Lecture 20: Binding Free Energy Calculations Dr. Ronald M. Levy Statistical Thermodynamics

focus on the binding problem of protein-ligand systems. Statistical theory of non-covalent binding equilibria Computer models for computing free energy of binding Binding energy distribution analysis method Examples from our research group Center for Biophysics and Computational Biology, Outline

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 (quasi- chemical 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. Statistical Theory of Non-Covalent Binding Equilibria

From general theory of chemical reactions, for receptor-ligand system: 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 roto- vibarational couplings, OK at physiological temperatures). Internal coordinates When inserting into the expression for K b (T), Λ's cancel because We get: [Gilson et al. Biophysical Journal 72, (1997)]

In the complex, RL, the “external” coordinates (translations, rotations) of the ligand become internal coordinates of the complex. R L Position of the ligand relative to receptor frame Orientation of the ligand relative to receptor frame External coordinates of the ligand relative to receptor 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.

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 and 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 K b 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 and the receptor are not interacting while the ligand is placed in the binding site. Standard free energy of binding: (analytic formulas) (numerical computation) Summary of Binding Free Energy Theory Binding energy: Binding Constant:

Interpretation in terms of binding thermodynamic cycle: RLR + L R(L) “Virtual” state in which ligand is in binding site without interacting with receptor Ligand and receptor in solution at concentration Cº Ligand bound to receptor in solution at concentration Cº Loss of translational, rotational freedom (to fit binding site definition) Binding while in receptor site (independent of concentration) BEDAM method and computer exercise will focus on the computation of by computer simulations.

Binding Free Energy Models [Gallicchio and Levy, Adv. Prot. Chem (2012)] Double Decoupling Method (DDM) Relative Binding Free Energies (FEP) Potential of Mean Force/ Pathway Methods MM/PBSA Mining Minima (M2) Exhaustive docking Docking & Scoring BEDAM (Implicit solvation) λ-dynamics Statistical mechanics theory Binding Energy Distribution Analysis Method

Free Energy Perturbation (FEP/TI)Double Decoupling (DDM) Jorgensen, Kollman, McCammon (1980’s – present) Jorgensen, Gilson, Roux,... (2000’s – to present) : Challenges: Dissimilar ligand sets Numerical instability Dependence on starting conformations Multiple bound poses Slow convergence Statistical mechanics based, in principle account for: Total binding free energy Entropic costs Ligand/receptor reorganization Free Energy Perturbation and Double Decoupling Methods

loss of conformational freedom, energetic strain translational/rotational entropy loss Interatomic interactions Reorganization Free Energy of Binding Consider the following thermodynamic cycle: Binding Free Energy= reorganization + interaction Docking/scoring focus on ligand-receptor interaction BEDAM accounts for both effects of interaction and reorganization

Binding Energy Distribution Analysis Method (Statistical Theory) [Gilson, McCammon et al., (1997)] Binding “energy” of a fixed conformation of the complex. W(): solvent PMF (implicit solvation model) Entropically favored Ligand in binding site in absence of ligand-receptor interactions

Binding Energy Distribution Analysis Method (Computing Method) P 0 ( ΔE) : encodes all enthalpic and entropic effects Solution: 1) treat binding energy as biasing potential = λ ΔE λ=0: uncoupled/unbound state,  weakly coupled states λ=1: full coupled /bound state 2) Hamiltonian Replica Exchange +WHAM ΔE [kcal/mol] P 0 ( Δ E ) [kcal/mol -1 ] P0(ΔE)P0(ΔE) Integration problem: region at favorable ΔE’s is seriously undersampled. Main contribution to integral Ideal for HPC cluster computing and distributed grid network ] Gallicchio, Lapelosa, & Levy, 2010; Xia, Flynn, Gallichio & et al, 2015

Hamiltonian Replica Exchange in λ-space Enhances conformational mixing Better convergence of conformational ensembles at each λ... λ= 0 λ = Translation/rotation of the ligand is accelerated when ligand- receptor interactions are weak (λ  Slow Conformational dof's Fast(er) Potential Energy MD Coupled Uncoupled

Reweighting techniques are necessary to recover the unbiased true observables (results) from more advanced sampling methods: Weighted Histogram Analysis Method (WHAM) Ferrenberg & Swendsen (1989) Kumar, Kollman et al. (1992) Bartels & Karplus (1997) Gallicchio, Levy et al. (2005) Unbinned Weighted Histogram Analysis Method (UWHAM) equivalent to Multistate Bennett Acceptance Ratio (MBAR) Shirts & Chodera J. Chem. Phys. (2008). Tan, Gallicchio, Lapelosa, Levy JCTC (2012). Reweighting Techniques in Free Energy Calculations

Results for Binding to Mutants of T4 Lysozyme L99A Hydrophobic cavity L99A/M102Q Polar cavity Brian Matthews Brian Shoichet Benoit Roux David Mobley Ken Dill John Chodera Graves, Brenk and Shoichet, JMC (2005) BEDAM: 20ns HREM, 12 replicas λ={10 -6, 10 -5, 10 -4, 10 -3, 10 -2, 0.1, 0.15, 0.25, 0.5, 0.75, 1, 1.2} IMPACT + OPLS-AA/AGBNP2

Binders vs. Non-Binders L99A T4 Lysozyme, Apolar Cavity L99A/M102Q T4 Lysozyme, Polar Cavity

SAMPL4

Large-Scale Screening by Binding Free energy Calculations: HIV-Integrase LEDGF Inhibitors SAMPL4 Ligand Candidates ~350 scored with BEDAM Docking + BEDAM Screening IN/LEDGF Binding Site HIV-IN is responsible for the integration of viral genome into host genome. The human LEDGF protein links HIV-IN to the chromosome Development of LEDGF binding inhibitors for novel HIV therapies SAMPL4 blind challenge: computational prediction of undisclosed experimental screens. Docking provides little screening discrimination: “everything binds”! Much more selectivity from absolute binding free energies BEDAM predictions ranked first among 25 computational groups in SAMPL4, 2.5 x fold enrichment factor in top 10% of focused library -5

Asynchronous Replica Exchange for Computational Grids Separate local file-based asynchronous exchanges and remote MD simulations Limited to large MD period (> 1 ps) but robust to failures of individual MD processes because no synchronizing process is required. Metroplis independence sampling approaching the infinite swapping limit (100s to 1000s swaps/cycle) Exchanges between all pairs of neighbors can be performed in a local CPU independent of MD jobs running remotely. Current Grid Computing Network: Temple University 450 CPUs Brooklyn 2000 CPUs World Community Grid at IBM 600,000+ CPUs Xia, Flynn, Gallicchio, Zhang, He, Tan, & Levy, J. Comput. Chem., Gallicchio, Xia, Flynn, Zhang, Samlalsingh, Mentes, &Levy, Comput. Phys. Comm., MD running remotely Exchange locally

Async REMD for β - cyclodextrin-heptanoate Host-Guest System β- cyclodextrin-heptanoate complex MD running remotely Exchange locally ) Converged binding energy distributions of λ=1 from1D Sync REMD (72ns x 16 replica)

2D Async RE Results for β - cyclodextrin-heptanoate Complex Binding energy distributions of λ=1 from different REMD simulations at T=300K MD running remotely Exchange locally ) Binding free energy as a function of λ from different REMD simulations at T=300K