Optimizing end-labeled free-solution electrophoresis by increasing the hydrodynamic friction of the drag-tag
OOptimizing end-labeled free-solution electrophoresisby increasing the hydrodynamic friction of thedrag-tag
Kai Grass, ∗ , † Christian Holm, ∗ , † , ‡ and Gary W. Slater ∗ , ¶ Frankfurt Institute for Advanced Studies, Goethe University, Ruth-Moufang-Strasse 1, D-60438Frankfurt am Main, Germany, Institute for Computational Physics, University of Stuttgart,Pfaffenwaldring 27, D-70569 Stuttgart, Germany, and Department of Physics, University ofOttawa, 150 Louis-Pasteur, Ottawa, Ontario K1N 6N5, Canada
E-mail: grass@fias.uni-frankfurt.de; [email protected]; [email protected]
Abstract
We study the electrophoretic separation of polyelectrolytes of varying lengths by means of end-labeled free-solution electrophoresis (ELFSE). A coarse-grained molecular dynamics simulationmodel, using full electrostatic interactions and a mesoscopic Lattice Boltzmann fluid to accountfor hydrodynamic interactions, is used to characterize the drag coefficients of different label types:linear and branched polymeric labels, as well as transiently bound micelles.It is specifically shown that the label’s drag coefficient is determined by its hydrodynamic † Frankfurt Institute for Advanced Studies, Goethe University, Ruth-Moufang-Strasse 1, D-60438 Frankfurt amMain, Germany ‡ Institute for Computational Physics, University of Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart, Germany ¶ Department of Physics, University of Ottawa, 150 Louis-Pasteur, Ottawa, Ontario K1N 6N5, Canada a r X i v : . [ c ond - m a t . s o f t ] F e b ai Grass et al. Optimizing ELFSEsize, and that the drag per label monomer is largest for linear labels. However, the addition ofside chains to a linear label offers the possibility to increase the hydrodynamic size, and thereforethe label efficiency, without having to increase the linear length of the label, thereby simplifyingsynthesis. The third class of labels investigated, transiently bound micelles, seems very promisingfor the usage in ELFSE, as they provide a significant higher hydrodynamic drag than the otherlabel types.The results are compared to theoretical predictions, and we investigate how the efficiency ofthe ELFSE method can be improved by using smartly designed drag-tags. Introduction
As known from experiments and theory, the free-solution mobility of a flexible polyelectrolytechain does not depend on the chain length N (number of monomers) anymore if the chain is longerthan a certain length N FD . The regime where N > N FD is called free-draining regime. In this regime,the counterions influence the inter-monomer hydrodynamic interactions and allow the fluid to drainthrough the polyelectrolyte coil. The effective friction Γ eff becomes linear in the chain length, asdoes the effective charge Q eff for longer chains, which leads to a constant, length-independentmobility µ = Q eff Γ eff . (1)It was shown that attaching a suitable uncharged molecule to an electrophoresis target can re-store the size-dependent mobility and overcome the free-draining property of long polyelectrolytechains. This method, which is known as end-labeled free-solution electrophoresis (ELFSE),is based on the alteration of the charge-to-friction ratio of the polyelectrolyte molecules by anuncharged drag label.The effect of the label can be compared to that of a parachute attached to a moving object.The additional friction provided by the parachute slows the object down. This effect is strongerfor smaller molecules with a lower effective friction and smaller charge to pull the drag-tag, as the2ai Grass et al. Optimizing ELFSEratio between charge and friction is changed more drastically than for larger molecules.Since the method’s introduction, finding suitable labels that provide a high hydrodynamic draghas been a major concern in this field.
A larger hydrodynamic drag enables the separation oflonger chain fragments, as the length-dependence of the electrophoretic mobility decreases withincreasing polyelectrolyte length. When the additional friction provided by the drag-tag becomesnegligible against the intrinsic effective friction of the polyelectrolyte, the chain becomes essen-tially free-draining again. Experimentally relevant is that the mobility of long polyelectrolytechains should differ by a factor large enough to allow for accurately separating them, althoughtheir lengths only vary by a single monomer. The maximum chain length resolvable in this way iscalled the read length .In general, the drag labels can be chosen from a wide range of molecules but they have to fulfillcertain requirements, such as being water-soluble at experimental conditions, having a uniqueattachment mechanism to the polyelectrolyte and showing minimal interaction with the walls of thecapillary. The read length is optimised by choosing a large molecule that imposes a high frictionaldrag. However, to fulfill resolution requirements, the labels must remain perfectly monodispersesince polydispersity will effectively be like an additional source of diffusion that would broadenthe peaks.As it poses an experimental challenge to produce large, monodisperse linear polymer labels,two recently proposed alternatives seem promising. Haynes et. al. proposed to use branchedpolymers with well-defined architectures. A first theoretical study on this method verified theapproach and concluded that, even though a branched polymer is more compact and thus providesa smaller hydrodynamic friction for a given molecular weight than a linear polymer, this draw-back is offset by the monodispersity of the branched labels created by assembling shorter linearchains. Grosser et. al. introduced nonionic surfactant micelles as drag labels with very largehydrodynamic friction. The inherent polydispersity of the micelles is overcome by using a PNAamphiphile that only provides a transient binding between the DNA fragments to be separated andthe micelles. Each fragment attaches to a different micelle every couple of seconds, which results3ai Grass et al. Optimizing ELFSEin an averaging procedure over the course of the elution time that remedies the need for perfectmonodispersity.The theoretical description of the methods discussed above is not complete and furthermorepredicts behaviours that are difficult to test experimentally, such as end of chain effects, the hydro-dynamic deformation of the label in high fields, or the steric segregation between the label and thechain. Since it is not possible to visualize DNA-label molecules in the lab, computer simulationscan support the understanding of the real physics as long as they include hydrodynamic interactionsbetween polyelectrolyte, label and solvent, as well as account for the influence of the electrostaticinteraction between the polyelectrolytes and its surrounding counterions. Of course, it is beyondthe scope of this article to cover all previous predictions. However, we will demonstratethat it is possible to study these factors, and that the standard theory appears to be sufficient for thecases treated here. More cases will be studied in future papers.Since the ELFSE method overcomes the main drawback of ordinary gel electrophoresis, thelong separation time due to the slow down by the applied gel matrix, it is a promising method onthe way to faster sequencing methods and, as such, of especial interest to the community.In this paper we will use coarse-grained MD simulations to study the electrophoretic separa-tion of fully flexible polyelectrolytes of varying lengths by end-labeling. After introducing thesimulation model, we confirm that the free-draining behaviour is correctly reproduced and testthe standard theory for ELFSE by attaching an uncharged linear label. In Section , we introducebranches to the drag label and test the predictions made by.
From the branched label, we willgo to micellar targets (Section ) and analyze the method proposed by.
We establish a relationbetween the average size of the micelle and its drag value. Our concluding remarks point out theefficiency of the ELFSE method and show the benefit of the different labeling methods.4ai Grass et al. Optimizing ELFSE
Theory
The theory for end-labeled free-solution electrophoresis is based on the interplay between hydro-dynamic and electrostatic forces, and it takes into account the stress that builds along the chainbackbone. In general, it is assumed that the electrostatic and frictional forces do not deform thehybrid molecule’s random coil conformation nor its cloud of counterions. Because of these as-sumptions, the theory used here is valid for low velocities and weak electric fields.Neglecting molecular end-effects, the electrophoretic mobility µ = v / E of the polyelectrolytewith an attached linear drag-tag can be described in terms of the effective friction of the polyelec-trolyte Γ PE , its effective charge Q PE and the hydrodynamic friction of the attached label Γ L : µ = Q PE Γ PE + Γ L = µ + Γ L / Γ PE , (2)where µ is the length independent free solution mobility without drag-tag.Equation 2 shows the importance of the ratio between Γ PE and Γ L . The electrophoretic mobility µ is a function of N for a fixed Γ L as long as Γ PE changes with N and the ratio between Γ L and Γ PE remains non-negligible.Since the electrophoretic friction coefficient Γ PE grows linear with the length of the polyelec-trolyte for long chains, as shown in a previous publication, Equation 2 can be reformulated asfollows: µ = µ + α L / N , (3)with a constant drag coefficient α L = Γ L Γ PE / N . (4)Here the ratio Γ PE / N is the friction per monomer of the polyelectrolyte. The chemistry and tem-perature dependent α L is a measure for the difference in hydrodynamic properties between thepolyelectrolyte and the label and represents the number of polyelectrolyte monomers that providea hydrodynamic friction equal to that of the label. α L = N (cid:18) µ µ − (cid:19) , (5)where α L is conveniently determined as the slope when plotting µ / µ versus 1 / N PE : µ µ = + α L / N . (6) Simulation model
We employ coarse-grained molecular dynamics (MD) simulations using the ESPResSo package to study the electrophoretic separation of fully charged linear flexible polyelectrolytes by end-labeled free-solution electrophoresis. The polyelectrolytes are modelled by a totally flexible bead-spring model as a set of spheres that represent the N individual monomers which are connected toeach other by finitely extensible nonlinear elastic (FENE) bonds: U FENE ( r ) = kR ln (cid:18) − (cid:16) rR (cid:17) (cid:19) , with stiffness k = ε , and maximum extension R = . σ , and r being the distance betweenthe interacting monomers. Additionally, a truncated Lennard-Jones or WCA potential is used forexcluded volume interactions: U LJ ( r ) = ε (cid:18)(cid:16) σ r (cid:17) − (cid:16) σ r (cid:17) + (cid:19) , with the cut-off being r cut = √ σ , at which U LJ ( r ) = ε and σ define the energy and length scale of the simulations. We use ε = k B T , i.e. the energy of the system is expressed in terms of the thermal energy. The length scale σ defines the size of the monomers. We set σ = . . σ represents a linear monomer distance of approximately 4.3 Å,the spacing of single-stranded DNA. Different polyelectrolytes can be mapped by changing σ .Unless mentioned otherwise, all observables are expressed in terms of the simulation units σ and ε , which we will not use explicitly from now on.Besides the dissociated counterions the system also contains additional monovalent salt. Thecounterions and the salt ions are modelled as charged spheres using the same WCA potential givingall particles in the system the same size.All chain monomers carry a negative electric charge q = − e , where e is the elementarycharge. For charge neutrality, N monovalent counterions of charge + e are added. Additionalmonovalent salt is added to the simulation. Full electrostatic interactions are calculated with theP3M algorithm using the implementation of Reference . The Bjerrum length l B = e / πε k B T = . ε r ≈
80. The appliedexternal field E = . We include hydrodynamics using a Lattice Boltzmann (LB) algorithm that is frictionallycoupled to the MD simulations via an algorithm introduced by Ahlrichs et al. The mesoscopic7ai Grass et al. Optimizing ELFSE a) b)c) d)
Figure 1: (a) Polyelectrolyte with surrounding counter- and co-ions. (b) with linear drag-tag, (c)with branched polymeric drag-tag, and (d) with micellar drag-tag.8ai Grass et al. Optimizing ELFSELB fluid is described by a velocity field generated by discrete momentum distributions on a spatialgrid, rather than explicit fluid particles. We use an implementation of the D3Q19 model with akinematic viscosity ν = .
0, a fluid density ρ = . The resulting fluid has a dynamic viscosity η = ρν = .
0. The space is discretized by a grid with spacing a = .
0. The fluid is coupled to theparticles by a frictional coupling with bare friction parameter Γ bare = .
0. Random fluctuationsfor particles and fluid act as a thermostat. The interaction between particles and fluid conservetotal momentum and are proved to yield correct long-range hydrodynamic interaction betweenindividual particles.The simulations are carried out under periodic boundary conditions in a rectangular simula-tion box. We investigate the behaviour of polyelectrolyte chains varying from N =
20 to N = L of the box is varied to realize a constant monomer density of n PE = − which corresponds to a concentration c PE = λ D ≈ .
2. A MD time step τ MD = .
01 and a LB timestep τ LB = .
05 are used. After equilibration of 10 steps, 10 steps are used for generating thedata. The time-series are analyzed using auto-correlation functions to estimate the statistical errorsas detailed in Reference. Error bars of the order of the symbol size or smaller are omitted in thefigures.The electrophoretic mobility is obtained by applying a constant electric field of reduced fieldstrength E = . v of the chain: µ = vE . (7)Before applying this method, it was ensured that the applied electric field strength E is smallenough not to distort chain conformations or counterion distributions. Therefore, the system is inthe linear response regime, i.e. , the measured mobility does not depend on the magnitude of theelectric field. 9ai Grass et al. Optimizing ELFSEUp to ten independent simulations are carried out for each data point, taking between one dayand two weeks on a single standard CPU depending on the chain length N and the type of labelinvestigated. Linear drag tags
In this section, the simulation model is applied to the electrophoresis of polyelectrolyte chainswith an attached linear polymeric drag-tag. The electrophoretic mobility for polyelectrolyte chainsis determined with and without different labels, and the results are compared to the theoreticalpredictions. We also examine how the effective friction of the drag-tag is influenced by the intrinsicstiffness and the salt concentration in the solution.
Testing the standard ELFSE theory
First, the free-solution electrophoretic mobility without an attached drag-tag, µ , is determined, asshown in Figure Figure 2. The measured mobility does not depend on the chain length, as expectedfor longer free-draining polyelectrolyte chains. The average mobility is determined to be µ = . ± . . (8)Additionally, in Figure Figure 2, the mobilities with attached drag-tags ranging from L = L =
70 monomers are measured, and it is confirmed that a length-dependence is achieved andthat the difference in mobilities, i.e. , the selectivity of the separation, is better the longer the at-tached label is. Equation 6 is used to calculate the hydrodynamic drag coefficients as shown inFigure Figure 3, resulting in values from α L = . ± . L =
30 to α L = . ± . L = α L based on the hydrodynamic size and shape of the labelis developed. The hydrodynamic friction Γ L of the uncharged label is related to the hydrodynamic Dual Core AMD Opteron(tm) Processor 270 m w/o labelL = 30L = 50L = 70 Figure 2: The free-solution electrophoretic mobility without label (black circles) shows no depen-dence on the chain length N . The free-draining mobility is µ = . ± . N -dependent behaviour. The label length L is varied from 30 to 70 monomers, with the largest labelresulting in the strongest slowdown. 11ai Grass et al. Optimizing ELFSE m / m L = 30L = 50L = 70
Figure 3: The hydrodynamic drag coefficient α L is given by the slope of the curve. For the linearlabels, α L ranges from 13 . ± . . ± . R h by means of the Stokes relation: Γ L = πη R h,L . (9)As in the previous chapter, the effective electrophoretic friction of the polyelectrolyte is expressedin terms of the free-solution mobility µ and the effective charge Q eff : Γ PE = Q eff / µ . (10)Using the Manning prediction Q eff = ( / ξ ) N for the effective charge, where the Manning con-densation parameter ξ = l B / b is a measure for the strength of the electrostatic potential of thepolyelectrolyte, finally yields α L = µ ξ πη R h,L . (11)With the system parameter used here, ξ = .
63, one obtains α L = ( . ± . ) R h,L . (12)Equation 11 will be shown to be valid for linear labels whose size is not exceeding the Debyelength λ D . When the label size becomes larger, the friction of the label is not anymore directlyrelated to the hydrodynamic radius, as the salt ions that penetrate the polymer coil influence theintermonomer hydrodynamic interactions and limit them to the electrostatic screening length. Asfor the polyelectrolyte itself, this screening length is of the order of the Debye length.For linear labels larger than the Debye length, McCormick et al. introduced a relation for thehydrodynamic drag coefficient, with which α L can be determined from the size of the polyelec-trolyte and label monomers, b PE and b L , and the corresponding Kuhn lengths, b k,PE and b k,L , whichdescribe the stiffness of the chains: α L = b L b k,L b PE b k,PE L . (13)13ai Grass et al. Optimizing ELFSEFigure 4: A schematic representation of the “blob” picture used to derive Equation 13.The derivation of Equation 13 assumes that the polyelectrolyte and the label can be representedby a series of hydrodynamically equivalent entities, called “blobs” as shown in Figure Figure 4.The number and the size of these blobs depend on the bond length and flexibility of the chains,resulting in the presented relation for α L .The total effective friction of the polyelectrolyte-label compound with the surrounding solventis linear in the total number of hydrodynamically equivalent monomers given by N = N PE + α N L . (14)This is true for long polyelectrolytes in the free-draining regime, where the size of the compoundis larger than the Debye length λ D , since the hydrodynamic interactions between the individualmonomers are screened on this length scale, as shown in a prior study on free-solution elec-trophoresis. Thus, the hydrodynamic drag α L can be directly calculated from the persistence lengths of thepolyelectrolyte and of the label using Equation 13. Here, l p,PE and l p,L are calculated from the bondcorrelation function: l p = b N / ∑ i = (cid:104) (cid:126) b N / · (cid:126) b N / + i + (cid:126) b N / · (cid:126) b N / − i (cid:105) , (15)14ai Grass et al. Optimizing ELFSEwhere (cid:126) b i is the i -th bond vector and b is the average bond length. The angular brackets (cid:104) . . . (cid:105) denotean ensemble average. Under the chosen conditions, the persistence length of the polyelectrolyte is found to be l p,PE = . ± . , and the label’s one l p,L = . ± . . The difference between these two values is due to the electrostatic repulsion between the monomersof the polyelectrolyte. In our model, all monomers have the same size, so that Equation 13 isreduced to α L = l p,L l p,PE L = ( . ± . ) L . (16)The comparison between the measured drag coefficient and the theoretical prediction in Fig-ure Figure 5 shows an agreement for the respective regimes of validity. The agreement betweentheory and experiments has been shown before. For labels with a hydrodynamic size smallerthan the Debye length, i.e. , R h < λ D , Equation 11 gives the correct prediction for the drag coeffi-cient α L . Longer labels, however, can no longer be seen as a single polymer coil with a hydrody-namic size R h , but instead the blob picture described by Equation 13 has to be used. This predictionis only valid when the hydrodynamic size becomes larger than the Debye length. For R h ≈ λ D theexpected cross-over between these regimes is observed.It remains to be emphasised that, by determining α L from the measurements of the persis-tence lengths and the hydrodynamic radius, there is no free fitting parameter and the quantitative agreement in Figure Figure 5 is noteworthy. For a discussion about different ways to determine the persistence length in computer simulations please referto. a L a L (b L b k,L /b PE b k,PE ) L xm R h Figure 5: The hydrodynamic drag coefficient α L for the linear label is compared to the theoreticalpredictions of Equations 11 and 13. Both predictions are valid in different regimes with the divisionbeing R h ≈ λ D indicated by the vertical dashed line. Note that neither theory has a free fittingparameter used to achieve the quantitative agreement with the simulation data.16ai Grass et al. Optimizing ELFSE Increasing the hydrodynamic drag coefficient
In Figure Figure 5, it is shown that the total drag coefficient α L for linear labels can be increasedby using longer labels, and that beyond the Debye length the increase is linear with the length L ofthe label. Unfortunately, the experimental requirement of strict monodispersity of the label limitsthe size of linear polymeric labels that can be synthesized and prepared. In this section, it will beshown how one can influence the total drag coefficient also by modifying the relative stiffness ofthe polyelectrolyte and drag-tag chains. Increasing the label stiffness
Equation 13 shows the dependence of α L on the persistence lengths of polyelectrolyte and label.Therefore, α L can be increased by either increasing the persistence of the label or decreasing thepersistence of the polyelectrolyte. Both ways will be investigated in this subsection.First, an additional harmonic bond angle potential, U BA = k BA ( φ − φ ) , (17)is added to the interaction between the label monomers, where φ is the angle between two consec-utive bonds. Here, k BA =
30 and φ = R h,L = . ± .
05 to R h,L = . ± . , and thus puts the label size into the regime where the blob picture is valid. The increased stiffnessdoubles the persistence length of the label from l p,L = . ± . l p,L = . ± . , α L = ( . ± . ) L ≈ . . m w/o labelL = 30L = 30, stiff Figure 6: The hydrodynamic drag coefficient of a stiff linear label is higher than that of a fullyflexible label with the same length. The slowdown of the stiff label is correctly predicted byEquation 13 (solid line).Figure Figure 6 compares the theoretical predicted slowdown of the stiffer 30 monomeric la-bel to the measured mobilities. As before, an excellent agreement to the theory is found for theinvestigated label lengths. Please note that there is no fitting parameter here.18ai Grass et al. Optimizing ELFSE
Reducing the polyelectrolyte stiffness
In practice, it is not easy to increase the stiffness of the drag-tag in order to to increase α L sincethis implies changing the chemistry of the label. However, the value of α L is a relative measureof the stiffness of the two components of the hybrid polyelectrolyte-label molecule (see Eq. 13).Therefore, an increase of α L can also be achieved if we reduce the stiffness of the polyelectrolyte.The persistence length of a polyelectrolyte can be reduced very effectively by increasing the ionicstrength of the buffer: this increases the screening of the electrostatic repulsive interactions thatstiffen the backbone of the charged polymer. In this subsection, we investigate the effect of chang-ing the concentration of salt from c S = mM to c S = M , which reduces the Debye length from λ D ≈ . λ D ≈ . l p,PE = . ± . , whereas the label persistence is unaffected. Thus, one predicts a hydrodynamic drag coefficient of α L = ( . ± . ) L , using Eq. 13, instead of α L = ( . ± . ) L for the previous salt concentration (see Eq. 16).The change in electrophoretic mobilities for labels of lengths L =
20 and L =
40 can be seenin Figure Figure 7. Please note that the free-draining mobility µ also changes due to the fact thatthe additional salt also increases the screening of the polyelectrolyte charge, thus reducing the netforce from the external field: µ = . ± . . m w/o labelL = 20L = 40 Figure 7: In the presence of 1 Mol additional salt, the persistence length of polyelectrolyte isreduced, changing the relative hydrodynamic drag α L of the label. The observed mobility of thepolyelectrolyte molecules for two linear labels of length 20 and 40 are compared to the predictionusing Equation 13. 20ai Grass et al. Optimizing ELFSEThe reduction in the absolute mobility µ together with the increase of the diffusion coefficientdue to the more compact conformation of the molecules at higher salt concentrations negativelyaffect the size-selectivity as can be seen when evaluating the resolution factor R as defined byMcCormick et al. If we keep the polyelectrolyte length N , the label length L , the electric fieldstrength E and the elution distance constant then one obtains for a given α L a resolution factor R that only depends on the diffusion coefficient D and the free-draining mobility µ : R ∼ (cid:112) D / µ . The way the resolution factor is defined a higher value indicates a lower size-selectivity and thusthe size-selectivity is decreased if the relative drag of the label is increased by increasing the saltconcentration. Consequently, an increased hydrodynamic drag coefficient is less effective whenachieved by adding additional salt.
Branched drag tags
In this section, we will investigate the use of branched labels as a possible way to synthesize moreefficient ELFSE drag-tags. First, the results obtained in a recent experimental study by Haynes etal. are briefly reviewed. The study compared a linear polypeptide drag-tag with 30 repeat unitsto two branched drag-tags, each with 5 side-chains spaced evenly along a 30 unit-long backbone.The two different branched labels had 4 and 8 monomer long side-chains. The drag coefficients α L were obtained by measuring the mobility of two different DNA fragments of 20 and 30 bases inlength. It was found that the value of α L increases roughly linearly with the total molecular weightof the branched label.This astonishing observation was theoretically analysed by Nedelcu et al. It was shown thatthe drag coefficient is directly related to the hydrodynamic radius (as one would expect from theblob picture), and that the linear dependence on molecular weight is only approximately true in thelimit of short side chains. 21ai Grass et al. Optimizing ELFSEAs a matter of fact, the drag provided by a linear label is always higher than that provided bya branched label of the same molecular weight. The reason for this is that, with a fixed lengthbackbone, a branched polymer is essentially a compact star polymer with a smaller hydrodynamicsize than the linear equivalent. Indeed, as the number of arms increases, the branched polymerbecomes even more compact and less favorable for ELFSE.Based on the observations, the following optimal design using branched polymeric labels forELFSE was proposed: I) side chains with length comparable to the distance between branchingpoints, or II) two long branches located near the ends of the molecule’s backbone.Here, the focus will be on investigating the effect of the length of the side chains for a polymericdrag-tag with a fixed backbone length. Similar to the structure of the label used by Haynes et al. ,the label has a backbone of L =
30 monomers to which 5 side chains are attached evenly spacedalong the backbone. The side-chain length is varied from 2 to 8 monomers, so that the total numberof monomers in the label ranges from 40 to 70. The drag coefficient of the labels is determined bymeasuring the electrophoretic mobility of polyelectrolyte chains from N =
20 to N = α L is determined according to Equation 6. The obtained α L values are compared to thecorresponding value of a purely linear drag-tag with the same number of monomers.Figure Figure 9 confirms the work by Nedelcu, showing that the label with the highest dragper monomer is the linear label. For the same number of monomers L , the hydrodynamic dragcoefficient α L of the linear label is higher than that of the branched one. But it also shows thatthe addition of side chains can be used to increase the hydrodynamic drag of the label. This isattributed to two effects: firstly, the hydrodynamic size of the label is increased as the side chainsextend from the label. Of similar importance is the second effect, namely that the side chains stiffenthe label due to steric repulsion with the backbone, increasing the overall persistence length andincreasing the linear length of the backbone. In fact, this is the main contribution to the increase forthe side chains of two and four monomers as we confirmed by measuring the change in persistence22ai Grass et al. Optimizing ELFSE m / m L = 30L = 30 + 5x4L = 30 + 5x8
Figure 8: The reduced mobility µ / µ for polyelectrolytes with an attached linear label with fiveside chains of length 4 and 8 shows a more pronounced slowdown than for the label without sidechains. 23ai Grass et al. Optimizing ELFSE a L a L (linear) xm R h a L (branched) xm R h Figure 9: The hydrodynamic drag coefficient α L of a branched polymeric label is compared tothe previously determined drag of a linear label. L is the total number of monomers. As long asthe hydrodynamic radius R h of the label is smaller than the Debye length λ D , the α L is given byEquation 11. The vertical lines indicate the number of monomers L for which R h ( L ) ≈ λ D obtainedfrom simulations. 24ai Grass et al. Optimizing ELFSElength of the backbone.Interestingly, the drag coefficients obtained for the labels show a scaling with the hydrodynamicradius R h , as given by Equation 11. Since the polymer coil formed by the branched label is morecompact, it is less penetrated by ions and, therefore, the prediction of Equation 11 remains validfor a higher number of monomers compared to the linear label.The experimentally observed linear scaling with L can be attributed to seemingly linear rela-tionship between R h and L , but, as Nedelcu et al. have shown before, this is only approximatelytrue in the case of side chains smaller or equal to the spacing along the backbone. The only relevantquantity in all cases is the hydrodynamic radius and its contribution to the hydrodynamic drag, asformulated in Equation 11.Although linear labels remain preferable as long as the pure hydrodynamic drag coefficient α L per molecular weight is concerned, branched polymers offer practical advantages because ofthe possibility of synthesizing larger and somewhat stiffer monodisperse molecules in a simple,stepwise way. Micellar drag tags
Recently, Grosser et al. proposed another promising class of drag-tags that in principle canprovide very large hydrodynamic drag coefficients α L . They used nonionic surfactant Triton X-100 micelles that attach to PNAA-taged (PNA amphiphile) DNA strands. The micelles are water-soluble and are created and destroyed on a timescale of milliseconds to seconds, forming a fairlymonodisperse populations of structures with a tunable size and morphology. During the wholeelectrophoresis time, a single DNA strand attaches to a large number of different micelles. Ofimportance for the ELFSE application is the fact that this leads to an averaging effect betweenmicelles of different sizes for the individual DNA strand, meaning the DNA can be though of ashaving a drag tag of fixed size (cid:104) R (cid:105) , where (cid:104) R (cid:105) is the average micelle size. Only with this averaging,the natural polydispersity of the micelles is overcome and a measurement with a size resolution up25ai Grass et al. Optimizing ELFSEto a couple of base pairs is possible.As a free DNA strand quickly attaches to a new micelle, the DNA is bound to a micelle mostof the electrophoresis time. Consequently, the transiently bound micelles provide about the samehydrodynamic drag as a covalently bound drag-tag of similar size would provide. The reported α L values range between 33 and 58 for a single micelle, depending on the micelle type and the PNAAmolecule used for connecting to the DNA strand. Savard et al. showed that dual-tagging of theDNA, i.e. , attaching a PNAA molecule to both ends of the DNA strand so that two micelles aretransiently bound can increase the hydrodynamic drag even further.In this study, four different micelles with radius R = R = α L are obtained as before and compared toEquation 11, which correctly predicts the observed behaviour. With the chosen micelle radius of R =
5, drag coefficients up to α L = . m w/o labelR = 2R = 3R = 4R = 5 Figure 10: The electrophoretic mobility µ as a function of the polyelectrolyte length N becomessize dependent when a micellar drag-tag is attached. The magnitude of the slowdown depends onthe radius R of the micelle. 27ai Grass et al. Optimizing ELFSE a L a L xm R Figure 11: The effective hydrodynamic drag coefficient α L of a micellar drag-tag is directly pro-portional to its radius R . Equation 11 (the solid line) gives a very good prediction of the dragcoefficient for all tested micelles. 28ai Grass et al. Optimizing ELFSEments to be analysed. Currently, we are investigating the usability of cationic micelles as drag-tagscarrying positive charges, which create an additional force on the polyelectrolyte-label compound,possibly enhancing the size separation. Conclusion
In this paper, we have presented a detailed study of end-labeled free-solution electrophoresis (us-ing various hydrodynamic drag-tags) by coarse-grained molecular dynamics simulations. Linear,branched and micellar drag-tags were investigated. The simulations support the theoretical predic-tions and can be matched quantitatively to it. This enables the use of computer simulation as a toolto support the design of improved hydrodynamic drag-tags usable for electrophoretic separation ofpolyelectrolytes in free-solution.It was specifically shown that the drag coefficient of the label is determined by its hydrody-namic size and not by its weight. The hydrodynamic drag per label monomer is largest for linearlabels, but experimental restrictions in the synthesis of such labels and the monodispersity require-ment limit their practical applicability.The addition of side chains to a linear label offers the possibility to increase the hydrodynamicsize without having to increase the linear length of the label. The synthesis process creates perfectlymonodisperse labels. It was shown that the label efficiency is increased with the length of the sidechains for the drag-tag sizes studied in this work. In addition to increasing the lateral size ofthe drag-tag, the side chains also increase the persistence of the backbone and thus contribute intwo different ways to the increased hydrodynamic size. Especially the steric stabilisation of thelinear backbone is responsible for an initial increase of the drag-coefficient with the total numberof monomers of the label, i.e. , with the molecular weight. For longer side chains, the lateralcontribution to the hydrodynamic radius becomes more important.The third class of labels investigated seems very promising for the future of ELFSE. Transientlybound micelles provide a significantly higher hydrodynamic drag, as they can be prepared with29ai Grass et al. Optimizing ELFSEa large hydrodynamic radius. Additionally, the time averaging by attaching to many differentmicelles over the electrophoresis time span helps to meet the monodispersity criteria. This studyshowed that the hydrodynamic drag is directly proportional to the hydrodynamic radius of themicelle. The efficiency of this method is, in principle, only limited by the size of labels that can beprepared.Our results demonstrate convincingly that theory and computer models can support the experi-mental progress towards the design of novel improved drag-tags, thereby extending the applicabil-ity of the ELFSE technique. The usability of charged drag-tags is currently under investigation.
Acknowledgement
Funds from the the Volkswagen foundation, and the DAAD are gratefully acknowledged. Allsimulations were carried out on the compute cluster of the Center for Scientic computing at GoetheUniversity Frankfurt. GWS would like to acknowledge the support from the Natural Science andEngineering Research Council of Canada.
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