# Coarse-graining bistability with the Martini force field ††thanks: Citation:
A.D.Muratov and V.A.Avetisov, Title. Pages…. DOI:000000/11111.

Alexander D. Muratov, Vladik A. Avetisov
Semenov Federal Research Center for Chemical Physics, Russian Academy of
Sciences, 119991 Moscow, Russia
Design Center for Molecular Machines, Moscow, Russia
<EMAIL_ADDRESS>(A.D.M<EMAIL_ADDRESS>(V.A.A.);

###### Abstract

An increasing interest in molecular structures whose long term dynamics
resemble those of bistable mechanical systems promotes the the search of
possible candidates that may operate as two-state switching units. Of
particular interest are the systems that are capable to exhibit such bistable
effects as spontaneous vibrations, stochastic resonance and spontaneous
synchronization. Previously, in all-atom molecular dynamics simulations it has
been demonstrated that short pyridine-furan springs show these bistable
phenomena. In this article we introduce a coarse-grained model of such springs
and investigate the possibility to observe the bistability effects, such as
spontaneous vibrations, in such model. Our findings allow the studies of short
pyridine-furan springs at large time an length scales and open field for
further computational research of large bistable systems.

_K_ eywords Duffing oscillators $\cdot$ bistability $\cdot$ nanosprings
$\cdot$ non-linear dynamics $\cdot$ spontaneous vibrations $\cdot$ stochastic
resonance $\cdot$ coarse-grained molecular dynamics $\cdot$ Martini

##  1 Introduction

The growing attention to nanoscale molecular structures with the dynamics
reminiscent of the bistable mechanical systems is driven by the increasing
demand for the development and implementation of various nanodevices. These
devices may act as switches and logic gates[1, 2, 3, 4, 5, 6], sensors and
actuators[7, 8, 9, 10, 11, 12], as well as mechano-electrical converters[13,
14] and energy harvesters[15, 16, 17, 18]. Nanoscale bistable systems are
equally important for validating the principles of stochastic
thermodynamics[19, 20, 21, 22]. The field is currently focused on expanding
thermodynamic theories to include nanoscale molecular machines[21, 23, 24, 25,
26].

This article introduces a coarse-grained model of spiral-like compound,
pyridine-furan (PF), which consists of 6- and 5-member hetherocycles (consult
Figure 1(a)), containing nitrogen and oxygen respectively[27, 28]. This
copolymer tends to form a helical shape stabilized by the interaction of
$\pi$-electrons between neighboring turns, as shown by quantum
calculations[29]. Our previous modeling reveals that oligo-PF springs exhibit
bistable dynamics of Duffing oscillators, accompanied by spontaneous
vibrations and stochastic resonance activated by thermal fluctuations[30, 31].
Moreover, dynamical synchronization of two or several oligo-PF springs is also
possible[32]. Nevertheless, a raise in the number of synchronized oligo-PF
springs inevitably leads to the increase of the simulation box and
subsequently to the slower simulation times. Moreover, our theoretical
findingssuggest an increase of lifetime of oligo-PF springs in each bistable
state. Thus if we desire to simulate synchronization of a large number of
oligo-PF springs, we have to compute more time steps while each time step is
calculated slower, making all-atom modeling almost impossible.

Figure 1: Pyridine-furan (PF) spring with five monomer units (oligo-PF-$5$
spring): (a) Chemical structure of a pyridine-furan monomer unit with
heterocyclic rings in cis-configuration. (b) Mapping scheme of oligo-PF
spring. (c) Two coarse-grained monomer units of oligo-PF spring.

An easy and elegant way to overcome this issue is coarse-graining (CG). There
are two approaches to the CG modeling: “bottom-up,” which concentrates on
accurately representing the underlying atomic structure of the compound, and
“top-down,” which focuses on reproduction of macroscopic properties of the
compound. The main disadvantage of "bottom-up" models, despite their accuracy,
is their non-transferability; that is, re-parameterization is required
whenever modeling conditions change. Also, they often use complex potential
forms that slow down calculations. At the same time, "top-down" approaches use
simple potential forms and usually make use of the idea of pre-parameterized
building blocks, making them easily transferable and computationally cheap.

Among various "top-down" CG force fields stand out Martini (which usually
deciphers as MARrink Toolkit INItiative) that uses water/oil partitioning free
energy to parameterize Lennard-Jones (LJ) potentials[33, 34]. The keystone of
the Martini force field is several discrete interaction levels between a
limited number of building blocks. First introduced for lipids[33], later it
has been updated to include more compounds in its second version[34] including
proteins[35], carbohydrates[36], DNA[37] and various other compounds.
Recently, a new version, coined Martini 3, with improved accuracy has been
released[38, 39]. The main difference is the introduction of so-called "tiny"
particles; this improvement plays a significant role for more accurate
description of ring structures. Thus, picking Martini 3 for oligo-PF springs
seems to be a straightforward choice.

In this paper we introduce a coarse-grained model of oligo-PF spring within
latest version of Martini force filed. The model captures the bistable dynamic
effects of a Duffing oscillator with thermally-activated spontaneous
vibrations and stochastic resonance.

##  2 Materials and Methods

###  2.1 Non-bonded parameterization

Martini $3$ incorporates four main types of beads: charged (Q), polar (P),
non-polar (N) and apolar (C). There are also special beads for water (W),
divalent ions (D) and groups containing halogen atoms (X). Most bead types
except for W-bead type and D-bead type are divided in subtypes. For the
simplicity, the LJ interaction levels $\epsilon$ are discretized; there are 22
possible interaction levels. Sizes of the beads $\sigma$ are also discretized
in Martini 3: there are regular (R, $\sigma=$0.47\text{\,}\mathrm{nm}$$),
small (S, $\sigma=$0.41\text{\,}\mathrm{nm}$$) and tiny (T,
$\sigma=$0.34\text{\,}\mathrm{nm}$$) beads. These beads are intended to be
used for $4$-to-$1$, $3$-to-$1$ and $2$-to-$1$ mappings respectively. Both
pyridine and furan are already parameterized within Martini $3$ force
field[39], and more thorough discussion will follow in the next subsection
2.2. Water is represented by a regular W bead.

###  2.2 Bonded parameterization

Aromatic ring structures are well-described by three T-beads:
$($TC$5)_{2}$-TN$2a$ is a representation for furan and $($TC$5)_{2}$-TN$6a$ \-
for pyridine (TN$2a$ bead includes oxygen, TN$6a$ \- nitrogen; consult Figure
1(b-c) for more details). It is important to emphasize that in furan TN$2a$
bead shares $-CH-$ groups with neighboring TC$5$ beads, while in pyridine
beads does not share atomic groups (see Figure 1(b))[39]. Alessandri et al.
[39] determined bonded parameters for standalone pyridine and furan, but bond
lengths will differ in compound. Here, an important difference from the
previous issue of Martini is that center of each bead corresponds to the
center of geometry (COG) of underlying molecular structure rather that center
of mass (COM). This is especially relevant for tiny beads.

To obtain bond potentials within the oligo-PF-$5$ molecule we adopt a strategy
for long chains of rigid fragments called "Divide and Conquer"[39, 40]. Within
this strategy we introduce a virtual dummy site (mass-less and not interacting
via non-bonded interactions) at the center of geometry of each pyridine or
furan ring. Then, we connect them with an harmonic bond and exclude the
nonbonded interactions between the two rings (see Figure 1(c)). Valence angle
potentials, such as $V_{i}$-$V_{j}$-$V_{k}$, $B_{i1}$-$V_{i}$-$V_{j}$ or
$B_{i2}$-$B_{i3}$-$V_{j}$ where $V_{i}$, $V_{j}$ and $V_{k}$ are the $i$-th,
$j$-th and $k$-th virtual sites and $B_{i1}$, $B_{i2}$, $B_{i3}$ are the beads
of the $i$-th ring, may also be introduced (here for $B_{i1}$ it is convenient
to take the bead describing oxygen/nitrogen; consult Figure 1(c) for more
details). Last, a dihedral potential $B_{i1}$-$V_{i}$-$V_{j}$-$B_{j1}$ can be
directly applied in such a construction.

###  2.3 Computational details

For modeling we use Gromacs $2023$[41] with the number of particles, volume
and temperature maintained constant (NVT) ensemble. The temperature is set to
$298\text{\,}\mathrm{K}$ and maintained by a velocity-rescale thermostat[42]
with $1.0\text{\,}\mathrm{ps}$ coupling time. For parameterization we perform
$10\text{\,}\mathrm{ns}$ atomistic simulations of oligo-PF-$5$ spring using
OPLS-AA[43] force field parameters for the oligomer, and the SPC/E model[44]
for water. After obtaining atomistic trajectories, we calculate reference
probability distributions for valence bonds and valence angles and get initial
CG potentials, which are used for a primary $25\text{\,}\mathrm{ns}$ CG
simulation run; then we compare CG probability distributions with reference
ones. We keep changing CG potentials until reasonable convergence between
probability distributions is obtained. For CG calculations we use straight
cutoff LJ potential that turns to zero at $1.1\text{\,}\mathrm{nm}$[45].

To study the bistable dynamics of the oligo-PF-$5$, we fixed one end of the
spring, while a pulling force was applied along the axis of the spring to
another end of the spring. The distance (denoted $R_{e}$) between the ends of
the oligo-PF-$5$ spring was considered a collective variable describing the
long-term dynamics of the spring. Bistability of the oligo-PF-$5$ spring was
specified in the agreement with two well-reproduced states of the spring with
the end-to-end distances equal to $R_{e}\sim$0.4\text{\,}\mathrm{nm}$$ and
$R_{e}\sim$0.75\text{\,}\mathrm{nm}$$. These states are referred to as the
squeezed and the stress-strain states, respectively.

##  3 Results

###  3.1 Non-bonded and bonded interactions

As mentioned in subsections 2.1 and 2.2, both pyridine and furan are already
parameterized in Martini $3$, but bonded potentials in a conjugated compound
differ from those in a single hetherocycle. To obtain these potentials, we
first perform $10\text{\,}\mathrm{ns}$ atomistic simulations of oligo-PF-$5$
using OPLS-AA force field with NVT ensemble. Temperature is maintained by
velocity-rescale themostat with $1.0\text{\,}\mathrm{ps}$ coupling time. Then,
we calculate probability distributions for all possible bonds and valence
angles and thus get primary CG potentials which are then used for a trial run.
The probabilities got from atomistic trajectory serve as target ones. We keep
modifying CG potentials until the centers and widths of the distributions
correspond the target ones. The obtained values are given in Table 1

Table 1: Intramolecular potentials of oligo-PF-$5$. Here, subscripts ${}_{\text{F}}$ and ${}_{\text{P}}$ denote belonging to furan and pyridine. An asterisk ($*$) denotes a bead from neighboring ring Valence bond | $r_{0},$\text{\,}\mathrm{nm}$$ | $k_{\text{b}},$\text{\,}\mathrm{kJ}\text{\,}{\mathrm{mol}}^{-1}\text{\,}{\mathrm{nm}}^{-2}$$ | Valence angle | $\theta_{0}$ | $k_{\theta},$\text{\,}\mathrm{kJ}\text{\,}{\mathrm{mol}}^{-1}$$
---|---|---|---|---|---
TN$2a$-TC$5_{\text{F}}$ | 0.1765 | rigid | V${}_{\text{F}}$-V${}_{\text{P}}$-V${}_{\text{F}}$ | 120 | 1500
TC$5_{\text{F}}$-TC$5_{\text{F}}$ | 0.2125 | rigid | V${}_{\text{P}}$-V${}_{\text{F}}$-V${}_{\text{P}}$ | 132 | 1500
TN$6a$-TC$5_{\text{P}}$ | 0.25 | rigid | TN$2a$-V${}_{\text{F}}$-($*$)V${}_{\text{P}}$ | 67 | 1000
TC$5_{\text{P}}$-TC$5_{\text{P}}$ | 0.255 | rigid | TN$6a$-V${}_{\text{P}}$-($*$)V${}_{\text{F}}$ | 80 | 1000
V${}_{\text{F}}$-V${}_{\text{P}}$ | 0.438 | 50000 | V${}_{\text{F}}$-($*$)V${}_{\text{P}}$-($*$)TN$6a$ | 40 | 1000
|  |  | TC$5_{\text{P}}$-TC$5_{\text{P}}$-($*$)V${}_{\text{F}}$ | 161 | 500
|  |  | V${}_{\text{F}}$-($*$)TC$5_{\text{P}}$-($*$)TC$5_{\text{P}}$ | 110 | 500
|  |  | TC$5_{\text{F}}$-TC$5_{\text{F}}$-($*$)V${}_{\text{P}}$ | 145 | 500

Since in pyridine atomic groups are not shared between beads, an assymetry is
observed in certain pairs of valence angles:
TN$6a$-V${}_{\text{P}}$-($*$)V${}_{\text{F}}$ and
V${}_{\text{F}}$-($*$)V${}_{\text{P}}$-($*$)TN$6a$ or
TC$5_{\text{P}}$-TC$5_{\text{P}}$-($*$)V${}_{\text{F}}$ and
V${}_{\text{F}}$-($*$)TC$5_{\text{P}}$-($*$)TC$5_{\text{P}}$ (consult Figure
1(b) for more details). Also, a dihedral potential
TN$6a$-V${}_{\text{P}}$-($*$)V${}_{\text{F}}$-($*$)TN$2a$ is directly taken
from OPLS-AA force field.

###  3.2 Spontaneous vibrations

Figure 2: (a) Computational models of the oligo-PF-$5$ system with applied
longitudinal load. The squeezed and the stress–strain states of the spring are
shown on the left and right, respectively. The yellow spheres at the lower end
of the spring indicate the fixation of the pyridine ring by rigid harmonic
force. The pulling force, $F$, is applied to the top end of the spring. (b)
The state diagram shows a linear elasticity of oligo-PF-$5$ spring up to
$F\approx$100\text{\,}\mathrm{pN}$$ and bistability of the spring in the
region from $F\approx$100\text{\,}$-$130\text{\,}\mathrm{pN}$$; (c)
Spontaneous vibrations of the oligo-PF-$5$ spring at
$F\approx$115\text{\,}\mathrm{pN}$$; (d) Evolution of the probability density
for the squeezed and stress–strain states when pulling force surpasses the
critical value.

To investigate the dynamics of the CG oligo-PF-$5$ spring under tension, we
followed the same strategy as for all-alom model[30]. After equilibrating the
oligo-PF-$5$ spring at $298\text{\,}\mathrm{K}$ with one end fixed and
applying a force $\vec{F}$ along the spring axis to pull the other end, the
spring’s initial state remained stable under weak tensile conditions. However,
once the pulling force reached a specific critical value, approximately
$F_{c}=$100\text{\,}\mathrm{pN}$$, the oligo-PF-$5$ spring began to exhibit
spontaneous vibrations between squeezed state and stress–strain state. The
average end-to-end distances of the spring between the states differ by
approximately $0.35\text{\,}\mathrm{nm}$, which is in accordance with both
$abinitio$ calculations[29] and atomistic simulations. Figure 2a) displays
coarse-grained level snapshots of these two states.

The evolution of the statistics of visits to the squeezed and stress-strain
states as the pulling force exceeds the critical point $F_{c}$ is shown on
Figures 2(b,d). Within the range of
$F=$110\text{\,}$-$120\text{\,}\mathrm{pN}$$, both the squeezed and stress-
strain states are visited almost equally. The mean lifetimes of the states in
the spontaneous vibration mode differed in the bistability region, ranging
from $\tau=$1\text{\,}$-$30\text{\,}\mathrm{ns}$$, depending on the tensile
force. In the symmetrical bistability region, neither the squeezed state nor
the stress-strain state dominates, resulting in the mean lifetimes of the two
states being approximately equal to $\tau=$3.75\text{\,}\mathrm{ns}$$. Figure
2(c) illustrates a typical trajectory of the long-term dynamics $R_{e}(t)$ of
the oligo-PF-$5$ spring within the symmetrical bistability region. Clear
spontaneous vibrations of the spring can be observed without any additional
random perturbations being applied to activate them. These vibrations are
activated only by the fluctuations of the thermal bath; on the other hand,
outside the bistability region, trajectories without vibrations prevail.

###  3.3 Stochastic resonance

Figure 3: Stochastic resonance of the oligo-PF-$5$ induces by an oscillating
field $E=E_{0}\cos(2\pi\nu t)=E_{0}\cos(\nicefrac{{2\pi t}}{{T}})$: (a) The
dynamic trajectory at $F=$115\text{\,}\mathrm{pN}$$,
$T=$7.5\text{\,}\mathrm{ns}$$, and
$E_{0}=$0.2\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$$; (b) Power
spectrum of spontaneous vibrations (red curve) and stochastic resonance (black
curve); (c) The dependence of the main resonance peak amplitude on $E_{0}$
($T_{0}=$7.5\text{\,}\mathrm{ns}$$); (d) The dependence of the main resonance
peak amplitude on the period $T$ of oscillating field
($E_{0}=$0.2\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$$).

We examined the stochastic resonance mode of the oligo-PF-$5$ spring by
applying an additional oscillating force that acted weakly on the pulling end
of the spring. This force was modeled by the application of an oscillating
electrical field, $E=E_{0}\cos(2\pi\nu t)$, on a unit charge preset on the
pulling end of the spring. Since CG particles are uncharged, such an addition
should not affect the dynamics of the system. Typical vibrations of the end-
to-end distance of the oligo-PF-$5$ spring in the stochastic resonance mode
and the power spectrum of the vibrations are shown in Figures 3(a-b).

According to the stochastic resonance theory[46, 47], the main resonance peak
was observed at the frequency $\nu=(2\tau)^{-1}$, that is, the period of the
applied oscillating field was equal to twice the mean lifetime of the state in
the spontaneous vibrations mode. In fact, we scanned a wide range of
oscillating fields to find the maximal resonance response defined in terms of
the spectral component at the resonance frequency. Corresponding results are
presented in Figure 3(c,d). Regarding the amplitude of the oscillating field,
the maximum resonance was found for
$E_{0}=$0.2\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$$ in full accordance
with the all-atom data[30]. It should be noted that outside the symmetric
bistability region, the lifetimes of the squeezed and the stress–strain states
became so different that the average lifetime was no longer a good guide to
the resonance frequency, and thus the resonance response was screened in the
region of symmetric bistability at $F=$115\text{\,}\mathrm{pN}$$. .

##  4 Discussion

The main finding of this work is that subtle bistable effects, such as
spontaneous vibrations and stochastic resonance, can be observed using coarse-
grained molecular dynamics. Indeed, our previous studies suggest that
spontaneous vibrations arise due to $\pi$-stacking between aromatic groups
located on the adjacent turns of the spring, which along with the stiffness of
the molecular backbone produces a two-well potential (see Avetisov et al. [30]
for more information). One can speculate how can $\pi$-stacking be addressed
in molecular dynamics (authors may suppose that it somehow emerges due to a
peculiar interplay between Lennard-Jones forces and Coulomb interaction of the
partial charges of the atoms of the rings, but such question is very far out
of the scope of the current work), but in the present Martini model adjacent
turns of the spring interact exclusively via LJ potential. In the meantime,
chain stiffness is kept by bonding between virtual dummy particles; both
factors seem strong arguments against observing bistability. Thus, such
emerging noticeably strengthens the credibility of Martini force field and
provides for its wider usability for the molecular modeling field.

Still, there is a certain difference between spontaneous vibrations observed
in the present work and in our previous atomistic modeling[30]. In CG
modeling, SV occur at smaller forces: $100\text{\,}$–$120\text{\,}\mathrm{pN}$
compared with $270\text{\,}$–$290\text{\,}\mathrm{pN}$ for all-atom modeling.
Nevertheless, we do not think is should be considered as significant
disadvantage of the model: the fact that the mean lifetimes in the states
$1\text{\,}$–$30\text{\,}\mathrm{ns}$ correspond to those obtained in fully
atomic simulation and that the widths of the force regions coincide seems to
be more important.

For the present work we have taken pre-parameterized pyridine and furan[39]
and as a consequence have determined only bonded potentials. Since TN$2a$ bead
in furan shares -$CH$\- group with adjacent TC$5$ beads while in pyridine
beads does not share molecular groups with their neighbors, we are obliged to
introduce differing valence angle potentials (for instance,
TN$6a$-V${}_{\text{P}}$-($*$)V${}_{\text{F}}$ and
V${}_{\text{F}}$-($*$)V${}_{\text{P}}$-($*$)TN$6a$)due to asymmetry of the
resulting construction. This could be overriden by a slight re-
parameterization of pyridine: if TN$6a$/TC$5$ beads representing pyridine
shared -$CH$\- groups, the construction would be symmetrical and valence angle
potentials would be the same. We plan to investigate more possible mappings in
our further work.

In our previous work concerning Martini[48] we noticed the ambiguity in
relation to bead sizes. Fortunately, Martini $3$ has addressed the issues we
mentioned by introducing tiny beads and strict definition when small and tiny
beads are to be used[38]. The latter preserves flexibility while implying more
accuracy.

##  5 Conclusion

We have performed coarse-grained simulations of oligo-PF-$5$ springs subject
to stretching and observed thermally-activated spontaneous vibrations and
stochastic resonance. We explored the conditions under which these effects are
present, indicated symmetrical bistability and determined mean lifetime of the
states. The results are in good accordance with atomistic simulations. Our
findings approve Martini force field as capable tool for molecular modeling.
Its combined flexibility and accuracy allows its applying for the modeling of
molecular machines since it is able to capture the necessary time- and
lengthscales.

## Acknowledgments

Authors thank Alexey Astakhov, Vladimir Bochenkov, Maria Frolkina, Vladislav
Petrovskii, Anastasia Markina and Alexander Valov for helpful discussions.

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