Abstract
A theoretical model for the translocation process of biomacromolecule is developed based on the self-consistent field theory (SCFT), where the biomacromolecule is regarded as a self-avoiding polymer chain actuated by the external potential. In this theoretical model, the external potential, the Coulomb electrostatic potential of the charged ions (the electrolyte effect), and the attractive interaction between the polymer and the nanopore (the excluded volume effect) are all considered, which have effects on the free energy landscape and conformation entropy during the translocation stage. The result shows that the entropy barrier of the polymer in the solution with high valence electrolyte is much larger than that with low valence electrolyte under the same condition, leading to that the translocation time of the DNA molecules in the solution increases when the valence electrolyte increases. In addition, the attractive interaction between the polymer and the nanopore increases the free energy of the polymer, which means that the probability of the translocation through the nanopore increases. The average translocation time decreases when the excluded volume effect parameter increases. The electrolyte effect can prolong the average translocation time. The simulation results agree well with the available experimental results.
Similar content being viewed by others
Abbreviations
- ω :
-
excluded volume effect parameter, m3
- l 0 :
-
length of the nanopore, m
- N :
-
Kuhn segment total number
- b :
-
Kuhn segment size, m
- ε :
-
average interaction potential between the nanopore and one chain segment, J
- ∆V 1,2 :
-
electric potential difference between two spheres, V
- E 1 :
-
electric field intensity, V·m−1
- R 1 :
-
radius of the donor sphere, m
- R 2 :
-
radius of the recipient sphere, m
- E 2 :
-
electric field intensity of the electric double layer, V·m−1
- u r :
-
velocity of the fluid, m·s−1
- r :
-
radial coordinate in the spherical coordinate system, m
- η :
-
viscosity of the Newtonian fluid, N·g·s·m−2
- ρ e(r):
-
charge density of the ions, C·m−3
- R :
-
radius of the sphere, m
- ϕ :
-
electric potential of the electric double layer, V
- σ w :
-
surface charge densities of the sphere wall, C·m−2
- ε f :
-
electrolyte permittivity, C2·m−1·J−1
- v :
-
velocity of the ions, m·s−1
- z i :
-
electro valence of the i-type ions, dimensionless
- f ieq (r, s):
-
equilibrium distribution function of the i-type ions, s3·m−6
- f M :
-
Maxwell distribution function, s3·m−6
- e :
-
electron charge, C
- t :
-
time, s
- m i :
-
mass of the i-type ions, kg
- f i :
-
distribution function of the i-type ions, s3·m−6
- A :
-
drift coefficient of the ion in the velocity space, m·s−2
- B :
-
diffusion coefficient of the ion in the velocity space, m2·s−3
- k 1 :
-
constant, m6·s−5
- k 2 :
-
constant, s−1
- v 1 :
-
velocity of the ions, m·s−1
- k 3 :
-
constant, m6·s−4
- R i :
-
radius of the i-type ions, m
- k B :
-
Boltzmann constant, J·K−1
- G(r, r 1, s):
-
probability density function, m−3
- s :
-
number of the chain segment, dimensionless
- ϕ(r):
-
external potential, J
- F(r, s):
-
probability distribution of the macromolecular, dimensionless
- T :
-
absolute temperature, K
- U(r):
-
external potential, J
- ρ(r):
-
density of the chain segment, m−3
- k 4 :
-
constant
- z D :
-
electro valence of the polymer chain, dimensionless
- F T(r, s):
-
probability distribution of the tethered polymer chain, dimensionless
- F a :
-
free energy of the state of Fig. 2(a), J
- S a :
-
entropy of the state of Fig. 2(a), J
- P a :
-
probability of the state of Fig. 2(a), dimensionless
- ∆V(s):
-
electric potential difference at s, dimensionless
- F b :
-
free energy of the state of Fig. 2(b), J
- S b :
-
entropy of the state of Fig. 2(b), J
- P b :
-
probability of the state of Fig. 2(b), dimensionless
- M :
-
length of the nanopore, dimensionless
- P T :
-
probability of the tethered polymer chain, dimensionless
- F (b→c) :
-
free energy of the state from Fig. 2(b) to Fig. 2(c), J
- S (b→c) :
-
entropy of the state from Fig. 2(b) to Fig. 2(c), J
- F (c→d) :
-
free energy of the state from Fig. 2(c) to Fig. 2(d), J
- S (c→d) :
-
entropy of the state from Fig. 2(c) to Fig. 2(d), J
- F (d→e) :
-
free energy of the state from Fig. 2(e) to Fig. 2(e), J
- S (d→e) :
-
entropy of the state from Fig. 2(d) to Fig. 2(e), J
- 〈τ 1〉:
-
average translocation time from Fig. 2(b) to Fig. 2(c), s
- 〈τ 2〉:
-
average translocation time from Fig. 2(c) to Fig. 2(d), s
- 〈τ 3〉:
-
average translocation time from Fig. 2(d) to Fig. 2(e), s
References
Branton, D. and Deamer, D. W. Membrane Structure, Springer-Verlag, Berlin, 1–70 (1972)
Akeson, M., Branton, D., Kasianowicz, J. J., Brandin, E., and Deamer, D. W, Microsecond time-scale discrimination among polycytidylic acid, polyadenylic acid, and polyuridylic acid as homopolymers or as segments within single RNA molecules. Microsecond time-scale discrimination among polycytidylic acid, polyadenylic acid, and polyuridylic acid as homopolymers or as segments within single RNA molecules 77, 3227–3233 (1999)
Kasianowicz, J. J., Brandin, E., Branton, D., and Deamer, D. W, Characterization of individual polynucleotide molecules using a membrane channel. Characterization of individual polynucleotide molecules using a membrane channel 93, 13770–13773 (1996)
Henrickson, S. E., Misakian, M., Robertson, B., and Kasianowicz, J. J, Driven DNA transport into an asymmetric nanometer-scale pore. Driven DNA transport into an asymmetric nanometer-scale pore 85, 3057 (2000)
Sung, W. and Park, P. J, Polymer translocation through a pore in a membrane. Polymer translocation through a pore in a membrane 77, 783 (1996)
Muthukumar, M, Translocation of a confined polymer through a hole. Translocation of a confined polymer through a hole 86, 3188 (2001)
Chuang, J., Kantor, Y., and Kardar, M, Anomalous dynamics of translocation. Anomalous dynamics of translocation 65, 011802 (2001)
Wong, C. T. A. and Muthukumar, M, Polymer translocation through a cylindrical channel. Polymer translocation through a cylindrical channel 128, 154903 (2008)
Wong, C. T. A. and Muthukumar, M, Scaling theory of polymer translocation into confined regions. Scaling theory of polymer translocation into confined regions 95, 3619–3627 (2008)
Wolterink, J. K., Barkema, G. T., and Panja, D, Passage times for unbiased polymer translocation through a narrow pore. Passage times for unbiased polymer translocation through a narrow pore 96, 208301 (2006)
Lubensky, D. K. and Nelson, D. R, Driven polymer translocation through a narrow pore. Driven polymer translocation through a narrow pore 77, 1824–1838 (1999)
Milchev, A., Binder, K., and Bhattacharya, A, Polymer translocation through a nanopore induced by adsorption: Monte Carlo simulation of a coarse-grained model. Polymer translocation through a nanopore induced by adsorption: Monte Carlo simulation of a coarse-grained model 121, 6042–6051 (2004)
Matsuyama, A., Yano, M., and Matsuda, A, Packaging-ejection phase transitions of a polymer chain: theory and Monte Carlo simulation. Packaging-ejection phase transitions of a polymer chain: theory and Monte Carlo simulation 131, 105104 (2009)
Matsuyama, A, Phase transitions of a polymer escaping from a pore. Phase transitions of a polymer escaping from a pore 17, S2847 (2005)
Gauthier, M. G. and Slater, G. W. A Monte Carlo algorithm to study polymer translocation through nanopores, I: theory and numerical approach. The Journal of Chemical Physics, 128, 065103 (2008)
Fyta, M., Melchionna, S., Succi, S., and Kaxiras, E, Hydrodynamic correlations in the translocation of a biopolymer through a nanopore: theory and multiscale simulations. Hydrodynamic correlations in the translocation of a biopolymer through a nanopore: theory and multiscale simulations 78, 036704 (2008)
Fyta, M., Melchionna, S., Succi, S., and Kaxiras, E, Hydrodynamic correlations in the translocation of a biopolymer through a nanopore: theory and multiscale simulations. Hydrodynamic correlations in the translocation of a biopolymer through a nanopore: theory and multiscale simulations 78, 036704 (2008)
Bhattacharya, A., Morrison, W. H., Luo, K., Ala-Nissila, T., Ying, S. C., Milchev, A., and Binder, K, Scaling exponents of forced polymer translocation through a nanopore. Scaling exponents of forced polymer translocation through a nanopore 29, 423–429 (2009)
Luo, K., Metzler, R., Ala-Nissila, T., and Ying, S. C, Polymer translocation out of confined environments. Polymer translocation out of confined environments 80, 021907 (2009)
Yu, W. C. and Luo, K. F, Chaperone-assisted translocation of a polymer through a nanopore. Chaperone-assisted translocation of a polymer through a nanopore 133, 13565–13570 (2011)
Alapati, S., Fernandes, D. V., and Suh, Y. K, Numerical and theoretical study on the mechanism of biopolymer translocation process through a nanopore. Numerical and theoretical study on the mechanism of biopolymer translocation process through a nanopore 135, 055103 (2011)
Kong, C. Y. andMuthukumar, M, Polymer translocation through a nanopore, II: excluded volume effect. Polymer translocation through a nanopore, II: excluded volume effect 120, 3460 (2004)
Yang, S. and Neimark, A. V, Adsorption-driven translocation of polymer chain into nanopores. Adsorption-driven translocation of polymer chain into nanopores 136, 214901 (2012)
Park, P. J. and Sung, W, Polymer translocation induced by adsorption. Polymer translocation induced by adsorption 108, 15–23 (1998)
Slonkina, E. and Kolomeisky, A. B, Polymer translocation through a long nanopore. Polymer translocation through a long nanopore 118, 7112–7118 (2003)
Mohan, A., Kolomeisky, A. B., and Pasquali, M, Effect of charge distribution on the translocation of an inhomogeneously charged polymer through a nanopore. Effect of charge distribution on the translocation of an inhomogeneously charged polymer through a nanopore 128, 125104 (2008)
Mohan, A., Kolomeisky, A. B., and Pasquali, M, Polymer translocation through pores with complex geometries. Polymer translocation through pores with complex geometries 133, 024902 (2010)
Wong, C. T. A. and Muthukumar, M, Polymer translocation through a cylindrical channel. Polymer translocation through a cylindrical channel 128, 154903 (2008)
Zhang, Y., Liu, L., Sha, J. J., Ni, Z. H., Yi, H., and Chen, Y. F, Nanopore detection of DNA molecules in magnesium chloride solutions. Nanopore detection of DNA molecules in magnesium chloride solutions 8, 1–8 (2013)
Gardiner, C. W. Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences, Springer-Verlag, Berlin (1983)
Lin, X. H., Zhang, C. B., Gu, J., Jiang, S. Y., and Yang, J. K, Poisson-Fokker-Planck model for biomolecules translocation through nanopore driven by electroosmotic flow. Poisson-Fokker-Planck model for biomolecules translocation through nanopore driven by electroosmotic flow 57, 2104–2113 (2014)
Fredrickson, G. H. The Equilibrium Theory of Inhomogeneous Polymers, Clarendon Press, Oxford, 34–97 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (No. 51375090)
Rights and permissions
About this article
Cite this article
Zhang, C., Lin, X. & Yang, H. Theoretical model of biomacromolecule through nanopore including effects of electrolyte and excluded volume. Appl. Math. Mech.-Engl. Ed. 37, 787–802 (2016). https://doi.org/10.1007/s10483-016-2082-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-016-2082-6