NANO LETTERS
Dispersion in Active Transport by Kinesin-Powered Molecular Shuttles
2005 Vol. 5, No. 7 1337-1342
Takahiro Nitta†,‡ and Henry Hess*,†,§ Department of Bioengineering, UniVersity of Washington, Seattle, Washington 98195, Department of Mathematical and Design Engineering, Gifu UniVersity, Gifu, 501-1193, Japan, and Department of Materials Science and Engineering, UniVersity of Florida, GainesVille, Florida 32611 Received March 25, 2005; Revised Manuscript Received May 23, 2005
ABSTRACT Active transport driven by molecular motors is a key technology for the continued miniaturization of lab-on-a-chip devices, because it is expected to enable nanofluidic devices with channel diameters of less than 1 µm and total channel lengths on the order of 1 mm. An important metric for a transport mechanism employed in an analytic device is dispersion, because it critically affects the sensitivity and resolution. Here, we investigate the mechanisms responsible for the dispersion of a swarm of “molecular shuttles”, consisting of functionalized microtubules propelled by surface-adhered kinesin motor proteins. Using a simple model and measurements of the path persistence length, motional diffusion coefficient, and the distribution of average velocities, we found that, at the time scale relevant in the envisioned nanobiodevices, variations in the time-averaged velocities between shuttles will make a stronger contribution to the dispersion of the swarm than both the fluctuations around the time-averaged velocity of an individual shuttle and the fluctuations in path length due to wiggling within the channel. Overall, the dispersion of such molecular shuttles is comparable to the dispersion of a sample plug transported by electroosmotic flow.
In nanodevices, scaling laws and practical considerations favor the replacement of pressure-driven fluid flow, electroosmotic or electrophoretic flow with molecular motordriven active transport.1 For example, the flow velocity achieved with a given pressure gradient falls with the square of the reduction in tube diameter, and the scaling down of devices using electric fields in liquids often experiences problems with the formation of bubbles at the electrodes or the shielding by electric double-layers in buffer solutions. Active transport by molecular motors does not suffer from these difficulties and has evolved in nature as a successful solution to the problem of nanoscale transport.2 In particular, in the axons of neurons, which have typically submicron diameters and can have lengths exceeding 1 m, specialized motor proteins efficiently transport a variety of molecules and nanoparticles.3,4 Inspired by such biological examples, a number of bionanodevices have been designed which utilize kinesin and myosin motor proteins to power active transport in microand nanofabricated channels.5-13 A general approach to the design of such “molecular shuttles” is to define a network of channels coated with motor proteins, which serve as tracks for the directed transport of either actin filaments (AFs) or microtubules (MTs) with speeds on the order of 1 µm/s. * Corresponding author. E-mail:
[email protected]. † University of Washington. ‡ Gifu University. § University of Florida. 10.1021/nl050586t CCC: $30.25 Published on Web 06/10/2005
© 2005 American Chemical Society
When functionalized with specific linkers, the AFs or MTs can bind to the complementary target analytes and carry them along the predefined track. Once bound, the analyte can be subjected to the sequence of purification, labeling, and detection steps typical for lab-on-a-chip devices. An open question is how well this sequence can be timed with this new transport method, or in other words how rapidly a sample plug formed by a swarm of shuttles disperses (Figure 1). For pressure-driven fluid flow, sample dispersion is an important limitation, since the parabolic velocity profile across the channel leads to rapid dispersion of the sample plug across the whole length of the channel.14 Electroosmotic flow, with its near uniform velocity profile, is in this respect superior, since in an ideal situation the dispersion stems only from the diffusion of the analyte out of the sample plug.15 Here, we develop a simple model for dispersion in active transport, and experimentally determine the parameters in the model. For active nanoscale transport we have to evaluate three contributions to the swarm dispersion: the path length dispersion along the channel due to the wiggling motion of shuttles, the fluctuation of the velocity around the timeaverage of each shuttle due to the stochastic action of the molecular motors, and the variation of the time-averaged velocity among shuttles (which are not necessarily identical). Assuming that the three contributions are independent, the total mean square deviation of the shuttle position from the
Figure 1. (Top) Schematics of the sample plug dispersion during transport by pressure-driven flow, electroosmotic flow, and active transport by molecular shuttles. (Bottom) Our design for a molecular shuttle relies on kinesin motors adsorbed to the surface of a channel. Dozens of motors attach to and propel a microtubule with a diameter of 24 nm and a length of several micrometers. Functionalization of the tubulin subunits of the microtubule permits selective attachment of different types of cargo.
average is given by 〈(∆x)2〉 ) 〈(∆xd)2〉 + 〈(∆xv-flu)2〉 + 〈(∆xv-avg)2〉
(1)
where ∆xd is the positional deviation due to the wiggling motion, ∆xv-flu is due to the velocity fluctuation of each MT, and ∆xv-avg is due to the variance of the time-averaged velocity among shuttles. Since the loading of different types of cargo, such as molecules,8 quantum dots,12,16 or microspheres,17 may impact the dispersion to a different degree, we focus here on the dispersion of unloaded shuttles, under the assumption that the principles governing the behavior of unloaded and loaded shuttles are the same, but that the parameters have to be adjusted. Contribution from Wiggling Motion. The wiggling path of the shuttles down the track causes fluctuations in the endto-end distance of the path, similar to fluctuations of the endto-end distance for a polymer confined in a narrow tube. Since no exact analytic solution describing these fluctuations exists, we developed a simple model outlined in the methods section to estimate this term as: w2 Vt Lp
〈(∆xd)2〉 ) 0.2
(2)
where Lp is the persistence length of the shuttle path, w is the width of the track, V is the shuttle velocity, t is the time, 1338
Figure 2. Trajectories of microtubules gliding over kinesin coated surfaces. (Left) A typical trajectory of a microtubule moving smoothly. (Right) A trajectory which contains two pinning events of the microtubule. The velocity fluctuates within a narrow band, unless a pinning event occurs. The angular change in the direction of microtubule trajectories between two frames also varies within a narrow band centered around zero. A pinning event causes large angular changes as the microtubule gets stuck, wiggles, and breaks free, heading in a new direction.
and w , Lp is assumed. For our MT-based shuttles we were able to determine the persistence length Lp from the analysis of 33 trajectories of MTs moving on unstructured kinesincoated surfaces (Figure 2). In the absence of the temporary pinning of the MT tip to the surface, which is possibly due to defective motors,18 the persistence length of the path (Figure 3) was 111 µm (92-132 µm). The temporary pinning, which occurred on average once every 750 µm traveled, leads to a drastic change in direction. If these events are included in the analysis, the persistence length dropped to 35 µm (28-44 µm). A theoretical analysis predicted that for high motor densities on the surface, as is the case here, the persistence length of the trajectory should equal the persistence length of the filament.19 However, here even the persistence lengths of smooth trajectories are much smaller than the persistence length of a free MT in solution, which ranges from 1 to 8 mm, depending on assembly conditions.20 Contribution from Velocity Fluctuation of Individual MT. The fluctuations in the velocity of individual MTs, which account for the second term in eq 1, have been measured previously over distances of up to a few micrometers to derive information about the inner workings of the associated motor proteins.23 In contrast, we are here concerned with total channel lengths on the order of 1 mm, Nano Lett., Vol. 5, No. 7, 2005
no defective motors introduce additional fluctuations. Consequently, this contribution cannot be reduced by the design of the channels or improved surface preparation, but ultimately limits the attainable resolution. Contribution from Variance in Time-Averaged Velocity among MTs. The variance of the time-averaged velocity among MTs contributes to the deviation of MT position from the swarm mean according to 〈(∆xv-avg)2〉 ) 〈(∆Vavg)2〉t2
Figure 3. Correlation functions of the directional fluctuations of smooth microtubule trajectories (solid squares, average of 16 trajectories ( sem), and trajectories with pinning events (open squares, average of 17 trajectories(sem). The linear regression fit included the region from 5 to 50 s. Deviations of the data points from a linear curve at later times are a consequence of the decreasing size of the available data set.21,22
Figure 4. The averaged mean-square deviation of the distance traveled by microtubules as a function of the lag time averaged over 16 smooth trajectories (solid squares, average ( sem) and 17 trajectories with pinning events (open squares, average ( sem). The linear regression fit included the region from 5 to 50 s.
which would be a typical length for a bionanodevice. As described previously,23 we found that the mean square displacement from the average distance traveled increases linearly with time (Figure 4) as given by 〈(∆xv-flu)2〉 ) 2Dv-flu‚t
(3)
where the motional diffusion coefficient Dv-flu is found to be equal to (2.0 ( 0.4) × 10-3 µm2/s for smooth trajectories, and (3.3 ( 0.7) × 10-2 µm2/s for trajectories with temporary pinning. The value for smooth trajectories matches the value of 2.6 × 10-3 µm2/s given by Imafuku et al.23 (our measurements were carried out for large MT lengths, 20 ( 8 µm), despite a 20-fold increase of the sampling time in our study. This is an exciting result, since it means that even over large distances the velocity fluctuations retain the signature of the stochastic movement of individual motors, provided Nano Lett., Vol. 5, No. 7, 2005
(4)
where ∆Vavg is the deviation of each MT velocity from the ensemble average. To determine 〈(∆Vavg)2〉, the variance of the distance traveled by smoothly gliding MTs was determined to be 0.5 ( 0.2 µm2 after 25 s, 0.9 ( 0.4 µm2 after 35 s, 1.6 ( 0.6 µm2 after 50 s, 2.4 ( 1.0 µm2 after 60 s, and 2.7 ( 1.5 µm2 after 75 s. These variances in the distance represent the sum of 〈(∆xv-flu)2〉, caused by fluctuations in the velocity of individual MTs over time, and 〈(∆xv-avg)2〉, caused by fluctuations of the time-averaged velocity between MTs. Fitting of eqs 3 and 4 to these data, using the previously determined motional diffusion coefficient Dv-flu, yields a variance of the average MT velocity of (5 ( 3) × 10-4 µm2/ s2, corresponding to a variation in the average velocity between microtubules of 3% around the mean. This variation of the time-averaged velocity among MTs can potentially arise from a number of effects, including different numbers of protofilaments, defects in the tubulin lattice arising from the polymerization process, or even defective tubulin subunits hindering kinesin motility. Summary. The overall time-dependent mean-square deviation of the MT position from the average can be expressed by 〈(∆x(t))2〉 ) 0.2
w2 Vt + 2Dv-flut + 〈(∆Vavg)2〉t2 Lp
if the MTs started at the same position at time zero. In the absence of temporal pinning, the measured parameters (Lp ) 111 µm, Dv-flu ) 0.002 µm2/s, 〈(∆vavg)2〉 ) 5 × 10-4 µm2/s2) suggest that the contribution from the variation in the average MT velocity will be larger than the contribution from the motional diffusion after around 10 s and becomes the dominant factor over longer times, and that the contribution from the wiggling motion in the channel becomes smaller than the contribution from the motional diffusion if the channel width is smaller than 2 µm. Thus, even narrower channels do not significantly reduce the dispersion. After a travel distance of 1 mm, the deviation of the MT shuttle position from the mean will be 0.03 mm, which is roughly the distance a protein of 100 kDa would diffuse in 3 s. Compared to systems employing pressure-driven fluid flow, the dispersion of the sample plug is remarkably small for active transport of microtubule shuttles by kinesin motors. The dispersion is roughly comparable to that of a system employing electroosmotic flow with a typical driving velocity of 0.3 mm/s. 1339
The presence of pinning of MT shuttles increases the contributions of wiggling and velocity fluctuations. This effect is just one example how fabrication challenges, such as defective motors or imperfect guiding at the sidewalls of the channels, will lead to deviations from the previously discussed ideal situation. Further deviations may arise from the loading of different cargo and potentially from interactions between shuttles. While the origin of the fluctuations in the velocity of an individual MT has been discussed at a fundamental level previously,24-26 we have only a very limited understanding of the origin of the persistence length of the MT path,19 which is an order of magnitude smaller than the persistence length of the MT, as well as the origin of the variance in the timeaverage velocity between MTs discovered here. Active transport driven by molecular motors is a bioinspired technology that is central for the continued miniaturization of fluidic systems. In addition to its advantages in energy utilization and systems integration, the uniformity of transport velocity compares favorably with established transport mechanisms. Materials and Methods. Kinesin and Microtubule Preparation. Kinesin preparation is described in detail elsewhere.27 Briefly, a kinesin construct consisting of the wild-type, fulllength Drosophila melanogaster kinesin heavy chain and a C-terminal His-tag was expressed in Escherichia coli and purified using a Ni2+-NTA column. The eluent contained functional motors with a concentration of ∼100 nM and was used as the stock solution. Fluorescently labeled tubulin (Cytoskeleton, Denver, CO) was polymerized into microtubules in BRB80 buffer (80 mM Pipes, 2 mM MgCl2, 1 mM EGTA, pH 6.85 with KOH) with 4 mM MgCl2, 1 mM GTP, and 5% DMSO for 30 min at 37 °C. Microtubules were 100-fold diluted and stabilized in BRB80 plus 10 µM paclitaxel (Sigma, St. Louis, MO). Motility Assay. Flow cells were assembled by sandwiching double-sided tape between a glass coverslip and a glass slide (Fisher Scientific, Pittsburgh, PA). The buffer for all solutions was BRB80. The flow cells were sequentially filled with a casein solution (0.5 mg/mL casein in BRB80), a kinesin solution (0.2 mg/mL casein, 1 mM ATP, and ∼5 nM kinesin), and finally a motility solution (∼1.6 µg/mL of tetramethylrhodamine-labeled microtubules, 1 mM ATP, 10 µM paclitaxel, oxygen-scavenging system [20 mM D-glucose, 20 µg/mL glucose oxidase, 8 µg/mL catalase, 1 mM DTT]). Flow cells were sealed with immersion oil to prevent evaporation. The assays were performed at room temperature (25 °C). Data Collection. Microtubules were tracked for more than 500 s by recording the position of the leading end using an epi-fluorescence microscope (Nikon TE200 with a 60× oil N.A. 1.4 objective, a Roper Cascade 650 CCD camera, Metamorph software) with a computer-controlled xy-stage (H128V3, Prior, Rockland, MA). Whenever a microtubule came close to the border of the field of view, stage movements to re-center the microtubule were carried out between image acquisitions (acquisition rate of 0.2 Hz, 1340
exposure time of 0.25 s). The position of the microtubule before and after the stage movement was calibrated by measuring a position of a small microtubule fragment stuck on the surface. The variance in repeated length measurements of moving, short MTs was 0.27 pixel2 (pixel size - 189 nm). Thus the position measurement error can be estimated as 70 nm. Calculation of Motional Diffusion Coefficient. Following Imafuku et al.,23 we calculated the motional diffusion coefficient from the variance of the sliding displacement: 〈(r(∆t) - 〈r(∆t)〉)2〉 ) 2Dv-flu∆t where r(∆t) is the net displacement of a filament along its trajectory during a given time interval ∆t ) tj+k - tj, and 〈 〉 denotes the average. Since the trajectories were not linear, the r(∆t) was obtained from the contour distance along the trajectory traveled by MT, as follows: k-1
r(∆t) )
∑ dr(tj+m+1,tj+m)
m)0
where dr(tj+m+1,tj+m) ) x(xj+m+1 - xj+m)2 + (yj+m+1 - yj+m)2 Calculation of Correlation Function of Fluctuation in Direction and Persistence Length. We calculated the persistence length Lp using
(
〈cos[∆θ(∆t)]〉 ) exp -
)
Vavg∆t 2Lp
(A1)
where ∆θ(∆t) is the difference between the tangent angles of the trajectory at two points of lag time ∆t. Calculation of Variance of Time-AVeraged Velocity among Microtubules. To exclude a drift of the average velocity over time due to temperature drift, we observed multiple MTs at the same time in a field of view within 75 s, without moving the xy-stage. We calculated the variance of the distance traveled by the MTs in the field of view after 25, 35, 50, 60, and 75 s. Such observations were made multiple times (5 for 50 s and less, 4 for 60 s and 3 for 75 s). The average and the standard error of the mean at each time for these observations were calculated. The total number of MTs observed was 53 (25 s), 50 (35 s), 30 (50 s), 17 (60 s), and 11 (75 s). Analytical Model of Dispersion due to Fluctuation in Direction. We developed a basic model for the deviation from the mean of the position of MTs due to the wiggling motions in the channel, based on the experimental data of Clemmens et al.28 MTs move along the channel in a wiggling motion, and wall collisions realign their direction of movement along the channel axis. The change in the direction of movement between collisions ∆θ is, according to eq A1, Nano Lett., Vol. 5, No. 7, 2005
Thus we obtain for the variance of distance traveled within a given time:
approximately 〈(∆θ)2〉 )
〈L〉 Lp
(A2) 〈(L|(t) - 〈L|(t)〉)2〉 )
for w , 〈L〉 and 〈L〉 , Lp, with w being the width of the channel, 〈L〉 being the average distance traveled by MT between collisions, Lp being the persistence length of the MT path on a planar surface coated homogeneously with kinesins. Assuming that the MTs travel on a circular path, ∆θ is geometrically related to 〈L〉 and w by ∆θ )
〈n2〉 - 〈n〉2 ≈ 〈n〉
3 ) 4w2Lp
With (A3) 〈n〉 )
Clemmens’ experimental data showed that the distribution around this mean distance is roughly Gaussian, with a standard deviation of ∼〈L〉/2. To build a simple analytical model, we replaced this Gaussian distribution of distances between collisions by two equally likely possibilities: The MT can travel either a short trajectory segment with length 1/2〈L〉 or a long segment with length 3/2〈L〉. Approximating the shape of these segments again as circular sections, we can calculate the velocity parallel to the channel axis as
) )
V|,short ) V 1 -
8w2 3 〈L〉2
V|,long ) V 1 -
8w2 27〈L〉2
1/2〈L〉 3/2〈L〉 + m‚ )t V|,short V|,long
(A5)
Eliminating m from eq A5 using eq A4,
(
) (
1 8w2 n 64w2 - 〈L〉 1 + L|(t) ≈ 〈L〉n + Vt 1 2 2 2 27〈L〉 27〈L〉2 Nano Lett., Vol. 5, No. 7, 2005
1/2〈L〉 + 3/2〈L〉
we obtain using eq A3: w2 〈(L|(t) - 〈L|(t)〉)2〉 ≈ 0.2 Vt Lp or 〈(∆xd)2〉 ) 0.2
w2 Vt Lp
Acknowledgment. We thank Viola Vogel for providing inspiration and equipment, Bruce Bunker and George Bachand for their support in our collaborative efforts, and gratefully acknowledge financial support from the U.S. Department of Energy, Office of Basic Energy Sciences under grant DE-FG03-03ER46024 and the DARPA Biomolecular Motors Program directed by Anantha Krishnan. References
(A4)
L|(t), the total distance MT traveled during t, is expressed by 3 1 L|(t) ) 〈L〉n + 〈L〉m 2 2
L|(t)
as listed above.
for the respective segments (V is velocity of movement along the MT axis). Within a given time t, each MT travels n short and m long segments:
n
Since the total number of steps (n + m) within time t depends on the random number of short segments n, the variance of n is not simply given by the variance of a binomial distribution for a fixed number of trials. However, we numerically evaluated the variance of n as a function of time and convinced ourselves that a good estimate is
2w 〈L〉
By substituting this into eq A2 we obtain
( (
( )
〈L〉2 64w2 2 2 (〈n 〉 - 〈n〉2) 4 27〈L〉2
)
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NL050586T
Nano Lett., Vol. 5, No. 7, 2005
