# Hill plot for binding of a ligand L to a protein

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marco corona on 18 Nov 2022
Answered: Star Strider on 20 Nov 2022
I'm trying to plot an hill function
I want to plot this logaritmic equation that is linear and calculate the n factor (Hill's coefficient=angular coeffient of the line)
I want to show that when K_1=10^-5 and K_2=10^-5 then 0<n<1 and when K_1=10^-3 and K_2=10^-5 then n>1

Star Strider on 20 Nov 2022
There’s too much missing information to provide a specific response. The K is not defined in the second equation, and there is not even a range or an order of magnitude for L.
Those considerations aside, it’s possible to get expressions for n (here ‘nval_1’ and ‘nval_2’) by simplifying and solving —
syms K K_1 K_2 L n
Y_L = (L^2+2*K_2*L) / (L^2 + 4*K_2*L + 2*K_1*K_2)
Y_L =
Eqn = log(Y_L / (1-Y_L)) == n*log(L)-n*log(K)
Eqn =
Eqn = exp(simplify(lhs(Eqn),500)) == simplify(exp(rhs(Eqn)),500)
Eqn =
Eqn = simplify(Eqn, 500)
Eqn =
ns(K_1,K_2,K,L) = solve(Eqn, n) % Solve For 'n'
ns(K_1, K_2, K, L) =
nval_1(K,L) = vpa(ns(1E-5,1E-5,K,L), 5)
nval_1(K, L) =
nval_2(K,L) = vpa(ns(1E-3,1E-5,K,L), 5)
nval_2(K, L) =
nfcn = matlabFunction(ns) % Anonymous Function Version
nfcn = function_handle with value:
@(K_1,K_2,K,L)-log((L.*(K_2.*2.0+L))./(K_2.*(K_1+L).*2.0))./(log(K)-log(L))
Use these functions with the appropriate values for K and L and you should be able to estimate the result for n.
The anonymous function ‘nfcn’ can be used outside the Symbolic Math Toolbox. Just copy it (you can do that directly from this page) and paste it to run it with the appropriate arguments to get estimates of n.
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