Capture rate (nanopore)

The capture rate of a nanopore is the number of molecules per second that enter it, usually quoted per unit of concentration, in events per second per nanomolar. Every translocation begins with a capture, so the rate sets how fast an instrument collects data and how little sample it needs. A capture has two stages, and the slower one governs. In the first, the molecule travels from the bulk solution to the mouth of the pore by diffusion and electrophoresis. In the second, it finds the end of its chain and threads in against a barrier. The first is called diffusion-limited and the second barrier-limited.

The volume from which a nanopore draws molecules is typically 10⁸ to 10¹⁰ times smaller than the volume of the sample.^[1]

Outside a pore the electric potential falls with distance. The current through a cylindrical pore of diameter d and length l spreads over a hemisphere, so the potential at distance r is V(r) = ΔV d²/(8lr). A charged molecule of mobility μ drifts down this gradient. Wanunu and colleagues defined a capture radius r*, the distance at which drift matches diffusion, by μV(r*) = D, where D is the diffusion coefficient of the molecule:^[2]

Inside that radius the molecule is carried in; outside it wanders. If every molecule that reaches r* is captured, the pore is a hemispherical sink, and the rate is 2πDr* per unit of concentration, which is πd²μΔV/(4l).^[2] The diffusion coefficient has cancelled. A longer DNA molecule diffuses more slowly, but its capture radius grows as 1/D, so the rate does not depend on the length of the molecule. It rises linearly with the voltage and the concentration, as the square of the pore diameter, and falls as the pore gets longer.

With the inputs of the simulation script in the open design of the MR1 Nanopore Reader, a pore 4 nm wide in a membrane 20 nm thick at 180 mV, a mobility of 4.5 × 10⁻⁸ m²/(V s) and a diffusion coefficient of 3 µm²/s for DNA of about 1,000 base pairs, the capture radius is 0.27 µm. At 10 nM the diffusion-limited rate is about 30 events a second.^[a]

Reaching the pore does not always decide the rate. A coil of DNA has to find an end, push it into the pore against the entropy of confinement, and draw the rest after it. Wanunu's group wrote the rate of this step in the form of Kramers' theory, as a prefactor times exp[(qΔV − U)/k_BT], where U is the height of the barrier at zero voltage and q the effective charge of the end of the chain. The rate rises exponentially with the voltage, and it governs the capture when it is lower than the diffusion-limited rate.^[2] In an α-hemolysin channel the blockade frequency is proportional to the concentration of polymer and rises exponentially with the applied potential.^[4] The Wanunu group found the same in silicon nitride pores 4 nm wide for short DNA: the rate for 400 base pairs rose exponentially with the voltage, with a slope of 9.6 V⁻¹, and for 3,500 base pairs with a slope of 7.1 V⁻¹. For 48,502 base pairs it rose linearly, which the authors read as capture with no barrier.^[2]

The theory also says why longer molecules do better, up to a point. A coil near the mouth of the pore sits about one radius of gyration from it, held there by the potential well, and is brought to the mouth many times before an attempt at threading succeeds. Pressed against the membrane, it has no electro-osmotic flow through it, so its effective charge grows in proportion to its length. In free solution charge and drag grow together and the mobility does not depend on length; at the pore it does. Between 800 and 8,000 base pairs the capture rate grew with length as e^CN/4N_p, where N is the number of base pairs, N_p is the persistence length, about 150 base pairs, and a fit gave C = 0.81 ± 0.15 for the pore and voltage used. Above 8,000 base pairs the rate no longer depended on length, as the diffusion-limited formula says. The authors called the preference for long molecules surprising in view of the length-independent mobility of DNA in bulk electrophoresis.^[2]

Captures are independent events, so the delays between them are exponentially distributed. The group's histograms of delay times fit P(t) = A exp(−t/τ), "as expected from a stochastic capture process with non-interacting molecules", and 1/τ is the capture rate.^[2] A count of N events carries a relative uncertainty of 1/√N, so 1% needs 10,000 captures, which at a rate of 0.4 s⁻¹, as in the table below, take about seven hours.

Raising the voltage raises the capture rate and shortens the translocation exponentially, which costs resolution.^[2] A salt gradient does better. Lowering the salt concentration on the side the DNA comes from strengthens the field outside the pore on that side, raises the capture rate and lengthens the translocation instead of shortening it.^[2]

The result of the last row was a continuous record of single molecules from about 4 attomoles of DNA in a chamber of 1 µL.^[2] Other methods change what pulls on the molecule. The field acts on the counterions of DNA as well as on the DNA, and the response of the DNA is altered by its electro-osmotic coupling to them; a calculation by Grosberg and Rabin gives the dependence of the capture radius and the diffusion-limited rate on voltage and molecular mass with that coupling included.^[5] Dielectrophoresis can gather DNA from a larger volume toward the tip of a metallic pore and detect single molecules at 5 fM.^[1] A molecule that has just left can be fetched back by reversing the voltage; see Ping-pong sensing.^[6]

At 1 fM a microliter holds about 600 molecules. Sheehan and Whitman worked out the flux of analyte onto sensors from the micrometer to the nanometer scale and concluded that reported femtomolar detection limits were "very likely an analyte transport limitation, not a signal transduction limitation", and that without directed transport "individual nanoscale sensors will be limited to picomolar-order sensitivity for practical time scales".^[7] At the 0.4 s⁻¹ nM⁻¹ of the first row of the table, a sample of 400 base pairs at 1 pM would deliver one event in about 40 minutes.

• Translocation (nanopore)
• Electrophoresis
• Brownian motion
• Dwell time