ligand binding energy James C. (JC) Gumbart Georgia Institute of - PowerPoint PPT Presentation

Accurate calculation of ligand binding energy James C. (JC) Gumbart Georgia Institute of Technology, Atlanta Chris Chipot U. Illinois, Urbana & CNRS, U. Lorraine France Computational Biophysics Workshop | DICP | July 11 2018

Outline I. What is an absolute binding energy? II. Using restraints to reduce the sampling problem III. Calculating the requisite PMFs IV. Comparing geometric with alchemical approach V. Illustration with barstar-barnase binding http://phdcomics.com/

Challenge: Absolute binding free energies K eq protein + ligand protein : ligand N ligands p i ∝ Z i Energy does not depend on position of ligand when 1, 2,…,N refer to each ligand unbound ( bulk is isotropic) , so can pick out a speci fi c 1 is bound (site) in num., point x 1* and hold it there unbound (bulk) in denom. C° is the standard concentration of 1 M → ∆ G 0 = − kT ln( K eq C ◦ ) C ◦ = 1 / 1661˚ 3 binding free energies are concentration A dependent!

Illustration using Abl SH3 domain A well known and conserved domain of Abl kinase Chosen ligand : APSYSPPPPP ( fl exible!) designed to bind with high a ffi nity peptide, so doesn’t require novel parametrization = -7.94 kcal/mol (exp) MM/PBSA estimate: -2.6 kcal/mol ! Pisabarro, M. T.; Serrano, L. Biochemistry 1996 , 35, 10634-10640 Hou, T. et al. PLoS Comput. Biol . 2006 , 2, 0046-0055

How to get K eq and Δ G ? Woo, H. J.; Roux, B. Proc. Natl. Acad. Sci. USA, 2005 , 102, 6825-6830 geometrical alchemical route route Forcibly separate the ligand from Make the ligand vanish from the the protein and calculate a PMF binding site and from bulk water Both approaches su ff er major sampling de fi ciencies when used on their own!!!

Overcoming sampling issues with restraints L 1 L 2 L 3 P 1 P 2 P 3 Bound state RMSD restrained Assorted spatial/rotational restraints -Design set of restraints to reduce conformational space needed to be sampled -Contributions of each restraint to free energy need to be rigorously computed Free state RMSD restrained Remember! Biasing is okay as long as we can unbias

Overcoming sampling issues with restraints Free Bound Conformational state - state - turn turn on/ o ff /on o ff restraints restraints Orientational Axial (un)binding Schematic of process From: Deng and Roux. (2009) J. Phys. Chem . 113 : 2234-2246.

Binding free energy (geometrical route) Z Z Z Z d x e − β ( U + u c + u o + u p ) d x e − β U d 1 d 1 site site K eq = × Z Z Z Z d x e − β ( U + u c ) d x e − β ( U + u c + u o ) d 1 δ ( x 1 − x ∗ 1 ) d 1 Φ x 1 * site bulk L 2 L 1 Z Z d x e − β ( U + u c ) d 1 site Θ × Z Z d x e − β ( U + u c + u Θ ) d 1 site L 3 Ψ L 2 L 1 r Z Z d x e − β ( U + u c + u Θ ) d 1 δ ( x 1 − x ∗ 1 ) bulk × Z Z d x e − β ( U + u c ) d 1 δ ( x 1 − x ∗ 1 ) θ L 3 bulk P 1 Z Z Z Z P 2 d x e − β ( U + u c ) d x e − β ( U + u c + u o ) d 1 δ ( x 1 − x ∗ 1 ) d 1 bulk site × × Z Z Z Z d x e − β U ϕ d x e − β ( U + u c + u o + u θ ) d 1 δ ( x 1 − x ∗ 1 ) d 1 bulk site P 3 Yu, Y. B. et al. (2001) Biophys . J. , 81 :1632-1642. Z Z d x e − β ( U + u c + u o + u θ ) d 1 Woo, H. J.; Roux, B. (2005) Proc. Natl. Acad. Sci. USA , site × Z Z 102 :6825-6830. d x e − β ( U + u c + u o + u θ + u ϕ ) d 1 Ma ff eo, C., Luan, B., Aksimentiev, A. (2012) Nucl. site Acids Res. 40 :3812-3821.

How to evaluate all of these integrals? Z Z d x e − β ( U + u c + u Θ + u Φ ) d 1 ratio of integrals can be = site related to a free energy Z Z d x e − β ( U + u c + u Θ + u Φ + u Ψ ) d 1 site Z Z d x e − β ( U + u c + u Θ + u Φ ) d 1 1 = e + β ∆ G site site = Ψ Z Z e − β ∆ G site d x ( e − β u Ψ )e − β ( U + u c + u Θ + u Φ ) Ψ d 1 site Z d Ψ e − β [ w site ( Ψ )] Potential of mean force, w site ( ѱ ) , e + β ∆ G site = Ψ encapsulates all degrees of freedom Z d Ψ e − β [ w site ( Ψ )+ u Ψ ] In practice, one determines the PMFs successively and then integrates them as prescribed above

Many PMFs are very straightforward θ θ Z Z d x e − β ( U + u c + u o ) d 1 site × Z Z d x e − β ( U + u c + u o + u θ ) d 1 site two windows used for ABF , 1 ns each sampling PMFs counts Φ Φ

Separation PMF from umbrella sampling entropic decay despite no interactions − kT ln r 2 Z Z d x e − β ( U + u c + u o + u p ) d 1 site × Z Z d x e − β ( U + u c + u o ) d 1 δ ( x 1 − x ∗ 1 ) bulk 37 windows used, spaced 0.5 - 1 Å apart -histograms are overlapping 1 ns/window 0.75 ns 0.5 ns PMF was already converged 0.25 ns within ~20 ns

Replica-exchange umbrella sampling (REUS) -helps to circumvent limitations in US by exchanging coordinates periodically between di ff erent windows -exchanges accepted with some probability: min(1 , e − ∆ E/kT ) where ∆ E = ( w i ( ξ j ) − w i ( ξ i )) + ( w j ( ξ i ) − w j ( ξ j )) (swapped) (original) (swapped) (original) See tutorial Methods for Calculating Potentials of Mean Force

What you get in the end (a big mess!)

Back to the Abl kinase story... ~30 ns ∆ G site Z Z Z Z = 3 . 52 kcal / mol d x e − β U d x e − β ( U + u c + u o + u p ) d 1 d 1 site site K eq = c × Z Z Z Z d x e − β ( U + u c ) d x e − β ( U + u c + u o ) d 1 δ ( x 1 − x ∗ 1 ) d 1 site bulk 6 ns ∆ G site = 0 . 71 kcal / mol Z Z d x e − β ( U + u c ) o d 1 site × Z Z d x e − β ( U + u c + u Θ ) d 1 4 ns ∆ G site = 0 . 20 kcal / mol site a 37 ns ∆ G sep = − 14 . 47 kcal / mol r Z Z d x e − β ( U + u c + u Θ ) d 1 δ ( x 1 − x ∗ 1 ) bulk × Z Z d x e − β ( U + u c ) d 1 δ ( x 1 − x ∗ 1 ) bulk ∆ G bulk (analytical) = 5 . 77 kcal / mol Z Z Z Z o d x e − β ( U + u c ) d x e − β ( U + u c + u o ) d 1 δ ( x 1 − x ∗ 1 ) d 1 bulk site × × Z Z Z Z d x e − β ( U + u c + u o + u θ ) d x e − β U d 1 δ ( x 1 − x ∗ 1 ) d 1 bulk site 30 ns ∆ G bulk = 5 . 43 kcal / mol Z Z d x e − β ( U + u c + u o + u θ ) d 1 c site × Z Z d x e − β ( U + u c + u o + u θ + u ϕ ) d 1 site ∆ G o = ( ∆ G bulk − ∆ G site ) + ( ∆ G bulk − ∆ G site ) ~ 120 ns c c o o required + ∆ G sep − ∆ G site = − 7 . 7 kcal / mol r a = -7.94 kcal/mol ( exp ) Agreement within 0.25 kcal/mol!

There’s more than one way to… Can use FEP to (de)couple the ligand to the binding site of the protein Δ G 0 “ Floating ligand ” problem protein + ligand 0 protein:ligand 0 site bulk Δ G c Δ G c bulk site bulk site Δ G a Δ G o Δ G o Δ G a Avoided through definition of a site bulk Δ G p Δ G p Δ G * set of restraints protein + nothing 0 protein + ligand * protein:nothing 0 protein:ligand * bulk site Δ G a Δ G a Follow a formalism akin to the reaction-coordinate (geometric) route protein + nothing * protein:nothing * - Alchemical transformations performed bidirectionally using FEP - Bennett acceptance ratio (BAR) estimator - Free-energy contributions due to restraints measured using TI - Most appropriate for buried ligands (no extraction pathway) Bennett, C. H. J. Comp. Phys. , 1976 , 22 , 245-268 Gilson, M. K. et al. Biophys. J., 1997 , 72 , 1047-1069

The alchemical (FEP) route Φ Θ r Ψ θ ϕ site bulk Deng, Y.; Roux, B. J. Phys. Chem. B 2009 , 113 , 2234-2246

Comparison of alchemical and geometric routes RMSD RMSD PMF 16 ns PMFs 12 ns 24 ns PMFs 8 ns PMF 20 ns decoupling 104 ns coupling 104 ns RMSD PMF RMSD 60 ns 48 ns

Comparison of alchemical and geometric routes = -7.8 kcal/mol = -7.7 kcal/mol

Error analysis Alchemical route Geometrical route - Low statistical errors - Estimates burdened by systematic error RMSD ±0.4 kcal/mol ±0.0 kcal/mol ±0.0 kcal/mol alchemy ±0.7 kcal/mol ±1.0 kcal/mol RMSD ±0.5 kcal/mol ±0.2 kcal/mol often very tedious but you should still do it! ±0.4 kcal/mol ( reviewers will often ±0.9 kcal/mol request it anyway! ) Hahn, A. M.; Then, H. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 2009 , 80, 031111 Rodriguez-Gomez, D. et al. J. Chem. Phys., 2004 , 120, 3563-3578 Pohorille, A. et al. J. Phys. Chem. B, 2010 , 114, 10235-10253 Hénin, J.; Chipot, C. J. Chem. Phys . 2004 , 121 , 2904-2914

Error analysis Geometrical route Alchemical route Advantages Advantages - Rigorous, formally correct framework - Rigorous, formally correct framework - Reasonably inexpensive - Reasonably inexpensive - Access to the statistical error for all terms - Access to the statistical error for all terms - In principle, applicable to protein:protein dimers - Embarrassingly parallelizable Shortcomings Shortcomings - Cumbersome - Cumbersome - Convergence of RMSD term; Degeneracy - Convergence of alchemical transformation - Convergence of separation term; ⊥ DoF’s ? - Convergence of restraint term - Limited to interfacial binding sites - In principle, limited to small ligands

Protein-protein binding free energy barstar - an inhibitor barnase - a ribonuclease interface is highly solvated = -19.0 kcal/mol (exp) Schreiber & Fersht. JMB , 248 :478-486. 1995. Gumbart, Roux, Chipot. E ffi cient Determination of Protein–Protein Standard Binding Free Energies from First Principles. JCTC 9 :3789-3798. 2013.

Numerous restraints needed Gumbart, Roux, Chipot. JCTC 9 :3789-3798. 2013. RMSD on barnase backbone RMSD on barstar backbone RMSD on barnase side chains RMSD on barstar side chains