Charge symmetry breaking in neutral polyzwitterions

Translocation experiments and charge symmetry breaking

The experimental setup (see “Methods”) is a flow cell consisting of two chambers (donor and receiver) with a 10 nm-thick silicon nitride membrane in between. We have fabricated a nanopore of desired pore diameter (3.5–3.7 nm) within the membrane using the controlled dielectric breakdown method^60,61. The choice of pore diameter was made to allow essentially single-file passage of extended conformations of poly(sulfobetaine methacrylate) (PSBMA) and poly(2-methacryloyloxyethyl phosphorylcholine) (PMPC) and to permit neither simultaneous occupancy of two chains inside the pore nor passage of the molecule as multiply folded blobs. Using the OVITO software⁶² the cross-sectional diameter of these molecules is  ~2.2 nm without accounting for hydration around the zwitterion moieties. Computer simulations⁶³ have shown that water molecules are bound to polyzwitterion backbone that can extend to multiple layers and that the mobility of water molecules near the backbone returns to the bulk solvent mobility only at about 1.8 nm from the backbone. Of course, most of the water molecules bound to the polymer are expected to be stripped away when the polymer undergoes translocation. However, as a conservative estimate, if we take the length of the pendant zwitterion group to be extended by an effective layer of even one strongly bound water (of length  ~0.28 nm), the effective cross-sectional diameter of the polymer is  ~2.8 nm. Therefore, the choice of pore diameter of 3.5–3.7 nm would prevent the simultaneous passage of more than one chain through the nanopore. The monomer length is estimated as 0.25 nm (using OVITO) and the average contour lengths of PSBMA and PMPC are estimated as 128 nm and 31.5 nm, respectively, which are longer than the pore length. After filling the flow cell with 1 M KCl, 10 mM HEPES, pH 7, the ionic current through the open pore is measured by applying an external electric potential difference ΔV across the two chambers of the flow cell. After adding 100 nM PSBMA (Fig. 1a) to the donor chamber, we have recorded the ionic current as a function of time.

a Chemical formula of PSBMA with positively charged group closer to the chain backbone and the negatively charged group away from the chain backbone. b Titration curve of PSBMA (80 mM in 1 mL of 1 M KCl) showing charge neutrality at the experimental pH range. In the positive configuration of the experimental setup (c), PSBMA does not show any translocation event (d). In the negative configuration of the experimental setup (e), PSBMA exhibits a series of translocation events, behaving like a polyanion (f). g Differential binding of counterions to the positive and negative groups of PSBMA, originating from the gradient of local dielectric constant, causes CSB. h Sketch of the local dielectric constant as a function of the distance away from the chain backbone. In (c–f), solution: 10 nM PSBMA in 1 M KCl buffered at pH 7 with 10 mM HEPES, pore diameter: 3.5 nm, voltage: 100 mV. In (d–f), sampling rate: 250 kHz, low-pass filter frequency: 10 kHz.

The main result is shown in Fig. 1 for the single-molecule electrophoresis of PSBMA.

At the experimental pH 7, the polymer is neutral (because pK_a of the sulfonic acid is around -7 so that it is fully deprotonated and the quaternary ammonium group carries a permanent positive charge) as evident from its pH titration curve (Fig. 1b). In the positive configuration (Fig. 1c) of the chambers, where the positive electrode is in the donor chamber, the measured ionic current through the nanopore without PSBMA is 0.95 nA at the applied voltage of 150 mV. Upon addition of 100 nM PSBMA into the donor chamber, the measured ionic current remains the same as the open pore current without any transient blockages that would be the earmark of PSBMA translocation (marked as “no effect” in Fig. 1d). This result is consistent with the thought that neutral polymers hardly can undergo electrophoretic translocation.

However, remarkably, in the negative configuration (Fig. 1e), where the negative electrode is in the donor chamber, upon addition of 100 nM PSBMA in the donor chamber, the molecules start translocating through the nanopore one at a time as evident from the ionic current blockages in the ionic current trace (Fig. 1f). As described below, most of these blockages are successful translocation events. Thus, PSBMA behaves like a polyanion even though it is neutral at the experimental pH (Fig. 1b). This means that the charge symmetry between positive and negative charges—where one positive and one negative charge balance each other to maintain neutrality (charge symmetry)—constituting the zwitterionic group is broken. We show below that this unexpected phenomenon of CSB is universal for all polyzwitterions independent of the zwitterion’s dipole orientation to the chain backbone, and counterion identity.

We attribute CSB to the dielectric heterogeneity around polymer backbones. As already alluded to, since ΔG is inversely related to ϵ_ℓ, the counterion binding is stronger in regions with lower dielectric constant. As expected for polymers in aqueous solutions, the local dielectric constant is low near the oil-like chain backbone (Fig. 1g), and continuously increasing to  ~80 in the bulk solution away from the chain (sketched in Fig. 1h). Therefore, the ionic group of the zwitterion closer to the backbone is subjected to more counterion binding (and thus more neutralized) compared to the ionic group further away from the chain backbone. This differential counterion binding makes the charge of the outer ionic group dominant (CSB).

The cause of the observed polyanionic behavior of PSBMA might, in principle, arise from several factors that include electrophoretic mobility due to differential counterion binding, EOF, stronger electric fields near the charged pore wall affecting peripheral ionic group more selectively, and stronger electric fields at the pore entrance. After careful scrutiny of these various potential causes given below, we conclude that the above-described mechanism of differential counterion binding across a gradient in the local dielectric constant is responsible for CSB.

The isoelectric point of our silicon nitride pore is about pH 6 (Supplementary Fig. 1, consistent with literature value⁶⁴), and the charge density outside pH 6 is only weak⁶⁵. In the experimental condition of pH 7 for PSBMA, the pore lumen is negatively charged so that the EOF is towards the negative electrode^10,47. Since PSBMA moves towards the positive electrode, the EOF is overwhelmed by the electrophoretic force resulting in the polyanionic behavior of PSBMA. Furthermore, the calculated radial electric field from the pore wall decays sharply within a very short distance comparable to the short Debye length corresponding to 1M KCl solution in the experiment⁴⁷. Since such electric field gradients do not depend on the externally applied voltage across the pore, the radial field gradient cannot be attributed as the origin for the observed voltage dependence of rectified mobility of the polyzwitterion. Also, the strength of the electric field along the field direction is uniform inside the nanopore (where translocation occurs) except at a very thin skin at the pore wall⁴⁷. Furthermore, stronger electric fields at the pore mouth result in enhanced localization of dipolar polymers at the pore mouths which increases the free energy barrier for the subsequent translocation. This will result in the opposite consequence on the voltage dependence of the translocation time compared to our experimental observation. Furthermore, such localized capture at the pore mouth, on its own, cannot lead to the observed voltage polarity dependence in the current blockades. Therefore, CSB cannot be attributed to any inhomogeneity of the electric field or EOF.

As a further validation of CSB, the frequency of arrival of molecules inside the pore entrance (capture rate R_c) increases with voltage (Fig. 2, left) in the negative flow cell configuration, consistent with PSBMA behaving like polyanions. Following previous works on single-molecule electrophoresis^11,14,38,42, R_c is obtained as the inverse of the average time duration τ_a (Fig. 2, top right) between the starting times of two consecutive blockage events (see “Methods”). Furthermore, to relate the magnitude of CSB with the effective charge of the dipoles and local dielectric constant, we have measured the blockage current ratio I_b/I₀ (where I₀ is the open pore current and I_b is the minimum current in the blockade) and the blockage duration (τ) as the full width at half maximum of the current blockage (Fig. 2, bottom right). τ is translocation time when the blockages correspond to successful translocation events (see below).

100 nM PSBMA was used in 1 M KCl buffered at pH 7 with 10 mM HEPES. Pore diameter: 3.7 nm, sampling rate: 250 kHz, low-pass filter frequency: 100 kHz. Gaussian filter frequency: 10 kHz (Left) Dependence of the ionic current trace, showing a series of temporary current blockages, on voltage (90 mV, 150 mV, and 200 mV, from top to bottom). (Right) Expanded view of selected current blockage events. τ_a is the inter-arrival time between two successive blockage events; I₀ is the open pore current, I_b is the blockage current corresponding to the minimum of the blocked current, and τ is the full-width at half maximum, identifying the blockage duration.

The dependence of R_c on the voltage is given in Fig. 3a, showing a linear dependence of R_c on ΔV. In general, R_c is controlled by thermal diffusion, electrophoretic drift, and free energy barrier at the pore^26,27. As a result, R_c is essentially zero at very low ΔV, then increases nonlinearly with ΔV in the barrier-dominated regime, and finally is directly proportional to ΔV in the drift-dominated regime. This general trend of R_c is seen for PSBMA. At voltages higher than 70 mV, R_c depends linearly on the applied voltage (Fig. 3a, error bars are from 95% confidence intervals) corresponding to the drift-dominated regime. By extrapolating the linear line to lower voltages until the intersection with zero R_c, the threshold voltage for the onset of the drift-dominated regime is obtained as 29 mV, below which R_c is essentially zero corresponding to the barrier dominated regime. The threshold value implies a free energy barrier for PSBMA capture, which arises from a combination of the entropic barrier for one chain end of jammed polymer coil to be inserted into the pore and polymer conformational entropy.

100 nM PSBMA in 1 M KCl buffered at pH 7 with 10 mM HEPES was used. Pore diameter: 3.7 nm, sampling rate: 250 kHz, low-pass filter frequency: 100 kHz, Gaussian filter frequency: 10 kHz. a Linear dependence of capture rate R_c on voltage showing a threshold for capture of PSBMA. Error bars are from 95% confidence intervals. b Three populations in the event scatter plot of blockage current ratio I_b/I₀ versus logarithm of blockage duration τ, identified as unsuccessful, single-file chain, and intra-pore-structured chain translocations (90 mV). Details of blockages for single-file and intra-pore-structured translocations in the inset. c Histogram of I_b/I₀ and deconvolution into unsuccessful, single-file, and intra-pore-structured translocation events (90 mV). Curves are Gaussian fittings. n is the number of events used for double Gaussian fitting. d Histograms of the translocation time for the single-file and intra-pore-structured translocations (90 mV). Curves are log-normal distribution functions. e Dependence of mean translocation time 〈τ〉 on voltage (magenta circle: intra-pore-structured; green square: single-file). n_ips and n_single are the numbers of events used for log-normal fitting of intra-pore-structured translocations and single-file translocations, respectively. f Expanded view of voltage dependence of 〈τ〉 for the single-file chain translocation. Curves in (e, f) are fittings to exponentials.

In the drift-dominated regime, the slope of capture rate versus voltage (4.7 × 10⁻³ Hz nM⁻¹ mV⁻¹ in Fig. 3a) is proportional to cμM⁻¹, where c is the concentration of PSBMA, μ is the electrophoretic mobility, and M is the pore length. This constant slope indicates that the effective charge of PSBMA chains is a constant and negative in the donor chamber.

The kinetics of translocation is as follows. The earmark of successful translocation of a molecule through a nanopore is the inverse dependence of the average translocation time (〈τ〉) on the applied voltage ΔV^{10,13,28,32,45}. τ is highly stochastic and broadly distributed (Fig. 3b). Conjugate to the distribution of τ, I_b/I₀ is also broadly distributed. The mutual distributions of these quantities for PSBMA at 90 mV is displayed in the event scatter plot (Fig. 3b) of a total of 2289 events. Please see Supplementary Fig. 2 for scatter plots at other voltages. The data in Fig. 3b fall into three populations marked by ellipses. The first population at the top of the event scatter plot (black-dashed ellipse) consists of shallow blockages of very short duration corresponding to collisions of the PSBMA molecules with nanopore entrance, as in the previous studies of polyelectrolyte translocation¹¹. These are labeled as unsuccessful events (Fig. 3b).

The second population (green-dashed ellipse) in the middle has moderate blockages, while the third population (magenta-dashed ellipse) at the bottom has deep and long blockages. Both of these populations correspond to translocation processes, because their average translocation times decrease with voltage (see below). Most of the moderate blockages have one blockage level as shown in Supplementary Fig. 3. The I_b/I₀ histograms fitted with multiple Gaussian distributions show that the mean I_b/I₀ for the moderate events is  ~0.4, as in the example at 90 mV given in Fig. 3c. Please see Supplementary Fig. 4 for other voltages and Methods for fitting details. This mean blockage current ratio corresponds to the cross-sectional diameter of polymer as  ~2.9 nm calculated by I_b/I₀ = 1 − (a/d)², where a is diameter of polymer and d is pore diameter (3.7 nm). Based on the effective cross-sectional diameter of the hydrated polymer ( ~2.8 nm), we assign the second population as the single-file translocation events. Even in this single-file mode, we expect weak conformational fluctuations due to chain flexibility and rearrangement of bound water, etc. as sketched in Supplementary Fig. 5. On the other hand, deep blockade events show complex features in the ionic current as in Supplementary Fig. 6. At least 93% of these events show two different blockage levels within one event as in the inset of Fig. 3b with one blockage level comparable to the blockage level of the moderate events. Although these two levels are not sharply quantized in contrast with dsDNA translocations of folded chains^16,29,44, these are translocation events, because of decrease of their average duration with voltage, which is shown later.

We conjecture that the population of events with deeper blockades corresponds to translocation events where the polymer adopts conformations with one or more local short hernia-like structures inside the nanopore, as cartooned in Supplementary Fig. 7. Such structures can easily arise in the present system due to the flexibility of the chain as well as local association of dipolar monomers. We call the mode of translocation of intra-pore conformations with local kinks or nano-cilia (or nano-blobs) as “intra-pore-structured (ips)” translocation. See Supplementary Note 1 for detailed discussions.

The proportion of the single-file translocation decreases with voltage (see Supplementary Fig. 8) indicating that higher voltages enhance intra-pore-chain structures during translocation. Furthermore, it is possible for the chain to additionally form doubly kinked conformations during translocation at higher voltages (175 mV and 200 mV) with even deeper and longer blockages (see Supplementary Fig. 9). During these translocations, most of the current blockages exhibit three different levels. In general, it is desirable to make an one-to-one correspondence between the polymer conformation inside the nanopore and the ionic current. This is a difficult task even for strong polyelectrolytes such as polystyrene sulfonate and DNA which have high charge density. The difficulty is more severe for flexible chains with very weak charge density that are investigated in the present study. Nevertheless, despite the impossibility to precisely identify the conformational details behind the very short-time features of the various ionic current traces, the large-scale behavior of translocation and the consequent CSB are evident from the data.

To separate and avoid the overlap of single-file and ips translocations in constructing τ histograms for each population, we have taken only the right half of the Gaussian distribution for the single-file translocation events and the left half of that for the ips translocation events. The histograms of the translocation time (τ) for the separated single-file translocation events and the ips translocation events at 90 mV are given in Fig. 3d as an example (for other voltages see Supplementary Figs. 10 and 11). While histograms for the single-file translocations are narrow with shorter translocation times, histograms for ips translocations at the same voltages are broader with very long translocation times. The longer translocation time for the ips translocation events is presumably due to enhanced friction of PSBMA against the pore. The shapes of the histograms are in conformity with that of generic drift-diffusion process^10,12,13. Following the previous practice in the literature^37,45, we have fitted these τ histograms with log-normal distributions in the range of 0 < τ < 10 ms, and 0 < τ < 100 ms for the single-file, and ips translocations, respectively. The average translocation times (〈τ〉) for these two processes are obtained from the log-normal fittings, and their dependence on the voltage for the single-file and ips translocations is portrayed in Fig. 3e f.

For both kinds of translocation, 〈τ〉 decreases roughly exponentially with voltage, implying that overcoming a free energy barrier arising from the loss of conformational entropy during translocation is a dominant factor in determining the voltage dependence of 〈τ〉, analogous to the behavior of single-stranded DNA translocation through solid-sate nanopores³⁰. As mentioned above, this decreasing dependence of 〈τ〉 with voltage is taken as the signature of successful translocation.

Universality of CSB in polyzwitterions

Since we attributed CSB in PSBMA to stronger counterion binding at the positively charged proximal ionic group due to lower local dielectric constant near the chain backbone, compared to the negatively charged distal ionic group, one strategy to validate the generality of CSB is to change the identity of counterion (LiCl instead of KCl) in the above PSBMA tranlocation experiments. Furthermore, CSB should still be present when the the dipolar orientation of the zwitterionic group is reversed, where the negatively charged group is proximal and positively charged group is distal. In this case, the polyzwitterion should move towards the negative electrode and there should not be any translocation towards the positive electrode.

In view of these strategies to validate the mechanism of CSB and its general occurrence, we have investigated the single-molecule electrophoresis of (1) PSBMA in 3.6 M LiCl, 10 mM HEPES, pH 6 and (2) PMPC, with proximal negatively charged phosphoryl group and distal positively charged choline group constituting the zwitterion group attached to the backbone, in 3.6 M LiCl, 10 mM HEPES, pH 3.7. Note that at pH 3.7, the pore surface is positively charged leading to EOF towards to the positive electrode with the MPC moiety being net charge neutral in the absense of CSB. As shown below, these two systems exhibit the predicted CSB demonstrating the universality of the CSB in polyzwitterions.

The independence of CSB on salt identity is found as follows. By changing the salt identity to 3.6 M LiCl (pH 6), instead of 1 M KCl (pH 7) in Figs. 1c–f, 2, and 3, examples of ionic current traces for PSBMA are in Fig. 4a, b at the illustrative voltage of 200 mV. Again, we did not find current blockages in the positive configuration (Fig. 4a), but find blockage events in the negative configuration (Fig. 4b). Following the same procedure of data analysis used above for PSBMA in 1 M KCl (see Supplementary Figs. 12, 13, 14, and 15 for event scatter plots, I_b/I₀ histograms, single-file translocation time histograms, and ips translocation time histograms, respectively), R_c depends linearly on ΔV indicating drift-dominated capture (Fig. 4c). The slope of the line in this regime is weaker (by a factor of about 3) than in the case of 1 M KCl, primarily due to weaker net charge arising from weaker extent of CSB and higher viscosity of the solution. Extrapolation of this linear relation to lower voltages yields a threshold voltage of 110 mV that is representative of the free energy barrier that needs to be crossed for successful capture. The higher value of the threshold voltage for 3.6 M LiCl, compared to the lower value for 1 M KCl, is due to a lower net charge after breaking the charge symmetry.

100 nM PSBMA in 3.6 M LiCl buffered at pH 6 with 10 mM HEPES was used. Pore diameter: 3.7 nm, sampling rate: 250 kHz, low-pass filter frequency: 100 kHz. a No translocation towards negative electrode in 3.6 M LiCl. b Occurrence of translocation towards positive electrode, whereby PSBMA acts like a polyanion as in Figs. 1 and 3. c Linear voltage dependence of capture rate R_c with a threshold value of voltage (110 mV). Error bars are from 95% confidence intervals. d Voltage dependencies of mean translocation time 〈τ〉 for single-file (main) and intra-pore-structured (inset) translocation events. Curves are fittings to exponentials.

As in the case of 1 M KCl solution, we find translocation events corresponding to both single-file single-file chain conformations and ips conformations of PSBMA. Using the same data analysis protocol to identify the single-file and ips translocations, the voltage dependencies of 〈τ〉 for the single-file (main curve) and ips translocations (inset) are given in Fig. 4d. In both cases, 〈τ〉 decreases roughly exponentially with the voltage confirming again that PSBMA undergoes successful translocation like a polyanion even with the different salt identity.

The independence of CSB on the dipole orientation of polyzwitterions is demonstrated as follows. After confirming that at the experimental pH, PMPC (with dipole orientation opposite to PSBMA) (Fig. 5a) is charge neutral as shown in its pH titration curve (Fig. 5b) due to low pK_a value of phosphoric acid group (≤1), we recorded ionic current traces in the positive and negative flow cell configurations with a pore diameter of 2.8 nm and 100 nM PMPC. In the negative configuration, where the negative electrode is in the donor chamber containing PMPC, there are no translocation events as shown in the example at 300 mV in Fig. 5c. However, current blockage events occur in the positive configuration (Fig. 5d) at the illustrative voltage of 300 mV. Here, PMPC molecules move toward the negative electrode acting like a polycation (in contrast with PSBMA acting like a polyanion) thus validating the general premise of CSB. Since the direction of electrophoretic mobility of PMPC is in the opposite direction to EOF, the role of EOF is overwhelmed by the electrophoretic force, as in the case of PSBMA. Further in-depth analysis of translocation kinetics of PMPC and other polyzwitterions is of high interest in our future work.

a Chemical formula of PMPC. b Titration curve of PMPC (63 mM in 1 mL of Milli Q water) showing that PMPC is neutral at the experimental pH 7. c PMPC translocates to the negative electrode acting like a polycation. d PMPC does not translocate to the positive electrode. In (c) and (d), 50 nM PMPC in 3.6 M LiCl buffered at pH 3.7 with 10 mM HEPES was used. Pore diameter: 2.8 nm, voltage: 300 mV, sampling rate: 250 kHz, low-pass filter frequency: 10 kHz.

Combination of the above results on PSBMA with different salt identities and PMPC comprehensively vouches for the universal occurrence of CSB in polyzwitterions.

Theory of effective charge and translocation time

In addition to exhibiting CSB, the above experiments provide quantitative values of 〈τ〉 which must be a function of the individual effective charges (due to differential counterion binding) of the ionic groups of the polyzwitterion, which in turn are due to gradients in local dielectric constant. To deduce the charges from 〈τ〉 and to estimate the local dielectric constant, we use theory. Understandably, the task of formulating a rigorous theory for these systems is difficult due to the complexities arising from a confluence of electrostatic and hydrophobic interactions, conformational entropy of polymer chains, solvent reorganization, and liquid crystal-like dipolar orientational correlations⁵³. Nevertheless, to explore the root cause of CSB, we present the following zeroth-order mean field theory and establish a relation between the macroscopically observed CSB and the microscopic nature of differential counterion binding within zwitterionic monomers of the polymer. The cross-sectional diameter that can be qualitatively inferred from the ratio of blocked current in single-file blockages to the open pore current is  ~2.9 nm, consistent with the value we obtained from molecular mechanics. Further, the radius of gyration R_g (estimated as  ~9.2 nm) is more than twice bigger than the pore diameter. Therefore, there is no possibility for the chain to translocate as either as a single blob or a sequence of blobs which would be unfavored due to conformational entropic penalty. In view of these considerations, we identified the mode of translocation as threading of extended conformations as in the well-established nanopore-based sequencing of polynucleotides and proteins.

There are four key elements in the theory. (1) We present an expression for the effective degree of ionization α (equivalently the effective charge q) of the ionic groups of the zwitterion in terms of the local dielectric constant ϵ_ℓ, based on counterion-binding equilibria (see Supplementary Note 2) as

where [A⁻] is the activity of the counterion.

(2) We present a zeroth-order model for the translocation kinetics, capable of capturing the essential physical concepts that contribute to the phenomenon, and yet mathematically simple enough to derive useful formulas to compare with experiments. The chemical details of the polyzwitterion molecule are many that include the size and orientation of the zwitterionic repeat unit with respect to the chain backbone, distance between the ionic groups and dipole moment of the zwitterionic unit, separation distance between two adjacent repeat units, thickness of the backbone, chain length of the polymer, pore length, pore diameter, surface charge density on the inner wall of the pore, ionic strength, pH, and identity of counterions, and physical quantities such as local dielectric constant and externally imposed electric field profile across the nanopore. Inevitably, these contributing factors require parametrization which is not arbitrary. As described in Supplementary Note 3, we have taken reasonable values of these parameters for the specific polyzwitterion systems studied here in setting up the model for the next step of the calculation.

(3) Using the model, we present the free energy landscape for the translocation of the molecule as sketched in Fig. 6 and described in Supplementary Notes 4 and 5. Briefly, the translocation occurs in three stages: crossing an entropic barrier to enter the pore, threading through the pore, and successful ejection from the pore. The free energy landscape F(m) arising from these three stages is a composite of four contributions: charge-electric field interaction (F_pE), conformational entropy of the chain (F_ent), pore wall-chain interaction (\({F}_{{{{\rm{pore}}}}}\)), and electrostatic energy (F_bE) of the chain in the bulk of the receiver chamber. The net result for the free energy is

where m denotes the translocation coordinate in units of monomer length. These contributions are presented in Supplementary Note 4 and summarized in Supplementary Tables (I and II).

a Three stages of translocation. b Sketch of free energy (F) landscape corresponding to the three stages. M and N are the pore length and chain length in units of monomer length ℓ. c Model of the conformation of a segment of a polyzwitterion chain inside the nanopore with its central axis along the x-coordinate in units of ℓ. Blue curve denotes the profile of the electric potential ψ due to the applied voltage. q₁ and q₂ denote the charges of the ionic groups closer and distant to the chain backbone, respectively. d is the dipole length in units of ℓ, and θ is the angle subtended by the zwitterionic group with respect to the chain backbone taken along the central axis of the nanopore. x_i is the x-coordinate of the center of the i-th zwitterionic group.

(4) Based on the free energy landscape, using the Fokker-Planck formalism⁶⁶ (Supplementary Note 6), we calculate the average translocation time as a function of the voltage in terms of the unassigned values of the charges of the ionic groups, given by

where F(y) is given in equation (2), N and M are the contour length of the polymer and pore length in units of monomer length (0.25 nm), and k₀ is the bare monomer diffusion coefficient inside the pore¹⁰. The important outcome of the theory is a fundamental understanding of the relative contributions from conformational entropic barrier, pore-polymer interaction, and electrophoretic drift to the translocation time. The calculated results are then compared with experimental results on the voltage dependence of the average translocation time, to determine the extent of CSB.

Comparison between theory and experiment to quantify CSB

The parameters entering the theoretical prediction of 〈τ〉 are the charge q₁ of the ionic group close to chain backbone, the charge q₂ of the ionic group away from the chain backbone, the local dielectric constant ϵ_ℓ, and ΔV, in addition to the pore length and chain length which are fixed. For polyelectrolytes in aqueous electrolyte solutions, it is known from a combination of Manning theory^54,55, counterion adsorption theory⁵⁷, and experiments^17,56, that the effective degree of ionization is around -0.25. Therefore, we have taken q₂ = −0.25 as reported in the literature with local dielectric constant around 80. We have estimated the pore wall-chain interaction energy per monomer ϵ₀ as 0.0009k_BT (see Supplementary Note 7). As given in equation (3), the average translocation time 〈τ〉 appears as k₀〈τ〉, where the monomer diffusion coefficient k₀ inside the pore is unknown. In view of this, to eliminate k₀, we have constructed the ratio of the calculated k₀〈τ〉 at a given voltage to that at a reference voltage (taken as 200 mV in Fig. 7a). k₀ is then determined by comparing the theoretical predictions and experimental data on 〈τ〉 (see below). The theoretically calculated value of 〈τ〉/〈τ〉_200mV is given in Fig. 7a as a function of the voltage for different values of q₁. To deduce the value of q₁ and hence the magnitude of CSB, these curves are then compared with experimental data on 〈τ〉/〈τ〉_200mV for different voltages (in the range of 70–200 mV) (black data points, Fig. 7a). By minimizing the ratio \(({({\langle \tau \rangle /\langle \tau \rangle }_{200{{{\rm{mV}}}}})}_{{{{\rm{theory}}}}}-{({\langle \tau \rangle /\langle \tau \rangle }_{{{{\rm{200mV}}}}})}_{{{{\rm{experiment}}}}})/{(\langle \tau \rangle /{\langle \tau \rangle }_{{{{\rm{200mV}}}}})}_{{{{\rm{theory}}}}}\) and its standard deviations (see “Methods”), we find the best value of the charge of the quaternary ammonium group as +0.10 using data of PSBMA translocation in 1 M KCl, 10 mM HEPES, pH 7 (Fig. 3f).

The experimental data is from Fig. 3(f). a Theoretical predictions of the ratio of mean translocation time at different voltages to that at the reference voltage 200 mV for different choices of q₁. Experimental values are black squares. Inset: expanded data sets for black squares and red circles for q₁ = +0.20. b Comparison between experimental data and theoretically predicted values for q₁ = +0.10.

For the LiCl data, both the experimental and theoretical values of 〈τ〉/〈τ〉_300mV exhibit the same common trend of 〈τ〉/〈τ〉_300mV decreasing with voltage, as shown in Supplementary Fig. 16. However, experimental values of 〈τ〉/〈τ〉_300mV decrease more rapidly with voltage compared to the theoretical values. Perfect quantitative fitting between the present theory and experiments cannot be expected due to the well-recognized ‘special ion’ effect of Li⁺ arising from its high electron density⁶⁷. The Fokker-Planck formalism used in the theory is not capable of describing the specific details of hydrated Li⁺ ion at different voltages. A more fundamental theoretical treatment of specificity of counterions in nano-confinement is beyond the scope of the present goal and is relegated to future investigations.

We note that the theoretically calculated translocation time is in units of the monomer friction coefficient 1/k₀. To estimate k₀, we have taken k₀ as an unknown constant independent of voltage and found its value as 1.41 × 10⁴ nm² ms⁻¹ from the best fit between the experimental values of 〈τ〉 (black squares) and theoretical values (red circles) (Fig. 7b). The procedure for finding the best fit is the same as for obtaining q₁, by minimizing the mean and STD of the error between theory and experiment. With this value of k₀, the experimental data (black squares) and theoretical values (red circles) for the voltage dependence of 〈τ〉 are given in Fig. 7b. The nice overlap between these two curves using a single value of k₀ indicates that all monomers of the chain in each translocation event have uniform velocity, without any additional nonlinear effects such as tension propagation forces seen in other systems⁴⁰.

The inference of local dielectric constant from CSB is performed as follows. The effective charge q₁ of the ionic group near the chain backbone, obtained using theory and experiments, is related to the local dielectric constant ϵ_ℓ through equation (3), where α = q₁. Assuming that the activity [A⁻¹] of the counterion can be adequately approximated by its mole fraction, equation (3) yields a relation between q₁ and ϵ_ℓ. Choosing r in the range of 0.3–0.5 nm, the local dielectric constant ϵ_ℓ is in the range of 20–30. This estimate is consistent with the counterion binding energy being comparable to k_BT, by equating the counterion binding energies for the internal ionic group, \({q}_{1}{e}^{2}/(4\pi \bar{\epsilon }{\epsilon }_{\ell }r)\) and for the outer ionic group, \({q}_{2}{e}^{2}/(4\pi \bar{\epsilon }\epsilon r)\), with k_BT. Since we have taken q₂ = −0.25 from literature^17,55 and q₁ = +0.1 from our experiments, the local dielectric constant is 28, when the same value of r is assumed for both groups and ϵ = 70⁶⁷ for 1 M KCl solutions. Thus, generally, the local dielectric constant is lower than the value in the bulk solution. A more accurate deduction of ϵ_ℓ is relegated to future work which might demand quantum-mechanical calculations. Nevertheless, the key quantity in CSB is its magnitude, that is the difference in the magnitude of the effective charges of the ionic groups of the zwitterion, instead of the absolute values of these effective charges.