free energy methods in drug...

Free Energy Methods in Drug Design

Jeff WereszczynskiNBCR Summer Institute

Friday, August 3, 12

There are multiple ways to estimate the free energy of ligand binding

• In general, we wish to compute the free energy of an arbitrary ligand binding to a binding site:

• Although docking is a powerful technique, there are other methods which may be more appropriate for some problems

Biomolecular Docking:Rigid Docking, Flexible Docking

End Point Calculations:MM-PBSA, MM-GBSA,

LIE, Mining Minima

Thermodynamic Pathways: Alchemical Transformations,

Potential of Mean Force

AccuracySpee

d

ΔGbind


MM-GBSA: A middle-ground in accuracy and cost

• To calculate ΔGbind, estimate the absolute free energy before and after binding, and take the difference

• The absolute free energy binding for each state can be estimated by:

• Estimating the total free energy of binding is therefore:

ΔGbind

GcomplexGreceptor+Gligand

ΔGbind = Gcomplex − Greceptor +Gligand( )

G =U −TS = EMM −T SMM + Gsolv + Gnp

ΔGbind = EMM complex − EMM ligand + EMM receptor( ){ }− T SMM complex − SMM ligand + SMM receptor( ){ }+ Gsolv complex − Gsolv ligand + Gsolv receptor( ){ }+ Gnp complex

− Gnp ligand+ Gnp receptor( ){ }


Calculating all those different terms

• Analyze solute conformations from MD simulations to compute:

• Molecular Mechanics term (<EMM>): force field energy

• Conformational entropy (T<SMM>): typically either harmonic or quasiharmonic approximations

• Polar solvation term (<Gsolv>): Generalized Born or Posson-Boltzmann solvation energy

• Non-polar solvation energy (<Gnp>): typically assumed to be proportional to solvent exposed surface area (SASA)

• Simulations are typically performed in one of two ways:

• Single trajectory method: run a simulation of the bound complex, extract the complex, receptor, and ligand conformations

• Three trajectory method: run different simulations of complex, receptor, and ligand


How well does this work?

• Each energy term has its limitations:

• MM: Noisy energy function

• Entropy: Both the quasiharmonic and harmonic approximations have several limitations and offer upper bounds to the entropy which may be significantly off from the true value

• Polar solvation term: Both PB and GB rely on several parameters, and don’t account for specific protein/water interactions

• Non-polar solvation term: Surface area has only a rough fit to non-polar solvation free energy

• Not all the terms are always included

• Energies tend to be exaggerated

• Despite all this, MM-GBSA can often provides much better correlation to binding energies then docking scores

• Typically works best for congeneric series


More Rigorous Methods: Computing the free energy along the binding pathway

• The free energy profile of brining a ligand into the active site can be computed:

• Umbrella sampling, metadyanmics, adaptive biasing force, etc.

• Integrating the PMF over the binding site gives ΔGbind:

• This method is rarely used for several reasons:

• Protein rearrangement along binding pathway (especially for buried binding sites)

• Sampling of large amounts of configuration space is required

• System sizes

ΔGbind

e−ΔG/kT = 4π r2e−W r( )/kT

site∫ dr

Distance from binding site (r)

Free

Ene

rgy

(W(r)

)


Alchemical free energy methods: Using a thermodynamics cycle to make your life easier

• By drawing a thermodynamic cycle, ΔGbind can be computed in an alternative way

• In these alchemical calculations, the binding energy can be calculated as the difference in free energies of removing a ligand from two systems, its binding environment and a solvent environment

• These calculations avoid many problems inherent to PMF based calculations

ΔGbind

ΔGwater ΔGprotein

ΔGbind = ΔGwater- ΔGprotein

ΔG = 0


Free Energy Perturbation (FEP) can (in theory) calculate alchemical free energy differences

• The Zwanzig equation can be used to compute free energy differences between two states:

• In practice, this only converges if the endpoints are close in phase space and there is sufficient “overlap”

• To improve convergence, a series of intermediate “windows” may simulated, which are defined by the variable λ:

• The overall free energy change can then be written as a sum of free energy changes between N windows:

Ha(r,p)

Hb(r,p)

ΔGa→b = −β−1 ln e−β U (r )b−U (r )a( )

a

H r, p;λ( ) = λHb r, p( )+ 1−λ( )Ha r, p( )

λ=0

λ=1

λ=0.5

λ=0.25

λ=0.75

0.75*Ha(r,p)+0.25*Hb(r,p)

0.5*Ha(r,p)+0.5*Hb(r,p)

0.25*Ha(r,p)+0.75*Hb(r,p)

ΔGa→b = −β−1 ln e−β U (r )i+1−U (r )i( )

ii=1

N−1

∑


What does it mean to mix Hamiltonians???

• The potential energy of the system can be partitioned into three parts:

• If the ligand is “decoupled” from its environment, this becomes:

• For intermediate values of λ this implies:

• This means, to compute the energy the potential energy at a specified λ value, the nonbonded potential energy terms between the ligand and surrounding atoms are scaled by (1-λ)

U r;λ = 0( ) =U r( )surrounding +U r( )ligand +U r( )ligand /surrounding

λ=1U r;λ =1( ) =U r( )surrounding +U r( )ligand

U r;λ( ) =U r( )surrounding +U r( )ligand + 1−λ( )U r( )ligand /surrounding1−λ( )U r( )ligand /surrounding = 1−λ( )UvdW i, j( )+ 1−λ( )Uelec i, j( )( )

j=surrounding∑

i=ligand∑

λ=0


Thermodynamic Integration: Avoiding that messy exponential

• Instead of calculating the average energy change between adjacent windows, calculate the average derivative of the energy with respect to λ at a window, and integrate:

• Proper integration techniques are important: the use of Simpson’s rule or Gaussian quadrature is recommended

ΔGa→b = dλ∂H λ( )∂λ λλa

λb

∫∂H λ( )∂λ

=∂Uelec λ( )∂λelec

+∂UvdW λ( )∂λvdW

Statistical Analysis of Uncertainties. Two alternativestatistical procedures were employed to evaluate the uncer-tainty ! for !GAfB or !GjAfB free-energy estimates.

First, a simulation standard error !sim(t) of the time-varyingHamiltonian derivative at a given " can be calculated as

with T being the total number of block averages41 throughoutthe single ith trajectory or all N concatenated independenttrajectories. (!Ht(")/!")" denotes the Hamiltonian derivative,block-averaged at time t, and 〈!HT(")/!"〉" is the ensembleaverage over the entire simulation time at a given ". As anexample, !sim(t) uncertainties are reported as error bars for〈!HT(")/!"〉" vs " in Figure 3 (solid black curve). Then, acorresponding free-energy uncertainty can be obtained as

This follows from the standard assumption that (!Ht(")/!")" values are statistically uncorrelated along the time over

different values of the coupling parameter ". However, the!!Gi uncertainty includes the physically based fluctuationsof (!Ht(")/!")", though corresponding noise is typicallyreduced by block-averaging.41 Therefore, despite its wideuse in the literature, !!Gi is a questionable measure ofuncertainty for a free-energy change of interest. For example,considering that overlap of phase space at neighboring "values is a requirement for smooth 〈!HT(")/!"〉" vs " curves(Figure 3), one could claim that (!Ht(")/!")" time series arestatistically correlated. Nonetheless, the abovementioneduncertainty defined in eq 4 is representing the lowest possibleuncertainty for a free-energy-change estimate from standardTI. Thus, it seems the fairest choice for this study comparingTI vs IT-TI results.

Second, for IT-TI, a statistics-based uncertainty !!Gj on agiven free-energy change !GjAfB from eq 2 can be calculatedas the standard deviation from the mean (standard error) ofthe N !Gi results

where !!G is the standard deviation of the free-energy changeover the N IT-TI trajectories employed. Importantly, !!Gj hasa clear statistical validity,42 because of its explicit dependenceon the repeated independent estimates.

Similarly, for a general overall free-energy change !GjAfB,calculated as the difference between two free-energy changes!Gj B and !GjA, a corresponding uncertainty can be obtainedby propagating the respective uncertainties as42

Then, the relative uncertainty for a given free-energychange A f B reads

In this study, IT-TI runs were extended to obtain suf-ficiently smooth curves of 〈!Hi(")/!"〉" vs " (Figure 3). IT-TI trajectories were independently equilibrated (0.5 ns) foreach of the 26 " points (from five initial equilibrated " ) 0configurations), followed by independent sampling periods(0.5 ns) used for free-energy estimation. Increased sampling(up to 2.5 ns) times were required in the ranges 0.12 e " e0.24 and 0.76 e " e 0.92. A summary of these calculationsis given in Supporting Information, Table S2. All annihilationand modification perturbations fulfilled the criterion !!Gj(%)< 6%. Only N1-TAIL1 and N1-TAIL2 modification per-turbations had larger !!Gj(%) values (up to 52%) due to thecorresponding small !GjAfB values (Supporting Information,Table S3).

Separation of Thermodynamic States. For !Gj N1(L)(Scheme 2a), the potential U(rL) ) -1/2k(rL - r0)2 was applied to harmonically restrain ligandtranslation and ensure its sampling of a finite phase-spacevolume V". An optimal k value of 246.5 kJ mol-1 nm-2

was estimated from the ensemble-averaged L root-mean-

Figure 3. 〈!H(")/!"〉" values and corresponding uncertainties!sim(t) vs ". (a) Annihilation of PVR in N1 binding site (!Gj N1(L);Scheme 2a). (b) Annihilation of PVR in water (!Gj wt(L);Scheme 2a). Black lines: average values over all N individualtrajectories. Gray lines: individual trajectory TI curves. Insetpanels highlight " regions where IT-TI averages outperformstandard individual TI calculations.

!sim(t) ) ! 1T - 1∑t)1

T [(!Ht(")

!" )"- 〈!HT(")

!" 〉"]

2

/√T

(3)

!!Gi) (""A

"B !sim2(t) d")1/2

(4)

!!Gj )!!G

√N(5)

!!GjAfB) √(!!GjA

)2 + (!!GjB)2 (6)

!!GjAfB(%) )

!!GjAfB

!GjAfB

× 100 (7)

1110 J. Chem. Theory Comput., Vol. 5, No. 4, 2009 Lawrenz et al.


Bennett Acceptance Ratio (BAR): Another option to improve convergence

• Unlike FEP, calculate the work to transition to windows both before and after the one being simulated

• Compute ΔF between adjacent windows using the maximum likelihood estimator:

• An extension of BAR also exists which uses work functions between all the simulated windows, called the multistate Bennet acceptance ratio (MBAR)

λ=i λ=i+1λ=i-1

Wi->i+1Wi->i-1

Wi-1->i Wi+1->i

11+ exp β ln(nF / nR )+Wi − ΔF( )( )i=1

nF

∑ − 11+ exp β ln(nR / nF )+Wj − ΔF( )( )j=1

nR

∑ = 0


Comparing FEP, TI, BAR, and MBAR: Identical simulations, different data collection and analysis

λ=0 λ=1λ=0.5λ=0.25 λ=0.75

FEP:Work in one

direction

W0->0.25 W0.25->0.5 W0.5->0.75 W0.75->1

BAR:Work in two

direction W0.75->0.5

W0->0.25 W0.25->0.5 W0.5->0.75 W0.75->1W0.25->0 W0.5->0.25

W1->0.75

TI:Derivatives of energy

∂U λ( )∂λ λ=0

∂U λ( )∂λ λ=0.25

∂U λ( )∂λ λ=0.5

∂U λ( )∂λ λ=0.75

∂U λ( )∂λ λ=1

MBAR:Work between

all windowsFriday, August 3, 12

Comparing FEP, TI, BAR, and MBAR: Which should you use?

• FEP gives biased estimates and has poor convergence, don’t use it.

• MBAR is seldom better then BAR

• TI and BAR give comparable results, BAR may be easier as it doesn’t require integration


Reasons you might not get the right answer

1.The force field

2.User error

3.Sampling issues


Improving your sampling

1.Run longer

2.Sample only the parts of phase space you are interested in

3.Change how you move through “λ-space”

4.Use a soft-core potential

5.Dynamically change your Hamiltonian with “λ-hopping”


Restraining your ligand when decoupling it from the protein

• You only want to sample when the ligand is in the protein binding site

• To prevent it from moving too much, restraints may be added to the ligand

• A common choice is to restrain the cartesian coordinates of a ligand atom (and three protein atoms)

• Pick the correct restraint:

• Account for the restraint in your free energy estimate:

• Other restraints choices are are possible and may be helpful

k = 3kbTrmsf( )2

V1 =2π kbTk

⎛⎝⎜

⎞⎠⎟3/2

ΔGrest = kbT ln CV1( )


Getting from λ=0 to λ=1It doesn’t matter how you get there, just get there

• Free energy is a state function, so it doesn’t matter how you get from λ=0 to λ=1

• Sampling is often improved by using different λ for van der Waals and electrostatics, and first decoupling the electrostatics

λvdWλ e

lec

0 10

1

λvdW= λelec


Soft-core potentials help to avoid catastrophes

• “Soft-core” potentials are typically used for vdW interactions

9026 Zacharias, Straatsma, and McCammon: Separation-shifted scaling

et aI.,14 since it allows the smooth transition in molecular simulations between real atoms and dummy atoms, for which all atomic interactions have vanished. This can be done with-out sprouting or desprouting of bonds. The approach was tested on the creation and annihilation of neon in water to optimize the shifting parameter. In addition, the free energy of hydration of a small molecule, ethanol, was determined.

II. METHODS A. Thermodynamic integration

Thermodynamic integration is a computational tech-nique to evaluate free energy differences from molecular dy-namics simulations. In the course of a molecular dynamics simulation, the Hamiltonian of a system initially in state A is slowly interconverted to a Hamiltonian representing state B. The two states are coupled by a variable }... In TI, an analytic expression for the derivative of the Hamiltonian with respect to }.. is evaluated at each time step of the simulation and integrated over the change in }... The free energy difference is then given by

(1)

In a computer simulation, this expression for the free energy difference between A and B is approximated by a sum over ensemble averages for the derivative of the Hamiltonian vs }.. at discrete steps in }..,2,3 .

dG= d}..i' , Ai

(2)

In the present article single configuration thermodynamic integrationS (SeTI) and multiconfiguration thermodynamic integration 15 (MeTI) were used. seTI is the extreme case of replacing the ensemble average in Eq. (2) by a single value for each molecular dynamics (MD) time step, meaning that}.. varies at each time step by an amount d}" which equals the inverse of the number of MD time steps performed for the TI. In MeTI, ensemble averages at a few values of }.. are evaluated independently for each }...IS This offers the possi-bility of systematically improving the calculations for each step in }...

B. Separation-shifted scaling of LJ potentials The LJ potential used in GROMOS (Ref. 16) and ARGOS

(Ref. 17) has the following form:

A B VU =12-:6 . r r (3)

Here, r is the interatomic distance, A and B are repulsive and attractive LJ parameters, respectively, calculated as a product of LJ parameters specific for atom types of the interacting partic1es.16 The LJ term of the Hamiltonian in TI can be coupled to }.. in the following way:

(4)

A and B correspond to the initial and final state of the TI, respectively. In case of the annihilation of an atom, which in practice is the transformation to a dummy atom, the LJ potential in state B is zero and Eq. (4) becomes

(5)

The LJ potential used in the present study to annihilate or create an atom has the following form:

(6)

For },,=O, the atom is present and the above expression is identical with Eqs. (3) or (5). At }"=1 the potential is zero everywhere, corresponding to state B where the atom is transformed to a dummy particle. The shift parameter {} comes into play for}.. larger than O. It allows a smooth tran-sition from the original LJ potential to nothing or vice versa, as can be seen from Fig. 1.

In addition, this potential and its derivative are continu-ous and finite for all r, except for }..=O and r=O. Note that the derivative of the potential vs r is zero at r=O, except when A=O. This can be an important feature because a finite derivative of the potential vs r at r=O could cause a discon-tinuous sign reversal of the force on an atom passing through the atom under study. For the above potential the sign rever-

: ..

:

Interatomic distance

FIG. 1. Comparison of linearly scaling (upper diagram) and separation-shifted scaling of LJ potentials (lower diagram). The LJ potentials, as given by Eqs. (5) and (6), are plotted as a function of interatomic distance for 10 values of lI. between 0 and 1. LJ parameters used are A = 1 and B =2, re-spectively, and the shift parameter 8=0.5. Potential energy and distance are in arbitrary units, but the same for (a) and (b).

J. Chem. Phys., Vol. 100, No. 12, 15 June 1994

Hard-corepotential

9026 Zacharias, Straatsma, and McCammon: Separation-shifted scaling

et aI.,14 since it allows the smooth transition in molecular simulations between real atoms and dummy atoms, for which all atomic interactions have vanished. This can be done with-out sprouting or desprouting of bonds. The approach was tested on the creation and annihilation of neon in water to optimize the shifting parameter. In addition, the free energy of hydration of a small molecule, ethanol, was determined.

II. METHODS A. Thermodynamic integration

Thermodynamic integration is a computational tech-nique to evaluate free energy differences from molecular dy-namics simulations. In the course of a molecular dynamics simulation, the Hamiltonian of a system initially in state A is slowly interconverted to a Hamiltonian representing state B. The two states are coupled by a variable }... In TI, an analytic expression for the derivative of the Hamiltonian with respect to }.. is evaluated at each time step of the simulation and integrated over the change in }... The free energy difference is then given by

(1)

In a computer simulation, this expression for the free energy difference between A and B is approximated by a sum over ensemble averages for the derivative of the Hamiltonian vs }.. at discrete steps in }..,2,3 .

dG= d}..i' , Ai

(2)

In the present article single configuration thermodynamic integrationS (SeTI) and multiconfiguration thermodynamic integration 15 (MeTI) were used. seTI is the extreme case of replacing the ensemble average in Eq. (2) by a single value for each molecular dynamics (MD) time step, meaning that}.. varies at each time step by an amount d}" which equals the inverse of the number of MD time steps performed for the TI. In MeTI, ensemble averages at a few values of }.. are evaluated independently for each }...IS This offers the possi-bility of systematically improving the calculations for each step in }...

B. Separation-shifted scaling of LJ potentials The LJ potential used in GROMOS (Ref. 16) and ARGOS

(Ref. 17) has the following form:

A B VU =12-:6 . r r (3)

Here, r is the interatomic distance, A and B are repulsive and attractive LJ parameters, respectively, calculated as a product of LJ parameters specific for atom types of the interacting partic1es.16 The LJ term of the Hamiltonian in TI can be coupled to }.. in the following way:

(4)

A and B correspond to the initial and final state of the TI, respectively. In case of the annihilation of an atom, which in practice is the transformation to a dummy atom, the LJ potential in state B is zero and Eq. (4) becomes

(5)

The LJ potential used in the present study to annihilate or create an atom has the following form:

(6)

For },,=O, the atom is present and the above expression is identical with Eqs. (3) or (5). At }"=1 the potential is zero everywhere, corresponding to state B where the atom is transformed to a dummy particle. The shift parameter {} comes into play for}.. larger than O. It allows a smooth tran-sition from the original LJ potential to nothing or vice versa, as can be seen from Fig. 1.

In addition, this potential and its derivative are continu-ous and finite for all r, except for }..=O and r=O. Note that the derivative of the potential vs r is zero at r=O, except when A=O. This can be an important feature because a finite derivative of the potential vs r at r=O could cause a discon-tinuous sign reversal of the force on an atom passing through the atom under study. For the above potential the sign rever-

: ..

:

Interatomic distance

FIG. 1. Comparison of linearly scaling (upper diagram) and separation-shifted scaling of LJ potentials (lower diagram). The LJ potentials, as given by Eqs. (5) and (6), are plotted as a function of interatomic distance for 10 values of lI. between 0 and 1. LJ parameters used are A = 1 and B =2, re-spectively, and the shift parameter 8=0.5. Potential energy and distance are in arbitrary units, but the same for (a) and (b).

J. Chem. Phys., Vol. 100, No. 12, 15 June 1994

Soft-corepotential

UvdW r,λ( ) = 1− λ( ) Ar2 +δγ( )6

− Br2 +δγ( )3

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Zacharias, Straatsma, & McCammon, J. Chem. Phys 1994Friday, August 3, 12

“λ-hopping”: swapping λ values may help improve sampling

Simulate

Check for swap

Simulate

Check for swap

Simulate

Check for swap

Simulate

•Instead of running a simulation at a constant λ value, the Hamiltonian can be swapped periodically through the use of a Metropolis criteria (Hamiltonian replica exchange)


Making smaller perturbations to your system with relative alchemical calculations

• The change in binding free energy between two ligands can be computed with relative alchemical calculations

• These calculations introduce less perturbation to the system

• Requires the modification of bonded terms in your λ schedule

ΔG1

ΔΔGwater ΔΔGprotein

ΔΔGbind = ΔG1-ΔG2=ΔΔGwater- ΔΔGprotein

ΔG2


Some other practical tips

• Decoupling of an entire ligand tends to require 15-20 windows of 2-5 ns in length each (for protein phase)

• Each window should be partitioned into an “equilibration” and “sampling” phase

• Running each window multiple times is a good idea to improve convergence and compute errors

• Use the appropriate strength for a restraint

• If possible, compute relative free energies of binding rather then absolute ones

• As with other MD methods, the more gentle you can be the better


Related alchemical methods

• λ-dynamics: Treats λ as a dynamical variable in the simulation

• Knight & Brooks, “λ-dynamics free energy simulations methods” J. Comp. Chem. (2009) 30, 1692

• Envelope distribution sampling: Sample at only a single, non-physical Hamiltonian

• Christ & van Gunsteren “Multiple free energies from a single simulation: extending enveloping distribution sampling to nonoverlapping phase-space distributions” J. Chem Phys. (2008) 128, 174112


Other Good References

• Lawrenz, Baron, Wang, & McCammon “Effects of biomolecular flexibility on alchemical calculations of absolute binding free energies” J. Chem. Theor. Comp. (2011) 7, 2224

• Wereszczynski & McCammon “Statistical mechanics and molecular dynamics in evaluating thermodynamic properties of biomolecular recognition” Quar. Rev. Biophys. (2012) 45, 1

• Christ, Mark, & Van Gunsteren “Basic ingredients of free energy calculations: A review” J. Comput. Chem. (2010) 31,8

• Chodera, Mobley, Shirts, Dixon, Branson, & Pande “Alchemical free energy methods for drug discovery: progress and challenges” Curr. Opin. Struct. Bio. (2011) 21, 150

• Zhou & Gilson “Theory of free energy and entropy in noncovalent binding” Chem. Rev. (2009) 109, 4092

• Shirts “Best practices in free energy calculations for drug design” Methods Mol. Biol. (2012) 819, 426


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