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When Translocation Dynamics Becomes Anomalous

2003; Elsevier BV; Volume: 85; Issue: 4 Linguagem: Inglês

10.1016/s0006-3495(03)74699-2

1542-0086

Ralf Metzler, J. Klafter,

Lipid Membrane Structure and Behavior

Recent single molecule experiments probing the passage process of a short single-stranded DNA (ssDNA) through a membrane channel (translocation) allow us to measure the passage time distribution. Building on a recent modeling approach, (Lubensky and Nelson, 1999Lubensky D.K. Nelson D.R. Driven polymer translocation through a narrow pore.Biophys. J. 1999; 77: 1824-1838Abstract Full Text Full Text PDF PubMed Scopus (374) Google Scholar), which has been demonstrated to be valid for chains of up to ≃300 nucleotides and therefore well applies to the system we have in mind, we discuss the consequences if the associated dynamics is not of Markov origin, but if strong memory effects prevail during the translocation. Motivation is drawn from recent results indicating that the distribution of translocation times is broader than predicted by simple Markovian models based on Brownian motion. The translocation of biomolecules through membrane pores (channels) is one of the most vital processes within or across biological cells, serving both delivery and signaling purposes (Alberts et al., 1994Alberts B. Roberts K. Bray D. Lewis J. Raff M. Watson J.D. The Molecular Biology of the Cell. Garland, New York1994Google Scholar). In (bio)chemistry, forced translocation is used in selection/purification of larger molecules, and in medicine, it plays an important role in drug delivery. Whereas the translocation of short, inflexible molecules is primarily determined by the properties of the pore (energy-driven transport, sticking events within the pore, etc.) and the difference of the chemical potential between the cis and trans sides of the pore, semiflexible and flexible molecules, in addition, have to cross an entropy barrier while being (partially) confined within the channel (Lubensky and Nelson, 1999Lubensky D.K. Nelson D.R. Driven polymer translocation through a narrow pore.Biophys. J. 1999; 77: 1824-1838Abstract Full Text Full Text PDF PubMed Scopus (374) Google Scholar; Muthukumar, 2001Muthukumar M. Translocation of a confined polymer through a hole.Phys. Rev. Lett. 2001; 86: 3188-3191Crossref PubMed Scopus (320) Google Scholar, Muthukumar, 1999Muthukumar M. Polymer translocation through a hole.J. Chem. Phys. 1999; 111: 10371-10374Crossref Scopus (459) Google Scholar; Slonkina and Kolomeisky, 2003Slonkina E. Kolomeisky A.B. Polymer translocation through a long nanopore.J. Chem. Phys. 2003; 118: 7112-7118Crossref Scopus (173) Google Scholar; Sung and Park, 1996Sung W. Park P.J. Polymer translocation through a pore in a membrane.Phys. Rev. Lett. 1996; 77: 783-786Crossref PubMed Scopus (520) Google Scholar). In the presence of a high external bias and for the rather short chains used in typical experiments, the entropic slowdown as well as the other interactions between chain and channel wall become negligible, the passage being dominated by the applied drift (Lubensky and Nelson, 1999Lubensky D.K. Nelson D.R. Driven polymer translocation through a narrow pore.Biophys. J. 1999; 77: 1824-1838Abstract Full Text Full Text PDF PubMed Scopus (374) Google Scholar). In what follows, we develop a scenario according to which the translocation dynamics is governed by slowly decaying memory effects, leading to a different behavior in the distribution of passage times which we believe can be measured experimentally. Experimentally, the translocation of ssDNA can be observed on a single molecular level, both voltage driven (Akeson et al., 1999Akeson M. Branton D. Kasianowicz J.J. Brandin E. Deamer D.W. Microsecond time-scale discrimination among polycytidylic acid, polyadenylic acid, and polyuridylic acid as homopolymers or as segments within single RNA molecules.Biophys. J. 1999; 77: 3227-3233Abstract Full Text Full Text PDF PubMed Scopus (824) Google Scholar; Kasianowicz et al., 1996Kasianowicz J.J. Brandin E. Branton D. Deamer D.W. Characterization of individual polynucleotide molecules using a membrane channel.Proc. Natl. Acad. Sci. U.S.A. 1996; 93: 13770-13773Crossref PubMed Scopus (2686) Google Scholar; Meller et al., 2001Meller A. Nivon L. Branton D. Voltage-driven DNA translocations through a nanopore.Phys. Rev. Lett. 2001; 86: 3435-3438Crossref PubMed Scopus (773) Google Scholar) and in the absence of an external electric field (Bates et al., 2003Bates M. Burns M. Meller A. Dynamics of DNA molecules in a membrane channel probed by active control techniques.Biophys. J. 2003; 84: 2366-2372Abstract Full Text Full Text PDF PubMed Scopus (129) Google Scholar). In such single-molecule translocation assays, fairly short chains are used, with some 60 bases corresponding to ∼12 persistence lengths, or six Kuhn lengths (Frank-Kamenetskii, 1997Frank-Kamenetskii M.D. Biophysics of the DNA molecule.Phys. Rep. 1997; 288: 13-60Crossref Scopus (93) Google Scholar). The width (≃50 Å) of the membrane amounts to about one persistence length (≃40 Å) of the ssDNA. A good measure for the translocation process is the distribution of passage times, i.e., the statistics of time spans the chain needs to cross from the entry (cis) side to the exit (trans) side of the pore. In the results, one observes two (or three) different timescales: the shortest corresponds to chains that retract from the pore back to the cis side, before completing the passage through the pore; the other (one or two) correspond(s) to real passage times (if there are two peaks, this can be explained by different orientations of the chain in respect to the passage direction (Lubensky and Nelson, 1999Lubensky D.K. Nelson D.R. Driven polymer translocation through a narrow pore.Biophys. J. 1999; 77: 1824-1838Abstract Full Text Full Text PDF PubMed Scopus (374) Google Scholar)). In a recent experiment (Bates et al., 2003Bates M. Burns M. Meller A. Dynamics of DNA molecules in a membrane channel probed by active control techniques.Biophys. J. 2003; 84: 2366-2372Abstract Full Text Full Text PDF PubMed Scopus (129) Google Scholar), it was noted that the first passage time distribution contains nonnegligible contributions over a large time range even in the presence of a low driving voltage, a case in which a Markovian model would predict exponentially fast decay. This may well indicate that additional mechanisms, so far neglected, play a role in the translocation dynamics, which might effect long-tailed first passage time distributions, and therefore imply a possible modeling by assuming a non-Markovian behavior of the system. In this note, we construct a framework in the limit of strong non-Markovian effects, taking into account anomalous translocation dynamics through long-tailed memory effects. Given the accuracy of the newly reported experiments in Bates et al., 2003Bates M. Burns M. Meller A. Dynamics of DNA molecules in a membrane channel probed by active control techniques.Biophys. J. 2003; 84: 2366-2372Abstract Full Text Full Text PDF PubMed Scopus (129) Google Scholar, it might well be possible to resolve such effects in log-log analyses of the presently available, or future data. We collect a number of possible sources for such anomalous dynamics. In the presence of a bias field and for chains with ≲300 nucleotides, the translocation dynamics in the Markov limit has been shown to follow the Smoluchowski-type equation (Lubensky and Nelson, 1999Lubensky D.K. Nelson D.R. Driven polymer translocation through a narrow pore.Biophys. J. 1999; 77: 1824-1838Abstract Full Text Full Text PDF PubMed Scopus (374) Google Scholar)∂P∂t=(−v∂∂x+K∂2∂x2P(x,t),(1) where P(x, t) is the probability density function (pdf) to find the chain at position x at time t, and v and K are the associated drift and diffusion constants, which may be determined from more microscopic models (Lubensky and Nelson, 1999Lubensky D.K. Nelson D.R. Driven polymer translocation through a narrow pore.Biophys. J. 1999; 77: 1824-1838Abstract Full Text Full Text PDF PubMed Scopus (374) Google Scholar). The translocation process without retraction from the pore can thus be described by the first passage time distribution F(t) from the point x = L to x = 0. In the presence of the external drift, this leads to the result (see, for instance, Redner, 2001Redner S. A guide to first-passage processes. Cambridge University Press, Cambridge, UK2001Crossref Google Scholar)F1(t)=L4πKt3exp−(L−vt)24Kt(2) Note the sharp exponential decay for longer passage times. In this picture of Markov advection-diffusion, the mean first passage time from the origin to an absorbing boundary a distance L away is given by T1≡∫0∞tF1(t)dt=L/v, i.e., the statistical mean corresponds exactly to a classical linear motion with the drift velocity v. Thus, the influence of the diffusivity in this average becomes negligible and the motion can be characterized by the mean 〈x〉 = vt. This picture dramatically changes in the presence of long-tailed memory, effected by a waiting time distributionψ(t)∼ταt1+α, (0<α 0, where the modulation factor A can follow logarithmic oscillations before eventually being cut off by an exponential (Nonnenmacher and Nonnenmacher, 1989Nonnenmacher T.F. Nonnenmacher D.J.F. A fractal scaling law for protein gating kinetics.Phys. Lett. A. 1989; 140: 323-326Crossref Scopus (25) Google Scholar), or be constant (Millhauser et al., 1988Millhauser G.L. Salpeter E.E. Oswald R.E. Diffusion-models of ion-channel gating and the origin of power-law distributions from single-channel recording.Proc. Natl. Acad. Sci. USA. 1988; 85: 1503-1507Crossref PubMed Scopus (131) Google Scholar). Within a finite time window, both are indistinguishable. It is therefore fair to say that gating events in a given time window in single ion channels follow power-law statistics, and typical values for β are ∼1.6. The distribution g translates into our waiting time distribution ψ(t) from Eq. 3 with α = β − 1.iii.For longer chains, Chuang et al., 2001Chuang J. Kantor Y. Kardar M. Anomalous dynamics of translocation.Phys. Rev. E. 2001; 65: 011802Crossref Scopus (274) Google Scholar argued that the diffusion of the chain becomes anomalous. Naively viewing the translocation as a waiting time process during which the monomers in the pore channel have to wait until they are given way by the vicinal monomers, and so on, creating a non-Markov process which, on some coarse-grained level, may well be described by Eq. 3; compare also Douglas, 2000Douglas J.F. Polymer science applications of path-integration, integral equations, and fractional calculus.in: Hilfer R. Applications of Fractional Calculus in Physics. World Scientific, Singapore2000Crossref Google Scholar. This list of scenarios is not meant to be complete. However, one might suspect that the sticking scenario (i) is most liable to be affected by the strength of the external bias, producing an effect similar to the recent experiments reported by Bates et al., 2003Bates M. Burns M. Meller A. Dynamics of DNA molecules in a membrane channel probed by active control techniques.Biophys. J. 2003; 84: 2366-2372Abstract Full Text Full Text PDF PubMed Scopus (129) Google Scholar, in which the dynamics exhibits the abovementioned turnover from broad to Brownian motion-type statistics on increase of the external bias field. In some translocation experiments, apart from the sharp initial peak in the first passage time density stemming from immediately retracting chains back to the cis side, there occurs another hump similar to the one of the translocated chains discussed above. It has been argued that this is due to the existence of second characteristic passage time, depending on the orientation of the chain to the membrane channel in respect to the cis-trans direction ("head or tail first") (Lubensky and Nelson, 1999Lubensky D.K. Nelson D.R. Driven polymer translocation through a narrow pore.Biophys. J. 1999; 77: 1824-1838Abstract Full Text Full Text PDF PubMed Scopus (374) Google Scholar). The same effect is expected in the case with long-tailed statistics following Eq. 3. However, it might well be that the associated power-law exponent α is different for the two orientations, as the nature of the effective interactions giving rise to the long-tailed waiting times may depend on this head-tail difference. One might speculate about the biological relevance of anomalous translocation dynamics. On the one hand, it might be the outcome of a tradeoff between lack of specificity, if the passage is too free and a large variety of molecules could pass the membrane, and too high suppression, which would require active transport through the pore, implying a fairly large energy cost for long molecules. On the other hand, it might be advantageous to have a large variation in the arrival times of translocated molecules on the trans side (and thereby very efficient retention of untranslocated molecules on the cis side). We have discussed possible changes arising in the distribution of first passage times in biopolymer translocation through a membrane channel, and listed a number of reasons that might give rise to such anomalous behavior. It should be possible to determine the quantity F(t) from experiments to sufficient accuracy, to be able to distinguish the normal (Brownian) dynamics result from its anomalous counterpart in both the presence and absence of an external drift. The large qualitative difference between exponential and power-law forms should be easily discernible on a double-logarithmic scale. It should, however, be stressed that the onset of the power-law trend depends on the strength of the drift, and might occur for fairly large times if the drift is weak. We finally mention that the proposed long-tailed effects may also pertain in other systems, like during the ejection of the DNA of bacteriophages from the capsid through a long pipe-like channel into the host cell (Alberts et al., 1994Alberts B. Roberts K. Bray D. Lewis J. Raff M. Watson J.D. The Molecular Biology of the Cell. Garland, New York1994Google Scholar; Muthukumar, 2001Muthukumar M. Translocation of a confined polymer through a hole.Phys. Rev. Lett. 2001; 86: 3188-3191Crossref PubMed Scopus (320) Google Scholar). We thank Amit Meller and Ophir Flomenbom for helpful discussions. J.K. acknowledges the support of the United States-Israel Binational Science Foundation and the Tel Aviv University Nanotechnology Center.