Letter Cite This: J. Phys. Chem. Lett. 2019, 10, 5189−5192
pubs.acs.org/JPCL
Oligomeric “Catastrophe Machines” with Thermally Activated Bistability and Stochastic Resonance Vladik A. Avetisov,* Anastasia A. Markina,* and Alexander F. Valov N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, 119991 Moscow, Russia
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S Supporting Information *
ABSTRACT: The desire to create nanometer-scale switching devices has motivated an active search for bistate macromolecular systems allowing for sharp conformational transitions in response to stimuli. Using full-atomic simulations, we found particular oligomers of thermosensitive polymers, themselves only a few nanometers in size, that possessed conformational bistability and reacted to power loads as nonlinear mechanical systems, termed “catastrophe machines”. We established the bifurcation and hysteresis effects, spontaneous vibrations, and stochastic resonance for these oligomers. It is important to note that the spontaneous vibrations and stochastic resonance were activated by thermal fluctuations. Because of such mechanic-like characteristics, short oligomers are a promising platform for the design of nanodevices and molecular machines.
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concave shape) and the back transitions occur at different critical values of the lateral force. It should be emphasized that a system does not have to be symmetrical to be bistable. For instance, bifurcation and hysteresis will be bistability markers even when an arch can bend in only one direction due to mechanical constraints. Besides deterministic transitions controlled by power loads, spontaneous jumps between two statesthat is, spontaneous vibrationsmay occur due to random perturbations of a system. The time intervals separating such random jumps are known to be widely distributed around a mean value, which depends on the parameters of bistability and random perturbations.14,15 However, the spontaneous vibrations can be transformed into regular jumps between two wells via “wiggling” of the double-well potential by a weak oscillating force. This phenomenon is known as stochastic resonance.14−16 Therefore, four basic effects are characteristic of classical bistable systems: the bifurcation effect, the hysteresis effect, spontaneous vibrations, and stochastic resonance. In macroscale mechanics, thermal fluctuations are too weak to activate spontaneous vibrationsthey may be activated by much stronger perturbations. However, in nanomechanics, this might not be the case. The barrier separating two states of a system can be so small that spontaneous vibrations and stochastic resonance might be possible even with thermal fluctuations. Therefore, the search is on for suitable macromolecular structures, several nanometers in size, that make it possible
anosystems capable of reversible transitions between two states in response to applied stimuli are attracting greater attention given the pursuit of miniaturization of trigger-type and switch-type elements. These elements are key units of many functional nanodevices (e.g., molecular and macromolecular switches, single molecular sensors, molecular actuators, electric and optoelectric devices, and logic gates).1−5 Mechanical and electromechanical nanodevices are also attracting attention because of the growing desire to make platforms for “algorithmic chemistry” with machinelike actions at the molecular level.6−11 Despite recent progress in microand nanomechanics, practical mastery of bistable systems with sizes ranging from a few nanometers to several tens of nanometers has not yet been achieved. Therefore, the search for nanoscale molecular structures with the dynamics inherent in bistable mechanical systems is of particular interest. In general, bistability refers to a nonlinear system whose dynamics are controlled by a double-well potential with two wells separated by a potential barrier. The so-called Euler’s arch assembled from two rigid rods joined by a hinge with an elastic spring is an impressively simple embodiment of bistable mechanics, which among those interested in the theory of nonlinear dynamical systems, is referred to as the mechanics of “catastrophe machines”.12,13 Indeed, if we slightly compress the arch in the longitudinal direction, it will remain straight. However, as soon as the compressive force exceeds a critical value, the straightened state will become unstable and will bifurcate into two arcuate states. With the addition of a lateral load, the arch will experience sharp transitions between the arcuate states as soon as the lateral load exceeds some critical value. In the latter case, hysteresis takes place; that is, the transitions to one direction (say, from a convex shape to a © XXXX American Chemical Society
Received: May 3, 2019 Accepted: August 22, 2019 Published: August 22, 2019 5189
DOI: 10.1021/acs.jpclett.9b01261 J. Phys. Chem. Lett. 2019, 10, 5189−5192
Letter
The Journal of Physical Chemistry Letters
Figure 1. Open and closed states of oligo-30s-NIPMAm equilibrated at temperatures T = 290 K and at T = 320 K, respectively. (a) The states that correspond to the equilibration dynamics in (b). (c) Longitudinal (F) and lateral (G) loads applied to oligo-30s-NIPMAm.
Figure 2. Response of oligo-30s-NIPMAm to a longitudinal load F (G = 0): (a) Time-averaged end-to-end distance, Re, vs compression F (Fc ≈ 400 pN is the critical compression). (b) Molecular dynamic trajectories far from the critical compression (black and blue curves) and near the critical compression (red curve); spontaneous vibrations clearly seen only near the critical compression. (c) The statistical weights, P(Re) for visits of open and closed states when the longitudinal load passes through the critical point: bimodal distribution P(Re) that corresponds to spontaneous vibrations clearly visible near the critical load.
simulation details, see S1). One could also take the oligomer in a closed state equilibrated at 320 K and apply the loads in the opposite directions. We also checked this setup, and the results were qualitatively the same. Figure 2 shows how oligo-30s-NIPMAm responds to longitudinal compression in the absence of a lateral load. One can clearly see the critical compression, which is equal to ∼400 pN (Figure 2a). The dynamic trajectories simulated sufficiently below the critical longitudinal load (subcritical compression), near this load, and sufficiently above it (supercritical compression) are shown in Figure 2b. Under subcritical compression, the oligomer remains in an open state, while it sharply transits to a closed state under supercritical compression. In other words, the oligomer acts as a mechanic-like switch when the longitudinal load quickly passes through critical compression. Near critical compression, the oligomer has no well-defined state: it clearly demonstrates spontaneous vibrations between open and closed states. To determine the appearance of bistability, we extracted from the trajectories the statistics P(Re) of visiting different states Re and studied how this statistic changed as longitudinal compression passed through the critical point. Under both subcritical and supercritical compressions, the oligomer dynamics were characterized by single-peak functions P(Re) related to open and closed states, respectively (Figure 2c). In contrast, near the critical point, P(Re) has a double-peak form in accordance with double-well bistability and spontaneous vibrations. Then, we estimated whether thermal fluctuations could activate spontaneous vibrations due to switching of HBs. Such estimates can be obtained from the phenomenological theory of stochastic resonance15,16 by using the mean value of the time intervals between the spontaneous jumps, τ = τ0 exp(ε/ kT), where ε is the barrier between two wells of the bistable
to realize mechanical bistability, thermally activated spontaneous vibrations, and stochastic resonance. We attempted to design such macromolecular structures using computer simulations. By studying the dynamics of short oligomers of thermosensitive polymers subject to power loads, we determined the oligomers, themselves only a few nanometers in size, which responded to power loads in the same way as bistable mechanical systems. These oligomers clearly demonstrated the bifurcation and hysteresis effects, as well as thermally activated spontaneous vibrations and stochastic resonance. In this letter, we announce the mechanic-like bistability of a syndiotactic N-isopropylmethacrylamide oligomer with N = 30 units (oligo-30s-NIPMAm). The oligo-30s-NIPMAm and the environmental water were modeled in a fully atomistic representation using GROMACS 5.1.1 software17−19 and the OPLS-AA force field20,21 (for simulation protocol, see the Supporting Information, Section S1). The oligomer states were specified by the distance between the oligomer ends, Re, and, also, by the oligomer gyration radius. It is known that poly-N-isopropylmethacrylamide in water solution undergoes a reversible coil−globule phase transition due to switching of hydrogen bonds (HB) from the polymer− water configuration to the polymer−polymer configuration.22−24 The low critical solution temperature of poly-Nisopropylmethacrylamide is close to 305 K.22−24 We first studied the oligo-30s-NIPMAm dynamics at temperatures of 290 and 320 K to confirm that the oligomer indeed had two well-distinguished conformational states. The “open” and “closed” shapes of these states with related equilibration trajectories are shown in Figure 1a,b, respectively. Next, taking oligo-30s-NIPAm in an open state equilibrated at 290 K, we began to bend it using the longitudinal and lateral loads applied to the oligomer as shown in Figure 1c (for the 5190
DOI: 10.1021/acs.jpclett.9b01261 J. Phys. Chem. Lett. 2019, 10, 5189−5192
Letter
The Journal of Physical Chemistry Letters
random fluctuations without any particular frequency (see Figure 3a), while under applied small periodic external load the system had periodic fluctuations with well-pronounced frequency (see Figure 3b). Then, we showed that the oligomer sharply transitioned from a closed state to open state with supercritical compression as soon as the lateral load attained some critical value. We also observed the hysteresis effect for the back transition (see Figure 4a). Figure 4b,c shows the hysteresis effect in terms of distributions P(Re) in the visiting open and closed states. Spontaneous vibrations (similar to those observed under longitudinal compression) were also observed near the critical lateral loads. The vibrational pulses ranged from 5 to 10 ns, which ensured that the same estimations for the bistability barrier were valid. Therefore, thermally activated switching of HB appears to be a common mechanism of spontaneous vibrations. In conclusion, our simulations show that short oligomers several nanometers in size may possess mechanic-like bistability and might realize thermally activated spontaneous vibrations and stochastic resonance. Oligo-30s-NIPMAm is one example of such an oligomer. Because of their nanomechanical characteristics, short oligomers are candidates for use as part of a new platform for designing nanodevices and molecular machines. The recent publication25 is of particular interest for the nanomechanics of short oligomers, demonstrating the implementation of Maxwell’s demon using bistability and spontaneous vibrations of a single DNA hairpin. For instance, similar techniques could be used for studies of bistability and spontaneous vibrations of oligomeric catastrophe machines.
potential, and τ0 is the collision time, which in our case ranges from 1 × 10−13 to 1 × 10−11 s. In our simulations, the time intervals in random jumps were distributed approximately from 5−10 ns. Consequently, the barrier ε was estimated to range from 10 to 15 kT, which meant switching about one HB. Complementary simulations showed strong correlations between spontaneous vibrations and switching of HB in the oligomer bending area (see Supporting Information, Section S2). Then, we examined the oligomer in the stochastic vibration regime and applied an additional weak oscillating force to simulate stochastic resonance. The oscillating force was realized by setting a charge (+1) at one end of the oligomer and a compensative charge (−1) at the other end; an external oscillating electrical field E = E0 cos ωt was directed along the compressive force F. Stochastic resonance was unambiguously observed with an intensity E0 varying from 0.01 to 1.00 V/nm, and the frequency ω ranged from 10 to 400 MHz (Figure 3; see also Supporting Information, Section S3).
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b01261. Figure 3. Stochastic resonance. Stochastic vibrations are shown together with the vibration’s power spectrum for the case of absence of later load (a) and with it (b). The lower row of panels shows the stochastic resonance under a weak oscillating force with E0 = 0.2 V/ nm and ω = 200 MHz (left) and the related power spectrum (right).
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In our in silico lateral load experiments (see Figure 1c), we first made sure that the oligo-30s-NIPMAm subjected to subcritical longitudinal compression responded smoothly to the lateral load. The system without lateral load exhibited
Simulation protocol, details of the force field for molecular dynamics simulations and input parameters; vibration mechanism, fluctuations of hydrogen bonds; stochastic resonance under different external loads (PDF)
AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. (V.A.A.) *E-mail:
[email protected]. (A.A.M.)
Figure 4. Responses of oligo-30s-NIPMAm to a lateral load G with supercritical longitudinal compression (F = 450pN). (a) Critical behavior and the hysteresis effect; the vibration effect appears in the vicinity of the critical points G1 = 100 pN and G2 = 200 pN. (b, c) The distributions P(Re) in the visiting open and closed states are transformed when the lateral load passes through the critical values G2 and G1, respectively. 5191
DOI: 10.1021/acs.jpclett.9b01261 J. Phys. Chem. Lett. 2019, 10, 5189−5192
Letter
The Journal of Physical Chemistry Letters ORCID
(18) Lindahl, E.; Hess, B.; van der Spoel, D. GROMACS 3.0: a Package for Molecular Simulation and Trajectory Analysis. J. Mol. Model. 2001, 7, 306−317. (19) Van Der Spoel, D.; Lindahl, E.; Hess, B.; Groenhof, G.; Mark, A. E.; Berendsen, H. J. C. GROMACS: Fast, Flexible, and Free. J. Comput. Chem. 2005, 26, 1701−1718. (20) Walter, J.; Sehrt, J.; Vrabec, J.; Hasse, H. Molecular Dynamics and Experimental Study of Conformation Change of Poly(Nisopropylacrylamide) Hydrogels in Mixtures of Water and Methanol. J. Phys. Chem. B 2012, 116, 5251−5259. (21) De Oliveira, T. E.; Mukherji, D.; Kremer, K.; Netz, P. Effects of Stereochemistry and Copolymerization on the LCST of PNIP. J. Chem. Phys. 2017, 146, 034904−034912. (22) Halperin, A.; Kröger, M.; Winnik, F. Poly(N-isopropylacrylamide) Phase Diagrams: Fifty Years of Research. Angew. Chem., Int. Ed. 2015, 54, 15342−15367. (23) Hoogenboom, R. Tunable Thermoresponsive Polymers by Molecular Design. In Complex Macromolecular Architectures: Synthesis, Characterization, and Self-Assembly; Wiley&Sons: New York, 2011. (24) Alaghemandi, M.; Spohr, E. Molecular Dynamics Investigation of the Thermoresponsive Polymer Poly(N-isopropylacrylamide). Macromol. Theory Simul. 2012, 21, 106−112. (25) Ribezzi-Crivellari, M.; Ritort, F. Large Work Extraction and the Landauer Limit in a Continuous Maxwell Demon. Nat. Phys. 2019, 15, 660−664.
Vladik A. Avetisov: 0000-0002-2516-8868 Anastasia A. Markina: 0000-0003-0699-7238 Author Contributions
V.A.A. initiated and supervised the work; A.A.M. designed and performed the computer simulations; V.A.A and A.A.M. prepared the final version of the manuscript. A.F.V. made noticeable contributions to the computer simulations and analysis of the vibration effect. Authors are listed in alphabetical order. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was conducted in the framework of a project supported by the Molecular Machine Corporation Ltd.
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ABBREVIATIONS HB - hydrogen bond; NIPMAm - N-isopropylmethacrylamide REFERENCES
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DOI: 10.1021/acs.jpclett.9b01261 J. Phys. Chem. Lett. 2019, 10, 5189−5192
