Journal of Computational Electronics

, Volume 13, Issue 4, pp 826–838 | Cite as

Modeling thermophoretic effects in solid-state nanopores

  • Maxim Belkin
  • Shu-Han Chao
  • Gino Giannetti
  • Aleksei Aksimentiev
Article

Abstract

Local modulation of temperature has emerged as a new mechanism for regulation of molecular transport through nanopores. Predicting the effect of such modulations on nanopore transport requires simulation protocols capable of reproducing non-uniform temperature gradients observed in experiment. Conventional molecular dynamics (MD) method typically employs a single thermostat for maintaining a uniform distribution of temperature in the entire simulation domain, and, therefore, can not model local temperature variations. In this article, we describe a set of simulation protocols that enable modeling of nanopore systems featuring non-uniform distributions of temperature. First, we describe a method to impose a temperature gradient in all-atom MD simulations based on a boundary-driven non-equilibrium MD protocol. Then, we use this method to study the effect of temperature gradient on the distribution of ions in bulk solution (the thermophoretic effect). We show that DNA nucleotides exhibit differential response to the same temperature gradient. Next, we describe a method to directly compute the effective force of a thermal gradient on a prototypical biomolecule—a fragment of double-stranded DNA. Following that, we demonstrate an all-atom MD protocol for modeling thermophoretic effects in solid-state nanopores. We show that local heating of a nanopore volume can be used to regulate the nanopore ionic current. Finally, we show how continuum calculations can be coupled to a coarse-grained model of DNA to study the effect of local temperature modulation on electrophoretic motion of DNA through plasmonic nanopores. The computational methods described in this article are expected to find applications in rational design of temperature-responsive nanopore systems.

Keywords

Molecular dynamics Thermophoresis Thermodiffusion Soret coefficient Plasmonic heating Nanofluidics Boundary-driven MD Non-equilibrium MD 

Notes

Acknowledgments

This work was supported by the Grants from the National Science Foundation (DMR-0955959) and the National Institutes of Health (R01-HG007406 and P41-RR005969).The authors gladly acknowledge supercomputer time provided through XSEDE Allocation Grant MCA05S028 and the Blue Waters petascale supercomputer system (UIUC).

References

  1. 1.
    Dekker, C.: Solid-state nanopores. Nat. Nanotechnol. 2, 209–215 (2007)CrossRefGoogle Scholar
  2. 2.
    Kasianowicz, J.J., Robertson, J.W.F., Chan, E.R., Reiner, J.E., Stanford, V.M.: Nanoscopic porous sensors. Annu. Rev. Anal. Chem. 1, 737–766 (2008)CrossRefGoogle Scholar
  3. 3.
    Howorka, S., Siwy, Z.: Nanopore analytics: sensing of single molecules. Chem. Soc. Rev. 38(8), 2360–2384 (2009)CrossRefGoogle Scholar
  4. 4.
    Branton, D., Deamer, D.W., Marziali, A., Bayley, H., Benner, S.A., Butler, T., Di Ventra, M., Garaj, S., Hibbs, A., Huang, X., Jovanovich, S.B., Krstic, P.S., Lindsay, S.: Xinsheng Sean, L., Mastrangelo, C.H., Meller, A., Oliver, J.S., Pershin, Y.V., Ramsey, J.M., Riehn, R., Soni, G.V., Tabard-Cossa, V., Wanunu, M., Wiggin, M., Schloss, J.A.: The potential and challenges of nanopore sequencing. Nat. Biotechnol. 26(10), 1146–1153 (2008)CrossRefGoogle Scholar
  5. 5.
    Timp, W., Mirsaidov, U.M., Wang, D., Comer, J., Aksimentiev, A., Timp, G.: Nanopore sequencing: electrical measurements of the code of life. IEEE Trans. Nanotechnol. 9(3), 281–294 (2010)CrossRefGoogle Scholar
  6. 6.
    Venkatesan, B.M., Estrada, D., Banerjee, S., Jin, X., Dorgan, V.E., Bae, M.-H., Aluru, N.R., Pop, E., Bashir, R.: Stacked graphene-\(\text{ Al }_{2}\text{ O }_{3}\) nanopore sensors for sensitive detection of DNA and DNA-protein complexes. ACS Nano 6(1), 441–450 (2012)CrossRefGoogle Scholar
  7. 7.
    Kasianowicz, J.J., Brandin, E., Branton, D., Deamer, D.W.: Characterization of individual polynucleotide molecules using a membrane channel. Proc. Natl. Acad. Sci. USA 93, 13770–13773 (1996)CrossRefGoogle Scholar
  8. 8.
    Luan, B., Aksimentiev, A.: Electro-osmotic screening of the DNA charge in a nanopore. Phys. Rev. E 78, 021912 (2008)CrossRefGoogle Scholar
  9. 9.
    Keyser, U.F.: Controlling molecular transport through nanopores. J. R. Soc. Interface 8(63), 1369–1378 (2011)CrossRefGoogle Scholar
  10. 10.
    Heng, J.B., Ho, C., Kim, T., Timp, R., Aksimentiev, A., Grinkova, Y.V., Sligar, S., Schulten, K., Timp, G.: Sizing DNA using a nanometer-diameter pore. Biophys. J. 87, 2905–2911 (2004)CrossRefGoogle Scholar
  11. 11.
    Comer, J., Dimitrov, V., Zhao, Q., Timp, G., Aksimentiev, A.: Microscopic mechanics of hairpin DNA translocation through synthetic nanopores. Biophys. J. 96(2), 593–608 (2009)CrossRefGoogle Scholar
  12. 12.
    Firnkes, M., Pedone, D., Knezevic, J., Doblinger, M., Rant, U.: Electrically facilitated translocations of proteins through silicon nitride nanopores: Conjoint and competitive action of diffusion, electrophoresis, and electroosmosis. Nano Lett. 10(6), 2162–2167 (2010)CrossRefGoogle Scholar
  13. 13.
    Jubery, T.Z., Prabhu, A.S., Kim, M.J., Dutta, P.: Modeling and simulation of nanoparticle separation through a solid-state nanopore. Electrophoresis 33(2), 325–333 (2012)CrossRefGoogle Scholar
  14. 14.
    Nadtochiy, A., Melnikov, D., Gracheva, M.: Filtering of nanoparticles with tunable semiconductor membranes. ACS Nano 7(8), 7053–7061 (2013)CrossRefGoogle Scholar
  15. 15.
    Meller, A., Nivon, L., Branton, D.: Voltage-driven DNA translocations through a nanopore. Phys. Rev. Lett. 86, 3435–3438 (2001)CrossRefGoogle Scholar
  16. 16.
    Wanunu, M., Morrison, W., Rabin, Y., Grosberg, A.Y., Meller, A.: Electrostatic focusing of unlabelled DNA into nanoscale pores using a salt gradient. Nat. Nanotechnol. 5(2), 169–165 (2010)CrossRefGoogle Scholar
  17. 17.
    Jung, Y., Bayley, H., Movileanu, L.: Temperature-responsive protein pores. J. Am. Chem. Soc. 128(47), 15332–15340 (2006)Google Scholar
  18. 18.
    Reber, N., Kuchel, A., Spohr, R., Wolf, A., Yoshida, M.: Transport properties of thermo-responsive ion track membranes. J. Memb. Sci. 193(1), 49–58 (2001)CrossRefGoogle Scholar
  19. 19.
    Schepelina, O., Zharov, I.: PNIPAAM-modified nanoporous colloidal films with positive and negative temperature gating. Langmuir 23(25), 12704–12709 (2007)CrossRefGoogle Scholar
  20. 20.
    Yameen, B., Ali, M., Neumann, R., Ensinger, W., Knoll, W., Azzaroni, O.: Ionic transport through single solid-state nanopores controlled with thermally nanoactuated macromolecular gates. Small 5(11), 1287–1291 (2009)CrossRefGoogle Scholar
  21. 21.
    Guo, W., Xia, H., Fan, X., Xu, H., Liuxuan, C., Wang, L., Xue, J., Zhang, G., Song, Y., Zhu, D., Wang, Y., Jiang, L.: Current rectification in temperature-responsive single nanopores. ChemPhysChem 11(4), 859–864 (2010)CrossRefGoogle Scholar
  22. 22.
    Nasir, S., Ali, M., Ensinger, W.: Thermally controlled permeation of ionic molecules through synthetic nanopores functionalized with amine-terminated polymer brushes. Nanotechnology 23(22), 225502 (2012)CrossRefGoogle Scholar
  23. 23.
    Jonsson, M.P., Dekker, C.: Plasmonic nanopore for electrical profiling of optical intensity landscapes. Nano Lett. 13(3), 1029–1033 (2013)CrossRefGoogle Scholar
  24. 24.
    Reiner, J.E., Robertson, J.W.F., Burden, D.L., Burden, L.K., Balijepalli, A., Kasianowicz, J.J.: Temperature sculpting in yoctoliter volumes. J. Am. Chem. Soc. 135(8), 3087–3094 (2013)CrossRefGoogle Scholar
  25. 25.
    Harata, A., Shen, Q., Sawada, T.: Photothermal applications of lasers: study of fast and ultrafast photothermal phenomena at metal-liquid interfaces. Annu. Rev. Phys. Chem. 50, 193–219 (1999)CrossRefGoogle Scholar
  26. 26.
    Min, H., Petrova, H., Hartland, G.V.: Investigation of the properties of gold nanoparticles in aqueous solution at extremely high lattice temperatures. Chem. Phys. Lett. 391, 220–225 (2004)CrossRefGoogle Scholar
  27. 27.
    Merabia, S., Shenogin, S., Joly, L., Keblinski, P., Barrat, J.-L.: Heat transfer from nanoparticles: A corresponding state analysis. Proc. Natl. Acad. Sci. USA 106(36), 15113–15118 (2009)CrossRefGoogle Scholar
  28. 28.
    Belkin, M., Maffeo, C., Wells, D.B., Aksimentiev, A.: Stretching and controlled motion of single-stranded DNA in locally heated solid-state nanopores. ACS Nano 7(8), 6816–6824 (2013)CrossRefGoogle Scholar
  29. 29.
    He, Y., Tsutsui, M., Scheicher, R.H., Bai, F., Taniguchi, M., Kawai, T.: Thermophoretic manipulation of DNA translocation through nanopores. ACS Nano 7(1), 538–546 (2013)CrossRefGoogle Scholar
  30. 30.
    Duhr, S., Braun, D.: Optothermal molecule trapping by opposing fluid flow with thermophoretic drift. Phys. Rev. Lett. 97(3), 038103 (2006)CrossRefGoogle Scholar
  31. 31.
    Jerabek-Willemsen, M., Wienken, C.J., Braun, D., Baaske, P., Duhr, S.: Molecular interaction studies using microscale thermophoresis. Assay Drug Dev. Technol. 9(4), 342–353 (2011)Google Scholar
  32. 32.
    Ludwig, C.: Diffusion zwischen ungleich erwärmten Orten gleich zusammengesetzter Lösungen. Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss. 20, 539 (1856)Google Scholar
  33. 33.
    Soret, C.: Sur l’état d’équilibre que prend, du point de vue de sa concentration, une dissolution saline primitivement homogéne, dont deux parties sont portées a des températures différentes. Arch. Sci. Phys. Nat. 2, 48–61 (1879)Google Scholar
  34. 34.
    Tanner, C.C.: The Soret effect. Part i. Trans. Faraday Soc. 23, 75–95 (1927)CrossRefGoogle Scholar
  35. 35.
    Agar, J.N., Turner, J.C.R.: Thermal diffusion in solutions of electrolytes. Proc. R. Soc. A 255(1282), 307–330 (1960)CrossRefGoogle Scholar
  36. 36.
    Snowdon, P.N., Turner, J.C.R.: The Soret effect in some 0.01 normal aqueous electrolytes. Trans. Faraday Soc. 56(10), 1409–1418 (1960)CrossRefGoogle Scholar
  37. 37.
    Debye, P., Bueche, A.M.: Remsen Press Division. Chemical Pub. Co., Brooklyn (1948)Google Scholar
  38. 38.
    Emery, A.H., Drickamer, H.G.: Thermal diffusion in polymer solutions. J. Chem. Phys. 23(12), 2252–2257 (1955)CrossRefGoogle Scholar
  39. 39.
    McNab, G.S., Meisen, A.: Thermophoresis in liquids. J. Colloid Interface Sci. 44(2), 339–346 (1973)CrossRefGoogle Scholar
  40. 40.
    Giglio, M., Vendramini, A.: Soret-type motion of macromolecules in solution. Phys. Rev. Lett. 38(1), 26–30 (1977)CrossRefGoogle Scholar
  41. 41.
    Caldwell, D.R., Eide, S.A.: Separation of seawater by Soret diffusion. Deep-Sea Res. 32(8), 965–982 (1985)CrossRefGoogle Scholar
  42. 42.
    Alexander, F.: Zur Theorie der Thermodiffusion in Flüssigkeiten. Akademische Verlag Ges, Leipzig (1954)Google Scholar
  43. 43.
    Wood, C., Hawksworth, W.: Thermal diffusion of 1:1 electrolytes in ordinary and in heavy water. Afr. Chem. Inst. 24, 170 (1971)Google Scholar
  44. 44.
    Caldwell, D.R.: Thermal and Fickian diffusion of sodium chloride in a solution of oceanic concentration. Deep-Sea Res. 20(11), 1029–1039 (1973)Google Scholar
  45. 45.
    Gaeta, F.S.: Radiation pressure theory of thermal diffusion in liquids. Phys. Rev. 182(1), 289–296 (1969)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Colombani, J., Bert, J., Dupuy-Philon, J.: Thermal diffusion in (LiCl, RH2O). J. Chem. Phys. 110(17), 8622–8627 (1999)CrossRefGoogle Scholar
  47. 47.
    Morozov, K.: Thermal diffusion in disperse systems. J. Exp. Theor. Phys. 88, 944–946 (1999)CrossRefGoogle Scholar
  48. 48.
    Duhr, S., Braun, D.: Thermophoretic depletion follows Boltzmann distribution. Phys. Rev. Lett. 96, 168301 (2006)CrossRefGoogle Scholar
  49. 49.
    Rusconi, R., Isa, L., Piazza, R.: Thermal-lensing measurement of particle thermophoresis in aqueous dispersions. J. Opt. Soc. Am. B 21(3), 605–616 (2004)CrossRefGoogle Scholar
  50. 50.
    Debuschewitz, C., Kohler, W.: Molecular origin of thermal diffusion in benzene plus cyclohexane mixtures. Phys. Rev. Lett. 87(5), 055901 (2001)CrossRefGoogle Scholar
  51. 51.
    Helfand, E., Kirkwood, J.G.: Theory of the heat of transport of electrolytic solutions. J. Chem. Phys. 32(3), 857–866 (1960)Google Scholar
  52. 52.
    Schimpf, M.E., Caldwell, K., Giddings, J.C.: Field-Flow Fractionation Handbook. Wiley, New York (2000)Google Scholar
  53. 53.
    Jiang, H.-R., Sano, M.: Stretching single molecular dna by temperature gradient. Appl. Phys. Lett. 91(15), 154104 (2007)Google Scholar
  54. 54.
    Gaeta, F.S., Bencivenga, U., Canciglia, P., Rossi, S., Mita, D.G.: Temperature gradients and prebiological evolution. Cell Biophys. 10, 103–125 (1987)CrossRefGoogle Scholar
  55. 55.
    Braun, R., Sarikaya, M., Schulten, K.: Genetically engineered gold-binding polypeptides: structure prediction and molecular dynamics. J. Biomater. Sci. 13, 747–758 (2002)CrossRefGoogle Scholar
  56. 56.
    Duhr, S., Braun, D.: Why molecules move along a temperature gradient. Proc. Natl. Acad. Sci. USA 103(52), 19678–19682 (2006)CrossRefGoogle Scholar
  57. 57.
    Parola, A., Piazza, R.: A microscopic approach to thermophoresis in colloidal suspensions. J. Phys.: Condens. Matter 17(45, SI), S3639–S3643 (2005)Google Scholar
  58. 58.
    Schimpf, M.E., Semenov, S.N.: Symmetric diffusion equations, barodiffusion, and cross-diffusion in concentrated liquid mixtures. Phys. Rev. E 70, 031202 (2004)Google Scholar
  59. 59.
    Ham, J.S.: Kinetic theory of thermal diffusion in dilute polymer solutions. J. Appl. Phys. 31(11), 1853–1858 (1960)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Khazanovich, T.N.: On the theory of thermal diffusion in dilute polymer solutions. J. Polymer Sci. : Part C 16, 2463–2468 (1967)Google Scholar
  61. 61.
    Mes, E.P.C., Kok, WTh, Tijssen, R.: Prediction of polymer thermal diffusion coefficients from polymer-solvent interaction parameters: Comparison with thermal field flow fractionation and thermal diffusion forced rayleigh scattering experiments. Int. J. Polym. Anal. Charact. 8(2), 133–153 (2003)CrossRefGoogle Scholar
  62. 62.
    Luettmer-Strathmann, J.: Two-chamber lattice model for thermodiffusion in polymer solutions. J. Chem. Phys. 119(5), 2892–2902 (2003)CrossRefGoogle Scholar
  63. 63.
    Zhang, M., Mueller-Plathe, F.: The Soret effect in dilute polymer solutions: Influence of chain length, chain stiffness, and solvent quality. J. Chem. Phys. 125(12), 124903 (2006)CrossRefGoogle Scholar
  64. 64.
    Ruckenstein, E.: Can phoretic motions be treated as interfacial tension gradient driven phenomena? J. Coll. Interface Sci. 83(1), 77–81 (1981)CrossRefGoogle Scholar
  65. 65.
    Andreev, A.F.: Thermophoresis in liquids. Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki 94(1), 210–216 (1988)Google Scholar
  66. 66.
    Semenov, S., Schimpf, M.: Thermophoresis of dissolved molecules and polymers: consideration of the temperature-induced macroscopic pressure gradient. Phys. Rev. E 69, 011201 (2004)Google Scholar
  67. 67.
    Piazza, v., Guarino, A.: Soret effect in interacting micellar solutions. Phys. Rev. Lett. 88(20), 208302 (2002)Google Scholar
  68. 68.
    Bringuier, E., Bourdon, A.: Colloid transport in nonuniform temperature. Phys. Rev. E 67(1, Part 1), 011404 (2003)Google Scholar
  69. 69.
    Rasuli, S.N., Golestanian, R.: Thermophoresis for a single charged colloidal particle. J. Phys.: Condens. Matter 17(14, SI), S1171–S1176 (2005)Google Scholar
  70. 70.
    Anderson, J.L.: Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21, 61–99 (1989)CrossRefGoogle Scholar
  71. 71.
    Macgowan, D., Evans, D.J.: Heat and matter transport in binary-liquid mixtures. Phys. Rev. A 34(3), 2133–2142 (1986)CrossRefGoogle Scholar
  72. 72.
    Simon, J.-M., Dysthe, D.K., Fuchs, A.H., Rousseau, B.: Thermal diffusion in alkane binary mixtures: A molecular dynamics approach. Fluid Phase Equilib. 150151, 151–159 (1998)CrossRefGoogle Scholar
  73. 73.
    Reith, D., Muller-Plathe, F.: On the nature of thermal diffusion in binary Lennard-Jones liquids. J. Chem. Phys. 112(5), 2436–2443 (2000)Google Scholar
  74. 74.
    Vogelsang, R., Hoheisel, C., Paolini, G.V., Ciccotti, G.: Soret coefficient of isotopic Lennard-Jones mixtures and the ar-kr system as determined by equilibrium molecular-dynamics calculations. Phys. Rev. A 36(8), 3964–3974 (1987)CrossRefGoogle Scholar
  75. 75.
    Artola, P.-A., Rousseau, B.: Microscopic interpretation of a pure chemical contribution to the Soret effect. Phys. Rev. Lett. 98(12), 125901 (2007)CrossRefGoogle Scholar
  76. 76.
    Phillips, J.C., Braun, R., Wang, W., Gumbart, J., Tajkhorshid, E., Villa, E., Chipot, C., Skeel, R.D., Kale, L., Schulten, K.: Scalable molecular dynamics with NAMD. J. Comput. Chem. 26, 1781–1802 (2005)CrossRefGoogle Scholar
  77. 77.
    Hafskjold, B., Ikeshoji, T., Ratkje, S.K.: On the molecular mechanism of thermal diffusion in liquids. Mol. Phys. 80(6), 1389–1412 (1993)CrossRefGoogle Scholar
  78. 78.
    Ikeshoji, T., Hafskjold, B.: Non-equilibrium molecular dynamics calculation of heat conduction in liquid and through liquid-gas interface. Mol. Phys. 81(2), 251–261 (1994)CrossRefGoogle Scholar
  79. 79.
    Bresme, F., Armstrong, J.: Note: local thermal conductivities from boundary driven non-equilibrium molecular dynamics simulations. J. Chem. Phys. 140(1), 016102 (2014) Google Scholar
  80. 80.
    Römer, F., Wang, Z., Wiegand, S., Bresme, F.: Alkali halide solutions under thermal gradients: Soret coefficients and heat transfer mechanisms. J. Phys. Chem. B 117(27), 8209–8222 (2013)CrossRefGoogle Scholar
  81. 81.
    Römer, F., Bresme, F.: Heat conduction and thermomolecular orientation in diatomic fluids: a non-equilibrium molecular dynamics study. Mol. Sim. 38(14–15), 1198–1208 (2012)CrossRefGoogle Scholar
  82. 82.
    Armstrong, J., Bresme, F.: Water polarization induced by thermal gradients: the extended simple point charge model (SPC/E). J. Chem. Phys. 139(1), 014504 (2013)CrossRefGoogle Scholar
  83. 83.
    Foloppe, N., MacKerrell Jr, A.D.: All-atom empirical force field for nucleic acids: I. Parameter optimization based on small molecule and condensed phase macromolecular target data. J. Comput. Chem. 21, 86–104 (2000)CrossRefGoogle Scholar
  84. 84.
    Yoo, J., Aksimentiev, A.: Improved parametrization of Li\(^+\), Na\(^+\), K\(^+\), and Mg\(^{2+}\) ions for all-atom molecular dynamics simulations of nucleic acid systems. J. Phys. Chem. Lett. 3(1), 45–50 (2012)CrossRefGoogle Scholar
  85. 85.
    Miyamoto, S., Kollman, P.A.: SETTLE: an analytical version of the SHAKE and RATTLE algorithm for rigid water molecules. J. Comput. Chem. 13(8), 952–962 (1992)CrossRefGoogle Scholar
  86. 86.
    Andersen, H.C.: RATTLE: a “velocity” version of the SHAKE algorithm for molecular dynamics calculations. J. Comput. Phys. 52(1), 24–34 (1983)MATHCrossRefGoogle Scholar
  87. 87.
    Skeel, R.D., Hardy, D.J., Phillips, J.C.: Correcting mesh-based force calculations to conserve both energy and momentum in molecular dynamics simulations. J. Comput. Phys. 225(1), 1–5 (2007)Google Scholar
  88. 88.
    Batcho, P.F., Case, D.A., Schlick, T.: Optimized particle-mesh Ewald/multiple-time step integration for molecular dynamics simulations. J. Chem. Phys. 115(9), 4003–4018 (2001) Google Scholar
  89. 89.
    Wells, D.B., Abramkina, V., Aksimentiev, A.: Exploring transmembrane transport through \(\alpha \)-hemolysin with grid-steered molecular dynamics. J. Chem. Phys. 127, 125101 (2007)Google Scholar
  90. 90.
    Koopman, E.A., Lowe, C.P.: Advantages of a Lowe-Andersen thermostat in molecular dynamics simulations. J. Chem. Phys. 124, 204103 (2006)CrossRefGoogle Scholar
  91. 91.
    Joung, I.S., Cheatham, T.E.: Determination of alkali and halide monovalent ion parameters for use in explicitly solvated biomolecular simulations. J. Phys. Chem. B 112(30), 9020–9041 (2008)CrossRefGoogle Scholar
  92. 92.
    Hart, K., Foloppe, N., Baker, C.M., Denning, E.J., Nilsson, L., MacKerell, A.D.: Optimization of the CHARMM additive force field for DNA: improved treatment of the BI/BII conformational equilibrium. J. Chem. Theory Comput. 8(1), 348–362 (2012)CrossRefGoogle Scholar
  93. 93.
    van Dijk, M., Bonvin, A.M.J.J.: 3D-DART: a DNA structure modelling server. Nucleic Acids Res. 37, W235–W239 (2009)CrossRefGoogle Scholar
  94. 94.
    Humphrey, W., Dalke, A., Schulten, K.: VMD–visual molecular dynamics. J. Mol. Graphics 14, 33–38 (1996)CrossRefGoogle Scholar
  95. 95.
    Maffeo, C., Ngo, T.T.M., Ha, T., Aksimentiev, A.: A coarse-grained model of unstructured single-stranded DNA derived from atomistic simulation and single-molecule experiment. J. Chem. Theory Comput. (2014). doi:10.1021/ct500193u
  96. 96.
    Pryor, R.W.: Multiphysics Modeling Using COMSOL 4: A First Principles Approach. Mercury Learning Series. Mercury Learning and Information, Dulles, VA (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Maxim Belkin
    • 1
  • Shu-Han Chao
    • 2
  • Gino Giannetti
    • 2
  • Aleksei Aksimentiev
    • 2
  1. 1.Beckman InstituteUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of PhysicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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