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1/27. Molecular Flexibility Esther Kellenberger Faculté de Pharmacie UMR 7200, Illkirch Tel: 03 68 85 42 21 e-mail: [email protected]. introduction. Force field. Geometry-based sampling. Energy-based sampling. conclusion. 2/ 27. Molecules have geometries …. - PowerPoint PPT Presentation
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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…
Geometry-basedsamplingForce field 3/27introduction Energy-based
sampling conclusion
unusual bond length, steric collisions, distorded ring, …
… and there are imposible conformations
The number of molecular conformations
Geometry-basedsamplingForce field 4/27introduction Energy-based
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?
Geometry-basedsamplingForce field 5/27introduction Energy-based
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
Geometry-basedsamplingForce field 6/27introduction Energy-based
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 0X
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
Geometry-basedsamplingForce field 8/27introduction Energy-based
sampling conclusion
Covalent contributions to E
Geometry-basedsamplingForce field 9/27introduction Energy-based
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
Geometry-basedsamplingForce field 10/27introduction Energy-based
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 wellσ ~ distance with minimum EVdW
Electrostatic term
Coulomb’s law
Ecoulomb = δ + δ - / 4πε0 rij
where δ = chargeε0 = solvent dielectric constant
Desolvation and hydrophobic term
Geometry-basedsamplingForce field 11/27introduction Energy-based
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
Geometry-basedsamplingForce field 12/27introduction Energy-based
sampling conclusion
Given a starting geometry, deterministic algorithms allow
the discovery of the adjacent local minimum.
Energy minimization
Energy
Conformational state
starting
starting
final
final
Amplitude of motioncontroled by heat
Geometry-basedsamplingForce field 13/27introduction Energy-based
sampling conclusion
Conformational state
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
Geometry-basedsamplingForce field 15/27introduction Energy-based
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
Geometry-basedsamplingForce field 16/27introduction Energy-based
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
Geometry-basedsamplingForce field 17/27introduction Energy-based
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 …
Geometry-basedsamplingForce field 18/27introduction Energy-based
sampling conclusion
EnergiEnergy
random modification of conformations combined with acceptation criteria
motion toward energetically favored regions
Conformational state
Geometry-basedsamplingForce field 19/27introduction Energy-based
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 steps Initial state
Better energyyes
no
Geometry-basedsamplingForce field 20/27introduction Energy-based
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 criteriaThe 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
Geometry-basedsamplingForce field 21/27introduction Energy-based
sampling conclusion
ççè
æçè
Geometry-basedsamplingForce field 22/27introduction Energy-based
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.
Evolutionenvironment 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
*
Geometry-basedsamplingForce field 23/27introduction Energy-based
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 »
Geometry-basedsamplingForce field 24/27introduction Energy-based
sampling conclusion
Conv
erge
nce:
evo
lutio
n of
the
aver
age/
best
fitn
ess
ma x
num
b er o
f ge n
e ra ti
o ns
Genetic operators
Selection fitness score (green), Survival rate (4)
Geometry-basedsamplingForce field 25/27introduction Energy-based
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.
Geometry-basedsamplingForce field 26/27introduction Energy-based
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!
Geometry-basedsamplingForce field 27/27introduction Energy-based
sampling conclusion