of 8
Using
Implicit-Solvent
Potentials
to Extract
Water
Contributions
to
Enthalpy
−
Entropy
Compensation
in Biomolecular
Associations
Shensheng
Chen
and Zhen-Gang
Wang
*
Cite This:
J. Phys.
Chem.
B
2023,
127,
6825−6832
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Supporting
Information
ABSTRACT:
Biomolecular
assembly
typically
exhibits
enthalpy
−
entropy
compensation
(EEC) behavior
whose molecular
origin remains
a long-standing
puzzle. While water restructuring
is believed
to play an important
role in EEC,
its contribution
to the entropy
and enthalpy
changes,
and how these changes
relate to EEC, remains
poorly
understood.
Here, we show that water
reorganization
entropy/enthalpy
can be obtained
by exploiting
the temperature
dependence
in effective,
implicit-solvent
potentials.
We find that the different
temperature
dependencies
in the hydrophobic
interaction,
rooted
in water
reorganization,
result in substantial
variations
in the entropy/enthalpy
change,
which
are responsible
for EEC. For lower-critical-solution-temperature
association,
water reorganization
entropy
dominates
the free-energy
change
at
the expense
of enthalpy;
for upper-critical-solution-temperature
association,
water reorganization
enthalpy
drives the process
at the
cost of entropy.
Other effects, such as electrostatic
interaction
and conformation
change
of the macromolecules,
contribute
much
less to the variations
in entropy/enthalpy.
■
INTRODUCTION
The association
between
biomolecules
such as proteins
and
nucleic acids in an aqueous
environment
is the molecular
basis
of life. The specific
interaction
between
biomolecules,
or
molecular
recognition,
guides
cellular
organization
and
functions.
Specific
interactions
in ligand
−
protein
binding
is
also the foundation
of pharmaceuticals.
Despite
decades
of
research
in the study of biomolecular
recognition,
a molecular
understanding
of the driving
force for their association
behavior
remains
unclear.
A long-standing
puzzle
in bio-
molecular
association
is the molecular
origin of the enthalpy
−
entropy
compensation
(EEC).
1
−
4
EEC refers to the con-
tributions
to the thermodynamic
driving
force in the binding
between
two biomolecules:
with small modifications
in the
chemical
structure
of the binding
species,
while the change in
Gibbs energy (
Δ
G
=
Δ
H
−
T
Δ
S
) of binding
remains
relatively
small, there are substantial
variations
in enthalpy
(
Δ
H
) and
entropy
(
Δ
S
), and the changes
in the two contributions
effectively
compensate
each other in the free-energy
con-
tribution.
Such a compensating
behavior
seems to hold in the
association
between
wide classes of biomolecules.
4
−
13
While
the magnitude
of the Gibbs free-energy
change
is most
probably
a consequence
of natural
evolution
14
−
17
�
biological
interactions
cannot be too strong or too weak
�
both
the wide
range of the entropy
(and enthalpy)
variation
and the origin of
the entropy
change
remain
poorly
understood.
An early
explanation
of the compensation
effect invoked
the intuitive
picture
where binding
results from the favorable
energetic
attraction
at the cost of conformation
entropy
loss.
18,19
However,
such an argument
does not explain
the prevalence
of entropy-driven
association
systems
and the large entropy
changes
in EEC cases where no significant
conformation
changes
are involved.
4,20,21
Water,
being the ubiquitous
solvent
for biomolecular
association,
is widely
believed
to play a key role in
EEC.
1,2,21
−
23
Water reorganization,
such as local structural
changes
in the hydrogen
bonding
network
and water release
from or adsorption
to the biomolecules,
is often invoked
as
being responsible
for the driving
forces in the biomolecular
associations.
It has been generally
recognized
that water
reorganization
plays a pivotal role in the organization
of living
matter,
24
including
protein
folding,
25
−
28
formation
of bio
condensates,
29
and formation
of membranes/micelles.
30,31
In
several experimental
reports
of EEC, water reorganization
has
been suggested
to be the major contributor
to the entropy/
enthalpy
change.
32
−
34
In a convincing
study, Breiten
et al.
examined
the association
between
a crystallized
protein
and a
series of rigid ligands in water;
20
since the binding
involves
no
obvious
changes
in the molecular
conformation,
they
concluded
that water reorganization
is the only explanation
for the observed
EEC behavior.
20,21
However,
the extent to
which water reorganization
contributes
to the entropy
and
Received:
June 5, 2023
Revised:
July 11, 2023
Published:
July 26,
2023
Article
pubs.acs.org/JPCB
© 2023
The Authors.
Published
by
American
Chemical
Society
6825
https://doi.org/10.1021/acs.jpcb.3c03799
J. Phys.
Chem.
B
2023,
127,
6825
−
6832
enthalpy
changes
in biomolecular
associations
and the
connection
between
these changes
and the observed
EEC
behavior
remain poorly understood.
Calculation
of the free-energy
contributions
from water
reorganization
in macromolecular
association
by quantum,
first-principles
methods
is currently
an impossible
task.
However,
in our recent
work, we showed
that water
reorganization
entropy/enthalpy
in an electrostatically
driven
association
can be obtained
by exploiting
the temperature
dependence
of the water-mediated
electrostatic
interaction.
35
For polyelectrolyte
complex
coacervation,
we found that water
reorganization
entropy,
rather than the commonly
believed
counterion
release
entropy,
is the primary
entropy-driving
force for many of the experimentally
relevant
polyelectrolyte
systems.
Thus, a promising
approach
to understanding
the
water reorganization
entropy/enthalpy
effects on biomolecular
association
is by analyzing
appropriately
constructed
temper-
ature-dependent,
effective
water-mediated
interaction
poten-
tials relevant
to biomolecular
systems.
Such temperature-
dependent,
coarse-grained
potentials
have been developed
by
Dignon
et al.
36
to successfully
predict the liquid
−
liquid
phase
separation
of disordered
proteins.
In this article, we present
a phenomenological
approach
to
understanding
EEC, in which the water reorganization
entropy
and enthalpy
are captured
by temperature-dependent,
coarse-
grained
potentials.
We illustrate
our approach
by studying
the
association
between
two model oppositely
charged
polymers
using molecular
dynamics
simulation.
From the analysis
of the
potential
of mean force (PMF),
we find that the entropy
and
enthalpy
change
from water reorganization
are the major
contributions
to the free-energy
change in the binding
process
and that these two components
of the driving
force
compensate
for each other. The substantial
variations
in the
entropy
and enthalpy
changes
observed
in systems
exhibiting
EEC behavior
arise from the different
temperature
dependence
of the hydrophobic/hydrophilic
interactions.
For association
in
lower-critical-solution-temperature
(LCST)
systems,
water
reorganization
entropy
dominates
the favorable
free-energy
change,
compensated
by unfavorable
enthalpy.
For association
in upper-critical-solution-temperature
(UCST)
systems,
water
reorganization
enthalpy
provides
the favorable
free-energy
driving
force, compensated
by unfavorable
entropy.
The
magnitude
of entropy/enthalpy
change from water reorganiza-
tion depends
strongly
on the difference
between
the
operational
temperature
T
(room temperature
in this study)
and the (mean-field)
θ
temperature
of the polymers.
■
METHODS
Water
Reorganization
Entropy/Enthalpy
from
Implic-
it-Solvent
Potentials.
The key to extracting
water reorgan-
ization entropy/enthalpy
from implicit-solvent,
coarse-grained
models
is the recognition
that the interaction
potentials
are
PMFs. As such, the PMF is the interaction
free energy that
contains
both entropic
and energetic
contributions,
37
with the
water degrees
of freedom
reflected
in the temperature
dependence
of the PMFs. (Strictly
speaking,
the free-energy
change refers to a Helmholtz
free-energy
change,
but for liquid
systems
under normal conditions,
it is approximately
the Gibbs
free-energy
change.
For this reason, we shall refer to the energy
change
as the enthalpy
change).
The PMF between
a pair of
solutes in water
w
(
r
) can be generally
written as
w
(
r
) =
u
(
r
) +
Δ
w
(
r
,
T
), where
u
(
r
) is the direct interaction
potential
in the
absence
of water and
Δ
w
(
r
,
T
) is the water-mediated
contribution.
38
Note that
Δ
w
(
r
,
T
) is temperature-dependent,
reflecting
the effects of integrating
the solvent
degrees
of
freedom.
Knowing
the exact form of
w
(
r
,
T
), we can calculate
the change
in entropy
and enthalpy
in molecular
association
as
37
=
T
s
T
w
T
(1)
and
=
+
h
w
T
s
(2)
where
Δ
refers to the change
before
and after solute
association.
The temperature
dependence
in
w
is due
exclusively
to the water-mediated
contribution,
Δ
w
(
r
,
T
).
Therefore,
the calculated
T
Δ
s
from eq 1 is the water
reorganization
entropy.
Consequently,
Δ
h
calculated
by eq 2
contains
the enthalpy
from solvent
reorganization.
From eqs 1
and 2, a strong temperature
dependence
in
w
will result in
significant
contributions
to the free-energy
change from water
reorganization.
The key to extracting
the water reorganization
entropy
and energy in biomacromolecular
association,
there-
fore, is knowing
the temperature
dependence
in the relevant
effective
potentials.
Biomolecular
association
in water is generally
considered
to
consist of two types of effective
interactions:
ionic electrostatic
interactions
and non-ionic
interactions.
Following
the common
usage,
39
−
41
we use the term hydrophobic
interaction
to refer to
all non-ionic
interactions.
Both electrostatic
and hydrophobic
interactions
in water have strong
temperature
depend-
ence.
42
−
44
Below, we discuss
the temperature
dependence
in
these two types of interactions
and how it relates to the water
reorganization
entropy/enthalpy.
We then use coarse-grained
molecular
dynamics
simulation
to study the association
between
two generic
oppositely
charged
polymers,
with the
aim of exploring
the role of water reorganization
in EEC.
Water
Reorganization
in Electrostatic
Interactions.
At
the coarse-grained
level, the effective
interaction
between
two
ions in water is described
by the Coulomb
potential
=
w
q q
r
4
i
j
r
ij
el
0
(3)
where
r
ij
is the distance
between
two charges
q
i
and
q
j
,
ε
r
is the
dielectric
constant
of the solvent,
and
ε
0
is the vacuum
permittivity.
In this treatment,
all solvent
degrees
of freedom
are subsumed
into the dielectric
constant
ε
r
. The dielectric
constant
of water has a strong temperature
dependence,
42
pointing
to the significant
role played
by entropy
in the
electrostatic
interaction
due to water reorganization.
45,46
In our
recent work,
35
using the experimentally
measured
temperature
dependence
of water dielectric
constant
given in ref 42, we
showed
that at room temperature,
the change
of water
reorganization
entropy
−
T
Δ
S
el
and energy (enthalpy)
Δ
H
el
in
electrostatic
interaction
calculated
by eqs 1 and 2 yields,
respectively,
=
T
S
U
1.36
el
el
(4)
and
=
H
U
0.36
el
el
(5)
where
⟨
U
el
⟩
is the average
of the sum of all effective
ionic pair
interactions
given by eq 3. Equations
4 and 5 highlight
the
surprising
result that electrostatic
assembly
in water at room
The Journal
of Physical
Chemistry
B
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Article
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J. Phys.
Chem.
B
2023,
127,
6825
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6832
6826
temperature
is primarily
entropy-driven.
The origin of this
entropic
driving
force can be understood
as arising from the
increased
orientational
freedom
of water molecules
upon
pairing
of two oppositely
charged
ions,
35,47
as illustrated
in
Figure
1A. This entropic
contribution
due to water
reorganization
in electrostatic
interactions
has been invoked
to successfully
explain
the predominantly
entropy-driven
nature
in polyelectrolyte
complex
coacervation
systems.
35
Moreover,
eqs 4 and 5 show that the entropic
and enthalpic
contributions
to the free-energy
change are of opposite
signs to
each other.
Thus, water reorganization
in electrostatic
interactions
exhibits
compensated
behavior.
In the studies of biopolymer
association,
screened
Coulomb
interaction
in the form of
=
w
q q
r
e
4
i
j
r
ij
r
el
0
ij
(6)
is often used to account
for the effects of salt ions, where
κ
is
the inverse
Debye
screen
length
given
by
=
k
T
C
(
/2
)
r
1
0
B
1/ 2
, with
e
the electronic
charge,
and
C
the concentration
of salt (assumed
to be monovalent
here). In
this work, we use eq 6 to model electrostatic
interactions
between
the biopolymers
with the salt concentration
at 0.1 M
to mimic the typical salt condition
in cellular
environments.
In
the wide temperature
range of interest
for this work, the Debye
screening
length changes
little,
48
so we take
κ
to be constant.
As a result, eqs 4 and 5 remain
valid for calculating
the
entropic
and enthalpic
contributions
from water reorganization
in biomolecular
association
with screened
electrostatics.
Water
Reorganization
in Temperature-Dependent
Hydrophobic
Interactions.
Hydrophobic
interaction
refers
to the effective
non-ionic
interaction
between
solutes
that
results
from the change
of local water structure
and the
adsorption/release
of water molecules.
44,49
These
water
reorganization
events
involve
both entropy
and enthalpy
changes.
38
As illustrated
in Figure 1B, depending
on whether
water molecules
tend to push two solutes together
or away, the
effective
hydrophobic
interaction
can be attractive
or repulsive
(here we consider
hydrophilic
interaction
as a special case of
the hydrophobic
interaction
where the solutes
repel each
other).
Hydrophobic
interactions
are known
to be strongly
temperature-dependent.
38,43,44,49
The temperature-dependent
hydrophobic
interactions
greatly
influence
the behavior
of
biomolecules,
such as protein
folding
43,50
and condensa-
tion.
36,51
Effective,
temperature-dependent
hydrophobic
inter-
action potentials
have been recently
developed
to model the
LCST and UCST liquid
−
liquid
phase separation
of disordered
proteins.
36,52
Following
a similar strategy
to that in ref 36, we
construct
a generic
temperature-dependent
hydrophobic
potential
that allows the water reorganization
entropy
and
energy to be easily extracted
using eqs 1 and 2. Our model
contains
a modified
Weeks
−
Chandler
−
Andersen
(WCA)
potential,
u
wca(50,49)
, to capture
the short-range
hard-core
repulsion
53
and a temperature-dependent
term
u
T
to describe
the effect of solvent
quality:
=
+
w
r
T
u
u
(
,
)
hp
wca(50,
49)
T
(7)
The generalized
WCA potential
has the form
=
+
i
k
j
j
j
j
j
y
{
z
z
z
z
z
Ä
Ç
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
i
k
j
j
j
y
{
z
z
z
i
k
j
j
j
y
{
z
z
z
É
Ö
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
u
r
r
wca(
,
)
r
r
a
r
a
/
r
a
a
r
a
r
a
(8)
with a cutoff
=
r
(
/
)
c
r
a
1/
r
a
, where
λ
r
= 50 and
λ
a
= 49.
ε
=
k
B
T
.
u
wca(50,49)
is a more faithful
representation
of the hard-
sphere repulsion
than the more traditional
u
wca(12,6)
,
53
as shown
in Figure S1, Supporting
Information.
Below,
we will write
u
wca(50,49)
as
u
wca
for simplicity.
As an interaction
mimicking
the
hard-sphere
repulsion,
u
wca
is primarily
entropic
in origin, and
for this reason, we choose
ε
=
k
B
T
in our simulation.
However,
as shown in Figure S2, Supporting
Information,
the change in
the interaction
due to this potential
in the binding
between
the
two polymers
studied
in this work is very small, so the precise
choice of
ε
is inconsequential.
For the temperature-dependent
term, we follow a commonly
assumed
form
36,54,55
=
u
T
r
(
)(
/
)
T
6
(9)
where the coefficient
λ
(
T
) captures
the different
solvent
conditions
for the solute. At the level of the second
virial
coefficient
B
2
, the solvent
condition
is characterized
by the
θ
temperature,
at which
B
2
= 0. For this reason,
we write the
coefficient
λ
(
T
) as
Figure
1.
Schematics
showing
how water reorganization
mediates
effective
interactions.
(A) Electrostatic
interactions
are mediated
by the
orientation
of water dipoles.
(B) Hydrophobic
interactions
are mediated
by local water restructuring
and/or release/adsorption
of water molecules.
The Journal
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Chemistry
B
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J. Phys.
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2023,
127,
6825
−
6832
6827
=
+
T
T
(
)
(
)
0
(10)
where
α
describes
the sensitivity
of the temperature.
In
principle,
it is possible
to include
higher-order
terms in
T
−
θ
for quantitative
accuracy.
Here, we use the linear form for
convenience
and simplicity;
for the temperature
range of
interest
in this work, the linear form is sufficient
to capture
the
temperature
response
of the hydrophobic
interaction.
λ
0
is a
constant,
whose value is determined
by the vanishing
of
B
2
at
T
=
θ
. From the expression
for
B
2
=
B
T
e
r
r
(
)
2
(
1)
d
w
r T
k
T
2
(
,
) /
2
hp
B
(11)
we determine
λ
0
≈
0.95. The sign of
α
in eq 10 distinguishes
between
LCST and UCST
behavior:
If
α
> 0, increasing
temperature
results
in a stronger
effective
solute
−
solute
attraction
that can eventually
lead to phase separation,
corresponding
to LCST;
if
α
< 0, the attraction
is enhanced
by decreasing
the temperature,
corresponding
to UCST.
We comment
that the concepts
of the
θ
temperature,
UCST,
and LCST,
are usually
employed
to describe
interactions
between
the same solute molecules.
However,
the general form
of
λ
(
T
) can be used to describe
cross-interactions
between
different
solute molecules.
In that case,
θ
refers to the
temperature
at which the second virial coefficient
between
two
different
molecular
species
vanishes.
Experimental
determination
of the
θ
temperature
for the
different
moieties
in the biomolecular
association
is challeng-
ing, and such data are at present
not available.
However,
in
most experiments
concerning
the temperature-dependent
biomolecular
association,
structural
changes
or phase tran-
sitions typically
take place in the range of (
−
100,
100)K from
room temperature.
50,51,56,57
Thus, in this study, we consider
(
T
−
θ
) in the range of (
−
100,
100)K to represent
the most
experimentally
relevant
conditions.
Figure 2 shows the second virial coefficient
calculated
from
eq 11, as a function
of the temperature
difference
T
−
θ
. For
both the LCST and UCST cases, in the range of parameters
that yield reasonable
values for
B
2
, its temperature
dependence
is approximately
linear, especially
close to the theta point, as
expected
on physical
grounds.
Using eqs 1 and 10, the entropic
contribution
to the free
energy
from water reorganization
in the hydrophobic
interaction
can be obtained
as
=
+
Ä
Ç
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
É
Ö
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
T
S
T
T
U
(
)
hp
0
hp
(12)
and the corresponding
enthalpic
contribution
calculated
using
eq 2 yields
=
+
Ä
Ç
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
É
Ö
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
H
T
T
U
1
(
)
hp
0
hp
(13)
where again
⟨
U
hp
⟩
refers to the average
of the sum of all pair
interactions
due to the potential
u
T
. From eqs 12 and 13, we
see that variations
in the parameter
α
and the temperature
difference
T
−
θ
result in variations
of the entropic
and
enthalpic
contributions
to the free-energy
change
and that
these two contributions
compensate
each other.
Other
Simulation
Details.
Our simulation
uses an
implicit-solvent
representation,
with dielectric
constant
ε
r
=
78 at
T
= 300 K. The monomer
size is set at
σ
= 6.0 Å for all
monomers
to represent
the generic
size of a protein
residue
in
coarse-grained
simulations.
36
Neighboring
monomers
along
the chains are subjected
to the harmonic
bond potential
given
by
=
U
r
K
r
r
(
)
(
)
bond
bond
0
2
, with
K
bond
= 100
k
B
T
/
σ
2
and
r
0
= 0.7
σ
. With
k
B
T
being the energy scale in the bond potential,
the bonding
interaction
should be considered
as entropic
in
nature. Using eqs 1 and 2, we get
=
T
S
U
bond
bond
(14)
and
=
T
H
0
bond
(15)
However,
similar
to
U
wca
, the change
in
U
bond
contributes
minimally
to the PMF (see Figure S2, Supporting
Informa-
tion); thus, the precise
designation
of
U
wca
and
U
bond
interaction
energies
is immaterial.
The total entropy
of complexation
is calculated
by the PMF
and all known enthalpy
contributions
=
+
+
+
T
S
H
H
H
H
(
)
P
M
F(0)
el
hp
wca
bo
nd
(16)
where PMF(0)
is the free energy of complexation.
The conformation
entropy
is then computed
by subtracting
all known
entropy
(that can be calculated
directly
from the
interaction
potentials)
from the total entropy
=
T
S
T
S
S
S
S
S
(
)
con
el
hp
wc
a
bond
(17)
Note that the translational
entropy
of the polymers
is not
considered
here.
All simulations
are performed
at
T
= 300 K with a Langevin
thermostat
using the large-scale
atomic/molecular
massively
parallel
simulator
(LAMMPS)
platform.
The simulation
time
scale is given by
=
m
k
T
/
2
B
where the mass of the
monomer
m
is set at 1. The positions
and velocities
of the
beads are updated
with an integration
time step
Δ
t
= 0.002
τ
.
The simulation
box has dimensions
100
σ
×
100
σ
×
100
σ
. Each
polymer
pair is equilibrated
for 10
6
τ
before performing
the
PMF calculations.
We calculate
the PMF of association
using the adaptive
bias
force algorithm
58,59
implemented
in LAMMPS.
60
The
coordinate
of the PMF is taken to be the center-of-mass
distance
between
the two chains. We sample the distance
r
in
the range 0
−
30
σ
.
The distance
range is divided
into 3
Figure
2.
Second
virial coefficient
due to the hydrophobic
interaction
as a function
of the difference
between
the room temperature
and the
theta temperature
(
T
−
θ
). (A) LCST system.
(B) UCST system.
The shaded
area in each figure shows the experimentally
relevant
conditions
for biomolecular
association.
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of Physical
Chemistry
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consecutive
windows
of 0
−
3
σ
, 3
−
10
σ
,
and 10
−
30
σ
to improve
the efficiency
of the PMF calculations.
59
Each window
is
further divided
into bins with equal widths of 0.5
σ
. For each
window,
we perform
the simulation
for 1
×
10
7
τ
to reach
convergence.
Other details in the PMF calculation
are the
same as in our previous
work.
35
■
RESULTS
AND
DISCUSSION
Free
Energy
and Pathway
of Association
between
Two
Oppositely
Charged
Polymers.
To illustrate
our
approach,
we simulate
the association
between
two generic
polyelectrolyte
chains
as a crude model
for charged
biomacromolecules.
The two model polymers
each have 60
beads connected
by harmonic
bonds.
The two chains are
oppositely
charged
with the same charge fraction
f
, which is
controlled
by placing
unit charges
regularly
spaced along the
chain backbone.
All beads have the same hydrophobic
interaction
with the same second
virial coefficient
B
2
. The
conformation
of the chains before association
depends
on the
charge fraction
f
and
B
2
. For
B
2
=
−
σ
3
, by tuning
f
, we can
model a chain that takes extended
(Figure
3A), coil-like
(Figure
3B), and globule-like
(Figure
3C) conformations,
mimicking
intrinsically
disordered
proteins
to folded proteins.
Below,
for simplicity
and to connect
with the literature
on
biomolecular
systems,
we call a chain in the extended
or coil-
like state a disordered
chain and a chain in the globule-like
state a folded chain.
The bottom
row of Figure 3 shows the free energy (PMF),
U
hp
, and
U
el
upon complexation
between
the two chains. On
the per residue basis, the binding
free energy is around
−
0.5 to
−
1.0 kcal/mol,
similar to the experimental
values for protein
−
protein
9
and protein
−
ligand
20
association.
The complexation
between
the disordered
chains is stronger
than between
the
folded chains. Despite
B
2
being the same in all three cases,
U
hp
decreases
more in the case of higher charge fraction
due to the
favorable
hydrophobic
contacts
brought
about by the electro-
static attractions
between
the two chains.
Interestingly,
the complexation
pathway
between
two folded
chains shows that
U
hp
first increases
when the two chains start
to contact
and then decreases
as they fuse together.
This
increase
in
U
hp
is a result of temporary
unfolding
of the chains
as they are drawn together
by the electrostatic
attraction;
see
the snapshots
in Figure 3B,C.
With the PMF,
U
hp
, and
U
el
calculated
this way, we are in a
position
to address
the issue of EEC by varying
the parameters
in the model, as detailed
below.
Enthalpy
−
Entropy
Compensation
Due
to Water
Reorganization.
To study the EEC, we perform
extensive
PMF calculations
for the complexation
between
two oppositely
charged
polymers
by varying
α
and
T
−
θ
in the hydrophobic
interaction
and the charge fraction
f
. To ensure the physical
relevance
of our simulations,
we consider
the range of
T
−
θ
to
be (
−
100,
100)K,
and we constrain
the second
virial
coefficient
due to the hydrophobic
interaction
to be within
(
−
1.1, 0.1)
σ
3
. Within the limits for
B
2
and
T
−
θ
, we vary
α
in
the range of
|
α
|
< 0.01. To cover the different
single-chain
conformations,
we choose
three charge fractions,
f
= 0.5, 0.3,
and 0.1. All simulations
are performed
at room temperature;
therefore,
by varying
T
−
θ
, we essentially
vary the
θ
temperature.
Figure 4A shows the total entropy
and enthalpy
changes
in
the associations
between
two disordered
chains,
with 150
combinations
of randomly
chosen
α
and (
T
−
θ
). In Figure
4A, the enthalpy
and entropy
show clear EEC behavior;
on a
per residue basis, while the variation
in
Δ
G
is within 0.28 kcal/
mol (out of
∼
1.5 kcal/mol
on average),
the variations
in
T
Δ
S
and
Δ
H
span about 6 kcal/mol,
with a slope close to 1 in the
T
Δ
S
−Δ
H
plot. The significant
changes
in enthalpy
and
entropy
are primarily
reflections
of water reorganization
from
hydrophobic
interactions
calculated
from eqs 12 and 13, as will
Figure
3.
PMF and the corresponding
changes
in electrostatic
and hydrophobic
interactions
for the association
of two oppositely
charged
chains at
different
charge fractions
f
. (A)
f
= 0.5: the two chains are extended.
(B)
f
= 0.3: the two chains are coil-like.
(C)
f
= 0.1: the two chains are globule-
like. All monomers
interact
with the same hydrophobic
attraction
corresponding
to
B
2
=
−
1
σ
3
.
The Journal
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Chemistry
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be discussed
further in Figure 5. These results are very similar
to the experimentally
observed
EEC reported
in ref 20 for
protein
−
ligand
binding
upon single residue substitutions
in the
ligand: the variations
in entropy
and enthalpy
were also about
6 kcal/mol,
significantly
larger than the variation
in
Δ
G
. The
authors
of ref 20 attributed
the EEC to the water
reorganization,
which is supported
by our simulation
and
analysis.
The association
between
two folded chains also exhibits
a
similar EEC behavior,
as shown in Figure 4B. Interestingly,
while the variation
in
Δ
G
is also small at 0.28 kcal/mol,
similar
to the disordered
chains, the variations
in entropy
and enthalpy
are about 3.5 kcal/mol,
less than the range for the disordered
chains
(6 kcal/mol).
These
results
are consistent
with
experimental
findings
by Huang
and Liu
9
that disordered
proteins
show more pronounced
EEC behavior
than folded
proteins.
The stronger
EEC behavior
in disordered
chains can
be explained
by the fact that unfolded
chains expose
more
residues
to water before
complexation,
resulting
in more
significant
water reorganization
upon binding,
thus leading
to
larger changes
in enthalpy
and entropy.
The EEC data in Figure 4 are naturally
separated
by the
LCST (
α
> 0) and UCST (
α
< 0) subgroups.
In the LCST
association,
the favorable
free-energy
change
is dominated
by
solvent
reorganization
entropy
in hydrophobic
interactions,
at
the expense
of energy.
In contrast,
for the UCST association,
the energetic
contribution
from solvent
reorganization
dominates
the free-energy
change,
at the cost of entropy
decrease.
To get further insight into the different
contributions
to the
enthalpy
and entropy
changes,
we show the separate
free-
energy components
from hydrophobic
interaction,
electrostatic
interaction,
and configuration
change
in Figure 5. Clearly,
water reorganization
is the primary
cause for EEC, which is
manifested
through
the large variations
in the entropy
and
enthalpy
changes
due to the temperature-dependent
hydro-
phobic interaction
U
hp
. Electrostatic
interaction
shows only a
modest
compensation
behavior,
with the changes
in
T
Δ
S
el
and
Δ
H
el
significantly
smaller compared
to those due to the water-
mediated
hydrophobic
interaction.
There
is appreciable
variation
in the conformation
entropy
change,
but this
variation
is 1 order of magnitude
smaller
than the variation
in the entropy
change
due to hydrophobic
interaction.
EEC in Biomolecular
Association.
Our phenomenolog-
ical model for EEC behavior
reported
in this work is based on
the interaction
between
a pair of homopolymers
that have the
same hydrophobicity
on all residues.
Here, EEC arises from
variations
in the
θ
point and the parameter
α
, which relates to
LCST/
UCST.
Experimentally,
EEC behavior
is observed
in
biomolecular
associations
involving
heteropolymers
such as
proteins.
Our analysis
can be easily generalized
to the
association
between
two heteropolymers
by allowing
different
temperature
dependencies
between
different
types of resi-
dues,
36,52
which has been used by Mittal and co-workers
to
successfully
describe
the temperature-dependent
liquid
−
liquid
phase separation
of disordered
proteins.
36
Once so generalized,
the interaction
between
the different
residues
will involve
different
values for
θ
and
α
. EEC is then manifested
through
the natural
variability
in the value of these parameters
upon
residue
substitutions.
We note that since the systems
in our study are relatively
small, it is in principle
possible
to compute
the PMF using
explicit
water models.
However,
to address
the EEC in
biomolecular
association,
we will have to perform
hundreds
of
such calculations
using different
monomer
structures,
which is
beyond
the reach of current
computational
resources.
More
importantly,
we are not aware of any simple and unambiguous
methods
for separating
the PMF into various
components
(electrostatic,
chain conformation,
and water reorganizations).
Our approach
provides
a general
and inexpensive
way to
extract water entropy
contributions
by exploiting
the temper-
ature-dependent
implicit-solvent
interaction
potentials.
Fur-
thermore,
our method
can be easily extended
to larger and
more complex
systems.
■
CONCLUSIONS
By exploiting
the temperature
dependence
in the implicit-
solvent,
coarse-grained
interaction
potentials,
we are able to
extract the enthalpic
and entropic
contributions
in the free-
energy change
for polymer
association
in water. We find that
under experimentally
relevant
conditions,
water reorganization
constitutes
a major source of the significant
variations
in the
entropy
and enthalpy
of association,
which is manifested
in the
different
temperature
dependence
of the hydrophobic
attractions.
Consequently,
water reorganization
results
in
pronounced
EEC behavior:
for the LCST association,
water
reorganization
entropy
dominates
the free-energy
change
at
the expense
of enthalpy;
for the UCST
association,
water
reorganization
enthalpy
dominates
the process
at the expense
of entropy.
Other effects, such as electrostatic
interaction
and
Figure
4.
EEC in the association
between
(A) disordered
chains and
(B) folded chains.
The disordered
chains mostly
have a charge
fraction
of 0.5 with some having a charge fraction
of 0.3, while the
folded chains mostly have a charge fraction
of 0.1 with some having a
charge fraction
of 0.3.
Figure
5.
Free-energy
components
from hydrophobic
interaction,
electrostatic
interaction,
and configurational
entropy
in the associa-
tion of (A) disordered
chains and (B) folded chains.
The Journal
of Physical
Chemistry
B
pubs.acs.org/JPCB
Article
https://doi.org/10.1021/acs.jpcb.3c03799
J. Phys.
Chem.
B
2023,
127,
6825
−
6832
6830