Theory for polymer analysis using nanopore-based single-molecule mass spectrometry

Abstract

Nanometer-scale pores have demonstrated potential for the electrical detection, quantification, and characterization of molecules for biomedical applications and the chemical analysis of polymers. Despite extensive research in the nanopore sensing field, there is a paucity of theoretical models that incorporate the interactions between chemicals (i.e., solute, solvent, analyte, and nanopore). Here, we develop a model that simultaneously describes both the current blockade depth and residence times caused by individual poly(ethylene glycol) (PEG) molecules in a single alpha-hemolysin ion channel. Modeling polymer-cation binding leads to a description of two significant effects: a reduction in the mobile cation concentration inside the pore and an increase in the affinity between the polymer and the pore. The model was used to estimate the free energy of formation for K(+)-PEG inside the nanopore (approximately -49.7 meV) and the free energy of PEG partitioning into the nanopore ( approximately 0.76 meV per ethylene glycol monomer). The results suggest that rational, physical models for the analysis of analyte-nanopore interactions will develop the full potential of nanopore-based sensing for chemical and biological applications.

Fig. 1.

PEG causes transient current blockades in a single α-hemolysin nanopore. (A) PEG reversibly partitions into and out of the pore causing well-defined current blockades. (B) A thresholding algorithm is used to detect the events. The blockade amplitude is defined by averaging the open channel current immediately adjacent to the event, 〈i₀〉 (green points), and the base of the blockade, 〈i〉 (orange points). The black data points represent transition states and are not used in the analysis. The residence time (t_res) is the difference between the onset and the termination of the event. (C) Current traces show that increasing the magnitude of the applied transmembrane voltage increases the frequency of blockade events and decreases blockade lifetimes. The trans solution contained a mixture of PEG with approximately equimolar concentrations of mean molecular masses M_w = 1,000 g/mol, 1,500 g/mol, 2,000 g/mol, and 3,000 g/mol and a chemically purified internal standard of PEG M_w = 1,294 g/mol in 4 M KCl, 10 mM tris adjusted to pH 7.5 with citric acid.

Fig. 2.

PEG-induced single channel current reduction distributions are voltage-dependent. Increasing the applied potential decreased the normalized current blockade amplitude. This effect is described by a model in which cations bind to PEG molecules (Eq. 5). The color-coded tick marks above the peaks for polymer n-mers with n = 20, 29 and 50 illustrate typical voltage-dependent shifts over a range of polymer size. The tick color corresponds to the magnitude of the voltage (green: -40 mV, orange: -50 mV, blue: -60 mV, and red: -70 mV). The shift was observed for all resolved peaks. Each distribution was formed with > 130,000 blockade events.

Fig. 3.

The residence time distribution for a given size PEG n-mer in the nanopore is exponential and voltage-dependent. The solid lines are least squares fits of P(t) = A exp(-t/〈τ_n〉) to the data for a PEG molecule of size n = 30, V_app = -70 mV, 〈τ₃₀〉 = (0.154〈 ± 〉0.03)ms (open squares) and V_app = -40 mV, 〈τ₃₀〉 = (0.78〈 ± 〉0.01)ms (solid circles). Similar results were obtained for all the polymers characterized here and all values of the applied potential.

Fig. 4.

The reaction scheme for the PEG, cation and nanopore interactions is described by two net reversible reactions: PEG-cation coordination and PEG-nanopore partitioning. In this simplified scheme m_b cations bind to a PEG n-mer with a free energy change of m_bΔG_o,bulk. The cation-PEG complex enters and binds to the pore, with confinement term, nΔG_c (22, 23, 38), and a cation-associated binding term, m_bΔG_eb. The adsorption of the PEG-cation complex to the nanopore wall causes electroosmotic flow via the anions (for simplicity the boundary regions are neglected see SI Text). The arrows indicate the direction of flow for anions (red), cations (green), the applied electric force , and viscous force on the entire PEG molecule. When m_b bound cations dissociate from the complex with a corresponding change in free energy, m_bΔG_o,pore, PEG exits the nanopore with a change in free energy nΔG_c. The total change in the free energy resulting from the exodus of PEG from the nanopore, used in Eq. 8, comes from the combination of the two steps highlighted by the blue arrow.

Fig. 5.

Mean current blockade amplitudes and polymer residence times as a function of polymer size and applied potential were simultaneously fit by the chemical reaction model. (A) Experimentally determined current blockade amplitudes (open orange circles) and least-squares fits from the model defined in Eq. 2 (solid orange line) for data obtained at V_app = -50 mV. See SI Text for the full dataset. (B) The normalized residuals calculated from current blockades measured at four different applied potentials [-40 mV (green), -50 mV (orange), -60 mV (blue), and -70 mV (red)], but with V_app held fixed at -40 mV in Eq. 2, show the explicit voltage dependence of the blockade amplitudes. (C) However, when the actual voltages are used in the model, the normalized residuals converge . (D) Experimentally determined PEG residence times in the nanopore (open circles) and least-squares fits from Eq. 8 (solid lines) along with normalized residuals above. The data and fits correspond to V_app values of -40 mV (green), -50 mV (orange), -60 mV (blue), and -70 mV (red) for each plot. (E) Normalized residuals between the residence time data and model (Eq. 8).