# Detection of Single Molecules Using Stochastic Resonance of Bistable Oligomers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Simulation Method

## 3. Results

#### 3.1. Oligomeric Templates for Molecular Dynamic Simulations

#### 3.2. Bistability and Spontaneous Vibrations

_{c}≈ 320 pN. Indeed, the open state remains the only state of the oligomer up to the critical compression. However, at the critical compression, a new branch of actually bent stationary states appears. Far from the critical point, the oligomer is found in the open or closed state, depending on compression. If the compressive force quickly crossed the critical point, then the oligomer would abruptly transition from the open state to the closed state, like a jump-like switching of a metal ruler. Somewhat above the critical compression, low-frequency spontaneous vibrations between the open and closed states are observed, i.e., the oligomer is bistable in this region. It should be noted that spontaneous vibrations are not observed just after the critical point, despite the fact that there are two branches of states. Spontaneous vibrations are observed with some offset from the critical point, i.e., in the region where the bistability barrier matches the thermal fluctuations. The bistability barrier should be neither too small nor too large relative to thermal fluctuations.

_{c}≈ 320 pN, the oligomer dynamics are characterized by single-peak distributions P(Re) at the open or closed state, respectively. Near the bifurcation point from above, there is an interval of compressions in which the distributions P(Re) have a double-peak form caused by spontaneous vibrations of the oligomer between the open and closed states. In our simulations, the mean lifetime of the states, i.e., the mean value of random time intervals between the jumps defined along dynamic trajectories, ranged from 5 to 10 ns, depending on the compression. Using Kramer’s exponential relation between the lifetime of the states and the bistability barrier, and assuming that the pre-exponent collision factor is equal to 10

^{−13}s, the bistability barrier was estimated as 10–15 k

_{B}T, where k

_{B}is the Boltzmann constant and T is the bath temperature. Following this estimate, we assumed that the reordering of hydrogen bonds between the oligomer and surrounding water could activate spontaneous vibrations of the oligo-30s-NIPMAm. If this were the case, the spontaneous vibrations would be caused precisely by the thermoresponsibility of the oligo-30s-NIPMAm, e.g., due to the switching of hydrogen bonds from the oligomer–water configuration to the oligomer–oligomer configuration [30,31,32].

#### 3.3. Stochatic Resonance

_{0}cosωt enacted on a charge (+1) set at one end of the oligomer, while a compensative charge was set at the opposite end. Following the well-known fact that the main resonance peak arises when the frequency of the oscillating field matches the mean value of the state lifetimes in the spontaneous vibrations mode [23], we tested the oci fields with the period of T = 5 ns (200 MGz in frequency); the amplitudes ranged from 0.1 V/nm to 1.0 V/nm. For more details, see Supplementary Materials, Note S1.

#### 3.4. Single Molecules Sensing via Spontaneous Vibrations Mode

#### 3.5. Single Molecules Sensing via Stochastic Resonance

^{−1}, and then studied how the oligomer dynamics changed with a molecular cargo attachment.

^{−1}. Then we attached an analyte and generated new trajectories. The analyte-induced transformation of the stochastic resonance mode is clearly seen in a comparison of these two sets of trajectories, shown in Figure 7a,d. Figure 7b,e show autocorrelation functions. One can see that upon attaching the analyte molecule, the autocorrelation function is blurred. This fact is also supported by the furrier spectrum of these autocorrelation functions (Figure 7c,f), where the main peak is significantly lower for the system with the analyte.

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bazargan, G.; Sohlberg, K. Advances in modelling switchable mechanically interlocked molecular architectures. Int. Rev. Phys. Chem.
**2018**, 37, 1–82. [Google Scholar] [CrossRef] - Luo, Q.; Guo, Z.; Zhang, S.; Yang, X.; Zou, X.; Hong, J.; You, L. Crackbased complementary nanoelectromechanical switches for reconfigurable computing. IEEE Electron Device Lett.
**2020**, 41, 784–787. [Google Scholar] [CrossRef] - Zhou, X.; Lee, S.; Xu, Z.C.; Yoon, J. Recent Progress on the Development of Chemosensors for Gases. Chem. Rev.
**2015**, 115, 7944–8000. [Google Scholar] [CrossRef] [PubMed] - McConnell, A.J.; Wood, C.S.; Neelakandan, P.P.; Nitschke, J.R. Stimuli-Responsive Metal-Ligand Assemblies. Chem. Rev.
**2015**, 115, 7729–7793. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lou, Z.R.; Li, P.; Han, K.L. Redox-Responsive Fluorescent Probes with Different Design Strategies. Acc. Chem. Res.
**2015**, 48, 1358–1368. [Google Scholar] [CrossRef] [PubMed] - Tamayo, J.; Kosaka, P.M.; Ruz, J.J.; San Paulo, A.; Calleja, M. Biosensors based on nanomechanical systems. Chem. Soc. Rev.
**2013**, 42, 1287–1311. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Van Dijk, L.; Tilby, M.J.; Szpera, R.; Smith, O.A.; Bunce, H.A.; Fletcher, S.P. Molecular machines for catalysis. Nat. Rev. Chem.
**2018**, 2, 0117–0120. [Google Scholar] [CrossRef] - Liang, J.; Huang, L.; Li, N.; Huang, Y.; Wu, Y.; Fang, S.; Oh, J.; Kozlov, M.; Ma, Y.; Li, F.; et al. Electromechanical Actuator with Controllable Motion, Fast Response Rate, and High-Frequency Resonance Based on Graphene and Polydiacetylene. ACS NANO
**2012**, 6, 4508–4519. [Google Scholar] [CrossRef] - Akarvardar, K.; Wong, H.-S.P. Nanoelectromechanical Logic and Memory Devices. ECS Trans.
**2009**, 19, 49–59. [Google Scholar] [CrossRef] - Ekinci, K.L.; Roukes, M.L. Nanoelectromechanical systems. Rev. Sci. Inst.
**2005**, 76, 061101. [Google Scholar] [CrossRef] [Green Version] - Zhang, L.; Marcos, V.; Leigh, D. Molecular machines with bio-inspired mechanisms. Proc. Nat. Acad. Sci. USA
**2018**, 115, 9397–9404. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cleland, A.N. Foundation of Nanomechanics. In Solid-State Theory to Device Applications; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar] [CrossRef]
- Eoma, K.; Park, H.S.; Yoonc, D.S.; Kwonc, T. Nanomechanical resonators and their applications in biological/chemical detection: Nanomechanics principles. Phys. Rep.
**2011**, 503, 115–163. [Google Scholar] [CrossRef] [Green Version] - Krakover, N.; Ilic, B.R.; Krylov, S. Displacement sensing based on resonant frequency monitoring of electrostatically actuated curved micro beams. J. Micromech. Microeng.
**2016**, 26, 115006. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hansen, J.C.; Skalak, R.; Hoger, A. An elastic network model based on the structure of the red blood cell membrane skeleton. Biophys. J.
**1996**, 70, 146–166. [Google Scholar] [CrossRef] [Green Version] - Togashi, Y.; Mikhailov, A.S. Nonlinear relaxation dynamics in elastic networks and design principles of molecular machines. Proc. Natl. Acad. Sci. USA
**2007**, 104, 8697–8702. [Google Scholar] [CrossRef] [Green Version] - Düttmann, M.; Mittnenzweig, M.; Mikhailov, A.S. Complex intramolecular mechanics of G-actin—An elastic network study. PLoS ONE
**2012**, 7, e45859. [Google Scholar] [CrossRef] [Green Version] - Arnol’d, V.I. Catastrophe Theory; Springer: Berlin, Germany, 1984. [Google Scholar] [CrossRef]
- Poston, T.; Stewart, I. Catastrophe Theory and Its Applications; Dover Publication, Inc.: New York, NY, USA, 1996; pp. 75–89. [Google Scholar]
- McNamara, B.; Wiensenfeld, K. Theory of stochastic resonance. Phys. Rev. A
**1989**, 39, 4854–4869. [Google Scholar] [CrossRef] - Benzit, R.; Sutera, A.; Vulpiani, A. The mechanism of stochastic resonance. J. Phys. A Math. Gen.
**1981**, 14, L453–L457. [Google Scholar] [CrossRef] - Wiesenfeld, K.; Jaramillo, F. Minireview of stochastic resonance. Chaos
**1998**, 8, 539–548. [Google Scholar] [CrossRef] - Gammaitoni, L.; Hänggi, P.; Jung, P.; Marchesoni, F. Stochastic resonance. Rev. Mod. Phys.
**1998**, 70, 223–287. [Google Scholar] [CrossRef] - Avetisov, V.A.; Markina, A.A.; Valov, A.F. Oligomeric “Catastrophe Machines” with Thermally Activated Bistability and Stochastic Resonance. J. Phys. Chem. Lett.
**2019**, 10, 5189–5192. [Google Scholar] [CrossRef] [PubMed] - Abraham, M.J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J.C.; Hess, B.; Lindahl, E. GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers. SoftwareX
**2015**, 1, 19–25. [Google Scholar] [CrossRef] [Green Version] - Jorgensen, W.L.; Maxwell, D.S.; Tirado-Rives, J. Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids. J. Am. Chem. Soc.
**1996**, 118, 11225–11236. [Google Scholar] [CrossRef] - Jorgensen, W.L.; Tirado-Rives, J. Potential energy functions for atomic-level simulations of water and organic and biomolecular systems. Proc. Natl. Acad. Sci. USA
**2005**, 102, 6665–6670. [Google Scholar] [CrossRef] [Green Version] - Jorgensen, W.L.; Tirado-Rives, J. The OPLS [optimized potentials for liquid simulations] potential functions for proteins, energy minimizations for crystals of cyclic peptides and crambin. J. Am. Chem. Soc.
**1988**, 110, 1657–1666. [Google Scholar] [CrossRef] - Halperin, A.; Kröger, M.; Winnik, F. Poly(N-isopropylacrylamide) Phase Diagrams: Fifty Years of Research. Angew. Chem. Int. Ed.
**2015**, 54, 15342–15367. [Google Scholar] [CrossRef] - Hoogenboom, R. Tunable Thermoresponsive Polymers by Molecular Design. In Complex Macromolecular Architectures: Synthesis, Characterization, and Self-Assembly; Wiley & Sons: New York, NY, USA, 2011. [Google Scholar]
- Alaghemandi, M.; Spohr, E. Molecular dynamics investigation of the thermoresponsive polymer poly(N-isopropylacrylamide). Macromol. Theory Simul.
**2012**, 21, 106–112. [Google Scholar] [CrossRef] - Liang, X.; Nakajima, K. Nanofishing of a single polymer chain: Temperature-induced coil–globule transition of poly(N-isopropylacrylamide) chain in water. Macromol. Chem. Phys.
**2017**, 219, 1700394. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Course of theoretical physics. In Statistical Physics; Pergamon Press: London, UK, 1952; Volume 5. [Google Scholar]
- Kondepudi, D.; Nelson, G. Weak neutral currents and the origin of biomolecular chirality. Nature
**1985**, 314, 438–441. [Google Scholar] [CrossRef] - Avetisov, V.A.; Kuz’min, V.V.; Anikin, S.A. Sensitivity of chemical chiral systems to weak asymmetric factors. Chem. Phys.
**1987**, 112, 179–187. [Google Scholar] [CrossRef]

**Figure 1.**Temperature-induced bistability of the oligo-30s-NIPMAm. (

**a**) Typical shapes of the oligomer in the “open” and “closed” states equilibrated at the temperatures 90 K and T = 320 K, respectively. (

**b**) Normalized distributions of fluctuations of the end-to-end distances Re at the open and closed states.

**Figure 2.**The response of the oligo-30s-NIPMAm to longitudinal compressions: (

**a**) Schematic presentation of the oligomer compression; (

**b**) The bifurcation diagram represents how the stationary states, Re, of the oligomer depend on compression F as a control parameter. A shadow interval of compression forces from 340–430 pN marks the area of spontaneous vibrations.

**Figure 3.**Correlation between the oligomer dynamics R

_{e}(t) and switching of hydrogen bonds located in the oligomer bending area. The compressions F are indicated on the panels. (

**a**) Spontaneous vibrations of the oligomer (black curves, left axes, Re nm) strongly correlate with the switching of hydrogen bonds from the oligomer–oligomer configuration to the oligomer–water configuration and back (red curves; right axes). (

**b**) No significant correlation between the oligomer dynamics and the fluctuations of hydrogen bonds is seen beyond the bistability region.

**Figure 4.**Stochastic resonance generated by harmonic electric field E = E

_{0}cosωt with the amplitude 0.2 V/nm and the frequency 200 MGz: (

**a**) The oligomer dynamic Re(t) in the stochastic resonance mode; (

**b**) The autocorrelation function of Re(t); (

**c**) The frequency spectrum of the autocorrelation function shown on panel (

**b**).

**Figure 5.**Sensitivity of the oligo-30s-NIPMAm spontaneous vibrations to the attachment of a single-molecule cargo (an analyte). (

**a**) Schematic presentation of the computer experiments. (

**b**) Spontaneous vibrations of the unloaded oligomer (black trajectory) at the compression of 375 pN, and non-vibrating dynamics of the oligomer loaded with an analyte at the same compressing force (rad trajectory). (

**c**) Evolution of statistical weights for visiting the open and closed states by the loaded oligomer vs. the compressive force. Molecular cargo (an analyte) shifts the spontaneous vibrations mode from the compression of 375 pN (see panel (

**b**)) to the compressive of 390 pN.

**Figure 6.**Stochastic resonance and the sensing regime. (

**a**) Spontaneous vibrations (red curve) and stochastic resonance (black curve) of the unloaded oligo-30s-NIPMAm under the compression of 375 pN. (

**b**) Dynamics of the oligomer loaded by a tryptophan molecule at the same conditions. Trajectories are marked as in panel (

**a**).

**Figure 7.**The analyte-induced transformation of the stochastic resonance mode. Vibrations of the oligo-30s-NIPMAm before (top row of panels) and after (bottom row of panels) the attachment of a tryptophan molecule: (

**a**–

**c**) Stochastic resonance mode of the oligo-30s-NIPMAm, its autocorrelation function, and the frequency spectrum. (

**d**–

**f**) Distortion of the spontaneous resonance mode caused by the attachment of a molecular cargo to the oligomer.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Markina, A.; Muratov, A.; Petrovskyy, V.; Avetisov, V.
Detection of Single Molecules Using Stochastic Resonance of Bistable Oligomers. *Nanomaterials* **2020**, *10*, 2519.
https://doi.org/10.3390/nano10122519

**AMA Style**

Markina A, Muratov A, Petrovskyy V, Avetisov V.
Detection of Single Molecules Using Stochastic Resonance of Bistable Oligomers. *Nanomaterials*. 2020; 10(12):2519.
https://doi.org/10.3390/nano10122519

**Chicago/Turabian Style**

Markina, Anastasia, Alexander Muratov, Vladislav Petrovskyy, and Vladik Avetisov.
2020. "Detection of Single Molecules Using Stochastic Resonance of Bistable Oligomers" *Nanomaterials* 10, no. 12: 2519.
https://doi.org/10.3390/nano10122519