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/2020 Distant Form of Teaching: Rev1
Lesson 25
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
Main summary:
➢ It is not possible to model microstates with the size of macrosystems.
➢ Thus, simplified models (in size) are employed instead.
➢ Th size of model determines the modeling approach.
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.
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 -5Revision:
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 -6Ergodic
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.
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.
=
=
K
i
iiMpM
1
=
tott
otot
dttM
t
M )(
1
the same outcome
Time average: Ensemble average:
https://en.wikipedia.org/wiki/Ergodic_hypothesis