Biomolecules Lecture 9 Binding and Acid Base PDF

B.Sc. Quantitative Biology Biomolecules Lecture 8 – Binding constants & acid/base equilibria Dr. James Birrell University of Essex, UK hhu.de  Last time  Thermodynamics is the study of energy changes  The 1st law states that t...

B.Sc. Quantitative Biology Biomolecules Lecture 8 – Binding constants & acid/base equilibria Dr. James Birrell University of Essex, UK hhu.de  Last time  Thermodynamics is the study of energy changes  The 1st law states that the total energy never changes i.e. ΔUsystem = -ΔUsurroundings  The 2nd law says that entropy always increases  ΔG = ΔH - TΔS  ΔG = ΔGo + RTlnΓ  Where Γ is the mass action ratio and Γ = Keq when ΔG = 0 2 hhu.de  Association/Dissociation reactions  Consider the following two processes: A + B → AB and AB → A + B Association reaction Dissociation reaction  There will be a ΔG, ΔGo and hence Keq for both of these reactions: [𝐴𝐵] 𝐴 [𝐵] ∆𝐺𝑎 = ∆𝐺𝑎𝑜 + 𝑅𝑇𝑙𝑛 ∆𝐺𝑑 = ∆𝐺𝑑𝑜 + 𝑅𝑇𝑙𝑛 𝐴 [𝐵] 𝐴𝐵 ∆𝐺𝑎𝑜 = −𝑅𝑇𝑙𝑛𝐾𝑒𝑞,𝑎 ∆𝐺𝑑𝑜 = −𝑅𝑇𝑙𝑛𝐾𝑒𝑞,𝑑 [𝐴𝐵]𝑒𝑞 [𝐴]𝑒𝑞 [𝐵]𝑒𝑞 𝐾𝑒𝑞,𝑎 = 𝐾𝑒𝑞,𝑑 = [𝐴]𝑒𝑞 [𝐵]𝑒𝑞 [𝐴𝐵]𝑒𝑞 1  It should be obvious that 𝐾𝑒𝑞,𝑎 = and therefore ΔGao = -ΔGdo and ΔGa = -ΔGd 𝐾𝑒𝑞,𝑑  It is only necessary to define one process and make that the conventional direction  We choose the dissociation direction  For a dissociation reaction we call Keq,d simply Kd – the dissociation constant 3 hhu.de  Association/Dissociation reactions  Dissociation reaction: AB ⇌ A + B [𝐴]𝑒𝑞 [𝐵]𝑒𝑞 𝐾𝑒𝑞,𝑑 = [𝐴𝐵]𝑒𝑞  We can use this model for many different processes  Ligand binding and dissociation from a protein or other macromolecule  Protonation/deprotonation  Conversion of two reactants to one product and conversion of one reactant to two products 4 hhu.de  Ligand binding to a protein  For a ligand (L) binding to a protein (P) PL ⇌ P + L [𝑃]𝑒𝑞 [𝐿]𝑒𝑞 𝐾𝑑 = [𝑃𝐿]𝑒𝑞  It is useful to define the number of ligands bound per protein: [𝑃𝐿]𝑒𝑞 𝑛𝐿 = [𝑃] 𝑒𝑞 +[𝑃𝐿]𝑒𝑞 [𝑃]𝑒𝑞 [𝐿]𝑒𝑞 [𝑃]𝑒𝑞 [𝐿]𝑒𝑞 𝐾𝑑  Since [𝑃𝐿]𝑒𝑞 = → 𝑛𝐿 = [𝑃]𝑒𝑞 [𝐿]𝑒𝑞 𝐾𝑑 [𝑃]𝑒𝑞 + 𝐾𝑑 𝐾  Multiply top and bottom by [𝑃]𝑑 gives: 𝑒𝑞 [𝐿]𝑒𝑞 𝑛𝐿 = 𝐾 Scatchard equation 𝑑 +[𝐿]𝑒𝑞  The fractional saturation (θ) is the number of occupied sites per protein (nL) over the total number of sites per protein (nT): 𝑛 𝜃 = 𝑛𝐿 𝑇  When there is only one site per protein: [𝐿]𝑒𝑞 𝜃 = 𝑛𝐿 = 𝐾 𝑑 +[𝐿]𝑒𝑞 5 hhu.de  The Hill coefficient  For binding of multiple ligands we can consider the situation as follows: PLn ⇌ P + nL 𝑛 [𝑃]𝑒𝑞 [𝐿]𝑒𝑞 𝐾𝑑,𝑜𝑏𝑠 = [𝑃𝐿𝑛 ]𝑒𝑞  Where the observed Kd is for saturation of all ligand binding sites  When considering an individual binding site we had: 𝑛𝑇 [𝐿]𝑒𝑞 𝑛𝐿 = 𝑛 𝑇 𝜃 = 𝐾 𝑑,𝑜𝑏𝑠 +[𝐿]𝑒𝑞  For binding of multiple ligands with one observed Kd value: 𝑛 𝑛𝑇 [𝐿]𝑒𝑞 𝑛𝐿 = 𝑛 𝐾𝑑,𝑜𝑏𝑠 + [𝐿]𝑒𝑞  Where n is the Hill coefficient  1 < n → positive cooperativity → binding of one ligand increase the affinity for additional ligands  n = 1 → no cooperativity → binding of one ligand does not affect binding of additional ligands  n < 1 → negative cooperativity → binding of one ligand decreases the affinity for additional ligands 6 hhu.de  Summary  Ligand binding to proteins is an association reaction  Defined, however, by a dissociation constant (Kd) [𝑃]𝑒𝑞 [𝐿]𝑒𝑞  𝐾𝑑 = [𝑃𝐿]𝑒𝑞  High Kd means lots of free protein and free ligand – weak binding  Low Kd means lots of protein:ligand complex – tight binding  ΔGo = -RTlnKd  Weak binding = high Kd = very negative ΔGo = ligand dissociation is spontaneous  Tight binding = low Kd = very positive ΔGo = ligand dissociation is not spontaneous = ligand binding  One protein, many ligands:  No cooperativity = ligand binding has no effect on other ligand binding  Positive cooperativity = ligand binding increases affinity for additional ligands  Negative cooperativity = ligand binding decreases affinity for additional ligands  Hill coefficient (n)  A measure of binding cooperativity  1 < n means positive cooperativity  n < 1 means negative cooperativity 7 hhu.de  Acids and bases  The ionic dissociation of water 2H2O ⇌ H3O+ + OH- 𝐻3 𝑂+ [𝑂𝐻 − ] 𝐻2 𝑂 2 𝐾= 𝐻3 𝑂+ 𝑜 [𝑂𝐻 − ]𝑜 𝐻2 𝑂 𝑜 2  [H2O] is close to that of a pure liquid in the standard state  ([H3O+]o and [OH-]o are 1 M  Therefore:  Kw = [H3O+][OH-]  This is the ionic dissociation constant or ionic product of water  Kw = 10-14 at 298 K  For pure water [H3O+] = [OH-] = 10-7 8 hhu.de  Arrhenius definition of an acid  A source of H+  Free H+ doesn’t really exist  Reacts with either OH- to make H2O or reacts with H2O to make H3O+ (hydronium ion)  Since Kw = [H3O+][OH-] = 10-14  If we increase [H3O+] we must decrease [OH-] and vice versa  For acidic solutions [H3O+] > [OH-]  For basic solutions [H3O+] < [OH-]  An acid can release H+ or consume OH-  A base can consume H+ or release OH- 9 hhu.de  Arrhenius definition  Definition of pH:  pH = -log10[H3O+]  [H3O+] = 10-pH  Acidic solutions  [H3O+] > 10-7 M → pH < 7  1 M H3O+ has a pH of 0  Basic solutions  [H3O+] < 10-7 M → pH > 7  1 M OH- has a pH of 14  Biological systems usually operate between pH 6 and 8 10 hhu.de  Conjugate acids and bases  Consider an acid (AH): AH + H2O ⇌ H3O+ + A-  A- is the conjugate base  Likewise, any base (B) will have a conjugate acid (BH+)  Acid dissociation: 𝐻3 𝑂+ [𝐴− ] [𝐻𝐴] 𝐻2 𝑂 𝐾𝑎 = 𝐻3 𝑂+ 𝑜 [𝐴− ]𝑜 𝐻𝐴 𝑜 𝐻2 𝑂 𝑜 𝐻2 𝑂  Again [H2O] is similar to a pure liquid so = 1 and [H3O+]o, [A-]o and [HA]o = 1 M 𝐻2 𝑂 𝑜 𝐻3 𝑂+ [𝐴− ] 𝐾𝑎 = [𝐻𝐴]  pKa = -log10Ka or Ka = 10-pKa  Likewise for a base (B): B + H2O ⇌ OH- + BH+ 𝑂𝐻 − [𝐵𝐻] 𝐾𝑏 = [𝐵]  pKb = -log10Kb or Kb = 10-pKb  Where pKb = 14 – pKa 11 hhu.de  Strong acids and bases  Strong acid:  pKa > 7  B + H2O → OH- + BH  Essentially 100% conversion of base to conjugate acid  E.g. NaOH, KOH 10−14  [OH-]eq = [B]initial, 𝐻3 𝑂+ 𝑒𝑞 = 𝐵 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 10−14  E.g. for 1 M NaOH, [OH-] = 1 M, 𝐻3 𝑂+ = = 10−14 and pH = -log10(10-14) = 14 𝑂𝐻 − 12 hhu.de  Weak acids and bases  Weak acid:  pKa < 7  AH + H2O ⇌ H3O+ + A-  Not 100% conversion of acid to conjugate base  E.g. Acetic (ethanoic) acid, citric acid  [H3O+]eq = Ka[HA]eq/[A-]eq ≈ √(Ka[HA]initial)  E.g. for 1 M acetic acid with pKa = 4.75 , [H3O+] ≈ √(10-4.75) = 4.22 x 10-3 M, pH = 2.38  Weak base:  pKa > 7  B + H2O ⇌ OH- + BH  Not 100% conversion of base to conjugate acid  E.g. Ammonia (NH3)  [OH-]eq = Kb[B]eq/[BH]eq ≈ √(Kb[B]initial)  [H3O+]eq = 10-14/[OH-]eq ≈ 10-14/√(Kb[B]initial)  E.g. for 1 M NH3 with pKa = 9.25, pKb = 14 – 9.25 = 4.75, [OH-] ≈ √(10-4.75) = 4.22 x 10-3 M, [H3O+] = 10-14/4.22 x 10-3 = 2.37 x 10-12 M, pH = 11.63 13 hhu.de  Composition as a function of pH  Henderson-Hasselbalch equation pH = pKa + log10([A-]/[AH])  The composition of an acid and conjugate base mixture will depend on the pH and the pKa value  Low pH = more H+ = more AH and less A-  High pH = less H+ = less AH and more A- 14 hhu.de  Polyprotic acids  An acid with multiple (de)protonation sites  E.g. phosphoric acid (H3PO4)  If the pKa are well separated then separate dissociations can be observed  At “equivalence points” the pH swaps from being determined by one equilibrium to being determined by another  If pKa values are close together then separate dissociations cannot be observed and a continuous change is observed 15 hhu.de  pI values and proteins  For molecules with multiple acidic and basic groups  E.g. proteins  Wide range of pKa values  At low pH there will be more protonated groups and overall more positive charge  At high pH there will be more deprotonated groups and overall more negative charge  In the middle there will be a point where the protein is completely uncharged  We call this the isoelectric point or pI value 16 hhu.de  pH buffers  A buffer is a solution that resists pH changes within a specific pH range  Equilibrium (1:1) mixture of the acid and conjugate base (or base and conjugate acid) AH + H2O ⇌ H3O+ + A-  +/- 1 pH unit around the pKa value  This can be made from essentially any molecule with a suitable pKa  What are they for?  Helping to maintain a constant pH  Some reactions are strongly pH dependent  Some proteins will denature under acidic/basic conditions  Buffering capacity (β)  Amount of acid/base added / change in pH  Highest value when pHinitial = pKa 17 hhu.de  Some other definitions of acids/bases  Brønsted-Lowry definition  Acids are H+ donors  Bases are H+ acceptors  This differs from the Arrhenius definition in that acids and bases do not require water to behave as acids or bases  E.g. NH3 + HCl → NH4Cl  All Arrhenius acids/bases are also Brønsted-Lowry acids/bases  Lewis definition  Acids are electron pair acceptors  Bases are electron pair donors  Consistent with MO theory  Acid can accept an electron pair into its lowest unoccupied molecular orbital (LUMO)  Base can donate an electron pair from its highest occupied molecular orbital (HOMO)  E.g. coordination of ligands to metals  Metal is Lewis acidic (accepts the electron pair), the ligand is Lewis basic (donates the electron pair) 18 hhu.de  Summary  Ligand binding/dissociation  Defined by dissociation constant (Kd)  Weak binding = high Kd, tight binding = low Kd  For many ligands can have cooperativity (negative or positive)  Hill coefficient (n > 1 is positive cooperativity, n < 1 is negative cooperativity)  Acids and bases  Arrhenius definition of an acid = source of H3O+ (hydronium ion)  pH = -log10[H3O+]  Acids increase [H3O+] and decrease pH  Bases decrease [H3O+] and increase pH  Kw = [H3O+][OH-] = 10-14  Strong acids = essentially fully deprotonated, strong bases = essentially fully protonated  [H3O+] = [acid] or 10-14/[base]  Weak acids/bases – need to consider the pKa value of the acid/base equilibrium 19 hhu.de  Summary  Acids and bases  Every acid has a conjugate base and every base has a conjugate acid  If an acid is strong then the conjugate base is weak and vice versa  Polyprotic acids = multiple pKa values  pI = isoelectric point = pH value where polyprotic acid is neutral  Buffers help maintain a relatively constant pH  Only useful around their pKa value  Buffer capacity tells us how good a buffer is at controlling the pH 20 hhu.de  Next time...  Redox reactions 21 hhu.de