Abstract
Using Langevin dynamics simulations, we study a simple model of interacting-polymer under a periodic force. The extension curves strongly depend on the magnitude of the amplitude (F) and the frequency (ν) of the applied force. In low frequency limit, the system retraces the thermodynamic path. At higher frequencies, response time is greater than the external time scale for change of force, which restrict the biomolecule to explore a smaller region of phase space that results in hysteresis of different shapes and sizes. We show the existence of dynamical transition, where area of hysteresis loop approaches to a large value from nearly zero value with decreasing frequency. The area of hysteresis loop is found to scale as F ανβ for the fixed length. These exponents are found to be the same as of the mean field values for a time dependent hysteretic response to periodic force in case of the isotropic spin.
We thank S. M. Bhattacharjee and D. Dhar for many helpful discussions on the subject. Financial supports from the DST and CSIR, India are gratefully acknowledged. We also acknowledge the generous computer support from IUAC, New Delhi.
I. INTRODUCTION
II. MODEL AND METHOD
III. EQUILIBRIUM PROPERTIES OF BIO-POLYMERS
IV. DYNAMICAL TRANSITION AT FINITE TEMPERATURE
V. FINITE SIZE SCALING
VI. CRITICAL FREQUENCY AND ITS DEPENDENCE ON N
VII. CONCLUSIONS
Figures
Schematic representations of an interacting polymer: (a) DNA in the zipped state, (b) a self-interacting polymer chain, and (c) extended form of an interacting polymer under the influence of applied force ( f ). In all these cases, one end of polymer is fixed and the other end may be subjected to a constant force or periodic stretching force. For DNA, the dashed lines represent base pairing interaction among complimentary nucleotides (say 1 to N/2 are made up of adenine (A) and (N/2 + 1) to N are made up of complimentary nucleotides, i.e., thymine (T)). In this case, base pairing interaction is restricted in such a way that the 1st monomer forms base pair with the Nth monomer and 2nd monomer forms base pair with (N − 1)th and so on. For a self-interacting polymer, the dashed lines show the attractive interaction among non-bonded monomers. In this case, any monomer of a chain can interact with the rest of monomers of the chain. Here, for example, we have shown the second monomer of the chain, which is interacting with rest of non-bonded monomers. Similarly, other monomers interact with rest of the non-bonded monomers.
Schematic representations of an interacting polymer: (a) DNA in the zipped state, (b) a self-interacting polymer chain, and (c) extended form of an interacting polymer under the influence of applied force ( f ). In all these cases, one end of polymer is fixed and the other end may be subjected to a constant force or periodic stretching force. For DNA, the dashed lines represent base pairing interaction among complimentary nucleotides (say 1 to N/2 are made up of adenine (A) and (N/2 + 1) to N are made up of complimentary nucleotides, i.e., thymine (T)). In this case, base pairing interaction is restricted in such a way that the 1st monomer forms base pair with the Nth monomer and 2nd monomer forms base pair with (N − 1)th and so on. For a self-interacting polymer, the dashed lines show the attractive interaction among non-bonded monomers. In this case, any monomer of a chain can interact with the rest of monomers of the chain. Here, for example, we have shown the second monomer of the chain, which is interacting with rest of non-bonded monomers. Similarly, other monomers interact with rest of the non-bonded monomers.
(a) Equilibrium force-extension ( f − ⟨y⟩) curve for DNA and (b) for a self-interacting polymer.
(a) Equilibrium force-extension ( f − ⟨y⟩) curve for DNA and (b) for a self-interacting polymer.
(a) Equilibrium force-temperature ( f − T ) diagram of DNA and (b) for a self-interacting polymer.
(a) Equilibrium force-temperature ( f − T ) diagram of DNA and (b) for a self-interacting polymer.
The averaged extension of DNA as a function of cyclic force of amplitude (a) 0.4 and (b) 1.0 at different ν. (c) and (d) are for the SIP for low (2.75) and high amplitude (5.0), respectively. It is evident from these plots ((a) and (c)) that at low amplitude and high frequency, the system remains in the zipped (collapsed) state, whereas at high amplitude and high frequency (b) and (d), the system remains in the extended state with a small hysteresis loop. As ν decreases, both the systems extend to the hysteretic state with a bigger loop. For ν → 0, the hysteresis loop vanishes and the system approaches its equilibrium path irrespective of the magnitude of amplitudes of the applied force.
The averaged extension of DNA as a function of cyclic force of amplitude (a) 0.4 and (b) 1.0 at different ν. (c) and (d) are for the SIP for low (2.75) and high amplitude (5.0), respectively. It is evident from these plots ((a) and (c)) that at low amplitude and high frequency, the system remains in the zipped (collapsed) state, whereas at high amplitude and high frequency (b) and (d), the system remains in the extended state with a small hysteresis loop. As ν decreases, both the systems extend to the hysteretic state with a bigger loop. For ν → 0, the hysteresis loop vanishes and the system approaches its equilibrium path irrespective of the magnitude of amplitudes of the applied force.
Figures show the variation of area of hysteresis loop with the frequency at different force amplitudes (a) for DNA and (b) for SIP. For both cases, the area of loop increases to its maximum with the frequency and then again approaches to zero. In these cases, the system approaches the equilibrium from the non-equilibrium as the frequency decreases.
Figures show the variation of area of hysteresis loop with the frequency at different force amplitudes (a) for DNA and (b) for SIP. For both cases, the area of loop increases to its maximum with the frequency and then again approaches to zero. In these cases, the system approaches the equilibrium from the non-equilibrium as the frequency decreases.
Variation of the area of hysteresis loop with force amplitude at different frequencies: (a) for DNA and (b) SIP. In this case, the system never approaches to equilibrium and always remains far away from the equilibrium as the amplitude of force increases.
Variation of the area of hysteresis loop with force amplitude at different frequencies: (a) for DNA and (b) SIP. In this case, the system never approaches to equilibrium and always remains far away from the equilibrium as the amplitude of force increases.
Figs. (a)–(d) show the scaling of loop area of hysteresis with respect to ν0.5 F 0.5 for different lengths. It is evident from all these plots that in low frequency limit, curves of different amplitudes collapse on a single line intricating that the dynamical transition may be observed in single molecule experiments.
Figs. (a)–(d) show the scaling of loop area of hysteresis with respect to ν0.5 F 0.5 for different lengths. It is evident from all these plots that in low frequency limit, curves of different amplitudes collapse on a single line intricating that the dynamical transition may be observed in single molecule experiments.
Same as Fig. 7 , but loop area of hysteresis has been plotted with respect to ν−1.0D(F) for different chain length. Here D(F) ∼ (F − f c )2.0±0.1. (a)–(d) In high frequency regime also, curves of different forces collapse on a single line.
Same as Fig. 7 , but loop area of hysteresis has been plotted with respect to ν−1.0D(F) for different chain length. Here D(F) ∼ (F − f c )2.0±0.1. (a)–(d) In high frequency regime also, curves of different forces collapse on a single line.
(a) Same as Fig. 7 , but for the SIP. In low frequency regime, curves of different amplitudes collapse on a single line. (b) At high frequency, curves for different F collapse on a straight line. Here D(F) ∼ (F − f c )2.0±0.1.
(a) Same as Fig. 7 , but for the SIP. In low frequency regime, curves of different amplitudes collapse on a single line. (b) At high frequency, curves for different F collapse on a straight line. Here D(F) ∼ (F − f c )2.0±0.1.
Figure shows the collapse of data for different lengths. In low frequency regime, curves of different amplitudes collapse on a single line.
Figure shows the collapse of data for different lengths. In low frequency regime, curves of different amplitudes collapse on a single line.
Figure shows that at high frequency, curves for different lengths collapse on a straight line. (For clarity we have plotted Fig. 10 in log-log scale.) Here, D( f ) has the same scaling form as mentioned in the caption of Fig. 8 .
(a) Area of the loop per bead with frequency at fixed amplitude of force 0.6. The peak of the curve shifts left with N. (b) Same as (a) but with scaled frequency ν* = Nν. The peak of all the curves collapsed at frequency ν c . (c) In low frequency regime, A loop /N scales as ν*0.5. The inset shows the scaling (ν*0.5 N 0.25) of x axis gives a better collapse.
(a) Area of the loop per bead with frequency at fixed amplitude of force 0.6. The peak of the curve shifts left with N. (b) Same as (a) but with scaled frequency ν* = Nν. The peak of all the curves collapsed at frequency ν c . (c) In low frequency regime, A loop /N scales as ν*0.5. The inset shows the scaling (ν*0.5 N 0.25) of x axis gives a better collapse.
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