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Lecture 10 1/31/14 University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2014 Lecture 10 1/31/14 A Non-Cooperative & Fully Cooperative Binding: Scatchard & Hill Plots • Assume N binding sitesWe have derived two equations for the extremes of behavior: Nk[ L] • Non-cooperative (i.e. Independent) Binding: ν = (10.1) 1+ kL[] • Fully-cooperative Binding: NkN [ L]NN NK[ L] ν == (10.2) 11++kLN []NN KL[] • In the contexts of practical experiments, a researcher may have no preconceived notion as to whether binding is non-cooperative or cooperative. There are two types of data plots however that are diagnostic for cooperative versus non-cooperative binding: • Equation 10.1 can be linearized into the Scatchard Equation ν =−Nk k ν (10.3) []L ν • Equation 10.3 means a plot of (i.e. y) as a function of ν (i.e. x) []L is a straight line with a slope of –N, a y-intercept of kN, and a x intercept of N. In Figure 10.1 shows a Scatchard Plot for N=4 and K=5. • • If a ligand binds cooperatively to a protein and obeys equation (10.2), a Scatchard plot will not be linear. Equation 10.2 can be linearized as follows. Calculate the quantity 1-fB: ν 1 11−=−=fB (10.4) N 1+ kL[]N • Now calculate the ratio of fB to 1-fB: NN fB KL[]/(1+ KL []) N ==N KL[] (10.5) 11/(1[])−+fKLb • Equation 10.5 is a form of the Hill Equation. The version of the Hill equation normally displayed as a plot is obtained by taking the logarithm of both sides of equation 10.5: ⎛⎞f ln⎜⎟B =+ lnKN ln[ L ] (10.6) ⎝⎠1− fB • Equation 10.6 means that if a ligand binds to a protein with full cooperativity, a plot of ln(fB/(1-fB) versus ln[L] yields a straight line with a slope of N and a y intercept of lnK. Such a plot is called a Hill Plot. B. Partial Cooperativity; Adair Equation • For non-cooperative binding a Hill plot will have slope = N=1. For fully cooperative binding the slope will be N>1. For Hemoglobin which is believed to bind 4 oxygen molecules cooperatively, we would expect a linear Hill plot with slope N=4. But the appearance of hemoglobin’s Hill plot is shown in Figure 10.2: • Myoglobin is an oxygen storage protein for which N=1. Myoglobin has a linear Hill plot with slope=1 as expected. • Hemoglobin (Hb) has a nonlinear Hill plot, shown in red in Figure 10.2. • The Hill plot of Hb has three distinct reagions, a situation that results because Hb binds oxygen with partial cooperativity. Figure 10.2: The oxygen storage protein myoglobin has N=1 and a linear Hill plot (black). Hemoglobin is an oxygen transport protein which has N=4 and a non-linear Hill plot (red). • Partial cooperativity means a solution of Hb is a mixture of unbound Hb, singly bound hemoglobin Hb-O2, doubly bound hemoglobin Hb-2O2, triply bound hemoglobin Hb-3O2 and filled hemobglobin Hb-4O2, but the binding constants for these Hb-O2 complexes are different. The binding affinity between the various forms of Hb and O2 increases as Hb fills its binding sites with oxygen, i.e. kkkk1234<<<. This means that Hb’s binding affinity is regulated by the amount of O2 bound, an effect called allosterism. • The Adair equation was the first equation to quantify Hb-O2 binding.The Adair equation is derived by writing out the binding polynomial for four binding sites with affinity constants kkkk1234< << Q=+ Hb14 k O + 6 kk O23 + 4 kkk O + kkkk O 4 (10.7) [ ]()12 [] 122 [] 1232 [] 12342 [] • In equation 10.7 the first term in the parenthesis is the amount of free hemoglobin [Hb]. The second term 4k1[L][Hb] is the amount of hemoglobin with one oxygen site bound etc. • From equation 10.7 we obtain the fraction of oxygen sites bound in Hb for a certain concentration of oxygen: kO++33 kkO23 kkkO + kkkkO 4 ν 1 []O ∂Q ( 1[] 2 12 [] 2 123 [] 2 1234 [] 2 ) f ==2 = (10.8) B 23 4 44QO∂[]2 14++k O 6 kk O + 4 kkk O + kkkk O ()1[] 2 12 [] 2 123 [] 2 1234 [] 2 • Equation 10.8 is the Adair equation and the constants kkkk1234< << can be adjusted to fit Hb’s non-linear Hill plot. This I smost easily seen by looking at the binding limits. • In the weak binding limit where [O2]<<1 equation 10.8 is ffBB⎛⎞ []OkOorkO21212<<1: ≈[] ln⎜⎟ = ln + ln[] (10.9) 11−−ffBB⎝⎠ • So in the WEAK binding limit the Hill plot is linear , slope=1, and the y- intercept is lnk1. • In the strong binding limit ffBB⎛⎞ []OkOorkO24242>>1: ≈[] ln⎜⎟ = ln + ln[] (10.10) 11−−ffBB⎝⎠ • So in the STRONG binding limit the Hill plot is linear , slope=1, and the y- intercept is lnk4. • In the intermediate region the Hill plot must be fit using the entire equation 10.8. The resulting slope is 2.9-3.5. C. Protein Allostery &: Pauling’s Sequential Model • The Adair equation can fit the Hill plot for Hb but it has four adjustable parameters and there is no physical insight as to why kkkk1234<<<. • Linus Pauling first proposed a sequential model for Hb allosterism where in Hb, O2 binding was enhanced as a result of pair-wise interactions between bound sites which are gradually increased in number by sequential binding of oxygen. • Pauling assumed that the oxygen binding sites occupied the vertices of a tetrahedron in Hb and thus are all equidistant. This allowed him to increased the O2 binding affinity of Hb as a simple function of the number of pair-wise interactions between occupied binding sites. • Assuming an equilibrium between Hb and Hb-O2, only a single site is bound in the product so no pair-wise interactions are present. Therefore ZZXk k1=k: Hb+⋅ O22YZZ Hb O • For the equilibrium between HbO2 and Hb-2O2, the product has one pair- wise interaction so that the affinity constant is enhanced by −ε0 /kTB −ε0 /kTB ke2 == kfk where fe= is the enhancement factor from a ZZZfkX single pair-wise interaction with energy ε0: Hb⋅+ O22 OYZZZ Hb ⋅2 O 2 • For the equilibrium between Hb-2O2 and Hb-3O2 −2/ε0 kTB 2 ke3 == kfkreflecting the two additional pair-wise interactions in ZZZXfk2 Hb3O2 versus Hb2O2: Hb⋅23 O22+⋅ OYZZZ Hb O 2 • For the equilibrium between Hb-3O2 and Hb-4O2 −3/ε0 kTB 3 ke4 == kfkreflecting the three more pairwise interactions in ZZZfk3 X Hb4O2 versus Hb3O2: Hb⋅+34 O22 OYZZZ Hb ⋅ O 2 Figure 10.3: The pairwise interactions between oxygen bound sites that enhance oxygen binding according to pauling’s Sequential Model. • Using Paulings hypothesis the four adjustable parameters in Adair’s equation are reduced to two adjustable parameters: k and f. The binding polynomial is QHbkOfkO=++14 623364234 + 4 fkO + fkO (10.11) [ ]() []22 [] [] 2 [] 2 o Note each term in the binding polynomial has f raised to the power of the number of pair-wise interactions in the Hb-O2 complex. For example, Figure 10.3 shows that in Hb where all four sites are filled with O2, there are 6 pairwise interactions so the fifth term in Q has contains f6. • With equation 10.11 the Adair equation becomes kO++33 fkO23364234 fk O + fk O ν ()[]22 [] [] 2 [] 2 fB == (10.12) 4 14++kO 6 fkO23364234 + 4 fk O + fk O ()[]22 [] [] 2 [] 2 • Equation 10.12 can be fitted to the Hill Plot in Figure 10.2 by adjusting f and k. D. Protein Allostery and Concerted Models a. Sequential Models assume oxygen binding sites are driven from weak to strong form by sequential addition of O2 to Hb. b. An alternative to sequential models are concerted models. Concerted models assume Hb exists in a form R where ALL binding sites are strong and form T where ALL binding sites are weak. R and T exist in equilibrium. All four O2 binding sites change together (i.e. in a concerted fashion) when R changes to T. Addition of O2 shifts the equilibrium from favoring T forms at low O2 levels to favoring R forms at high O2 levels. c. Monod-Wyman-Changeaux (MWC) Theory is a concerted model that was proposed as an explanation of cooperative oxygen binding in hemoglobin. X-ray studies have identified some intermediates ppredicted by MWC which indicate this model has validity. d. According to MWC theory, in the absence of oxygen , Hb exists in two K [T ] forms T and R that are in dynamic equilibrium RTKYZZZZZXZ ; = , []R • In the T state, all sites bind O2 weakly • In the R state, all binding sites bind O2 tightly o In the absence of oxygen , the T form is favored, i.e. K>>1 o As oxygen is added, the RL and RL2 forms are favored over TL and TL2. o MWC theory was demonstrated for 4 binding sites in Hb. For simplicity we only show results for two binding sites. In the two binding site model there are o K six protein forms: R YZZZZZXZ T TTLTLRRLRL,,22 ,,, . The MWC 77kkRT model proposes a dynamic exchange cK RLTLYZZZZZXZ between R and T forms as shown 77 below for two oxygen binding sites: kkRT cK2 RLTLYZZZZZZX 22 • The equilibria between the various forms of bound and unbound T are characterized by the equilibrium constant kT. The equilibria between the various forms of bound and unbound R are characterized by the equilibrium kT constant kR.
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