Decomposing Complex Cooperative Ligand Binding into Simple Components: Connections between Microscopic and Macroscopic models.

Transcription

1 Decomposing Complex Cooperative Ligand Binding into Simple Components: Connections between Microscopic and Macroscopic models. Alexey Onufriev 1 and G. Matthias Ullmann, 1 Department of Computer Science, 660 McBryde Hall, Virginia Tech, Blacksburg, VA onufriev@cs.vt.edu; Fax: (540) Structural Biology / Bioinformatics, University of Bayreuth, Universitätsstr. 30, BGI, Bayreuth, Germany, Matthias.Ullmann@uni-bayreuth.de; Fax: submitted to J. Phys. Chem. B. January 5, 004 Supporting Information 1

2 Details of the Derivation of Cooperativity Measure Ξ Here we investigate how the slope at the inflection point of decoupled binding curves can be used as a measure of cooperativity. We consider two limiting cases. The maximum cooperativity is achieved if either none or all N ligands bind, i.e. only two states exist: no ligands bound or all ligands bound. X = Ne β(gk Nµ L ) 1 + e β(gk Nµ L ) The first and second derivative of eq 1 are given in eq and 3, respectively. = βn e β(gk NµL) (1 + e β(gk Nµ L) ) () = β N 3 e β(gk NµL) (1 e β(gk NµL) ) (1 + e β(gk Nµ L) ) 3 (3) From eq 3 the inflection point µ I of eq 1 is given by eq 4. (1) = 0 : µ µ L = Gk I N (4) The slope at the inflection point is consequently given by eq 5 (µ I ) = βn 4 (5) Eq 5 gives the maximum possible slope for a binding curve of N ligands and one receptor. On the other extreme is a non-cooperative system of N non-interacting equivalent sites. The total binding curve of such a system is given by eq 6 The first and the second derivative of eq 6 are given by eqs 7 and 8. The slope at the inflection point is then given by eq 9. X = N e β(go i µ L) 1 + e β(go µ L ) (6) = N βe β(go µl) (1 + e β(go µ L) ) (7) = N β e β(go µl) (1 e β(go µl) ) (1 + e β(go µ L) ) 3 (8) (µ I ) = βn 4 (9)

3 The combination of eq 5 and eq 5 suggests the following cooperativity measure Ξ: Ξ = 4 (µ I ) βn (10) at the inflection point where the normalization factor is chosen so that for non-cooperative binding: Ξ = 1 and for fully cooperative binding: 1 < Ξ N Cooperativity in the system of two interacting sites. The total binding curve is X = e β(go 1 µ L) + e β(go µ L) + e β(go 1 +Go +W µ L) 1 + e β(go 1 µ L) + e β(go µ L) + e β(go 1 +Go +W µ L) (11) which can be rewritten with A = e βgo 1 + e βg o B = e β(go 1 +Go +W ) (1) Total binding curve λ = e βµ L X = First derivative of the Total binding curve is Second derivative of the Total binding curve is Aλ + Bλ (13) 1 + Aλ + Bλ = βλ(a + 4Bλ + ABλ ) (1 + Aλ + Bλ ) (14) = β λ(1 Bλ )(A B λ + 8Bλ + ABλ ) (1 + Aλ + Bλ ) 3 (15) The total binding curve has only one inflection point for cooperative binding curves µ I = 1 ln B = 1 β(go 1 + G o W ) = λ I = 1 B (16) The slope of the binding curve eq 13 at its inflection point is λi = βλ I(A + 4Bλ I + ABλ I ) (1 + Aλ I + Bλ I ) = The cooperativity measure is then given by β(4b + A B) 4B + 4A (17) B + A 4B + A B Ξ = 4B + 4A (18) B + A 3

4 Relation of Ξ to the Hill coefficient. For two sites one can show that Ξ is exactly equivalent to the Hill coefficient. Note, however, that this statement is generally not valid for N > as seen from the examples in the current work. Let Y be saturation of the molecule, i.e. Y = 1 X (19) N and consider only two interacting sites. The Hill coefficient n Hill is defined as slope of the Y/(1 Y ) curve when the receptor is half saturated. From eq 11 one can find that n Hill = Y 1 Y at µ I = 1 ln B λ I = 1 B (0) Y 1 Y = Aλ + Bλ + Aλ (1) From comparison with eq 18 one see that Y 1 Y = λ(a + 4Bλ + ABλ ) ( + Aλ) () n Hill = Y 1 Y µi (3) = (A B + 4B) 4B + 4A B + A (4) n Hill = Ξ for N =. (5) Adair equation re-written using microscopic constants. For the sake of simplicity, we restrict the discussion to the N=4 ligand binding sites. We choose N=4 mainly due to the importance of the hemoglobin test case. The total ligand binding curve of a molecule that can bind N ligands of the same type is given by: R + 4 L K 1 RL + 3 L K RL + L K 3 RL 3 + L K 4 RL 4 (6) The species RL i is the macrostate of the receptor with i ligands bound. The total binding curve X is in general described by eq 7. X = a 1[L] + a [L] + 3 a 3 [L] a 4 [L] a 1 [L] + a [L] + a 3 [L] 3 + a 4 [L] 4 (7) 4

5 In terms of macroscopic binding constants, we have a 1 = K 1 a = K 1 K a 3 = K 1 K K 3 (8) a 4 = K 1 K K 3 K 4 In terms of microscopic binding constants, we have a 1 = k k k k a = k1000k k1000k k1000k k0100k k0100k k0010k a 3 = k0001k k k0001k k k0001k k k1000k k a 4 = k1000k k k (9) Different combinations of microscopic constants can yield the same coefficients a i. Adair 1 3 developed a model in which it is assumed that all of the ligands bind with the same affinity to the same macrostate, i. e., k 1 = k = k , k = k = k =..., etc. This assumption implies that all binding sites are equivalent. The coefficients in eq 7 are then given by a 1 = 4k 1 a = 6k 1 k a 3 = 4k 1 k k 3 (30) a 4 = k 1 k k 3 k 4 Comparing eq 8 with eq 30, one finds k 1 = 1 4 K 1 k = 3 K k 3 = 3 K 3 (31) k 4 = 4K 4 The Adair model is equivalent to eq 7 considering eq 31. Thus, the Adair model with N binding sites can always fit a binding curve of Nth degree. However, from a good fit to the Adair equation one can not conclude that binding constants for binding of the ith ligand are all equal. References [1] Adair, G. S. (195) J. Biol. Chem. 63,

6 [] Chein, J. C. W & Mayo, K. H. (1980) J. Biol. Chem. 5, [3] Voet, D & Voet, J. G. (1995) Biochemistry. (New York), edition. 6