Origin of the Electrophoretic Force on DNA in Nanopores. Biological and Soft Systems - Cavendish Laboratory

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1 Origin of the Electrophoretic Force on DNA in Nanopores Ulrich F. Keyser Biological and Soft Systems - Cavendish Laboratory

2 Acknowledgements Delft Cees Dekker, Nynke H. Dekker, Serge G. Lemay R. Smeets, S. Van Dorp, D. Krapf, B. Koeleman, M. Y. Wu, H. Zandbergen Leipzig Friedrich Kremer C. Gutsche, G. Stober, M. Salomo, J. H. Peters Cambridge L. Steinbock, O. Otto, C. Chimerel, L. Hild DFG Forschergruppe 877

3 Outline Early experiments revisited (Nature Physics (2006)) Force on charged wall in solution Force on DNA in nanopores: numerical modeling and new experiments

4 What is the force on DNA in a nanopore? Bare double stranded DNA has 2e - /bp at ph= in aqueous solution Counter ions in salt solution lead to part screening of charge Literature: wide spread values for effective charge of DNA : <0.1 to 1e - /bp Many major publications claim to have measured THE effective charge of DNA ~2.2 nm See e.g. Schellman et al. Biopolymers 16, 1415 (1977) -- Manning Q Rev Biophys 11, 179 (1978) -- Laue J Pharm Sci 85, 1331 (1996) -- Long Phys Rev Lett 76, 3858 (1996) -- Gurrieri Proc Natl Acad Sci USA 96, 453 (1999) -- Stellwagen Biophys J 84, 1855 (2003) -- Smeets et al. Nano Letters 6 (2006)

5 Molecular Coulter Counters: Nanopores A nanopore is a small hole with diameter <20 nm Electrical field in salt solutions is confined nanopore is a spatial filter Possible applications for nanopores: Single molecule detectors Label-free detection Analysis of biopolymers Lab-on-a-chip Model systems for biological pores Current (pa) Since 1996 Kasianowicz, Branton, Bayley, Deamer, Akeson, Meller

Single DNA Translocations Type 2 Type 21 Several

in a few minutes DNA conformation and length can be

Golovchenko et al.(2003), Storm et al.(2005), R. M.

6 Single DNA Translocations Type 2 Type 21 Several 1000 single molecule measurements can be conducted in a few minutes DNA conformation and length can be detected We extracted using a simple model Type 1 Golovchenko et al.(2003), Storm et al.(2005), R. M. M. Smeets, U. F. Keyser et al., Nano Letters 6, 89 (2006)

7 Variation of Ionic Strength Effective DNA charge 500 mm KCl 100 mm KCl

8 Extract Effective DNA-Charge DNA charge appears as constant at 0.58(±0.02) e - /bp Contradiction to measurements by gel electrophoresis π G d ( µ + µ ) n 2 ~ DNA K Cl KCl 4 ( ) G q q + ~ µ K l, DNA µ K l, DNA ql, DNA = 0.58 ± 0.02 e bp 71% R. M. M. Smeets, U. F. Keyser et al., Nano Letters 6, 89 (2006)

9 Effective Charge in Nanopore leads to a Force Potential drops over nanopore Force on DNA F = F q eff = ( q = V E( z) dz ( q / a ) E ( z ) dz eff eff / a) V effectivecharge/bp In this model: gradient effective charge

Optical tweezers allow to adjust translocation speed,

10 Directly Measure Force on DNA in a Nanopore Combine optical tweezers with nanopores and current detection Optical tweezers allow to adjust translocation speed, force and position U. F. Keyser et al. Nature Physics (2006)

11 Optical Tweezers and Nanopores (1) (2) A bead coated with DNA above a biased nanopore When the DNA enters the pore: (1) the current changes (2) the bead position changes

Measurements 40 30 DNA in Pore Controlled insertion of DNA strands one by one Time (m ms) 20 10 0 3.55 3.50 0-0.1-0.

12 Measurements DNA in Pore Controlled insertion of DNA strands one by one Time (m ms) Current (na) Position (µm) F = k ( Z 1 Z0) Exact number of DNA in the nanopore is known from ionic current measurement U. F. Keyser et al. Nature Physics 2, 473 (2006)

13 Pull DNA Out of the Nanopore Pull λ-dna (48.5 kb) out of the nanopore DNA is pulled at 30 nm/s five orders of magnitude slowed down U. F. Keyser et al. Nature Physics 2, 473 (2006)

14 Force on DNA Linear force-voltage characteristic Force does not depend on distance nanopore-trap Extract the gradient and vary salt concentration Effective charge? U. F. Keyser et al. Nature Physics 2, 473 (2006)

15 Salt Dependence of Force Slope (pn/mv) Nanopore diameter around ~10 nm KCl concentration (M) Force is constant as ionic strength is varied From literature force is expected to decrease with increasing salt concentration Force/voltage conversion 0.23±0.02 pn/mv Effective charge~0.5 e - /bp BUT See e.g. Manning Q Rev Biophys 11, (1978), Laue et al. J.Pharm. Sci. 85, (1996), Long, Viovy, and Ajdari Biopolymers (1996), Keyser et al. Nature Physics 2, 473 (2006)

16 Hydrodynamics Should Matter Hydrodynamic interactions matter

17 What is the force on a charged wall in solution?

18 Poisson Boltzmann equation describes screening Distribution of ions Boltzmann distributed When we have n ( x) n e φ ± = 0 eφ ( x) / kt 1 Taylor expansion yields = ± 0 e ( x) / kt ( φ ) n ( x) n 1 e ( x) / kt Calculate f (x) self consistently with the Poisson eq. 2 φ( r) = ρ( r) / ε w [ n ] ( r) = e n ( r) ( r) ρ +

19 Poisson Boltzmann equation describes screening This yields a simple differential equation φ( ) ktε φ( ) = φ( x) = dx 2e n λ 2 d x w x And we have the Debye screening length λ ( ktε / 2 e n ) w Solution for the differential equation Boundary conditions: 0 2 1/ 2 φ( x) = Ae + Be 0 x / λ + x / λ dφ( x) σ dφ( x) = ; = 0 dx ε dx x= 0 w x= d

20 Poisson Boltzmann equation describes screening This yields the solution for f(x) σλ φ( x) = ε w Assuming that d>>λ we get φ σλ e e e x / λ 2 d / λ x / λ 1+ e 2 d / λ / ( ) x λ x = e ( d λ) ε w σ n ( x) = n e ± 0 2eλ x / λ Uncharged wall does not influence the screening layer!

21 Potential for a slightly charged wall E d E 2d 20 mm KCl σ = C/m 2 Uncharged wall does not influence the screening layer!

22 Ion distribution for a slightly charged wall E d E 2d 20 mm KCl σ = C/m 2 Uncharged wall does not influence the screening layer!

23 Electroosmotic flow along charged wall Excess of counterions near surface leads to electroosmotic flow 2 d v ( x) ρ( x) E z dx 2 + = η ρ(x)e force exerted by electric field E, η viscosity of water This leads to velocity of water v(x) assuming no-slip boundaries Eσλ v( x) 1 η 0 x / λ = e x d E d E 2d Uncharged wall DOES influence electroosmotic flow!

24 Electroosmotic flow along charged wall E d E 2d 20 mm KCl σ = C/m 2

25 Which forces do we have to take into account? Bare force F bare is just product of area A, charge density and electric field E Fbare Aσ E The drag force F drag exerted by the flowing liquid x= 0 The force required to hold the charged wall stationary is thus = dv( x) λ λ Fdrag = Aη = AEσ 1 = 1 F dx d d ( ) λ F = F = F + F = AEσ d mech elec bare drag bare

26 Part of the force goes to the uncharged wall F mech depends on the distance d between the walls

27 High charge densities For high charge densities linearized PB does not work: ( ) 2en ( ) = sinh 2 dx ε 2 d φ x 0 eφ x w kt With two infinite walls can still be solved: φ( x) = 2kT 1+ γ e ln e 1 γ e Introducing the Gouy-Chapman length λgc = 2 ktε w / e σ x / λ x / λ GC ( 2 2 ) 1/ 2 GC γ = λ / λ λ / λ

28 Gouy-Chapman solution of PB equation σ (C/m 2 ) DNA 0.1 σ ~0.16C/m

29 Lots of interest recently Analytical calculations S. Ghosal Phys. Rev. E 74, (2006) Phys. Rev. E 76, (2007) Phys. Rev. Lett. 98, (2007) Molecular dynamics simulations A. Aksimentiev et al. Phys. Rev. E (2008)

30 PB in cylindrical coordinates nanopore with DNA Electrostatic potential Φ and distribution of ions n ± : with as natural potential Boundary conditions: Insulating nanopore walls (uncharged) Simplification: access resistance is neglected Only possible to solve numerically on DNA surface

31 In cylindrical coordinates nanopore with DNA PB can be solved analytically only by linearizing again Combining Poisson Boltzmann and Stokes equation yields: Potential Φ(a) on DNA surface, Φ(R) Nanopore wall S. Ghosal PRE 76, (2007) Logarithmic dependence of F mech on nanopore radius R slow variation as function of R

32 Finite Element Calculation Combining Poisson Boltzmann and Stokes Main result: Force on DNA depends on pore diameter Change in pore diameter by factor 10 increases drag by a factor of two

33 Hydrodynamics Should Matter Here! Test hydrodynamic interactions by increasing nanopore diameter S. van Dorp, U. F. Keyser et al. Nature Physics (2009)

robust, ph, solvents, no control on atomic level (yet)

34 Solid-State Nanopores Top-Down (Nanotechnology) 20 nm Solid-State Nanopores SiN 20 nm diameter: variable very robust, ph, solvents, no control on atomic level (yet) Golovchenko Group (2001) Dekker Group (2003) Timp Group (2004)... and many more now

35 Increase Nanopore Diameter relative DNA area ~ 1:25 relative DNA area ~ 1: nm 80 nm DNA Detection of a single DNA molecule still possible? YES

36 applied voltage (mv) DNA in a D=80 nm Nanopore cu urrent (na) time (s) Nanopore diameter ~80 nm Salt concentration M KCl current (na) data 20 point average time (s)

11 pn/mv 11 pn/mv 2 in for rce (pn) 8 4 1 in Force on

37 Force Measurements pn/mv 0.24 pn/mv 0.11 pn/mv 11 pn/mv 2 in for rce (pn) in Force on two DNA strands is doubled as expected DNA strands do not interact in large nanopores voltage (mv)

Change in Conductance G Nanopore diameter ~80 nm - Salt concentration 0.

38 Change in Conductance G Nanopore diameter ~80 nm - Salt concentration M KCl Usually 100 events are measured S. van Dorp et al. Nature Physics 5, 347 (2009)

39 Force Dependence on Nanopore Radius Force is proportional to voltage as expected For larger nanopore force is roughly halved S. van Dorp, U. F. Keyser et al. Nature Physics (2009)

40 Comparison: Model Data Force depends on nanopore radius R Data can be explained by numerical model solving full Poisson Boltzmann and Navier-Stokes equations Hydrodynamics matters S. van Dorp, U. F. Keyser et al. Nature Physics (2009)

41 Explanation: Newton s Third Law

42 Summary Electrophoretic forces are due to electrical forces AND hydrodynamic interactions Effective charge of DNA DEPENDS on the experimental conditions and the model used to extract it

Origin of the Electrophoretic Force on DNA in a Nanopore

Origin of the Electrophoretic Force on DNA in a Nanopore Stijn van Dorp 1 Ulrich F. Keyser 2, *Nynke H. Dekker 1, Cees Dekker 1, Serge G. Lemay 1 1 Kavli Institut of Nanoscience, Delft University of Technology,