Appendix

A. Corrections for missed events and their impact on estimated rate constants

Here, “true” is used to describe actual events and their characteristics when unperturbed by the recording instrumentation. The estimation of the mean lifetime of the true events was carried out according to the procedure of Blatz and Magleby (1986). Based on the analysis of dwell-time histograms, we used a kinetic scheme with two different occupied states with the PEG polymer in the lumen of the pore. L0, L1 and L2 are defined as the mean lifetimes of states 0 (unoccupied state), 1 (short occupied state) and 2 (long occupied state). The dead time (Td) is the lifetime of a true event that gives a response of half the true amplitude under the recording conditions.

The fractions of missed true events (Fmiss) less than the dead time (Td) for each category are given by:

(A1)

Fractions of recorded true events (Fcap) are given by:

(A2)

The mean lifetime of all missed true events (Tmiss) for each species is (Colquhoun and Sigworth, 1995):

(A3)

Let us denote by: N, the number of true unoccupied states; N1, the number of true short occupied states; N2, the number of true long occupied states (N=N1+N2); Na1, the number of missed true short occupied states above threshold; Na2, the number of missed true long occupied states above threshold; Nb0, the number of missed true unoccupied states below threshold; Nobs0, the number of observed unoccupied states; Nobs1, the number of observed short occupied states; Nobs2, the number of observed long occupied states; Lobs0, the mean lifetime of observed unoccupied states; Lobs1, the mean lifetime of observed short occupied states; Lobs2, the mean lifetime of observed long occupied states.

The mean lifetime of observed states is given by the total observed time for each event category divided by the number of observed states:

(A4)

Assuming that the contribution of missed unoccupied states below threshold to the total observed time of each category of occupied states is proportional to the relative frequency of occupied states for each category, then the mean lifetimes of observed occupied states are given by the following expressions:

(A5)

The observed time constants of the states are dependent on both the dead time and mean lifetime of the observed states (obtained directly from the dwell time histogram, Colquhoun and Sigworth, 1995):

(A6)

The number of observed unoccupied, short occupied and long occupied states may be calculated according to the procedure of Blatz and Magleby (1986), but here extended to a three-state model:

(A7)

The sum from (A7) was calculated according to a geometric progression. The coefficient (1/2) located in front of both sums, was introduced as a normalization factor for the sum of two independent probabilistic processes. We can also evaluate the observed number of short and long occupied states:


(A8)

Similarly, the number of missed true unoccupied states below threshold are given by the expression:

(A9)

The numbers of missed true short and long occupied states above threshold are the following:

(A10)

(A11)

If the mean lifetimes of missed true unoccupied and occupied states (Tmiss0, Tmiss1 and Tmiss2) are negligible compared with the mean lifetime of true unoccupied states (L0) (which is our case, since Tmiss0, Tmiss1, T miss2 < Td < L0), then the mean lifetime of observed unoccupied states from relation (A4) is the following:

(A12)

As the mean lifetime of true unoccupied states is much longer than the dead time (L0 > D), this also implies that:

or in other words (A13)

Under these conditions, the mean lifetime of observed unoccupied states is given by:

(A14)

Or the mean lifetime of true unoccupied states is:

(A15)

As a simple example, if the mean lifetime of observed unoccupied states is 5000 s, the dead time is 10 s, the mean lifetimes of true occupied states are 43 and 110 s, respectively, then the mean lifetime of true unoccupied states is 4264 s.

If we introduce the relations (A1-A3), (A9-A11) and (A6) into relation (A5), then the mean lifetime of true short and long occupied states may be written as follows:

(A16)

(A17)

As Fmiss0 < 1 and Tmiss0 < Td, the error in measuring the mean lifetime of occupied states is very small. The effect of the missed brief and long occupied states is more important for determining the mean lifetime of true unoccupied states.

L0obs0(A18)

L1 obs1(A19)

L2 obs2(A20)

Finally, we can obtain an expression for the mean lifetime of true unoccupied states as a function of the time constants of observed unoccupied, short occupied and long occupied intervals:

(A21)

B. Comparison of partition coefficients and formation constants

We consider a bimolecular interaction between a polymer (pol) and a protein pore (HL):

(B1)

and

(B2)

where: Kf, the formation constant of the HLpol complex; PHL, the probability of the unoccupied state; PHLpol, the probability of the occupied state:

The partition coefficient () is defined:

(B3)

where: [pol]sol, the polymer concentration in solution; [pol]HL, the polymer concentration within the pore.

The occupancy of the pore (F) is given by:

(B4)

At low concentrations of the polymer, Kf [pol]sol < 1. So that:

(B5)

At F=1, let the effective polymer concentration in the barrel domain of the HL pore (with an approximate volume V104 Å3) be:

(B6)

where NAV is the Avogadro number.

At low concentrations of a polymer with a tight binding site (e.g. cyclodextrin at F1), the apparent partition coefficient is given by:

(B7)

Alternatively, for a weak binding polymer (such as PEG) the true partition coefficient can be given in terms of an apparent formation constant (Kfapp):

(B8)

By using  and app (or Kf and Kfapp), weak and tight binding polymers, such as PEG and CD, can be compared (see e.g. Table 5).

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