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			<b>The Polymer Barrier Crossing Problem</b>

			<p>Wokyung Sung<br>

			Dept of Physics, Pohang University of Science and Technology<br>

			Pohang, 790-784, South Korea</p>

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		<p>Dynamics of polymers crossing barriers is not only a basic problem in soft matter physics but also important in biological applications such as polymer transport across membranes, DNA gel electrophoresis, etc. In this paper we review our theoretical investigations on (1) polymer translocation through a narrow pore in a membrane [1-4], and (2) dynamics of a polymer surmounting a potential barrier. In dealing with the dynamics of complex systems, the first thing to do is to determine the primary, relevant dynamical variable(s) which can effectively the dynamics in question. In terms of the variable, the dynamics is described by a stochastic equation of motion involving the free energy associated with the variable, which is, in the Markovian approximation, the overdamped Langevin equation. For the case of polymer tanslocation , the variable is chosen as the number of translocated segments (n) across the membrane, and the free energy for a given n is calculated from polymer conformational entropy [1]. The translocation time, now viewed as the mean first passage time of the diffusion process crossing the free energy barrier, can be readily calculated. We considered [1] the effects of chain flexibility as well as the asymmetry caused by transmembrane chemical potential difference. With no asymmetries, the translocation times scales as N<sup>3</sup> irrespective of chain flexibility, where N is the total segment number. From the prefactors, however, an ideal chain translocation rate is found to be slower compared with that of a rigid chain of the same length by 23%. With a chemical potential difference per segment as small as kT/N, the translocation time shows a pronounced change; for a forward bias, the scaling behavior is changed to N Regardless of chain flexibility, this extreme sensitivity is a cooperative phenomena rising from chain connectivity; the segments respond all hand in hand rather than as individuals to a driving asymmetry. As another type of asymmetry mechanism we studied the Brownian Ratchet (BR), which was originally suggested by Simon, Peskin and Oster[5] as a nonspecific driving mechanism for biased diffusion arising from chemical binding of chaperons on the chains entering the trans side of the membrane. It is found that the strong ratchet (many chaperon binding) not only rectify the motion but also suppresses the chain flexibility to rigid rod like behavior.</p>

		<p>We also have investigated other types of driving asymmetries, such as curvature effect considering polymer release out of spherical vesicle[2], and polymer adsorption on the trans side[3]. It turns out that all of these driving forces induce the crossover on the translocation time behavior from Nto Nin the same manner as mentioned before. Since the membrane is subject to ceaseless non-equilibrium fluctuations from the background, the chemical potential of each segment can be modulated in a random fashion. The effect of the fluctuations to the transport of single Brownian particle have been extensively studied [6]. An important finding is the phenomena of resonant activation, where the fastest translocation occurs when the time-scale of a non-equilibrium fluctuation matches a certain optimal time. It is found that, compared with a single particle transport, the polymer resonant translocation become much more rapid as the magnitude of the fluctuation in chemical potential difference increases, again due to chain connectivity [4].</p>

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			<p>In the second part of the paper we consider a polymer crossing the one-dimensional Kramers potential. The potential varies over a large distance so that the polymer dimension is much smaller than the distance between the well bottom and the barrier top. As an application, an efficient device can be suggested to separate polymers with different lengths by preparing potential barrier of a macroscopic distance. For this case, the relevant variable is the center of mass position of the chain, for which the Langevin equation is set up. The driving force in the equation is given by the gradient of the polymer free energy for a given center of mass position, which is obtained by using the Rouse model. As the barrier crossing of the center of mass is a quasi-stationary process, we adopt the Kramers rate theory of a single Brownian particle by considering both the friction and the potential to be N-times that of the single particle and replacing the potential barrier in the Arrhenius factor by the free energy barrier. For small N or the well and barrier of small enough curvatures, the polymer keeps the globular shape throughout the process of barrier crossing, having essentially the same crossing dynamics of a globular Brownian particle of N segments. For the other cases, however, the crossing rate tends to be enhanced beyond the globular limit, due to chain flexibility that manifests confined and extended conformations at the well and barrier respectively and reduces the free energy barrier. A surprising result is that the rate tends to be infinitely large when the spring constant of the first Rouse mode matches 2N times the barrier curvature. This resonance behavior is due to infinite chain extensibility in the bead spring (Rouse) model; the coil-to-stretch transition at the barrier top of a large enough curvature results in infinitely negative chain energy and thus an infinite Arrhenius factor! To avoid the infinite extensibility in describing the stretching behaviors of polymers, we employ the transfer matrix method which enables us to compute the polymer free energy at the barrier top. It is found that a chain of intermediate length can exhibit a novel minimum in crossing rate due to the competition between the potential barrier and the finite free energy decrease by stretching. It is shown again that due to conformational flexibility the unfolded chain has the rate always larger than the case for the folded (globular) state. This is in a sharp contrast to the entropic barrier crossing (the polymer translocation discussed earlier) where the flexibility happens to retard the rate.</p>

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			<ol>

				<li>W. Sung and P. J. Park, Phys. Rev. Lett. 77,783 (199
6)</div>

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				<li>P .J .Park and W Sung, Phys. Rev. E 57,730 (199
8)</div>

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				<li>P. J. Park and W. Sung, J. Chem. Phys. 108,3013 (199
8)</div>

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				<li>P. J. Park and W. Sung, Int. J. Bifur. and Chaos 8, 927 (199
8)</div>

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				<li>S. M .Simon, C. S. Peskin, G. F. Oster, Proc. Natl. Acad. Sci. USA 89,3770 (199
2)</div>

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				<li>C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69, 2318 (199
2)

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