure 4.15 Reaction scheme of a two‐state allosteric homodimer that binds two ligands at different binding sites. The dimer binds two molecules of ligand X and one of ligand Y, at different sites.
16. Figure 4.16 A ligand‐binding isotherm is symmetric if the lower half of the curve can be perfectly superimposed to the upper half by 180° rotation around the point log(X_1/2),0.5. If this occurs, the areas A1 and A2 are equal.
17. Figure 4.17 Definition of the Xm: the shadowed areas are equal.
Chapter 05
1. Figure 5.1 Quaternary enhancement and quaternary constraint. Panel A: Oxygen‐binding isotherm of human hemoglobin at pH = 7.4 and T = 25 °C, and its isolated α and β subunits (redrawn after Imai, 1982). This is a case of quaternary constraint, because the oligomer has lower affinity than its isolated monomers. Panel B: Oxygen‐binding isotherm of the hexameric hemocyanin from the lobster Panulirus interruptus at pH = 9.6 and its isolated subunits. In the presence of 10 mM CaCl₂ the hexamer (Hcy) is populated; in its absence the oligomer dissociates into its constituent subunits (redrawn after Kuiper et al., 1977). This is a case of quaternary enhancement, because the oligomer has higher affinity than its isolated monomers.
2. Figure 5.2 Reaction scheme for a dissociating homodimer.
3. Figure 5.3 Simulation of the oxygen‐binding isotherms of a non‐cooperative homodimer in equilibrium with its constituent monomers, in the presence of quaternary constraint (J_d>K_d). Panel A: versus log ([X]) plot; the different curves for each family represent different protein concentrations (here expressed as multiples of the homodimer dissociation K₀: [M]_tot = 0.02 K₀, 0.2 K₀, K₀, 2 K₀, 5 K₀, 10 K₀). In this set of curves, calculated for J_d = 5 K_d, dilution of the protein causes a left shift of the ligand binding isotherm (i.e., an increase of the apparent ligand affinity), given that the monomers have higher affinity than the homodimers. Panel B: derivative of the two isotherms from panel A, for the two extreme values of protein concentration ([M]_tot = 0.02 K₀ and 10 K₀). Both curves present a small degree of positive cooperativity, more evident at higher protein concentration. Cooperativity depends on the change of the oligomerization state as a function of ligation, because both the dimer and the monomer in themselves are non‐cooperative. Panel C: plot of the half saturating ligand concentration X_1/2 as a function of protein concentration (from the data in panel A). Panel D: changes of the degree of dissociation α of the homodimers at constant protein concentration, as a function of ligand concentration (from the data in panel A).
4. Figure 5.4 Simulation of the oxygen‐binding isotherms of a non‐cooperative homodimer in equilibrium with its constituent monomers, in the presence of quaternary enhancement (J_d<K_d). Panel A: versus log ([X]) plot; the different curves for each family represent different protein concentrations (here expressed as multiples of the homodimer dissociation K₀: [M]_tot = 0.02 K₀, 0.2 K₀, K₀, 2 K₀, 5 K₀, 10 K₀). In this set of curves, calculated for J_d = 0.2 K_d, dilution of the protein causes a right shift of the ligand binding isotherm (i.e., a decrease of the apparent ligand affinity), given that the monomers have lower affinity than the homodimers. Panel B: derivative of the two isotherms from panel A, for the two extreme values of protein concentration ([M]_tot = 0.02 K and 10 K). Both curves present a small degree of positive cooperativity, more evident at lower protein concentration. Cooperativity depends on the change of the oligomerization state as a function of ligation, because both the dimer and the monomer in themselves are non‐cooperative. Panel C: plot of the half saturating ligand concentration X_1/2 as a function of protein concentration (from the data in panel A). Panel D: changes of the degree of dissociation α of the homodimers at constant protein concentration, as a function of ligand concentration (from the data in panel A).
5. Figure 5.5 Effect of ligand concentration and total monomer concentration on a protein that forms homodimers only in the unliganded state (see text). The curves in panel A are calculated for [M]_tot = 10 K, 5 K, 2 K, K, 0.2 K and 0.02 K, using eqns. 5.12 and 5.13; the corresponding degree of dissociation of the dimer, in the absence of the ligand, is α = 0.2, 0.27, 0.39, 0.5, 0.77, 0.96, whereas at saturating ligand concentration no homodimer is present. Panel B: derivatives of two isotherms from panel A, at [M]_tot = 10 K and 0.02 K, highlighting the cooperativity observed at concentrations of [M]_tot that exceed K. Panel C: dependence of X_1/2 on the protein concentration, calculated using eqn. 5.14; the points are the X_1/2 values of the isotherms reported in panel A. Panel D: dependence of the degree of dissociation α on ligand concentration, at constant [M]_tot for the same experimental conditions as in the isotherms in panel A, calculated using eqn. 5.15.
6. Figure 5.6 Effect of ligand concentration and total monomer concentration on a protein that forms homodimers only in the liganded state (see text). The curves in panel A are calculated for [M]_tot = 10 K, 5 K, 2 K, K, 0.2 K, and 0.02 K, using eqns. 5.17, 5.18, and 5.19; the corresponding degree of dissociation of the dimer, in the presence of saturating concentrations of the ligand, is α = 0.2, 0.27, 0.39, 0.5, 0.77, 0.96, whereas in the absence of the ligand no homodimer is present. Panel B: derivatives of two isotherms from panel A, at [M]_tot = 10 K and 0.02 K; cooperativity and affinity increase as the concentration of [M]_tot increases. Panel C: dependence of X_1/2 on the protein concentration, as calculated from the isotherms reported in panel A. Panel D: dependence of the degree of dissociation α on ligand concentration, at the same [M]_tot of the isotherms in panel A, calculated from the binding polynomial (eqn. 5.17).
7. Figure 5.7 Effect of ligand concentration and total monomer concentration on a cooperative homodimer that reversibly dissociates into monomers. The curves in panel A are calculated for [M]_tot = 10 K₀, 5 K₀, 2 K₀, K₀, 0.2 K₀ and 0.02 K₀, using eqns. 5.20, 5.21 and 5.22. The ligand dissociation constants are: K_d=0.5, J_d,1=3, J_d,2=0.33 in arbitrary units. In all the equations used in this plot the ligand dissociation constants appear in the dimensionless [X]/K_d or [X]/J_d ratio; thus it is unnecessary to give a definite concentration unit, provided that the same units are used for K_d, J_d,1, J_d,2; as a frame of reference the reader may assume the units to be μM. Panel B: derivatives of two isotherms from panel A, at [M]_tot = 10 K₀ and 0.02 K₀; at the lowest protein concentration very little dimer is present and binding is essentially non‐cooperative, whereas at the highest protein concentration significant cooperativity is present because the cooperative homodimer is the predominant species. Panel C: dependence of X_1/2 on the protein concentration, as calculated from the isotherms reported in panel A. Panel D: dependence of the degree of dissociation α on ligand concentration, at the same [M]_tot as the isotherms in panel A, calculated from the binding polynomial (eqn. 5 .20).
8. Figure 5.8 Dimerization of a protein induced by a ligand that binds at the intersubunit interface. The figure highlights the asymmetric structure of both the protein subunits and the ligand; by contrast, the homodimer is symmetric. The unbound dimeric protein (square brackets) is not considered in the treatment reported in the text.
9. Figure 5.9 Binding of a single molecule of effector to a self‐associating receptor (see text).
10. Figure 5.10 Ligand‐linked dissociation within the framework of the allosteric two‐state model. The protein is a homodimer, but only the first ligand‐binding step is represented in the figure. The scheme assumes pure quaternary constraint, that is, the monomers have the same ligand affinity as the R‐state dimer.
Chapter 06
1. Figure 6.1 Simulated time courses of ligation intermediates and fractional ligand saturation for the non‐cooperative homodimer.
2. Figure 6.2 Simulations of the time courses of ligation to three homodimeric proteins.
3. Figure 6.3 Simulated time course of ligand binding to a heterodimeric protein P_aP_b.
4. Figure 6.4 Time courses of oxygen dissociation from hemoglobin, as a function of inital saturation (oxygen pulse). Experimental conditions: observation wavelength 438 nm; T=20 °C; 0.05 M phosphate buffer pH=7; hemoglobin concentration 17 μM (after mixing); oxygen concentrations, after mixing: 150 μM (circles); 63.5 μM (square); 26.6 μM (asterisks); dithionite concentration (before mixing) 0.1%. In an oxygen pulse experiment a solution of deoxy hemoglobin containing sodium dithionite is rapidly mixed with an oxygen containing solution; the protein transiently forms an oxygen complex, which subsequently dissociates because the gas is consumed by excess dithionite. The experiment demonstrates that the higher the fraction of oxyhemoglobin formed, the slower the dissociation of the complex. This is a consequence of cooperativity: at high saturation the high affinity, slowly dissociating state of the complex is prevalent, whereas at low saturation the low affinity, rapidly dissociating state predominates. Data from Gibson (1973).
5. Figure 6.5 Simulated time courses of ligand binding to a cooperative homodimer: extent of autocatalysis.
6. Figure 6.6 Simulated time courses for a photolysis experiment.
7. Figure 6.7 Thermodynamic basis of the T‐jump experiment.
Chapter 07
1. Figure 7.1 The heme.
2. Figure 7.2 Oxygen‐binding isotherms of human Hb in different experimental conditions (from Imai, 1982).
3. Figure 7.3 Ribbon representation of the backbone of sperm whale myoglobin.
4. Figure 7.4 The αβ dimer of human deoxyhemoglobin.
5. Figure 7.5 The human deoxy‐hemoglobin tetramer and the a1b2 intersubunit interface. α subunits are shown in green, β subunits in pink. The FG corner and C helices of each subunit, which constitute the α1β2 interface are highlighted using stronger colors.
6. Figure 7.6 Superposition of the α subunit of deoxy‐hemoglobin (green) and carbon monoxy‐hemoglobin (cyan). The figure shows only residues 1‐93 (helices A through E) in order to better visualize the C‐helix (bottom) and F‐helix and FG corner (top). Superposition was optimized for the region where ligand‐linked structural changes are minimal, corresponding to the α1β1 contact region (helices B, G and H, residues 2‐35 and 94‐138), and the orientation emphasizes the region where the ligand‐linked structural changes are largest, corresponding to the α1β2 contact region.
7. Figure 7.7 Ligand‐dependent quaternary structural change at the αC‐βFG contact region of the α1β2 interface of human hemoglobin (detail). Upon oxygenation the FG corner of the β subunit slides by one helical turn over the C helix of the α subunit. The α C helices of the structures of HbO₂ (light blue, PDB file 2dn1) and Hb (green, PDB file 2dn2) were superimposed to make evident the relative movement of the β FG corner.
8. Figure 7.8 Time courses of CO combination to human hemoglobin by stopped flow (open circles) and flash photolysis (closed squares). Experimental conditions: T = 20 °C; buffer 0.1 M Bistris plus 0.1 M NaCl pH = 7; Hb concentration 2.8 μM (on a heme basis); CO concentration 25 μM. The time course obtained by rapid mixing of unliganded Hb and CO (open circles) is autocatalytic; that obtained by photolysis is described by two exponentials, one for the quickly reacting dimers, the other for the slowly reacting tetramers. Redrawn from Bellelli et al. 1987.
9. Figure 7.9 Effect of concentration on the oxygen affinity of human hemoglobin (modified from Mills et al., 1976). Experimental conditions: 0.1 M Tris‐HCl buffer pH = 7.4 containing 0.1 M NaCl and 1 mM EDTA; T = 21.5 °C; hemoglobin concentrations, on a heme basis (from right): 382 μM, 270 nM, 80 nM, and 40 nM. The leftmost curve indicates the calculated binding isotherm for pure αβ dimers. The dependence of ligand affinity on target concentration is the hallmark of ligand‐linked dissociation (see Chapter 5).
10. Figure 7.10 The model‐free (left) and two‐state model (right) approach to describe ligand binding and dissociation into dimers of hemoglobin, limited to the binding of the first ligand molecule to the tetramer. The suffix indicates the number of bound ligands; for example, Hb₀ indicates an unliganded hemoglobin tetramer, and D₁ a singly liganded dimer. The two‐state model is written under the assumption of perfect quaternary constraint, that is, the assigning the same ligand affinity to the dimer and to the R‐state tetramer.
Chapter 08
1. Figure 8.1 The Michaelis‐Menten plot, showing the hyperbolic dependence of V on [S] in a typical enzyme‐catalyzed reaction.
2. Figure 8.2 Lineweaver‐Burk plot of the data from Figure 8.1.
3. Figure 8.3 Flow chart of the experiments aimed at testing the applicability of the pseudo‐equilibrium approximation.
4. Figure 8.4 Pure competitive inhibition.
5. Figure 8.5 Pure non‐competitive inhibition.
6. Figure 8.6 Heterotropic inhibition, mixed type.
7. Figure 8.7 Dependence of K_{M app} on inhibitor concentration (panel A) and of IC₅₀ on substrate concentration (panel B). Curve 1 (both panels): competitive inhibition; curve 2: non‐competitive inhibition; curve 3: uncompetitive inhibition, with negative linkage between I and S (^SK_I = 5 K_I); curve 4: uncompetitive inhibition, with positive linkage between I and .
Chapter 09
1. Figure 9.1 Simulated steady‐state experiments for a two‐substrate enzyme that does not form a ternary complex. Panel A: normalized rate of product formation (expressed as the fraction of k_{cat, max}) as a function of the concentration of variable substrate XT. Velocity data were simulated at five concentrations of constant substrate Y, over the range 0.2K_Y, 0.5K_Y, 1K_Y, 2K_Y, 5K_Y. Parameters of the simulation: k_XT = 1 s⁻¹, k_Y = 1 s⁻¹, k_{cat, max} = 0.5 s⁻¹, K_XT = 1 mM, K_Y = 1 mM. Panel B: Lineweaver‐Burk plot of the same data as in panel A.
2. Figure 9.2 Dependencies of the Michaelis‐Menten parameters for substrate XT obtained at constant concentration of substrate Y.
3. Figure 9.3 Simulated Michaelis‐Menten (panel A) and Lineweaver‐Burk plots (panel B) for an inhibitor that competes with the two substrates of an enzyme that does not form a ternary complex (see text). Simulation conditions: constant concentration of substrate Y = 5 K_Y; K_IT = 5 K_I. Notice that the inhibitor changes both k_{cat XT, [Y]=const.} and K_{M XT, [Y]=const.}.
4. Figure 9.4 Simulations of the rate of product formation according to eqn. 9.18.
5. Figure 9.5 Inhibition of mouse liver thioredoxin reductase by dimethylarsonous diiodide during the fast, reversible binding process (original data by Lin et al., 1999; redrawn and reanalyzed by Saccoccia et al., 2014). The enzyme has two substrates, DTNB (or thioredoxin) and NADPH; the inhibitor competes with the former, but not with the latter.
6. Figure 9.6 Michaelis‐Menten (panel A) and Lineweaver‐Burk (panel B) plots of the rate of product formation simulated according to eqn. 9.25. ^XK_Y = 5K_Y; ^YK_X = 5K_X; variable substrate is X, and six concentrations of the constant substrate Y are explored: 0.2K_Y, 0.5K_Y, K_Y, 2K_Y, 5K_Y, and 10K_Y. In this mechanism the k_cat and K_M vary as a function of the constant substrate concentration, but with different laws (see eqns. 9.26 and 9.28), hence the kinetic efficiency exhibits a dependence on the constant substrate concentration.
7. Figure 9.7 Dependency of k_{cat X, [Y]=const.} (panel A) and K_{M X, [Y]=const.} (panel B) on the concentration of substrate Y. Same data and parameters as in Figure 9.6. Points represent the values of k_{cat X, [Y]=const.} (panel A) and K_{M X, [Y]=const.} for the curves in Figure 9.6; lines are drawn according to eqns. 9.26 or 9.28.
8. Figure 9.8 Simulation of the Michaelis‐Menten (panel A) and Lineweaver‐Burk (panel B) plots of a two‐substrate enzyme in the presence of an inhibitor that competes with both substrates. Simulation parameters as in Figure 9.6, plus five inhibitor concentrations equalling 0, K_I, 2K_I, 5K_I and 10K_I.
9. Figure 9.9 Michaelis‐Menten and Lineweaver‐Burk plots for a two‐substrate enzyme that forms a ternary complex in the presence of an inhibitor that competes with only one of the substrates (see text). Simulation parameters as in Figure 9.6, ^IK_X = 5K_X; inhibitor concentrations: 0, K_I, 2K_I, 5K_I, 10K_I. Panels A and B: rates of product formation at constant concentration of the competing substrate Y. Panels C and D: rates of product formation at constant concentration of the non‐competing substrate X. Notice that in experiments at variable concentration of the competing substrate (panels C and D) one obtains plots identical to those of competitive inhibitors of single‐substrate enzymes.
Chapter 10
1. Figure 10.1 Catalytic scheme of a copper containing amine oxidase: order of substrate addition and product release.
2. Figure 10.2 The catalytic cycle of Lentil seedling amine‐oxidase. Schematic representation of the active site, showing the TPQ group and the adjacent copper ion. Absorbance spectra of TPQ in its different functional states: spectrum (i) is characteristic of the fully oxidized intermediates (a) and (b) of the upper panel; spectrum (ii) is characteristic of the reduced intermediates (c) and (d); spectrum (iii) is characteristic of intermediates containing the semiquinolamine radical (e).
3. Figure 10.3 Time course of the reaction of bovine serum amine oxidase (BSAO) with benzylamine. The experiment was run in a stopped‐flow apparatus, by rapidly mixing a 33.4 μM solution of BSAO in the presence of 1.25 mM benzylamine with an equal volume of buffer containing 280 (circles) or 150 μM O₂ (squares). Concentrations after mixing: reduced BSAO (species ER in the scheme) 16.7 μM; benzylamine 625 μM; O₂ 140 or 75 μM. Potassium phosphate buffer pH 7.2; T = 25 °C. The time courses present three clearly separated processes: the pre‐steady state rapid oxidation of the enzyme, marked by an increase of absorbance; the steady state process, without changes of absorbance; and the exhaustion of oxygen, with return of the enzyme to the reduced state. Lines were calculated by numerical integration of the set of differential equations reported in the text, and a non‐linear least‐squares minimization routine was used to find the best estimates of the eight kinetic rate constants, as reported in Table 10.1.
4. Figure 10.4 Application of the King‐Altman method to the case of bovine serum CuAO (see text).
Chapter 11
1. Figure 11.1 Simulated time courses of binding of a reversible (panel A) and an irreversible inhibitor (panel B).
2. Figure 11.2 Simulation of the binding of an irreversible enzyme inhibitor.
3. Figure 11.3 Effect of the slowly binding pure competitive inhibitor.
4. Figure 11.4 Simulation of the time course of the incubation of a slowly binding reversible inhibitor with an enzyme.
5. Figure 11.5 Titration of an enzyme with a rapidly binding irreversible inhibitor. The closed point marks the stoichiometric equivalence (see text). Because the irreversible inhibitor lowers the total concentration of the active enzyme, its effect is evident as a reduction of the apparent V_max (V_max,I), which reaches zero when all the enzyme has been inactivated.
6. Figure 11.6 Simulated time courses of enzyme inactivation by irreversible inhibitors.
7. Figure 11.7 Proposed mechanism of gold transfer from auranofin to thioredoxin reductase. Auranofin is a bidentate complex of gold, whose ligands are acetoxy‐thioglucose and triethyl phosphine. A seleno‐Cys residue of the target enzyme displaces acetoxy‐thioglucose, presumably by forming a transient tridentate complex; then an adjacent Cys residue displaces triethyl phosphine, forming a bidentate complex in the C‐terminal Cys‐Sec couple (protein adduct 3). Since the C‐terminal Cys‐Sec couple is redox active, and is responsible for the electron transfer from the active site (which contains a FAD and a Cys couple) to thioredoxin, this adduct is already inactive. The enzyme contains other redox‐active Cys couples and the gold ion can be transferred to these from the flexible C‐terminal arm (to form complexes 5 and 7, both observed by X‐ray crystallography). From Angelucci et al., 2009.
8. Figure 11.8 Typical cycle of a mechanism based inhibitor. The average number of cycles required for inactivation is 1+k_cat/k_inact.
9. Figure 11.9 Simulation of the time course of inactivation of an enzyme reacting with a suicide substrate as in Figure 11.8.

Theory and Experiment