molecular flexibility esther kellenberger faculté de pharmacie umr 7200, illkirch tel: 03 68 85 42...

Molecular Flexibility

Esther Kellenberger

Faculté de PharmacieUMR 7200, Illkirch

Tel: 03 68 85 42 21 e-mail: [email protected]

1/27

Molecules have geometries…

Geometry-basedsamplingForce field 2/27introduction Energy-based

sampling conclusion

Methotrexate, used in treatment of cancer, autoimmune diseases

methotrexate bound to therapeutcal targets (dihydrofolate reductase and thymidilate synthase)

… « good » geometries in bioactive conformations

Molecules have geometries…


sampling conclusion

unusual bond length, steric collisions, distorded ring, …

… and there are imposible conformations

The number of molecular conformations


sampling conclusion

= Number of rotatable bonds (NROT)

Appr. number of simple bonds between

two non-hydrogen atoms.

For methotrexate, NROT= 10

Considering 3 possible angular values for each

NROT yields 310 = 59 049 different conformations

… depends on the molecular degrees of freedom

How to evaluate the conformations?


sampling conclusion

In physics, potential energy exists when a force acts

upon an object that tends to restore it to a lower

energy configuration.

Potential energy is the energy stored in a body or in a

system due to its position in a force field or due to its

configuration (SI unit= Joules, common unit = kcal/mol,

1 cal = 4.1868 J)

A force field is a vector field that describes a non-

contact force acting on a particle at various positions in

space.

potential energy

unstable (bad) conformationhigh energy

stable (good) conformationlow energy

Experimental properties of a molecular is an mean of properties of populated conformers


sampling conclusion

Boltzmann’s probability distribution

P (conformer of energy E) ~ exp ( - E / kb T)

Boltzmann averaging for the observed property

Property (molecule) = Σ P(conformer) X property(conformer)

Boltzmann’s probability distribution

0

0.2

0.4

0.6

0.8

1

-1000 -800 -600 -400 -200 0

X

EXP(X/T)

T=50

T=273

T=1000

Chapter1:

Evaluation

of the potential

energy

of conformers

7/27

Molecular systems are modeled using Newton’s

laws:

• each atom is simulated as a single particle

• each particle is assigned a radius (van der

Waals), polarizability, and a constant net charge

• bonded interactions are treated as "springs"

with an equilibrium distance equal to the bond

length

Molecular system's potential energy (E) in a given

conformation as a sum of individual energy terms:

E = E covalent + E non covalent

Molecular mechanics


sampling conclusion

Covalent contributions to E


sampling conclusion

Bond

stretching

Angle

stretching

Ex.of « standard » values:r0=1.53Å for Csp3‐Csp3

r0=1.09Å for C‐H

Torsion correction term

Ex. of « standard » values:θ0= 109.5° for Csp3

θ0= 120° for Csp2

θ0= 180° for Csp

Ex.of values:for Csp3‐Csp3

n= 3, γ= 0Etors = 0 at 60°, 180° & -60°

Non covalent contributions to E


sampling conclusion

Van der Waals term

Lennard Jones potential (6-12)

EVdW = A / rij12 – B/rij

6

where A = 4 εσ12 B = 4 εσ6

ε = depth of the well

σ ~ distance with minimum EVdW

Electrostatic term

Coulomb’s law

Ecoulomb = δ + δ - / 4πε0 rij

where δ = charge

ε0 = solvent dielectric constant

Desolvation

and

hydrophobic

term


sampling conclusion

Energy

Local minimum

Global minimum

Conformational state

Local minimum

high barrier

low barrier

Key points on the energy surface

« good » geometries

« ugly » geometries


sampling conclusion

Given a starting geometry, deterministic algorithms allow

the discovery of the adjacent local minimum.

Energy minimization

Energy


starting

starting

final

final

Amplitude of motioncontroled by heat


sampling conclusion


Energy

Molecular dynamics trajectory may be seen as an exchange of potential and kinetic energy, with total energy being conserved. The dynamic system consists of moving particles (i.e. molecular atoms with coordinates and velocities). Particle position as a function of time is obtained by solving equation from the Newton’s laws.

The limits of conformational exploration by molecular dynamics

starting

heating

minimisation

sampling depends on the number of frames (time)

Chapter2:

exploration of the

molecular energy

landscape

14/27

Torsions : the gateway to conformational sampling


sampling conclusion

Energy surface with respect to two torsions

ONH

OHO

NH2

O

angular incremental or random change

of selected rotatable bonds

Solutions sorted by Energy (relative)

Systematic Search and random search


sampling conclusion

1. Enumerating ring conformations and invertible nitrogen atoms (fragment library)2. Torsion alteration3. Reassembly4. Evaluation

MMFF force fieldKnowledge based Tables

Generation of haloperidol 3D conformers by omega

http://www.eyesopen.com/products/applications/omega.html

pairwise rmsd>2.5Å, Energy threshold 28 conformers


sampling conclusion

Geometry-based sampling methods:

• a systematic search is possible if NROT < 4-5

• Enumeration restricted to a fixed number of conformers for flexible

compounds (Ex: 200 in omega)

Energy-based sampling methods:

• (molecular dynamics )

• stochastic sampling: Monte-Carlo and Genetic algorithm

Increasing complexity of energy hypersurface …


sampling conclusion

EnergiEnergy

random modification of conformations combined with acceptation criteria

motion toward energetically favored regions



sampling conclusion

Monte Carlo

ONH

OHO

NH2

O

O

NH

OHO

NH2

O

yes,

Perform move

Evaluate E(x)

acceptance test

replace state

no, restore

previous state

X stepsInitial state

Better energyyes

no


sampling conclusion

Monte carlo algorithm

Randomly chosen torsional axisRandom rotation around that axis

Χ11 Χ1

2 … Χ1n

Χ21 Χ2

2 … Χ2 n

Test

if Ef < Ei new pose is accepted

if Ef > Ei calculate probability

P of acceptance

Compare P with random number h

if h < P new pose accepted

if h > P restart based on last accepted pose

Acceptation criteria

The Boltzmann statistics: P is also called the Bolzmann factor

= eP = exp ÷÷ççè

æ- ÷÷çç

è

æ-

kT

Ef -Ei k: boltzman constantT: temperature

Large energy differences and

low temperature lower the

Boltzmann factor P

acceptance range goes down


sampling conclusion

ççè

æçè


sampling conclusion

Genetic algorithm

Genetic in the real world

Genotype : ensemble of genes contained

in chromosomes. Diploid organism : 2

copies of each gene.

Phenotype : ensemble of individual

features, resulting from gene

expression.

Evolution

environment selection pressure

survival if adapted phenotype

parent 1 parent 2

+

Reproduction

Chromosomesgeneration 1

gene2 copies

child 1 child 2 child 3

genera-tion 2

generation 3

&

&

evolutiondominant genes adapted phenotype

recessive genes inadapted phenotype

increased diversity after:

Cross-over mutation *

parent 1 parent 2

+

Reproduction

generation 1

child 1

generation 2

child 2

*


sampling conclusion

Genetic in the real world (continued)

parent 1 1101100100110110parent 2 1100111000011110

« crossover » : mixing 2 chromosomes (random position)

parent 1 11011 | 00100110110parent 2 11001 | 11000011110child 1 11011 | 11000011110child 2 11001 | 00100110110

« mutation » : random modification of one (or more) string

parent 1 1101111000011110parent 2 1100100100110110child 1 1101011000011110child 2 1101101100110110

« selection »: energy below a selection threshold (fitness)

« chromosome »: fingerprint which codes ligand conformation (e.g., Torsions: binary coding of the angle value)

« virtual genetic »


sampling conclusion

Conv

erge

nce:

evo

lutio

n of

the

aver

age/

best

fitn

ess

max

num

ber o

f gen

erati

ons

Genetic operators

Selection fitness score (green), Survival rate (4)


sampling conclusion

Χ11 Χ1

2 … Χ1n

Χ21 Χ2

2 … Χ2n

Χ31 Χ3

2 … Χ3n

Χ11 Χ1

2 … Χ1n

Χ21 Χ2

2 … Χ2n

Χ41 Χ4

2 … Χ4n

Χ31 Χ3

2 … Χ3n

Χ41 Χ4

2 … Χ4n

Χ51 Χ5

2 … Χ5n

Χ61 Χ6

2 … Χ6n

Χ71 Χ7

2 … Χ7n

Χ81 Χ8

2 … Χ8n

random

crossoverrate

mutationrate

Χ11 Χ1

2 … Χ1n

Χ21 Χ2

2 … Χ2n

Χ31 Χ3

2 … Χ3n

Χ41 Χ4

2 … Χ4n

initial population

Size (4) individuals sorted by energy (color: high fitness low fitness)

Intermediate population

Final population

Genetic algorithm is an optimization method:

How to preserve the diversity?

• Selection pressure: child chromosome replace the worst members of the population / bias in

the selection of parent chromosomes (towards high fitness or favoring torsion values seen in in

previous populations)

• Multiple islands model: population split into sub-populations, with parallel simulations and

occasionally swapping solutions (migration)

• Discard of redundant chromosomes (requires a metric to evaluate the similarity of individuals)

the niche model: a niche is a ensemble of similar individuals in a population (as estimated by

RMSD). If there a more than niche size individuals in the niche, then the new individual is

replaces the worst individual of the niche rather than the worse individual of the population, in

order to preserve diversity within the population.


sampling conclusion

CONCLUSION

• Conformational Sampling is the key element for understanding of molecular

behavior

• It may range from very simple to extremely difficult, to impossible

• If you don’t do it well, better don’t do it at all: empirical methods based on

molecular topology only may be more accurate than 3D models based on wrong –

or too few – conformations

• Two main sources of errors: A.) wrong calculated energy- geometry landscape

(poor Force Field parameterization) and B.) – insufficient sampling!

Thanks to Dragos Howarth!


sampling conclusion

molecular flexibility esther kellenberger faculté de pharmacie umr 7200, illkirch tel: 03 68 85 42...

Documents