1758 Nan Shi Victor M. Ugaz Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX, USA
Received December 23, 2013 Revised March 12, 2014 Accepted March 25, 2014
Electrophoresis 2014, 35, 1758–1765
Research Article
Noise-enhanced gel electrophoresis Macromolecules confined within a nanoporous matrix experience entropic trapping when their dimensions approach the average pore size, leading to emergence of anomalous transport behavior that can be beneficial in separation applications. But the ability to exploit these effects in practical settings (e.g., electrophoretic separation of DNA) has been hindered by additional dispersion introduced as a consequence of the uncorrelated process by which the embedded macromolecules discretely hop from pore to pore. Here, we show how both the source and solution to these difficulties are intimately linked to the inherent dynamics of the underlying activated transport mechanism. By modulating the applied electric field at a frequency tuned to the characteristic activation timescale, a resonance condition can be established that synergistically combines accelerated mobility and reduced diffusion. This resonance effect can be precisely manipulated by adjusting the magnitude and period of the driving electric field, enabling enhanced separation performance and bi-directional transport of different-sized species to be achieved. Notably, these phenomena are readily accessible in ordinary hydrogels (as opposed to idealized nanomachined topologies) suggesting broad potential to apply them in a host of useful settings. Keywords: DNA / Entropic trapping / Gel electrophoresis / Stochastic resonance DOI 10.1002/elps.201300644
Additional supporting information may be found in the online version of this article at the publisher’s web-site
1 Introduction Macromolecules confined within nanoporous surroundings experience entropic trapping (ET) when their dimensions approach the average pore size [1–3]. ET is an activated process characterized by a sequence of hops between neighboring large pores linked by confined interconnecting spaces that impose an energy barrier. In the context of DNA gel electrophoresis, ET-dominated transport is potentially advantageous owing to favorable size dependencies of mobility and diffusion D (i.e., M −(1+␣) , D M −(2+) , where ␣,  ⬎ 0 and M is DNA length in base pairs (bp); ET is inherently a low electric field phenomenon) that are stronger than in the conventionally accessed reptation regime ( M−1 , D M−2 in the low field limit) [2, 4]. But these effects are difficult to isolate in conventional hydrogel matrices because
Correspondence: Professor Victor M. Ugaz, Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX 77843-3122, USA E-mail: [email protected] Fax: +1-979-845-6446
Abbreviations: bp, base pairs; ET, entropic trapping; SR, stochastic resonance C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
their inherently heterogeneous nanoporous architecture frustrates selection of formulations possessing a range of pore sizes comparable to that of the DNA coil. The situation is further complicated by the noisy uncorrelated process by which the embedded macromolecules travel between neighboring large pores joined by confined interconnecting spaces, where additional dispersion is introduced as a consequence of the thermally activated dynamics [5–8]. Stochastic resonance (SR) has emerged as a framework to explain the seemingly counterintuitive ability for a noisy background to become synchronized with and even amplify a weak periodic signal. Initially developed in the context of modeling abrupt global climate change events [9], the concept has since been refined and applied to successfully describe a diverse array of dynamic phenomena in fields including predatory sensing [10], hearing and tactile perception [11, 12], and ion channel transport [13, 14]. The origin of this resonance effect can be understood by considering an archetypical thresholding process whereby a transition between two states is triggered when a system’s output amplitude exceeds a specific value (i.e., the threshold). A sinusoidal input signal whose amplitude is too weak to trigger a
Colour Online: See the article online to view Figs. 1–5 in colour.
www.electrophoresis-journal.com
Microfluidics and Miniaturization
Electrophoresis 2014, 35, 1758–1765
A
B
1759
C
Figure 1. (A) SR is illustrated by a process whereby a change in a system’s output state (blue trace) occurs each time an input signal’s amplitude (red trace) exceeds a given threshold. A switch in output value from −1 to +1 or from +1 to −1 is triggered when the combined input and noise amplitude becomes greater than +1 or less than −1, respectively (units of amplitude and time are arbitrary). (B) Resonance occurs at intermediate ⌫ where periodicity of the system response becomes correlated to that of the input. Synchronization is evident by a characteristic peak at half the resonant period in the distribution of elapsed time between successive switching events. The insets show the distribution of timescales within a single period associated with switching in each direction (top, switching from +1 to −1 occurs at an input phase of 3⌫/4; bottom, switching from −1 to +1 occurs at ⌫/4). (C) This SR phenomenon can be applied to DNA transport by exploiting the activated nature of ET. The distribution of activation times between successive hops from pore-to-pore (migration events) acts analogously to the noise in (A), suggesting that synchronization can be achieved by modulating the applied electric field at a period tuned to a characteristic activation timescale (although activation and migration times are comparable, migration events are discreetly depicted to emphasize the governing role played by the distribution of activation timescales).
switching event can spontaneously acquire the ability to do so by adding a Gaussian noise component of sufficient strength, albeit with a randomly varying elapsed period between switching events that mirrors the inherently stochastic character of the noise. But the output changes dramatically when either the frequency of the sinusoidal signal or the amplitude of the superimposed noise is modulated within a specific window that enables switching events to be triggered with a periodicity synchronized to that of the input (Fig. 1A). This synchronization between the signal and noise is a hallmark of SR. We have recently succeeded in rationally identifying conditions favorable for ET-dominated transport in photopolymerized cross-linked polyacrylamide hydrogels [15] by quantitatively characterizing their pore size distributions [16] and directly probing the scalings of mobility and diffusion with DNA size [17]. In this paper, we describe how these insights can be applied to link ET and SR, two seemingly unrelated phenomena, in a new way that enables greatly enhanced electrophoretic separations. This idea, inspired by the aforementioned activated nature of ET, introduces the possibility of globally synchronizing the ensemble of discrete hops (migration events) between pores throughout the entire matrix by applying a weak periodic driving force (e.g., by cyclically switching the electric field on and off) at a frequency correlated with the system’s characteristic distribution of activation timescales (the noise; Fig. 1B). This actively induced resonant coupling therefore offers a previously unexplored avenue to counteract undesirable dispersion ordinarily encountered in the ET regime under constant electric fields where C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
migration events become progressively uncorrelated over time.
2 Materials and methods DNA gel electrophoresis experiments were performed using a previously described microchip-based platform [17]. Photocurable cross-linked polyacrylamide gels were prepared using Duracryl (30%T, 2.6%C; Proteomic Research Services) and the photoinitiator (Solution B) from a ReproGel DNA sequencing gel kit (GE Healthcare). For example, a 30 L quantity of 6%T gel contained 6 L of the 30%T gel stock solution, 4 L of deionized water, 18.5 L of Solution B, and 1.5 L of stock 10x TBE buffer (Extended Range; Bio-Rad). An electrophoresis microchip (300–500 m wide, 40 m deep microchannel) was filled with the gel mixture, and the gel interface was defined by masking with opaque tape. Polymerization was performed under UV light (EXFO OmnniCure S1000) for 10–12 min. Separations were performed using a 100 bp dsDNA ladder (Bio-Rad). Samples were prepared by mixing 5 L of the DNA ladder, 8.5 L of YOYO-1 intercalating dye (Invitrogen) diluted to one-tenth of the stock concentration, and 1.5 L of the antiphotobleaching agent -mercaptoethanol (Sigma-Aldrich). Time-varying electric fields with a square wave profile were applied using a function generator (Agilent 33220A) interfaced with a voltage amplifier (Trek Model 603). Data analysis was performed using MATLAB. Plotted data are averages over an ensemble of at least three experiments. www.electrophoresis-journal.com
N. Shi and V. M. Ugaz
A
C
3
1 0
200
400
M (bp)
B
800 1000
400
600
M (bp)
5 300 600
1 0.5 1
3
5
7
(ms)
9
11
reverse: N2 2
C
300
1
600
0.2
0
5
10
(ms)
const.
Figure 2. (A) Microchip gel electrophoresis of dsDNA in the ET regime (imposed by UV curing the gel matrix at 625 mW/cm2 ) shows enhanced separation resolution under RET conditions when the electric field is cyclically switched on and off at ⌫ = 5 ms ( , Eavg = 15 V/cm; Eavg represents the time average electric field amplitude) versus application of a continuous field ( , E = 15 V/cm). An inverted dependence of R on DNA size is also evident, as compared with the constant or decreasing size dependence of R in gels where transport is reptation-dominated (, ♦; UV cured at 5 and 100 mW/cm2 , respectively, E = 15 V/cm). (B) Experimental measurements of mobility versus switching period display a distinct peak in the vicinity of 5 ms that is most pronounced at DNA fragment lengths 500 bp and above. (C) Experimental data reveal that the actuation period associated with the mobility peak, ⌫max , is dependent on DNA fragment length (line denotes linear regression fit to the data). (D) The periodically applied electric field also yields a reduction in experimentally measured diffusion coefficients relative to the constant field case.
3 Results and discussion When the ET regime is accessed under a constant electric field, an inverted trend of increasing separation resolution with DNA size is observed (the opposite of what is conventionally seen), with a caveat that the absolute resolution values are below those obtainable in a gel matrix where transport is reptation dominated in the low field limit [15] (Fig. 2A, blue points; the resolution parameter R expresses the difference in distance traveled by two neighboring species relative to the sum of the half-widths of their associated peaks [18]). But R increases significantly when the electric field is cyclically switched on and off at a period of ⌫ = 5 ms, reaching levels that match or exceed those obtained in the reptation regime while simultaneously preserving the trend of increasing size dependence (Fig. 2A, red points; the time average electric field amplitude, Eavg , was held close to the value applied in our continuous field experiments). The enhancement coincides with a distinct mobility maximum at ⌫ 5 ms that is most pronounced in the 500–1000 bp range, shifting toward lower ⌫ for smaller fragments (Fig. 2B). The modulation period, associated with the mobility peak, displays a dependence C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
600 0
500
1000
1500
0
500
1000
1500
t (s)
2.1 2.0 1.9 1.8
0.0
2
300
1.4
0.48 0.44
300
800
D D (x 107 cm2/s)
µ (x 105 cm2/V·s)
600
200
1.8
R
4
2 1
–E
Intensity (a.u.)
(ms) max
R
2
5
...
+E
6
3
B
forward: N1 1
600
7
ET (G = 5 ms) ET (constant field) Reptation (constant field)
...
A
Electrophoresis 2014, 35, 1758–1765
x (mm)
1760
0.5
1.0
x (mm)
1.5
2.0
1.7
t (s)
Figure 3. (A) A time sequence of intensity traces shows that bi-directional transport of 300 and 600 bp dsDNA fragments is achieved by applying alternating forward and backward pulses of an oscillatory electric field selected to optimize resonance in each species (1.1 min forward pulse at ⌫1 = 5 ms; 1.0 min backward pulse at ⌫2 = 2 ms; Eavg = 15 V/cm in both directions). The net result is (B) positive mobility of the leading species (300 = 1.5 × 10−5 cm2 V−1 s−1 ) and negative mobility of the trailing species (600 = −0.5 × 10−6 cm2 V−1 s−1 ), (C) yielding continually increasing separation resolution within a short distance. Intensity traces in (A) were acquired at t = 0 and 1400 s. Note that the electric field direction is only changed in the bidirectional transport cases shown in Fig. 3. Unidirectional fields are applied in all other experiments.
on DNA fragment length (Fig. 2C). This effect is synergistically coupled with reduced diffusion relative the continuous field case (Fig. 2D). Emergence of a size-dependent optimal modulation period ⌫ max makes it possible to induce bi-directional transport of different sized DNA fragments. Conventionally, bidirectional transport has been achieved by designing nonsymmetric systems, the operation of which usually involves applying backward and forward driving forces of different strengths [19]. In contrast, our approach involves modulating the timescales over which the electric field driving force is applied such that they are correlated with molecular-level transport phenomena (i.e., the activation time of hopping events). Specifically, the bi-directional transport presented here is achieved by periodically alternating between an electric field in the forward direction with ⌫ tuned to match resonance of the faster (leading) species, and a field in the reverse direction at a resonant period associated with the slower (trailing) analyte (Fig. 3A). The combined effect drives forward transport of a 300 bp DNA fragment (positive mobility) while a 600 bp fragment simultaneously travels backward (negative mobility; Fig. 3B). This bi-directional transport makes it possible to achieve a state in which separation resolution continually increases over time—an effect that can be precisely tuned to isolate DNA in a specific size range within a very short separation distance (Fig. 3C). To show that these anomalous observations are manifestations of SR, we must establish a connection between the optimal ⌫ and a characteristic system timescale associated with macromolecular transport. Our view of the nanoporous gel matrix architecture shown in Fig. 1C provides a www.electrophoresis-journal.com
Microfluidics and Miniaturization
Electrophoresis 2014, 35, 1758–1765
A
C (t )
U (x)
off
(t ) (x 100 s-1)
Field off
∆WS,off x
t
4 2 0
U (x)
Field on
(t )
( t ) (x 100 s-1)
on
∆WS,on x
B
t
E
( t ) (x 100 s-1)
(t )
t
7 /2
4
6
8
t (ms)
10
= 5 ms
2 0
2
4
6
8
t (ms)
10
= 10 ms
6 4 2 0
5 /2
2
4
0
3 /2
0
6
0
/2
= 1 ms
6
0
2
4
6
t (ms)
8
10
Figure 4. (A) The free-energy landscape U(x) associated with electrophoretic DNA transport between neighboring gel pores is represented by a double potential well with an entropic energy barrier height ⌬WS and characteristic Kramers escape rate ␥. In the absence of an applied electric field (top), the well displays a symmetric profile [9] with respect to xm described U(x) by 4⌬W = − 21 ( xxm )2 + 14 ( xxm )4 with barrier height ⌬WS,off = ⌬WS . S The profile’s symmetry is broken when an electric field is imU(x) posed (bottom), becoming tilted according to 4⌬W = − 21 ( xxm )2 + S 1 x 4 elec xm ( ) − ( F4⌬W ) xxm with an accompanying decrease in barrier 4 xm S height to ⌬WS,on = ⌬WS − Felec xm that also reduces the breadth of the activation time (residence time) distribution (t). (B) A periodically applied electric field causes (t) to alternate between the field-on and field-off states, yielding a distribution curve with peaks occurring at odd multiples of ⌫/2. (C) The spacing between peaks relative to the overall breadth of the distribution can be tuned by adjusting the modulation period to achieve an optimal state of synchronization whereby the majority of pore-to-pore hops occur within a single field-on interval (⌫ 5 ms).
starting point, enabling the fundamental framework developed to describe ET-dominated DNA transport in idealized nanoslit arrays to be applied by integrating over the pore size distribution [15, 20]. The free-energy landscape U(x) encountered by the DNA can therefore be expressed in terms of a series of double potential wells [9] whose depths represent the energy barrier at the entrance to the narrow interconnecting space between adjacent large pores (Fig. 4A). Assuming these transport events are uncorrelated [2,9,21], an activation timescale is obtained from the instantaneous probability that a well is occupied for a time t, P˙ (t ) = −␥ (t ) P (t ), where P(t) is the probability of adopting a given energy state and ␥ (t) is the escape rate [22]. This quantity is more conventionally expressed in terms of a trapping time distribution: t ˙ ␥ (t )dt . (t) = − P(t) = ␥ (t) exp −
(1)
0
Time-dependent escape rates ␥ (t) corresponding to fieldoff and field-on conditions, respectively, are obtained from C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1761
Kramers rate analysis [20,23] as ␥ off exp(−⌬ U/kB T), where ⌬ U = −T⌬ S; and ␥ on E2 exp(−⌬ U/kB T), where ⌬ U = −T⌬ S − Felec L*. Here, T is absolute temperature, kB is Boltzmann’s constant, E is the electric field strength (V/cm), ⌬ S and Felec L*, respectively, represent the entropic and electric potential contributions to the free-energy change associated with traveling a distance L* through a pore (1.9 × 10−2 kB T here [15]), and end-activation is dominant [15, 24]. In the interest of simplicity, our model does not explicitly consider thermally activated “backward” transport events (i.e., in the direction opposite to the driving force). These events occur with exponentially decaying probability [22], and are indirectly accounted for in our fits to experimental data (i.e., to refine our estimates of the constants in Eq. (9). For simplicity, we consider a square wave (i.e., on-off) oscillatory electric field with amplitude E(t) given as follows: 2 E0 1 − sgn sin t . E (t) = 2 ⌫
(2)
The initially symmetric double well energy landscape becomes distorted when a driving potential is applied, tilting along the electric field direction and experiencing a corresponding reduction in barrier height. Cyclically switching the electric field on and off therefore causes (t) to alternate between these states, introducing multiple peaks in the distribution curve spaced with a periodicity coinciding with the electric field actuation (Fig. 4B and C). The DNA migration path through a surrounding gel matrix is viewed as a series of pore units (pairs of neighboring pores of radius ri and rj representing a large pore and narrow interconnecting space, respectively [15]), randomly dispersed in space, within which the pore sizes are distributed following Gaussian statistics with mean b and variance 2 [16]:
(r − b )2 f (r i ) = f (r j ) = √ e xp − 2 2 2 2 1
.
(3)
We express electrophoretic mobility between two neighboring pores of radius ri and rj relative to the reptation model prediction in tight gels N [4] (the limiting low field value in the absence of ET) in terms of a ratio of characteristic timescales trap and mig , respectively, representing the time to enter the small interconnecting space and the time to migrate through this region [20]: mig i,j = . N mig + trap
(4)
We assume that N = 0 /3Nk is independent of pore size since the characteristic Kuhn segment length of 2p = 100 nm is larger than the average gel pore size [16] (p is the persistence length) [4], 0 is the free solution mobility (3.3 × 10−4 cm2 V−1 s−1 ) [17], and Nk is number of Kuhn segments (Nk = LD /lk ; LD and lk are the contour and Kuhn segment lengths, respectively; LD = 0.34 nm per bp) [15]. A quantity corresponding to the experimentally observed www.electrophoresis-journal.com
1762
N. Shi and V. M. Ugaz
Electrophoresis 2014, 35, 1758–1765
mobility can therefore be obtained by integrating over both the residence time distribution associated with an isolated double potential well (the local energy landscape) and the distribution of pore sizes within the gel matrix (the global energy landscape). The effect of applying a time-varying electric field (i.e., cyclically switched on and off with period ⌫) is incorporated by noting that mig / trap 1 in the ET regime [15], implying that sequential activation and migration events rarely occur during the same field-on period. Instead, activated molecules remain trapped at the entrance to the narrow interconnecting region until the next field-on interval, yielding a residence time distribution characterized by peaks at odd multiples of the half period. Our assumption that the trap time distribution is peaked at odd multiples of ⌫/2 enables (t) to be expressed in terms of the cumulative probability that activation will occur between the two possible trap time values: ⌫/2 (t)dt {trap = ⌫/2} = 0 (2n+1)⌫/2 (5) (t)dt, n = 1, 2, 3 . . . . . {trap = (2n + 1)⌫/2} = (2n−1)⌫/2
This simplifies the analysis by allowing the ratio of migration and trap times in the mobility equation to be factored outside the integral over the trap time probability distribution: mig i,j = (t ) dt N mig + trap ⌫ 2 mig (t ) dt + · · · = ⌫ mig + 2 0 +
∞ n=1
mig mig + (2n + 1)
⌫ 2
(2n+1) ⌫2
(2n−1) ⌫2
(t ) dt.
(6)
Global transport throughout the entire gel can therefore be captured by integrating Eq. (6) over the distribution of all possible adjacent pore size combinations (i.e., pore size distribution), yielding the following expression for overall mobility : ⌫ 2 mig = (t ) dt + · · · ⌫ N mig + 2 0 pore size distribution ∞
mig + (2n + 1) ⌫2 + n=1 mig
f (r i ) f r j dr i dr j .
(2n+1) ⌫2
(2n−1) ⌫2
(t ) dt (7)
(We assume Gaussian pore size distribution with mean 9 nm, and variance 0.25 nm2 based on our previous characterization studies [16].) The migration time is mig = Rg /(N E) = C1 M(1+) /E, where Rg is the DNA’s radius of gyration, and is a fitted exponent (a value of 0.82 is assumed [15]). The trapping times embedded within the expression for (t) C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
are obtained from the reciprocal of the corresponding escape rates (i.e., ␥ off = ( trap,off )−1 and ␥ on = ( trap,on )−1 ), yielding: trap = trap,off + trap,on = C2 e xp(−T ⌬ S/kB T ) 1 −T ⌬ S − Felec L ∗ . (8) + C3 2 e xp E kB T The three remaining scaling constants are determined as follows: 3Nk 3L D /l k = 0.22 ≈ 0.7425 × 10−6 0 0 2 C2 = xm2 ≈ 0.5 × 10−3 4⌬ Ws 2 kB T C3 = D0−1 ≈ 0.69, Felec /E C1 = 0.22
(9)
where D0 = kB T/ 10−10 m2 /s is the Zimm model diffusion coefficient, xm is a scaling constant (chosen to be the average DNA radius of gyration (Rg = 30 nm for a 500 bp fragment length), and = 6s Rg is the DNA friction coefficient (s is the solvent viscosity) [25]). We also assume that ⌬ WS kB T, consistent with the physical scenario of an activated process. In subsequent calculations, the constants C1 , C2 , and C3 were adjusted slightly to reflect a global best fit over the entire experimental dataset, yielding C1 = 1.5 × 10−6 , C2 = 1 × 10−3 , and C3 = 0.7. Substituting these results into Eq. (7) enables the mobility to be calculated as a function of DNA size. We numerically evaluated this expression over the first 300 periods of electric field modulation (beyond which (t ) dt → 1), and found remarkably good agreement with mobility versus DNA length data obtained from microchip electrophoresis experiments performed over an ensemble of electric field actuation periods (Supporting Information Fig. 1; some deviation is evident with 100–200 bp fragments whose size likely falls outside the ET regime). The MATLAB function dblquad was applied to execute the double integration between 0 and 50 nm pore size limits (values outside this range either are either not realistic or do not significantly change the results). The pore size averaged escape rate was obtained from ␥ off = exp(⌬ WS )/C2 . Our transport model can now be applied to understand the mobility peak observed at ⌫ 5 ms (Fig. 2B). Analysis of the summation term in Eq. (6) reveals two contributions in each term: a timescale ratio (outside the braces), and a trap time probability (inside the braces). The emergence of a mobility peak becomes evident by separating the first term of the summation from the subsequent terms as follows:
mig i,j ⌫ 1 − e xp(− ␥off ) = N mig + ⌫/2 2
mig ⌫ e xp(− ␥off ) (10) + mig + 3⌫/2 2 ⌫ ⌫ − e xp − ␥off − (␥off + ␥on ) + · · · . 2 2
www.electrophoresis-journal.com
Microfluidics and Miniaturization
Electrophoresis 2014, 35, 1758–1765
A
B
1763
C
D
Figure 5. (A) Predicted dependence of mobility on ⌫ for a 600 bp DNA fragment in hydrogel matrices incorporating three different pore size distribution breadths (expressed in terms of their Gaussian variance 2 ). The pronounced peak at ⌫ = 5 ms is evident at 2 = 0.25 nm2 , with good agreement between the predicted (filled symbols) and experiment (open symbols) values. The mobility peak shifts to higher ⌫ and decreases in height as the pore size distribution broadens. SR is evident in each case owing to the fact that ␥off ⌫/2 1 is satisfied (␥off −1 = 3.2, 4.3, and 5.2 ms for 2 = 0.25, 0.5, and 0.75 nm2 , respectively). (B) Comparison with experiment data in Fig. 2C shows that our transport model also predicts the DNA size dependence of the actuation period associated with the mobility peak, ⌫max . (C) Computed histograms of the entropic energy barrier height show a broadening of the distribution and increase in average value with increasing pore size polydispersity (⌬WS,avg = 2.7, 3.7, and 4.6 kB T for 2 = 0.25, 0.5, and 0.75 nm2 , respectively). The insets show the predicted cumulative activation time probability ⌽(t) versus the number of field actuation periods NP at small (green), optimal (blue), and large (red) ⌫ (in units of ms). (D) Uncertainty in the trap time distribution yields a parameter proportional to the observed diffusion coefficient (overbar denotes integration over the pore size distribution). Predicted values are below the constant field condition in all cases, in agreement with experiment data in Fig. 2D.
Increasing ⌫ in the first term (representing the probability of an activation event [26]) simultaneously decreases the timescale ratio and increases the trap time probability, consistent with emergence of a maximum value (the value of this term is maximized when ␥ off ⌫ max /2 = 1.145 1; obtained by evaluating d/d(⌫/2) = 0 for this term and noting that ␥ off mig = mig / trap,off 1 in the ET regime). In contrast, the magnitude of both contributions becomes smaller in all subsequent terms as ⌫ increases, acting to smooth out the local maximum generated by the first term. The mobility peak therefore corresponds to intermediate ␥ off , and its prominenence at DNA lengths 500 bp and above is explained by the size-dependent escape rate (␥ off increases as the molecules become shorter). In addition to displaying good agreement with experimentally measured mobilities, evaluating our transport model as a function pore size distribution breadth reveals critical fundamental insights (Fig. 5A). First, in each case, the mobility peak is positioned such that ␥ off ⌫/2 1, shifting to larger ⌫ as the distribution becomes more polydisperse. This clear link between the electric field actuation period and the characteristic trap time (the system timescale), combined with their importance in dictating placement of the mobility peak, compellingly confirms the governing role of SR (further echoed by the simulations of Tessier and Slater [26]). C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The DNA size dependence of the actuation period associated with the mobility peak, ⌫ max , is also captured (Fig. 5B). Our model shows how the gel pore morphology shapes the global energy landscape, and hence macromolecular transport (Fig. 5C). As the pore size becomes increasingly polydisperse, the corresponding entropic energy barrier distribution broadens accompanied by an increase in the average well depth (trap strength) ⌬ WS due to the growing population of small-sized pores (very large and very small pores are excluded since the experimentally measured pore sizes are well-fit by a Gaussian distribution [15, 16]). The cumulative t residence time probability ⌽ (t ) = 0 (t ) dt increases most slowly at small ⌫ (where multiple actuation cycles are needed to cross the energy barrier) and in gels with broader pore size distributions (reflecting higher average ⌬ WS ; Fig. 5B, insets). The distribution of entropic energy barriers was generated by sampling a Gaussian ensemble of 104 pore units (i.e., ri and rj combinations), after which ⌬ WS was calculated for each as follows with C4 = 0.85 [15, 27]: 1 1 1 1 ⌬ WS = MC4 kB T − . ri rj
(11)
Separation performance is ultimately determined by the combined action of mobility and diffusion. In conventional www.electrophoresis-journal.com
1764
N. Shi and V. M. Ugaz
reptation-dominated transport, a characteristic diffusion coefficient Dx is obtained by considering variation (uncertainty) in the migrating molecule’s center-of-mass position x (i.e., Dx ∝ var (x) = x2 − x2 ), but the variation in trap time (i.e., Dt ∝ var (t ) = tr2 ap − tr ap 2 ) [28,29] also becomes significant in the ET regime. A new diffusion coefficient can 2 − trap 2 , where the therefore be expressed as Dt ∝ trap overbar denotes an average over the gel pore size distribution. Recall that the residence time distribution curve under a periodically applied electric field is qualitatively different from the continuous field case, distinguished by a series of peaks coinciding with the actuation period (Fig. 4). These multiple peaks act to cluster the highest probability events at discrete intervals, yielding a narrower overall statistical variation. Our transport model captures this effect, where decreasing values 2 2 − trap are predicted as ⌫ becomes small (Fig. 5D). of trap Synchronization of macromolecular transport therefore reduces the undesirable dispersion ordinarily encountered in the ET regime, enabling enhanced separation resolution to be achieved relative to the continuous field case (Fig. 2A). We remark that the mode of electric field actuation we apply here (i.e., cyclically switching the electric field on and off) is distinct from conventional PFGE where the electric field direction is changed in a way that periodically induces re-orientation of very long DNA fragments [27]. The characteristic timescales associated with electric field actuation are also distinct from molecular diffusion, as evident by considering a characteristic self-diffusion time tc = lc 2 /kB T [25], where lc is a characteristic length scale (100 nm for the DNA fragments of interest here) and is the friction coefficient defined in the discussion accompanying Eq. (9). This analysis yields a characteristic timescale of 0.1 ms, an order of magnitude smaller than the period of the applied electric field. Our method is also distinct form zero-integrated field approaches where the electric field strength and duration are simultaneously varied but the period is not tuned to match a system timescale [30–40]. It is worthwhile to point out that there is a rich history of exploration of forward/reverse electric field actuation mechanisms to enhance size-dependent electrophoretic transport and trapping. The most important way in which our method differs from these previous approaches is that we select timescales for modulation of the electric field driving force that are correlated with molecular-level transport phenomena (i.e., the activation timescale of hopping events). This is further highlighted by the fact that in examples where bi-directional transport is illustrated (Fig. 3), we apply the same time averaged electric field strength in the forward and reverse directions. This level of control is enabled by rational selection of the gel pore size distribution and operating conditions in the low electric field limit to access a synergistic coupling of ET and SR.
4 Concluding remarks
Electrophoresis 2014, 35, 1758–1765
to execute a synchronized physical process (i.e., macromolecular transport). This is achieved through the coupled action of SR and ET, two distinct phenomena that have been studied individually but not experimentally combined. From a practical standpoint, the ability to access these nanoscale processes in ordinary polymer gels (as opposed to idealized planar nanomachined topologies [8]) offers a direct pathway to implement them in a host of useful settings. The inherent size dependence of SR also introduces possibilities to employ these phenomena as a sensitive probe of biomolecular conformation [41]. Looking ahead, our transport model establishes clear rules to guide design of matrix architectures optimally tailored to exploit this unique transport mode (e.g., by self-assembly of DNA or protein-building blocks [42]). More broadly, we envision that SR may enable new manipulation and sorting functionalities when coupled with other emerging adaptations of nanoscale potential well traps [43]. V.M.U. gratefully acknowledges support from the US National Science Foundation under grant CBET-1160010 monitored by Dr. Rosemarie Wesson, the Camille & Henry Dreyfus Foundation, and the K.R. Hall Professorship at Texas A&M. The authors have declared no conflict of interest.
5 References [1] Hoagland, D. A., Muthukumar, M., Macromolecules 1992, 25, 6696–6698. [2] Rousseau, J., Drouin, G., Slater, G. W., Phys. Rev. Lett. 1997, 79, 1945–1948. [3] Smisek, D. L., Hoagland, D. A., Science 1990, 248, 1221–1223. [4] Viovy, J.-L., Rev. Mod. Phys. 2000, 72, 813–872. [5] Dorfman, K. D., Rev. Mod. Phys. 2010, 82, 2903–2947. [6] Dorfman, K. D., Brenner, H., Biomed. Microdevices 2002, 4, 237–244. [7] Slater, G. W., Desruisseaux, C., Hubert, S. J., Mercier, ´ J.-F., Labrie, J., Boileau, J., Tessier, F., Pepin, M. P., Electrophoresis 2000, 21, 3873–3887. [8] Han, J., Craighead, H. G., Science 2000, 288, 1026–1029. [9] Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F., Rev. Mod. Phys. 1998, 70, 223–287. [10] Russell, D. F., Wilkens, L. A., Moss, F., Nature 1999, 402, 291–294. [11] Collins, J. J., Imhoff, T. T., Grigg, P., Nature 1996, 383, 770. [12] Morse, R. P., Evans, E. F., Nat. Med. 1996, 2, 928–932. [13] Bezrukov, S. M., Vodyanoy, I., Nature 1995, 378, 362–364. [14] Lee, C. Y., Choi, W., Han, J.-H., Strano, M. S., Science 2010, 329, 1320–1324. [15] Shi, N., Ugaz, V. M., Phys. Rev. Lett. 2010, 108, 108101.
Although SR is well known to act passively in nature, our results show how these dynamics can be actively harnessed C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
[16] Wang, J., Gonzalez, A. D., Ugaz, V. M., Adv. Mater. 2008, 20, 4482–4489.
www.electrophoresis-journal.com
Electrophoresis 2014, 35, 1758–1765
[17] Lo, R. C., Ugaz, V. M., Lab Chip 2008, 8, 2135–2145. [18] Giddings, J. C., Unified Separation Science, Wiley, New York 1991. [19] Thomas, J. D. P., Joswiak, M. N., Olson, D. W., Park, S.-G., Dorfman, K. D., Lab Chip 2013, 13, 3741–3746.
Microfluidics and Miniaturization
1765
[31] Torres, F. A., Gauthier, M. G., Slater, G. W., Phys. Rev. E 2008, 78, 065701. [32] Carle, G. F., Frank, M., Olson, M. V., Science 1986, 232, 65–68.
[20] Fu, J. P., Yoo, J., Han, J., Phys. Rev. Lett. 2006, 97, 018103.
[33] Griess, G. A., Rogers, E., Serwer, P., Electrophoresis 2000, 21, 859–864.
[21] Olson, D. W., Ou, J., Tian, M., Dorfman, K. D., Electrophoresis 2011, 32, 573–580.
[34] Noolandi, J., Slater, G. W., Lim, H. A., Viovy, J.-L., Science 1989, 243, 1456–1458.
[22] Zhou, T., Moss, F., Jung, P., Phys. Rev. A 1990, 42, 3161–3169.
[35] Turmel, C., Brassard, E., Slater, G. W., Noolandi, J., Nucleic Acids Res. 1990, 18, 569–575.
[23] Kramers, H. A., Physica 1940, 7, 284–304.
[36] Zimm, B. H., Phys. Rev. Lett. 1988, 61, 2965–2968.
[24] Panwar, A. S., Kumar, S., Macromolecules 2006, 39, 1279–1289.
[37] Zimm, B. H., J. Chem. Phys. 1991, 94, 2187–2206.
[25] Doi, M., Edwards, S. F., The Theory of Polymer Dynamics, Oxford University Press, Oxford 1986.
[38] Griess, G. A., Harris, R. A., Serwer, P., Appl. Theor. Electrophor. 1993, 3, 305–315.
[26] Tessier, F., Slater, G. W., Appl. Phys. A 2002, 75, 285–291.
[39] Griess, G. A., Serwer, P., Anal. Chim. Acta 1998, 372, 299–306.
[27] Duke, T. A. J., Austin, R. H., Cox, E. C., Chan, S. S., Electrophoresis 1996, 17, 1075–1079.
[40] Griess, G. A., Rogers, E., Serwer, P., Electrophoresis 2001, 22, 981–989.
[28] Nixon, G. I., Slater, G. W., Phys. Rev. E 1996, 53, 4969–4980.
[41] Shi, N., Ugaz, V. M., 10.1002/smll.201303046.
[29] Olson, D. W., Dutta, S., Laachi, N., Tian, M., Dorfman, K. D., Electrophoresis 2011, 32, 581–587.
[42] Um, S. H., Lee, J. B., Park, N., Kwon, S. Y., Umbach, C. C., Luo, D., Nat. Mater. 2006, 5, 797–801.
[30] Slater, G. W., Guo, H. L., Nixon, G. I., Phys. Rev. Lett. 1997, 78, 1170–1173.
[43] Krishnan, M., Mojarad, N., Kukura, P., Sandoghdar, V., Nature 2010, 467, 692–695.
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Small
2014,
10,
DOI:
www.electrophoresis-journal.com
Received December 23, 2013 Revised March 12, 2014 Accepted March 25, 2014
Electrophoresis 2014, 35, 1758–1765
Research Article
Noise-enhanced gel electrophoresis Macromolecules confined within a nanoporous matrix experience entropic trapping when their dimensions approach the average pore size, leading to emergence of anomalous transport behavior that can be beneficial in separation applications. But the ability to exploit these effects in practical settings (e.g., electrophoretic separation of DNA) has been hindered by additional dispersion introduced as a consequence of the uncorrelated process by which the embedded macromolecules discretely hop from pore to pore. Here, we show how both the source and solution to these difficulties are intimately linked to the inherent dynamics of the underlying activated transport mechanism. By modulating the applied electric field at a frequency tuned to the characteristic activation timescale, a resonance condition can be established that synergistically combines accelerated mobility and reduced diffusion. This resonance effect can be precisely manipulated by adjusting the magnitude and period of the driving electric field, enabling enhanced separation performance and bi-directional transport of different-sized species to be achieved. Notably, these phenomena are readily accessible in ordinary hydrogels (as opposed to idealized nanomachined topologies) suggesting broad potential to apply them in a host of useful settings. Keywords: DNA / Entropic trapping / Gel electrophoresis / Stochastic resonance DOI 10.1002/elps.201300644
Additional supporting information may be found in the online version of this article at the publisher’s web-site
1 Introduction Macromolecules confined within nanoporous surroundings experience entropic trapping (ET) when their dimensions approach the average pore size [1–3]. ET is an activated process characterized by a sequence of hops between neighboring large pores linked by confined interconnecting spaces that impose an energy barrier. In the context of DNA gel electrophoresis, ET-dominated transport is potentially advantageous owing to favorable size dependencies of mobility and diffusion D (i.e., M −(1+␣) , D M −(2+) , where ␣,  ⬎ 0 and M is DNA length in base pairs (bp); ET is inherently a low electric field phenomenon) that are stronger than in the conventionally accessed reptation regime ( M−1 , D M−2 in the low field limit) [2, 4]. But these effects are difficult to isolate in conventional hydrogel matrices because
Correspondence: Professor Victor M. Ugaz, Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX 77843-3122, USA E-mail: [email protected] Fax: +1-979-845-6446
Abbreviations: bp, base pairs; ET, entropic trapping; SR, stochastic resonance C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
their inherently heterogeneous nanoporous architecture frustrates selection of formulations possessing a range of pore sizes comparable to that of the DNA coil. The situation is further complicated by the noisy uncorrelated process by which the embedded macromolecules travel between neighboring large pores joined by confined interconnecting spaces, where additional dispersion is introduced as a consequence of the thermally activated dynamics [5–8]. Stochastic resonance (SR) has emerged as a framework to explain the seemingly counterintuitive ability for a noisy background to become synchronized with and even amplify a weak periodic signal. Initially developed in the context of modeling abrupt global climate change events [9], the concept has since been refined and applied to successfully describe a diverse array of dynamic phenomena in fields including predatory sensing [10], hearing and tactile perception [11, 12], and ion channel transport [13, 14]. The origin of this resonance effect can be understood by considering an archetypical thresholding process whereby a transition between two states is triggered when a system’s output amplitude exceeds a specific value (i.e., the threshold). A sinusoidal input signal whose amplitude is too weak to trigger a
Colour Online: See the article online to view Figs. 1–5 in colour.
www.electrophoresis-journal.com
Microfluidics and Miniaturization
Electrophoresis 2014, 35, 1758–1765
A
B
1759
C
Figure 1. (A) SR is illustrated by a process whereby a change in a system’s output state (blue trace) occurs each time an input signal’s amplitude (red trace) exceeds a given threshold. A switch in output value from −1 to +1 or from +1 to −1 is triggered when the combined input and noise amplitude becomes greater than +1 or less than −1, respectively (units of amplitude and time are arbitrary). (B) Resonance occurs at intermediate ⌫ where periodicity of the system response becomes correlated to that of the input. Synchronization is evident by a characteristic peak at half the resonant period in the distribution of elapsed time between successive switching events. The insets show the distribution of timescales within a single period associated with switching in each direction (top, switching from +1 to −1 occurs at an input phase of 3⌫/4; bottom, switching from −1 to +1 occurs at ⌫/4). (C) This SR phenomenon can be applied to DNA transport by exploiting the activated nature of ET. The distribution of activation times between successive hops from pore-to-pore (migration events) acts analogously to the noise in (A), suggesting that synchronization can be achieved by modulating the applied electric field at a period tuned to a characteristic activation timescale (although activation and migration times are comparable, migration events are discreetly depicted to emphasize the governing role played by the distribution of activation timescales).
switching event can spontaneously acquire the ability to do so by adding a Gaussian noise component of sufficient strength, albeit with a randomly varying elapsed period between switching events that mirrors the inherently stochastic character of the noise. But the output changes dramatically when either the frequency of the sinusoidal signal or the amplitude of the superimposed noise is modulated within a specific window that enables switching events to be triggered with a periodicity synchronized to that of the input (Fig. 1A). This synchronization between the signal and noise is a hallmark of SR. We have recently succeeded in rationally identifying conditions favorable for ET-dominated transport in photopolymerized cross-linked polyacrylamide hydrogels [15] by quantitatively characterizing their pore size distributions [16] and directly probing the scalings of mobility and diffusion with DNA size [17]. In this paper, we describe how these insights can be applied to link ET and SR, two seemingly unrelated phenomena, in a new way that enables greatly enhanced electrophoretic separations. This idea, inspired by the aforementioned activated nature of ET, introduces the possibility of globally synchronizing the ensemble of discrete hops (migration events) between pores throughout the entire matrix by applying a weak periodic driving force (e.g., by cyclically switching the electric field on and off) at a frequency correlated with the system’s characteristic distribution of activation timescales (the noise; Fig. 1B). This actively induced resonant coupling therefore offers a previously unexplored avenue to counteract undesirable dispersion ordinarily encountered in the ET regime under constant electric fields where C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
migration events become progressively uncorrelated over time.
2 Materials and methods DNA gel electrophoresis experiments were performed using a previously described microchip-based platform [17]. Photocurable cross-linked polyacrylamide gels were prepared using Duracryl (30%T, 2.6%C; Proteomic Research Services) and the photoinitiator (Solution B) from a ReproGel DNA sequencing gel kit (GE Healthcare). For example, a 30 L quantity of 6%T gel contained 6 L of the 30%T gel stock solution, 4 L of deionized water, 18.5 L of Solution B, and 1.5 L of stock 10x TBE buffer (Extended Range; Bio-Rad). An electrophoresis microchip (300–500 m wide, 40 m deep microchannel) was filled with the gel mixture, and the gel interface was defined by masking with opaque tape. Polymerization was performed under UV light (EXFO OmnniCure S1000) for 10–12 min. Separations were performed using a 100 bp dsDNA ladder (Bio-Rad). Samples were prepared by mixing 5 L of the DNA ladder, 8.5 L of YOYO-1 intercalating dye (Invitrogen) diluted to one-tenth of the stock concentration, and 1.5 L of the antiphotobleaching agent -mercaptoethanol (Sigma-Aldrich). Time-varying electric fields with a square wave profile were applied using a function generator (Agilent 33220A) interfaced with a voltage amplifier (Trek Model 603). Data analysis was performed using MATLAB. Plotted data are averages over an ensemble of at least three experiments. www.electrophoresis-journal.com
N. Shi and V. M. Ugaz
A
C
3
1 0
200
400
M (bp)
B
800 1000
400
600
M (bp)
5 300 600
1 0.5 1
3
5
7
(ms)
9
11
reverse: N2 2
C
300
1
600
0.2
0
5
10
(ms)
const.
Figure 2. (A) Microchip gel electrophoresis of dsDNA in the ET regime (imposed by UV curing the gel matrix at 625 mW/cm2 ) shows enhanced separation resolution under RET conditions when the electric field is cyclically switched on and off at ⌫ = 5 ms ( , Eavg = 15 V/cm; Eavg represents the time average electric field amplitude) versus application of a continuous field ( , E = 15 V/cm). An inverted dependence of R on DNA size is also evident, as compared with the constant or decreasing size dependence of R in gels where transport is reptation-dominated (, ♦; UV cured at 5 and 100 mW/cm2 , respectively, E = 15 V/cm). (B) Experimental measurements of mobility versus switching period display a distinct peak in the vicinity of 5 ms that is most pronounced at DNA fragment lengths 500 bp and above. (C) Experimental data reveal that the actuation period associated with the mobility peak, ⌫max , is dependent on DNA fragment length (line denotes linear regression fit to the data). (D) The periodically applied electric field also yields a reduction in experimentally measured diffusion coefficients relative to the constant field case.
3 Results and discussion When the ET regime is accessed under a constant electric field, an inverted trend of increasing separation resolution with DNA size is observed (the opposite of what is conventionally seen), with a caveat that the absolute resolution values are below those obtainable in a gel matrix where transport is reptation dominated in the low field limit [15] (Fig. 2A, blue points; the resolution parameter R expresses the difference in distance traveled by two neighboring species relative to the sum of the half-widths of their associated peaks [18]). But R increases significantly when the electric field is cyclically switched on and off at a period of ⌫ = 5 ms, reaching levels that match or exceed those obtained in the reptation regime while simultaneously preserving the trend of increasing size dependence (Fig. 2A, red points; the time average electric field amplitude, Eavg , was held close to the value applied in our continuous field experiments). The enhancement coincides with a distinct mobility maximum at ⌫ 5 ms that is most pronounced in the 500–1000 bp range, shifting toward lower ⌫ for smaller fragments (Fig. 2B). The modulation period, associated with the mobility peak, displays a dependence C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
600 0
500
1000
1500
0
500
1000
1500
t (s)
2.1 2.0 1.9 1.8
0.0
2
300
1.4
0.48 0.44
300
800
D D (x 107 cm2/s)
µ (x 105 cm2/V·s)
600
200
1.8
R
4
2 1
–E
Intensity (a.u.)
(ms) max
R
2
5
...
+E
6
3
B
forward: N1 1
600
7
ET (G = 5 ms) ET (constant field) Reptation (constant field)
...
A
Electrophoresis 2014, 35, 1758–1765
x (mm)
1760
0.5
1.0
x (mm)
1.5
2.0
1.7
t (s)
Figure 3. (A) A time sequence of intensity traces shows that bi-directional transport of 300 and 600 bp dsDNA fragments is achieved by applying alternating forward and backward pulses of an oscillatory electric field selected to optimize resonance in each species (1.1 min forward pulse at ⌫1 = 5 ms; 1.0 min backward pulse at ⌫2 = 2 ms; Eavg = 15 V/cm in both directions). The net result is (B) positive mobility of the leading species (300 = 1.5 × 10−5 cm2 V−1 s−1 ) and negative mobility of the trailing species (600 = −0.5 × 10−6 cm2 V−1 s−1 ), (C) yielding continually increasing separation resolution within a short distance. Intensity traces in (A) were acquired at t = 0 and 1400 s. Note that the electric field direction is only changed in the bidirectional transport cases shown in Fig. 3. Unidirectional fields are applied in all other experiments.
on DNA fragment length (Fig. 2C). This effect is synergistically coupled with reduced diffusion relative the continuous field case (Fig. 2D). Emergence of a size-dependent optimal modulation period ⌫ max makes it possible to induce bi-directional transport of different sized DNA fragments. Conventionally, bidirectional transport has been achieved by designing nonsymmetric systems, the operation of which usually involves applying backward and forward driving forces of different strengths [19]. In contrast, our approach involves modulating the timescales over which the electric field driving force is applied such that they are correlated with molecular-level transport phenomena (i.e., the activation time of hopping events). Specifically, the bi-directional transport presented here is achieved by periodically alternating between an electric field in the forward direction with ⌫ tuned to match resonance of the faster (leading) species, and a field in the reverse direction at a resonant period associated with the slower (trailing) analyte (Fig. 3A). The combined effect drives forward transport of a 300 bp DNA fragment (positive mobility) while a 600 bp fragment simultaneously travels backward (negative mobility; Fig. 3B). This bi-directional transport makes it possible to achieve a state in which separation resolution continually increases over time—an effect that can be precisely tuned to isolate DNA in a specific size range within a very short separation distance (Fig. 3C). To show that these anomalous observations are manifestations of SR, we must establish a connection between the optimal ⌫ and a characteristic system timescale associated with macromolecular transport. Our view of the nanoporous gel matrix architecture shown in Fig. 1C provides a www.electrophoresis-journal.com
Microfluidics and Miniaturization
Electrophoresis 2014, 35, 1758–1765
A
C (t )
U (x)
off
(t ) (x 100 s-1)
Field off
∆WS,off x
t
4 2 0
U (x)
Field on
(t )
( t ) (x 100 s-1)
on
∆WS,on x
B
t
E
( t ) (x 100 s-1)
(t )
t
7 /2
4
6
8
t (ms)
10
= 5 ms
2 0
2
4
6
8
t (ms)
10
= 10 ms
6 4 2 0
5 /2
2
4
0
3 /2
0
6
0
/2
= 1 ms
6
0
2
4
6
t (ms)
8
10
Figure 4. (A) The free-energy landscape U(x) associated with electrophoretic DNA transport between neighboring gel pores is represented by a double potential well with an entropic energy barrier height ⌬WS and characteristic Kramers escape rate ␥. In the absence of an applied electric field (top), the well displays a symmetric profile [9] with respect to xm described U(x) by 4⌬W = − 21 ( xxm )2 + 14 ( xxm )4 with barrier height ⌬WS,off = ⌬WS . S The profile’s symmetry is broken when an electric field is imU(x) posed (bottom), becoming tilted according to 4⌬W = − 21 ( xxm )2 + S 1 x 4 elec xm ( ) − ( F4⌬W ) xxm with an accompanying decrease in barrier 4 xm S height to ⌬WS,on = ⌬WS − Felec xm that also reduces the breadth of the activation time (residence time) distribution (t). (B) A periodically applied electric field causes (t) to alternate between the field-on and field-off states, yielding a distribution curve with peaks occurring at odd multiples of ⌫/2. (C) The spacing between peaks relative to the overall breadth of the distribution can be tuned by adjusting the modulation period to achieve an optimal state of synchronization whereby the majority of pore-to-pore hops occur within a single field-on interval (⌫ 5 ms).
starting point, enabling the fundamental framework developed to describe ET-dominated DNA transport in idealized nanoslit arrays to be applied by integrating over the pore size distribution [15, 20]. The free-energy landscape U(x) encountered by the DNA can therefore be expressed in terms of a series of double potential wells [9] whose depths represent the energy barrier at the entrance to the narrow interconnecting space between adjacent large pores (Fig. 4A). Assuming these transport events are uncorrelated [2,9,21], an activation timescale is obtained from the instantaneous probability that a well is occupied for a time t, P˙ (t ) = −␥ (t ) P (t ), where P(t) is the probability of adopting a given energy state and ␥ (t) is the escape rate [22]. This quantity is more conventionally expressed in terms of a trapping time distribution: t ˙ ␥ (t )dt . (t) = − P(t) = ␥ (t) exp −
(1)
0
Time-dependent escape rates ␥ (t) corresponding to fieldoff and field-on conditions, respectively, are obtained from C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1761
Kramers rate analysis [20,23] as ␥ off exp(−⌬ U/kB T), where ⌬ U = −T⌬ S; and ␥ on E2 exp(−⌬ U/kB T), where ⌬ U = −T⌬ S − Felec L*. Here, T is absolute temperature, kB is Boltzmann’s constant, E is the electric field strength (V/cm), ⌬ S and Felec L*, respectively, represent the entropic and electric potential contributions to the free-energy change associated with traveling a distance L* through a pore (1.9 × 10−2 kB T here [15]), and end-activation is dominant [15, 24]. In the interest of simplicity, our model does not explicitly consider thermally activated “backward” transport events (i.e., in the direction opposite to the driving force). These events occur with exponentially decaying probability [22], and are indirectly accounted for in our fits to experimental data (i.e., to refine our estimates of the constants in Eq. (9). For simplicity, we consider a square wave (i.e., on-off) oscillatory electric field with amplitude E(t) given as follows: 2 E0 1 − sgn sin t . E (t) = 2 ⌫
(2)
The initially symmetric double well energy landscape becomes distorted when a driving potential is applied, tilting along the electric field direction and experiencing a corresponding reduction in barrier height. Cyclically switching the electric field on and off therefore causes (t) to alternate between these states, introducing multiple peaks in the distribution curve spaced with a periodicity coinciding with the electric field actuation (Fig. 4B and C). The DNA migration path through a surrounding gel matrix is viewed as a series of pore units (pairs of neighboring pores of radius ri and rj representing a large pore and narrow interconnecting space, respectively [15]), randomly dispersed in space, within which the pore sizes are distributed following Gaussian statistics with mean b and variance 2 [16]:
(r − b )2 f (r i ) = f (r j ) = √ e xp − 2 2 2 2 1
.
(3)
We express electrophoretic mobility between two neighboring pores of radius ri and rj relative to the reptation model prediction in tight gels N [4] (the limiting low field value in the absence of ET) in terms of a ratio of characteristic timescales trap and mig , respectively, representing the time to enter the small interconnecting space and the time to migrate through this region [20]: mig i,j = . N mig + trap
(4)
We assume that N = 0 /3Nk is independent of pore size since the characteristic Kuhn segment length of 2p = 100 nm is larger than the average gel pore size [16] (p is the persistence length) [4], 0 is the free solution mobility (3.3 × 10−4 cm2 V−1 s−1 ) [17], and Nk is number of Kuhn segments (Nk = LD /lk ; LD and lk are the contour and Kuhn segment lengths, respectively; LD = 0.34 nm per bp) [15]. A quantity corresponding to the experimentally observed www.electrophoresis-journal.com
1762
N. Shi and V. M. Ugaz
Electrophoresis 2014, 35, 1758–1765
mobility can therefore be obtained by integrating over both the residence time distribution associated with an isolated double potential well (the local energy landscape) and the distribution of pore sizes within the gel matrix (the global energy landscape). The effect of applying a time-varying electric field (i.e., cyclically switched on and off with period ⌫) is incorporated by noting that mig / trap 1 in the ET regime [15], implying that sequential activation and migration events rarely occur during the same field-on period. Instead, activated molecules remain trapped at the entrance to the narrow interconnecting region until the next field-on interval, yielding a residence time distribution characterized by peaks at odd multiples of the half period. Our assumption that the trap time distribution is peaked at odd multiples of ⌫/2 enables (t) to be expressed in terms of the cumulative probability that activation will occur between the two possible trap time values: ⌫/2 (t)dt {trap = ⌫/2} = 0 (2n+1)⌫/2 (5) (t)dt, n = 1, 2, 3 . . . . . {trap = (2n + 1)⌫/2} = (2n−1)⌫/2
This simplifies the analysis by allowing the ratio of migration and trap times in the mobility equation to be factored outside the integral over the trap time probability distribution: mig i,j = (t ) dt N mig + trap ⌫ 2 mig (t ) dt + · · · = ⌫ mig + 2 0 +
∞ n=1
mig mig + (2n + 1)
⌫ 2
(2n+1) ⌫2
(2n−1) ⌫2
(t ) dt.
(6)
Global transport throughout the entire gel can therefore be captured by integrating Eq. (6) over the distribution of all possible adjacent pore size combinations (i.e., pore size distribution), yielding the following expression for overall mobility : ⌫ 2 mig = (t ) dt + · · · ⌫ N mig + 2 0 pore size distribution ∞
mig + (2n + 1) ⌫2 + n=1 mig
f (r i ) f r j dr i dr j .
(2n+1) ⌫2
(2n−1) ⌫2
(t ) dt (7)
(We assume Gaussian pore size distribution with mean 9 nm, and variance 0.25 nm2 based on our previous characterization studies [16].) The migration time is mig = Rg /(N E) = C1 M(1+) /E, where Rg is the DNA’s radius of gyration, and is a fitted exponent (a value of 0.82 is assumed [15]). The trapping times embedded within the expression for (t) C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
are obtained from the reciprocal of the corresponding escape rates (i.e., ␥ off = ( trap,off )−1 and ␥ on = ( trap,on )−1 ), yielding: trap = trap,off + trap,on = C2 e xp(−T ⌬ S/kB T ) 1 −T ⌬ S − Felec L ∗ . (8) + C3 2 e xp E kB T The three remaining scaling constants are determined as follows: 3Nk 3L D /l k = 0.22 ≈ 0.7425 × 10−6 0 0 2 C2 = xm2 ≈ 0.5 × 10−3 4⌬ Ws 2 kB T C3 = D0−1 ≈ 0.69, Felec /E C1 = 0.22
(9)
where D0 = kB T/ 10−10 m2 /s is the Zimm model diffusion coefficient, xm is a scaling constant (chosen to be the average DNA radius of gyration (Rg = 30 nm for a 500 bp fragment length), and = 6s Rg is the DNA friction coefficient (s is the solvent viscosity) [25]). We also assume that ⌬ WS kB T, consistent with the physical scenario of an activated process. In subsequent calculations, the constants C1 , C2 , and C3 were adjusted slightly to reflect a global best fit over the entire experimental dataset, yielding C1 = 1.5 × 10−6 , C2 = 1 × 10−3 , and C3 = 0.7. Substituting these results into Eq. (7) enables the mobility to be calculated as a function of DNA size. We numerically evaluated this expression over the first 300 periods of electric field modulation (beyond which (t ) dt → 1), and found remarkably good agreement with mobility versus DNA length data obtained from microchip electrophoresis experiments performed over an ensemble of electric field actuation periods (Supporting Information Fig. 1; some deviation is evident with 100–200 bp fragments whose size likely falls outside the ET regime). The MATLAB function dblquad was applied to execute the double integration between 0 and 50 nm pore size limits (values outside this range either are either not realistic or do not significantly change the results). The pore size averaged escape rate was obtained from ␥ off = exp(⌬ WS )/C2 . Our transport model can now be applied to understand the mobility peak observed at ⌫ 5 ms (Fig. 2B). Analysis of the summation term in Eq. (6) reveals two contributions in each term: a timescale ratio (outside the braces), and a trap time probability (inside the braces). The emergence of a mobility peak becomes evident by separating the first term of the summation from the subsequent terms as follows:
mig i,j ⌫ 1 − e xp(− ␥off ) = N mig + ⌫/2 2
mig ⌫ e xp(− ␥off ) (10) + mig + 3⌫/2 2 ⌫ ⌫ − e xp − ␥off − (␥off + ␥on ) + · · · . 2 2
www.electrophoresis-journal.com
Microfluidics and Miniaturization
Electrophoresis 2014, 35, 1758–1765
A
B
1763
C
D
Figure 5. (A) Predicted dependence of mobility on ⌫ for a 600 bp DNA fragment in hydrogel matrices incorporating three different pore size distribution breadths (expressed in terms of their Gaussian variance 2 ). The pronounced peak at ⌫ = 5 ms is evident at 2 = 0.25 nm2 , with good agreement between the predicted (filled symbols) and experiment (open symbols) values. The mobility peak shifts to higher ⌫ and decreases in height as the pore size distribution broadens. SR is evident in each case owing to the fact that ␥off ⌫/2 1 is satisfied (␥off −1 = 3.2, 4.3, and 5.2 ms for 2 = 0.25, 0.5, and 0.75 nm2 , respectively). (B) Comparison with experiment data in Fig. 2C shows that our transport model also predicts the DNA size dependence of the actuation period associated with the mobility peak, ⌫max . (C) Computed histograms of the entropic energy barrier height show a broadening of the distribution and increase in average value with increasing pore size polydispersity (⌬WS,avg = 2.7, 3.7, and 4.6 kB T for 2 = 0.25, 0.5, and 0.75 nm2 , respectively). The insets show the predicted cumulative activation time probability ⌽(t) versus the number of field actuation periods NP at small (green), optimal (blue), and large (red) ⌫ (in units of ms). (D) Uncertainty in the trap time distribution yields a parameter proportional to the observed diffusion coefficient (overbar denotes integration over the pore size distribution). Predicted values are below the constant field condition in all cases, in agreement with experiment data in Fig. 2D.
Increasing ⌫ in the first term (representing the probability of an activation event [26]) simultaneously decreases the timescale ratio and increases the trap time probability, consistent with emergence of a maximum value (the value of this term is maximized when ␥ off ⌫ max /2 = 1.145 1; obtained by evaluating d/d(⌫/2) = 0 for this term and noting that ␥ off mig = mig / trap,off 1 in the ET regime). In contrast, the magnitude of both contributions becomes smaller in all subsequent terms as ⌫ increases, acting to smooth out the local maximum generated by the first term. The mobility peak therefore corresponds to intermediate ␥ off , and its prominenence at DNA lengths 500 bp and above is explained by the size-dependent escape rate (␥ off increases as the molecules become shorter). In addition to displaying good agreement with experimentally measured mobilities, evaluating our transport model as a function pore size distribution breadth reveals critical fundamental insights (Fig. 5A). First, in each case, the mobility peak is positioned such that ␥ off ⌫/2 1, shifting to larger ⌫ as the distribution becomes more polydisperse. This clear link between the electric field actuation period and the characteristic trap time (the system timescale), combined with their importance in dictating placement of the mobility peak, compellingly confirms the governing role of SR (further echoed by the simulations of Tessier and Slater [26]). C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The DNA size dependence of the actuation period associated with the mobility peak, ⌫ max , is also captured (Fig. 5B). Our model shows how the gel pore morphology shapes the global energy landscape, and hence macromolecular transport (Fig. 5C). As the pore size becomes increasingly polydisperse, the corresponding entropic energy barrier distribution broadens accompanied by an increase in the average well depth (trap strength) ⌬ WS due to the growing population of small-sized pores (very large and very small pores are excluded since the experimentally measured pore sizes are well-fit by a Gaussian distribution [15, 16]). The cumulative t residence time probability ⌽ (t ) = 0 (t ) dt increases most slowly at small ⌫ (where multiple actuation cycles are needed to cross the energy barrier) and in gels with broader pore size distributions (reflecting higher average ⌬ WS ; Fig. 5B, insets). The distribution of entropic energy barriers was generated by sampling a Gaussian ensemble of 104 pore units (i.e., ri and rj combinations), after which ⌬ WS was calculated for each as follows with C4 = 0.85 [15, 27]: 1 1 1 1 ⌬ WS = MC4 kB T − . ri rj
(11)
Separation performance is ultimately determined by the combined action of mobility and diffusion. In conventional www.electrophoresis-journal.com
1764
N. Shi and V. M. Ugaz
reptation-dominated transport, a characteristic diffusion coefficient Dx is obtained by considering variation (uncertainty) in the migrating molecule’s center-of-mass position x (i.e., Dx ∝ var (x) = x2 − x2 ), but the variation in trap time (i.e., Dt ∝ var (t ) = tr2 ap − tr ap 2 ) [28,29] also becomes significant in the ET regime. A new diffusion coefficient can 2 − trap 2 , where the therefore be expressed as Dt ∝ trap overbar denotes an average over the gel pore size distribution. Recall that the residence time distribution curve under a periodically applied electric field is qualitatively different from the continuous field case, distinguished by a series of peaks coinciding with the actuation period (Fig. 4). These multiple peaks act to cluster the highest probability events at discrete intervals, yielding a narrower overall statistical variation. Our transport model captures this effect, where decreasing values 2 2 − trap are predicted as ⌫ becomes small (Fig. 5D). of trap Synchronization of macromolecular transport therefore reduces the undesirable dispersion ordinarily encountered in the ET regime, enabling enhanced separation resolution to be achieved relative to the continuous field case (Fig. 2A). We remark that the mode of electric field actuation we apply here (i.e., cyclically switching the electric field on and off) is distinct from conventional PFGE where the electric field direction is changed in a way that periodically induces re-orientation of very long DNA fragments [27]. The characteristic timescales associated with electric field actuation are also distinct from molecular diffusion, as evident by considering a characteristic self-diffusion time tc = lc 2 /kB T [25], where lc is a characteristic length scale (100 nm for the DNA fragments of interest here) and is the friction coefficient defined in the discussion accompanying Eq. (9). This analysis yields a characteristic timescale of 0.1 ms, an order of magnitude smaller than the period of the applied electric field. Our method is also distinct form zero-integrated field approaches where the electric field strength and duration are simultaneously varied but the period is not tuned to match a system timescale [30–40]. It is worthwhile to point out that there is a rich history of exploration of forward/reverse electric field actuation mechanisms to enhance size-dependent electrophoretic transport and trapping. The most important way in which our method differs from these previous approaches is that we select timescales for modulation of the electric field driving force that are correlated with molecular-level transport phenomena (i.e., the activation timescale of hopping events). This is further highlighted by the fact that in examples where bi-directional transport is illustrated (Fig. 3), we apply the same time averaged electric field strength in the forward and reverse directions. This level of control is enabled by rational selection of the gel pore size distribution and operating conditions in the low electric field limit to access a synergistic coupling of ET and SR.
4 Concluding remarks
Electrophoresis 2014, 35, 1758–1765
to execute a synchronized physical process (i.e., macromolecular transport). This is achieved through the coupled action of SR and ET, two distinct phenomena that have been studied individually but not experimentally combined. From a practical standpoint, the ability to access these nanoscale processes in ordinary polymer gels (as opposed to idealized planar nanomachined topologies [8]) offers a direct pathway to implement them in a host of useful settings. The inherent size dependence of SR also introduces possibilities to employ these phenomena as a sensitive probe of biomolecular conformation [41]. Looking ahead, our transport model establishes clear rules to guide design of matrix architectures optimally tailored to exploit this unique transport mode (e.g., by self-assembly of DNA or protein-building blocks [42]). More broadly, we envision that SR may enable new manipulation and sorting functionalities when coupled with other emerging adaptations of nanoscale potential well traps [43]. V.M.U. gratefully acknowledges support from the US National Science Foundation under grant CBET-1160010 monitored by Dr. Rosemarie Wesson, the Camille & Henry Dreyfus Foundation, and the K.R. Hall Professorship at Texas A&M. The authors have declared no conflict of interest.
5 References [1] Hoagland, D. A., Muthukumar, M., Macromolecules 1992, 25, 6696–6698. [2] Rousseau, J., Drouin, G., Slater, G. W., Phys. Rev. Lett. 1997, 79, 1945–1948. [3] Smisek, D. L., Hoagland, D. A., Science 1990, 248, 1221–1223. [4] Viovy, J.-L., Rev. Mod. Phys. 2000, 72, 813–872. [5] Dorfman, K. D., Rev. Mod. Phys. 2010, 82, 2903–2947. [6] Dorfman, K. D., Brenner, H., Biomed. Microdevices 2002, 4, 237–244. [7] Slater, G. W., Desruisseaux, C., Hubert, S. J., Mercier, ´ J.-F., Labrie, J., Boileau, J., Tessier, F., Pepin, M. P., Electrophoresis 2000, 21, 3873–3887. [8] Han, J., Craighead, H. G., Science 2000, 288, 1026–1029. [9] Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F., Rev. Mod. Phys. 1998, 70, 223–287. [10] Russell, D. F., Wilkens, L. A., Moss, F., Nature 1999, 402, 291–294. [11] Collins, J. J., Imhoff, T. T., Grigg, P., Nature 1996, 383, 770. [12] Morse, R. P., Evans, E. F., Nat. Med. 1996, 2, 928–932. [13] Bezrukov, S. M., Vodyanoy, I., Nature 1995, 378, 362–364. [14] Lee, C. Y., Choi, W., Han, J.-H., Strano, M. S., Science 2010, 329, 1320–1324. [15] Shi, N., Ugaz, V. M., Phys. Rev. Lett. 2010, 108, 108101.
Although SR is well known to act passively in nature, our results show how these dynamics can be actively harnessed C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
[16] Wang, J., Gonzalez, A. D., Ugaz, V. M., Adv. Mater. 2008, 20, 4482–4489.
www.electrophoresis-journal.com
Electrophoresis 2014, 35, 1758–1765
[17] Lo, R. C., Ugaz, V. M., Lab Chip 2008, 8, 2135–2145. [18] Giddings, J. C., Unified Separation Science, Wiley, New York 1991. [19] Thomas, J. D. P., Joswiak, M. N., Olson, D. W., Park, S.-G., Dorfman, K. D., Lab Chip 2013, 13, 3741–3746.
Microfluidics and Miniaturization
1765
[31] Torres, F. A., Gauthier, M. G., Slater, G. W., Phys. Rev. E 2008, 78, 065701. [32] Carle, G. F., Frank, M., Olson, M. V., Science 1986, 232, 65–68.
[20] Fu, J. P., Yoo, J., Han, J., Phys. Rev. Lett. 2006, 97, 018103.
[33] Griess, G. A., Rogers, E., Serwer, P., Electrophoresis 2000, 21, 859–864.
[21] Olson, D. W., Ou, J., Tian, M., Dorfman, K. D., Electrophoresis 2011, 32, 573–580.
[34] Noolandi, J., Slater, G. W., Lim, H. A., Viovy, J.-L., Science 1989, 243, 1456–1458.
[22] Zhou, T., Moss, F., Jung, P., Phys. Rev. A 1990, 42, 3161–3169.
[35] Turmel, C., Brassard, E., Slater, G. W., Noolandi, J., Nucleic Acids Res. 1990, 18, 569–575.
[23] Kramers, H. A., Physica 1940, 7, 284–304.
[36] Zimm, B. H., Phys. Rev. Lett. 1988, 61, 2965–2968.
[24] Panwar, A. S., Kumar, S., Macromolecules 2006, 39, 1279–1289.
[37] Zimm, B. H., J. Chem. Phys. 1991, 94, 2187–2206.
[25] Doi, M., Edwards, S. F., The Theory of Polymer Dynamics, Oxford University Press, Oxford 1986.
[38] Griess, G. A., Harris, R. A., Serwer, P., Appl. Theor. Electrophor. 1993, 3, 305–315.
[26] Tessier, F., Slater, G. W., Appl. Phys. A 2002, 75, 285–291.
[39] Griess, G. A., Serwer, P., Anal. Chim. Acta 1998, 372, 299–306.
[27] Duke, T. A. J., Austin, R. H., Cox, E. C., Chan, S. S., Electrophoresis 1996, 17, 1075–1079.
[40] Griess, G. A., Rogers, E., Serwer, P., Electrophoresis 2001, 22, 981–989.
[28] Nixon, G. I., Slater, G. W., Phys. Rev. E 1996, 53, 4969–4980.
[41] Shi, N., Ugaz, V. M., 10.1002/smll.201303046.
[29] Olson, D. W., Dutta, S., Laachi, N., Tian, M., Dorfman, K. D., Electrophoresis 2011, 32, 581–587.
[42] Um, S. H., Lee, J. B., Park, N., Kwon, S. Y., Umbach, C. C., Luo, D., Nat. Mater. 2006, 5, 797–801.
[30] Slater, G. W., Guo, H. L., Nixon, G. I., Phys. Rev. Lett. 1997, 78, 1170–1173.
[43] Krishnan, M., Mojarad, N., Kukura, P., Sandoghdar, V., Nature 2010, 467, 692–695.
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Small
2014,
10,
DOI:
www.electrophoresis-journal.com
