Lecture 2: Receptor-ligand binding and cooperativity Paul C Bressloff (Spring 209) A biochemical receptor is a protein molecule that receives a chemical signal in the form of ligand molecules. The ligands bind to the receptor and change its conformational state, initiating some form of cellular response. Hence, receptor-ligand interactions play a major role in environmental sensing, signal transduction, and cell to cell signaling. 2. Cooperative binding Consider a set of receptors R in the plasma membrane of a cell that can bind to ligand L in the extracellular environment, see Fig. 4. (This is mathematically identical to the problem of transcription factor binding to operator sites of a gene s promoter.) For the moment we assume that the receptor has a single binding site, and that the ion channel is only open when the binding site is occupied by a ligand. The corresponding reaction scheme is taken to be R + L k + k RL, where α and β are the binding and unbinding rates, respectively. If the number of ligands in the environment is large relative to the number of receptors, then c = [L] is approximatively constant and this reaction scheme is well approximated by R ck + k RL. If the number of receptors is sufficiently large and fluctuations in the ligand concentration are negligible, then one use equilibrium thermodynamics and the law of mass action to determine the fraction of open ion channels. In the case of the simple reaction scheme R + L LR, the law of mass action gives [LR] [R][L] = K d. k + k - Figure 4: Population of ligand-gated ion channels in the plasma membrane of a cell. Ligands in the extracellular domain can bind to a receptor at a rate k + and unbind at a rate k.
fraction of open receptors.0 n = 4 n = 3 n = 2 n = 0.5 K n ligand concentration Figure 5: Cooperative binding model. Plot of fraction of open receptors as a function of ligand concentration for various n, where n is the number of binding sites. Assuming that the total number of receptors is fixed, [R] + [LR] = [R tot ], we have which on rearranging gives [LR] (R tot ] [LR])[L] = K d, [LR] [R tot ] = [L]. (2.) [L] + K d The fraction of bound receptors increases linearly with [L] at low ligand concentrations but saturates at high concentrations for which [L] K d. A sharper dependence on [L] can be obtained if there is some form of cooperative binding. The latter refers to situations in which a receptor has multiple binding sites, which can influence each other. An extreme example is when a receptor has n binding sites such that mutual interactions force all of the binding sides to be either simultaneously occupied or simultaneously empty. This can be represented by the reaction scheme R 0 + nl R n, where R 0 denotes a receptor with empty binding sites and R n denotes a receptor with all sites filled. The law of mass action shows that at equilibrium [R n ] [L] n [R 0 ] = K n, where K n is an effective dissociation rate. Note that since the forward reaction involves n ligands, one has to include the factor [L] n. Again setting [R n ] + [R 0 ] = [R tot ] and rearranging gives [R n ] [R tot ] = [L] n [L] n. (2.2) + K n The dependence of the fraction of open ion channels as a function of [L] and n is illustrated in Fig. 5. Page 8
2.2 Monod-Wyman-Changeux (MWC) model The above model of receptor-ligand binding is unrealistic in at least two aspects: i) The binding to multiple sites is not all-or-none, that is, a fraction of sites can be occupied at any one time. ii) It is possible for the ion channel to be either open or closed in each binding state - changes in binding state shift the balance between open and closed. A more realistic model of a ligand-gated ion channel with cooperative binding has been developed for the nicotinic acetylcholine receptor, which is found at the neuromuscular junction, and also cgmp ion channels that amplify signals in photoreceptors. It is analogous to the classical Monod- Wyman-Changeux (MWC) model of dimoglobin. The nicotinic receptor has two binding sites for acetylcholine and the equilibrium between the open and closed state of the channel is shifted to the open state bybinding of acetylcholine, see Fig. 6: a) T j denotes a closed receptor with j occupied sites and R j denotes a receptor in the corresponding open state. b) K T and K R are the equilibrium constants (inverse of the dissociation constant) for binding of an acetylcholine molecule to an individual site of a closed and an open receptor, respectively. The additional factor of 2 takes into account the fact that there are two unoccupied sites in the forward reaction T 0 T, whereas the additional factor of /2 takes into account the fact that there are two occupied sites in the backward reaction T 2 T (and similarly for R j ). c) Finally, Y j is the equilibrium constant associated with the opening and closing of a receptor with j occupied sites. Applying the law of mass action to each of the reversible reactions leads to the following set of a b 2K T K T /2 T 0 + L T + L T 2 Y 0 Y Y 2 R 0 + L R + L R 2 2K R K R /2 CLOSED OPEN Figure 6: The MWC model of nicotinic acetylcholine receptor with two binding sites. (a) Schematic illustration of different conformational states distinguished by the number of occupied binding sites and whether the ion channel is open or closed. (b) Reaction diagram. Page 9
equations for the concentrations: [R i ] [T i ] = Y i (2.3a) [T ] [L][T 0 ] = 2K T, [T 2 ] [L][T ] = K T /2, [R ] [L][R 0 ] = 2K R [R 2 ] [L][R ] = K R/2 We are interested in the fraction of receptors that are in the open state, which is [R 0 ] + [R ] + [R 2 ] [R 0 ] + [R ] + [R 2 ] + [T 0 ] + [T ] + [T 2 ]. Equations (2.3b,c) can be used to express [T j ] and [R j ] in terms of [T 0 ] and [R 0 ]: [T ] = 2K T [L][T 0 ], [T 2 ] = (K T [L]) 2 [T 0 ], [R ] = 2K R [L][R 0 ], [R 2 ] = (K R [L]) 2 [R 0 ]. Substituting these results into the formula for p open and using (2.3a) gives We now observe that when [L] = 0, (2.3b) (2.3c) Y 0 ( + K R [L]) 2 Y 0 ( + K R [L]) 2 + ( + K T [L]) 2. (2.4) p open (0) = + /Y 0, whereas when [L] is large p open ([L]) + (K T /K R ) 2 (/Y 0 ). Figure 7: Plot of fraction of open receptors as a function of ligand concentration for the MWC model with two binding sites. Shown is an example of a curve fitted to data from a cgmp ion channel. Page 0
It follows that if the open receptor has a higher affinity for binding acetylcholine than the closed receptor (K R > K T ) then p open ([L]) > p open (0). An interesting feature of MWC type models is that activation of the receptor, as specified by p open, is a sigmoidal function of ligand concentration [L]. Thus binding is effectively cooperative even though there are no direct interactions between binding sites. Finally, note that it is straightforward to generalize the MWC model to the case of n binding sites. Defining the fraction of open receptors according to n j=0 [R j] n j=0 [R j] + n j=0 [T j] the law of mass action gives Y 0 ( + K R [L]) n Y 0 ( + K R [L]) n + ( + K T [L]) n. (2.5) The MWC model has emerged as a general mechanism for receptor-ligand interactions within a diverse range of applications, including ion channel gating, chemotaxis, and gene regulation. 2.3 Ising model Consider a D array of N receptors such that receptor-ligand complexes on neighboring sites can interact, see Fig. 8. (An analogous problem arises for transcription factors binding to operator sites of a promoter.) If a receptor is bound by a ligand but the adjacent receptors are unbound, then there is a reduction in free energy given by a µ, where a is the binding energy and µ is the chemical potential of the ligand. However, due to interactions, when two adjacent receptors are simultaneously occupied by ligands, there is an additional reduction in free energy given by J. We can express this mathematically by introducing the occupation states σ n, n =,..., N, with σ n = if the nth receptor is occupied and σ n = 0 it is is unoccupied. The total free energy for a given configuration σ = (σ,..., σ N ) is then given by E[σ] = (a + µ) N σ n J n= The corresponding Boltzmann-Gibbs distribution is N n= σ n σ n+. p(σ) = Z e E[σ]/k BT, (2.6) -2a - J -a -a Figure 8: D array of membrane receptors with nearest neighbor interactions. An isolated bound receptor has lower free energy a, whereas a pair of adjacent bound receptors each have a lower free energy 2a J with J representing the strength of cooperative interactions. Page
and the partition function is Z = σ =0, σ N =0, e α N j= σ j+γ N j= σ jσ j+, (2.7) with α = (a + µ)/k B T and γ = J/k B T. The partition function can be treated as a generating function for the mean receptor occupancy, that is, σ := ( ) N p(σ) σ n = d ln Z[α]. (2.8) N N dα σ n= The above model is equivalent to the classical Ising model of a D lattice of magnetic spins s n = σ n /2 = ±. (The latter can also be interpreted as a model of a D polymer). A well-known result from statistical mechanics is that the D Ising model can be solved exactly. In particular, one can derive an exact expression for Z using transfer matrices. First, it is convenient to take the sites to be arranged on a ring by making the identification σ N+ = σ. (This yields a good approximation for large N, since boundary effects at the ends of a D array are negligible.) We then rewrite Z in the more suggestive form Z =... [ ] [ ] ] e α(σ +σ 2 )/2+γσ σ 2 e α(σ 2+σ 3 )/2+γσ 2 σ 3... [e α(σ N +σ )/2+γσ N σ σ σ N We can view each term on the right-hand side as the element of a matrix T with matrix elements labeled by σ, σ 2 etc., that is, T σ σ 2 = e α(σ +σ 2 )/2+γσ σ 2. Hence. ( ) ( T T T = 0 e α+γ e = α/2 ) T 0 T 00 e α/2. In terms of the transfer matrix T Z = σ... σ N T σ σ 2 T σ2 σ 3... T σn σ = Tr[T N ], where we have used the standard rules of matrix multiplication. The eigenvalues of T satisfy the characteristic equations (e α+γ λ)( λ) e α = 0, which gives the pair of eigenvalues λ = λ ± with λ ± = [(e α+γ + ) ± ] (e 2 α+γ + ) 2 + 4e α. Since T is a symmetric matrix, it can be shown that Z = Tr[T N ] = λ N + + λ N λ N + Since λ + > λ, it follows that for large N, we have λ N + λ N. Hence, σ = N d dα ln λn + = d dα ln λ + = dλ + λ + dα. A plot of σ against k B T α = a + µ yields a sigmoid function, indicative of cooperativity - recall from lecture that e µ/k BT is proportional to the ligand concentration. Page 2