C7790 Introduction to Molecular Modelling -1Petr
Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
JS/2022 Distant Form of Teaching: Rev1
Lesson 16
Reaction Energy I (general)
C7790 Introduction to Molecular
Modelling
TSM Modelling Molecular Structures
C9087 Computational Chemistry for Structural Biology
C7790 Introduction to Molecular Modelling -2-
Context
bridge
C7790 Introduction to Molecular Modelling -3Revision:
Thermodynamics & Modelling
Δ𝐺𝑟
0
= −𝑅𝑇 ln 𝐾
𝐾 =
𝐶 𝑟
𝑐 𝐷 𝑟
𝑑
𝐴 𝑟
𝑎
𝐵 𝑟
𝑏
Fundamental relation
Δ𝐺𝑟
0
= 𝑐Δ𝐺𝑓,𝐶
0
+ 𝑑Δ𝐺𝑓,𝐷
0
− 𝑎Δ𝐺𝑓,𝐴
0
+ 𝑏Δ𝐺𝑓,𝐵
0
𝜟𝑮 𝒓
𝟎
is the standard reaction free energy.
➢ its value determines the composition of
the reaction mixture at the chemical
equilibrium.
aA + bB cC + dD
C7790 Introduction to Molecular Modelling -4Revision:
Thermodynamics & Modelling
Δ𝐺𝑟
0
= −𝑅𝑇 ln 𝐾
𝐾 =
𝐶 𝑟
𝑐
𝐷 𝑟
𝑑
𝐴 𝑟
𝑎
𝐵 𝑟
𝑏
Fundamental relation
Δ𝐺𝑟
0
= 𝑐Δ𝐺𝑓,𝐶
0
+ 𝑑Δ𝐺𝑓,𝐷
0
− 𝑎Δ𝐺𝑓,𝐴
0
+ 𝑏Δ𝐺𝑓,𝐵
0
What do we need to know?
solution at
equilibrium
A
B
C
D
We only need to know
the properties of individual
components involved in the
reaction at standard conditions (or
at different conditions, which are
well defined).
We need to know the composition of
solution at equilibrium.
easier for modelling It is hard or impossible to model.
A B
C
D
D
CD
D B
C
C7790 Introduction to Molecular Modelling -5Revision:
Partition function and modelling
Helmholtz energy F:
QTkF B ln−==
−
=
K
j
E j
eQ
1

Canonical partition function:
Consider only the most important microstate
approximation
E1, E2, E3, …
The most important microstate is the
microstate with the lowest energy.
𝐹 = 𝐸1
It is very often used for qualitative
consideration or when computationally
demanding methods are employed (typically
quantum chemical calculations).
C7790 Introduction to Molecular Modelling -6Reaction
Energy
E(x)
xreaction coordinate
(usually not known)
REMEMBER: This is 1D projection of E(R), which is a function of 3N variables (N-number of atoms).
reactant
configuration
product
configuration
𝐸(𝜉 𝑃)
𝐸(𝜉 𝑅)
Δ𝐸𝑟 = 𝐸 𝜉 𝑃 − 𝐸 𝜉 𝑅
reaction energy
the sign convention:
always use the thermodynamics convention
difficult description
(rarely used in
practice)
Δ𝐸𝑟
➢ potential energy is a state function,
thus the reaction path is not
necessary for reaction energy
calculation
➢ only reactant and product states
need to be characterized
C7790 Introduction to Molecular Modelling -7Reaction
Energy
E(x)
xreaction coordinate
(usually not known)
REMEMBER: This is 1D projection of E(R), which is a function of 3N variables (N-number of atoms).
reactant
configuration
product
configuration
E(R)
states
reactant
state
product
state
𝐸(𝜉 𝑃)
𝐸(𝜉 𝑅) 𝐸 𝑅
𝐸 𝑃
Δ𝐸𝑟
Δ𝐸𝑟 = 𝐸 𝜉 𝑃 − 𝐸 𝜉 𝑅 = 𝐸 𝑃 − 𝐸 𝑅
reaction energy the sign convention:
always use the thermodynamics convention
difficult description
(rarely used in
practice)
only important states
are described
C7790 Introduction to Molecular Modelling -8Binding
Energy
BB+
A + B AB
A
A
Δ𝐸 𝑏 = Δ𝐸𝑟 = 𝐸𝐴𝐵 − (𝐸𝐴 + 𝐸 𝐵)
Binding energy:
Δ𝐸 𝑏
Binding is the special case of reaction.
Reactant state:
➢ model of non-interacting components
Product state:
➢ complex between A and B, usually non-covalently bound
C7790 Introduction to Molecular Modelling -9Binding
Energy
BB+
A + B AB
A
A
Δ𝐸 𝑏 = Δ𝐸𝑟 = 𝐸𝐴𝐵 − (𝐸𝐴 + 𝐸 𝐵)
Binding energy:
Source of energies (model chemistry):
➢ potential energy of geometry optimized states (QM, MM, QM/MM, etc.)
➢ REMEMBER: it does not cover ZPVE (zero-point vibrational energy)
➢ no thermal contributions
➢ ….
➢ various improvements to E(R): ZPVE, ideal gas model, etc. …
➢ ….
➢ free energy
➢ ….
➢ scoring functions (docking)
complex of A and B,
usually non-covalently
bound
Δ𝐸 𝑏
C7790 Introduction to Molecular Modelling -10Binding
Energy
BB+
A + B AB
A
A
Δ𝐸 𝑏 = Δ𝐸𝑟 = 𝐸𝐴𝐵 − (𝐸𝐴 + 𝐸 𝐵)
Binding energy:
Source of energies (model chemistry):
➢ potential energy of geometry optimized states (QM, MM, QM/MM, etc.)
➢ REMEMBER: it does not cover ZPVE (zero-point vibrational energy)
➢ no thermal contributions
➢ ….
➢ various improvements to E(R)
➢ ….
➢ free energy
➢ ….
➢ scoring functions (docking)
complex of A and B,
usually non-covalently
bound
Δ𝐸 𝑏
Sometimes it is convenient to decompose the
binding energy into deformation and interaction
energies.
C7790 Introduction to Molecular Modelling -11Binding
vs Interaction Energy
BB+
A + B AB
A
A
B'
A'
+
Δ𝐸 𝑏=Δ𝐸 𝑑+Δ𝐸𝑖
Δ𝐸𝑖Δ𝐸 𝑑
deformation energy interaction energy
binding energy
complex of A and B,
usually non-covalently
bound
Binding energy is the difference between the energy of the complex minus the energy of
the isolated monomers in their minima configuration.
!!! though (hypothetical) pathway, experimentally non-observable !!!
its main purpose is to better understand the binding process
(also the preparation energy
or the strain energy)
C7790 Introduction to Molecular Modelling -12Binding
vs Interaction Energy
BB+
A + B AB
A
A
B'
A'
+
Δ𝐸 𝑏=Δ𝐸 𝑑+Δ𝐸𝑖
Δ𝐸𝑖Δ𝐸 𝑑
deformation energy interaction energy
Deformation energy is energy required to deform monomers from their relaxed geometry
into the geometry observed in the complex. This energy is ALWAYS positive.
Interaction energy is the difference between the energy of the complex minus the energy
of the isolated monomers in the geometry of the complex.
binding energy
complex of A and B,
usually non-covalently
bound
Binding energy is the difference between the energy of the complex minus the energy of
the isolated monomers in their minima configuration.
(also the preparation energy
or the strain energy)
C7790 Introduction to Molecular Modelling -13Binding
vs Interaction Energy
➢ Terms "binding energy" and "interaction energy" are very often used interchangeably.
Therefore, you have to follow the intended context of a work.
➢ With increasing model size, the calculation of the binding or interaction energy
employing potential energy becomes difficult.
C7790 Introduction to Molecular Modelling -14Large
models
E(x)
x
configurationslocal minima representing some conformational changes
nearest "global" minimum
reactant state
global minimum representing a given state
product state
REMEMBER: This is 1D projection of E(R),
which is a function of 3N
variables (N-number of atoms).
Δ𝐸𝑟
➢ Increasing degrees of freedom (model size) result in increased roughness of PES.
➢ The nearest "global" minimum representing the state could be difficult or impossible
to select.
➢ Then, the only reasonable solution is to use statistical weighting using the free energy
calculations.
C7790 Introduction to Molecular Modelling -15Large
models, cont.
Helmholtz energy F:
QTkF B ln−==
−
=
K
j
E j
eQ
1

Canonical partition function:
All microstates representing a thermodynamic state need to be considered:
E1, E2, E3, …
➢ Then, the only reasonable solution is to use statistical weighting using the free energy
calculations.
Typical scenario:
➢ models employing explicit solvent models
Approches:
➢ molecular dynamics
➢ Monte Carlo simulations
➢ etc.
C7790 Introduction to Molecular Modelling -16Uncharacterizable
compounds
Δ𝐺𝑟
0
= −𝑅𝑇 ln 𝐾
Fundamental relation
Δ𝐺𝑟
0
= 𝑐Δ𝐺𝑓,𝐶
0
+ 𝑑Δ𝐺𝑓,𝐷
0
− 𝑎Δ𝐺𝑓,𝐴
0
+ 𝑏Δ𝐺𝑓,𝐵
0
A
B
C
D
We only need to know
the properties of individual
components involved in the
reaction at standard conditions (or
at different conditions, which are
well defined).
easier for modelling
What to do if some compound is difficult
to describe by modeling?
Typical example: pKa calculations
C7790 Introduction to Molecular Modelling -17Example:
pKa calculation
HA + H2O H3O+ + A𝐾
𝑎 =
{H3O+} 𝑟 {A−} 𝑟
{HA} 𝑟 {H2O} 𝑟
𝑝𝐾 𝑎 = −log(𝐾𝑎)
{H2O} 𝑟 = 1 (activity of bulk water is one)
Definitions:
This reaction describes the most important chemical
change, but the structure of solvated proton is
more complex than H3O+ in reality.
C7790 Introduction to Molecular Modelling -18pKa
calculation
HA + H2O H3O+ + A𝐾
𝑎 =
{H3O+} 𝑟 {A−} 𝑟
{HA} 𝑟 {H2O} 𝑟
𝑝𝐾 𝑎 = −log(𝐾𝑎)
{H2O} 𝑟 = 1 (activity of bulk water is one)
approximation
HA H+
solv + A𝐾
𝑎 =
{H+
solv} 𝑟 {A−} 𝑟
{HA} 𝑟 How to model (characterize) H+
solv?
(this eq. is already simplified
description of reality)
they can be modelled
difficult or impossible
to model
Definitions:
C7790 Introduction to Molecular Modelling -19pKa
calculation
HA H+
solv + A𝐾
𝑎 =
{H+
solv} 𝑟 {A−} 𝑟
{HA} 𝑟
they can be modelled
difficult or impossible
to model
Solution:
HA H+
solv + AHB
H+
solv + B-
experimentally
known pKa(HA)
prediction of
pKa(HB)modelled compounds
C7790 Introduction to Molecular Modelling -20pKa
calculation - "training"
HA H+
solv + A-
experimentally
known pKa(HA)
𝐾 𝑎 =
{H+
solv} 𝑟 {A−} 𝑟
{HA} 𝑟
Δ𝐺𝑟
0
= −𝑅𝑇 ln 𝐾 𝑎 𝑒−
Δ𝐺 𝑟
0
𝑅𝑇 = 𝐾 𝑎
Definitions:
𝑝𝐾 𝑎 = −log(𝐾𝑎)
Solution:
Δ𝐺𝑟
0 = −𝑅𝑇 ln 𝐾 𝑎
Δ𝐺𝑟
0
𝑅𝑇
log(𝑒) = 𝑝𝐾 𝑎
𝑝𝐾 𝑎 =
Δ𝐺𝑟
0
2.303𝑅𝑇
≈
Δ𝐸𝑟
2.303𝑅𝑇
approximation
Δ𝐸𝑟 = 𝐸 𝐻 𝑠𝑜𝑙𝑣
(+) + 𝐸 𝐴
(−) − 𝐸 𝐻𝐴
𝐸 𝐻 𝑠𝑜𝑙𝑣
(+) = Δ𝐸𝑟 − 𝐸 𝐴
− + 𝐸 𝐻𝐴
𝐸 𝐻 𝑠𝑜𝑙𝑣
(+) = 2.303𝑅𝑇𝑝𝐾 𝑎(𝐻𝐴) − 𝐸 𝐴
− + 𝐸 𝐻𝐴
C7790 Introduction to Molecular Modelling -21pKa
calculation - "prediction"
𝐾 𝑎 =
{H+
solv} 𝑟 {A−} 𝑟
{HA} 𝑟
𝑝𝐾 𝑎 = −log(𝐾𝑎)
Solution:
Δ𝐺𝑟
0 = −𝑅𝑇 ln 𝐾 𝑎 Δ𝐸𝑟 = 𝐸 𝐻 𝑠𝑜𝑙𝑣
(+) + 𝐸 𝐴
(−) − 𝐸 𝐻𝐴
𝐸 𝐻𝑠𝑜𝑙𝑣
(+) + 𝐸 𝐵
− − 𝐸 𝐻𝐵 = 2.303𝑅𝑇𝑝𝐾 𝑎(HB)
HB H+
solv + Bprediction
of
pKa(HB)Definitions:
𝐸 𝐻 𝑠𝑜𝑙𝑣
(+) = 2.303𝑅𝑇𝑝𝐾 𝑎(𝐻𝐴) − 𝐸 𝐴
− + 𝐸 𝐻𝐴
𝑝𝐾 𝑎 𝐻𝐴 +
−𝐸
𝐴
− +𝐸 𝐻𝐴 +𝐸
𝐵
− −𝐸 𝐻𝐵
2.303𝑅𝑇
= 𝑝𝐾 𝑎(HB)
}
the difference between two pKa is obtained from the modelling
C7790 Introduction to Molecular Modelling -22-
Summary
➢ For small size models, it is rather simple to characterize all components unambiguously
on PES.
➢ Calculation of reaction (binding) energy is then straightforward (BUT NOT IN QM ).
➢ Binding energy (as special case of reaction energy) quantifies the binding strength of
two or more components, for example: protein and inhibitor.
➢ The binding energy can be conveniently decomposed into the deformation and
interaction energies.
➢ The deformation energy is ALWAYS positive (Why?).
➢ For strong binders, the interaction energy must be then large enough to
counterbalance the deformation energy.
➢ Calculation of reaction energy can be supplemented by experimental data (typical use:
pKa calculations).
➢ For large models, the simple approach employing PES for reaction (binding) energy
calculation is not suitable and more advance sampling techniques must be employed to
get reasonable data.