1. Atomistic simulations of liquid crystals
by
Luis Moux-Dom´ınguez
A masters project report submitted in partial fulfillment of
the requirements for the degree of
Master of Science
(Chemical Engineering)
at the
UNIVERSITY OF WISCONSIN–MADISON
2008
2. c Copyright by Luis Moux-Dom´ınguez 2008
I hereby place the content of this report into public domain.
3. i
Abstract
This report describes the efforts invested, from September 2006 till April 2008, to-
wards the elucidation of some fundamental phenomenological aspects inherent to the
problem of using liquid crystals in biosensors. Computational techniques, in the form of
molecular dynamics, were employed in modeling of the system. The work is novel in the
extend to which it was carried —over 50 ns of mealtime were simulated. Furthermore, it
reiterates the truism that emerging technological advances and the current state of the
art provide henceforth adequate resources to realistically model materials and accurately
predict their macroscopic properties from first principle calculations.
In particular, an atomistic force field, product of an ongoing collaboration between
groups in Greece and USA, describing at the molecular level the idiosyncrasies of 5CB
—a canonic liquid crystal– was characterized as to its macroscopic predictions on phase
behavior of said compound.
Molecular dynamics simulations were performed on 5CB utilizing this and other atom-
istic models with the hope of studying its phase behavior in confined geometries. We also
assessed how achievable/sensitive was predicting conformational changes in polypeptides
with the same techniques and level of description.
4. ATOMISTIC SIMULATIONS OF LIQUID CRYSTALS
Luis Moux-Dom´ınguez
Under the supervision of Professor Juan J. de Pablo
At the University of Wisconsin-Madison
This report describes the efforts invested, from September 2006 till April 2008, towards
the elucidation of some fundamental phenomenological aspects inherent to the problem
of using liquid crystals in biosensors. Computational techniques, in the form of molecular
dynamics, were employed in modeling of the system. The work is novel in the extend to
which it was carried —over 50 ns of mealtime were simulated. Furthermore, it reiterates
the truism that emerging technological advances and the current state of the art provide
henceforth adequate resources to realistically model materials and accurately predict
their macroscopic properties from first principle calculations.
In particular, an atomistic force field, product of an ongoing collaboration between
groups in Greece and USA, describing at the molecular level the idiosyncrasies of 5CB
—a canonic liquid crystal– was characterized as to its macroscopic predictions on phase
behavior of said compound.
Molecular dynamics simulations were performed on 5CB utilizing this and other atom-
istic models with the hope of studying its phase behavior in confined geometries. We also
assessed how achievable/sensitive was predicting conformational changes in polypeptides
with the same techniques and level of description.
Date Juan J. de Pablo
5. ii
Acknowledgments
I would like to thank Jose Moreno-Razo and Rajshekhar for their help and for the
many interesting discussions. Without them my time in Madison would have been fruit-
less and miserable.
Additionally, I would like to mention my appreciation and acknowledge the help of
Manan Chopra, Francisco Hung, Rohit Malshe, Clark Miller, A. Santosh Reddy, Edward
Sambrinski, De-Wei Yin and Yu Zhang. Certainly, one can not ask for better colleagues.
This work was accomplished through the extensive use of GNU software.[113, 117, 124,
83] Simulations were performed with GROMACS while the visualizations were prepared
utilizing VMD and its ray-tracing capabilities.[128, 58, 114] The document per se was
prepared utilizing LATEX.[71]
15. ix
List of Symbols
α Molecular polarizability
χ, χ GB: Molecular anisotropic parameters χ ≡ κ2−1
κ2+1
, χ ≡ κ 1/µ−1
κ 1/µ+1
∆t Time step
(ω), Dielectric response function and static dielectric constant
0 Permittivity of free space
γ Surface tension
ˆı
√
−1
κ, κ GB: Aspect and energetic ratios σe/σs , s/ e
F Helmholtz free energy
H Hamiltonian
Hijk Hamaker constant between surfaces i and j across k
ω, λ Frequency and Wavelength (λω = c)
[¯z], [¯z] Real and imaginary parts of complex number ¯z
ρ Density
16. x
F i Force “felt” by ith
particle
rij Relative intermolecular distance between ith
and jth
particles
θ Orientational angle between liquid crystal molecules
ˆui Molecular orientation of ith
particle
A Area
c Velocity of light in vacuum
h Planck constant (2π = h)
kB Boltzmann constant
mi, ri, vi, ai Mass, Position, Velocity and Acceleration of ith
particle
Pn nth
Legendre polynomial
S2 Nematic order parameter
T, β Temperature and inverse temperature (kBT)−1
TNI Nematic-Isotropic transition temperature
U(r) Energy potential
U(i)
(r1, . . . , rN ) ith
contribution to total potential energy
Q,Q , Qij Tensor quantity Q, its transpose and its ijth
value (2nd
order)
19. 1
Chapter I: Liquid crystals
Introduction and History
Some materials, mostly organic in nature[25], show one or more intermediate phases
between their solid and liquid states. In general, these so-called mesophases have features
characteristic of both solids and liquids. In particular, they are named liquid crystalline
phases if they have long-range orientational order (associated with solids) but lack trans-
lational order (typical of liquids). On the other hand, plastic crystals lack a preferred
orientation but maintain translational order.
In perspective, the study of liquid crystals began when Reinitzer reported multiple
melting points on cholesteryl benzoate. He noticed that when melting from a solid
phase, the sample became a opalescent liquid. Upon further heating the liquid became
transparent. Later Friedel published on the classification of liquid crystals. In particular,
credit is due to him for the terminology —both mesophase and mesogen.
Liquid crystals (LCs) are versatile soft materials with many applications from fields
as diverse as optoelectronics [34, 22], stem cell research [?], biomolecular modeling [75,
108, 85], and cosmology [?]. We are particularly interested in their study because of
their application in biosensors [16] —an interesting situation that stems from liquid
crystals’ capacity to amplify changes in surfaces and interfaces [?]. In fact, there are
already patents [?] of devices that employ nematogens within their main characterization
mechanisms.
20. 2
Properties and idiosyncrasies
The key idiosyncrasy of liquid crystals is their anisotropic shape. This is commonly
encountered as a rigid core containing σ or π bonds (defining the long axis) with flex-
ible side-chains. As would be expected from this description, strong dipoles and easily
polarizable groups within the molecule are a common features.[?]
Furthermore, like liquids LCs are characterized with showing high fluidity, with being
unable to support shear, and with forming droplets. Like crystals, they exhibit anisotropy
in their properties —e.g., electromagnetic and optical properties [?], head conduction [80],
and others.
Applications
It is well known that LCs amplify changes in surfaces. This capability has applica-
tions in sensor manufacturing. One such instance is detecting protein conformational
changes.[?, 16, 75, ?]
Brake [15] reports, that mesogens communicated their orientation at interfaces to
fellow mesogens in the bulk over distances of up to 100 µm from the interface.1
He goes
on to explain that these processes occur within tenths of seconds and that LCs provide
a high level of sensitivity to a range of events. Fortunately, the changes can be easily
identified with polarized light microscopy.
In particular, an interesting case is the detecting conformational changes due to phos-
phorylation of EGFR (epidermal growth factor receptor) since this event activates several
biochemical pathways including, according to Kitano et al. [92], important pathways that
regulate growth, survival, proliferation and differentiation in mammalian cells; according
to Lynch et al. [?], its mutations has linked to cancer; and Herbs et al. [?], relate it to
transmission signals regulating tumor growth and metastasis. Even though conforma-
tional changes in proteins are detectable with LCs, it is not completely understood how
1
*This is the traditional picture, but, as results from research carried out by Ankit and Abbott (as
of yet unpublished work) point out, it may well be bulk that effects the changes in the interface.
21. 3
this comes about and, thus, its elucidation warrants the intellectual effort. Our goal is
to elucidate some of the phenomenological aspects inherent to this problem.
Characterization
Mesogens are classified on at least two levels, namely their molecular shape and the
driving force for their phase transitions. Prolate and oblate, two molecular shapes tradi-
tional commonly encountered when dealing with these, compounds are called calamitic
and discotic particles, respectively. There are instances in which temperature is the driv-
ing force and there are others this role is played by concentration instead. The former
behavior is called thermotropic while the latter is known as lyotropic.
There are various liquid crystalline phases. For instance, there are nematic (from the
Greek word for “thread”), smectic phases, columnar, their chiral versions, and banana
among others. The nematic phase has a high degree of long-range orientational order
but no translational order —meaning that, on average, molecules prefer to orient in a
particular direction, which is commonly known as the director. Smectic phases are more
ordered than the nematic ones.
Figure 1.1 5CB molecule
This preferred orientation of the nematic
phase is characterized by the nematic order
parameter. Defined as the ensemble average
of the second Legendre polynomial (refer to
on page 75) of the cosine of the angle be-
tween each molecular pair. That is,
S2 ≡ P2(cos θ) .
It is worth mentioning that, according to the modern classification (as oppose to the
Ehrenfest classification), the nematic-isotropic transition is a first order one.
In this work we were concerned with modeling 4-pentyl-4’-cyanobiphenyl (5CB) This
may well be a canonical liquid crystal since so much attention has been invested in
its study. Having such a low-molecular weight (249.35 g/mol) —it consists of merely 38
22. 4
atoms—, it is the first in its homologous series (nCB) to show liquid crystalline properties
and its only known mesophase is the nematic phase.2
Ergo, we will concentrate our discussion on the nematic phase and disregard the
other mesophases for the most part, except where there might be something to gain in
describing other mesophases. In particular, we would like to derive examples and rules
from the more general and apply them to our particular case.
Theoretical background
Lattice as well as “hard” models provide simple, theoretical frameworks with which to
approximate problems. As such, they generally provide conceptual insights that enhance
our physical understanding of the phenomena of interest.
The Lebwohl-Lasher is a particularly simple lattice model. It divides the space into a
lattice and assigns a unit vector to each lattice point to represent the molecular orientation
of the corresponding particle. Then the Hamiltonian of the system is calculated by
summing the interactions over nearest neighbors; H = − i,j
ˆdi · ˆdj
p
. For a three-
dimensional system and quadratic contributions (p = 2), this model predicts a first-order
transition from an isotropic to a nematic phase.3
What’s more, studies of ellipsoids and cut spheres[10, 3] with hard walls4
showed
that short range interactions are sufficient to create nematic phases provided that the
anisotropy inherent to the molecules is reflected in their shape. Yet, for these particular
models, the phase behavior is controlled by density and molecular shape. That is, the
materials are lyotropic.
Attractive contributions and long range interactions were fundamental aspects lack-
ing in these studies for very simple, generic liquid crystals. Since the models predicted
2
Refer to Appendix A for a compendium of 5CB’s properties.
3
There is controversy as to what its predictions are for two-dimensional systems. In fact, for the
special case of quadratic contributions to the Hamiltonian, the model reduces to the 2-D XY model
–which in 1973 was shown to have a phase transition (known as Kosterlitz-Thouless). Refer to [?] for
elaboration.
4
With the potential defined as U(r) =
∞ r ≤ σ
0 r > σ
23. 5
lyotropic but not thermotropic liquid crystallinity per se, it follows that these additional
contributions, coupled with the aforementioned peculiarities (namely, short-ranged re-
pulsive energetic contributions in molecules of anisotropic shape) ought to be studied to
understand thermotropic behavior.
Thus we learn that it is possible to explain some macroscopic properties of mesogens
by just incorporating broken symmetries into the molecular picture. We would then
expect any reasonable molecular representation to incorporate an anisotropic shape into
the description. Essentially, no other aspect apart from the anisotropic molecular shape
is needed to account for liquid crystallinity.[?] Furthermore, any description aiming to
model thermotropic liquid crystals should also account for attractive and long-range
interactions —i.e van der Waals forces proper[59].
Onsager utilized a variational approach for modeling the isotropic-nematic transition
of hard-rod lyotropic liquid crystals. He argued that upon truncating the Helmholtz free
energy F expansion in terms of the anisotropic density profile ρ(r, θ) one would expect
F ≡ drdθρ(r, θ) [ln ρ(r, θ) − 1]
−
kBT
2
dr1dθ1 dr2dθ2ρ(r1, θ1)ρ(r2, θ2)f(r1 − r2, θ1, θ2)
As it turns out, this, and Maier-Saupe [78] theory, are asymptotic limits describing the
nematic-isotropic transition.[93] The Onsager theory describes lyotropic mesogens while
thermotropic nematogens are described by Maier-Saupe theory. The former explains
that the systems entropy is minimized when molecules align in a preferred orientation
(parallel to each other) while the latter describes how a mean field produced by other
molecules coming closer would induce a mean orientation across the system.
24. 6
Chapter II: Simulation techniques & Algorithms
Introduction to molecular dynamics
Molecular dynamics simulations aim to simulate experiments as relativistic as possible
and determine the statistical predictions. However, this is done at the molecular scale
rather than at a macroscopic one. A basic assumption is the ergodic hypothesis: namely,
that the (long) time averages are equivalent to the ensemble averages.
Just as in actual experiments, the sample of material is prepared and its time evolution
monitored. In “preparing the sample” we locate, say, N particles in a subset of space
(which we will refer to as box) and set the appropriate environmental variables to match
the desired criteria for our observations —for instance, at temperature T of pressure P.
Then we express mathematically the interactions between the molecules and we let time
take its course.
In doing so, molecular dynamics formalisms provide a direct way to compare experi-
ments with molecular perspectives. Moreover, it is a mean by which theoretical models
may be tested without, in general, many approximations and without the hindrance, and
possible limitations, of having to correct for experimental factors. That is, molecular dy-
namics provide the most straightforward way to benchmark molecular the implications
of molecular models.
Furthermore, a relatively simple way of measuring properties that would, otherwise,
be very difficult to measure empirically. For instance, Cheung employed molecular dy-
namic simulations to calculate flexoelectric coefficients es and eb for a nematic liquid
crystal, which, while being increasingly important in several areas, are difficult to deter-
mine experimentally.[25] In fact, any quantity that can be expressed in terms of ensemble
25. 7
statistics (e.g., averages for free energies and deviations for Cp) is a property that can be
calculated with this family of techniques.
Technically, molecular dynamics (MD) simulations seek to evolve Newton’s equations
of motion in time for many-body problems. This is done by expressing them in terms of
total potential interaction each of the N particles feels
mi
d2
ri
dt2
= −
∂UT
∂ri
= F i
UT ≡ U(i)
(r1, . . . , rN ),
where U(i)
is the ith
contribution to the total potential energy (UT ) which is a function
of the particles’ positions {r1, . . . , rN } and F i is the force “felt” by said particle. These
equations are solved concurrently with relative tiny time steps. The connection with
physics is established by imposing physical restrictions like conservation of energy.
The Newton’s equations shown in II.1 were expressed utilizing vector notation and
may be formulated in generalized coordinates. As a basis, throughout this work we will
assume we are dealing in Cartesian unless otherwise specified. Moreover, in their formu-
lation we assumed there were no constraints. Had there been constraints we would have
had to include them in the set of equations and solve them simultaneously. Definitely,
the fact that we are considering classical behavior follows immediately from the fact that
deal with Newton’s equations. This implies that quantum effects are neglected which is
the case for all but the lightest atoms (e.g., hydrogen) and for situations where we are
well above kBT.
It is important to note that the edges of our system (the subset of the universe we
are simulating, heretherein referred to as box) are not physically meaningful. Thus it
is important to minimize their effects. This is commonly done by introducing what is
known as periodic boundary conditions. The technique simulates an infinite system by
replicating the box all around itself. Andersen [4] gives the implied analogy that, given
the tiny amount of particles normally simulated1
, systems without periodic boundary
1
due, understandably, to scarcity of computational resources or, paraphrased, the large effort required
to do model more
26. 8
conditions would resemble droplets rather than bulk fluids and their properties would
be strongly affected by its surface. Thus we are effectively simulating an infinitely large
bulk materials2
whenever we incorporate them into the model.
Many algorithms exist to integrate these equations. Schemes based on the Verlet
approach tend to have little overall energy long-term drifts. This is important because
it assures energy conservation. Our simulations used the Leap Frog algorithm and it
is equivalent to the Verlet algorithm. This scheme evaluates the particles’ velocities at
half-integer time steps and uses them to obtain the updated positions:
vi(t + 0.5∆t) = vi(t − 0.5∆t) +
F i(t)
mi
∆t
ri(t + ∆t) = ri(t) + vi(t + 0.5∆t)∆t
However efficient an algorithm may be, it is basically impossible to accurately predict
MD trajectories. This is called the Lyapunov instability and states (for a derivation see
Frenkel et al. [46] and the extensive literature on the subject [69])3
that two relatively
similar initial configurations will diverge exponentially as time progresses. Yet this prob-
lem is circumvented once one realizes that we are, in molecular dynamics, only concerned
with the statistical predictions and not the actual trajectories.
A trick used to speed up the computations is to use cut-off radii. These have to be
considered, for instance, when computing the system’s potential energy. Also, implemen-
tations often rely on neighbor lists4
to accelerate the calculations.
We refer the reader to Frenkel and Allen books [46, 2] for the theory behind electro-
static computations, their implementation and tricks of the trait rather than take more
space here. Also, we would neglect discussing about the constraint algorithms heavily
2
Take this statement cum grano salis for the time being because, as we will discuss later, there are size
effects inherent in the calculations that are particularly important when dealing with phase transitions.
3
This is a measure of the system’s chaoticity and it is maximized at phase transitions.[36]
4
A list of the neighboring particles within a cut-off radius is dynamically maintained in memory per
particle.
27. 9
utilized to speed the simulations. The reader may want to refer to G.R. Kneller’s ha-
bilitation thesis for a rigorous development of commonly used techniques as well as how
they may be extended.
Yet the is a topic tht ought to be discussed here. And, that is that there are times that
ensembles other than NV E need to be simulated. For instance, one may want to couple
the system to an heat reservoir or perhaps keep the pressure constant. The algorithm we
described above is particular to the NV E ensemble and needs to be modified to work in
others. The next few sections deal with how constant temperature and constant pressure
is achieved.
Thermostats
Temperature is taken to mean the instantaneous kinetic temperature and is computed
through the system’s kinetic energy:
T =
N
i miv2
i
NdegkB
=
N
i miv2
i
kB (3N − Nc − Ncomm)
,
Nc is the degrees of freedom lost due to constraints, N the number of particles and Ncom
the degrees of freedom related to the center-of-mass. The aim of “constant temperature”
simulations is to keep the average temperature constant. This can be done in several
ways and each has its particular appeal.[2, 46]
Probably the must simple way to control the average temperature of the system
in question is by scaling the velocities (momenta) at each time step. There are two
main ways of accomplishing this. The first, called the iso-kinetic thermostat, calculates
the instantaneous temperature TI with the equipartition theorem and rescale the linear
momenta by a factor λ defined as λ = T
TI
. The second, called the Berendsen thermostat,
has
λ = 1 +
∆t
τT
T
TI
.
Berendsen et al. [11] explained a simple way with which a system could be coupled
to an external bath. They developed the particular cases of pressure (barostat) and
temperature (thermostat) but described how the scheme could be extended to other
28. 10
variables. This scheme provides, in the authors’ words, weak coupling to an external bath
via the least local perturbation consistent with the required global coupling. In fact, it
mimics weak coupling to the external bath at temperature T via first-order kinetics.[125]
Although these schemes are relatively simple and serve the purpose they were de-
signed for, they do not strictly reproduce the canonical ensemble. More advanced tech-
niques are required to maintain thermodynamic rigor. Nevertheless, their deviations from
the correct ensembles are rather small and often reproduce the expectations accurately
enough.[39]
More formal techniques —that ensure correctness– include stochastic approaches
as well as extending the system to including additional degrees of freedom to repre-
sent the coupling.[2] Stochastic methods such as Andersen [4] and Langevin [49] add
stochastic forces or velocities to control the temperature. The former assigns momenta
from a Gaussian distribution (at the target temperature) to a certain number of par-
ticles so as to mimic collisions with the bath’s particles. The latter extends the La-
grangian/Hamiltonian formalism —hence the names extended-ensemble and extended
approaches.
In particular, the Nos´e-Hoover modifies the system’s Hamiltonian by introducing an
additional degree of freedom (in the form of a friction contribution) with a mass Υ
associated with it that dictates the coupling strength.[90] Thus the equations of motion
(now N + 1) become:
mi
d2
ri
dt2
= −
∂UT
∂ri
− miξ
dri
dt
and
dξ
dt
=
TI − T
Υ
, where Υ =
τ2
T T
4π2
,
where we follow GROMACS’ convention in defining the reservoir’s mass in terms of the
kinetic energy period of oscillation τT . As it is evident, this is not a stochastic process
and it is time-reversible. There is an additional technique, based on the Nos´e-Hoover
approach, that implements a chain of these thermostats (each of these “demons” coupled
to the previous one) and it is as rigorous as one can get.[90]
It is common practice to use the Berendsen scheme to relax the system to the
target temperature and then utilize the Nos´e-Hoover coupling to probe the ensemble
29. 11
correctly.[125] This is exactly what we did for most of the jobs. The idea, of course,
is that Berendsen’s coupling is less computational intensive than the other schemes de-
scribed here. Furthermore, the former scheme results in a strongly damped exponential
relaxation while the latter produces an oscillatory relaxation so that it requires the system
much more time to relax.
Barostats
Similarly to thermal coupling, a system may also be coupled to a pressure reservoir
to simulate iso-baric ensembles. The Berendsen scheme can be used for this case as well
but with certain considerations pertaining to the idiosyncrasies of the pressure —namely,
what it represents and that it is a tensor rather than a scalar quantity. Following the same
convention specified in the thermostat section, the instantaneous pressure is denominated
PI and the target pressure P.
Utilizing Berendsen’s approach, coordinates and box vectors would be rescaled at
every integration step with a tensor factor µ such that a first-order kinetic relaxation is
achieved (the argument is the same used in the thermostat version):
∂P
∂t
=
P − PI
τP
and µij = δij −
∆t (Pij − PI,ij)
3τP
ηij,
where η refers to the isothermal compressibility. Needless to say, the specific implemen-
tation of the coordinates rescaling depends on the box geometry.
Again, weak coupling result in inaccuracies when simulating ensembles. Thus, a
scheme similar to the Nos´e-Hoover thermostat, was developed by Parrinello et al. [99]
and later extended by Nos´e et al. [91]. Again, following GROMACS’ conventions [125],
the equations of motions must be modified as follows
mi
d2
ri
dt2
= −
∂UT
∂ri
− miM
dri
dt
and
d2
b
dt2
= V W−1
b
−1
(PI − P)
M = b−1
b
db
dt
+
db
dt
b b
−1
where W−1
ij
=
4π2
3τ2
P L
ηij,
where V represents the box current volume, b represents the box vectors, W represents
the mass parameter matrix, τP is the time constant, and L ≡ max b.
30. 12
To simulate the ensemble the NPT one would have to introduce both a barostat and
a thermostat and, in case Nos´e-Hoover thermostat and Parrinello-Rahman barostat were
to be implemented, both additional degrees of freedom would be included.
31. 13
Chapter III: Molecular models for mesogens
In Chapter I we discussed several LC models. In particular, we discussed Onsager’s
limit, Lebwohl-Lasher’s model and generalized hard representations. Here we concentrate
on a particularly popular single-site model often used when representing mesogens.1
Then
we discuss shortcomings for such models and explain the need, for the level of under-
standing our particular research problems require, for more detailed models.
In doing so, atomistic descriptions of molecules are introduced. Their history is
mentioned with a discussion of common contributions and the concepts behind them.
Finally, their use in the field of nematogen simulations is discussed with emphasis on the
subset most closely related to the present work.
Coarse-grained representations: single-site models
σ
ε
1.0
0.5
1.2 1.4 1.6 1.8 2.0 2.2 2.4
Energy
Internuclear distance
−0.5
−1.0
1.0
0.0
Figure 3.1 Lennard-Jones 12-6 potential
It is well known that neutral atoms —
and, by extension, neutral particles— at-
tract through long-range interactions and re-
pel at close distances. The attractive con-
tribution originates from instantaneous in-
duced dipole moments due to electron fluc-
tuations and is inversely proportional to the
sixth power of the intranuclear distance.
These long-range interactions are the van
der Waals dispersion contributions —often
1
Curiously, it is related to the boids flocking model for simulating collective motion in fauna.[105]
32. 14
called London forces. The repulsion, on the other hand, is a manifestation of Pauli’s
exclusion principle —electron density clouds may not overlap.
The Lennard-Jones potential [72] captures this behavior accurately. This model (semi-
empirical in nature) relies on expressing an attractive contribution with sixth power de-
pendence on distance, and a repulsive short-ranged part —usually taken to be a function
of the twelfth power.
It assumes spherical symmetry and takes as parameters the molecular diameter σ
(defined as the distance from nucleus to nucleus) as well as the potential well depth .
As we can conclude from its formula
U(LJ)
(r; σ, ) = 4
σ
r
12
−
σ
r
6
,
σ is the separation at which the contributions cancel out. Furthermore, the most stable
separation corresponds to re = 21/6
σ.
It has been shown to accurately describe ideal gases. What’s more, neutral spheric
particles are often modeled utilizing the LJ potential.2
It therefore provides an adequate
basis to base coarse-grained models for LCs.
Yet, from what we have learn so far about LCs, one would expect that if one were to
describe the interactions between mesogens solely by a pairwise potential, then this po-
tential ought to be a function of both their relative intermolecular distance (rij ≡ |rij| ˆrij)
as well as their respective orientations (ˆui, ˆuj). Succinctly, one expects the functional
form of the potential to be Uij = U (ˆui, ˆuj, rij) , in accordance with the aforementioned
nomenclature. That is, the LJ potential lacks the quintessential requirements to model
LCs.
The Gay-Berne (GB) potential[48] is a single site (a.k.a. Corner) potential that ac-
curately represents a range of mesogen behavior. It is a generalization of the 12-6 LJ
2
The phase behavior is summarized as going from an isotropic liquid to cubic close-packing upon
cooling and to hexagonal close-packing upon further cooling.[?]
33. 15
interaction potential in which particles are modeled as ellipsoids of revolution with orien-
tation ˆui along the major axis of ith
particle. Thus, it is generic potential that includes
the essential characteristic for liquid-crystalline behavior —namely, anisotropy[7]
Even though GB was originally designed for calamitic molecules, it has since been
extended to discotic molecules. It is generally considered computationally appealing and,
historically, it has revealed great value.[8, ?, ?, ?] It incorporates a molecular anisotropy
parameter χ that is defined as χ ≡ κ2−1
κ2+1
, where κ is the ratio (σe/σs) of the major to
minor distances (defined in terms of contact distances) for the ellipsoid —thus χ = 0
when dealing with spheres. Additionally, well depth anisotropy is incorporated into the
χ parameter that is defined as χ ≡ κ 1/µ−1
κ 1/µ+1
, where κ is the ratio ( s/ e) between the
well depth corresponding to molecules aligned side-by-side and aligned in a tip-to-tip,
respectively.
Its angular dependent description of the contact distance is done by defining σ0 to cor-
respond to the side-by-side contact distance —e.g., twice the minor, and σ (ˆui, ˆuj, ˆrij) =
σ0 [g(χ)]− 1
2 . Furthermore, angular dependence of the well depth, (ˆui, ˆuj, ˆrij), is ex-
pressed in terms of the product of two functions
(ˆui, ˆuj, ˆrij) = [ 1(ˆui, ˆuj)]υ
[ 2 (ˆui, ˆuj, ˆrij)]µ
,
where the first of these functions 1(ˆui, ˆuj) favors parallel alignment aiding liquid crys-
talline formations and is defined as 1(ˆui, ˆuj) ≡ 1 − χ2
(ˆui · ˆuj)2 − 1
2
. The second func-
tion 2 (ˆui, ˆuj, ˆrij) has a analogous form to σ (ˆui, ˆuj, ˆrij) and promotes side-side align-
ments over tip-to-tip and favors smectic phases. It is defined as 2 (ˆui, ˆuj, ˆrij) ≡ g(χ ),
and incorporates a depth anisotropy parameter χ parameter that is defined as χ ≡
κ 1/µ−1
κ 1/µ+1
, where κ is the ratio ( s/ e) between the well depth corresponding to molecules
aligned side-by-side and aligned in a tip-to-tip, respectively. Here g( ) is defined as
g(ζ) ≡ 1 − ζ
(ˆui · ˆrij)2
+ (ˆui · ˆrij)2
− 2ζ (ˆui · ˆrij) (ˆuj · ˆrij) (ˆui · ˆuj)
1 − ζ2 (ˆui · ˆuj)2 .
Combined, the full potential is U (ˆui, ˆuj, rij) = 4 0 (ˆui, ˆuj, ˆrij) (R−12
− R−6
) , where
R was defined as R ≡
|rij|−σ(ˆui,ˆuj,ˆrij)+σ0
σ0
and corresponds to a corrected separation.
34. 16
Figure ?? summarizes graphically what we have just described and it might be useful in
clarifying any doubt that may remain.
The parameters {σ0, 0} incorporate information about scaling in size and energy,
respectively. The adjustable parameters {κ, κ , µ, ν} control the interaction’s anisotropy.
Thus, Gay-Berne’s potential includes four adjustable parameters as well as two scaling
parameters and may be expressed in Luckhurst’s notation GB(κ, κ , µ, ν).[9]
The shortcomings inherent to this type of coarse-graining are explained below. Yet,
this particular description it still very popular and useful —particularly when build upon
it, like, for instance, developing a model for polymers with liquid crystalline side chains.
Atomistic models
Empirical force fields (FFs) provide an intermediate approximation between utilizing
quantum mechanics directly and using “single-site representations”. They take into ac-
count both intermolecular and intramolecular interactions. These latter class is usually
divided into bond bending/stretching, dihedral, among other potentials. Moreover, they
assume that the Bonn-Oppenheimer approximation holds, that electronic structure is
static and that pairwise additivity holds
Warshel3
and Lifson [131] initiated the field of atomistic simulations as it is cur-
rently understood with their pioneer consistent force field (CFF).[51] In 1983 Karplus
[17] and his group in Harvard University published CHARMM (Chemistry at Harvard
Macromolecular Mechanics), a force field originally conceived to simulate proteins.
As history tells, documented in the CHARMM’s website, towards the end of the
1970’s there was quite a large interest in describing the contributions to the potential
energy of small molecules. They aimed to produce a program that would take as input
an amino acid sequence and their respective coordinates to calculate the system’s energy.
They borrowed ideas and acknowledged the work of Warshel (CFF) and others.
3
He also was the first to perform free energy perturbations simulations in biomolecules.[130, 132]
35. 17
In 1984, van Gunsteren et al. [52] from Groningen University in collaboration with
ETH Z¨urich published a general purpose molecular dynamics simulation force field and
corresponding software named GROMOS (GROningen MOlecular Simulations). It was
later [126] updated. Its goals, as of the latest version, are: analysis of conformations
and studying the effect of environment on them; calculating binding constants through
free energy calculations; studying the effect of modifications of molecular building blocks
(be them amino or nucleic acids) on their respective polymers (e.g., structural and ener-
getic changes); three-dimensional structure elucidation; and properties prediction under
conditions experimentally inaccessible. Its parameters were optimized against empirical
results of condensed phase alkanes.
Figure 3.2 General force fields genealogy
In 1984, Kollman’s group at California
University - San Francisco published what
would become the family of force fields called
AMBER (Assisted Model Building and En-
ergy Refinement).[67] This is in fact merely
a functional formalism (with the correspond-
ing theory behind each contribution) and re-
quires parameterization for specific applica-
tions. There are set of these parameteriza-
tions that have been optimized for partic-
ular materials. For instance, there exist a
so called GAFF (Generalized AMBER force
field) that aims to facilitate simulations of
small organic molecules in drug-ligand with
biomolecules; sets describing biomolecules (with naming convention: ff and two digit year
36. 18
number); and others describing carbohydrates. It is this form (the so called AMBER-like)
of force fields that we used.4
In 1988 Jorgensen et al. [63, 62, 32] published their optimized potential for liquid
simulations (OPLS) that, as the name implies was optimized to reproduce liquid prop-
erties. This in turn influenced subsequent parameterizations of AMBER —in particular
ff94 which is also known as “Cornell et al. force field” and reported a set of parameters
for proteins all-atom simulations in condensed phase.[?] Competing force fields include
one developed by Siepmann’s group (TraPPE) [81, 82, 24, 133, 23, 115, 134, 77] that
began with the objective of predicting accurate phase behavior for alkanes as well as one
developed within UW-Madison (NERD) [88, 87] to simulate alkanes.
When Berger and coworkers [12] began their simulation work on lipid, they noticed
that, with the then available forcefields, the volume per lipid predictions were inaccu-
rate. The best results, according to them, were those predicted by the OPLS force field.
Jorgensen and Tirado-Rives (OPLS’ original authors) had discussed in their work that
inaccurate predictions in volume (too low) and heat of vaporizations (too high) resulted
when simulating larger alkane chains. Thus, Berger et al. adjusted the Lennard-Jones pa-
rameters by optimizing them against the heat of vaporization of pentadecane and thence
improved their predictions (BergerFF).
The geneology of general purpose force fields is summarized in Figure 3.6. The ener-
getic contributions regularly associated with AMBER (or AMBER-like) force fields are
discussed next.
Regular contributions
There are, in general, two kinds of contributions to the Hamiltonian of the system:
bonded and non-bonded interactions —i.e., UT = Ubonded
+ U!bonded
. Within the bonded
interactions we would expect to account for bond stability (i.e., stretching), and geomet-
rical factors such as preferred proper angles as well as dihedral angles —i.e., torsions. On
4
The force fields discussed here are the so-called classical force fields and atomistic. There are various
other kinds.
37. 19
the other hand, we expect to account for dispersion forces and electrostatic interactions.
Thus, U!bonded
= ULJ
+ UUlectro
, while Ubonded
= Ubond
+ Udihedral
+ Uangle
.
Figure 3.3 Bond stretching
One accounts for the stability of a bond by penalizing
how much said bond can stretch. For our work, we utilize
a simple harmonic potential of the form
U
(b)
ij (rij) =
1
2
kb
ij(rij − re
ij)2
.
When this functional form is used, the average energy of a bond equals 1
2
kBT. In the
more general case, the Morse potential
U
(b)
ij (rij) = De
ij 1 − e−
√
ke
ij/2De
ij(rij−re
ij)
2
could be used since it accounts for both the anharmonicity of real bonds and for bond
breaking. The last version of GROMOS also used a “fourth-power” potential.
Figure 3.4 Angle
contributions
The force fields studied adopted a similar functional
form to account for bond angle vibrations. That is, we
used the harmonic angle potential and its functional form
is U
(a)
ijk (θijk) = 1
2
(θijk − θe
ijk)2
. As does the harmonic bond
stretching contribution, this potential corresponds to a
parabola when one plots it.
The torsion contribution is commonly approximated
with either a periodic potential or with Ryckaert-
Bellemans (RB) functions. The periodic form is used in GROMOS and OPLS while
the other is used in CHARMM. These functions have the following forms, with first one
being the harmonic,
kφ
ijkl 1 + cos[nφijkl − φs
ijkl] v.s.
5
n=0
Cn,ijkl (cos[φijkl − 180◦
]).
As it turns out, both forms are equivalent and after some trigonometry one can express
one in terms of the other.
38. 20
Figure 3.5 Dihedral
contributions
Non-bonded interactions account for dispersion forces
as well as for electronic interactions. Dispersion forces
(a.k.a. van der Waals interactions) are due to induced
dipole moments in atoms formed from fluctuations in the
electronic structure, and a short-range repulsive force per-
taining to overlapped regions and Pauli’s exclusion prin-
ciple as discussed above. Electronic interactions, in the
sense meant here, are limited to permanent partial charges —i.e., Coulombic. Gener-
ally, the dispersion contributions take the form of the above mentioned Lennard-Jones
potential but sometimes, instead of the power function with 12th power, the repulsive
part is modeled with an exponential decay —in which case the potential is known as
Buckingham potential. Some argue that using an exponential decay provides greater
flexibility and is more realistic (theoretically sound), but the results are comparable for
thermodynamic properties. Therefore, it is no surprise that in practice the LJ potential
is usually preferred, since the Buckingham potential is more computationally expensive
than the LJ potential and produces comparable results.[98]
Atomistic models for LCs
Even though nematogens have a rigid core, more often than not, they are also com-
posed of flexible side-chains. This molecular flexibility greatly influences the macroscopic
properties of the materials. For instance, it is currently understood that flexible side-
chains induce liquid-crystalline behavior in low-molecular mass nematogens by inhibit-
ing crystallization. What’s more, ever since hard-models were generalized to incorporate
non-spherical shapes[10, 3] we know that molecular shape has a large effect on the phase
behavior of materials. Current, single-site coarse-grained models fail to reflect this in-
trinsic flexibility.
Molecular dynamics by Komolkin et al. [104] show that much flexibility needs to be
represented in terms of the groups incorporated into the molecular description. By that
we mean that in their work (which included both experimental and simulation results)
39. 21
they compared the predictions of AA and UA FF about the dihedral angle. Their results,
shown in Table ?? show how the more groups explicitly simulated, the more accurate the
prediction becomes. This rather intuitive result suggests, ceteris paribus, that all-atoms
representations ought to be used to tweak molecular properties in important regions of
the molecule in question.
Figure 3.6 Force fields for LCs
Annotated transitions pertain to 5CB.
Given the current state of computational
resources –one could argue that emerging
technologies, that work more efficiently than
previous ones, provide a surplus of computa-
tional power—, it is reasonable to seek more
realistic models of everything —including
liquid crystals. In doing so, the community
is guided by the truism “realistic predictions
of properties follow from accurate descrip-
tions of the molecular interactions”.
Yet, traditional, general force fields
with their particular parameterizations fail
to reproduce accurately the macroscopic
properties of liquid crystals. This situ-
ation prompted the development of force
fields particularly designed to model them.
The Hybrid Force Field (HFF) provided
a scheme by which atomistic nematogens
could be developed systematically. The
functional form correspond to a combination
of the regular contributions enunciated above. In particular, it borrowed the contributions
included in the Dreiding II FF with RB torsional contributions. For both Lennard-Jones
parameters, pair values correspond to the geometric average of individual ones —i.e., not
40. 22
Lorentz-Berthelot combination rules. The particular equilibrium parameter values per-
taining to molecular geometry were calculated from ab initio simulations utilizing density
functional theory.[47]
In particular, HFF grouped hydrogens with carbons when these were sp3
carbons. Yet
hydrogens were explicitly modeled for sp2
, sp1
and aromatic carbons. This is common
practice. Partial charges were calculated using DFT. Ergo, the HFF has percolated into
the community to the extend of being the de facto standard as to how force fields for
liquid crystals ought to be developed.
Later, Cheung and Wilson [26] simulated it in bulk using their own AMBER-like
united-atoms model. More recently Tani et al proposed an united-atoms force field of
the same form for the nCB family from first principles using their FRM methodology
[14, 18, ?].
Kim et al. [65, 64] utilized Cheung’s parameters in order to carry out umbrella
sampling simulations. They wished to obtain the potential mean force of 5CB and another
nematogen molecules partitioning into a lipid bilayer. It was this implementation that
we inherited and heretherein, whenever we mention Kim and coworkers’ work we are
referring to this implementation of Cheung’s FF (heretherein called CheungFF). As it
turns out, Cheung’s (as implemented by Kim et al.) parameters predict incorrectly the
nematic-isotropic transition as well as the density as a function of temperature.
We carried out simulations utilizing this implementation for about a semester —both
in bulk to characterize the transition and studying the effect of surfactants (in the form of
DPPC) on the preferred orientation. Its predicted TNI was close to 77 ◦
C (13% absolute
error above the correct transition of approximately 37 ◦
C; refer to Appendix A) plus it
resulted in a density of 1.029 g/cc at 87 ◦
C when we would expect it (extrapolating from
Tintaru’s regression in Figure 1.3 but noting that that temperature would be beyond the
range for which the regression is valid) to be much less than that.
Going over Cheung and coworkers’ methodology [26, 25], it was found that they
had used a dubious DFT (Density functional theory) implementation —they used the
41. 23
CASTEP [31, 107] program instead of using a more rigorous and established imple-
mentation like GAUSSIAN [123]— with a rather lax level of theory —they utilized the
pseudo-potential-plane wave method PW91 [5]— for their parameterization. It has since
been challenged in literature. However, it is acknowledged for setting the standard of
using biphenyl parameters for the rings.
Thus we set out to find or develop a better force field for 5CB. Tani et al. [18]
proposed force field aces the density (1.085 at 27◦
C [6%]), transition temperature (approx.
37◦
C [0%]) and the order parameter (0.54 at 27◦
C [0.4%]). However, they utilize an
extended Lennard-Jones term with an extra parameter ξi to vary the potential well’s
width independently from depth and position.5
Figure 3.7 Our FF’s atom types
This schematic representation of 5CB shows our nomencla-
ture. Refer to this image and its annotations when using
table C.2(a).
This additional parameter is rather un-
conventional, even though it is reasonably
justified, so that we decided against using
it. They also inserted extra artifacts to
the dihedral contributions in order to tweak
the molecular geometry, but apart from that
they developed a force field from first prin-
ciples. Furthermore, they have thoroughly
proved that the head group can be modeled
as pure biphenyl, and they have extended
they parameterization to other members of
the nCB homologous series.
Theodorou’s group [84], as part of an ongoing collaboration, combined Cheung’s de-
scription of 5CB’s head (which is consistent with Tani’s parameterization) with Berger’s
description of lipids for 5CB’s tail. The bulk of the work presented here was carried out
using this force field —which was sent towards the end of last summer. Its parameters
5
Uij = 4 ij
ξij σij
rij +σij (ξij −1)
12
+
ξij σij
rij +σij (ξij −1)
6
, where ij =
√
i j, σij =
(σi+σj )
2 , ξij = ξiξj.
42. 24
and idiosyncrasies are detailed in the next section and summarized in table C.2(a). From
now on, when no other force field is mentioned by name, we refer to this one (TheoFF).
2 4 6 8 10 12 14 16 18 20 22 24 26
Molecular group number
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
0.50
Partialcharge[e]
DFT [AMI]
TheoFF
TaniFF
Figure 3.8 Partial charges comparison
Refer to table C.2(a) and figure 3.7 for mapping
the group numbers to the molecule.
TheoFF is an AMBER-like united/all-
atom hybrid. It partitions the 5CB molecule
into 27 groups. All the atoms in the
head group are represented independently,
while the tail is represented with five groups
—four CH2 groups and one CH3 groups.
The groups in the tail correspond to the
LP2 (CH2) and LP3 (CH3) atom types in
Berger’s nomenclature. On the other hand,
the groups in the head correspond to Che-
ung’s parameters.
This FF consistently utilizes harmonic
potentials for angle and bond vibrations
throughout the molecule. The tail’s dihedral angles (δi=1,2,3) are modeled with RB po-
tential while the ones within the head group are modeled with the periodic potential.
That is, for TheoFF:
Ubonds
=
1
2 ij
kb
ij(rij − ro
ij)2
Harmonic: everywell
Uangles
=
1
2 ijk
kφ
ijk(φijk − φo
ijk)2
Harmonic: everywhere
Unon-bonded
ij = 4 ij
σij
rij
12
−
σij
rij
6
+
zizje2
4π 0r2
ij
Udihds
=
All
Periodic: head
kφ(1 + cos(nφ − φo
))
5
n=0
Cncos(φ − 180)
RB: tail
43. 25
The specific atom types and parameters we used are defined in tables C.1(b)-C.2(a).
Lorentz-Berthelot (LB) combination rules6
were utilized in our work —other rules7
were
originally used by the originating group but, after carrying out benchmarks, we since
decided to use LB rules. Additionally, the dihedral angles, proper angles and bond pa-
rameters are summarized in tables ?? through ??. No improper dihedral angle penalties
were utilized. The functional form as well as the contributions we include in the model
are summarized (for the intramolecular case) in Figure ??. The resulting dihedral angle
distribution (refer to ??) agrees with experimental values [70, 102, 104, 6, 21, 1] as well
as what Tani et al report for their force field [18].
Figure 3.8 illustrates the differences between Tani’s FF and TheoFF in terms of partial
charges. The partial charges predicted by a DFT calculations performed by Dr. Moreno-
Razo utilizing the AM1 (Austin Model 1) theory [38] in GAUSSIAN are also included as
a reference.
We also, during the course of our work, tried to modify TheoFF to better capture
5CB’s density profile. For instance, we modified it by modeling the tail with the NERD
[?] parameters —safe for dihedral contributions. This modified FF proofed worse in its
prediction of density than the original one so we discarded it. Additionally, we studied
the possibility of using TraPPE to model the whole molecule. But to use TraPPE we
would need to determine the partial charges with the CM4 method.[?] Because of the
transferability objective of the FF, it wouldn’t include information about the dihedrals
—proper or otherwise.
Dihedral angles are geometrical characteristics of utmost importance.[47] From what
we learned in Tani’s paper [14] —their FF based on first principles— that dihedral penal-
ties are necessary to constraint the flexibility of the tail to avoid unphysical situations,
we would expect a prudent representation for liquid crystals to include dihedral contri-
butions to the energy. Thus, TraPPE, which to have even more flexibility, would seem to
6
LB rules: σij =
σi+σj
2 and ij =
√
i j
7
Original rules: σij =
√
σiσj and ij =
√
i j
44. 26
be an unlikely candidate. From this period we () Hess and Lindahl’s aphorism8
: “force
fields must be self-consistent”.
CheungFF vs TheoFF
Theodorou
10000
15000
20000
25000
30000
−150 −100 −50 0 50 100 150Energy(kJ/mol)
Torsional angle (deg)
Torsional potential profile for proper dihedral angles
Kim
5000
Figure 3.9 Torsional overall differences
This profile correspond to the hypothetical case
in which all the bonds share the same angle.
Theodorou’s group shared their FF
(TheoFF) with us. We had been told that
this FF accurately predicted density. Never-
theless we assessed the differences in molec-
ular description between it and the one we
had inherited from Kim. We also wished to
compare it with the one propose by Tani.[14]
Each term was compared with its coun-
terpart in the FF previously used. The nam-
ing convention for the angles was borrowed
from Tani’s paper and Figure 1.1 depicts
it. We found that equilibrium values for
all contributions, safe electrostatics, directly
matched, in both FFs, what DFT calcula-
tions predicted.
Figure portrays the torsional potential comparison. We have only accounted for
proper dihedrals. Our old force field only had periodic terms but the new one included
three RB terms and describe the tail of the mesogen.
These profiles aren’t physically meaningful per se, since they were prepared summing
all the contributions while applying the same angles/length to each bond —which cer-
tainly is highly improbable. However, they serve to grasp the overall picture of the FF’s
differences.
8
As expressed during their talks at the GROMACS Workshop, Spring 2007 @ CSC. Refer to the
videos available at http://www.csc.fi/chem/course/gmx2007/.
45. 27
Theodorou
2e+10
4e+10
6e+10
8e+10
1e+11
1.2e+11
1.4e+11
1.6e+11
1.8e+11
−150 −100 −50 0 50 100 150
Energy(kJ/mol)
r (nm)
Bond streching potential profiles
Kim
0
Figure 3.10 Bond overall differences
This profiles correspond to the hypothetical case
in which all the bonds share the same angle.
Figures 3.11(a), 3.12(a) and from
3.13(a)-3.13(c) show how the ring-ring and
chain-ring interactions behave. These other
set offers a more detailed taste of what is go-
ing on. They are presented in the same for-
mat as Tani et al.[14] present their results.
The improper dihedral potential used by
the FF’s has the form of 1
2
kξ (ξ − ξ0)2
. These
FF’s have kξ,i=0 —i.e., this contribute noth-
ing. On the other hand, they still have bond
stretching contributions which, as discussed
above, may be harmonic, “fourth-power” or Morse. Both FF’s use the harmonic approx-
imation.
Figure compares the bond stretching contribution to the intramolecular energy.
Again, this was calculated with an unphysical situation. All bonds were changed so
as to have equal stretching. Figures 3.11(b) and 3.12(b) show the potentials for the
ring-ring and chain-ring bonds, respectively.
Torsional potential profile for˙ dihedral
200
400
600
800
1000
1200
1400
1600
1800
−150 −100 −50 0 50 100 150
Energy(kJ/mol)
Torsional angle (deg)
Kim
Theodorou
0
(a) Torsional potential
Theodorou
5e+08
1e+09
1.5e+09
2e+09
2.5e+09
3e+09
3.5e+09
4e+09
−150 −100 −50 0 50 100 150
Energy(kJ/mol)
r (nm)
Bond streching potential profiles
Kim
0
(b) Bond potential
Figure 3.11 Ring-ring (φ) potentials comparison
These sets of graphs portray a general picture of what each interaction contributes to
the overall situation. However, their cumulative effect can be better understood via their
47. 29
predictions of the dihedral angle distributions. This is done in Figures 3.14(a)-3.14(b).
The next sections pertain to this force field’s complete characterization and a series of
applications.
48. 30
Figure 3.14 Dihedral angle distribution
Compared with published data collected in Table A.2 and with the work of Tani et al. [18], TheoFF
reproduces 5CB torsional angle distributions.
49. 31
Chapter IV: Characterization of bulk predictions
We used the GROMACS [128, 74] package for all simulations. In particular, we
utilized its implementation of the Nos´e-Hoover and Berendsen thermostats [90, 11] (at
various temperatures) as well as the Parrinello-Rahman barostat [99] (at 1 bar). Fur-
thermore, we simulated cubic boxes of 4.4 nm initial sides in the NPT ensemble with
three-dimensional periodic boundary conditions (that is, on ˆx, ˆy and ˆz axes) and 2 fs
time steps. Finally, we report the results for the cooling process from 400K to 310K
of 1000, 500 and 216 particles —each temperature data corresponding to at least 10 ns
with output frequency of 5 ps.
N T NI Converged? Solid?
216 ? Yes ?
500 55-60 Yes Yes
1000 66-72 No No
Table IV.1 Apparent TNI
The condensed results shown in Fig-
ure 4.1 present how the amount of the simu-
lated particles affects the macroscopic prop-
erties. The experimental values of density
as a function of temperature are included so
that they may serve as a reference. By uti-
lizing more particles we expected to decrease
the fluctuations since they scale with the reciprocal of the square root of particle number,
yet we expected computation time of each run to increase. We were also probing size-
effects since N = 216 were predicting no clear isotropic-nematic transition and possible
a smectic phase even though there should be none for 5CB. That is, we were probing
whether the predicted properties were due to artifacts of the system.
The detailed concatenated results for the 500 and 1000 molecules cases are shown
from Figure 4.2(a) through 4.3(c). We also include the case corresponding to our test
of Tani’s et al. [18] charges, and compare it accordingly in figures 4.4(a) through 4.5(c).
50. 32
The density behavior seems to improve using Tani’s charges, yet the order parameter
definitely worsens.
303K 313K 323K 333K 343K
70
80
90
100
E/N(kJ/mol)
Bulk 5CB: MD-NPT with P=1bar, began isotropically
0.98
1.00
1.02
1.04
1.06
1.08
ρ(g/mL)
Tintaru [Exp]
N=216
N=500
N=1000
30 40 50 60 70
T (ºC)
0.00
0.40
P2
(-)
Note: Only values for the temperatures' last runs are shown
Not all temperatures have run the same amount of time
Figure 4.1 Condensed size effect results
Including N = 216, 500, 1000
Figures 4.6 and 4.7 show the averaged (as a function of temperature) and concatenated
(in terms of simulation time) results for the N = 500 and N = 1000 cases. Within them,
we show the approximate amount of time spent simulating the systems. Most of these
simulations were performed in parallel with four CPUs per job. All of them ran the MPI
[?] implementation of GROMACS on the Condor [?] scheduling system.
These results show how sensitive to system size is the phase behavior. This is con-
sistent with what is expected for first-order transitions.[13] Furthermore, scaling theory
arguments[45] could be used to better characterize the true transition from the informa-
tion we have collected so far. This would be a very interesting project, but we leave it
for a future occasion.
Certainly other methodologies to determining the phase behavior could be used.[27]
Seeing how easily and fast the ExE-DOS scheme predicts phase transitions [103, 64,
86] (particularly its parallel implementations and dynamical adjustments enhancements
51. 33
currently being developed), it would be advantageous to implement our atomistic models
and try to elucidate the correct thermodynamic properties. In particular, the Shekhar’s
CExE-DOS approach, an improvement on the former, seems a promising technique for
even faster results. In fact, this problem, of simulating atomistically liquid crystals, as
opposed to with the Gay-Berne coarse-graining, could greatly use the boost. What’s
more, ExE-DOS could, in principle, be employed as a test bed to iteratively develop
accurate molecular models.
54. 36
0 10 20 30 40 50 60
Real Time (ns)
0.0
0.2
0.4
0.6
0.8
1.0
P2
5 10 15 20 25 30
CPU Time (Days)
42 C
47 C
52 C
57 C
62 C
67 C
72 C
77 C
0 10 20 30 40 50 60
Real Time (ns)
1.00
1.01
1.02
1.03
1.04
1.05
ρ(g/cc)
5 10 15 20 25 30
CPU Time (Days)
40 50 60 70 80
Temperature (ºC)
0.0
0.2
0.4
0.6
0.8
1.0
P2
20 30 40 50 60 70 80
Temperature (ºC)
1.01
1.02
1.03
1.04
1.05
ρ(g/cc)
Sim
Exp [Tintaru]
Figure 4.6 N=500 results
0 5 10 15 20 25 30
Real Time (ns)
0.0
0.2
0.4
0.6
0.8
1.0
P2
5 10 15 20 25 30
CPU Time (Days)
37 C
42 C
47 C
52 C
57 C
62 C
67 C
72C
77 C
0 5 10 15 20 25 30
Real Time (ns)
1.00
1.01
1.02
1.03
1.04
1.05
ρ(g/cc)
5 10 15 20 25 30
CPU Time (Days)
40 50 60 70 80
Temperature (ºC)
0.0
0.2
0.4
0.6
0.8
1.0
P2
20 30 40 50 60 70 80
Temperature (ºC)
1.01
1.02
1.03
1.04
1.05
ρ(g/cc)
Sim
Exp
Figure 4.7 N=1000 results
55. 37
Chapter V: Understanding a LC based biosensor
The aim of our project was to better understand how liquid crystals amplify events
on their interfacial surface. This is a fundamental phenomenological aspect of exploiting
the properties of liquid crystals to manufacture sensors. In particular, we were interested
in studying the phenomena from the perspective of its application to characterize a
conformational change on EGFR. In order to do this, however, the problem has to be
taken in stride and divided into smaller mini-projects.
First, it was necessary to develop molecular model ( III on page 13) that could re-
produce accurately the macroscopic properties —e.g., phase behavior and anchoring con-
ditions — of a liquid crystal ( on page 31) and validate it against both theoretical and
empirical predictions ( on page 41). Also, it was necessary to ascertain that configura-
tional changes in the target polypeptide could be perceived with our technique and level
of theory ( on page 49). Then the effect of surfactants ( on the current page) and of con-
finement ( on page 65) ought to be characterized, since both effects would, in principle,
affect our target system.
Finally, all of these parts would have to be combined and a simulation system con-
figured. This would incorporate everything and provide closure. However, we are still
not at that level. The pillars are set and other can build upon our results. We estimate
that the project’s original goals may be accomplished extending what is included in this
report.
Surfactant
We studied how including a single molecule of surfactant (in the form of DPPC)
into a “sea” of liquid crystal (5CB, albeit with CheungFF) would affect the the local
56. 38
orientations. We also studied how adding an interface of water would affect the anchoring
on the mesogen-aqueous interface.
We were asked to implement and study a subset of the lyotropic behavior in ther-
motropic 5CB system analyzed by Blanc and coworkers. Blanc’s paper[122] reports the
phase behavior for the quaternary system of 5CB, water, DDAB (as surfactant) and
DTAB (as co-surfactant). These are some of the components that were studied in Prof.
Abbott’s original paper.[16] We used SPC model[35] for the water molecules. There are
currently no published models for DDAB or DTAB.
However, we found a series of already published models for phospholipids in The
Karttunen Group of the Department of Applied Mathematics at The University of West-
ern Ontario web site.1
From those available, DPPC was selected on account that it has
a structure similar to DDAB and because it was also studied by Prof. Abbott’s group
—the paper discusses its effects.[16]
The model for DPPC used by Manolis Doxastakis et al.[40] used the GROMOS force
field for the head groups and NERD force field for the aliphatic tails. We used the model
used by Ilpo Vattulainen and coworkers.[100, 101] In their studies, Ilpo Vattulainen et
al. checked the effect of truncation distances on DPPC’s lipid bilayer properties. The
results published on their website are those of 20 and 50 ns of simulation that coincide
with experimental observations. The parameters they used were those of Tieleman et
al.[119] and have been extended, as well as validated, by Berger et al.[12] They also
validated the results.[100]
This force field is also based on the GROMOS forcefield. Furthermore, the corrob-
oration was done by testing the resulting lipid density so as to obtain the correct area
per lipid —Berger et al. report that it agrees up to 3% of the experimental values. They
point out that the head group is pretty much resolved but that the rest of the molecule is
very sensitive to the details of the forcefield. What’s more, these parameters have been
1
http://www.apmaths.uwo.ca/∼mkarttu/
57. 39
extensively used by themselves, checked and even used to develop forcefields for other
molecules.[55, 111, 96]
The NERD potential was developed by the group specially to simulate alkanes, alpha-
olefins, etc. It is in great agreement with phase equilibrium data. Tieleman et al.[120]
discusses that analysis of dihedral transitions in the lipid chains coincide with the idea
that isomerization rates in alkanes are similar to those of liquid crystalline bilayers.
Furthermore, the fast motions present in DPPC are comparable to those of hexadecane.
While the model used by Manolis incorporates the NERD parameters for the tail, the
Tieleman-Berger parameters form a consistent set which have been fine-tuned to simulate
DPPC and recuperate its properties precisely. Thus, we don’t see the need to change
and would probably continue using it as it is very specific to our molecule of interest and
has been extensively tested/fine-tuned.
Since CheungFF was so off in its prediction of TNI and we wanted to model the inter-
face with liquid water, we had to increase the pressure so as to avoid water vaporization
—since at the time we were doubtful about the concept of individual temperature cou-
pling. The following set of results 5.1 on the next page correspond to 77 ◦
C and 100 bar
simulations after running about 2 and 4.2 ns, respectively. Both runs began with water
randomly dispersed within and around a bulk isotropic cluster of 5CB, but differ in the
position of the surfactant. They show that the surfactant effects the local orientation of
the mesogen by inducing homeotropic anchoring.
Thus we learned that interfacial circumstances that affect the nematic behavior of
our system can be readily analyzed in the context of molecular dynamics with atomistic
descriptions. However, the model we used was less accurate in its prediction of 5CB’s
properties. Furthermore, we notice the importance of using consistent descriptions.
That is, it is imperative to use force fields that at the very least can tolerate each
other. What’s more, having a force field that predicts exactly all the properties of interest
for a pure substance is superb. But, and this is a big issue, it is useless when trying to
incorporate other materials with inconsistent parameters into the picture.
58. 40
(a) (300:1 ⊥:4362)
(b) (300:1 ||:3780)
Figure 5.1 Final configurations for 5CB/DPPC/H2O systems
Here blue represents water, green 5CB and yellow DPPC.
59. 41
This issue therefore serves to enhance our confidence on TheoFF. The fact that it
models 5CB’s tail with BergerFF almost automatically makes it compatible with the
model of DPPC we employed in this part of the project.
Interactions between liquid crystals across water
There have been considerable inquiries as to why liquid crystals (LC) adopt a prof-
fered anchoring at interfaces. Okano and Murakami [95] discussed that elastic theory
successfully explains LCs’ parallel alignment along a grooved surfaces. Thus they ven-
tiored to try to developing an argument that could explain why this happens at flat
surfaces since no theory at the time (1979) could do it. Needless to say they succeeded
in doing so and we adopt their technique for elucidation of a similar problem.
Figure 5.2 Scheme of H20/5CB system
In particular, experiments [?] show that
liquid crystals interact across so called spac-
ers (layers of a different material in between
bulk regions of the nematogen). This inter-
action is rather unintuitive because the ne-
matogens at either side of the spacer seem
to have their orientations coupled. However,
we know that van der Waals dispersion con-
tributions scale as the reciprocal of distance
to second power (Vdisp ∝ 1
D2 ) and would
expect this coupling to die quicker than it
seems to do empirically.
This project (originally procured for
Prof. Abbott’s advanced colloids course) required me to measure the van der Waals
(vdW) interactions of bulk liquid crystal (in the form of 5CB) across a layer of water of
varying lengths. Figure 5.2 portrays a schematic representation of our system. The idea
behind the project was to study how the LC’s orientations at either face coupled. That
60. 42
is, in Prof. Abbott’s own words, trying to answer what is the nature of this long-range
force that is orientational dependent.
Okano and Murakami [95] developed a formalism, based on Lifshitz theory [41], to
explain the preferred anchoring orientations for LCs. They followed the theory proposed
by Israelachvili et al. [60] but with the extension of having an anisotropic dielectric
tensor. This extension, based on a special case (since the authors and no us are only
concern with non-retarded forces ) of surface mode analysis.[127, 89]
On Lifshitz theory
Lifshitz theory is macroscopic in nature and regards materials as continuous media.
It is based on quantum field theoretic calculations. Probably its simplest task is to calcu-
late Hamaker constants.[57] It utilizes the dielectric responses of materials to determine
interaction free energies.[59]
If we consider [57] two half-spaces with dielectric responses 1(ω) and 3(ω), respec-
tively, separated by a material of dielectric response 2(ω) then Lifshitz theory [41] pre-
dicts that the non-retarded approximation (D < 5nm) to the free energy per unit area
U123(D)/A is
U123(D)/A =
kBT
2π
∞
n=0
∞
0
kdk ln (1 − ∆12∆32 exp{−2kD})
∆kj(ˆıξn) ≡
k(ˆıξn) − j(ˆıξn)
k(ˆıξn) + j(ˆıξn)
and ξn ≡ n
2πkBT
(ω) is the macroscopic manifestation of the microscopic polarisability of the con-
stituent atoms.[57] Its physical significance is well understood and one intuitively expects
its presence in macroscopic theories of vdW interactions.
Defining the electric displacement vector D(r, t) (in terms of a time varying elec-
tric field E(r, t) and the polarization density at a given position P (r, t)) as D ≡
E + 4πP . All of these are real and finite quantities. Assuming a linear response,
P (r, t) = 1
4π
∞
0
dτf(τ)E(r, t − τ) where f(τ) describes the degree of decay if an induced
polarization and must tend to nullity when τ → ∞ to be physically sound. Moreover,
61. 43
it contains all information pertaining to relaxations (electronic, vibrational, etc.) of the
molecules and, following the treatment of Hough et al. we take it to be isotropic.[57]
Defining a complex electronic field and noting that only its real contribution has
physical significance E(r, r) = [E0(r)eitω
] we find that the dielectric response function
of a given material is defined as
(ω) ≡
E(r; t)
D(r, t)
= 1 +
∞
0
dτf(τ)eˆıtω
,
which for a static electric field becomes 0 = 1 +
∞
0
dτf(τ)
Thus, the dielectric response function is an extension of the static dielectric to a time
varying field. Now, the Kramer-Kronig relation serves to relate (iω) precisely with the
regular dielectric response function just defined above: (ˆıξ) = 1 + 2
π
∞
0
dxx ( (ξ))
x2+ξ2 for
which we find in the limit that ξ → ∞, (ˆıξ) = 1. Furthermore, (ω) = [n(ω)]2
when
[ (ω)] = 0. [57]
The Ninham-Parsegian representation [89], used by Okano et al.[95], follows from the
mathematical relations enunciated above after making a series of assumptions. Their
derivation and relation with spectroscopy, IR in particular, are explained by Hough et
al.[57] and would take more space to give proper justice to their treatment than we are
able to provide here. We refer the reader to “The calculation of Hamaker constants from
Lifshitz theory with applications to wetting phenomena” by Hough and White [57], to
the original papers [127, 89] and to Jone’s book[61].
On the extended Israelachvili’s approach
Okano and Murakami [95], following Israelachvili’s approach to calculate surface ten-
sions values, modeled a process (`a la thermodynamic integration2
) to calculate the energy
difference between the final states and related it to the surface tension γ. They argued
that γ is the work (per area) necessary to form the interface so that it is equal to the
2
i.e.,
∆U =
ξ2
ξ1
dξ
∂U
∂ξ
62. 44
difference of bringing the slabs to molecular contact from being infinitely separated. The
algorithm is summarized in below.
n2
(ω) ⇒ (here given in terms of extended Cauchy equations)
Calculate Hamaker’s constants from dielectric tensor f [ (ω)] → H121
Utilize Hamaker constant to calculate energy per area as a function of slab separation:
U(D)/A = −H121/12πD2
Relate cycle to surface tension: γ = − 1
2A
{U(0) − U(∞)}
One of several models may be used for representing LC’s anisotropic dielectric. Some
of these are summarized below:
• Cauchy-models:
n(λ) = n∞ +
α
λ2
But these are only valid near λvis and not theoretically sound.[109, 73]
• Oseen’s model of cholesterics:
Not really applicable to 5CB.
• Ninham-Parsegian representation[89]:
[ (ω)] = 1 +
N
i=1
Ci
1 − (ω/ωi)2
Method chosen by Okano et al.[95] and it is a general approximation that follows
(as discussed in text) from mathematical relations modeling electric fields and di-
electric. We didn’t find, in time, parameters appropriate for 5CB.
• Relating polarizabilities to dielectric through different equations[59]:
α5CB =
66.15 −2.65 −0.90
−2.65 26.67 −2.07
−0.90 −2.07 27.64
calculated via ab initio methods.[28] But this was not given as a function of ω
63. 45
• Extended Cauchy models[73]: (described in detail within the text)
Theoretically sound models that provide an accurate description for 5CB. This is
what we used.
Figure 5.3 5CB’s refractive index
From Li and Wu[73] with solid lines cor-
responding their extended Cauchy model.
n||(λ) = Ai +
Bi
λ2
+
Ci
λ4
− GS
λ2¯λ2
λ2 − ¯λ2
= A|| +
B||
λ2
+
C||
λ4
≈ 1.5187 +
0.0016
λ2
+
0.0011
λ4
n⊥(λ) = Ai +
Bi
λ2
+
Ci
λ4
+ GS
λ2¯λ2
λ2 − ¯λ2
= A⊥ +
B⊥
λ2
+
C⊥
λ4
≈ 1.6795 +
0.0048
λ2
+
0.0027
λ4
niso(λ) = Ai +
Bi
λ2
+
Ci
λ4
≈ 1.5721 +
0.0021
λ2
+
0.0016
λ4
We used Li & Wu’s [73] extended Cauchy model
and regression parameters at 25.1 ◦
C. It is worth
noting that they reported their model’s dependence
on temperature and order parameter S2 but that
we utilized only the values corresponding to room
temperature for simplicity. Figure shows their re-
sults juxtaposed with empirical data. Their param-
eters are enunciated below the figure for the paral-
lel, perpendicular and isotropic cases (meaning no
preferred orientation), respectively.
Now that we have an appropriate model for
the refractive indexes, we relate them to the
dielectric responses through the approximation
(ω) ≈ n(ω).[59, 57] The Hamaker constants
H121[ 1(ω), 2(ω)] may then be calculated from
these approximate dielectric responses.
Following Okano et al, we define ∆iso2(ω, φ),
∆⊥2(ω, φ) and ∆||2(ω, φ) to be used as our particu-
lar ∆kj(ˆıξn) but instead of j corresponding to vac-
uum we take it to be water. Thus, ∆iso2(ω, φ) ≡
iso(w)− 2(ω)
iso(w)+ 2(ω)
2
, ∆⊥2(ω, φ) ≡
√
||(w) ⊥(ω)
[
√
||(w) ⊥(ω)+ 2(ω)]
2 ,
and
∆||2(ω, φ) ≡
||(w) ⊥(ω) cos2 φ + 2
⊥(ω) sin2
φ)
||(w) ⊥(ω) cos2 φ + 2
⊥(ω) sin2
φ) + 2(ω)
2 .
64. 46
This is the extension they utilized to incorporate the anisotropy when calculating the
Hamaker constant with the added modification of being performed in a medium different
than vacuum —here it is explicitly incorporated through 2(ω). Once these are defined,
one can calculate the Hamaker constants for the different orientations via
H12[ 1(ω), 2(ω)] =
3kBT
4π
2π
0
dφ
∞
0
∆12(ˆıυ, φ)dυ
=
3kBT
8π
2π
0
dφ∆12(0, φ) +
3h
8π2
2π
0
dφ
∞
1
∆12(ˆıυ, φ)dυ.
The energy per unit area as a function of slab separation D for the particular for dielectric
mediums disregarding retarded forces [60] becomes
U(D; 1, 2)
A
=
D
0
H121[ 1(ω), 2(ω)]
6πX3
dX.
Case H121 γ/kBT
isotropic 1.06 ×10+2
-2.8 ×10−0
⊥ 2.86 ×10−2
-3.5 ×10−4
|| 4.63 ×10−2
-6.0 ×10−4
Table V.1 Surface tension results
Finally, after integrating the equations
with GNU/Octave [42] (refer to page 80
for the code) we obtain the results summa-
rized in Table V.1. In particular, this model
does not dictates restrictions as to how they
molecules ought to orient with respect to the
interface safe that they should be parallel (in
the case of parallel calculation). That is, they do not say they should be aligned in the
same parallel orientations on both sides of the layer. This could be studied further but
is not included in this work. Figures 5.4(a) through 5.4(b) show the energy profile for
the particular cases studied.
We were originally going to measure this forces through the use of Molecular Dynamics
as described elsewhere in this thesis. In particular, we setup various systems with the
layer length varying from 1nm to 10nm —in steps of 1nm each. Some of these are shown
below. However, due to issues with our computational usage quota, the production stages
were lost —among other jobs like final efforts to converge the bulk N5CB = 1000 jobs.
There were N5CB = 250 at either side of the layer. The number of water molecules
present depended on the length of the layer and we had NH2O ∝ D (2857 water molecules
65. 47
Isotropic
0.5
1
1.5
2
2.5
1 2 3 4 5 6 7 8 9 10
Energy/kT
Distance [µm]
Energy profiles per anchoring
0
(a) No preferred orientation
Perpendicular
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
1 2 3 4 5 6 7 8 9 10
Energy/kTperarea
Distance [µm]
Energy profiles per anchoring
Parallel
0
(b) With preferred orientation
Figure 5.4 Results for preferred orientations
Perpendicular
1e−05
0.0001
0.001
0.01
0.1
1
10
1 2 3 4 5 6 7 8 9 10
Energy/kTperarea
Distance [µm]
Energy profiles per anchoring
Isotropic
Parallel
1e−06
Figure 5.5 Comparison between all results
when D=10nm). Figures 5.6(a)-5.6(c) give a graphical representation of what we have
just described. Furthermore, they correspond to the initial configurations of systems
with 1, 5 and 10 nm.
Our cut of radius, rc, was set to 4nm. We utilized the SPC model [35] for water and
Hybrid United/Full Atoms description for 5CB that we have been using through this
reports. However, due to issues with our computer usage quota within de Pablo’s group,
66. 48
we only obtained results for the equilibration stages —which implies that not much can
be gained from their analysis, in terms of what we were aiming to study. If they are
finished and converge, they would proof a simple way to validate the model described
above.
One problem we consistently encountered was that we predicted extremely large,
and certainly unrealistic, pressures. All of the tensor elements were equally unphysical.
Noting that the pressure depend both on the temperature (ideal contribution) and the
interactions (virial contribution), we propose using independent temperature coupling.
Our argument is that,
• in the first place the SAM dissociates at high temperatures, so it is unrealistic to
hold it on a lattice.
• our current force field for 5CB predicts a TNI too high, so that a displacement could
help account and counteract this energetic excess. Arguable, a similar circumstance
may be occurring with the alkanes as well. Thus it would be prudent to validate
those force fields as well.
(a) D=1nm (b) 5nm (c) 10nm
Figure 5.6 System’s initial configurations
67. 49
Observing conformational changes in polypeptides
Our project was more or less a joint effort with Prof. Abbott’s group, in particular
with Mrs. Bai, to study the conformational changes experimentally and and through
simulations concurrently.
The system in question is a 14-residue fragment of Epidermal Growth Factor Re-
ceptor (EGFR) with cysteine attached to a surface that collapses onto the surface once
phosphorylated and would, otherwise, stand erected. Its importance is related to the
EGFR per se and it relates to its phosphorylation. You see, EGFR is a cell membrane
protein and the phosphorylation of several of its tyrosine residues educes various signal
transduction cascades. This signaling pathway is of consequence in growth regulation,
mammalian cell differentiation and proliferation.[92]
When we began to study the system (the polypeptide sequence CTAENAEYLRVAPQ
with the peripheral surface) we wanted to make sure that its conformational change could
be assess through molecular dynamics utilizing atomistic FFs, and, equally important,
understand why such a change would occur from a theoretical perspective.
Stultz et al. [116] explored the effect of phosphorylation on a 10-residue MAP kinase
substrate. They found, through a combination of fluorescence resonance energy transfer
(FRET) and molecular dynamics, that phosphorylation causes that particular peptide
backbone to fold into a more compact conformation, as well as to change its direction.
Their polypeptide sequence was KQAEAVTSPR.
Thus it became clear that these type of changes in the structure of polypeptides
may be predicted with our methodologies. Moreover, Smart et al. [112] (who studied
SAAAAAAAAA) determined that phosphorylation of this poly-alanine sequence stabi-
lized the n-terminal of the alpha-helix.
All these results imply that phosphorylation of key residues induces conformational
changes. Moreover, they show that this phenomena 1) is relatively well understood, and
2) that our level of molecular description succeeds in predicting it. Thus we can build
68. 50
upon this knowledge for our applications. The first step, then, was to check whether
phosphorylation of our sequence would induce a change and be predictable with our
methods.
Figure 5.7
Schematic of the
polypeptide
Our molecule of interest is artificial and no experiment had
been done to determine its three-dimensional structure. Thus, we
did a similarity (homology) search (using BLAST) by querying
the ExPASy server and found that the fragment was located at
the end of UniProtKB/Swiss-Prot’s entry P00533.3
Our fragment
of interest corresponds to the 120th
and part of the 121th
fragments
of EGRF.
Using this information, we determined4
that it ought to have
extended-coils at the ends connected by an intermediate sheet
region. Furthermore, using I-Sites/HMMstr/ROSETTA (IHR)
we prepared an estimate for the three-dimensional position of the residues.
Actually, the IHR service requires a sequence of at least 20 amino acids residues so
we opted to add a subset of the EGFR sequence. That is, CYS was incorporated at
the extreme and the extra residues of the 121th
fragment were attached to reach that
minimum. Also we tried using two of Bai’s polypeptide in series but that did not result
in the predicted secondary structure we were looking for. Additionally, since IHR only
produces a pdb file of the polypeptide’s back-chain, we incorporated the residues’ side
groups with DeepView/Swiss-PdbViewer, which does an energy minimization using the
GROMOS 43B1 forcefield[129, 33, 106] in vacuo.
We then used this preliminary conformation to do an energy minimization in water
for a couple of picoseconds. The result is shown in Figure 5.7 and it is consistent with
the secondary structure originally predicted.
3
http://ca.expasy.org/uniprot/P00533
4
Various secondary and tertiary structure prediction were used. These tools are available through
the Proteomics Tools section of the ExPASSY website. URL is: http://ca.expasy.org/tools/
69. 51
The un-phosphorylated (ordinary) polypeptide was simulated. Two approaches have
been employed: (1) walls feature in CVS version with position restrains in the topology
and (2) including some atoms while employing “freeze groups” feature.
The “freeze group” feature (and its related commands as “freezedim”, etc.) signal
GROMACS to not update the position of the specified groups. All three coordinates for
the wall atoms were not updated while only the z-coordinate of the CYS residue was
fixed. That is, the polypeptide was able to move in the x- and y-coordinates —relative
to the CYS residue, of course. This is the same feature we later used in VI on page 65.
The wall feature is supposed to tell GROMACS that there is a surface of a certain
density of atoms (of a specified type). One chooses whether the energy contribution is
calculated based on the area or on the volume behind the wall (assuming it is composed
of the same atoms).
While invoking the wall feature the simulation box collapsed. So instead of using it
we generated a surface of several equally spaced atoms at z=0 and then froze them as
well as the CYS residue, which was located very close to z=0. That is, we manually
added a certain number of atoms (of specific type), all of which have their z-coordinate
equal to null (that, in our reference frame implies that they are located on the bottom
of the simulation box), and the CYS residue is initially located very close to that surface
(and remains so because we do not update its z-coordinate). Refer to Figure 5.8(b) to
see how the system is initially configured.
Furthermore, the phosphorylated residue was then substituted into the minimized
polypeptide with the wall. Afterwards a NPT were run at 27 ◦
C and 1 bar. It used the
modified G43a1 force field contributed to GROMACS by Graham Smith.5
Condor/Kelvin, at the time, was holding the extensions of the 5CB systems (Che-
ungFF surfactant project). We had to put them on hold because the trajectory files were
huge (over a GB each for each simulation) and we had been having issues with burning
5
http://www.gromacs.org/contributed by users/task,doc details/gid,39/
