Lecture 20: Binding Free Energy Calculations

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1 Statistical Thermodynamics ecture 20: Binding Free Energy Calculations Dr. onald M. evy

2 Outline 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,

Sta$s$cal Theory of Non- Covalent Binding Equilibria General chemical reaction: ν A A+ν B B ν C C+ν D D Very

dimerization/association l Supramolecular complexes l Protein-ligand binding l Protein-protein

.. *Note*: We are implicitly assuming above that we can describe the system as being composed of 3 distinct

3 Sta$s$cal Theory of Non- Covalent Binding Equilibria General chemical reaction: ν A A+ν B B ν C C+ν D D Very important type of reaction : bimolecular non-covalent binding (sol) + (sol) (sol) l Small molecule dimerization/association l Supramolecular complexes l Protein-ligand binding l Protein-protein binding/dimerization l Protein-nucleic acids interactions l... *Note*: We are implicitly assuming above that we can describe the system as being composed of 3 distinct chemical species,,, and (quasichemical description). If interactions between and are weak/non-specific then it would be more appropriate to treat the system as a non-ideal solution of and.

4 From general theory of chemical reactions, for receptor-ligand system: ( ) o ( C / C ) o o ( / ) ( / ) ΔA o b ( T)/ KT eq. Kb T = Binding =e = C C C C constant eq. ϕ i ( ) ν A = ν n u j 1 3n 3 βψ u i u 3n Λ u uj T = dx e B = 1 ;ν = 1;ν = 0 C D ν C ( ) D( ) A ( T) ϕ ( T) ν D D C A o B C b ( ) ( ) ν ν B ϕa B ( xu ) eq. ( ) ( T) ϕ ( T) ν +ν ν ν ϕ T ϕ T o ϕ T K T = V =C ϕ Ψi( x u ) Effective potential energy of solute i in solution.

5 If the solution is isotropic ( ϕ Ψ i ( x ) u 8π invariant upon rotation of solute), integrate analytically over rotational degrees of freedom (ignoring rotovibarational couplings, OK at physiological temperatures). ( x 2 u ) 8 π Z' 2 2 3n 6 βψ u i i i ( T ) = 8π ϕθi = n u u dx e = n u 3n u 3n u Λuj Λuj j j Internal coordinates When inserting into the expression for K b (T), Λ's cancel because We get: Z' = dx 3n 6 e βψ ( x ) C Z' ( ) Kb T = 2 8 π Z' Z' o Z' =??? [Gilson et al. Biophysical Journal 72, (1997)] Z' = dx 3n 6 e n βψ ( x ) = n + n

6 In the complex,, the external coordinates (translations, rotations) of the ligand become internal coordinates of the complex. Z' = dx 3n 6 dx 3n 6 dζ 6 e βψ ( x,x,ζ ) External coordinates of the ligand relative to receptor 6 ζ = {( r, θ, φ),( ω, ω, ω )} Position of the ligand relative to receptor frame Orientation 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 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.

7 It is convenient to introduce an indicator function for the complex: then: I ( ζ ) = 1 0 ligand is in binding site ligand is outside binding site Z' = = d x 3n 6 3n 6 dx dζ 6 I( ( βψ x,x,ζ ) ζ )e Next, define binding energy of a conformation of the complex: u=u ( x,x,ζ ) =Ψ ( x,x,ζ ) Ψ ( x ) Ψ ( x ) basically, change in effective energy for bringing ligand and receptor together at fixed internal conformation: +

8 In terms of binding energy: then: C 8π 2 Ψ ( x,x,ζ ) =Ψ ( x ) +Ψ ( x ) +u( x,x, ζ ) K b ( T )= C 8π 2 Z' Z' Z' = d x 3n 6 3n 6 dx dζ 6 I( ( βψ x ) ( βψ x ) βu x ζ )e e e (,x,ζ ) ( ) ( βψ x ) e βψ d x 3n 6 3n 6 x dx e Now: we are not very good at computing partition functions. We are much better at computing ensemble averages: < O > = dxo βu ( x) ( x) e = dxo( x) ρ( x) βu ( x) dxe

9 To transform the expression for K b so that it looks like an average: multiply and divide by: d ζ 6 I ζ ( )=V site Ω site then: C 8π 2 V site V K b ( T )= C 8π 2 d x 3n 6 3n 6 dx dζ 6 βψ I( x ζ )e Ω site 8π 2 Z' Z' Z' = βψ d x 3n 6 3n 6 x dx e d x 3n 6 3n 6 dx dζ 6 βψ I( x ζ )e βψ ( ) x e ( ) βu x e,x,ζ βψ ( ) x e ( ) d x 3n 6 3n 6 dx dζ 6 βψ I( x ζ )e ( ) = βψ ( ) x e ( ) βu x e,x,ζ βψ ( ) x e ( ) ( ) or: K b ( T )= V site V Ω site ( 8π < e βu x,x,ζ ) >++I 2 ( )

10 Binding energy: Binding Constant: o site site 2 o 2 (,x,ζ ) C Z' V Ω βu x K ( ) b T = = < e > 8π Z' Z' V 8π ++I 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: ΔA ( T ) = ktlnk ( T ) ΔA b ( translational) u=u ( x,x,ζ ) =Ψ ( x,x,ζ ) Ψ ( x ) Ψ ( x ) Summary of Binding Free Energy Theory b b (,x,ζ ) Vsite Ω βu x o site ΔA ( ) ln ln ln b T = kt kt kt < e > o 2 V 8π ++I ( ) ( ) (analytic formulas) (numerical computation)

11 Interpretation in terms of binding thermodynamic cycle: igand and receptor in solution at concentration Cº + oss of translational, rotational freedom (to fit binding site definition) ΔA b ( t+r) ΔA b ΔA exc. b( ) igand bound to receptor in solution at concentration Cº Binding while in receptor site (independent of concentration) () Virtual state in which ligand is in binding site without interacting with receptor ΔA b = ΔA b ( t + r) + ΔA ( exc. ) BEDAM method and computer exercise will focus on the computation of by computer simulations. ( ) ΔA b exc. b

12 Binding Free Energy Models [Gallicchio and evy, Adv. Prot. Chem (2012)] Double Decoupling Method (DDM) elative Binding Free Energies (FEP) λ-dynamics Potential of Mean Force/ Pathway Methods Docking & Scoring K AB = C Z AB 8 π 2 Z A Z B Statistical mechanics theory BEDAM (Implicit solvation) Exhaustive docking MM/PBSA Mining Minima (M2) Binding Energy Distribution Analysis Method

Kollman, McCammon (1980 s present) Double Decoupling (DDM) Jorgensen, Gilson, oux,.

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

eorganiza$on Free Energy of Binding Consider the following thermodynamic cycle: ΔG conf loss of conformational freedom, energetic strain ΔG b ΔG reorg Interatomic interactions

14 eorganiza$on Free Energy of Binding Consider the following thermodynamic cycle: ΔG conf loss of conformational freedom, energetic strain ΔG b ΔG reorg Interatomic interactions translational/rotational entropy loss ΔS + V = kln V site b, t r Ω 8π site 2 ΔE b Binding Free Energy= reorganization + interaction ΔG b = ΔG reorg + ΔE b Docking/scoring focus on ligand-receptor interaction Δ E b BEDAM accounts for both effects of interaction and reorganization

15 Entropically favored Binding Energy Distribu$on Analysis Method (Sta$s$cal Theory) [Gilson, McCammon et al., (1997)] K AB = e βδf b C = 8π 2 Z Z A AB Z B ΔE = ΔU + ΔW K AB = C Vsite βδe Binding energy of a fixed conformation of the complex. W(): solvent PMF (implicit solvation model)... 0 : igand in binding site in absence of ligand-receptor interactions e 0 Area = β K AB = e ΔF AB Δ F AB = ΔE " TΔSconf "

16 Binding Energy Distribu$on Analysis Method (Compu$ng Method) ΔE = E E = bound unbound ΔU + ΔW P 0 (ΔE): encodes all enthalpic and entropic effects Integration problem: region at favorable ΔE s is seriously undersampled. Solution: 1) treat binding energy as biasing potential = λ ΔE λ=0: uncoupled/unbound state, weakly coupled states λ=1: full coupled /bound state K P0(ΔE ) [kcal/mol-1] βδe ( ΔE) e P ( ΔE) βδe AB = C Vsite e = C Vsite d 0 0 e βδe P 0 (ΔE) 2) Hamiltonian eplica Exchange +WHAM Main contribution to integral ΔE [kcal/mol] Ideal for HPC cluster computing and distributed grid network ] Gallicchio, apelosa, & evy, 2010; Xia, Flynn, Gallichio & et al, 2015

Hamiltonian eplica Exchange in λ- space Translation/rotation of the ligand is accelerated when ligandreceptor interactions are weak (λ» 0) Potential Energy Slow

17 Hamiltonian eplica Exchange in λ- space Translation/rotation of the ligand is accelerated when ligandreceptor interactions are weak (λ» 0) Potential Energy Slow Fast(er) Conformational dof's Uncoupled Coupled λ= λ = 1 Enhances conformational mixing Better convergence of conformational ensembles at each λ MD

18 eweigh$ng Techniques in Free Energy Calcula$ons eweighting 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, evy et al. (2005) Unbinned Weighted Histogram Analysis Method (UWHAM) equivalent to Multistate Bennett Acceptance atio (MBA) Shirts & Chodera J. Chem. Phys. (2008). Tan, Gallicchio, apelosa, evy JCTC (2012).

Shoichet, JMC (2005) 99A/M102Q Polar cavity BEDAM: 20ns HEM, 12 replicas λ={10-6,

19 esults for Binding to Mutants of T4 ysozyme 99A Hydrophobic cavity Brian Matthews Brian Shoichet Benoit oux David Mobley Ken Dill John Chodera Graves, Brenk and Shoichet, JMC (2005) 99A/M102Q Polar cavity BEDAM: 20ns HEM, 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 + OPS-AA/AGBNP2

20 99A T4 ysozyme, Apolar Cavity Binders vs. Non- Binders 99A/M102Q T4 ysozyme, Polar Cavity

21 SAMP4 utgers/temple E. Gallicchio, N. Deng, P. He,. evy Scripps - A. Perryman, S. Forli, D. Santiago, A. Olson,

arge-scale Screening by Binding Free energy Calculations: HIV-Integrase EDGF Inhibitors HIV-IN is responsible for the

The human EDGF protein links HIV-IN to the chromosome Development of EDGF binding inhibitors for novel HIV therapies

......... -5 SAMP4 blind challenge: computational prediction of undisclosed experimental screens.

22 arge-scale Screening by Binding Free energy Calculations: HIV-Integrase EDGF Inhibitors HIV-IN is responsible for the integration of viral genome into host genome. The human EDGF protein links HIV-IN to the chromosome Development of EDGF binding inhibitors for novel HIV therapies IN/EDGF Binding Site 450 SAMP4 igand Candidates ~350 scored with BEDAM Docking + BEDAM Screening SAMP4 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 SAMP4, 2.5 x fold enrichment factor in top 10% of focused library

Asynchronous eplica Exchange for Computa$onal Grids Separate local file-based asynchronous exchanges and remote MD simulations imited to large MD period (> 1 ps) but robust to failures of individual

23 Asynchronous eplica Exchange for Computa$onal Grids Separate local file-based asynchronous exchanges and remote MD simulations imited to large MD period (> 1 ps) but robust to failures of individual MD processes because no synchronizing process is required. MD running remotely 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 College@CUNY 2000 CPUs World Community Grid at IBM 600,000+ CPUs Exchange locally Xia, Flynn, Gallicchio, Zhang, He, Tan, & evy, J. Comput. Chem., Gallicchio, Xia, Flynn, Zhang, Samlalsingh, Mentes, &evy, Comput. Phys. Comm., 2015.

24 Async EMD for β- cyclodextrin-heptanoate Host-Guest System MD running remotely ) β- cyclodextrin-heptanoate complex Exchange locally Converged binding energy distributions of λ=1 from1d Sync EMD (72ns x 16 replica)

25 2D Async E esults for β- cyclodextrin-heptanoate Complex MD running remotely ) Exchange locally Binding energy distributions of λ=1 from different EMD simulations at T=300K Binding free energy as a function of λ from different EMD simulations at T=300K

Large scale free energy calculations for blind predictions of protein ligand binding: the D3R Grand Challenge 2015

J Comput Aided Mol Des (2016) 30:743 751 DOI 10.1007/s10822-016-9952-x Large scale free energy calculations for blind predictions of protein ligand binding: the D3R Grand Challenge 2015 Nanjie Deng 1 William