MECHANICS (WITH LAB)
MECHANICS (WITH LAB) PHYS 111
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PRL 96 118103 2006 PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 Dynamics of Polymer Translocation through Nanopores Theory Meets Experiment Silvina Matysiakl Alberto Montesi2 Matteo Pasquali1392 Anatoly B Kolomeisky1392 and Cecilia Clementi 12gtxlt 1Department of Chemistry Computer and Information Technology Institute 6100 Main street Rice University Houston Texas 77005 USA 2Department of Chemical and Biomolecular Engineering Computer and Information Technology Institute 00 Main street Rice University Houston Texas 77005 A Received 5 October 2005 published 22 March 2006 The dynamics of translocation of polymer molecules through nanopores is investigated via molecular dynamics We nd that an offlattice minimalist model of the system is suf cient to reproduce quantitatively all the experimentally observed trends and scaling behavior Speci cally simulations show i two translocation regimes depending on the ratio of pore and polymer length ii two different regimes for the probability of translocation depending on applied voltage iii an exponential dependence of translocation velocity upon applied voltage and iv an exponential decrease of the translocation time with temperature We also propose a simple theoretical explanation of mch of the observed trends within a free energy landscape framework DOI 101103PhysRevLett96l18103 The translocation of a polymer molecule through a highly con ned geometry is relevant in both chemical and biological processes 173 Many experiments have been performed in recent years on singlestranded D A and RNA molecules driven through an ahemolysin mem brane channel via electrophoresis the translocation time and probability have been measured over a wide range of conditions 14713 DNA and RNA translocation is im portant in the characterization of viral injection and in the design of A 39 39 thus quot this phenomenon may lead to new and improved biotechno logical applications Because of the universality of poly mer behavior DN experiments should yield gener alizable results on the dependence of the dynamics of translocation on the physical parameter for generic linear macromolecules In such a polymer physics perspective the chemical details of the DNA and RNA chains and of the pore are less relevant thus coarsegrained computational models can provide insight and bridge experimental results and theoretical understanding Many simpli ed computa tional and theoretical approaches have been proposed 147 24 however a simple model that can recover all the key experimental results is still missing In this Letter we propose a coarsegrained model for the polymer chain and the nanopore The model reproduces quantitatively all the trends and scaling laws observed in experiments by introducing three free parameters which are t to experimental data Moreover we interpret the results within a theoretical framework 20 and explain all the experimental results and relate them to relevant phe nomenological parameters The present work is a step forward towards the understanding of polymer transloca tion through nanopores and lays a base for further model ing which might include chemical and physical details of the pore and polymer Such model improvements may yield ab initio predictions of the free parameters 0031900706961111810342300 1181031 PACS numbers 8715La 8715Aa The linear polymer molecule Fig 1 is represented as a semiflexible chain ofN beads with contour length l Na where a is the beadbead equilibrium distance A single nanopore is considered as a structureless cylindrical tube of length L 12a and diameter D 37511 in agreement with the dimensions of the narrowest part of the ahemolysin membrane channel 1 The potential energy of the system is N N U krrii1 12 k90i71ii1 golz 11 12 6p 2 ltgt12 Zqua ri 1174 1 4mz1ltgt 151 where the rst two terms are the harmonic bond potential and the bending potential respectively the third term is the excluded volume the fourth term is the potential energy of FlGl color Schematic representation of the simulation model for polymer and nanopore 2006 The American Physical Society PRL 96 118103 2006 PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 the polymer in the external electric eld and the last term accounts for the van der Waals interaction between the polymer and the pore or membrane walls see caption of Fig 2 for the de nition of all the constants The electric eld increases linearly across the pore and is smoothed at the entrance and at the exit of the pore to avoid any discontinuities in the forces Fig 1 We use the package AMBER6 25 properly modi ed to include the effects of nanopore and electric eld to perform molecular dynam ics simulations at constant temperature with implicit sol vent 26 The dynamics is described by a Langevin equation including the coupling with the thermal bath 26 a T mi i U 071t 2 8r 270 T We perform hundreds of realizations of the translocation event for each set of simulation conditions In each initial con guration one end of the polymer is placed at the pore entrance Fig 1 whereas the positions of the remaining beads are sampled randomly from the equilibrium distri bution of the free polymer The simulation conditions are de ned through three effective parametersistrength of the porepolymer attraction 5U unit charge 1 and cou pling time constant To or equivalently hydrodynamic friction coef cient 5 267and two external parame tersiapplied voltage V and temperature T All the other terms in Eq 1 are polymer properties that can be evalu ETH E N FIG 2 color Translocation time vs polymer length com parison between simulation an experimental results polydA 712 Inset distribution of translocation times or all the simulation results shown the monomer size is a 4 mono mer mass m 312 amu elastic constant between monomers k 27366LJa2 bending stiffness k5 86M 1824 equilib rium angle between successive connectors 60 7139 Lennard Jones parameters 6P 054 kcalmol and UL 039 a 28 The three effective parameters 6U regulating the pore polymer stickiness q the charge per monomer and T0 the coupling time constant related to the hydrodynamic friction coef cient 5 are extracted from the comparison of simulation and experimental data as detailed in the text ated from known data see Fig 2 The parameter To regu lates the time scale of the simulations the three energy scales of the system are RT qV and EL ln dimensionless terms two energy ratios are relevant eg qVELJ and RTeu Each of the intrinsic parameters is estimated by comparing iteratively simulations and experimental re sults To is evaluated by comparing results at different polymer length and xed temperature and voltage The parameter To is then held xed to the value so determined and q is obtained by varying V while keeping temperature and polymer length xed Finally simulations at different temperatures at xed L and V and xed values of To and q yiel 5U Two different regimes of translocation dynamics have been observed in experiments where the polymer length is varied molecules shorter than the pore length move much faster than molecules longer than the pore These two regimes have been predicted theoretically using a simple free energy landscape argument 20 Essentially the ex istence of these two regimes is mainly ascribable to the free energy contribution corresponding to the con gurational entropy associated with the polymer ends hanging out of the pore this entropy is signi cant only for polymers longer than the pore Our simulation results Fig 2 agree with both theory and experiment and con rm the existence of two different translocation regimes depending on the ratio lL of poly mer and nanopore length when lL S 1 the velocity in side the pore drops with growing I whereas when lL gt 1 the polymer moves at constant speed and the translocation time grows linearly with polymer length The simulation and experimental data fully agree within the error bars once the effective parameter To is properly adjusted values for the friction coef cient in the range 7 X 1079 14 X 1078 kg 3 1 give a 2 lt 10 with an optimal value Opt m 10 8 kg 371 This is consistent with both experi mental and theoretical estimates 1520 The simulations reproduce correctly the asymmetric fattailed distribution of translocation times which is captured well by a Weibull distribution Fig 2 inset over the whole range of parame ters investigated see also Ref 18 Because the distribu tion is positively skewed the average translocation time is larger and less signi cant than the most probable trans location timeithe latter is reported here as commonly done in experimental reports Consistently with experi mental results the standard deviation of the time distribu tion of the shorter chains is much larger than that of the longer ones Clearly porepolymer interactions and poly mer conformation at the pore entrance are far more im portant for shorter chains and this accounts for the larger scatter of translocation times Figures 3 and 4 show that the simulations reproduce quantitatively the experimentally measured dependence of translocation probability and translocation time on the ap plied voltage Speci cally the simulations clearly show two different regimes for the translocation probability P depending on the voltage V Fig 3 P is computed as the 1181032 PRL 96 118103 2006 PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 Translucation probability FIG 3 color Effect of applied voltage on translocation probability The different symbols corresponds to the different identi ed regimes fraction of simulations leading to a successful translocation at given conditions Experimental results 12 show that the capture rate as a function of applied voltage exhibits two regimes Both regimes are well approximated by exponen tial ts Fig 3 in agreement with a previous theoretical estimate 20 The translocation probability computed in our simulations is qualitatively comparable to the capture rate measured in experiments de ned as the inverse of the time lag between two capture events a more quantitative comparison of the data is not possible as the capture rate depends also on the polymer concentration Figure 3 shows that at low voltage the translocation probability increases steeply with the applied voltage while it slowly approaches saturation at high voltage This change of scaling behavior can be explained in terms of change in translocation mechanism from barrier cross ing to downhill The two regimes barrier crossing and downhill are very different and this translates into a clear change of the trend of lnP vs V For a xed polymer length and temperature as the applied potential is increased the regime is a barriercrossing one until the barrier com pletely disappears where V memr upon further in crease the process becomes downhill If the applied voltage is smaller than the free energy barrier associated with the translocation process the translocation mechanism can be considered as a diffusion over a free energy barrier Increasing the applied voltage reduces the effective barrier and therefore signi cantly increases the probability of translocation On the other hand when the applied voltage is comparable to the free energy barrier associated with the translocation the process becomes essentially downhill Clearly increasing the applied voltage in the downhill regime does not signi cantly affect the probability of trans location as the probability is already close to 1 For a xed applied voltage the free energy barrier becomes larger as the polymer length and temperature are increased Thus the change in regime meovar depends on the polymer length and the temperature This argument can be more rigorously quanti ed by using simple polymer physics concepts 27 Figure 4 shows the comparison between the transloca tion time obtained from experiments and simulations for a wide range of applied voltage and two different polymer lengths By tting our results to the experimental data with the friction coef cient xed to its optimal value Opt as determined above a 2 lt 10 is obtained for values of q in the range 0352 0562 with an optimal value 10pt 042 An effective partial charge for DNA monomer is consistent with the mechanisms of condensation or screening that have been proposed by different studies 2124 The agreement is extremely good showing an 1 quot 1 39 39meonthe applied voltage when choff lt V lt chsovar We can refer to the simple comparison with a chemical reaction with AF qt 0 to explain the observed trend The rate constant in such a reaction is proportional to expAFRT and AF decreases with increasing V Therefore in the range chwff S V lt chmm the translocation velocity should grow exponentially with V as observed in our simulations At lower V such scaling does not hold because the trans location event is energetically unfavorable thus when V lt choff the translocation time quickly increases as the translocation probability drops At higher V the translo cation event becomes energetically downhill rather than barrier crossing this transition yields a different scaling of translocation time versus voltage The effect of temperature on the translocation time has been investigated experimentally only in a smaller range of T and less systematically than the dependence on L and V However our simple computational model is able to re produce the experimental results correctly as shown in Fig 5 and clearly con rm an exponential dependence of 1n all ii l simulation experiment AN80 ll 9 91 6 FIG 4 color Dependence of 7P with the applied potential The experimental data correspond to polydA and were ex tracted from Ref 12 1181033 PRL 96 118103 2006 PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 I simulation results i I experu nental result 1 e 111111111 11 111111 32 34 i 36 1T units of lOOOK FIG 5 color Dependence of 7P with temperature for a polymer of length 100 The experimental data correspond to polydA and were extracted from Ref 5 TI on T Values of the parameter EU in the range 05 07 kcalmol t our results to experimental data with a 2 lt 10 The best t is obtained for euopt 06 kcalmol 1RT at room temperature This is con sistent with the typical strength of van der Waals interac tions The experimental data have been obtained with V 120 mV with N 100 12 when the translocation process is barrier crossing therefore the exponential de pendence on the temperature can be explained theoreti cally once more through the chemical reaction comparison where the reaction rate is proportional to expAF RT Simulations show quantitative agreement with the experi mental results This Letter reports a simple coarsegrained computa tional model for polymer translocation through a nanopore for the rst time such a model can reproduce all the experimentally observed trends Moreover we provide a theoretical framework for explaining simulations and ex perimental results The present work clearly indicates the relevant time and energy scales of the process The actual friction coef cient inside the pore or equivalently the coupling time To depends on the strong con nement the equivalent unit charge of the monomer q is affected by the counterion distribution and the LennardJones potential between polymer and pore walls EL depends on the pore chemistry Future work may focus on ab initio prediction of these three parameters The authors wish to thank Giovanni Fossati for his help with computer related issues and insightful discussions Support for this project was provided in part by grants CHECAREER0349303 CC CHE CAREER0237105 ABK CTSCAREER0134389 M P the Robert A Welch Foundation Norman Hackerman grant and C1570 CC C1559 ABK and the Sloan Foundation BR4418 A B K Author to whom correspondence should be addressed Email address ceciliarice Phone 17133483485 Fax 17133485155 1 A Meller J Phys Condens Matter 15 R581 2003 2 J Nakane M Akenson and A Maiziali J Phys Condens Matter 15 R1365 2003 3 H Lodish D Baltimore A Berk S L Zipursky P Matsudaira and J Darnell Molecular Cell Biology W 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