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P ROTEIN -L IGAND S TANDARD B INDING F REE -E NERGY C ALCULATIONS P ROTEIN -L IGAND S TANDARD B INDING F REE -E NERGY C ALCULATIONS Chris Chipot Laboratoire International Associ CNRS-UIUC, Unit Mixte de Recherche n 7565, Universit de

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SLIDE 1

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS

Chris Chipot

Laboratoire International Associé CNRS-UIUC, Unité Mixte de Recherche n° 7565, Université de Lorraine Beckman Institute for Advanced Science and Technology, Department of Physics University of Illinois at Urbana-Champaign

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 2

THE LONG-STANDING PROTEIN-LIGAND PROBLEM RELATIVE BINDING FREE ENERGIES BEYOND PROTEIN-LIGAND BINDING

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS OUTLINE

THE GEOMETRICAL ROUTE THE ALCHEMICAL ROUTE ALANINE SCANNING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 3

THE LONG-STANDING PROTEIN-LIGAND PROBLEM

RELATIVE BINDING FREE ENERGIES BEYOND PROTEIN-LIGAND BINDING

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS OUTLINE

THE GEOMETRICAL ROUTE THE ALCHEMICAL ROUTE ALANINE SCANNING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 4

Chipot, C. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2014, 4, 71-89. Shirts, M. R.; Mobley, D. L.; Chodera, J. D. Annual Reports Comput. Chem. 2007, 3, 41-59. Chipot, C.; Rozanska, X.; Dixit, S. B. J. Comput. Aided Mol. Des. 2005, 19, 765-770.

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE LONG-STANDING PROTEIN-LIGAND PROBLEM rigid docking flexible docking free energy constant computational investment shape recognition

drugs

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 5

which can readily be determined by experiment: — A single event is evidently not enough.

Buch, I.; Giorgino, T.; Fabritiis, G. D. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 10184-10189 Gilson, M. K. et al. Biophys. J. 1997, 72, 1047-1069 Kollman, P.A. Chem. Rev. 1993, 93, 2395-2417 Karlsson, R.; Larsson, A. Methods Mol. Biol. 2004, 248, 389-415 Chipot, C.; Pohorille, A. Free-energy calculations. Springer 2007.

Kd = koff kon

response units

The computational microcalorimeter

THEORETICAL BACKGROUND

Keq = [protein : ligand] [protein][ligand]

protein + ligand protein : ligand Keq

— Brute-force simulations are limited by kon and koff.

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE LONG-STANDING PROTEIN-LIGAND PROBLEM

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 6

Shoup, D.; Szabo, A. Biophys. J. 1982, 40, 33-39 Woo, H. J.; Roux, B. Proc. Natl. Acad. Sci. USA 2005, 102, 6825-6830

N ligands

Keq = 1 [ligand] 8 > > < > > : Z

site

d1 Z

bulk

d2 . . . Z

bulk

dN Z dx e−βU Z

bulk

d1 Z

bulk

d2 . . . Z

bulk

dN Z dx e−βU + Z

bulk

d1 Z

site

d2 . . . Z

bulk

dN Z dx e−βU Z

bulk

d1 Z

bulk

d2 . . . Z

bulk

dN Z dx e−βU + · · · + Z

bulk

d1 Z

bulk

d2 . . . Z

site

dN Z dx e−βU Z

bulk

d1 Z

bulk

d2 . . . Z

bulk

dN Z dx e−βU 9 > > = > > ; Keq = [protein : ligand] [protein][ligand] Keq = p1[protein]tot [ligand] p0 [protein]tot = 1 [ligand] p1 p0

[protein] = p0 [protein]tot [protein : ligand] = p1 [protein]tot

protein + ligand protein : ligand Keq

THEORETICAL BACKGROUND

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE LONG-STANDING PROTEIN-LIGAND PROBLEM

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 7

Shoup, D.; Szabo, A. Biophys. J. 1982, 40, 33-39 Woo, H. J.; Roux, B. Proc. Natl. Acad. Sci. USA 2005, 102, 6825-6830

alchemical route geometrical route

Keq = 1 [ligand] N Z

site

d1 Z

bulk

d2 . . . Z

bulk

dN Z dx e−βU Z

bulk

d1 Z

bulk

d2 . . . Z

bulk

dN Z dx e−βU = 1 [ligand] N Z

site

d1 Z dx e−βU Z

bulk

d1 Z dx e−βU = 1 [ligand] N Z

site

d1 Z dx e−βU Vbulk Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU

Keq = Z

site

d1 Z dx e−βU Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU

x∗

1

[ligand] = N/Vbulk

THEORETICAL BACKGROUND

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE LONG-STANDING PROTEIN-LIGAND PROBLEM

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 8

THE LONG-STANDING PROTEIN-LIGAND PROBLEM RELATIVE BINDING FREE ENERGIES BEYOND PROTEIN-LIGAND BINDING

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS OUTLINE

THE GEOMETRICAL ROUTE

THE ALCHEMICAL ROUTE

ALANINE SCANNING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 9

protein + ligand* protein:ligand* protein + nothing* protein:nothing*

ΔG*

protein + nothing0 protein:nothing0 protein + ligand0 protein:ligand0

ΔG0 ΔGa

site bulk

ΔGa ΔGc

site bulk

ΔGc

Couple reversibly the ligand to the binding site of the protein

ΔGo

site bulk

ΔGo ΔGp

site bulk

ΔGp ΔGa

site bulk

ΔGa

  • Floating ligand problem.
  • Corpora non agunt nisi fixata.

Paul Ehrlich

  • Definition of a set of restraints.
  • The loss of translational, orientational and conformational

entropies contributes to the free energy.

Gilson, M. K. et al. Biophys. J., 1997, 72, 1047-1069

THE DOUBLE-ANNIHILATION STRATEGY

Hermans, J.; Wang, L. J. Am. Chem. Soc. 1997, 119, 2707-2714

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE ALCHEMICAL ROUTE

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 10

Deng, Y.; Roux, B. J. Phys. Chem. B 2009, 113, 2234-2246

Φ

Ψ

ϕ

θ

Θ

r

site bulk THE DOUBLE-ANNIHILATION STRATEGY

Keq = Z

site

d1 Z dx e−βU1 Z

site

d1 Z dx e−β(U1+uc) × Z

site

d1 Z dx e−β(U1+uc) Z

site

d1 Z dx e−β(U1+uc+uo) × Z

site

d1 Z dx e−β(U1+uc+uo) Z

site

d1 Z dx e−β(U1+uc+uo+up) × Z

site

d1 Z dx e−β(U1+uc+uo+up) Z

site

d1 Z dx e−β(U1+uc+uo+up+ur) × Z

site

d1 Z dx e−β(U1+uc+uo+up+ur) Z

site

d1 Z dx e−β(U0+uc+uo+up+ur) × Z

bulk

d1 Z dx e−β(U0+uc+uo+up+ur) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U0+uc+uo) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U0+uc+uo) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U0+uc) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U0+uc) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U1+uc) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U1+uc) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU1

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE ALCHEMICAL ROUTE

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-11
SLIDE 11

THE LONG-STANDING PROTEIN-LIGAND PROBLEM RELATIVE BINDING FREE ENERGIES BEYOND PROTEIN-LIGAND BINDING

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS OUTLINE

THE GEOMETRICAL ROUTE

THE ALCHEMICAL ROUTE ALANINE SCANNING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

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SLIDE 12

P3 P2 P1 L1 L

2

L

3

Φ Ψ Θ r

Keq = Z

site

d1 Z dx e−βU Z

site

d1 Z dx e−β(U+uc) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU Woo, H. J.; Roux, B. Proc. Natl. Acad. Sci. USA 2005, 102, 6825-6830 Yu, Y. B. et al. Biophys. J. 2001, 81, 1632-1642 Gumbart, J. C.; Roux, B.; Chipot, C. J. Chem. Theory Comput. 2013, 9, 794-802 × Z

site

d1 Z dx e−β(U+uc) Z

site

d1 Z dx e−β(U+uc+uΘ) × Z

site

d1 Z dx e−β(U+uc+uΘ) Z

site

d1 Z dx e−β(U+uc+uΘ+uΦ) × Z

site

d1 Z dx e−β(U+uc+uΘ+uΦ) Z

site

d1 Z dx e−β(U+uc+uΘ+uΦ+uΨ) × Z

site

d1 Z dx e−β(U+uc+uo) Z

site

d1 Z dx e−β(U+uc+uo+uθ) × Z

site

d1 Z dx e−β(U+uc+uo+uθ) Z

site

d1 Z dx e−β(U+uc+uo+uθ+uφ) × Z

site

d1 Z dx e−β(U+uc+uo+up) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uo) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ+uΦ+uΨ) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ+uΦ) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ+uΦ) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ)

x1

*

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE GEOMETRICAL ROUTE

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

Z

site

d1 Z dx e−β(U+uc) Z

site

d1 Z dx e−βuΘe−β(U+uc) = e+β∆Gsite

Θ

= dΘ e−βwsite(Θ) dΘ e−β

  • wsite(Θ)+uΘ
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SLIDE 13

+ +

∆G0

bind

∆Gsite

c

∆Gbulk

c

∆Gbulk

  • ∆Gsite
  • ∆Gsite

a,r

Robust experimental data. Sampling constitutes the primary source of error. The protein: Abl Src homology domain 3. The binder: APSYSPPPPP (p41). = -7.94 kcal/mol (experimental).

∆G0

Fully geometrical route: -7.8 ± 0.9 kcal/mol Fully alchemical route: -7.7 ± 1.0 kcal/mol

+

Θ Φ Ψ r Φ Θ Ψ

−(1/β) ln(S⇤I⇤C)

bulk site

A GEOMETRICAL ROUTE

Gumbart, J. C.; Roux, B.; Chipot, C. J. Chem. Theory Comput. 2013, 9, 794-802 Pisabarro, M. T.; Serrano, L. Biochemistry 1996, 35, 10634-10640 Pisabarro, M. T.; Serrano, L.; Wilmanns, M. J. Mol. Biol. 1998, 281, 513-521 Fu, H.; Cai, W.; Hénin, J.; Roux, B.; Chipot, C. J. Chem. Theory Comput. 2017, 13, 5173-5178 Fu, H. and Gumbart, J. C. and Chen, H. and Shao, X. and Cai, W. and Chipot, C. J. Chem. Inf. Model. 2018, 18, 556-560

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE GEOMETRICAL ROUTE

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-14
SLIDE 14

r

' µ Θ Φ Ψ

xL yL zL zP yP xP

  • Colvars scripted functions

Euler angles: Polar angles:

  • Handled by the extended adaptive biasing force algorithm (eABF)

# Euler angles # Phi namespace eval eulerPhi { } proc calc_eulerPhi { args } { global eulerPhi::q0 global eulerPhi::q1 global eulerPhi::q2 global eulerPhi::q3 set q0 [ lindex [ lindex $args 0 ] 0 ] set q1 [ lindex [ lindex $args 0 ] 1 ] set q2 [ lindex [ lindex $args 0 ] 2 ] set q3 [ lindex [ lindex $args 0 ] 3 ] set f [ expr 180 / 3.1415926 * atan2(2 * ($q0 * $q1 + $q2 * $q3), 1 - 2 * ($q1 * $q1 + $q2 * $q2)) ] return $f }

NEW COLLECTIVE VARIABLES

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE GEOMETRICAL ROUTE

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-15
SLIDE 15

r

' µ Θ Φ Ψ

xL yL zL zP yP xP

NEW COLLECTIVE VARIABLES

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS THE GEOMETRICAL ROUTE

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-16
SLIDE 16

THE LONG-STANDING PROTEIN-LIGAND PROBLEM

RELATIVE BINDING FREE ENERGIES

BEYOND PROTEIN-LIGAND BINDING

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS OUTLINE

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2017

THE GEOMETRICAL ROUTE THE ALCHEMICAL ROUTE ALANINE SCANNING

slide-17
SLIDE 17

THE LONG-STANDING PROTEIN-LIGAND PROBLEM

Chipot, C.; Pohorille, A. Free-energy calculations. Springer 2007.

ΔGa

site bulk

ΔGa

protein + ligand0 protein:ligand0

A A

ΔG0

A

ΔG0

B

protein + ligand0 protein:ligand0

B B

Relative binding affinity — alternate guests

ΔGa

site bulk

ΔGa

protein + ligand0 protein :ligand0

A A

ΔG0

A

ΔG0

B

protein + ligand0 protein :ligand0

B B

Relative binding affinity — alternate hosts ∆∆G0 = ∆G0

B − ∆G0 A = ∆Gsite a

− ∆Gbulk

a

  • Easier to carry out than standard binding free-energy calculations
  • Cheaper than standard binding free-energy calculations
  • Well-suited to series of congeneric compounds
  • May require the introduction of geometric restraints

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS RELATIVE BINDING FREE ENERGIES

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-18
SLIDE 18

conformational

6.6 kcal/mol 6.6 kcal/mol

EVALUATING THE CONTRIBUTIONS (P5)

∆G

bind(exp) = −5.52 kcal/mol

  • rientational

0.4 kcal/mol 0.3 kcal/mol 0.3 kcal/mol 0.2 kcal/mol 15.6 kcal/mol 0.2 kcal/mol

positional

  • 6.6
  • 0.4
  • 0.3
  • 0.3
  • 0.2
  • 0.2
  • 15.6

11.5 6.6

  • 5.5

p5: APTYPPPLNP

Abl-SH3 + p41 Abl-SH3:p41 Abl-SH3 + p5 Abl-SH3:p5 ∆Gunbound ∆Gbound ∆G

p41

∆G

p5

∆G

p41 − ∆G p5 = ∆Gbound − ∆Gunbound =125.7-125.7=0

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS RELATIVE BINDING FREE ENERGIES

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-19
SLIDE 19

THE LONG-STANDING PROTEIN-LIGAND PROBLEM RELATIVE BINDING FREE ENERGIES BEYOND PROTEIN-LIGAND BINDING

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS OUTLINE

THE GEOMETRICAL ROUTE THE ALCHEMICAL ROUTE

ALANINE SCANNING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-20
SLIDE 20

ALANINE-SCANNING EXPERIMENTS

Ramadoss, V.; Dehez, F. ; Chipot, C. J. Chem. Info. Model. 2016, 56, 1122-1126

Protein-substrate binding Thermal-shift assay PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS ALANINE SCANNING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-21
SLIDE 21

THE LONG-STANDING PROTEIN-LIGAND PROBLEM RELATIVE BINDING FREE ENERGIES

BEYOND PROTEIN-LIGAND BINDING

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS OUTLINE

THE GEOMETRICAL ROUTE THE ALCHEMICAL ROUTE ALANINE SCANNING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-22
SLIDE 22

PROTEIN-PROTEIN ASSOCIATION Measuring binding constants from one-dimensional separation potentials of mean force is justified in the limit of all other degrees

  • f freedom being appropriately sampled. This is true for small, fast-relaxing molecular species.

∆A(kcal/mol)

r(˚ A)

Ka = 4π Z Rc dr r2 exp[−β∆A(r)]

Shoup, D.; Szabo, A. Biophys. J. 1982, 40, 33-39 Woo, H. J.; Roux, B. Proc. Natl. Acad. Sci. USA 2005, 102, 6825-6830

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS BEYOND PROTEIN-LIGAND BINDING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-23
SLIDE 23

Measuring binding constants from one-dimensional separation potentials of mean force is justified in the limit of all other degrees

  • f freedom being appropriately sampled. This is true for small, fast-relaxing molecular species.

Ka = 4π Z Rc dr r2 exp[−β∆A(r)]

Shoup, D.; Szabo, A. Biophys. J. 1982, 40, 33-39 Gumbart, J. C.; Roux, B.; Chipot, C. J. Chem. Theor. Comput. 2013, 9, 3789-3798

In more complex molecular assemblies, e.g., protein-ligand complexes, the partners acquire upon separation additional configurational - i.e., conformational, positional and orientational entropy, not easily captured over timescales amenable to molecular dynamics.

∆A(kcal/mol)

r(˚ A)

PROTEIN-PROTEIN ASSOCIATION

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS BEYOND PROTEIN-LIGAND BINDING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-24
SLIDE 24

Keq = 1 [barstar] N Z

site

d1 Z

bulk

d2 . . . Z

bulk

dN Z dx e−βU Z

bulk

d1 Z

bulk

d2 . . . Z

bulk

dN Z dx e−βU = 1 [barstar] N Z

site

d1 Z dx e−βU Z

bulk

d1 Z dx e−βU Keq = 1 [barstar] N Vbulk Z

site

d1 Z dx e−βU Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU Keq = Z

site

d1 Z dx e−βU Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU

[barstar] = N Vbulk

x∗

1

alchemical route geometrical route

Gilson, M. K.; Given, J. A.; Bush, B. L.; McCammon, J. A. Biophys. J. 1997, 72, 1047-1069 Woo, H. J.; Roux, B. Proc. Natl. Acad. Sci. USA 2005, 102, 6825-6830

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS BEYOND PROTEIN-LIGAND BINDING

PROTEIN-PROTEIN ASSOCIATION

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-25
SLIDE 25

× Z

site

d1 Z dx e−β(U+uc+uo+up) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uo) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ+uΦ+uΨ) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ+uΦ) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ+uΦ) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uΘ) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uBS,res+uBN,res) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uBS,res) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc+uBS,res) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uc) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uBS,c+uBN,c) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uBS,c) × Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−β(U+uBS,c) Z

bulk

d1 δ(x1 − x∗

1)

Z dx e−βU

Gumbart, J. C.; Roux, B.; Chipot, C. J. Chem. Theory Comput. 2013, 9, 794-802 Woo, H. J.; Roux, B. Proc. Natl. Acad. Sci. USA 2005, 102, 6825-6830

Theoretical underpinnings

Keq = Z

site

d1 Z dx e−βU Z

site

d1 Z dx e−β(U+uBS,c) × Z

site

d1 Z dx e−β(U+uBS,c) Z

site

d1 Z dx e−β(U+uBS,c+uBN,c) × Z

site

d1 Z dx e−β(U+uc) Z

site

d1 Z dx e−β(U+uc+uBS,res) × Z

site

d1 Z dx e−β(U+uc+uBS,res) Z

site

d1 Z dx e−β(U+uc+uBS,res+uBN,res) × Z

site

d1 Z dx e−β(U+uc,all) Z

site

d1 Z dx e−β(U+uc,all+uΘ) × Z

site

d1 Z dx e−β(U+uc,all+uΘ) Z

site

d1 Z dx e−β(U+uc,all+uΘ+uΦ) × Z

site

d1 Z dx e−β(U+uc,all+uΘ+uΦ) Z

site

d1 Z dx e−β(U+uc,all+uΘ+uΦ+uΨ) × Z

site

d1 Z dx e−β(U+uc,all+uo) Z

site

d1 Z dx e−β(U+uc,all+uo+uθ) × Z

site

d1 Z dx e−β(U+uc,all+uo+uθ) Z

site

d1 Z dx e−β(U+uc,all+uo+uθ+uφ)

Ψ

P3 P2 P1 P’

1

P’

2

P’

3 Φ

Θ

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS BEYOND PROTEIN-LIGAND BINDING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-26
SLIDE 26

ΔG°= -19.0 kcal/mol

∆Gsite

BS,c

∆Gsite

BN,c

∆Gsite

BN,res

∆Gsite

BS,res

∆Gsite

Θ

∆Gsite

Φ

∆Gsite

Ψ

∆Gsite

θ

∆Gsite

φ

∆Gbulk

BS,res

∆Gbulk

BN,res

∆Gbulk

BN,c

∆Gbulk

BS,c

−1/β ln(S∗I∗c0)

  • 2.0±0.3
  • 3.1±0.1
  • 1.9±0.8
  • 3.5±0.6
  • 0.1±0.4
  • 0.4±0.1
  • 0.2±0.1
  • 0.1±0.3
  • 0.1±0.1
  • 37.1±0.3

+6.6 ∆Gbulk

  • +5.2±0.4

+4.2±0.5 +3.2±0.2 +8.1±0.3 6 12 12 24 8 4 8 4 4 212 15 18 24 21 ∆G0

bind

  • 21.0±1.4

372 component free energy (kcal/mol) time (ns)

backbone: +2.5 kcal/mol interface: +8 kcal/mol

  • rientation: +5.8 kcal/mol

ΔVeff = 12.8 Å3 ~15 kcal/

Gumbart, J. C.; Roux, B.; Chipot, C. J. Chem. Theory Comput. 2013, 9, 3789−3798

PROTEIN-LIGAND STANDARD BINDING FREE-ENERGY CALCULATIONS BEYOND PROTEIN-LIGAND BINDING

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018

slide-27
SLIDE 27

School of Physics Georgia Institute of Technology Department of Biochemistry and Molecular Biology Gordon Center for Integrative Science The University of Chicago Centre National de la Recherche Scientifique Laboratoire International Associ´ e CNRS-UIUC Universit´ e de Lorraine University of Illinois at Urbana-Champaign Beckman Institute for Advanced Science and Technology Theoretical and Computational Biophysics Group

Protein:ligand standard binding free energies: A tutorial for alchemical and geometrical transformations

James Gumbart Benoˆ ıt Roux Christophe Chipot

July 4, 2013

Please visit www.ks.uiuc.edu/Training/Tutorials/ to get the latest version of this tutorial, to obtain more tutorials like this one, or to join the tutorial-l@ks.uiuc.edu mailing list for additional help.

standard binding free energies

geometric free-energy calculations alchemical free-energy calculations

advanced tutorial

Contributors: Gumbart, J. C.; Hénin, J.; Fajer, M.; Roux, B.; Chipot, C.

INTRODUCTION TO FREE-ENERGY CALCULATIONS TUTORIALS

HANDS-ON WORKSHOP ON ENHANCED SAMPLING AND FREE-ENERGY CALCULATIONS

NIH CENTER FOR MACROMOLECULAR MODELING & BIOINFORMATICS, URBANA, ILLINOIS, SEPTEMBER 2018