Molecular Dynamics Simulations Illuminate the Role of Counterion

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Molecular dynamics simulations illuminate the role of counterion condensation in the electrophoretic transport of homogalacturonans Amir Hossein Irani, Jessie L. Owen, Davide Mercadante, and Martin A. K. Williams Biomacromolecules, Just Accepted Manuscript • DOI: 10.1021/acs.biomac.6b01599 • Publication Date (Web): 06 Jan 2017 Downloaded from http://pubs.acs.org on January 9, 2017

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Molecular dynamics simulations illuminate the role of counterion condensation in the electrophoretic transport of homogalacturonans Amir H. Irani,† Jessie L. Owen,† Davide Mercadante,‡ and Martin A.K. Williams∗,†,¶ †Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand ‡Heidelberger Institut for Theoretische Studien, Heidelberg, Germany ¶The MacDiarmid Institute of Advanced Materials and Nanotechnology, Wellington, New Zealand E-mail: [email protected] Abstract Homogalacturonans (HGs) are polysaccharide co-polymers of galacturonic acid and its methylesterified counterpart. The inter- and intra-molecular distributions of the methylesterifed residues are vital behaviour-determining characteristics of a sample’s structure and much experimental effort has been directed to their measurement. While many techniques are able to measure the sample-averaged degree of methylesterification (DM), the measurement of inter- and intra-molecular charge distributions are challenging. Here, molecular dynamics (MD) simulations are used to calculate the electrophoretic mobilities of HGs that have different amounts and distributions of charges placed along the backbone. The simulations are shown to capture experimental results

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well even for low-DM samples that possess high charge densities. In addition they illuminate the role that local counterion condensation can play in the determination of the electrophoretic mobility of heterogeneous blocky polyelectrolytes that cannot be adequately described by a single chain-averaged charge spacing.

Introduction Pectins, Homogalacturonans, and the Plant Cell Wall Pectin, a macromolecular complex of the plant cell wall, can consist of some 17 different types of sugar residue, and has been called ‘probably the most complex macromolecule in nature’ 1 . While the details of the arrangements of certain predominant motifs, and the nature and positioning of the attachment of these to each other, and to other elements of the plant cell wall, are not yet settled, it is largely accepted that the structure consists of an assembly of distinct, structurally well-characterized domains 2–4 . Homogalacturonan, an extensive linear region of 1-4 linked galacturonic acid residues, each of which can exist in a state of methylesterification or not, is probably the most studied domain and is most associated with the mechanical functionality of pectin, both in the cell wall 5–8 and in its technological applications as a gelling agent 9–12 . Certainly the ability of this portion to assemble in the presence of divalent ions or in acidic conditions is key to its functionality in-muro, and typically pectin manufacturers seek to maximise the proportion of HG in commercially available pectins (typically achieving between 60 and 85%).

The Importance of Patterning in Biological Polyelectrolytes Clearly there are many examples of how the spatial patterning of substituent groups displayed along biological-polymer backbones is exploited in nature, including the iconic replication of DNA 13 . But the nucelotide base sequence not only stores information in its 1-D sequence, it also determines the local mechanical and assembly properties of the chains, as exemplified 2 ACS Paragon Plus Environment

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by the emergence of such biologically relevant structures as G-quadruplexes 14,15 and the exploitation of DNA as a construction material in bionanotechnology 16,17 . Homogalacturonans offer another example of the importance of the patterning of substituent groups and of how 1-D sequence information can be manifest in 3-D mechanical function. The patterning of the methylester groups (and thus charge) plays a key role in determining local polymer mechanics 18 , in controlling the propensity of the chains to self assemble into functional networks mediated by ions, hydrogen bonding and hydrophobicity 9–12 , and in mediating the binding of proteins 19–22 . The biological function of many other polysaccharides, including several that are important in mammalian physiology, from the chondroitin and dermachondan sulphate strings that hold collagen fibrils in perfect register 23–26 to the multi-functional heparin 27–29 also depend on the patterning of charged residues. Detailed structure-function understanding then depends on the reliable measurement of the distribution of charge-carrying groups, so that for HGs the sequence that denotes the state of methylesterification of the individual residues should be ascertained. But with polysaccharide fine structure being routinely re-modeled to optimize its function, as a function of its position or time (in-muro or in the processing plant) , each HG chain in a sample presented for analysis can possess a different pattern of methylesterification.

Intermolecular Charge Distributions and Their Measurement It has previously been argued that measuring the intermolecular distribution of DM (the relative numbers of chains possessing slightly different chain-averaged DM) has some advantages in sample characterization over the more traditionally used fragmentation approaches that pursue information regarding intramolecular sequences and that by measuring and modeling this intermolecular distribution, significant progress in generating faithful ensembles of chains can be made 30,31 . The success of this methodology clearly relies on having an experimental technique that can faithfully measure the intermolecular distribution of 3 ACS Paragon Plus Environment

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DM. This has been successfully demonstrated using HG samples that had been randomly demethylesterified from a high-DM starting substrate to obtain a series of samples with different sample-averaged DM values 31 . The agreement obtained, not just for the measured average DM values, but also for the known form of the DM distributions, attests to the fact that the empirical relationship used between the mobility and the fraction of charged residues in a chain (Figure 1) is a reasonable description of the actual physical relationship.

Figure 1: Electrophoretic mobilities, µ, measured for pure HGs and a number of pectin samples, gathered from the literature, as a function of the fraction of the sugar rings charged, z, and the dimensionless polyelectrolytic charge density parameter, ξ, described in the text.

Counterion Condensation and Electrophoretic Mobility It is clear from the data in figure 1 that, in the region in which the majority of previous work has been carried out, corresponding to fractional charge density (the number of charged residues divided by the total number of residues comprising the chain), z ∼ 0.1 - 0.6, there is a well-defined consistent relationship, wherein simple linear regression analysis based on the mobility of surrounding standards can yield a reasonable estimate of the DM of an unknown sample from its electrophoretic mobility 32 . However it is also abundantly clear that at higher


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vales of z (lower DM) the dependence of electrophoretic mobility on the fractional charge (as calculated naively from the structure as 1-(DM/100)) is modified. In this regime chains migrate consistently with lower charge densities than might be naively anticipated from their structures if an extrapolation of the linear relationship is assumed, reminiscent of the postulates of counterion condensation theory 33–37 . It should be noted that although in detail both the chain conformation and the pK a of the galacturonic acid residues will change with DM the effect of these changes on the electrophoretic mobility are expected to be minor owing to the relatively high intrinsic stiffness of the uncharged chains (figure 2c) and the experimental pH respectively 38 . Conceptually, counterion condensation in polyelectrolyte solutions can be understood by considering of the distribution of ions around a charged line as described by PoissonBoltzmann theory. The fine structure of the polyelectrolyte (the precise patterning of the charged and uncharged residues) is simply mapped onto a charged-line model by setting a uniform spacing of charges on the line that corresponds to the average spacing found on the polyelectrolyte. The abstracted charged lines are subsequently characterized by a so-called polyelectrolyte charge parameter, ξ, defined by

ξ=

lB b

(1)

where lB is the Bjerrum length and b is the uniform distance between charges. The Bjerrum length is defined as lB =

e2 4π0 kB T

(2)

where is the dielectric constant of solvent and all other symbols have their usual meanings. Physically this length represents the distance at which the electrostatic interaction energy between two single charges is equivalent to the thermal energy, and in water at 298 K is around 0.7 nm. The distance between charges placed on the line is defined as described as b =< li > where li is distance between the centre of mass of two neighbouring charges on


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the polyelectrolyte. The electric field, E, and concomitant potential around such a uniformly charged line in a continuum solvent with a uniform dielectric constant can be calculated by Gauss’ theorem. Subsequently, with the position-dependent potential-energy in hand, the expected probability distribution of counterions around such a uniformly charged line can be described as a function of radial distance from the line, r, simply using the Boltzmann equation

P = P0 r−2ξ

(3)

where P0 is a constant that ensures the cumulative probability sums to 1. In the case where the spacing between charges on the line used to model the polyelectrolyte is significantly greater than the Bjerrum length then ξ can be extracted from equation 3 if the distribution of the counterions about a polyelectrolyte is known. Such strategies have been pursued experimentally by techniques such as anomalous small angle x-ray scattering (ASAXS) that aim to measure the ion distribution directly 39–42 . The basic argument for the existence of more complex behaviour at increased charge densities, and the proposal of the so-called condensation of counterions, originates from attempting to integrate equation 3 in order to calculate the number of counterions inside a cylinder of fixed radius and unit length. In this case the integrand contains the radius to the exponent (1-2ξ) so that when ξ > 1 (that is; the charge spacing becomes less than the Bjerrum length) the integral diverges at the charged line, signaling that such a form for the counterion distribution is unstable under these conditions, and suggesting that in reality a tight binding of a certain number of counterions to the chain effectively limits ξ to values less than 1. Indeed, under these conditions, when a single average charge spacing, b, can realistically be used to define ξ then it has been shown that a fraction of counterions, 1-(1/ξ), become condensed 37 . It is noteworthy that in figure 1 the simple linear relationship between electrophoretic mobility and naively-calculated charge density indeed appears to break down at a polyelectrolyte charge parameter close to that of 1. Although there continues to be debate and 6 ACS Paragon Plus Environment

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legitimate concerns around the ad hoc nature of some of its assumptions 43 , Manning’s counterion condensation theory has previously been found to be useful for the description of the behavior of a number of polyelectrolytes of both synthetic and biological origin, 44–47 and while more complicated and rigorous models do exist, 48–50 these do not presently appear to be able to capture the available experimental data. In previous studies using HGs as model polyelectrolytes the key relationship that was used to map a chain’s electrophoretic mobility to the fraction of methylesterified residues it contained, (1-z) where z is the fraction of charged groups, was obtained empirically from the data shown in figure 1 31 . Hypothesizing that the form of the relationship shown in figure 1 does indeed signal that counterion condensation is important at low degrees of methylesterification, raises the question of what role local condensation might play in determining the mobility of samples with high degrees of methylesterification, but with non-random, blocky, fine structures. How many singlycharged sugar residues can be placed next to one another, sandwiched between neutral sugar residues, before local condensation effects modify the ion distribution and effect the measured electrophoretic mobility? Herein CE experiments have been carried out comparing the electrophoretic mobilities of just such locally blocky samples with sister-samples of similar average DM but with random DM distributions. It is shown that Molecular Dynamics (MD) simulations can not only explain the differences found in this case, but can also provide a prediction of the form of figure 1, for random patterns, that agrees well with the experimental data, both illuminating the role of counterion condensation and paving the way to allowing fine structure predictions for non-random samples to be mapped to predicted experimental electrophoretic mobility distributions.


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Experimental Sample Details and Capillary Electrophoresis Samples. Pectin samples were a gift from CP Kelco. Two pairs of samples were used in this study: R47 and B48, and R57 and B58, each pair comprising of two samples that had a similar DM (denoted by the number following the letter) but a different fine structure, with the methylesters arranged in a random (prefix R) or blocky (prefix B) pattern. (This was achieved by applying different pectinmethylesterase enzymes with either random or processive modes of action to the same high DM starting material and was assessed qualitatively through a so-called degree of blockiness measurement 51 that involves measuring the susceptibility of the sample to degradation by block-preferring enzymes). These pectin samples are quoted as being 85% galacturonic acid, and have minimal sidechain sugars, and as such are expected to exhibit similar transport phenomena to pure homogalacturonans 31 . DP 30 DNA oligomers used as electrophoretic mobility markers were purchased from DNA Technologies. Capillary Electrophoresis. Experiments carried out in this work used an automated CE system (HP 3D), equipped with a diode array detector. Electrophoresis was carried out in a fused silica capillary of internal diameter 50 µm and a total length of 46.5 cm (40 cm from inlet to detector). The capillary incorporated an extended light-path detection window (150 µm) and was thermostatically controlled at 25◦ . Phosphate buffer at pH 7.0, prepared by mixing 50 mM solutions of NaH2 PO4 and Na2 HPO4 as appropriate, was used as a CE background electrolyte (BGE). All new capillaries were conditioned by rinsing for 30 min with 1 M NaOH, 30 min with a 0.1 M NaOH solution, 15 min with water, and 30 min with BGE. Between runs, the capillary was typically washed for 2 min with 1M NaOH, 2 min with 0.1 M NaOH, 1 min with water, and 2 min with BGE. Detection was carried out using UV absorbance at 191 nm with a bandwidth of 2 nm. 0.25 % pectin samples were loaded hydrodynamically (various injection times at 5000 Pa, typically giving injection volumes of the order of 10 nL) and were typically


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electrophoresed across a potential difference of 20 kV. All experiments were carried out at normal polarity (inlet anodic) unless otherwise stated. Electrophoretic mobilities, µ, are related to the migration times of the injected samples relative to a neutral marker, t and t0 , respectively:

µ = µobs − µeo =

1 lL 1 ( − ) V t t0

(4)

where L is the total length of the capillary, l is the distance from the inlet to detector, V is the applied voltage, µobs is the observed mobility, and µeo is the mobility of the electroosmotic flow (EOF). The EOF position was determined by a dip in the intensity of transmitted light corresponding to the refractive index change associated with the passage of excess solvent through the detection window (this was confirmed to be the case in additional experiments using mesityl oxide as a UV absorbing maker). In addition short DNA oligomers were coinjected with the samples of interest in some runs in order to provide a marker species that migrated after the pectic polymers and thus ensure that any observed changes in peak shape did not result from time dependent changes in the EOF.

Molecular Dynamics Simulations Molecular Dynamics (MD) simulations were performed using GROMACS 4.6.5 52 . Simulations were carried out based on the GLYCAM force field (version 06h) 53 after assigning the partial charges of atoms as calculated using density functional theory (DFT).

DFT Schematic representations of a single galacturonic acid residue and its methyl-esterified counterpart are shown in figure 2a, which displays the atomic nomenclature used hereafter. All homogalacturonan (HG) substrates of interest are simple co-polymers comprised of different arrangements of these two kinds of residues i) galacturonic acid (G), which was negatively


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Figure 2: a) Galacturonic acid (left) and methylesterified galacturonic acid (right); b) and c) fully charged trimer and 25-mer respectively after 5ns MD simulation.


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charged both in our CE experiments at pH 7 and in our simulation, and ii) uncharged methylesterified galacturonic acid (M). In order to consider the effects of the chemical identity (G or M) of the nearest neighbour residues on the partial charges of a central (G or M) residue, we performed exhaustive calculations on the 8 possible trimer configurations. DFT calculations on these substrates were performed at the B3LYP/6-31G* level of theory in water using Gaussian09 54 . RESP fitting 55 (using a weighting factor of 1.0, and a value of 0.001 for the restraints applied using the hyperbolic function) was used to derive partial charges from canonical force fields that are conformation independent. In this way the partial charges assigned to the atoms in residues in the MD simulations could be selected according to the local (nearest neighbor) pattern of residue type. The partial charges used for the first and last residues of chains of interest in MD were assigned using the results obtained for the atoms in the terminal residues of the trimer, shown in figure 2b. These calculations were performed on a Intel(R) Xeon(R) 3.50GHz desktop computer and took around 18 hours to complete for each configuration. Tables 1 and 2 show the partial charges of atoms calculated respectively for the central and end residues for the different trimer configurations. M labels methylesterified galacturonic acid residues; and G, galacturonic acid residues; while non-r labels the non-reducing end of the oligomer and r, the reducing end. The electrophoretic mobility simulations were carried out in phosphate buffer, in order to match the experimental conditions as closely as possible, and as such the partial charges of the electrolyte atoms were also calculated and are shown in Table 3.

MD Armed with reliable partial charges obtained from the quantum mechanical calculations carried out for a set of trimer sequences as described, larger HGs were constructed in-silico (figure 2c) and MD simulations carried out. SPC/E water molecules 56 were used. The temperature was maintained at 298 K using a Berendsen thermostat 57 , all bonds were kept


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constrained, and the pressure was kept fixed at 1.0 bar using the Parrinello-Rahman algorithm 58 . Periodic boundary conditions were applied and the integration step was set to 2 fs. Coordinates were recorded every 500 steps for the first nanosecond in order to calculate a molecular diffusion coefficient, and subsequently an external electric field was applied, with coordinates recorded every 5000 steps in this part of the simulation. The Particle Mesh Ewald summation method 59 was used in order to treat the long-range electrostatics. The simulations were performed using supercomputing facilities available through NeSI (https://www.nesi.org.nz) and took around 15 hours to complete for each individual case. The initial configurations of the HG chains were generated based on those available at glycam.org, with bonded and Lenard-Jones parameters retrieved from the GLYCAM force field 53 , which has been specifically developed to simulate the dynamics of sugars and sugar-like molecules. PyMol software 60 was used to perform methylesterification of selected galacturonic acid residues. MD visualization was performed using VMD 61 . Phosphate buffer was simulated by mixing HPO4 −2 and H2 PO4 − anions to obtain a specific ionic strength, and finally the whole system was neutralised by Na+ cations. 200 ps of simulation was performed for both the NVT and NPT ensemble prior to the start of the simulation proper, which was sufficient to achieve equilibrium of temperature, pressure and density. After 1 ns of simulation without the presence of any electric field, simulations were run in which external electric fields were applied, typically with E = 0.01, 0.02, 0.03 and 0.04 V nm−1 . Additional details regarding the MD simulations are provided in Table S.1 presented in the Supplementary Information.


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Results and Discussion MD Box Size, the Viscosity of SPC/E Water, and Field-Induced Perturbations of the Ion Atmosphere In principle the electrophoretic mobility of the in-silico simulated species can be obtained simply by monitoring the drift velocity (the movement of the centre-of-mass of the molecule with time) versus the applied electric field. However, in order to obtain electrophoretic mobilities that can be compared with experiment, a number of factors must be take into account. Firstly, because of the periodic boundary conditions used, transport phenomena will be affected by the finite size of the MD box. This is well known, and easily corrected for, by performing a series of MD simulations in boxes of different sizes and observing how the self-diffusion coefficient of the modeled molecule scales. The results of simulations carried out using tri-galacturonic acid are shown in the Supplementary Information. Figure S.1. shows the mean squared displacement of the centre-of-mass of the molecule versus time, from which self-diffusion coefficients, D, were extracted from the slope of the linear regime (Equation 5). ∂ h|r(t) − r(0)|2 i t−→∞ ∂t 6

D = lim

(5)

Figure S.2 displays how the calculated diffusion coefficients scale with box size, and how, by applying the following equation, values calculated with any size of box can be corrected to give a physically realistic value (that would obtained from a very large box)

Dmd (L) = D0 −

kB T ξEW 6πηL

(6)

where Dmd is the diffusion coefficient calculated from MD simulation, D0 is the diffusion coefficient, T is the absolute temperature, η is the solvent viscosity, ξEW = 2.83729 62 is the self-term for a cubic lattice and L is the size of simulation box. Appealing to the 13 ACS Paragon Plus Environment

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Nernst-Einstein relationship to provide a relationship between diffusion coefficient and electrophoretic mobility, the scaling relationship can equally provide a reliable method for correcting the results of the electrophoretic mobility simulations for the effects of finite box size. In addition it is well known that while SPC/E water performs well in MD in many regards, it does not exactly reproduce its viscosity and, as such, when transport properties are of interest, the calculated mobility must be scaled by the ratio of the viscosity of SPC/E water and the viscosity of the actual physical solution in which the experiments being modeled were carried out.

Dη =

ηSP C/E D0 ηH2 O

(7)

Finally, it should be noted that, owing to the short lengths of time for which MD can be realistically simulated, to observe reasonable displacements of the molecules under study requires that the electric fields applied in MD are significantly greater than those applied in the experimental situation. It is therefore pertinent to ensure that the field used in the simulations is not so large that it perturbs the ion atmosphere significantly during its application. Figure S.3 shows the radial distribution function of sodium counterions around an HG molecule comprising of 25 charged residues upon application of external electric fields with different strengths. It can be seen that while the application of fields larger than 0.1 V nm−1 substantially modify the distribution of the ions, smaller fields can be safely applied with minimum perturbation. Having established the strength of electric fields that could safely be applied in the simulations, and how the results could be corrected to account for the effects of the finite box size and the underestimated viscosity of the simulated water, simulations of the dependence of the electrophoretic mobility of HGs on fine structure were carried out. Figure 3 shows how a typical simulation was performed by monitoring the position of the centre-of-mass of the molecule in question as a function of time, extracting a drift velocity, and repeating 14 ACS Paragon Plus Environment

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the simulation while incrementing the strength of the applied electric field. From this electrophoretic mobilities were extracted (using µ =

v ), E

and subsequently scaled as described

to produce quantitative experimental predictions.

Figure 3: Illustration of the simulation procedure: Drift velocities, v, for molecules of interest are obtained from the slope of plots of the centre-of-mass coordinate, z, movement versus time, and plotted against applied field strength, E, (Inset), from which the electrophoretic mobility was obtained.

Calculation of electrophoretic mobility as a function of the patterning of charged groups and the role of counterion condensation HG samples, which provide a substantial amount of the prior experimental data 31 , typically have a degree of polymerisation of around 100 residues, while pectin samples (comprising of several HG domains that are linked together) are larger still. Molecules this large are currently challenging to study in-silico becoming prohibitively time consuming to simulate as their size increases. However it is known that above a length of around 20-25 residues fully charged oligosaccharides with different degrees of polymerization exhibit experimentally indistinguishable electrophoretic mobilities. 63 This is a general phenomena and it is, in fact,


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crucial to the demonstrated ability of CE to extract DM distributions from electrophoretic mobilities that the measurements are unperturbed by differences in the molecular weight of the chains. This means that MD-tractable calculations performed on molecules of reduced DP are still expected to be relevant to larger HGs and to pectins. The results of preliminary simulations that considered how the electrophoretic mobilities of fully charged oligoglacturonides change with their degrees of polymerisation, show that indeed simulating 25-mers is a good compromise between species containing enough residues for patterning to be investigated while gleaning results relevant to even larger HGs in a reasonable simulation time.

Randomly Methylesterified Substrates Table 4 shows the fine-structure of several (DP=25) HGs, each with different specific sequences of methylesterification, whose electrical transport was investigated using MD. Simulations were first carried out for several different molecules possessing the same chainaveraged DMs: typically two different sequences that had been generated randomly and one with a regular placement of methylesterification were investigated (Shown in Table 4). Figure 4 shows the results of these molecular dynamics simulations superimposed on experimental data of electrophoretic mobility versus fractional charge, as shown in Figure 1. The quantitative agreement obtained with only one variable parameter is remarkably good, and suggests that the MD simulations capture the important physics of the transport of these polyelectrolytes well. (This variable parameter can be incorporated as a scaled viscosity and in this case would correspond to an increase compared to water to a value of around 1.06 mPa s, some 20% larger than that of pure water, but eminently reasonable considering the local concentration of polymer present in the sample zone of the separation.) Motivated by this observation we examined more closely the spatial distribution of ions around the HGs and their dynamics in order to illuminate how these changes ultimately yield the observed transport behavior particularly for systems with z > 0.6.


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Figure 4: Electrophoretic mobilities, µ, measured for HGs and a number of pectin samples, as a function of the fraction of the sugar rings charged, z, compared with the results of MD simulations described herein. The lines are splines, added as a guide to the eye. Figures 5 (a and b) show the number density of counterions as a function of the radial distance from the chain, R, obtained for several HGs. In all cases, a regime is observed where the ion concentration varies as a ‘diffuse layer’, that is; as predicted by the PoissonBoltzmann (PB) equation described in the introduction in equation 3. Fitting the data in this region to the PB model allows the extraction of a polyelectrolyte parameter, ξ, which gives the ratio of an average charge spacing along the backbone to the Bjerrum length (equation 1). In cases where the HG samples were highly methylesterfied (>40%) the average charge spacing predicted in this fashion was indeed found to be close to that which would have been predicted simply from counting every unmethylesterified carboxyl group as a single electronic charge and using inter-charge spacings obtained directly from the molecular structure. However, as the putative charge spacing calculated in this way decreases and becomes less than the Bjerrium length, the ions in the diffuse regime appear from their distribution to be surrounding a polymer of significantly lower charge density than expected. These more sparsely methylesterifed samples have an chain-averaged polyelectrolyte parameter of greater than 1 and as such are indeed predicted to exhibit counterion condensation yielding 17 ACS Paragon Plus Environment

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Figure 5: Number density (ρ) distributions for counterions as a function of closest distance from the chain, R, obtained for several randomly methylesterifed HGs with different DMs. Left, linear; Right, Logarithmic. Lines show fits to equation 3 allowing the extraction of ξ. an ion-decorated chain with reduced charge density compared to the naked polyelectrolyte. Details of the results of these simulations are given in Table 5, including a comparison of the linear charge density, λ, (electronic charges per nm) both naively calculated from the structure and extracted from the fits to the simulation results as described. It has been suggested previously that, by monitoring a point of inflection in a plot of the fraction of ions found within cylinders of certain radii as a function of the radii, a socalled Manning radius can be defined within which ions are taken as condensed 37 . However, in these simulations, which were primarily concerned with the local environment around charged regions of increasing but limited extents, such an approach is complicated by the heterogeneity of the polyelectrolytes of interest. In addition the HG systems investigated here have a maximum ξ of around 1.6, and are exclusively for monovalent ions. Under these conditions the inflections clearly seen at higher values of ξ or for multivalent ions 37 are difficult to locate. For these reasons we turn to the examination of the restricted nature of the thermal fluctuations of the ions relative to the Bjerrium length as a pragmatic demarcation between those that are free or condensed. In order to provide further evidence that our simulations capture counterions condensing onto the more highly charged HGs, and that it is this phenomena that explains both the changes in the electrophoretic mobility and the


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modified distribution of ions observed in the diffuse layers, the mobilities of ions at different proximities from the chain were examined. Figure 6 (a-c) shows how the radial distance of ions from the chain backbone changes over the timecourse of the simulations; the data points represent the average, and the bars represent the standard deviation of the fluctuations in the position. When the degree of methylesterification is high (c: 80%) (the charge density is low) all ions appear relatively mobile and none of them spend long times within a Bjerrium length of the HG backbone (indicated by the dashed line). In contrast, as the degree of methylesterification decreases (b: 40% and a: 0%) (so that the charge density increases and the naively calculated polyelectrolyte parameter approaches or passes one respectively) the motion of several counterions clearly become constrained. In particular for the completely charged chain several ions are now found only in the proximity of the chain (closer than the Bjerrium length) over the entire length of the simulation. These are identified as the condensed ions. While this assignment is somewhat pragmatic it finds justification by observing that the linear charge density of ions selected in this manner closely resembles the amount by which the linear charge density of the polyelectrolyte, λ, is found to be modified (5). That is; selecting condensed ions according to this definition and using their linear charge density to calculate the appropriate value for the ion-decorated chain is consistent with the reduced value of linear charge density extracted using the simulated ion distributions.

Substrates Exhibiting Blockwise Demethylesterification Finally attention is focused on results from samples where, although the overall chainaveraged degree of methylesterifcation is not low enough for significant condensation to take place if the distribution of the charged groups along the chain was random, high localized charge densities can nevertheless occur. In this case these locally blocky charge distributions have been introduced into the fine structure by a processive enzyme 64 . Figure 7(a) shows the electropherograms obtained from experiments carried out on two


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Figure 6: The radial distance of each ion from the chain backbone over the timecourse of the simulations (a, b and c; DM 0, 40 and 80 % respectively). Data points show the average position, and the bars represent the standard deviation of the fluctuations in the position over whole simulation. The dashed line denotes the Bjerrum length, and the greyscale-darkness of the region of space in closer proximity to the chain represents the fraction of the backbone charge effectively neutralised by the counterions.


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pairs of samples, each pair with similar sample-averaged degrees of methylesterification (∼ 47 or ∼ 57%) but with different intramolecular charge distributions. These samples were generated by starting with the same highly methylesterified (∼78%) mother sample and using different methods of demethylesterification, i) a method that removes methylesters randomly and ii) an enzymatic method that removes methylesters processively; as described in the experimental section. The EOF can be seen just before 4 minutes, and the migration of the co-injected DNA-oligomer markers at around 19 mins; (the broad baseline feature that starts just after 14 minutes arise from the phosphate buffer 32 ). It can clearly be seen that within each pair of similar DM samples the partner that contains localised blocks of charge has a lower mobility, attesting to a lower overall charge density of the migrating species. It is hypothesised that local counterion condensation takes place in the case of the blocky samples, negating some of the charge density and producing a difference in the electromigration of the HGs and their ion-decorated counterparts. In order to investigate whether this effect could be predicted from the results of our MD simulations we calculated the electrophoretic mobility of HGs with fine structures that were designed to mimic the blocky samples studied in our experiments. These fine structures were designed by modeling the action of the processive enzyme in question as described in detail elsewhere 30 in order to produce likely methylester sequences of similar overall DMs to those studied in the experiments. As can be seen in Table 4, such sequences do possess local blocks of unmethylesterified regions. Figure 7 (b) and (c) shows the experimental electropherograms, with the time-axis transformed to electrophoretic mobility, overlayed with several different fine structures that have been placed in the figure according to their predicted mobilities that have been obtained through MD simulation. It is clear that in the DM range examined the differences in the electrophoretic mobility of samples possessing random or blockwise intramolecular distributions of charge can indeed be explained by local counterion condensation, and is captured by the MD. The results of the MD simulations performed on these blocky fine structures are also shown in figure 4.


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Figure 7: (a) Electropherograms obtained from experiments carried out on pairs of blocky and random fine structures of similar DM; (b) and (c) Measured electrophoretic mobility distributions of DM∼47 and DM∼57% respectively, along with pictures of fine-structures that have been investigated in MD, placed at a position corresponding to the resulting calculated mobility.


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So, how blocky does an HG have to be to induce local counterion condensation, thereby modifying the local distribution of ions enough to affect the measured electrophoretic mobility of the chain? Figure 8 shows how the positions of ions (relative to the charged block) change over the timecourse of the simulations, specifically in the proximity of charged blocks of different lengths sandwiched between two runs of methylesterified residues. Once again the points show the average position of ions, while bars show the standard deviations of the fluctuations. Note that in previous figures R denotes the closest distance between an ion and any backbone atom while here r denotes the closest distance between an ion and any atom within the charged block. It should be noted that, in contrast to the simulations carried out previously, in some of these cases partially-condensed species could be observed; that is, ions that spend a considerable amount of time within a Bjerrum length of the chain before escaping, as shown in figure 9. In such cases the degree of charge that such a block contributes to that reflected by the electrophoretic mobility of the species is approximated by subtracting the charge of the condensed ions averaged over the timecourse of the simulation away from that of the residues themselves. Finally, figure 10 shows the fraction of the galacturonic acid residues within a continguous block that host a condensed ion, as the local block-length increases. Once the block-length reaches around ten residues the ion-decorated block only manifests some 65% of the nominal charge of the galacturonic residues, consistent with the behaviour observed for longer galacturonic oligomers, and in line with the 1 − ( 1ξ ) predicted 37 . In addition an empirical fit to the data below block lengths of ten enables an approximate prediction of the electrophoretic mobility of any fine structure, from its block length distribution, taking counterion condensation into account without the necessity of carrying out MD simulations.


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Figure 8: Representative results of the distance of counter-ions and co-ions, r, from charged blocks of length: 1 (a), 2 (b), 5 (c), 10 (d), and 13 (e), sandwiched between small methylesterfied blocks as shown. (Note that in previous figures R denotes the closest distance between an ion and any backbone atom while here r denotes the closest distance between an ion and any atom within the charged block ). The dashed line denotes the Bjerrum length, and the greyscale-darkness of the region of space 24 in closer proximity to the chain represents the Paragonneutralised Plus Environment fraction of the backbone chargeACS effectively by the counterions.

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Figure 9: The behaviour of ‘free’, ‘partially condensed’ and ‘condensed’ ions over the course of the simulation shown in figure 8 (e) (gray, dark gray and black respectively). Data points show the average position, and the bars represent the standard deviation of the fluctuations in the position over the presented time.

Figure 10: The fraction of charge of galacturonic blocks that is negated by counterion condensation (the fraction of the galacturonic acid residues within a continguous block that host a condensed ion) as a function of block length, together with a fitted empirical relationship to data for block lengths less than 10 residues (A Log(Bx) where A = 0.34 ± 0.02 and B = 0.81 ± 0.07), dashed line is 1 − 1ξ .


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Conclusion By using carefully designed anionic polysaccharides, with different degrees and patterns of charge, and performing both experimental measurements and MD simulations of their respective electrophoretic mobilities, the role that counterion condensation plays in determining the electrical transport properties of these polyelectrolytes has been investigated. When the DM of HGs whose charged groups are randomly positioned along the polymer backbone becomes low enough counterion condensation takes place. In addition, HGs with non-random distributions of charged groups can exhibit local counterion condensation regardless of the average charge spacing if the patterning of charged groups becomes blockwise enough. In these cases both the number density distribution of mobile diffuse ions and the measured electrophoretic mobility reflect the charge density of the ion-decorated polyelectrolyte. These simulations not only give a clearer understanding of the physics in play but also permit that, if independent information is available from other measurements regarding the charge patterning motif of the sample (for example something is known of the demethylesterfication process), then the number of charges on the pre-condensed substrate can be ascertained from the charge density of the ion-decorated species. That is; even in cases where counterion condensation is prominent, (for samples of low DM or a blocky nature) experiments that measure the electrophoretic mobility can be still be useful indicators of the polyelectrolyte fine structure.

Supporting Information Figure S.1. shows the mean squared displacement of the centre-of-mass of the molecule versus time, from which self-diffusion coefficients, D, were extracted. Figure S.2 displays how the calculated diffusion coefficients scale with box size, and how calculated with any size of box can be corrected to give a physically realistic value. Figure S.3 shows the radial distribution 26 ACS Paragon Plus Environment

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function of sodium counterions around an HG molecule comprising of 25 charged residues upon application of external electric fields with different strengths. Table S.1. gathers details of the MD simulations.

Author Contributions A.H.I performed MD simulations; J.L.O carried out CE experiments; D.M. advised on computational aspects; and all authors designed the research and contributed to writing the paper.

Acknowledgments Lisa Kent is acknowledged for help with the selection and supply of the DNA oligomeric standards in the CE experiments. Fu-Guang Cao is acknowledged for helpful discussions. New Zealand eScience Infrastructure (NeSI) are acknowledged for help with the provision of supercomputing facilities and support.


C1 C2 C3 C4 C5 C6 C7 O2 O3 O4 O5 O6a O6b H C1 HC2 HC3 HC4 HC5 HC7a,b,c HO2 HO3

Atoms

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28 b

a

Configuration of trimers GMM MGG 0.210043 0.252289 0.141114 0.010878 0.108965 0.339527 0.360802 −0.012004 −0.649044 0.051279 0.994454 0.757562 −0.104429 −0.581491 −0.653183 −0.633724 −0.595536 −0.337687 −0.350809 −0.258792 −0.418866 −0.576614 −0.745235 −0.367808 −0.745235 0.11751 0.095236 0.11006 0.09468 0.069077 −0.010498 0.072211 0.101878 0.20975 0.046812 0.104443 0.392374 0.434081 0.419946 0.396357

MGM MMG MMM 0.218816 0.353304 0.234788 0.055789 −0.043442 0.090265 0.303079 0.380371 0.248778 −0.02664 −0.007495 0.17443 −0.091423 −0.097512 −0.504228 0.807265 0.854788 0.98798 0.075765 −0.000708 −0.562525 −0.626992 −0.570358 −0.580636 −0.593728 −0.580762 −0.328754 −0.32173 −0.340898 −0.377406 −0.424225 −0.316513 −0.750284 −0.562269 −0.575778 −0.750284 −0.43451 −0.424128 0.114181 0.086154 0.117785 0.124973 0.115228 0.120726 0.004016 −0.01246 0.026812 0.129669 0.119146 0.10861 0.096481 0.112453 0.215869 0.059373 0.079496 0.378744 0.420522 0.382058 0.375051 0.391431 0.39196

G: Galacturonic acid residue (charged), M: Methyl-esterified galacturonic acid residue (uncharged).

GGMb

GMG 0.295411 0.028503 0.375167 0.004091 0.211784 −0.015165 0.363802 0.25631 0.247381 −0.00033 0.091063 0.081301 0.059998 −0.168396 −0.066966 0.762103 0.839591 0.837277 −0.039288 −0.666086 −0.626173 −0.649277 −0.62102 −0.604337 −0.647126 −0.336961 −0.345825 −0.272239 −0.436404 −0.308606 −0.42838 −0.752341 −0.75608 −0.567394 −0.752341 −0.75608 −0.398075 0.077065 0.107689 0.075339 0.088749 0.10881 0.097866 −0.021592 −0.000831 0.037281 0.096458 0.103224 0.089566 0.035448 0.111034 0.078769 0.090176 0.437909 0.383868 0.435538 0.403485 0.388749 0.434768

GGGa

Table 1: The partial charges of atoms for the central residue in different trimers

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Atoms C1 C2 C3 C4 C5 C6 C7 O1 O2 O3 O4 O5 O6a O6b H C1 H C2 H C3 H C4 HC5 HC7a,b,c HO1 HO2 HO3 HO4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 G MN GM GG GM GG r r on−r N on−r N on−r 0.138035 −0.081060 0.217976 0.305962 0.237162 0.142353 0.332360 0.126433 0.060131 0.061565 0.276928 0.200103 0.275893 0.393768 0.358668 0.171072 0.309286 −0.024045 −0.153665 0.235028 0.015365 −0.270597 −0.079502 0.055059 −0.220026 0.674528 0.716553 0.785623 0.736887 0.836663 −0.086742 −0.638295 −0.642703 −0.691993 −0.656639 −0.629435 −0.619936 −0.677417 −0.643421 −0.640147 −0.618586 −0.63462 −0.668783 −0.656472 −0.699794 −0.339676 −0.359699 −0.658769 −0.369253 −0.235501 −0.376872 −0.429101 −0.379368 −0.716826 −0.709020 −0.747840 −0.733502 −0.558800 −0.716826 −0.709020 −0.747840 −0.733502 −0.379480 0.100269 0.132331 0.082742 0.062306 0.093441 0.047674 0.063912 0.114644 0.130765 0.092074 −0.013474 −0.008133 0.028955 0.006821 −0.018590 0.021453 0.013464 0.119266 0.144321 0.024851 0.023728 0.131926 0.092657 0.062229 0.113132 0.105382 0.466898 0.461373 0.441763 0.400868 0.415905 0.406515 0.439672 0.411355 0.397842 0.399467 0.399775 0.425189 0.381925 0.415211 0.436962

M MrM MrG MN on−r 0.171910 0.325665 0.336403 0.171742 0.040349 0.017793 0.225641 0.317615 0.368305 0.356168 −0.073884 −0.112143 −0.507068 −0.112010 −0.098054 0.956310 0.830591 0.824933 −0.106765 0.094453 0.092716 −0.627780 −0.627664 −0.611177 −0.604182 −0.599261 −0.657305 −0.598190 −0.603951 −0.659229 −0.322105 −0.308813 −0.306195 −0.409427 −0.414181 −0.576409 −0.550237 −0.545287 −0.391783 −0.439973 −0.436369 0.109623 0.075920 0.076089 0.112790 0.140270 0.143740 0.025683 0.032605 0.028649 0.030397 0.140252 0.151294 0.184131 0.137802 0.137354 0.109680 0.056190 0.057937 0.464362 0.464691 0.393186 0.405752 0.404966 0.415795 0.400892 0.398364 0.434955

Table 2: The partial charges of atoms for end residues

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Table 3: The partial charges for atoms of the phosphate ions used in the MD simulations. H2 PO− 4 atoms partial charges P 1.008824 O1 ,O2 −0.759102 O3 , O4 −0.664733 HO3 , HO4 0.419422

HPO−2 4 atoms partial charges P 0.910761 O1 ,O2 ,O3 −0.856866 O4 −0.714562 HO4 0.374399


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sample Sequence of Sample 1 G G G Sample 2 M G M Regular G G M Sample 1 M M M Sample 2 G G M Regular G M G M M M Sample 1 M G G Sample 2 G G G Regular M G M Sample 1 G M M Sample 2 G M M Regular M M G

random G G G M G G M G G M G M G M M G M M M G G G M G M M

residues G M G G G G G G G M G M M M G G M G M G M M M M M M

DM% Sample Sequence of blocky residues 48 % M M G G G G G 60 % M M G M M M M

80 %

60 %

48 %

40 %

20 %

DM%

G M

G G M G G G M M G M G G G

G G G G G M G M M G M G M

G G M M

G M G G M G G M M M M M M

G M

G G G M G G G G M M M M M

G M

G M G G G M G M M G M M M

G G G G M G G M G M M M M

M G G M M M G M G G M M M

G G G G G G M G M M M M M

G G G G G M G M G G M M M

G M G M M M G M G G G G

G G M M M G M M M M M M G

G G G G G G M G M M M M M

M G G M M M M M G G M M M

M M M G G G

M G M M G G M M M M M M G

G G G G M M M G M G M G M

M G G M

M G G G G G G G G M M M M

G G G M G G G M M M G M M

M M M M

G G M G G G G M M M M M G

G G G G M M G M M G M M M

Table 4: The sequence of residues for different DM substrates on which MD simulations were carried out (G; charged, M; uncharged).

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Table 5: Linear charge density calculated from the structure λ = eb (e nm−1 ), polyelectrolyte charge parameter, ξ, external electric field applied during calculation in V nm−1 , effective charge parameter, ξef f , calculated by fitting ρ0 R−2ξ to the simulation results in the Gouy region, corresponding effective linear charge density, λef f , and the linear charge density of ’condensed’ ions within the Bjerrum length of the chain. DM%

Sample

0%

-

2.016

1.608

Sample1

1.605

1.280

Sample2

1.597

1.273

Regular

1.608

1.282

Sample1

1.182

0.942

Sample2

1.187

0.947

Regular

1.223

0.975

-

1.021

0.814

Sample1

0.79

0.630

Sample2

0.788

0.628

Regular

0.753

0.601

Sample1

0.369

0.294

Sample2

0.357

0.285

Regular

0.355

0.283

20 %

40 %

48 %

60 %

80 %

λ

ξ = lB λ

Eext 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01

ξef f 0.98±0.06 0.95±0.04 0.91±0.04 0.91±0.05 0.90±0.03 0.89±0.07 0.99±0.04 0.90±0.05 0.65±0.07 0.64±0.03 0.65±0.07 0.70±0.04 0.65±0.04 0.71±0.07 0.64±0.03 0.64±0.03 0.55±0.06 0.53±0.05 0.54±0.06 0.49±0.09 0.41±0.07 0.51±0.05 0.30±0.02 0.29±0.07 0.27±0.05 0.28±0.06 0.29±0.07 0.27±0.03

ξ

λef f = lefBf 1.20±0.08 1.19±0.05 1.14±0.05 1.14±0.06 1.13±0.04 1.12±0.09 1.24±0.05 1.13±0.06 0.81±0.09 0.80±0.04 0.82±0.09 0.88±0.05 0.82±0.05 0.89±0.09 0.80±0.04 0.80±0.04 0.69±0.08 0.64±0.06 0.67±0.08 0.6±0.1 0.51±0.09 0.65±0.06 0.38±0.03 0.36±0.09 0.34±0.06 0.35±0.08 0.36±0.09 0.34±0.04

λQions 0.727 0.755 0.437 0.482 0.486 0.496 0.357 0.436 0.292 0.310 0.327 0.296 0.294 0.302 0.189 0.181 0.116 0.147 0.140 0.172 0.057 0.146 -0.024 0.031 0.041 -0.009 -0.020 -0.017

References (1) Vincken, J.-P.; Schols, H. A.; Oomen, R. J.; McCann, M. C.; Ulvskov, P.; Voragen, A. G.; Visser, R. G. Plant Physiol. 2003, 132, 1781–1789.


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(2) Yapo, B. M. Carbohydr. Polym. 2011, 86, 373–385. (3) Round, A. N.; Rigby, N. M.; MacDougall, A. J.; Morris, V. J. Carbohydr. Res. 2010, 345, 487–497. (4) Yapo, B. M.; Lerouge, P.; Thibault, J.-F.; Ralet, M.-C. Carbohydr. Polym. 2007, 69, 426–435. (5) Peaucelle, A.; Braybrook, S.; Höfte, H. Front. Plant. Sci. 2012, 3, 1–6. (6) Braybrook, S. A.; Hofte, H.; Peaucelle, A. Plant Signaling Behav. 2012, 7, 1037–1041. (7) Levesque-Tremblay, G.; Pelloux, J.; Braybrook, S. A.; Müller, K. Planta 2015, 242, 791–811. (8) Wolf, S.; Greiner, S. Protoplasma 2012, 249, 169–175. (9) Mansel, B. W.; Chu, C.-Y.; Leis, A.; Hemar, Y.; Chen, H.-L.; Lundin, L.; Williams, M. A. Biomacromolecules 2015, 16, 3209–3216. (10) Cameron, R. G.; Kim, Y.; Galant, A. L.; Luzio, G. A.; Tzen, J. T. Food Hydrocolloids 2015, 47, 184–190. (11) Assifaoui, A.; Lerbret, A.; Uyen, H. T.; Neiers, F.; Chambin, O.; Loupiac, C.; Cousin, F. Soft matter 2015, 11, 551–560. (12) Ström, A.; Ribelles, P.; Lundin, L.; Norton, I.; Morris, E. R.; Williams, M. A. Biomacromolecules 2007, 8, 2668–2674. (13) Watson, J. D.; Crick, F. H. Nature 1953, 171, 737–738. (14) Gómez-Márquez, J. FEBS J. 2010, 277, 3451–3451. (15) Sun, B.; Wang, M. D. Crit. Rev. Biochem. Mol. Biol. 2015, 1–11. (16) Rothemund, P. W. Nature 2006, 440, 297–302. 33 ACS Paragon Plus Environment

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Graphical TOC Entry


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Molecular Dynamics Simulations Illuminate the Role of Counterion

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