Electrophoretic Mobility of Double-Stranded DNA in Polymer Solutions

May 19, 2014 - widely used in the life sciences for the size separation of uniformly charged ... 1 for DNA in polymer networks under the influence of ...
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Electrophoretic Mobility of Double-Stranded DNA in Polymer Solutions and Gels with Tuned Structures Xiang Li,† Kateryna Khairulina,† Ung-il Chung,† and Takamasa Sakai†,* †

Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan S Supporting Information *

ABSTRACT: We report a systematic experimental study on the migration behavior of double-stranded DNA (dsDNA) in polymer networks with precisely controlled network structures. The electrophoretic mobility (μ) appeared to be a power law function of the number of base pairs (n), μ ∼ n−γ, with 0.36 < γ < 1.46. The variance in γ has been commonly explained using the reptation model with constraint release or using the entropic trapping (ET) model. However, our results indicated that the μ values were expressed as products of a power law function and an exponential function of n, which differs from any of the existing models. The power law function terms corresponded to the existing models, the Rouse model or the reptation model. In polymer gels, we observed a crossover from the Rouse to the reptation model with an increase in n, while the migration behavior in polymer solutions always obeyed the Rouse model. These results revealed that the continuous change in γ was accommodated by the exponential function terms.

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into the reptation model.11 In contrast, when γ > 1, the migration behavior was explained with a different process, known as entropic trapping (ET).12 While these models provided qualitative explanations for the experimental results, a quantitative agreement with the results was lacking.7,9,10 To fully understand migration behavior in polymer networks, a systematic study based on experiments is required. However, systematic investigations have been hindered by the heterogeneity of conventional polymer gels, which results in uncontrolled structural parameters, such as the degree of strand polymerization between cross-links (Ns) and the polymer volume fraction (ϕ).13,14 To resolve this problem, we recently fabricated a homogeneous gel system (Tetra-PEG gels) through the A−B type cross-link coupling of two mutually reactive tetra-arm poly(ethylene glycol) units (Tetra-PEG prepolymers).15−17 The degree of heterogeneity was extremely lower than that of conventional polymer gels. In addition, we can precisely control ϕ and Ns by tuning the amount of prepolymer and the molecular weight of the prepolymer, respectively. In this article, we report a systematic study of the electrophoretic migration behavior of double-stranded DNA (dsDNA) in polymer networks with controlled network structures. We observed a continuous change of γ in the range 0.36 ≤ γ ≤ 1.46 by changing ϕ and Ns. Through

INTRODUCTION The dynamics of polymer chains in polymer networks are important in various applications, including gel electrophoresis, size exclusion chromatography, controlled-release drug delivery systems, and cell culture. One of the oldest applications of polymer chain dynamics is gel electrophoresis, which has been widely used in the life sciences for the size separation of uniformly charged polymer chains (e.g., DNA, proteins). During electrophoresis, the interaction between the polymer network and the charged polymer chains is the key factor in differentiating the electrophoretic mobility (μ) of the charged polymer chains according to length. However, the mechanisms governing the migration behavior of charged chains, which are extremely relevant to both applied and fundamental polymer physics, remain unclear.1,2 Southern et al. discovered the relationship μ ∼ n−γ, where γ = 1 for DNA in polymer networks under the influence of a moderate electric field, and n is the number of base pairs.3 To explain this experimental result, two different groups extended the reptation model4 to gel electrophoresis, which predicts μ ∼ n−2(1−ν), where ν is the scaling parameter (ν is 0.5 for an ideal chain and 0.6 for a real chain).5,6 The reptation model prediction for ideal chains corresponds well to the results of Southern et al. Although this good agreement strongly suggests the applicability of the reptation model to the migration of charged polymer chains in polymer networks, numerous experimental results with γ ≠ 1 have been reported.7−10 When γ < 1, the migration behavior was explained using a model that incorporates the relaxation of the polymer network © 2014 American Chemical Society

Received: March 31, 2014 Revised: May 11, 2014 Published: May 19, 2014 3582

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experimental analyses not based on existing models, we obtained, for the first time, empirical equations of μ that were products of a power law and an exponential function of n. Only two relationships were observed with respect to the power law function terms, in agreement with the Rouse and reptation models. The continuous change in γ results from the exponential function term.

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EXPERIMENTAL SECTION

Tetra-PEG gels with defined degrees of strand polymerization between cross-links (Ns: 113, 227, 454) and polymer volume fractions (ϕ: 0.034−0.081) were prepared inside fused glass capillaries with inner diameters of 75 μm, as described elsewhere.15,18−21 Linear poly(ethylene glycol) (PEG) solution with a molecular weight of 200 kDa and polymer volume fraction (ϕ: 0.034−0.081) were also prepared. dsDNA molecular markers ranging in size from 100 to 8,000 bp (BioRad) were dissolved at a final concentration of 66.6 mg/L in electrophoresis buffer (25 mM BisTris, 25 mM N-(2-acetamido)-2aminoethanesulfonic acid, 1 mM 2Na-EDTA, 11.3 mM NaOH, pH 7.0)22 that contained 1 mg/L ethidium bromide. Capillary electrophoresis was conducted under different electric field strengths (E: 25− 100 V/cm) at 25 ± 0.5 °C. The detection of dsDNA was achieved by monitoring the fluorescence intensities of dsDNA−ethidium bromide complexes at a fixed spot on the capillary with a custom-made capillary electrophoresis apparatus.23

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RESULTS AND DISCUSSION We performed capillary electrophoresis of dsDNA (100−8000 bp) in Tetra-PEG gels and PEG solutions with different Ns and ϕ. Figure 1 has the double-logarithmic plots indicating the ndependence of μ. With increasing n, μ first decreased and then became independent of n; n-dependent and n-independent migration regions were observed. The boundary of these two regions was approximately 1000 bp. We confirmed that the electric field strength did not significantly influence the value of μ in the n-dependent region (SFigure 1, Supporting Information), suggesting that the conformational change in the dsDNA caused by the electric field is negligible under this experimental condition. The crossover from n-dependent to nindependent migration qualitatively agrees with the biased reptation model with fluctuation (BRF) model,1,24 signifying the former as a reptation of unoriented chains and the latter as a reptation of oriented chains. In this article, we focus on and only discuss the n-dependent region, and refer to the “reptation of unoriented chains” as “reptation” for simplification. In the n-dependent migration region, all the existing models based on the reptation concept, including the BRF model, predict the power law relationship between μ and n for an ideal chain as μ ∼ n−γ with γ = 1. However, as shown in Figure 1 (dotted lines), γ increased from 0.36 to 1.46 with increasing ϕ or decreasing Ns. This discrepancy with the reptation models was also observed experimentally in previous studies with single-stranded DNA (ssDNA) and polystyrenesulfonate (PSS) in polyacrylamide gels7,10 and with ssDNA and dsDNA in various polymer solutions,9,25 and also in a systematic study of diffusion coefficient of polystyrenes.26 These data indicate that the simple reptation concept cannot predict the migration behavior of dsDNA in polymer networks under these experimental conditions. Figure 2 illustrates the semilogarithmic plots of μ as a function of ϕ for Tetra-PEG gels and PEG solutions (the original data are the same as in Figure 1). As shown in Figure 2, we can clearly observe linear relationships between log10μ and ϕ. The deviation observed with the relatively large dsDNA

Figure 1. Double-logarithmic plots of μ as a function of n in TetraPEG gels (a−c) with varied ϕ and Ns and in PEG solutions (d) with varied ϕ at E = 50 V/cm. The error bars were inside the size of the symbols. The dashed lines are the guides showing the steepest power law slopes.

samples in the low ϕ region may be due to the orientated migration occurring via the electric field (SFigure 1). In our previous study involving the migration of shorter dsDNA,23 we obtained the same exponential relationship between μ and ϕ. This agreement indicates that μ is an exponential function of ϕ in the range 20 ≤ n ≤ 1000, as shown below μ = μ0 exp( −Bϕ) ,

(1)

where μ0 and B are functions that depend only on n and Ns. Notably, μ0 does not stand for the mobility in free solution but the extrapolation value of μ at ϕ = 0, which is influenced by the presence of polymer gels or linear polymers. Similar exponential relationships were observed in previous studies for the electrophoretic migration of proteins and dsDNA in both gels and polymer solutions.27−29 Furthermore, the same relationship was observed in studies of diffusion coefficients in polymer solutions and gels.30 The form of eq 1 may thus be universal for the dynamics of substances in polymer networks. We estimated μ0 and B from the fit of eq 1 (dotted lines in Figure 2). Figure 3 shows the double-logarithmic plot of μ0 as a function of n. The filled inverted triangles are the data from our previous study,23 where the values of μ0 at each n were universal for the PEG solution and Tetra-PEG gels regardless of Ns. In general, μ0 decreased with increasing n. A clear difference was observed between the PEG solution and Tetra-PEG gels; 3583

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200−300 regardless of Ns. These data indicate that, although permanent cross-links are vital for this crossover, the mesh size of the polymer network that is defined by Ns, does not influence the crossover. Notably, the crossover is close to the number of base pairs np (np ≈ 150 bp) that corresponds to the persistence length of dsDNA (∼50 nm). The other parameter obtained from the fit of eq 1 (dotted lines in Figure 2) is B. Figure 4 displays the double-logarithmic

Figure 4. A double-logarithmic plot of B as a function of n. The filled symbols are the data from a previous study.23 The dotted lines demonstrate the fitting curves B = αnβ, where α and β are functions that depend only on Ns. The dashed lines indicate the guide B = αn0.3. The solid vertical line indicates np.

plot of B as a function of n. The filled symbols indicate the data from our previous study.23 B increased with increasing n and decreasing Ns. For each Ns, two different power law relationships (B = αnβ) were observed with a crossover at approximately np. Notably, β in the range n < np increased with decreasing Ns, while β in the range n > np was a universal value (∼0.3) regardless of Ns. Chen et al. investigated the migration of dsDNA in polyacrylamide solutions. Although the authors did not mention, the transition from B ∼ n0 to B ∼ n0.3 at approximately n = 200−400 bp was observed.28 This crossover may be universal for dsDNA migration in polymer networks. Through the analyses, we obtained the following empirical equations for polymer solutions:

Figure 2. Semilogarithmic plots of μ as a function of ϕ in Tetra-PEG gels (a−c) and in PEG solutions (d). dsDNA with different sizes (n: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000 bp) are displayed from top to bottom in each graph. The original data are the same as in Figure 1. The dotted lines illustrate the fitting curves of eq 1

μ = A1n−0.17 exp( −12.7n0ϕ),

n < np

(2a)

μ = A1n−0.17 exp( −2.9n0.3ϕ),

n > np

(2b)

The following equations were obtained for gels: μ = A1n−0.17 exp( −α1n β1ϕ),

n < np

μ = A 2n−0.81 exp( −α2n0.3ϕ), Figure 3. Double-logarithmic plot of μ0 as a function of n. The filled inverted triangles are the data from our previous study,23 where the values of μ0 are universal for each n in the PEG solution and TetraPEG gels regardless of Ns. The dotted lines illustrate the fitting curves μ0 ∼ n−0.17 and μ0 ∼ n−0.81. The solid vertical line indicates the number of base pairs np (np ≈ 150 bp) that corresponds to the persistence length of dsDNA.

−4

−1 −1

(2c)

n > np

(2d) −2

where A1 (=5.4 × 10 cm V s ) and A2 (=2.0 × 10 cm2 V−1 s−1) are constants and α1, α2, and β1 are functions that depend only on Ns. The values of α1 was increasing function, and β1 was decreasing function of Ns.23 The values of α2, which was decreasing function of Ns, is shown in SFigure 2. Let us compare these empirical equations with the existing models for electrophoretic mobility. Considering the equation forms, we can roughly classify the existing models into two types based on either power law function models (the Rouse and reptation models) or exponential function models (the Ogston and ET models).1,12 Each of the empirical equations

μ0 scales as μ0 ∼ n−0.17 in the PEG solution, while the scaling relationship changed from μ0 ∼ n−0.17 to μ0 ∼ n−0.81 in the Tetra-PEG gels. A crossover was observed at approximately n = 3584

2

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relationship with n (ΔS ∼ n0.3) in both the gels and polymer solutions, indicating that the origin of the entropy loss does not result from mesh-dsDNA interactions and might be due to the instantaneous conformational change of dsDNA that accompanies migration in polymer networks. From this viewpoint, a common bottleneck-like structure must exist both in the gels and the polymer solutions. The origin of the bottleneck structure might be concentration fluctuations within the polymer networks, which are commonly observed for both gels and polymer solutions16 and may cause temporarily sparse or condensed polymer regions in the networks. By assuming a cavity-bottleneck network,12 the ET model and simulation predict ΔS ∼ n1,12 the exponent of which is higher than the 0.3 that was observed in this study. This deviation from the model or the simulation is most likely because the definite boundaries assumed by the model and simulation do not exist in the real polymer networks. This ambiguous polymer network boundary may result in a weaker n-dependence of ΔS. Indeed, this discrepancy was predicted by Daoud and de Gennes, who proposed the basic concept for the ET model.34

(eqs 2a−2d) is neither a simple power law nor an exponential function but is a product of both. This observation strongly suggests that the migration of dsDNA in polymer networks involves the mechanisms of both types of models. The power law function models originate from the manner in which charged chains migrate in polymer networks.1 The Rouse model assumes that chains migrate in a freely drain manner and predicts μ ∼ n0. In contrast, the reptation model assumes that chains reptate in a narrow tube formed by the polymer network and predicts μ ∼ n−2(1−ν), where ν is the scaling parameter (ν is 0.5 for an ideal chain and 0.6 for a real chain). The exponential function models originate from the entropy loss of the chains during migration. The equations in the Ogston and ET models are derived by calculating the change in the number of states due to the geometric confinement of the network; both models are expressed as μ ∼ exp(ΔS/kB), where ΔS denotes the entropy loss, and kB represents the Boltzmann constant.12,31 We will first focus on the power law terms in eqs 2a−2d. For migration in the gels, the power law term changed from n−0.17 to n−0.81 with a crossover at approximately np. The term n−0.81 clearly corresponds to the reptation model, especially to the BRF model in tight gels,32 which predicts μ ∼ n−0.8 for real chains (ν = 0.6) and that the mobility is independent of the mesh size. This correspondence is reasonable, because the mesh sizes of Tetra-PEG gels (several nm), which is calculated as Gaussian chains with polymerization degree of Ns, is much smaller than the persistence length of dsDNA (∼50 nm). In contrast, the term n−0.17 was close to the prediction of the Rouse model, although the finite difference in the power was observed over a wide range of n. These similarities indicate that in gels, stiff chains (n < np) migrated via a Rouse-like manner, while semiflexible chains (n > np) migrated by reptation. The crossover from the Rouse to the reptation model was observed in a simulation study of the diffusion coefficient (D),33 where a similar degree of deviation from the Rouse prediction was also observed in the range n < np. Contrary to what was observed with the gels, in the polymer solutions, the power law term was always Rouse-like (n−0.17) without a crossover. This result clearly indicates that reptation-type migration did not occur in polymer solutions under our experimental conditions. The entanglements of polymers in the non-cross-linked system may not be able to form a reptation tube that restricts the migration of dsDNA by the electric field. The requirements for reptation in the semidilute polymer networks under electric field may be the flexibility of the migrating chain and the presence of the permanent cross-links to form stable reptation tubes. Last, we will focus on the exponential terms. According to the origin of the exponential function models, the exponential terms of eqs 2a−2d may express entropy loss. In eqs 2a−2d, ΔS is proportional to ϕ regardless of n and Ns; the geometric confinement is proportional to the polymer volume fraction.

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CONCLUSION In conclusion, we systematically studied the migration behavior of dsDNA in polymer gels and solutions. By tuning ϕ and Ns, the apparent power law slope γ, which scales as μ ∼ n−γ, was changed from 0.36 to 1.46, which agreed with previous experimental studies. Through experimental analyses not based on the existing models, we obtained empirical equations of μ for the first time that described values of μ as products of a power law and an exponential function of n. Notably, our empirical equations are qualitatively different from the equation proposed by van Winkle et al.35 Their equation is a phenomenological equation with three adjustable parameters; while our equations are equations with intrinsic parameters including ϕ, Ns, and n, which provides us a molecular picture of the migration. The resulting power law function terms reduced into two types, n−0.17 and n−0.81. The former type is similar to the Rouse model prediction, and the latter type corresponds to the reptation model for real chains. The power law function terms may express the basic manner of migration. In polymer gels, the basic migration changed from Rouse-like migration to reptation with a crossover at approximately np, while in polymer solutions, the migration behavior was always Rouse-like. These results suggest that the presence of cross-links is a key factor in forcing chains to reptate in the polymer networks. The exponential function terms caused the continuous change in γ, and may express the entropy loss that accompanies migration in polymer networks. Because the characteristics of the exponential terms drastically changed near np, chain flexibility may be a key factor in the type of entropy loss. We envision that this systematic study will be useful in further understanding the dynamics of chains in polymer networks and that a thorough theoretical interpretation of the entropy loss term will occur in the future.

1

However, the n-dependence of ΔS changed from ΔS ∼ nβ to ΔS ∼ n0.3 with a crossover at approximately np (Figure 4). This result indicates that the origin of the entropy loss changed drastically at approximately np. As for stiff dsDNA with n < np, the origin of the entropy loss may be due to an interaction of dsDNA with the mesh size of the polymer network that is defined by Ns. As illustrated in Figure 4, the n-dependence of ΔS became weaker with increasing Ns and asymptotically approached the relationship of the PEG solution for which ΔS ∼ n0. In contrast, regarding the semiflexible dsDNA with n > np, ΔS had a constant scaling

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ASSOCIATED CONTENT

S Supporting Information *

Double-logarithmic plots of μ as a function of n and plots of α2 as a function of Ns. This material is available free of charge via the Internet at http://pubs.acs.org. 3585

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(24) Duke, T. A. J.; Semenov, A. N.; Viovy, J. L. Phys. Rev. Lett. 1992, 69, 3260−3263. (25) Heller, C. Electrophoresis 1999, 20, 1962−1977. (26) Wheeler, L. M.; Lodge, T. P. Macromolecules 1989, 22, 3399− 3408. (27) Ferguson, K. Metabolism 1964, 13, 985−1002. (28) Chen, N.; Chrambach, A. J. Biochem. Biophys. Methods 1997, 35, 175−184. (29) Stellwagen, N. C.; Stellwagen, E. J. Chromatogr A 2009, 1216, 1917−1929. (30) Masaro, L.; Zhu, X. X. Prog. Polym. Sci. 1999, 24, 731−775. (31) Ogston, A. G. Trans. Faraday Soc. 1958, 54, 1754−1757. (32) Semenov, A.; Duke, T.; Viovy, J. L. Phys. Rev. E 1995, 51, 1520− 1537. (33) Bulacu, M.; van der Giessen, E. J. Chem. Phys. 2005, 123, 114901. (34) Daoud, M.; de Gennes, P. G. J. Phys. (Paris) 1977, 38, 85−93. (35) Van Winkle, D. H.; Beheshti, A.; Rill, R. L. Electrophoresis 2002, 23, 15−19.

AUTHOR INFORMATION

Corresponding Author

*(T.S.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors appreciate the advice of Dr. K. Sumitomo and Dr. K. Mayumi. This work was supported by the Japan Society for the Promotion of Science (JSPS) through the Grants-in-Aid for Scientific Research, the Center for Medical System Innovation (CMSI), the Graduate Program for Leaders in Life Innovation (GPLLI), the International Core Research Center for Nanobio, Core-to-Core Program, A. Advanced Research Networks and the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST program); the Ministry of Education, Culture, Sports, Science, and Technology in Japan (MEXT) through the Center for NanoBio Integration (CNBI); and the Japan Science and Technology Agency (JST) through the S-innovation program and COI STREAM; and Grants-inAid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology (No. 12J07977 to X.L., No. 23700555 to T.S., and No. 24240069 to U.C.).

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