Statistical Thermodynamics Lecture 15 Binding equilibria II DDM
Statistical Thermodynamics Lecture 15: Binding equilibria II: DDM, FEP, BEDAM
General chemical reaction: Very important type of “reaction”: bimolecular non-covalent binding R(sol) + L(sol) RL(sol) Small molecule dimerization/association Supramolecular complexes Protein-ligand binding Protein-protein binding/dimerization Protein-nucleic acids interactions. . . *Note*: We are implicitly assuming above that we can describe the system as being composed of 3 distinct chemical “species”, R, L, and RL (quasichemical description). If interactions between R and L are weak/non-specific then it would be more appropriate to treat the system as a non-ideal solution of R and L.
Based on the earlier results for general reactions: Effective potential energy of solute i in solution.
If the solution is isotropic ( invariant upon rotation of solute), integrate analytically over rotational degrees of freedom (ignoring rotovibarational couplings, OK at physiological temperatures. Internal coordinates When inserting into the expression for Kb(T), L's cancel because We get: [Gilson et al. Biophysical Journal 72, 1047 -1069 (1997)]
In the complex, RL, the “external” coordinates (translations, rotations) of the ligand become internal coordinates of the complex. L R External coordinates of the ligand relative to receptor Position of the ligand relative to receptor frame It is up to us to come up with a reasonable definition of “BOUND”. That is we need to define the RL species before we can compute its partition function. The binding constant will necessarily depend on this definition. Must match experimental reporting. If the binding is strong and specific the exact definition of the complexed state is often not significant. Orientation of the ligand relative to receptor frame
It is convenient to introduce an “indicator” function for the complex: then: Next, define “binding energy” of a conformation of the complex: basically, change in effective energy for bringing ligand receptor together at fixed internal conformation: +
In terms of binding energy: then: Now: we are not very good at computing partition functions. We are much better at computing ensemble averages:
To transform the expression for Kb so that it looks like an average: multiply and divide by: then: or:
We can see that binding constant can be expressed in terms of an average of the exponential of the binding energy over the ensemble of conformations of the complex in which the ligand the receptor are not interacting while the ligand is placed in the binding site. Standard free energy of binding: (analytic formulas) (numerical computation)
Interpretation in terms of binding thermodynamic cycle: Ligand in solution at concentration Cº R+L RL Loss of translational, rotational freedom (to fit binding site definition) Ligand bound to receptor Binding while in receptor site (independent of concentration) R(L) “Virtual” state in which ligand is in binding site without interacting with receptor A future lecture (BEDAM) and computer exercise will focus on the computation of by computer simulations.
Binding Energy Distribution Analysis Method (BEDAM) (I) • Binding Site: square well potential • Excess binding free energy: The function W represents the solvent potential of mean force, which describes solvent-mediated interactions (implicit solvent model). J. Chem. Theory Comput. Vol. 6 2967 (2010)
Binding Energy Distribution Analysis Method (BEDAM) (II) ΔAb(exc. ) is measured by Hamiltonian Replica Exchange Sampling Replica Exchange is essential to obtain the correct binding free energy if multiple binding modes exist. Two binding modes of β-Cyclodextrin heptanoate binding complex. J. Phys. Chem. B Vol. 120 8289 (2016)
Free energy perturbation method • The main idea: One starts with an initial system, called the unperturbed or reference system. The system of interest, called the target system, is represented in terms of a perturbation to the reference system.
Probability density function P 0 of finding the reference system in a state defined by x: Fundamental formula for the transformation 0 1: Similarly, the free energy difference can also be written in terms of an average over the ensemble of the final state corresponding to target system with potential U 1(x)
Multiple State Simulations and UWHAM (I) Suppose M parallel simulations in the canonical ensemble are run at M state. The probability of observing Xi at the γ state: Let uαi to be the reduced (energy) coordinate of a microstate Xαi observed at the α state. The probability of observing an observation with the reduced coordinate uαi where Ω(uαi) is the density of state.
Multiple State Simulations and UWHAM (II) The likelihood of the observed data ensemble is Maximum likelihood estimate: solve Ω(uαi) and Zα by maximizing the likelihood function UWHAM: unbinned weighted histogram analysis method. The UWHAM estimate of the probability of observing uγi at the αth state is
Double Decoupling Method (DDM) Protein (P) Ligand (L) Complex (PL) ΔF 0 Lelec+vdw ΔFwater PLelec+vdw ΔFprotein, restraint L PLrestraint+elec+vdw ΔFgas, restraint ΔFprotein ΔFgas= 0 Lrestraint PLrestraint
Calculation of binding free energy with restraint
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