Analysis and Biological Significance of Bivalent Ligand Binding Reactions. Duane W. Sears

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1 Duane W. Sears 9/12/2004 Analysis and Biological Significance of Bivalent Ligand Binding Reactions Duane W. Sears Department of Molecular, Cell, and Developmental Biology, University of California Santa Barbara TABLE OF CONTENTS I. Biological significance of reversible ligand binding reactions... 1 II. BIVALENT equilibrium ligand binding reactions: Definitions and relationships... 2 A. Bivalent receptors with distinct ligand binding sites...2 B. Acid/base titration profile of glycine and other bivalent amino acids...3 C. Determining the pi (isoelectric ph) of glycine and percentages of 4 equilibrium microstates...4 D. Bivalent receptors with identical or very similar ligand binding sites...4 E. Saturation analysis of length-dependent, anti-cooperative proton binding by di-carboxylates...6 F. Hill analysis of the length-dependent anti-cooperative proton binding by di-carboxylates...7 G. The Hill plot slope at 50% saturation and guidelines for interpreting Hill plots...8 H. Ionization properties of interacting catalytic carboxyl groups in aspartyl proteases...9 I. Biological significance of reversible ligand binding reactions The dynamic chemistry of living organisms stems from complex networks of selectively-linked and highly-regulated chemical reactions, many of which involve reversible interactions between receptors and their ligands. Generally a receptor (R) is a protein, enzyme, or other macromolecule and a ligand (L) may be an ion, small molecule, substrate, co-factor, or even another macromolecule (i.e., co-receptor ). Reversible receptor/ligand binding reactions are typically weak interactions with receptor and ligand continuously and reversibly engaging and disengaging, usually very rapidly over short time spans. Such interactions are ideal in terms of biological regulation because reaction reversibility allows biochemical processes to adjust rapidly and appropriately to chemical fluctuations in the reaction environment. Consider the now classic example of oxygen binding by hemoglobin, the oxygen transport protein of the blood. In arterial blood passing through the high oxygen environment of the lungs, each hemoglobin molecule carried in a red blood cell tends to bind up to a maximum of 4 O 2 molecules. However, in venous blood passing through oxygen-depleted tissues, the arterially oxygenated hemoglobin tends to release its bound O 2 thereby delivering O 2 to low oxygen environments where it can be consumed for energy production by mitochondria. The reversible binding and release of O 2 by hemoglobin, which repeats itself over and over as blood circulates through the body, is highly regulated by several other hemoglobin ligands in the blood (e.g., H + and Cl - ions, 2,3-diphosphoglycerate, etc.). These also reversibly bind hemoglobin in proportion to their concentrations and thereby fine-tune the release or binding of O 2 by hemoglobin. The nuanced properties of hemoglobin echo a recurrent theme in contemporary studies of biochemical processes and their underlying reactions. The molecular elucidation of such processes often hinges on characterization and analysis of reversible ligand binding reactions linked to such processes. The primary aim biivalent-ligand-binding.doc Page 1 of 9 Pages MCDB 108A, Fall 2004

2 of the following sections is to forge a blueprint, as were, for effective analysis of reversible ligand binding reactions. Topics are systematically ordered according to the increasing levels of reaction complexity. Thus, simple monovalent receptor/ligand reactions (including weak acid/base interactions) are considered first, followed by the analysis of more complex multivalent receptor/ligand reactions where receptors have more than one binding site for the same type of ligand. Bivalent receptor/ligand reactions are analyzed in detail because exact mathematical solutions are possible and because general guidelines can be developed for identifying whether a multivalent ligand binding reaction is non-cooperative, anti-cooperative, or cooperative in nature. Non-cooperative ligand binding reactions, for example, would involve receptors that have the same affinity for ligand whether or not ligand already occupies other sites on the same receptor. In contrast anti-cooperative or cooperative ligand binding reactions, would involve receptors that have different affinity for ligand depending on occupancy state of other sites on the same receptor. Thus, the receptor would bind ligand anti-cooperatively if its ligand binding affinity decreases after other sites become occupied. Conversely, the receptor would bind ligand cooperatively if its ligand binding affinity increases after other sites become occupied, as found for the hemoglobin binding reaction with O 2, for example. Receptor/ligand reactions are evaluated here in terms of reversible equilibrium reactions, even though the corresponding reactions in a biological system would most likely occur as non-equilibrium, possibly steadystate reactions. However, as illustrated by some of the examples considered below, it is still possible to gain deep insights into non-equilibrium biological reactions by appropriate analysis of the equilibrium reactions. Also, because chemists tend to emphasize the dissociation reaction in discussions of reversible reactions (e.g., acid/base dissociation constants) and biologists tend to emphasize the association reaction (e.g., ligand binding to receptors, substrates binding to enzymes), most of the reactions discussed below are formulated in both directions. Obviously, the equilibrium thermodynamics doesn t change but it is conceptually easier to discuss some reactions in terms of dissociation constants (Kdn) and others in terms of association constants (Kan). II. BIVALENT equilibrium ligand binding reactions: Definitions and relationships A. Bivalent receptors with distinct ligand binding sites Assume that () 2 R 1 () is a receptor (e.g., protein, macromolecule, enzyme, etc.) with two distinct binding sites (1 & 2) for ligand, L (e.g., ion, small molecule, co-receptor, etc.) that each undergo association/association reactions over very different ranges of ligand concentration. Assume Kdn1 >> Kdn2. Equilibrium dissociation reactions for a bivalent receptor with distinct ligand binding sites, 1 & 2: (L) 2 R 1 (L) (L) 2 R 1 () + L Step 1 (L) 2 R 1 () () 2 R 1 () + L Step 2 Equilibrium dissociation constants, assuming Kdn1 >> Kdn2: Kdn1 = [(L) 2 R 1 ()][L]/[(L) 2 R 1 (L)] = Empirical definition pkdn1 = -log Kdn1 = -log Kdn1 = 10 -pkdn1 Kdn2 = [() 2 R 1 ()][L]/[(L) 2 R 1 ()] = Empirical definition pkdn2 = -log Kdn2 = -log Kdn2 = 10 -pkdn2 Equilibrium association reactions for a bivalent receptor with distinct ligand binding sites, 1 & 2: () 2 R 1 () + L (L) 2 R 1 () Step 1 (L) 2 R 1 () + L (L) 2 R 1 (L) Step 2 Equilibrium association constants, assuming Kan2 >> Kan1: biivalent-ligand-binding.doc Page 2 of 9 Pages MCDB 108A, Fall 2004

3 Kan2 = [(L) 2 R 1 ()]/[() 2 R 1 ()][L] =1/Kdn2 = 1/ ; Empirical definition pkan2 = +log Kan2 = -log Kdn2 = -log Kan2 = 10 +pkan2 Kdn2 = 10 -pkdn2 Kan1 = [(L) 2 R 1 (L)]/[(L) 2 R 1 ()][L] = 1/Kdn1 = ; Empirical definition pkan1 = +log Kan1 = -log Kdn1 = -log Kan1 = 10 +pkan1 Kdn1 = 10 -pkdn1 Equilibrium dissociation fraction, Yd, assuming that Kdn1 >> Kdn2; or that Kan2 >> Kan1 Yd = (0*[(L) 2 R 1 (L)] + 1*[(L) 2 R 1 ()] + 2*[() 2 R 1 ()]) / 2*([() 2 R 1 ()] + [(L) 2 R 1 ()] + [(L) 2 R 1 (L)]) Yd = (1*[(L) 2 R 1 ()] + 2*[() 2 R 1 ()]) / 2*Co Yd = (Kdn1*[L] + 2*[L] 2 ) / (Kdn1*Kdn2 + Kdn1*[L] + 2*[L] 2 = 0.25, = Kdn1; = = Kdn2 Equilibrium association fraction, Ya, assuming that Kdn1 >> Kdn2; or that Kan2 >> Kan1 Ya = (2*[(L) 2 R 1 (L)] + 1*[(L) 2 R 1 ()] + 0*[() 2 R 1 ()]) / 2*([() 2 R 1 ()] + [(L) 2 R 1 ()] + [(L) 2 R 1 (L)]) Ya = (2*[(L) 2 R 1 (L)] + 1*[(L) 2 R 1 ()]) / 2*Co Ya = (Kan2*[L] + 2* Kan2*Kan1*[L] 2 ) / (1 + Kan2*[L] + 2* Kan1*Kan2 *[L] 2 ) B. Acid/base titration profile of glycine and other bivalent amino acids Pure glycine, NH 2 CH 2 COOH, is essentially a chemical composite of methylamine and acetic acid (minus one the methyl carbon). When dissolved in water, the weakly acidic carboxylic acid group completely ionizes releasing protons while the weakly basic amino group binds nearly an equivalent number of protons. In effect, the basic group soaks up protons released by the acidic group resulting in a zwitterionic molecule with a negatively-charged carboxyl group (α-coo - ) and positively-charged amino group almost (α-nh + 3 ). The acid/base titration profile for glycine is shown below with two distinct inflection points corresponding to its two ionizable groups Ya % 50% Acid/Base Titration of Amino Acids with 2 Ionizable Groups pkdn(c) = 2.3 Glycine or Valine (pkdn-c = 2.3, pkdn-n = 9.6) pkdn(n) = 9.6 ph = E E-06 1.E E-10 Selected AA Ala (2.4, 9.7) Asn (2.0, 8.8) Gln (2.2, 9.1) Gly (2.3, 9.6) Leu (2.4, 9.6) Met (2.3, 9.2) Phe (1.8, 9.1) Pro (2.1, 10.6) Ser (2.2, 9.2) Thr (2.6, 10.4) Trp (2.4, 9.4) ph = % E E E E E E-06 [H+] 1.0E E E E E E-12 biivalent-ligand-binding.doc Page 3 of 9 Pages MCDB 108A, Fall 2004

4 C. Determining the pi (isoelectric ph) of glycine and percentages of 4 equilibrium microstates When pure glycine is dissolved in pure water (initially, ph = 7.0), the ph will drop a little because the acid strength of the α carboxylic acid group (pkdn = 2.3, 4.7 ph units below neutral ph of 7.0) is greater than the base strength of the α amino group (pkdn = 9.6, only 2.6 ph units above neutral ph of 7.0). Just what will the final ph be? This is easy to figure out remembering that biological solutions are always electrically neutral. That is, the net concentration of positive charges always equals the net concentration of negative charges. For this bivalent system, the following condition holds: Positive charge concentration = [H + ] + [α NH + 3 ] = [OH - ] + [α COO - ] = negative charge concentration If the concentration of glycine, Co >> [H + ] or [OH - ] (initially 10-7 ), the equation above simplifies to [α NH + 3 ] [α COO - ], or Co*Ya(N) = Co*Yd(C), ph = pi, the isoelectric ph where the average charge concentration for all ionization states of a molecule equals zero. pi is found using the following relationships for Ya(N) and Yd(C): Ya(N) = 1/(1+10 pi-pkdn(n) ) = 1/(1+10 pkdn(c)-pi ) = Yd(C) Solving for pi: (1+10 pkdn(c)-pi ) = (1+10 pi-pkdn(n) ), and 10 pkdn(c)-pi = 10 pi-pkdn(n), and pkdn(c)-pi= pi-pkdn(n), and 2*pI = pkdn(c) + pkdn(n), and pi = ½ (pkdn(c) + pkdn(n)), the average of the dissociation pkdn values. For glycine, pi = ½ ( ) = 5.9, which is slightly acidic as predicted. Note: The fractions or percentages of each of four glycine microstates in an aqueous equilibrium solution can be calculated from the dissociation and association fractions of the two ionizable groups. These fractions can also be used to calculate the average charge of glycine at any given ph. Unionized or ionized groups Association/ dissociation fraction Distribution of glycine microstates at ph = pi Assume that ph = pi = 6.1, pkdn(c) = 2.4, and pkdn(n) = 9.8 Unionized / ionized group percentages Glycine microstates in solution Net charge q Microstate fractions in solution Microstate percentages in solution [α NH 3 + ] Ya(N) = % [NH 3 + CH2 COO - ] 0 Ya(N)*Yd(C) = % [α NH 2 ] Yd(N) = % [NH 3 + CH2 COOH] +1 Ya(N)*Ya(C) = % [α COOH] Ya(C) = % [NH 2 CH 2 COO - ] -1 Yd(N)*Yd(C) = % [α COO - ] Yd(C) = % [NH 2 CH 2 COOH] 0 Yd(N)*Ya(C) = % D. Bivalent receptors with identical or very similar ligand binding sites Assume that () 2 R 1 () is a receptor (e.g., protein, macromolecule, enzyme, etc.) with 2 identical or very similar binding sites (1 & 2) for ligand, L (e.g., ion, small molecule, co-receptor, etc.) that undergo association/association reactions over very identical or very similar ranges of ligand concentration. Equilibrium dissociation reactions for a bivalent receptor with identical or very similar ligand binding sites: k dn1 (step 1) k dn2 (step 2) L + (L) 2 R 1 () (L) 2 R 1 (L) () 2 R 1 ()+ L dissociation reactions and constants L + () 2 R 1 (L) biivalent-ligand-binding.doc Page 4 of 9 Pages MCDB 108A, Fall 2004

5 k dn2 (step 1) k dn1 (step 2) step 1 step 2 For the 1 st step of the dissociation reaction, the receptor may dissociate ligand from either binding site (site 1, top, or site 2, bottom). For the 2 nd step of the dissociation reaction, the receptor dissociates the ligand remaining in the other site (site 2, top, or site 1, bottom). For this analysis, it is assumed that the dissociation constants for sites 1 and 2 may be different when the other site is occupied as compared to being empty. In other words, the equilibrium constant for L dissociation from site 1 might differ when site 2 is occupied (step 1, top) as compared to when site 2 is empty (step 2, bottom). In particular, step 1 and step 2 equilibrium constants are likely be different IF the ligand binding sites interact directly with each other and IF this interaction depends on the presence of bound ligand. Also, step 1 and step 2 equilibrium constants might be different if the receptor structure changes around the binding sites after the 1st ligand dissociates. This complex ligand-binding scenario is greatly simplified IF the ligand binding sites are identical and symmetrically placed in the receptor, or IF the binding sites 1 and 2 have similar ligand binding properties when the other site is occupied or empty. Thus, it will be assumed that site 1 and site 2 are equivalent in their properties. In other words, it will be assumed from this point on that: k dn1 (step 1) = k dn2 (step 1) = kdn-1st k dn1 (step 2) = k dn2(step 2) = k dn-2nd IF sites 1 and 2 don t interact, or IF the ligand binding properties of both sites do not change with ligand bound or not bound by the opposite site, then k dn-1st = k dn-2nd., IF, however, the sites interact in a ligand-dependent fashion or IF their ligand binding properties are modified when ligand binds or dissociates, then k dn-1st π k dn-2nd. Equilibrium association reactions for a bivalent receptor with identical or very similar ligand binding sites k an2 (step 1) k an1 (step 2) +L (L) 2 R 1 () +L () 2 R 1 () (L) 2 R 1 (L) association reaction and constants +L () 2 R 1 (L) +L k an1 (step 1) k an2 (step 2) step 1 step 2 Equilibrium association constants For this analysis, it is assumed that the two binding sites are equivalent in that both have the same affinity for binding the 1 st ligand and both have the same affinity for binding the 2 nd ligand after one site is already occupied. Thus, it is assumed that: k an1 (step 1) = k an2(step 1) = k an-1st = 1/ k dn-2nd k an1 (step 2) = k an2 (step 2) = k an-2nd = 1/ k dn-1st Note that for this two-step reaction, k an-1st = 1/ k dn-2nd and k an-2nd = 1/ k dn-1st. IF sites 1 and 2 don t interact, or IF the ligand binding properties of both sites do not change with ligand bound or not bound by the opposite site, then k an-1st = k an-2nd. IF, however, the sites interact in a ligand-dependent fashion or IF their ligand binding properties are modified when ligand binds or dissociates, then k an-1st π k an-2nd. Equilibrium association fraction, Ya, assuming that k an-1st not equal to k an-2 nd biivalent-ligand-binding.doc Page 5 of 9 Pages MCDB 108A, Fall 2004

6 Because it is easier to conceptualize the significance of different association constants, the following analysis, equilibrium association reactions and association constants are used to derive an expression for the association fraction, Ya. By definition, Ya = (2*[(L) 2 R 1 (L)]+1*[(L) 2 R 1 ()]+1*[(L) 2 R 1 ()]+0*[() 2 R 1 ()]) / 2*([() 2 R 1 ()]+[(L) 2 R 1 ()]+[(L) 2 R 1 ()]+[(L) 2 R 1 (L)]) Ya = (2*[(L) 2 R 1 (L)]+1*[(L) 2 R 1 ()]+1*[(L) 2 R 1 ()]) / 2*Co Replacing the receptor concentrations using the equations for equilibrium association constants (and assuming ligand binding site symmetry), the expression for Ya can be re-written in terms of only two variables: [H + the H + concentration at 50% saturation, and Ran, which is defined as follows: Ran = k an-2nd / k an-1st, the ratio of the equilibrium association constants for receptors with either one (numerator) or two (denominator) unoccupied binding sites. Ran = k dn-1st / k dn-2nd = 10 pkdn2-pkdn1 = 10 pkdn Thus, Ya = (Ran -1/2 *[H + ]*[H + + [H + ] 2 ) / ([H + ] 2 + 2*Ra -1/2 *[H + ]*[H + + [H + ] ) Yd = 1 - Ya Yd = (Ran -1/2 *[H + ]*[H + + [H + ] ) / ([H + ] 2 + 2*Ra -1/2 *[H + ]*[H + + [H + ] ) E. Saturation analysis of length-dependent, anti-cooperative proton binding by di-carboxylates Technically, the equations above can be used to predict the [H + ]-dependent saturation and dissociation fractions as functions of [H + ] for a bivalent ligand binding receptor. However, have one limitation: they include only one experimentally-determined parameter, [H + For a bivalent system, there are two unknowns. k an-2nd and k an-1st. and two independent equations with two independent parameters are required in order to determine two unknowns. For a monovalent ligand binding system, [H + = Kdn, but for multivalent systems, [H + may or may not correspond to a physical dissociation constant. As discussed in more detail in later sections, [H + = Kdn for a multivalent receptor only when the receptor binds ligand non-cooperatively. Thus, for non-cooperative bivalent receptor ligand binding, k an-2nd = k an-1st, Ran = 1, and the equations for Ya and Yd above simplify to the following: Ya = [H + ]/([H + ] + [H + ) and Yd = [H + /([H + ] + [H + ) Note that these equations are identical to those describing a monovalent ligand binding receptor. However, when a bivalent receptor binds ligand either anti-cooperatively or cooperatively, the midpoint of the titration, [H + does not correspond to a physical equilibrium constant. It can be shown that [H + = (1/k an-2nd *k an-1st ) 1/2 = (k dn-2nd *k dn-1st ) 1/2, sort of an average equilibrium constant. In this case, the only way to find exact values for k an-2nd and k an-1st with the expressions above is to generate a series of theoretical titration curves for different Ran values until one curve superimposes on the observed experimental titration data as shown on the next page. Here the ionization properties of a series of HOOC-(CH 2 ) x -COOH dicarboxylic acids of varying lengths ( x ) are compared based on pka values from R. P. Bell, "The Proton in Chemistry," 2nd ed. p. 96, Cornell University Press, Ithica, NY, 1973 Note that the values for Ran, pkdn1, kdn1, pkdn2, and kdn2 as indicated on this plot are identified as those describing the titration behavior of succinate (open triangles), which has 2 methylene carbons biivalent-ligand-binding.doc Page 6 of 9 Pages MCDB 108A, Fall 2004

7 separating the two carboxyl groups of this di-carboxylic acid. Titration profiles for several other dicarboxylates lead to following patterns emerge. A di-carboxylate with only one intervening methylene carbon (malonic acid, x = 1) exhibits a saturation curve with the greatest curvature (open squares) as compared to succinate (open triangles). The curve rises sharply as the ph rises from low ph values but flattens out as ph values increase to very high values. However, the saturation curves for di-carboxylates with greater separation (i.e., with x > 2 ) between carboxyl groups show less and less curvature in inverse proportion to x and these approach ( x >> 1) the saturation curve expected for acetic acid with only a single carboxyl group. (line marked with x symbols). However, even the titration curve for adipate, with x = 7 ((open circles) shows a small amount of curvature relative to the acetic acid titration curve. Ya as a function of [H+] for HOOC-(CH2)x-COOH Dicarboxylic Acids Length-Dependent, Anti-Cooperative Titration Behavior of HCOO-(CH 2 ) x -COOH Di-Carboxylic Acids Succinate, x = 2 H+ ion saturation of succinate: Ya Malonate, n = 1 Succinate, n = 2 Azelate, n = 7 Dicarboxylate, n >> 1 Ya (monovalent) Ya (R = 0.200) pkdn1 = < Ran < 1, Anti-Cooperative Ran = E (CH2) 2 - (COOH) 2 The predicted titration parameters are indicated E-06 pkdn2 = E E E E E E-03 [H+] F. Hill analysis of the length-dependent anti-cooperative proton binding by di-carboxylates An more rigorous and informative way to find values for k an-2nd and k an-1st is to generate a Hill plot by charting the value for log (Ya/Yd) against log [H + ]. The resulting line will have a slope at 50% saturation ), which is mathematically related to k an-2nd and k an-1st. As discussed in several of the following sections, the value of on a Hill plot is an extremely informative parameter for analyzing multivalent ligand binding reactions. For a bivalent receptor, the (Ya/Yd) ratio yields the following Hill equation: (Ya/Yd) = (Ran -1/2 *[H + ]*[H + + [H + ] 2 ) / (Ran -1/2 [H + ]*[H + + [H + ] ) biivalent-ligand-binding.doc Page 7 of 9 Pages MCDB 108A, Fall 2004

8 On a log-log plot with log (Ya/Yd) recorded along the Y-axis and log [H + ] recorded along the X-axis, it can be shown that the slope at 50% saturation for a line on the Hill plot for a bivalent receptor will equal: = 2*Ran +1/2 /(Ran +1/2 + 1) at Ya/Yd = 1 Using the mathematical relationships for the two experimentally determined parameters, and [H + if can be shown for a bivalent receptor that in general, k an-1st = (2 - *[H + ); and k an-2nd = /((2 - )*[H + ) Ya as a function of [H+] for HOOC-(CH2)x-COOH Dicarboxylic Acids Length-Dependent, Anti-Cooperative Titration Behavior of HCOO-(CH2)x-COOH Di-Carboxylic Acids Malonate, n = 1 Succinate, n = 2 Azelate, n = 7 Dicarboxylate, n >> 1 Acetic acid Ya/Yd (R = 0.200) slope (S@50% = 0.618) 0 < S@50% < 1, Anti-Cooperative S50% = 0.62 Succinate, x = 2 H+ ion saturation of succinate: (CH2) 2 - (COOH) 2 Ya/Yd < Ran < 1, Anti-Cooperative 0.60 pkdn1 = E E-05 pkdn2 = The predicted titration parameters are indicated E E E E E E-03 [H+] The line for is also shown by the red line. G. The Hill plot slope at 50% saturation and guidelines for interpreting Hill plots Several significant conclusions about a bivalent receptor can be made from the value for : 1. When > 1, Ran > 1 and k an-2nd /k an-1st > 1. By definition, ligand binding is cooperative with the 2 nd association constant for binding being greater than the 1 st association constant. 2. When < 1, Ran < 1 and k an-2nd /k an-1st < 1. By definition, ligand binding is anti-cooperative with the 2 nd association constant for binding being less than the 1 st association constant. 3. When = 1, Ran = 1 and k an-2nd /k an-1st = 1. By definition, ligand binding is non-cooperative with the 2 nd association constant for binding being equal to the 1 st association constant. 4. <= 2. Specifically, the Hill plot saturation can never exceed the valence (2) of a bivalent receptor. In effect, the valence of the receptor is the upper limit for maximum cooperativity for the receptor because the closer is to the valence, the more cooperative the reaction. biivalent-ligand-binding.doc Page 8 of 9 Pages MCDB 108A, Fall 2004

9 5. > 0. In other words, the Hill plot slope at 50% saturation is always > 0 but never = 0. In effect, the closer is to zero, the more anti-cooperative the reaction. H. Ionization properties of interacting catalytic carboxyl groups in aspartyl proteases biivalent-ligand-binding.doc Page 9 of 9 Pages MCDB 108A, Fall 2004

CHMI 2227 EL. Biochemistry I. Test January Prof : Eric R. Gauthier, Ph.D.

CHMI 2227 EL. Biochemistry I. Test January Prof : Eric R. Gauthier, Ph.D. CHMI 2227 EL Biochemistry I Test 1 26 January 2007 Prof : Eric R. Gauthier, Ph.D. Guidelines: 1) Duration: 55 min 2) 14 questions, on 7 pages. For 70 marks (5 marks per question). Worth 15 % of the final