Prospects for single-molecule electrostatic detection in molecular motor gliding motility assays

A key frontier in nanotechnology in recent decades has been the development of functional nanodevices featuring biomolecular motors, with a strong focus on motile ‘gliding’ assays using the actin-myosin and kinesin-microtubule molecular motor systems [1, 2, 3, 4]. Gliding assays were initially developed to better understand the underpinning biophysics of the actin-myosin [5, 6] and kinesin-microtubule [7] motor systems. Directed motion of actin filaments [8, 9] and microtubules [10] via lithographically-patterned guiding structures opened the path towards motile biosensors [11] and associated diagnostics [12], nanoscale cargo-trafficking systems [13, 2, 14], bio-inspired optical devices [15], and most recently, structures for efficiently solving combinatorial problems by having molecular motors explore a complex network [16].

Network-based biocomputation [16] is a strong motivator to extend beyond microscope-based detection of filaments because scale-up inevitably leads to the need to detect filament passage at a very large number of points spread across an area much larger than the field-of-view for an appropriate magnification objective. This need for new detection strategies led us to consider an electrical approach where a nanoscale transistor made with, e.g., carbon nanotubes, graphene or semiconductor nanowires, spans the bottom of a lithographically-patterned guiding structure. Both actin filaments [19] and microtubules [20] carry negative charge that can in principle electrostatically gate a nanoscale transistor providing it passes within a distance smaller than the buffer’s Debye length. Such an electrostatic sensor for gliding molecular motor assays could have more widespread applications, for example, as a trigger for optical [17] or thermal [18] actuation of motility, direction of traffic at junctions [21], addition/removal of cargo [2, 4], or addition/removal/read-out of chemical ‘tags’ added to filaments to improve information carrying capacity [22].

Our goal here is to establish the feasibility and develop some design guidelines for such a nanoscale transistor detector. We do this via basic ‘first principles’ calculations accounting for key aspects of the sensor geometry and design, combined with detection estimates based on published experimental data for detection of other biological targets using nanoscale transistor devices. Our calculations show that electrostatic detection of passing filaments is certainly possible in principle. We define and address key experimental parameters, e.g., buffer ionic strength, protein charge, etc., and discuss how they influence the limits of detection for this type of sensor. Notably, we find that electrostatic detection is likely to be a significantly easier prospect for microtubules than it is for actin.

The development of nanoscale transistors featuring carbon nanotube, graphene or semiconductor nanowire conducting channels for biosensing applications has a long history [23]. We have chosen to focus on a transistor featuring one single-walled carbon nanotube as the transistor channel for two main reasons. First, the low diameter ($1-2$ nm) provides a strongly confined but highly exposed conduction channel that enables strong gate response [23]. Semiconductor nanowires are much larger ($10-100$ nm diameter) with less strongly confined carriers that are set slightly back from the surface due to band-bending and surface oxidation effects [24]. Graphene falls between the two but involves an additional free parameter, channel width, which adds complexity to the calculations. Second, there are numerous experimental works characterizing the response of single-walled carbon nanotube transistors to single biomolecules [25, 26, 27] and aspects related to Debye screening effects in these devices [28, 29]. These help make our task considerably easier than it might otherwise be.

From this point onwards we focus entirely on the output of our electrostatic modelling for actin filaments and microtubules. Figure 2 shows plots of effective transient gate voltage step $\Delta V_{g}$ that we expect a passing filament to produce versus the height $h_{p}$ of the pedestal that raises the carbon nanotube above the guiding channel floor. Figure 2(a) covers the full range of $h_{p}$ for both filament types for a broader comparative picture. Figures 2(b/c) present the same calculations with finer $h_{p}$ resolution and tighter focus on the possible detection regime for actin (Fig. 2(b)) and microtubules (Fig. 2(c)). Data for actin is presented as diamonds and microtubules as circles with colour assigned to corresponding buffer composition. The horizontal dashed lines indicate detection limits with labelling as designated in the figure caption. The background shading indicates our expectation of the likelihood of electrostatic detection being viable, and ranges from unlikely (white) to likely (green) and highly likely (blue). The vertical dashed lines indicate the natural elevations for microtubules (green, marked MT) and actin (blue, marked A). We limit our analysis to $h_{p}<g$ as a natural outcome of our assumption of infinite persistence length (all parts of the filament are always at elevation $g$).

Looking at Fig. 2(a), there are two obvious behaviours in the data. The first is that the signal $\Delta V_{g}$ strengthens as the carbon nanotube is raised closer to the filament. This is a natural consequence of reduced distance $d$ between the charge elements $dQ$ and the point of action (0, 0, 0) via Eqn. 1. The second is that $\Delta V_{g}$ at a given $h_{p}$ increases as the buffer ionic strength is reduced, arising from the $\lambda_{D}$ contribution to Eqn. 1. The slope $\Delta V_{g}/h_{p}$ also decreases with reduced ionic strength, however, slope is a deceptive concept here due to the logarithmic axis for $\Delta V_{g}$. Note that even with the logarithmic axis, the $\Delta V_{g}$ versus $h_{p}$ is not truly linear in these plots because there is a $1/d^{2}$ Coulomb term in addition to the exponential Debye screening contribution, i.e., the data curves gently even with the log axis for $\Delta V_{g}$. Care needs to be taken with linear extrapolation for data presented as we have it in Fig. 2.

Regarding detection prospects, we start by considering actin filaments and return to microtubules afterwards. The data in Fig. 2 makes it clear that detection of actin is impossible without the nanotube on a pedestal of at least $h_{p}=30$ nm. Improvement in detection prospects can be achieved by reducing buffer ionic strength – the stronger improvement in moving from a40 to a20 compared to moving from a60 to a40 in Fig. 2(b) arises from $\lambda_{D}\sim 1/\sqrt{I}$, as is visually evident in Fig. S5. However, there is a limit to reducing ionic strength due to the biochemical reasons outlined earlier. Thus there is no escaping the need for a pedestal, and the associated fabrication challenges in achieving it, for detecting passing actin filaments at the charge density expected based on the literature [59, 19]. We explore the effects of increasing the filament charge density later in the paper.

Turning to microtubules, there are three notable aspects compared to actin that are vital to the discussion. First, microtubules travel closer to the guiding channel floor, i.e., have lower $g$, which means our scope for possible detection without a pedestal structure, i.e., with $h_{p}=0$, is substantially improved. Second, the charge density for microtubules is an order of magnitude higher. Thirdly, an important aspect of microtubule assays that we have so far ignored is the use of a casein coating on the substrate [66]. The casein layer serves to anchor the kinesin tail and position the kinesin head to better interact with microtubules [67, 68]. Actin assays have no corresponding layer; the myosin tails bind directly to the silanized SiO₂ surface [51]. The data in Fig. 2 ignores the casein layer, enabling us to make a more direct comparison with actin purely on the basis of elevation, charge density and buffer ionic strength. We will add the casein layer for microtubules to our model momentarily (Fig. 3).

The data in Fig. 2 with a bare SiO₂ surface for both filament types also points to the need to raise the nanotube on a pedestal for microtubules, albeit to a much lesser extent. For the commonly used PIPES buffers BRB-80 and BRB-20, a pedestal with $h_{p}=13-16$ nm would be required. In contrast, the imidazole and HEPES buffer results, particularly for I50Mg05-3$\times$ and HEPES are such that one could potentially imagine detection occurring without a pedestal at all, e.g., due to fluctuations arising from finite persistence length. Thus before we even consider the casein layer, microtubules have an obvious advantage from a device fabrication perspective owing to charge density and buffer ionic strength range alone.

Adding the casein layer to our model is relatively straightforward and we use an approach similar to how we deal with exclusion of the buffer by the body of the filament itself. We consider the casein layer to have thickness $t_{cas}$, and use geometric arguments (see Fig. S3/4) to work out what this corresponds to as a path length $d_{cas}$ on the direct line from $dQ$ to the point of action $(0,0,0)$. We then switch the screening term off for that segment of the path, as we do with the segment passing through the body of the filament. Note that this results in a slight underestimate of signal because we cannot account for the change in dielectric strength for reasons described earlier. The greater challenge is working out a meaningful value to assign for $t_{cas}$. Whole casein is a mixture of four casein sub-groups – $\alpha_{s1}$, $\alpha_{s2}$, $\beta$ and $\kappa$ – in proportions that vary based on species of origin and processing [69]. The composition of the surface-adsorbed casein layer for a microtubule assay can vary and has a significant effect on microtubule binding [67] and motility [70], yet is rarely precisely specified/known for many assays in the literature (e.g., often non-specific descriptors like ‘casein-containing buffer’ appear in literature protocols). Of the four caseins, only $\beta$-casein is well characterized in terms of formation of a layer on solid surfaces. Our view of the literature is that there is general agreement that $\beta$-casein forms a bilayer on a hydrophilic surface such as SiO₂. $\beta$-casein is amphiphilic [69] and the bilayer forms as a two-step process: a) first the hydrophilic domain adsorbs to the SiO₂ surface giving a tightly-packed monolayer presenting a hydrophobic outer surface, then b) a second loosely-packed monolayer forms as hydrophobic domains adsorb to the monolayer presenting a hydrophilic outer surface for the final bilayer [67, 71, 72, 73]. The disagreement in the literature is more related to the thickness and packing density for the $\beta$-casein bilayer. Verma et al. [67] point to Nylander et al. [71] as evidence for $t_{cas}=15$ nm. However, that measurement is obtained by compressive force measurements between a pair of $\beta$-casein bilayers on opposing hydrophilic surfaces [71]. As such, it detects the maximal steric extent of the bilayer, and given the looser packing of the outer monolayer and its hydrophilicity, assuming full suppression of screening over $15$ nm is probably an over-estimate. We thus consider this an absolute upper-bound on our calculations, setting the plotted range in Fig. 3. Most papers on $\beta$-casein layers suggest $t_{cas}$ for our estimate should be smaller. Tiberg et al. [72] report neutron reflectometry data for $\beta$-casein on SiO₂ and suggest the bilayer is best modelled as a trilayer with a) $3$ nm thickness at $63\%$ volume fraction, b) $3.5$ nm thickness at $51\%$ volume fraction and c) $4.0$ nm thickness at $28\%$ volume fraction. We treat this as an effective fully non-screening layer of thickness $4.8$ nm for simplicity. Ozeki et al. [73] report quartz crystal microbalance (QCM) measurements for $\beta$-casein on a SiO₂-functionalized QCM crystal and suggest an inner layer of $2.1$ nm thickness with a density of $279$ ng/cm² and an outer layer that is either $1/3$ of the thickness at the same density or $1/3$ of the density at the same thickness. As an effective single non-screening layer the corresponding thickness would be $2.75$ nm.

In Figure 3, we plot $\Delta V_{g}$ versus the casein thickness $t_{cas}$ (fully non-screening) assuming a pedestal height $h_{p}=0$. In other words, we focus on the question of how thick the casein layer needs to be to achieve detection without needing a pedestal at all. We run the model out to $t_{cas}=15$ nm to allow Verma et al. [67] as a very optimistic upper-bound. We indicate the more realistic scenarios in Ozeki et al. [73] (effective $t_{cas}=2.75$ nm) and Tiberg et al. [72] (effective $t_{cas}=4.8$ nm) with blue and green dashed vertical lines respectively. The $\Delta V_{g}$ for the I50Mg05-3$\times$ buffer is very close to the lower-bound detection limit at the $t_{cas}\sim 2.5-5$ nm range corresponding to the realistic effective $t_{cas}$ thickness [72, 73] that one might expect using purely $\beta$-casein. The HEPES and I50Mg05-2$\times$ buffer results are also reasonably close to the detection limit in this range.

However, $\beta$-casein is only a small part of a larger picture. We focus on it in Fig. 3 because it is the only casein sub-group that is well characterised as an adsorbed film on SiO₂ in terms of thickness and volume fraction. Maloney et al. [70] performed studies of microtubule motility in assays using pure $\beta$-casein surface passivation and found a) few motile filaments, b) that the filaments that were motile tended to be quite long, and c) weaker attachment at the ends of the microtubule, which led to poor tracking and a short time before microtubules detached and floated away. They instead found that whole casein ($49\%$ $\alpha$, $37\%$ $\beta$, $14\%$ $\kappa$) was a much better option, supporting stable motility for a wide range of filament lengths with good filament velocity and reproducibility, with pure $\alpha$-casein performing nearly as well [70]. The thickness and volume fraction for $\alpha$-casein and whole casein passivation layers on SiO₂ is not presently available, but if they increase the effective thickness of the casein layer from a buffer displacement perspective without increasing the filament elevation, i.e., move us further to the right of the two scenarios in Fig. 3, then the prospects of electrostatic detection of microtubules with $h_{p}=0$ only improve. Given this, we suggest a characterisation of whole casein and $\alpha$-casein on SiO₂ using neutron reflectometry or QCM studies would be a worthwhile endeavour.

Before moving on, we make three final observations regarding casein passivation. First, $\beta$-casein has a net negative charge of $-7.5~{}e$/molecule at pH $6-7$ [74]; the other caseins are likely also charged. This charge would cause an offset in the transistor’s maximum sensitivity point (i.e., peak transconductance), which would need to be corrected by applying a back-gate voltage $V_{BG}$. It is presently unclear whether the required $V_{BG}$ would: a) be within the available range for nanotube transistors, as too high a $V_{BG}$ leads to dielectric breakdown of the SiO₂ layer, destroying the device, or b) is compatible with retaining casein adsorption, kinesin binding and microtubule motility. The former could be established simply by comparing nanotube transistors before/after casein passivation, without the need for other structures, e.g., guiding channels, or a full motility assay. The latter is readily testable without the carbon nanotube device structure. Both would be worthwhile experiments in their own right. We note that relatively strong transverse electric fields ($10-100$ kV/m) were not a significant issue for microtubule gliding assays [21], so there may well not be major problems with the vertical electric fields associated with gate-tuning of the nanotube transistor either. Second, carbon nanotubes are generally [75] hydrophobic [76] and casein will bind with its hydrophilic domains down against the SiO₂ substrate. This means there is some chance the casein layer is affected/interrupted local to the nanotube, e.g., incomplete, thinner than usual, etc. We would not immediately expect this to affect motility, but it might mean that the screening is slightly stronger than expected based on Fig. 3. Some focussed studies might also be required to establish this in an experimental context. Third, one could ask why a similar strategy could not be used to improve the prospects for actomyosin assays. A critical feature of why casein works in the microtubule assay is that the kinesin tails bind to the lower half of the casein bilayer with the kinesin heads positioned just above the casein bilayer surface [67, 68]. The crucial aspect for this work is that the addition of this layer does not increase filament elevation $g$; simply putting in an underlayer that coats the nanotube but at the same time increases $g$ will not have the same effect in terms of reduced screening. We know of no such layer presently available for actomyosin assays.

As a last consideration in this paper, we turn to the possibility of varying the charge of the filaments used in motility assays. This could be due to uncertainty in the filament charge or it could be part of a deliberate charge-enhancement strategy, e.g., by binding short-strand DNA to the outside of a microtubule [77, 78]. Figure 4(a) shows a plot of the pedestal height required to reach the detection limit, i.e., $\Delta V_{g}$ at the lower-bound (LB), versus the linear charge density $\lambda$ on the filament in $e/\mu$m. The data presented is for the lowest ionic strength buffers, a20 for actin and I50Mg05-3$\times$ for microtubules, in the latter case without the casein layer. The default charge densities are indicated by vertical dashed lines (blue for actin/green for microtubules) with two notable values from earlier discussion, namely $16,500$ e/$\mu$m for actin [60] and $270$ e/$\mu$m for microtubules [56, 57], indicated by vertical dotted lines as points of note (see also Fig. S6/S7 for underlying data). For actin, increasing the charge only really reduces $h_{p}$. Rather extreme amounts of added charge would be needed to eliminate the pedestal entirely. In contrast, for microtubules, only a $33\%$ increase in $\lambda$ is required to achieve detection with $h_{p}=0$ before we even consider the casein layer. We add the casein layer to our consideration in Fig. 4(b)and focus on the $t_{cas}$ required to achieve the detection limit with $h_{p}=0$ for all six buffers used for microtubule assays. Decreasing the ionic strength too much can lead to poor motility, so knowledge of the highest ionic strength buffer viable for detection is useful. Our data in Fig. 3(b) already showed that microtubules should be observable for the I50Mg05-3$\times$ buffer at $h_{p}=0$ and the data in Fig. 4(b) confirms that the charge density would need to be less than $1/3$ of that measured by van den Heuvel et al. [20] to not be detectable. A similar situation holds for the HEPES and I50Mg05-2$\times$ buffers, however increasing the charge density would certainly help. For the undiluted I50Mg05 buffer, $\lambda$ would need to increase by a factor of $5-10$ but might be within reasonable possibility. In contrast the PIPES buffers, BRB-80 and BRB-20 are still a long way from detectability even with large increases in $\lambda$ unless a commensurate improvement in the density/thickness of the casein layer can be obtained somehow. Ultimately, the data in Figs. 3 and 4 convinces us that the prospects for electrostatic detection using microtubules are strong with the carbon nanotube on the guiding channel floor ($h_{p}=0$) providing lower ionic strength buffers than those commonly used for microtubule assays, i.e., BRB-80 and BRB-20, are deployed. The prospects can be further improved if the effective thickness/density of the casein layer can be improved and the microtubule charge can be increased slightly using an approach similar to that demonstrated by Isozaki et al. [77, 78] and further work in those two directions is also encouraged.

Finally, we note two aspects that we have not accounted for in our calculations that might improve the detection prospects slightly for both assay types. The first is the finite persistence length of the filaments and thermal fluctuations. These lead to vertical movement of the filament segments between attachments to surface-bound motors. This effect can be enhanced by reducing the bound motor density on the substrate [79]. The second is that we have neglected the nanotube diameter, which at $1-2$ nm is already $25-50\%$ of the small pedestal height ($h_{p}=4$ nm) required for microtubules when the casein layer was not taken into account. It would slightly improve detection prospects beyond our estimates for $h_{p}=0$ when the casein layer is accounted for. This second contribution will be much less substantial for actin.

We have addressed the prospects for electrostatic detection of passing actin filaments and microtubules in a molecular motor motility assay by combining a basic electrostatic model for determining the effective gate potential at a carbon nanotube transistor due to a filament overhead with realistic detection scenarios from recent state-of-the-art nanotube transistor sensors [26, 27]. On a bare consideration of magnitudes, we suggest that both actin and microtubules are likely detectable electrostatically in a device context with sufficient effort. Actin requires the carbon nanotube to be raised on a pedestal of height $28-33$ nm above the guiding channel floor depending on buffer ionic strength; a possible fabrication route for achieving this is given in the Supplementary Information. Microtubules carry three distinct advantages that make them a much better prospect for electrostatic detection: a) higher charge density, b) lower natural travel elevation, and c) the presence of a casein passivation layer [66], which incorporates/supports the kinesin and partially displaces buffer without changing the microtubule elevation. In contrast, for an actomyosin assay, the myosin molecules are bound directly to a silanized-SiO₂ surface with no equivalent to the casein passivation layer [51]. We suggest microtubules are likely detectable without the nanotube on a pedestal for the lowest ionic strength buffers when the casein passivation layer is accounted for. Another aspect that might improve detection prospects is increasing the charge density of the filament. Surface charge density modification for microtubules using short-strand DNA has already been demonstrated [77, 78] and might be fruitful towards electrostatic detection.

Our results suggest that attempts at electrostatic detection would have a greater likelihood of success if focussed on kinesin/microtubules. Low ionic strength buffers, e.g., imidazole or HEPES, should be used rather than higher ionic strength PIPES buffers, e.g., BRB-80, and the casein layer should be prepared to optimise its thickness and volume fraction as much as possible to reduce screening contributions. Some obvious experiments that our work here also points to are a more detailed characterisation of the structural aspects, e.g., thickness and volume fraction, of $\alpha$-casein and whole casein passivation layers on SiO₂ to facilitate comparison with $\beta$-casein-on-SiO₂, which is better characterised [72, 73] but less suited for motility assay performance [70]. Studies of how casein passivation shifts the gate threshold voltage in carbon nanotube transistors and how motility depends on the back-gate voltage for an assay structure with an incorporated back-gate, e.g., SiO₂-on-$n^{+}$-Si, would also be useful towards further development of devices for electrostatic detection of filaments in gliding motility assays.

This work was funded by the Australian Research Council (ARC) under DP170104024 and DP210102085, the Volkswagen Foundation and the European Union’s Horizon 2020 Programme under grant agreement No. 732482 (Bio4Comp). A.P.M. was a Japan Society for the Promotion of Science (JSPS) Long-term Invitational Fellow during the drafting of this manuscript. The process development work reported in the Supplementary Information was performed using the NSW node of the Australian National Fabrication Facility (ANFF). We gratefully acknowledge helpful discussions with Paul Curmi, Mercy Lard, Lawrence Lee, Frida Lindberg and Cordula Reuther in the course of this work.