Electric field induced coil-stretch transition of DNA molecule

Journal of Molecular Liquids 288 (2019) 110966

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Electric field induced coil-stretch transition of DNA molecule Ayesha Sahreen, Adeel Ahmad ⁎ Departments of Mathematics, COMSATS University Islamabad, Pakistan

a r t i c l e

i n f o

Article history: Received 4 March 2019 Received in revised form 26 April 2019 Accepted 12 May 2019 Available online 21 May 2019

a b s t r a c t The dynamics of single DNA molecules in a homogeneous planar elongational electric field is investigated in the framework of the Hookean Dumbbell model. For a Gaussian, rapidly changing advecting flow the Fokker-Planck equation for time-dependent probability density function of DNA elongation is obtained. The analytical solution for probability density function of time independent problem is derived. The coil-stretch transition is studied when the elongation of the DNA molecule is very small than the maximum length. The characteristic relaxation time is calculated as a function of the Weissenberg number. It is observed that the coil-stretch transition and characteristic relaxation time of DNA molecule strongly depend on the angle of the principal axis of molecule with respect to axis of elongation. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Understanding the dynamics of DNA stretching and engineering DNA stretching processes are the main stages in the development of biological microfluidic devices [1]. Although there are many other methodologies, the most common choice to stretch DNA continuously that researchers and experimentalist prefer is to adopt the uniform flow or uniform electric field. Several experimental studies have been conducted to investigate the stretching of DNA in extensional hydrodynamic flows (like extensional flow, mixed flow or shear) and electric fields. Electric field induced DNA stretching is one of the vital phenomena that characterize the deformation of flexible polymers in a fluid flow and has many practical applications namely separation of DNA molecules by size and moving fluids or analytes inside the micro channels in microfluidics. Electric field can be used over flow field to avoid shear because electric field induced electrophoretic velocity is purely elongational or shear-free and DNA in electric field experiences pure elongational deformation. Furthermore, they have the additional advantages over pressure-driven flows at nanoscopic dimensions of avoiding high pressure-related channel failure and viscoelastic instabilities such as vortex formation [2,3]. Coil-stretch transition is one of the phenomena, predicted in 1974 for shear and hyperbolic flows [4], that describe the deformation of a polymer in a flow. It consist an increase in the polymer extension as the intensity of the velocity gradient exceeds a critical value. The related dimensionless parameter for this transition is the Weissenberg number (Wi); it is the product of the characteristic rate of deformation and the greatest relaxation time [5]. Studies that concern coil-stretch transition ⁎ Corresponding author. E-mail address: [email protected] (A. Ahmad).

https://doi.org/10.1016/j.molliq.2019.110966 0167-7322/© 2019 Elsevier B.V. All rights reserved.

mainly consider the Hookean Dumbbell model to understand the behavior of polymers in random flows. This model is a very basic model and is appropriate only for the coiled state i.e. when the extension of the molecule is much smaller than its maximum length and it can be replaced by a non-linear elastic force when the transition parameter exceeds a critical value. There are many other models which can be used for the study of coil-stretch transition and some examples are FENE dumbbell model [6], FENE-L [7], FENE-LS model [8], FENE-P model [9] and FENE-P-α model [10]. Balkovsky et al. [11] have worked on the elastic dumbbell model to show that the coil-stretch transition also happens in chaotic or random flows. The dumbbell approximation is the origin of the most common models of single polymers dynamics and viscoelastic models of dilute polymer solutions [12]. The coil-stretch transition for a planar extensional flow was predicted by De Gennes [4]. An abrupt rise in the steady-state DNA elongation was noticed near a critical value of the velocity (electric field) gradient. The aim of this work is to study the coil-stretch transition of DNA molecule in the presence of electric field. We wish to follow the approach of Celani et al. [13], where they investigated the dynamics of polymers for a Hookean Dumbbell model in a random flow. In this work, we have derived the Fokker-Planck equation for probability density function (pdf) of the elongation of the DNA to study the dynamics of DNA molecule in the planar homogeneous extensional electric field using the theorem of electrodynamics equivalence presented by Long et al. [14].

Nomenclature Wi Weissenberg number K Boltzmann constant T Temperature

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A. Sahreen, A. Ahmad / Journal of Molecular Liquids 288 (2019) 110966

R Elongation of the molecule vn Eigenvalues E(xi, t) Electric field of the bead J[P(r)] Probability current functional R0 Equilibrium size of the molecule N Normalization constant P(r, T) Non-dimensional probability density function pdf Probability density function k Stiffness of spring E Elastic energy ζ Bead drag coefficient μ Electrophoretic mobility θ Angle of principal axis w.r.t. the axis of elongation τ Relaxation time η(t) Vectorial white noise ϵ Strain rate

The bead's size is considered to be very small so that their dynamics can be affected by thermal noise, which can be seen from the Gaussian pffiffiffiffiffiffiffiffiffiffiffi process 2ζKT ξi such that k

l

bξi ðt ÞN ¼ 0; bξi ðt Þξ j ðt 0 ÞN ¼ δij δkl δðt−t 0 Þ;

ð1Þ

In Eq. (1), K represents the Boltzmann constant and T is used for temperature. The equilibrium size of the dumbbell is not zero owing pffiffiffiffiffiffiffiffiffiffiffi to molecular noise, but it can be expressed as Ro ¼ KT=k by taking the DNA's elastic energy as E = kR2/2, and its equilibrium value at temperature T as E = KT/2. By considering the elastic force, the electrophoretic stretching force and the thermal noise, the equations governing the dynamics of the beads take the form: mx€1 ¼ −kðx1 −x2 Þ−ζ ðx_1 −μEðx1 ; t ÞÞ þ

pffiffiffiffiffiffiffiffiffiffiffi 2ζKT ξ1

ð2Þ

2. Mathematical model

mx€2 ¼ −kðx2 −x1 Þ−ζ ðx_1 −μEðx2 ; t ÞÞ þ

pffiffiffiffiffiffiffiffiffiffiffi 2ζKT ξ2

ð3Þ

We consider a Brownian dumbbell model, a simplification of Rouse's model, to study the dynamics of electric field induced DNA stretching. DNA is considered as two beads that are charged and joined through an elastic Hookean spring, appropriate when the elongation of the molecule is very small compared to its maximum length, representing the elasticity of DNA molecule. We take into account only the fundamental oscillation mode. Two beads are connected having radius l from their centers with a linear spring having stiffness k [5]. The vectors x1 and x2 represent the positions of the two beads where they represent the ends of the DNA. From the theorem of electrohydrodynamic equivalence presented by Long et al. [14], the electrophoretic stretching force in a bead is −ζ ðx_i −μEðxi ; tÞÞ i.e. drag force exerted on the bead by a hydrodynamic flow in which the flow velocity is the same with the bead electrophoretic velocity in the electric field, where x_i represents the velocity of the bead i and E(xi, t) is for the electric field at the position of the i-th bead. Further, ζ is used for the bead drag coefficient and μ is the bead's electrophoretic mobility (Fig. 1).

In the frame of reference of the center of mass, the configuration of a dumbbell is specified by the separation vector between the beads R, R = x2 − x1, which represents the extension of the DNA molecule. Neglecting the inertial effects, we derive a stochastic ordinary differential equation for the dumbbell end-to-end distance: R R_ ¼ μ ðEðx2 ; t Þ−Eðx1 ; t ÞÞ− þ τ

sffiffiffiffiffiffiffiffiffiffiffiffi 2R2o ξ; τ

ð4Þ

pffiffiffi where τ = ζ/2k is the time of relaxation and ξ ¼ ðξ2 −ξ1 Þ= 2 has the same properties as ξ1 and ξ2. The extension of the DNA in physical applications, however large it is, never approaches the viscous scale of the turbulent flow, where dissipation and advection balance [15]. Below this scale, viscosity reduces the velocity fluctuations and the flow becomes smooth. Therefore, the DNA always moves in a regular velocity field, which at the scale R can be estimated by a uniform gradient flow [16]: vðx2 ; t Þ ¼ vðx1 ; t Þ þ ðx2 −x1 Þ  ▽vðt Þ

ð5Þ

For the small electric field (E b 103 V/cm), drift velocity is either in the direction parallel or anti-parallel to the direction of the electric field i.e. v = μE, where μ is the bead's electrophoretic mobility. Therefore, Eq. (5) may be written as: Eðx2 ; t Þ ¼ Eðx1 ; t Þ þ ðx2 −x1 Þ  ▽Eðt Þ Thus, the stochastic ordinary differential Eq. (4) for R gets the form: R R_ ¼ R  ∇μE− þ τ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2R20 ηðt Þ; τ

ð6Þ

where η(t) is vectorial white noise. For the planar homogeneous extensional electric field, the first term on the right hand side of Eq. (6) can be expressed as [17]: R  ∇μE ¼

Fig. 1. Physical interpretation of elastic dumbbell model. The DNA is demonstrated by two beads connected by a linear spring. Each bead experiences the hydrodynamic drag force and thermal noise and FS represents the spring force.

  ∂ 1 2 εR cos ð2θÞ ; ∂R 2

ð7Þ

where θ is the angle of the dumbbell principal axis with respect to the axis of elongation (−π/2 b θ b π/2), ε is the strain rate and R = ∣ R∣. Eq. (6) governs the dynamics of the elongation of the DNA transported by the nonhomogenous flow. The electric field gradient elongates the molecule, which behaves elastically and then tries to return to the equilibrium spherical shape.

A. Sahreen, A. Ahmad / Journal of Molecular Liquids 288 (2019) 110966

3

3. Fokker-Planck equation Fokker-Planck equation is a partial differential equation that describes the diffusion processes. It describes the time evolution of the pdf of the velocity of a particle under the influence of random and drag forces. In this section, our aim is to derive the pdf of the elongation p(R, t) that follows the Fokker-Planck equation. The Fokker-Planck equation related to Eq. (6) takes the form:  ∂t p þ ∂Ri

   1 2 R2 2 εR cos ð2θÞ−f ðRÞRi p ¼ 0 ∂Ri p: 2 τ

∂Ri

ð8Þ

For the coil-stretch transition, we consider the pdf for the norm of R, P(R,t) = ∫ P(R,t)Rd−1dΩ, where dΩ denotes the integration over the angular variables and d is the dimension of the flow. Taking T = t/τ, r = R/ R0, the one dimensional Fokker-Planck equation for P(r,t) may be written as: 2

∂T P ¼ −∂r ½C 1 ðr ÞP  þ ∂r ½C 2 ðr ÞP ;

ð9Þ

where C 1 ðr Þ ¼ Wið1−dÞ cos ð2θÞ r=2−r þ ðd−1Þ=r; C 2 ðr Þ ¼ −Wi cos ð2θÞ r 2 =2 þ 1

ð10Þ

and Wi = ετ is the Weissenberg number. 4. Results and discussion As the coefficients of Eq. (9) are independent of time because of the stationarity of advecting velocity field, the pdf P(r,T) can be reduced to stationary pdf P(r). We supplement the associated steady equation 2

−∂r ½C 1 ðrÞP  þ ∂r ½C 2 ðr ÞP  ¼ 0;

ð11Þ

with the reflective boundary conditions (the probability does not flow through the boundaries of the domain of definition) that are generally used for the Fokker-Planck equation's solution on an infinite domain: J ð0Þ ¼ lim J ðr Þ ¼ 0; r→∞

ð12Þ

where J ½P ðr Þ ¼ C 1 ðr ÞP ðr Þ−∂r ½C 2 ðr ÞP ðrÞ

Fig. 2. Stationary pdf of dimensionless length r for θ = 3π/8 and Wi = 0.01 (Solid line), Wi = 0.5 (Dashed line) and Wi = 0.86 (Dot-dashed line).

pffiffiffi of any involved parameter. The maximum of all the pdf occurs at r ¼ 2 irrespective of values of any parameter. The probability of molecules of pffiffiffi length r ¼ 2 is maximum for all Wi. From Eq. (10), it can be seen that the coefficients C1(r) and C2(r) of Fokker-Planck equation become independent of Wi for θ = π/4. Thus, the solution of Eq. (9), i.e. the pdf for P (r) becomes independent of Wi and there is no variation in P(r) for different Wi. Also, it is observed that for θ N π/4 the probability of highly elongated molecules increases with an increase in Wi i.e. the elongational forces that stretches the DNA become dominant for high value of the Wi and entropic forces becomes weak due to which the molecules get stretched and opposite behavior is observed for θ b π/4, but with very small difference. This implies that the position of the dumbbell in the flow has an important effecton the stretching of the molecule. In Fig. 4, it is observed that the probability of molecules pffiffiffi with length r ¼ 2 is higher for small Wi when the inclination of

ð13Þ

is the probability current functional. Under the reflective boundary conditions, the stationary solution of Eq. (9) is (e.g. Ref. [18]) Z P ðr Þ∝ exp

 dζ C 1 ðζ Þ=C 2 ðζ Þ =C 2 ðr Þ;

r

r1

ð14Þ

where r1 is fixed by the normalization condition. Placing the values of C1 (r) and C2(r) in above equation, we obtain  P ðr Þ ¼ Nrd−1

h r2 −2 ; 1þh

ð15Þ

1 −1 and N is the normalization constant. For Wi cos ð2θÞ small extensions compared to the equilibrium size i.e. r ≈ 1 the stationary pdf scales as rd−1. For long elongations, P(r) is proportional to rd+2h−1. In Figs. 2 and 3, the stationary pdf for dimensionless length r is plotted against Wi for different values of θ, where θ is the angle of the dumbbell principal axis with respect to the axis of elongation as shown in pffiffiffi pffiffiffi Fig. 1. The stationary points of P(r) are − 2; 0; 2, and are independent where h ¼

Fig. 3. Stationary pdf of dimensionless length r for θ = 2π/9 and Wi = 0.01 (Dot-dashed line), Wi = 0.1 (Solid line) and Wi = 0.5 (Dashed line).

4

A. Sahreen, A. Ahmad / Journal of Molecular Liquids 288 (2019) 110966

molecule is greater than π/4 whereas for inclination less than π/4, the pffiffiffi probability of molecules with length 2 is less for low Wi. Now we obtain the solution of the Fokker-Planck equation that depends on time by the method of separation of variables. The solution is written in the form p(r, T) = pv(r)e−Tvn, then Eq. (9) becomes an eigenvalue problem. LFP pv ¼ −vn pv

ð16Þ

Table 1 Rescaled relaxation time for different Wi. Wi

T for θ = π/2

T for θ = π/4

T for θ = π/6

0.01 0.03 0.05 0.07 0.09 0.1

0.390430667 0.386558883 0.382515593 0.378360474 0.374139953 0.372016775

0.315591408 0.315591408 0.315591408 0.315591408 0.315591408 0.315591408

0.393176622 0.394910501 0.396551089 0.398082758 0.399487859 0.400136821

where LFP is the differential operator for the Fokker-Planck equation. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 − 4 þ Wi ðd−2Þ cos ð2θÞ þ 4Wiðd þ 2vn −2Þ cos ð2θÞ β¼ Wicos ð2θÞ 4

2

∂ ∂ LFP ≡ − ½Aðr Þ þ 2 ½Bðr Þ ∂r ∂r

−2 þ ð2 þ dÞWicos ð2θÞ þ Wicos ð2θÞ

The solution of Eq. (9) can be written now as an expansion in terms of pv where the initial condition fixes the coefficient. The eigenvalue equation related to Fokker-Planck Eq. (9) is given by 

   2 −1 d p −1 d−1 dp r2 þ 1 ð5−dÞr þ r− þ 2 2ðh þ 1Þ 2ðh þ 1Þ r dr dr   −1 d−1 ð3−dÞ þ 1 þ 2 þ vn p ¼ 0 þ 2ðh þ 1Þ r

ð17Þ

The above equation can be solved explicitly. Hence, the solution of Eq. (17) fulfilling the first reflecting boundary condition in Eq. (12) gives C1 = 0 and takes the form:  Wicos ð2θÞ 2 r Þ pv ðr Þ ¼ C 2 r d−1 F α; β; γ; 2

ð18Þ

Wicos ð2θÞ 2 r Þ denotes the hypergeometric functions 2 with parameters:

where Fðα; β; γ;

1 α¼ 4

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 þ Wi2 ðd−2Þ cos ð2θÞ2 þ 4Wiðd þ 2vn −2Þcos ð2θÞ Wicos ð2θÞ −2 þ ð2 þ dÞWicos ð2θÞ þ Wicos ð2θÞ

d and γ ¼ . 2 The values of vn, the eigenvalues, are found such that the solution in Eq. (18) satisfy the second condition for large r. The relaxation time is 1 thus defined as t rel ¼ τ, where T = trel/τ is the rescaled time [19]. v1 The rescaled time for different Wi is tabulated in Table 1. When the DNA is placed in a flow, it gets stretched due to the elongational forces. Here the elongational forces are due to the electric field gradient which is not that large hence the elastic forces become dominant over the viscous forces. In Table 1, relaxation time for different values of θ against Wi is computed. It can be observed from the table that when θ N π/4, relaxation time becomes a decreasing function of Wi which implies that the entropic force is stronger than the viscous forces due to which the relaxation time decreases for increasing wi and opposite behavior is noticed when the inclination is less than π/4 i.e. relaxation time becomes an increasing function of Wi. Also, when θ = π/4 there is no effect of Wi on the solution of the Fokker-Planck equation. 5. Conclusion The aim of this paper was to demonstrate and convey the idea that how a DNA molecule gets stretched in a turbulent flow in the presence of a homogeneous planar electric field. We investigated the statistics of DNA extension in the Hookean Dumbbell model and the flow was chosen to have the Batchelor-Kraichnan statistics. The analytical expression for the pdf of elongation of the DNA that obeys the Fokker-Planck equation was derived. It should be noted that pdf strongly depends on the angle of the dumbbell principal axis with respect to the axis of elongation. For the stationary pdf of the elongation of the molecule, we find that the maximum of all the probability density function occurs at same value irrespective of values of any parameter. For θ = π/4, there is no effect of Wi on the solution of the Fokker-Planck equation. When the inclination is greater than π/4, the probability of molecules with maximum length is greater for small Wi and opposite behavior is observed when the inclination is less than π/4. Also, in presence of the electric field it is observed that the probability of highly elongated molecules increases with an increase in Wi for θ N π/4 and opposite behavior is observed for θ b π/4. Lastly, we discussed the effect of electric field on the relaxation time of molecule as a function of Wi. The relaxation time becomes an increasing function of Wi when the inclination is less than π/4 and it is a decreasing function of Wi when the inclination is greater than π/4. References

Fig. 4. Stationary pdf of dimensionless length r for Wi = 0.1 and θ = π/2 (Dashed line), θ = 3π/8 (Solid line) and θ = π/4 (Dot-dashed line).

[1] C.G. Randall, K.M. Schultz, S.P. Doyle, Methods to electrophoretically stretch DNA: microcontractions, gels, and hybrid gel-microcontraction devices, Lab Chip 6 (4) (2006) 516–525.

A. Sahreen, A. Ahmad / Journal of Molecular Liquids 288 (2019) 110966 [2] H.C. Lee, C.C. Hsieh, Stretching DNA by electric field and flow field in microfluidic devices: an experimental validation to the devices designed with computer simulations, Biomicrofluidics 7 (1) (2013), 014109. [3] A. Wakif, Z. Boulahia, R. Sehaqui, A semi-analytical analysis of electro-thermohydrodynamic stability in dielectric nanofluids using Buongiorno's mathematical model together with more realistic boundary conditions, Results Phys. 9 (2018) 1438–1454. [4] P.G. de Gennes, Coil-stretch transition of dilute flexible polymer under ultrahigh velocity gradients, J. Chem. Phys. 60 (1974) 5030. [5] A. Ahmad, D. Vincenzi, Polymer stretching in the inertial range of turbulence, Phys. Rev. E 93 (5) (2016), 052605. [6] M. Martins Afonso, D. Vincenzi, Nonlinear elastic polymers in random flow, J. Fluid Mech. 540 (2005) 99. [7] G. Lielens, P. Halin, I. Jaumain, R. Keunings, V. Legat, New closure approximations for the kinetic theory of finitely extensible dumbbells, J. Non-Newtonian Fluid Mech. 76 (1–3) (1998) 249–279. [8] G. Lielens, R. Keunings, V. Legat, The FENE-L and FENE-LS closure approximations to the kinetic theory of finitely extensible dumbbells, J. Non-Newtonian Fluid Mech. 87 (2–3) (1999) 179–196. [9] R. Keunings, On the Peterlin approximation for finitely extensible dumbbells, J. NonNewtonian Fluid Mech. 68 (1) (1997) 85–100. [10] M. Kruger, E. De Angelis, An extended FENE dumbbell model theory for concentration dependent shear-induced anisotropy in dilute polymer solutions: addenda, J. Non-Newtonian Fluid Mech. 125 (1) (2005) 87–90.

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[11] E. Balkovsky, A. Fouxon, V. Lebedev, Turbulent dynamics of polymer solutions, Phys. Rev. Lett. 84 (2000) 4765. [12] R.B. Bird, O. Hassager, R.C. Armstrong, C.F. Curtiss, Dynamics of Polymeric Liquids, vol. 2, Wiley, New York, 1987. [13] A. Celani, S. Musacchio, D. Vincenzi, Polymer transport in random flow, J. Stat. Phys. 118 (2005) 531. [14] C. Desruisseaux, D. Long, G. Drouin, W.G. Slater, Electrophoresis of composite molecular objects. 1. Relation between friction, charge, and ionic strength in free solution, Macromolecules 34 (1) (2001) 44–52. [15] J.L. Lumley, Drag reduction in turbulent flow by polymer additives, J. Polym. Sci. Macromol. Rev. 7 (1973) 263. [16] K.G. Batchelor, Small-scale variation of convected quantities like temperature in turbulent fluid part 1. General discussion and the case of small conductivity, J. Fluid Mech. 5 (1) (1959) 113–133. [17] J. Tang, W.D. Trahan, S.P. Doyle, Coil-stretch transition of DNA molecules in slitlike confinement, Macromolecules 43 (6) (2010) 3081–3089. [18] H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer, Berlin/Heidelberg/New York, 1989. [19] A. Celani, A. Puliafito, D. Vincenzi, Dynamical slowdown of polymers in laminar and random flows, Phys. Rev. Lett. 97 (2006), 118301.