Lecture 10 1/31/14 University o Washington Department o Chemistry Chemistry 453 Winter Quarter 014 A on-cooperative & Fully Cooperative inding: Scatchard & Hill Plots Assume binding siteswe have derived two equations or the extremes o behavior: k [ L] on-cooperative (i.e. Independent) inding: ν = (10.1) 1 + k[ L] Fully-cooperative inding: k L K L ν = = k L K L (10.) In the contexts o practical experiments, a researcher may have no preconceived notion as to whether binding is non-cooperative or cooperative. There are two types o data plots however that are diagnostic or cooperative versus non-cooperative binding: Equation 10.1 can be linearized into the Scatchard Equation ν = k k ν (10.3) L ν Equation 10.3 means a plot o [ L] (i.e. y) as a unction o ν (i.e. x) is a straight line with a slope o, a y-intercept o k, and a x intercept o. In Figure 10.1 shows a Scatchard Plot or =4 and K=5.
I a ligand binds cooperatively to a protein and obeys equation (10.), a Scatchard plot will not be linear. Equation 10. can be linearized as ollows. Calculate the quantity 1- : ν 1 1 = 1 = (10.4) 1 + k[ L] ow calculate the ratio o to 1- : KL /(1 + KL ) = = KL (10.5) 1 b 1/(1 + K[ L] ) Equation 10.5 is a orm o the Hill Equation. The version o the Hill equation normally displayed as a plot is obtained by taking the logarithm o both sides o equation 10.5: ln = ln K + ln[ L] (10.6) Equation 10.6 means that i a ligand binds to a protein with ull cooperativity, a plot o ln( /(1- ) versus ln[l] yields a straight line with a slope o and a y intercept o lnk. Such a plot is called a Hill Plot.. Partial Cooperativity; Adair Equation For non-cooperative binding a Hill plot will have slope = =1. For ully cooperative binding the slope will be >1. For Hemoglobin which is believed to bind 4 oxygen molecules cooperatively, we would expect a linear Hill plot with slope =4. ut the appearance o hemoglobin s Hill plot is shown in Figure 10.: Myoglobin is an oxygen storage protein or which =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.. The Hill plot o Hb has three distinct reagions, a situation that results because Hb binds oxygen with partial cooperativity. Figure 10.: The oxygen storage protein myoglobin has =1 and a linear Hill plot (black). Hemoglobin is an oxygen transport protein which has =4 and a non-linear Hill plot (red).
Partial cooperativity means a solution o Hb is a mixture o unbound Hb, singly bound hemoglobin Hb-O, doubly bound hemoglobin Hb-O, triply bound hemoglobin Hb-3O and illed hemobglobin Hb-4O, but the binding constants or these Hb-O complexes are dierent. The binding ainity between the various orms o Hb and O increases as Hb ills its binding sites with oxygen, i.e. k1 < k < k3 < k4. This means that Hb s binding ainity is regulated by the amount o O bound, an eect called allosterism. The Adair equation was the irst equation to quantiy Hb-O binding.the Adair equation is derived by writing out the binding polynomial or our binding sites with ainity constants k1 < k < k3 < k4 ( 3 4 1 1 1 3 1 3 4 ) 1 4 6 4 Q= Hb + k O + kk O + kkk O + kkkk O (10.7) In equation 10.7 the irst term in the parenthesis is the amount o ree hemoglobin [Hb]. The second term 4k 1 [L][Hb] is the amount o hemoglobin with one oxygen site bound etc. From equation 10.7 we obtain the raction o oxygen sites bound in Hb or a certain concentration o oxygen: [ O ] 3 4 ( k1[ O] 3k1k[ O] 3k1kk3[ O] k1kk3k4[ O] ) ν 1 Q + + + = = = 4 4 Q [ O ] 4k1O + 6kk 1 O + 4kkk 1 3O + kkkk 1 3 4O 3 4 ( ) Equation 10.8 is the Adair equation and the constants k1 < k < k3 < k4 can be adjusted to it Hb s non-linear Hill plot. This I smost easily seen by looking at the binding limits. In the weak binding limit where [O ]<<1 equation 10.8 is [ O] << 1: k1[ O] or ln = ln k ln[ O] (10.9) 1 So in the WEAK binding limit the Hill plot is linear, slope=1, and the y- intercept is lnk 1. In the strong binding limit [ O] >> 1: k4[ O] or ln = ln k4 + ln[ O] (10.10) 1 So in the STOG binding limit the Hill plot is linear, slope=1, and the y- intercept is lnk 4. In the intermediate region the Hill plot must be it using the entire equation 10.8. The resulting slope is.9-3.5. C. Protein Allostery &: Pauling s Sequential Model The Adair equation can it the Hill plot or Hb but it has our adjustable parameters and there is no physical insight as to why k1 < k < k3 < k4. Linus Pauling irst proposed a sequential model or Hb allosterism where in Hb, O binding was enhanced as a result o pair-wise interactions (10.8)
between bound sites which are gradually increased in number by sequential binding o oxygen. Pauling assumed that the oxygen binding sites occupied the vertices o a tetrahedron in Hb and thus are all equidistant. This allowed him to increased the O binding ainity o Hb as a simple unction o the number o pair-wise interactions between occupied binding sites. Assuming an equilibrium between Hb and Hb-O, only a single site is bound in the product so no pair-wise interactions are present. Thereore k 1 =k: Hb + O k Hb O For the equilibrium between HbO and Hb-O, the product has one pairwise interaction so that the ainity constant is enhanced by ε0 / kt 0 k = e k = k where / kt = e ε is the enhancement actor rom a k single pair-wise interaction with energy ε 0 : Hb O + O Hb O For the equilibrium between Hb-O and Hb-3O ε0 / kt k3 = e k = krelecting the two additional pair-wise interactions in k Hb3O versus HbO : Hb O + O Hb 3O For the equilibrium between Hb-3O and Hb-4O 3 ε0 / kt 3 k4 = e k = k relecting the three more pairwise interactions in Hb4O versus Hb3O : Hb 3O + O Hb 4O 3 k Figure 10.3: The pairwise interactions between oxygen bound sites that enhance oxygen binding according to pauling s Sequential Model. Using Paulings hypothesis the our adjustable parameters in Adair s equation are reduced to two adjustable parameters: k and. The binding polynomial is 3 3 3 6 4 4 Q= Hb 4k O + 6k O + 4 k O + k O (10.11) ( ) o ote each term in the binding polynomial has raised to the power o the number o pair-wise interactions in the Hb-O complex. For example,
Figure 10.3 shows that in Hb where all our sites are illed with O, there are 6 pairwise interactions so the ith term in Q has contains 6. With equation 10.11 the Adair equation becomes 3 4 ν ( k[ O ] + 3k ) O + 3 3 k 3 O + 6 k 4 O = = (10.1) 3 4 4 ( 4k[ O ] + 6k ) O + 4 3 k 3 O + 6 k 4 O Equation 10.1 can be itted to the Hill Plot in Figure 10. by adjusting and k. D. Protein Allostery and Concerted Models a. Sequential Models assume oxygen binding sites are driven rom weak to strong orm by sequential addition o O to Hb. b. An alternative to sequential models are concerted models. Concerted models assume Hb exists in a orm where ALL binding sites are strong and orm T where ALL binding sites are weak. and T exist in equilibrium. All our O binding sites change together (i.e. in a concerted ashion) when changes to T. Addition o O shits the equilibrium rom avoring T orms at low O levels to avoring orms at high O levels. c. Monod-Wyman-Changeaux (MWC) Theory is a concerted model that was proposed as an explanation o cooperative oxygen binding in hemoglobin. X-ray studies have identiied some intermediates ppredicted by MWC which indicate this model has validity. d. According to MWC theory, in the absence o oxygen, Hb exists in two K [ T ] orms T and that are in dynamic equilibrium T; K =, [ ] In the T state, all sites bind O weakly In the state, all binding sites bind O tightly o In the absence o oxygen, the T orm is avored, i.e. K>>1 o As oxygen is added, the L and L orms are avored over TL and TL. o MWC theory was demonstrated or 4 binding sites in Hb. For simplicity we only show results or two binding sites. o In the two binding site model there are K six protein orms: T TTLTL,,, LL.,, The MWC k kt model proposes a dynamic exchange ck L TL between and T orms as shown below or two oxygen binding sites: k kt ck L TL The equilibria between the various orms o bound and unbound T are characterized by the equilibrium constant k T. The equilibria between the
various orms o bound and unbound are characterized by the equilibrium kt constant k. The ratio C = << 1 because binds O more strongly than T. k ote that as more and more oxygen is added more L, L, TL, and TL are ormed. ut because C<1 then C K<CK<K, and the equilibria between T and orms shits rom avoring T over to avoring L over TL. Assume two binding sites on T and we start with the raction o sites bound: [ L] + [ L] + [ TL] + [ TL] ν = (10.13) [ ] + [ T] + [ L] + [ L] + [ TL] + [ TL] [ T ] kt Substitute K = and C [ ] = k to obtain ater some algebra the Hill equation or the MWC model: Ck [ O ] KC k [ O ] = k [ O ] (10.14) 1 Ck [ O ] K k [ O ] Equation 10.14 seems complicated but it also explains the Hill Plot or Hb. O <<. Then: Assume the weak binding limit where 1 KC k [ O ] 1 K (10.15) ote K>>1 because the T orm is avored. Then 1 kco (10.16) or ln = ln ( kc ) + ln[ O] = ln kt + ln[ O] (10.17) According to equation 10.17 the Hill Plot is linear at low [O ] with slope = 1 and intercept ln k T In the high oxygen concentration limit [ O ] >> 1 KC k [ O ] 1 KC (10.18) has a higher binding ainity so C<<1. Thereore the Hill equation O >> : becomes in the limit [ ] 1
ln = ln k + ln[ O ], i.e. slope = 1 and intercept ln k These limiting equations, together the general equation that is eective at intermediate ligand concentrations, yields the Hill plot below. Although the sequential and concerted models both explain the Hill Plot data or Hb-O binding, the simple orm o the MWC concerted theory and the act that many o its proposed intermediates have been identiied by crystallography have caused this theory to be avored over sequential theories. y the late 1990 s some o the intermediates proposed on sequential models had been detected with the result that the real model or binding between O and Hb is likely a hybrid theory o the sequential and concerted models.