Elementary Binding Equations and Related Equations in Biochemistry

Transcription

1 Elementary Binding Equations and Related Equations in Biochemistry Lerson Tanasugarn Department of Biochemistry, Faculty of Science, Chulalongkorn University, Bangkok, Thailand. 1

2 Equations Covered in this Lecture Hill equation (cooperativity) Scatchard equation (binding) Sips plot (homogeneity of binding constants) Michaelis-Menten equation (enzyme kinetics) Langmuir adsorption isotherm (adsorption, binding) Lineweaver-Burk plot Equations not covered Eadie-Hofstee plot Adair equation Klotz equation Pauling equation etc. 2

3 Other Applications of the Hill equation that are Not Covered Here renal uptake of aminoglycosides by proximal tubular cells tubuloglomerular feedback of the glomerular filtration in the kidney ligand binding on the conductance of voltagedependent ion channels pharmacokinetic-pharmacodynamic (PK-PD) modeling 3

4 References: Dahlquist, F.W The meaning of Scatchard and Hill plots. Methods in Enzymology 48: Goulelle, S. et al The Hill equation: a review of its capabilities in pharmacological modeling. Fundamental & Clinical Pharmacology 22: Nelson, D. L. & M. M. Cox Lehninger Principles of Biochemistry, 4th Edition. W. H. Freeman & Co. Chapter 5, Section 5.1: Reversible Binding of a Protein to a Ligand: Oxygen-binding Protein, pp Sabouri, A.A. & A.A. Moosavi-Movahedi Evaluation of the Hill Coefficient from Scatchard and Klotz Plots. Biochem. Educ. 22(1): Scatchard, G The attraction of proteins for small molecules and ions. Annals of the New York Academy of Sciences. 51: Sips, R On the structure of a Catalyst Surface. J. Chem. Phys. 16(5): Voet, D. and J. G. Voet Biochemistry. Chapter 10: Hemoglobin: Protein function in microcosm, 10.1 Hemoglobin and myoglobin function, pp Weiss, J.N The Hill equation revisited: Uses and misuses. The FASEB Journal 11:

5 Myoglobin binds oxygen at its heme group. 5

6 Oxygen Saturation Curve of Myoglobin Dissociation constant K = Mb MbO 2 Fractional saturation θ= θ = Mb + O 2 MbO 2 [ ][ O 2 ] [ ] [ MbO 2 ] [ Mb]+ [ MbO 2 ] = [ Mb] [ O 2 ] K [ Mb]+ [ MbO 2 ] [ O 2 ] [ O [ K + K MbO = 2 ] 2 ] K + [ O 2 ] [ Mb] { } Since O 2 is a gas; Y O2 = θ= po 2 K + po 2 Define p 50 as the value of po 2 when Y O2 equals = p 50 K + p 50 ; K = p 50 po So, Y O2 = θ= 2 p 50 + po 2 6

7 1904 Christian Bohr s Concept of Cooperative Binding Christian Bohr ( ) Danish physician, physiologist father of Niels Bohr (Nobel prizefor studies in atomic structure and quantum mechanics Wikipedia: the Free Encyclopedia studied the oxygen binding to hemoglobin Wikipedia: the Free Encyclopedia The Bohr Effect : H + and CO2 heterotropically decreases the O2 binding affinity of hemoglobin 7

8 Nelson & Cox Lehninger Biochemistry 4th ed. The Bohr Effect In the muscle, H + from the bicarbonate formation is soaked up by hemoglobin, which is forced to release more oxygen to the muscle. In the lung, the high po2 causes more and more oxygen binding to hemoglobin, which is now forced to release the Bohr protons so that the reaction is reversed and bicarbonate is broken down to carbon dioxide. Nelson & Cox Lehninger Biochemistry 4th ed. 8

9 Archibald Vivian Hill ( ) exercise physiology energy metabolism in muscle chemical -> mechanical energy conversion A.V. Hill ( ) English physiologist 1922 Nobel Prize in Physiology or Medicine (shared with Meyerhof) Founders of Biophysics A.V. Hill Bassett, D.R. Jr Scientific contribution of A. V. Hill exercise physiology pioneer. J. Appl. Physiol. 93: Herman von Helmholtz ( ) 9

10 1910 The Hill Equation Mechanistic - Not really Curvefitting - Yes fitting to the Sigmoidal Curve Fractional saturation Hill coefficient Rearrangement yields 10

11 Oxygen Saturation Curves of Hemoglobin is Sigmoidal Fractional saturation Hill coefficient = slope Rearrangement yields 11

12 log-log plot of the logarithmic form of the Hill Equation n = Hill coefficient = slope positive cooperation, n>1 negative cooperation, n<1 no cooperation, n=1 dissociation constant = 10-6 In case of hemoglobin, n is between 2.8 and 3.0 This is okay since Hill equation was used as a graph fitter. 12

13 Myoglobin: n=1 hyperbola 13

14 General Form of Hill Equation General Myoglobin Hill 1 n fractional saturation ymax 1 1 x This form of equation will appear again and again for systems under dynamic equilibrium Goutelle et al The Hill equation: a review of its capabilities in pharmacological modelling. Fundamental & Clinical Pharmacology. 22:

15 A Simple Case (1) Fractional saturation (2) (2)/(1) fraction bound divided by free ligand concentration Ka slope= -Ka 1 fraction bound 15

16 Scatchard Plot change 1 on the righthand side of the equation from last slide to n, and we have the Scatchard Plot slope= -Ka n association constant number of binding sites Do not forget the Equilibrium Assumption 16

17 Examples of Applicable Systems Generic Symbol general Ab-Ag Receptor Cyclodextrin [P] protein antibody receptor cyclodextrin [L] [PL] [P]+[PL], Y free small molecule bound small molecule total protein concentration fractional saturation free antigen c free ligand bound antigen bound ligand total antibody concentration r r/c total receptor concentration free guest molecule bound guest molecule total CD concentration 17

18 Experimental Separation of free and bound species by Equilibrium dialysis Sedimentation Column chromatography etc. 18

19 An example: Ab-Ag If excess antigen is added, [antigen]=[l]>>0 ɵ/[l] 0 ɵ=n = number of antigen binding sites per one antibody molecule The y-intercept in the graph is about 10 6 l/mole. n=2 In general, 10 5 < Ka <10 10 l/mole. Why do you think Ka from the graph is not very high? 19

20 Karush, Scatchard Plot is Not Always a Straight Line There may be several kinds of receptors with different binding constants. 20

21 Let K be a distribution of values about a mid value called K0 Normal distribution ends up with a messy integral equation that needs to be solved numerically Sips first distribution function (Sips 1948) leads to a simple solution (next slide)

22 Sips Plot - Uniformity of Bindings a=index of dispersion of K about K0 This graph shows the Sips plots of the serum samples taken at four different times from a single rabbit immunized with the Type III pneumococcal polysaccharide. The average association constant, K0, rose from 1.6 x 104 M 1 at week 1 to 6 x 105 M 1 at week 25. Affinity maturation reflects an adaptive response to antigen exposure. As time goes by, the antibodies produced are able to bind antigen more tightly and to deal more efficiently with the antigen. BiologyPages/A/Affinity.html Hohman et al PNAS USA 77(12):

23 Sips Plot ~ Hill Plot Sips plot Index of dispersion a=1 means uniform K Recall Hill plot Hill coefficient n=1 means no cooperativity 23

24 1916 Langmuir Adsorption Isotherm An adsorption isotherm relates adsorption of molecules on a solid surface to gas pressure or concentration of the molecules in the medium that is in contact with the surface. equilibrium between empty surface sites, S* particles, P and filled particle sites, SP Association constant compare this with Rewrite this equation as fractional saturation or fractional coverage Irving Langmuir American chemist & physicist doctoral advisee of Walther Nernst GE Research Lab Nobel Prize in Chemistry for work in surface chemistry (Wikipedia Encyclopedia) gas pressure or concentration 24

25 Other Isotherms 25

26 Isotherms for Cyclodextrins 26