C7790 Introduction to Molecular Modelling -1C7790
Introduction to Molecular
Modelling
TSM Modelling Molecular Structures
Petr Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
PS/2021 Present Form of Teaching: Rev2
Lesson 22
Large Models - Ensembles Averages
C7790 Introduction to Molecular Modelling -2-
Context
microworldmacroworld
equilibrium (equilibrium constant)
kinetics (rate constant)
free energy
(Gibbs/Helmholtz)
partition function
phenomenological thermodynamics
statistical thermodynamics
microstates
(mechanical properties, E)
states
(thermodynamic properties, G, T,…)
microstate ≠ microworld
Description levels (model chemistry):
• quantum mechanics
• semiempirical methods
• ab initio methods
• post-HF methods
• DFT methods
• molecular mechanics
• coarse-grained mechanics
Structure
EnergyFunction
Simulations:
• molecular dynamics
• Monte Carlo simulations
• docking
• …
C7790 Introduction to Molecular Modelling -3Revision:
Statistical thermodynamics
Statistical approach:
Statistical physics (statistical mechanics) relates two levels of description of physical
reality, namely the macroscopic and microscopic levels. In a more traditional sense, it
deals with the study of the properties of macroscopic systems or systems, considering
the microscopic structure of these systems (statistical thermodynamics). The founders
were Ludwig Boltzmann and Josiah Willard Gibbs.
Level of description:
▪ particles and interactions between them
▪ equations of motions
wikipedia.cz, simplified
C7790 Introduction to Molecular Modelling -4Revision:
PES
➢ Increasing degrees of freedom (model size) result in increased roughness of PES.
➢ The only reasonable description of large models is to use statistical weighting using
right sampling technique.
Small models
E(x)
xreaction coordinate
reactant
configuration
product configuration
𝐸(𝜉 𝑃)
𝐸(𝜉 𝑅)
Δ𝐸𝑟
transition state
configuration
𝐸≠
This is a single point approach. Each state is
characterized by only ONE configuration.
C7790 Introduction to Molecular Modelling -5Revision:
PES
➢ Increasing degrees of freedom (model size) result in increased roughness of PES.
➢ The only reasonable description of large models is to use statistical weighting using
right sampling technique.
E(x)
xconfigurations
reactant
state
product
state Δ𝐸𝑟
transition
state
𝐸≠
?
?
?
Large modelsSmall models
E(x)
xreaction coordinate
reactant
configuration
product configuration
𝐸(𝜉 𝑃)
𝐸(𝜉 𝑅)
Δ𝐸𝑟
transition state
configuration
𝐸≠
C7790 Introduction to Molecular Modelling -6Revision:
System properties
t
)(tM
Time average: Ensemble average:
=
=
K
i
iiMpM
1
2/6 2/6 1/6 1/6
We can run Monte Carlo simulations to
get value of property by molecular
modelling.
The observable value ( ഥ𝑀) of the property M can be determined by two approaches:
=
tott
otot
dttM
t
M )(
1
We can run molecular dynamics
simulations to get value of property by
molecular modelling.
snapshot of the system at time t is called
a microstate
𝑀𝑖
C7790 Introduction to Molecular Modelling -7Ergodic
Hypothesis
The ergodic hypothesis is often assumed in the statistical analysis of computational
physics. It postulates that the average of a process parameter over time and the average
over the statistical ensemble are the same.
=
=
K
i
iiMpM
1
=
tott
otot
dttM
t
M )(
1
the same outcome
Time average: Ensemble average:
https://en.wikipedia.org/wiki/Ergodic_hypothesis
In special cases: However, this assumption—that it is as good to simulate a system over a
long time as it is to make many independent realizations of the same system—is not
correct for all physical systems.
C7790 Introduction to Molecular Modelling -8Two
sampling approches
=
=
K
i
iiMpM
1
=
tott
otot
dttM
t
M )(
1
Time average: Ensemble average:
Typically, a very large number of microstates (> 106) is needed in BOTH approaches
to get converged thermodynamical properties.
Molecular dynamics
simulations
Monte Carlo
simulations
C7790 Introduction to Molecular Modelling -9-
)()()(ˆ
ekkek EH rRr RR
 =
Simplified models are needed
However, solution of SE is too computationally expensive to provide a suitable number
of microstates for converged thermodynamical properties.
We need simplified computational models (chemistry models)!
Solution of Schrodinger equation (SE) can provide mechanical properties (for example,
potential energy) describing microstates.
C7790 Introduction to Molecular Modelling -10Chemistry
Models
QM (Quantum mechanics) MM (Molecular mechanics) CGM (Coarse-grained mechanics)
)(RE )(RE )(RE
R - position of atom nuclei R - position of atoms R - position of beads
approximations approximations
Potential energy surface can be calculated by various method (model chemistry)!
C7790 Introduction to Molecular Modelling -11-
Summary
➢ Microstates of macrosystems are too large to model. Two levels of simplifications is
necessary:
➢ model (small systems containing enough atoms to represent the studied
phenomenon)
➢ theory (chemistry model)
➢ Altogether, they must provide enough information about the system in reasonable
computational time.