Theory of DNA translocation through narrow ion channels and nanopores with charged walls

A DNA molecule in a water solution carries negative charge. With the help of applied voltage $V$, it can translocate through an ion channel located in a lipid membrane or through a solid state nanopore in a semiconductor film. In this paper we are interested in the cases when DNA barely fits into a narrow pore leaving only a small gap for water with the width $d<l_{B}$, where $l_{B}=e^{2}/\kappa k_{B}T$ is the Bjerrum length and $\kappa$ is the dielectric constant of water. An intensively studied example is the translocation of a single stranded DNA (ssDNA) molecule through an $\alpha$-hemolysin ($\alpha$-HL) channel  Henrickson ; Meller2001 ; Meller2002 ; Meller2003 ; Sauer ; Mathe ; Ambj rnsson ; Nakane ; Bonthuis . With the average internal diameter $\sim 1.7\,$nm the channel can accomodate ssDNA molecule with $\sim 1\,$nm diameter. In this case, $d=0.35~$nm$<l_{B}$.

Our theory also should be applicable to a double helix DNA (dsDNA) with $\sim 2\,$nm diameter translocating through a narrow solid state nanopore with $2a\sim 3\,$nm diameter Li ; Min . On one hand, no experimental data is available for that narrow nanopores. On the other hand, there is impressive progress in making and studying wider nanopores Li ; Min ; Lemay ; Aks ; Aks1 .

The peculiarity of narrow channels is related to the fact that the dielectric constants of the channel stem and lipids and the dielectric constant of the body of DNA are much smaller than dielectric constant of water. When the water filled gap between DNA and the channel wall is narrow ($d<l_{B}$), the electric field of small ions is squeezed in the gap. Potential of interaction of charges becomes logarithmic. This creates the electrostatic barrier for the ion current similar to one which was intensively studied for the ion transport through narrow DNA free channels Parsegian ; Zhang ; Kamenev .

Previous discussion of the role of this barrier for ion transport in the case of DNA translocation Jing was narrowly focused on neutral channels, because the wild type $\alpha$-HL channel can be considered practically neutral. A good measure of neutrality of a channel is its cation/anion selectivity, measured by the ratio of cation to anion currents. For the wild type $\alpha$-HL channel this ratio is equal 1.1, while it is equal to unity for a exactly neutral channel. It is known that at pH 7 the $\alpha$-HL channel is neutral in the body of stem, but has the ring of 7 negative charges near the bottom of the the narrow cylindrical part of the channel (stem) HLimage . These charges are screened by the salt in the water outside the stem and, therefore, do not determine the transport through the channel which remains weakly selective. Solid state nanopores can be neutral, too. For a neutral narrow channel Ref. Jing addressed several challenging problems posed by the experimental data Henrickson ; Meller2001 ; Meller2002 ; Meller2003 ; Sauer ; Mathe ; Ambj rnsson ; Nakane ; Bonthuis .

First, Ref. Jing explained how the electrostatic barrier makes the blocked by DNA current $I_{b}$ at least 10 times smaller than the open current $I_{0}$. Second, the effective stall charge $q_{s}$ of the piece of DNA residing in the channel was calculated. This charge determines the force $F_{s}=q_{s}V/L$ stalling DNA against the voltage $V$ ($L$ is the length of the channel). It was shown that for a neutral channel with small $I_{b}/I_{0}$ the charge $q_{s}\simeq q_{b}I_{b}/I_{0}$, where $q_{b}$ is the bare charge of the piece of DNA occupying the channel (for $\alpha$-HL channel $q_{b}=-12e$). In agreement with experiments this results in very small absolute value of the stall charge, namely $q_{s}\sim-1e$. Third, the origin of an the exponentially small and growing with the salt concentration DNA capture rate was elucidated.

Recently, genetically modified $\alpha$-HL channels became available Aka ; Mer . In this paper we concentrate on those of them, which have amino acids with positive residues on the internal wall of the narrow cylindrical part of the channel (stem). Internal walls of solid state nanopores may also be charged. The charge density of these walls can be tuned by different chemical treatments or just by a change of the solution pH. Thus, our theory for charged $\alpha$-HL channels simultaneously addresses narrow charged nanopores used for dsDNA translocation experiments Min .

We assume below that the fraction $x$ of the bare charge $q_{b}$ of DNA piece fitting into the channel is compensated by positive internal wall charges, which are roughly speaking randomly distributed on internal wall of the channel (Fig. 1). We predict below that this simple assumption leads to a number of dramatic changes of DNA translocation in comparison with a neutral channel. Let us list these predictions:

(i) The blocked ion current $I_{b}$ becomes even smaller than in the neutral channel particularly at small concentrations of salt.

(ii) The effective charge $q_{s}$ of the piece of DNA residing in the channel grows with $x$ as $q_{s}\simeq xq_{b}$. At $x=1$ the stall and bare charges of DNA are almost equal. The large effective charge will makes possible DNA manipulation with the help of small voltages.

(iii) The barrier for the DNA capture decreases with $x$. As a result the DNA capture rate exponentially grows with $x$ and the number of translocation events observed in a given experiment increases. This should lead to much more effective averaging of the noise and may prove helpful in attempts of DNA sequencing. At some $x=x_{c}$ the capture barrier vanishes. At $x_{c}<x\leq 1$ DNA is attracted to the channel. The capture rate then is only diffusion limited and independent on $x$. On the other hand, for a captured DNA the probability to escape from the channel becomes activated. The escape barrier grows with $x$ at $x>x_{c}$.

The structure of our paper is simple. It consists of three sections leading to conclusions (i), (ii) and (iii) respectively.

In the case of $\alpha$-HL channel we assume the ssDNA molecule is a rigid cylinder coaxial with the channel. The inner radius of the $\alpha$-HL channel is $a\!\simeq\!0.85\,$nm, and the radius of the ssDNA molecule is $r\!\simeq\!0.5\,$nm. Salt ions are located in the water-filled gap between them, with thickness $b\!\simeq\!0.35\,$nm. The length of the channel is $L\!\simeq\!5\,$nm. This kind of model is even more appropriate for double helix DNA in a wider (say $4\,$nm in diameter) cylindrical solid state nanopore Li ; Min .

The dielectric constant of the channel or the ssDNA molecule ($\kappa^{\prime}\!\sim\!2$) is much smaller than that of water ($\kappa\!\simeq\!80$). So if ssDNA is neutralized by cations and there is an extra charge $e$ in the thin water-filled gap between the channel internal wall and ssDNA, the electric field lines starting from this charge are squeezed in the gap. This results in a high self energy of the charge Parsegian ; Zhang ; Jing . According to the estimate of Ref Jing for the case of ssDNA in $\alpha$-HL channel, the self energy of the charge in the middle of the channel is $\sim 5k_{B}T$. Here and everywhere in this paper $T$ is the room temperature.

Because of the large self energy of a charge in the narrow water gap, the piece of ssDNA inside the wild type neutral $\alpha$-HL channel is neutralized by counterions, say K⁺ in KCl solution Jing . ssDNA covered by cations presents a conducting DNA backbone wire responsible for the blocked ion current $I_{b}$ at small concentration of salt $c<1$M. In this range of concentrations $I_{b}$ is practically $c$ independent Bonthuis . At larger concentration $c\geq 1$M additional pairs of anions and cations in the channel provides a parallel to DNA backbone wire mechanism of conductivity. (Recall that DNA backbone wire occupies only a small fraction of the water filled gap.) The linear growth of $I_{b}$ with $c$ at $c\geq 1$M is an experimental evidence for the second mechanism of conductivity Bonthuis

In a mutated positively charged channel situation is rather different. Let us consider the channel with uniformly distributed 12 positive charges ($x=1$). We argue that in this case both ssDNA and internal wall charges release their counterions into the surrounding salt solution. The net charge of the channel is still zero and, thus, there is practically no price in the Coulomb energy. On the other hand, counterion release leads to the large gain in their entropy. As a result DNA backbone wire looses its carriers and becomes an insulator. Therefore, $I_{b}$ is determined only by contribution of additional pairs of salt ions. This should lead to linear dependence of $I_{b}$ on $c$ in the whole range of salt concentrations. In other words, $I_{b}$ becomes much smaller than in the wild type channel at small $c<1$M, but is not strongly changed at larger concentration of salt.

So far we talked about wall charges totally compensating the bare charge of DNA ($x=1$ ). At $x<1$ DNA counterions are only partially released and conductance of DNA backbone wire is only partially depleted. Although the number of counterions on the DNA wire is proportional to $x$, their mobility may somewhat grow with decreasing $x$ due to the increase of the number of empties. It is possible, but seems unlikely that this growth leads to the conductance maximum at $x\sim 0.5$.

As we mentioned above for the wild type channel the stall charge of DNA $q_{s}$ is much smaller than the bare charge of DNA $q_{b}$. Let us remind why this happens. Counterions neutralizing DNA in the channel receive from electric field momentum with the same absolute value as DNA, but in the opposite direction. Most of the time counterions are bound to DNA charges and transfer all received momentum to DNA. During this time, the net electric field force acting on DNA vanishes. At rare moments when counterions get free and move along the channel contributing to $I_{b}$, they transfer half of their momentum to the internal channel wall. This deficit of momentum transfer to DNA results in a small net average force on DNA and its small effective charge $q_{s}$ Jing .

In a channel where positively charged walls compensate the bare charge of DNA, the balance of forces is completely different. When countrerions of DNA and walls are released, electric field provides opposite momentums to DNA and to the wall charges. The latter are static and, therefore, transfer all their momentum to the wall. Thus, DNA gets its momentum only directly from electric field. This means that $q_{s}=q_{b}$.

So far we talked about the channel which totally compensates charges of DNA ($x=1$). Similar logic leads to the result $q_{s}=xq_{b}$ for any $x<1$.

Besides the blocked current and the stall charge one can measure the average time between the two successive translocation events, $\tau$, or the capture rate $R_{c}=1/\tau$ of a DNA molecule into the channel. It is natural to compare the observed value of $R_{c}$ with the diffusion limited rate $R_{D}$ of ssDNA capture. For the wild type neutral $\alpha$-HL channel this comparison shows that $R_{c}\ll R_{D}$. The capture rate at zero voltage $R_{c}(0)$ is so small that all experiments are actually done with a large applied voltage $V=50-200$ mV. Apparently there is a large barrier for the DNA capture. A large part of this barrier is due to the loss of the conformational entropy of ssDNA. The capture barrier, however, depends on the salt concentration, what means that a part of it has an electrostatic origin. The reason for such an electrostatic barrier is as follows Jing . When a DNA molecule enters the channel, the DNA counterions are squeezed in the narrow water-filled space surrounding the DNA. Due to this compression the total free energy of DNA and ions is higher for DNA in the channel than for DNA in the bulk. In agreement with experiment this barrier decreases with growing $c$ because the entropy of counterions in the bulk solution decreases and, therefore, the price for compression is smaller.

In the case of a channel with positively charged walls the ssDNA does not need to bring all its counterions into the channel, because there are already some positive charges. Thus, the charge $xq_{b}$ is released by DNA to the bulk of solution making the electrostatic barrier for DNA smaller. Additional, roughly speaking, equal gain is provided by release of counterions of the wall charges, which screen the walls in the absence of DNA. Thus, due to the counterion release the electrostatic barrier becomes $1-2x$ times smaller. At $x=1/2$ the electrostatic barrier vanishes, but the conformation barrier remains intact and the capture rate is still activated. At $x>1/2$ the electrostatic contribution to the total barrier becomes negative and at small enough concentration of salt, when $x=x_{c}<1$, it eventually compensates the conformation barrier, so that $R_{c}=R_{D}$. At $x>x_{c}$ the capture rate saturates at $R_{D}$, but the escape rate has an activation energy. The linear dependence of the barrier on $x$ can be measured.

To summarize, in this paper we studied DNA translocation through narrow channels with positively charged walls. Our predictions for the stall effective charge and capture rate are dramatically different from the case of neutral channels. Interpretation of the stall effective charge theory also becomes much simpler. This paper extends theory of narrow channels started in Ref. Jing . Meanwhile, the detail, theoretical description of wider channels became available Lemay . Therefore, one can ask when our theory crossovers to results Lemay . The crossover happens when the width of water gap $d$ becomes larger than $l_{B}$. For double helix DNA this happens for nanopores with diameter $3.5~$nm or larger.

We are grateful to A. Aksimentiev, T. Butler, J. Gundlach, Jingshan Zhang, J. J. Kasianovich, O. V. Krasilnikov and A. Meller for useful discussions.