Typical examination questions (with answer notes)

Chemistry with Medicinal Chemistry (CMC)-3 Biophysical Chemistry Module Biomolecular Interactions (Professor Alan Cooper) Typical examination questions (with answer notes) The following questions are adapted from previous class/degree examinations in this topic (both 3 rd year and Finals papers). Answer notes do not necessarily comprise the full solution, and are provided for your guidance only. You are strongly advised to attempt each of the questions before consulting the answer notes. Please be aware that slight variations in course content from year to year may be reflected in these sample questions. AC: November 2000

Qu. 1 (a) What is the Levinthal Paradox in the context of protein folding? Describe how it arises and discuss its implications for the mechanism of protein folding. Thermal stability studies have given the following (partial) thermodynamic data for unfolding of a protein in aqueous solution at ph 7.4 at different temperatures: t K G H S / C /kj mol -1 /kj mol -1 /J K -1 mol -1 45 0.133 5.33 150.0? 50? 2.86 175.0? 55? 0 200.0 609.8 60 3.22? 225.0? (i) Complete this table by supplying the missing data (?) where possible. (ii) What fraction of the protein molecules would be unfolded at 50,55 and 60 C, respectively, under these conditions? (iii) What does the temperature dependence of the unfolding enthalpy ( H ) suggest about the forces responsible for stabilizing the folded protein conformation? [10] The complete genome sequence of a simple nematode worm (C. elegans) has just been completed. One of the major tasks now is to identify the function of many of the gene products. Glasgow scientists have identified one protein that might have metal-binding properties. Give three different biophysical techniques that might be used to investigate the binding of metal ions to this protein in solution. In each case describe the theoretical basis of the method and indicate how thermodynamic information may be derived. [9] [Gas constant R = 8.314 J K -1 mol -1 ; zero of the Celsius scale = 273.15 K] Qu. 2 (a) Describe the molecular basis for some of the anomalous properties of liquid water and explain the significance of this with regard to hydrogen bonding and hydrophobic interactions in biomolecules. The dimerization of N-methylacetamide in solution has frequently been used as a model system for inter-peptide hydrogen bonds in proteins. Some data for the dimerization equilibrium constant (K) and enthalpy of dimerization at 25 C in various solvents are given below: Solvent K G H S /M -1 /kj mol -1 /kj mol -1 /J K -1 mol -1 CCl 4 4.7? -17.6? Dioxane 0.52? - 3.3? Water 0.005? 0? Supply the missing thermodynamic data in the table (?) and explain what these data might suggest about the role of H-bonding in biomolecular interactions. [7]

Describe three different experimental techniques that may be used to monitor the unfolding of a protein molecule with change in temperature. In each case explain the physical basis for the method and the nature of the results that might be observed. How could the Gibbs free energy of unfolding be determined from such measurements? [7] What is meant by the heat capacity increment ( Cp) for protein unfolding? How might it be measured and what is it significance to understanding the forces responsible for protein folding stability? [5] Qu. 3 (a) List and explain some of the anomalous properties of water, and describe how they might relate to the forces responsible for stabilising protein and other biomolecular structures. [7] Describe experimental techniques using model compounds that might be used to obtain information about the thermodynamics of hydrophobic interactions and hydrogen bonding in the context of protein folding. The following experimental data have been obtained for the fluorescence intensity (F) and circular dichroism intensity (CD) of a protein solution at different temperatures. T / C F (arbitrary units) CD (arbitrary units) 20 65.0-1310 30 65.0-1310 40 64.7-1304 46 58.8-1186 50 40.0-810 56 17.8-366 60 15.5-320 70 15.0-310 80 15.0-310 What is the T m of this protein? What fraction of the protein might be unfolded at 46 C, and what is the Gibbs free energy of unfolding at this temperature? [7] Explain what molecular properties are being monitored by the two different sets of data in. Do the transitions monitored by fluorescence and CD necessarily have to occur at the same temperature? If not, explain why not. [5] [Gas constant R = 8.314 J K -1 mol -1 ; zero of the Celsius scale = 273.15 K] Qu. 4 (a) Describe, with appropriate examples and definitions, how the thermodynamics of hydrogen bonding and hydrophobic interactions can be studied experimentally by use of small model compounds. [8]

Discuss the role of hydrogen bonding and hydrophobic interactions in stabilising the folded conformations of globular proteins. What is the current view regarding the relative importance of the contributions from these two interactions? [5] There is currently considerable concern regarding the environmental levels of plasticisers and their potential effects on male sexual development. Why might these compounds be of concern? [3] A series of organic compounds with increasing levels of methyl group substitution, X-(CH 2 ) n -CH 3, are under investigation as potentially more environmentally-friendly plasticisers. Using groupadditivity data, the following values have been obtained for the predicted free energy of transfer of such compounds from cyclohexane to water at 25 C :- n G transfer (cyclohexane water) / kj mol -1 2-0.2 4 + 6.9 8 + 21.1 Calculate the partition (distribution) coefficient, D, for each of these compounds, defining carefully what you mean by this quantity. Discuss how these data might affect your choice of which compound might be the best choice for its use as a plasticiser, and how such molecules might bind to hydrophobic binding sites on transport proteins and hormone receptors. [3] [Gas constant R = 8.314 J K -1 mol -1 ; zero of the Celsius scale = 273.15 K] Qu. 5 (a) List and describe briefly the different non-covalent interactions thought to be involved in stabilizing protein folding and protein-ligand binding interactions in solution. In each case, discuss the role of solvent water and how it might affect the interaction. What are the typical thermodynamic features for thermal unfolding of a globular protein in solution and how are they determined experimentally? What do they suggest about the dominant interaction(s) responsible for stabilizing the folded conformation? [7] Describe the equilibrium dialysis method for determining protein-ligand binding affinities. For binding of a ligand (L) to a protein (P) to form a 1:1 complex (PL), show that: c p /[PL] = 1 + 1/K[L], where K is the equilibrium constant and c p is the total protein concentration. Explain how this expression is used to analyse equilibrium dialysis data. In an equilibrium dialysis experiment to study the binding of a new organic ligand to a soluble receptor protein, the following data were obtained:

Left-hand (protein + ligand) compartment: Total protein concentration Total ligand concentration Right-hand (ligand only) compartment: Total ligand concentration = 8.3 x 10-9 M = 3.9 x 10-8 M = 3.5 x 10-8 M What is the equilibrium binding constant for this process? Qu. 6 (a) Sequence analysis in the Human Genome Project has identified a new class of proteins with hitherto unknown properties. List the different biophysical techniques (at least three) that might be used to investigate the folding stability and interactions of these proteins in solution. In each case describe the theoretical basis of the method and indicate how thermodynamic information may be derived. [9] Thermal stability studies of one of the proteins in this family have given the following (partial) thermodynamic data for unfolding of the protein in aqueous solution at ph 7.4 at different temperatures: t K G H S / C /kj mol -1 /kj mol -1 /J K -1 mol -1 35 0.28 3.26 75.0? 40? 2.02 100.0? 45? 0 125.0 392.9 50 2.85? 150.0? (i) Complete this table by supplying the missing data (?) where possible. (ii) What fraction of the protein molecules would be unfolded at 40, 45 and 50 C, respectively, under these conditions? (iii) What does the temperature dependence of the unfolding enthalpy ( H ) suggest about the forces responsible for stabilizing the folded protein conformation? [10] What is the Levinthal Paradox in the context of protein folding? Describe how it arises and discuss its implications for the mechanism of protein folding. [Gas constant R = 8.314 J K -1 mol -1 ; zero of the Celsius scale = 273.15 K] Qu. 7 (a) List some of the anomalous properties of liquid water and how they can be rationalised in terms of what we know about the molecular structure and interactions of water. Explain the significance of this with regard to hydrogen bonding and hydrophobic interactions in biomolecules. The dimerization of carboxylic acids in solution has sometimes been used as a model system for - C=O...HO- hydrogen bonds in proteins. Some data for the dimerization equilibrium constant (K) and enthalpy of dimerization at 25 C of a particular carboxylic acid in various solvents are given below:

Solvent K G H S /M -1 /kj mol -1 /kj mol -1 /J K -1 mol -1 CCl 4 10.3? -15.6? Dioxane 0.7? - 3.5? Water 0.001? 0? Supply the missing thermodynamic data in the table (?) and explain what these data might suggest about the role of H-bonding in biomolecular interactions. [7] Describe three different experimental techniques that may be used to monitor the unfolding of a protein molecule with change in temperature. In each case explain the physical basis for the method and the nature of the results that might be observed. How could the Gibbs free energy of unfolding be determined from such measurements? Discuss briefly what effects might be relevant in designing a drug for delivery across biological membranes and how this might be studied experimentally with simple model systems. Qu. 8 (a) A new, virulent form of teenage acne has struck the West of Scotland. As senior biophysical chemist at PanGalacticDrugCo you have been charged with the task of identifying possible compounds that might bind to the viral protein involved. List the various techniques you might use to determine the binding of small molecular ligands to proteins. In each case, describe briefly the basis of the method and the sort of information that may be obtained. [8] One of the proteins responsible for the effects of teenage mutant acne is thought to contain a hydrophobic binding site that might be a potential target for drugs. Organic chemists in your company have synthesized three possible compounds, and your junior colleagues have measured the partitioning of these molecules between cyclohexane and water at 25 C as follows: Drug # D = [Drug] cyclohex /[Drug] water ZitBlast-A 9.7 x 10-4 ZitBlast-B 4.3 ZitBlast-C 7.6 x 10 3 Estimate the Gibbs free energy of transfer of each drug from water to cyclohexane. Suggest which of these drugs might be the best candidate for binding to this protein, and why. [8] Meanwhile, your old granny suggests that simply washing the spots with vinegar worked well enough in her day. Measurement of the tryptophan fluorescence of the relevant protein at different ph s gives the following data: ph Fluorescence at 335nm (arbitrary units) 1.5 5 2.0 5 2.5 10 3.0 35 3.5 70 4.0 95 5.0 100 7.0 100

Is your granny right? Interpret the above data in terms of protein unfolding. Estimate the free energy of unfolding at ph 2.5 and ph 3.5. What protein groups might be responsible for this ph behaviour? [9] Qu. 9 (a) Describe, briefly, the various kinds of interactions that may be involved in stabilising folded protein structures. In each case, explain how the interaction may be affected by the presence of water. [7] The dipolar properties of -NH and -C=O groups in polypeptides might be represented as follows in terms of partial charges: -0.28 +0.28 +0.39-0.39 N H C O 102 pm 124 pm Calculate the dipole moments of each of these groups. If these dipoles were placed in line, 4.5 Å apart in vacuum, what might be the interaction potential energy between them? How might the presence of solvent water molecules affect this? [8] For a particular protein at ph 6.3 the fraction unfolded at different temperatures is found to be:- t / C Fraction unfolded 58.5 0.012 63.5 0.101 68.5 0.500 73.5 0.896 78.5 0.985 What is T m for this protein and what is the free energy of unfolding ( G unf ) at this temperature? Estimate G unf at 63.5 C and at 73.5 C. Hence estimate the enthalpy and entropy of unfolding of the protein under these conditions. [10] [ε 0 = 8.85 x 10-12 J -1 C 2 m -1 ] Qu. 10 (a) Explain the practical and theoretical basis of the equilibrium dialysis method for measuring ligand binding to biological macromolecules. What other methods might be used to measure protein-ligand binding, and how do they differ fundamentally from equilibrium dialysis? [7] Starting from the equilibrium expression for binding of a ligand(l) to a protein (P) to form a 1:1 complex (PL), show that: c p /[PL] = 1 + 1/K[L] where K is the equilibrium constant and c p is the total protein concentration. How might this expression be used to obtain ligand binding data graphically from equilibrium dialysis data?

In an equilibrium dialysis experiment to study the binding of a putative anti-cancer drug (Q) to a receptor protein (R), the following data were obtained: Left-hand (protein + ligand) compartment: Total protein concentration = 9.2 x 10-9 M Total ligand concentration = 4.06 x 10-8 M Right-hand (ligand only) compartment: Total ligand concentration = 3.60 x 10-8 M What is the equilibrium binding constant for this process? Discuss briefly what effects might be relevant in designing a drug for delivery across biological membranes and how this might be studied experimentally with simple model systems. [Answer notes start on next page]

Answer Notes [Do not consult these until you have attempted the questions.] 1. (a) Levinthal paradox - too many polypeptide conformers to explore in realistic timescale. Numerical estimate: e.g. assume 3 x 3 = 9 possible Φ-Ψ angles per peptide, leads to typically 100 9 possible conformers for typical protein. Implies that protein folding must follow specific pathways. (i) Use G = -RT.ln(K) = H -T. S to complete:- t K G H S / C /kj mol -1 /kj mol -1 /J K -1 mol -1 45 0.133 5.33 150.0 454.9 50 0.345 2.86 175.0 532.9 55 1 0 200.0 609.8 60 3.22-3.24 225.0 685.4 (ii) Fraction unfolded = K/(1+K) = 0.26, 0.5, 0.76 (respectively) (iii) Increase in H with temperature (confirmed by increase in S with T) signifies a positive heat capacity increment ( Cp), characteristic of hydrophobic stabilizing interactions. Spectroscopic methods (UV, fluorescence, CD)-changes in environment/conformation on binding Hydrodynamics (viscosity, sedimentation)-changes in gross macromolecular properties Calorimetry (DSC, ITC)-direct measure of energy changes on binding Equilibrium dialysis-direct measure of ligand binding - all covered in lectures. Thermodynamic information may be obtained indirectly from temperature dependence, G = -RT.ln(K) = H -T. S, etc., or directly ( H, Cp) by microcalorimetry. 2. (a) Standard lecture material. High m.p./b.p., high density of liquid (ice floats), high dielectric, 4 C max density, heat capacity, etc... H-bonding, residual open tetrahedral lattice structure... hydrogen bonding between biomolecular groups generally unfavourable in water (water mols compete for H-bonding sites), but hydrophobic interactions (arising from structural propoerties of liquid water) more significant.. Use K = [NMA 2 ]/[NMA] 2 ; G = -RT.ln(K) = H - T. S Results (for 25 C): Solvent K G H S /M -1 /kj mol -1 /kj mol -1 /J K -1 mol -1 CCl 4 4.7-3.8-17.6-46 Dioxane 0.52 + 1.6-3.3-16.5 Water 0.005 + 13.1 0-44 Suggests H-bonding unfavourable in polar solvents.

Anything sensible from: UV difference, fluorescence, CD, nmr, viscosity, hydrodynamic effects, DSC, etc. (all done in lectures). Use G = -RT.ln(K), where K = [U]/[N] = (F-F 0 )/(F inf - F), for any observable F. Positive increase in excess heat capacity of unfolded protein w.r.t. folded. Observed from temperature dependence of H or calorimetric (DSC) studies. Characteristic of hydrophobic contribution to protein folding. 3. (a) Standard lecture material: High mp/bp, density increase on melting (ice floats), high dielectric, 4 C max density, high heat capacity... Water dipole, H-bonding, residual open tetrahedral lattice structure hydrogen bonding between biomolecular groups generally unfavourable in water (water molecules compete for H-bonding sites), but hydrophobic interactions (arising from structural properties of liquid water) more significant. [7] Partitioning between solvents, gaseous dimers Liquid phase dimerization of peptide analogues (e.g. N-methyl acetamide) Different results in aqueous/non-aqueous systems Polar vs. non-polar environments in folded/unfolded proteins - [lecture material] Both F and CD follow the same transition in this case, so may use either. T m = 50 C (mid-point of unfolding transition) Fraction unfolded = (F - F 0 )/(F inf - F 0 ) = (58.8-65)/(15-65) = 0.124 Equilib.const. for unfolding K = (F - F 0 )/(F inf - F) = (58.8-65)/(15-58.8) = 0.142 G unf = -RT.lnK = -8.314 x (273 + 46) x ln(0.142) = + 5.18 kj mol -1 [7] Fluorescence is probing the polarity of the environment of aromatic amino acid residues (primarily tryptophan), which changes (non-polar --> polar) as the protein unfolds. CD measures secondary structure changes (α-helix, β-sheet, etc.). These do not necessarily occur simultaneously, since protein unfolding may take place in two (or more) steps, e.g. change in tertiary structure exposing aromatic groups but retaining secondary structure, followed by melting of the secondary structure at higher temperatures. [5] 4. (a) Gas phase dimerization, e.g. formic acid Liquid phase dimerization of peptide analogues (e.g. N-methyl acetamide) Different results in aqueous/non-aqueous systems - [lecture material] [8] Hydrogen bonds required for structure, but do not necessarily contribute to stability, because of competing H-bonds with water. Hydrophobic interactions are most likely source of stability, supported by evidence from thermodynamics ( C p, etc.) - [lecture material] [5]

Plasticisers are hydrophobic compounds that can mimic endocrines and act as feminizing hormones by binding to receptor and fatty-acid transport proteins - [tutorial material] [3] D = [A] water /[A] cyclohexane (or equivalent) G transfer (cyclohexane water) = -RT.lnD hence D... n G transfer / kj mol -1 D = exp(- G transfer x 1000/8.314x298) 2-0.2 1.1 4 + 6.9 0.06 8 + 21.1 0.0002 n = 8 probably best since, even though this would probably also act best as an endocrine disrupter, it will have the least propensity to leach into aqueous environment. (Or anything equally sensible, or better!) [3] [25 marks total] 5. (a) Standard lecture material - electrostatics, charge-charge, dipole-dipole, etc., H-bonds, hydrophobic interactions, London dispersion forces. Water can affect indirectly (via dielectric constant effect) or directly (via H-bonding to water mols.). Water is essential for hydrophobic interaction! For globular protein unfolding: H unf normally positive (endothermic), but temperature dependent (positive Cp) S unf normally positive (favourable), again depends on T. G unf positive below T m, negative above T m, T m = H / S Need to overcome large configurational entropy of unfolded polypeptide chain to stabilise folded conformation. G per residue much less than RT, therefore folding must be cooperative. Positive heat capacity increment ( Cp) conventionally taken to be characteristic of hydrophobic interactions (from small model compounds). Determined by microcalorimetry or temperature dependence of spectroscopic properties, etc. Equilibrium dialysis (from lectures) - two compartments, semi-permeable membrane, protein/macromolecule confined to one side, (small) ligand free to move across membrane so that, at equilibrium, measurements of total protein and total ligand either side of membrane will give all info necessary to determine K. Appropriate sketch diagram. P + L PL c p = [PL] + [P] ; K = [PL]/[P][L] c p /[PL] = 1 + [P]/[PL] = 1 + 1/K[L] (QED) Slope of d-r plot = 1/Kc p. Useful because only equilibrium dialysis (and related methods) gives free ligand concentration [L] directly. For most other methods need to make approximations, or fit to complete binding expression.

c p = [PL] + [P] = 8.3 x 10-9 M (from left hand compartment) c L = [PL] + [L] = 3.9 x 10-8 M.... Free ligand [L] = 3.5 x 10-8 M (from right hand compartment) [PL] = 3.9 x 10-8 - 3.5 x 10-8 = 4.0 x 10-9 M [P] = 8.3 x 10-9 - 4.0 x 10-9 = 4.3 x 10-9 M Hence K = [PL]/[P][L] = 2.7 x 10 7 M -1 6. (a) Spectroscopic methods (UV, fluorescence, CD) -changes in environment/conformation on folding or binding Hydrodynamics (viscosity, sedimentation) -changes in gross macromolecular properties Calorimetry (DSC, ITC) -direct measure of energy changes on unfolding or binding Equilibrium dialysis -direct measure of ligand binding - all covered in lectures. Thermodynamic information may be obtained indirectly from temperature dependence, G = -RT.ln(K) = H -T. S, etc., or directly ( H, Cp) by microcalorimetry. (i) Use G = -RT.ln(K) = H -T. S to complete:- t K G H S / C /kj mol -1 /kj mol -1 /J K -1 mol -1 35 0.28 3.26 75.0 232.8 40 0.46 2.02 100.0 312.9 45 1.0 0 125.0 392.9 50 2.85-2.81 150.0 472.9 (ii) Fraction unfolded = K/(1+K) = 0.22, 0.5, 0.74 (respectively) (iii) Increase in H with temperature (confirmed by increase in S with T) signifies a positive heat capacity increment ( Cp), characteristic of hydrophobic stabilizing interactions. Levinthal paradox - too many polypeptide conformers to explore in realistic timescale. Numerical estimate: e.g. assume 3 x 3 = 9 possible Φ-Ψ angles per peptide, leads to typically 100 9 possible conformers for typical protein. Implies that protein folding must follow specific pathways. 7. (a) Standard lecture material. High m.p./b.p., high density of liquid (ice floats), high dielectric, 4 C max density, heat capacity, etc... H-bonding, residual open tetrahedral lattice structure... hydrogen bonding between biomolecular groups generally unfavourable in water (water mols compete for H-bonding sites), but hydrophobic interactions (arising from structural properties of liquid water) more significant..

Use K = [dimer]/[monomer] 2 ; G = -RT.ln(K) = H - T. S Results (for 25 C): Solvent K G H S /M -1 /kj mol -1 /kj mol -1 /J K -1 mol -1 CCl 4 10.3-5.8-15.6-32.9 Dioxane 0.7 +0.9-3.5-14.8 Water 0.001 +17.1 0-57.4 Suggests H-bonding unfavourable in polar solvents. Anything sensible from: UV difference, fluorescence, CD, nmr, viscosity, hydrodynamic effects, DSC, etc. (all done in lectures). Use G = -RT.ln(K), where K = [U]/[N] = (F-F 0 )/(F inf - F), for any observable F. Discussed in lectures/tutorials. Polarity, relative solubility in aqueous/organic phases. Measure distribution/partition coeffs. 8. (a) Equilibrium dialysis: direct method --> no. of binding sites (n), binding const. (K) UV/fluorescence changes: indirect method --> n, K, polarity of binding site? In both of above: K --> G, temperature dependence --> H and S Microcalorimetry methods --> n, K and direct measure of H and S D G(water-->cyclohexane) = RT.ln(D) ZitBlast-A 9.7 x 10-4 -17.2 kj mol -1 ZitBlast-B 4.3 3.6.. ZitBlast-C 7.6 x 10 3 22.1.. ZitBlast-A possibly best candidate for binding to hydrophobic site, assuming cyclohexane is a reasonable model for hydrophobic environment. Partitions favourably into cyclohexane, negative G transfer. Protein is folded (presumably) at neutral ph and unfolds at low ph. Buried tryptophans become exposed to more polar environment, hence decrease in relative fluorescence. ph of vinegar (dilute ethanoic acid) 2-3, granny possibly right. For a 2-state unfolding transition: N U, K = [U]/[N] K = (F - F0)/(Finf - F) ; G unf = -RT.lnK Assuming F 0 (high ph) = 100, and F inf (low ph) = 5, T = 25 C = 298 K ph 2.5: K = (10-100)/(5-10) = 18 G unf = -7.2 kj mol -1 ph 3.5: K = (70-100)/(5-70) = 0.46 G unf = +1.9 kj mol -1 Mid-point ph in region 2-3, therefore protein groups titrating in this region most likely responsible, aspartic/glutamic/c-terminal -COOH

9. (a) Standard lecture material - electrostatics, charge-charge, dipole-dipole, etc., H-bonds, hydrophobic interactions, London dispersion forces. Water can affect indirectly (via dielectric constant effect) or directly (via H-bonding to water mols.). Water is essential for hydrophobic interaction! Dipole moment µ = charge x distance ; 1 Debye 3.336 x 10-30 Cm µ NH = 0.28 x 1.6 x 10-19 x 102 x 10-12 = 4.6 x 10-30 Cm 1.37 D µ CO = 0.39 x 1.6 x 10-19 x 124 x 10-12 = 7.7 x 10-30 Cm 2.32 D V µµ (in line) ± 2.µ 1 µ 2 /4πε 0 ε r r 3 ± 7.0 x 10-21 J 4.25 kj mol -1 (in vacuo) In water: (i) continuous dielectric effect would reduce this by factor 80 (ii) in reality, H-bonds with water might compete N U ; K = [U]/[N] = f/(1-f) where f = [U]/([N]+[U]) = fraction unfolded. G unf = -RT.lnK = H - T. S (R=8.314 J K -1 mol -1, temp. in K) T m = 68.5 C ; G unf = 0 63.5 C : G unf = + 6.1 kj mol -1 73.5 C : G unf = - 6.2 kj mol -1 H and S at T m from simultaneous equations of G = H - T. S at 2 temperatures (e.g. 63.5 and 73.5 C )... Answer: H unf = +420 kj mol -1 ; S unf = +1230 J K -1 mol -1 (note signs). 10. (a) Equilibrium dialysis - anything sensible... (done in lectures) Other methods (UV, fluorescence, CD, etc.) are indirect methods, e.g. do not give free ligand concs directly. P + L PL c p = [PL] + [P] ; K = [PL]/[P][L] c p /[PL] = 1 + [P]/[PL] = 1 + 1/K[L] (QED) Slope of d-r plot = 1/Kc p. Useful because only equilibrium dialysis (and related methods) gives free ligand concentration [L] directly. For most other methods need to make approximations, or fit to complete binding expression.

c p = [RQ] + [R] = 9.2 x 10-9 M c L = [RQ] + [Q] = 4.06 x 10-8 M Free ligand [Q] = 3.60 x 10-8 M [RQ] = 4.06 x 10-8 - 3.60 x 10-8 = 4.6 x 10-9 M [R] = 9.2 x 10-9 - 4.6 x 10-9 = 4.6 x 10-9 M Hence K = [RQ]/[R][Q] = 2.8 x 10 7 M -1 Discussed in lectures/tutorials. Polarity, relative solubility in aqueous/organic phases. Measure distribution/partition coeffs. END