Abstract
Although its importance is widely accepted recently, force measurement has been difficult in living biological systems, mainly due to the lack of the versatile non-invasive force measurement methods. The fluctuation theorem, which represents the thermodynamic properties of small fluctuating non-equilibrium systems, has been applied to the analysis of the thermodynamic properties of motor proteins in vitro. Here, we extend it to the axonal transport of endosomes. The fluctuation of their displacement distributed with several distinct peaks at multiples of a unit value, which the fluctuation theorem can convert into the drag force exerted on the endosomes. The results demonstrated that a single cargo vesicle is often conveyed by two to three force producing units. Furthermore, this method enabled us to compare the force-velocity relations in vitro and in vivo. These results showed the versatility of this approach for general non-invasive force measurements in vivo.
Introduction
One of the technical hurdles in mechanobiology, a growing field of science at the interface of biology and physics, has been the methods to measure force in living cells non-invasively. The force or stress on the outer surface of the cells, or the plasma membrane, can be measured by the traction force microscopy (Polacheck & Chen, 2016). Fluorescent protein-based biosensors for force or tension at the cellular levels are also actively developed using Förster resonance energy transfer (Guo, Wang, Sachs, & Meng, 2014; Meng & Sachs, 2012; Meng, Suchyna, & Sachs, 2008). Optical tweezers have been used to measure force exerted on the lipid droplets in cultured cells or in Drosophila embryos (Jun, Tripathy, Narayanareddy, Mattson-Hoss, & Gross, 2014; Mas, Farre, Sancho-Parramon, Martin-Badosa, & Montes-Usategui, 2014; Shubeita et al., 2008), but its application to other organelles or subcellular structures is difficult. The Stokes relation can be theoretically used to estimate the drag force on the organelles moving at the velocity v as F=6πηrv. But, the viscosity η and the diameter of the organelle r need to be measured, the latter of which is often difficult for small organelles whose size is close to or below the diffraction limit of the microscope resolution. One may think that a friction coefficient Γ=6πηr can be estimated from the diffusion coefficient of an organelle, D via the Einstein relation D=kBT/Γ, which is a one kind of the fluctuation-dissipation theorem (FDT) valid in near equilibrium. However, FDT is reported to be violated in various living cellular systems (Dinis, Martin, Barral, Prost, & Joanny, 2012; K. Hayashi et al., 2013; Turlier et al., 2016). We have previously reported that the Einstein relation is violated for organelle transports which was regarded as a complex non-equilibrium phenomena far from equilibrium (K. Hayashi et al., 2013). Therefore, force measurement in living cells needs essentially new physics beyond equilibrium statistical mechanics and the linear response theories.
A good example that needs force measurements is the axonal transport of vesicles. They are transported mainly by kinesins from the cell body to the periphery (anterograde) and dynein for the reverse direction (retrograde) (Hirokawa, Noda, Tanaka, & Niwa, 2009). Although many studies to date have elucidated their biological or functional importance, many physical or biophysical properties are still unclear (Encalada & Goldstein, 2014). For example, there is still controversy regarding the relationship between motor number, velocity and force. The in vitro velocity of purified kinesin-1 is around 1 μm/s and does not change with increasing motor number, while the cargo velocity in the axon is often much faster, close to 5 μm/s (Allen, Metuzals, Tasaki, Brady, & Gilbert, 1982; Chiba et al., 2014) and apparently is dependent on the motor number (Reis et al., 2012). Mammalian dynein is more controversial. Many studies report its maximum force as only around 1 pN (A. G. Hendricks, Holzbaur, & Goldman, 2012; Mallik, Carter, Lex, King, & Gross, 2004; Rai, Rai, Ramaiya, Jha, & Mallik, 2013), though a few studies report values close to that of kinesin (Belyy et al., 2016; Nicholas et al., 2015; Toba, Watanabe, Yamaguchi-Okimoto, Toyoshima, & Higuchi, 2006).
To investigate these questions, here we propose a non-invasive force measurement by using the fluctuation theorem (FT). FT is a new universal law for entropy production in small nonequilibrium systems actively studied in the field of physics, and is considered as a generalized form of FDT, which connects energy dissipation to fluctuation (Ciliberto, Joubaud, & Petrosyan, 2010). In previous studies, for example, we have established that the FT can be applied to estimate molecular energies from the fluctuation property of bio-molecules in vitro (Kumiko Hayashi, Ueno, Iino, & Noji, 2010). This approach enables the estimation of the energy or force from only the passive measurement, the fluctuation of the movement. Thus, it is a fully passive and non-invasive method, potentially suitable for measurements in the living cells.
Results
Overview of the Theory
The quantity to be discussed in this study is force (F) generated by motors hauling a single cargo. In the stationary state, it is balanced by the drag force. The major cause of energy dissipation during the transport is the drag force, which would be increased with friction due to collisions or interactions with various cytoplasmic components such as water molecules, macro molecules, other organelles or cytoskeletal elements. Because the increase of friction slows diffusion, there should be an inverse relation between the drag force and the fluctuation (Figure 1), which is formulated by the FT as follows (see supplementary text for theoretical details): where ΔX is the displacement of the cargo calculated from the center position of a cargo as ΔX=X(t+Δt)−Δ(t), P(ΔX) is the probability distribution of ΔX, kB is the Boltzmann constant. Teff is the effective temperature, which is a generalized measure of temperature in a non-equilibrium system (Cugliandolo, 2011; Dieterich, Camunas-Soler, Ribezzi-Crivellari, Seifert, & Ritort, 2015; Kumiko Hayashi & Sasa, 2004; Mizuno, Tardin, Schmidt, & MacKintosh, 2007).
Validation of FT with kinesin-driven beads in vitro
Before applying this theory to a real biological target, we first experimentally characterized the empirical parameter, effective temperature Teff, by analyzing the bead-movement along a microtubule conveyed by a single or two kinesin molecules in vitro. The beads were prepared in a condition that most of the beads showed processive runs with stall forces at around 8 pN or 16 pN (Figure 2-figure supplements 1), indicating that each run was conveyed by one or two kinesin molecules (dimers) (Furuta et al., 2013). The net force F exerted on the bead was controlled by optical tweezers as F=Fm–Fopt (=Γν) where Fm and Fopt are the forces produced by the kinesin motors and the load by the optical tweezers, respectively.
The fluctuation of the bead displacement was analyzed as follows. The traces of bead movement were divided into several segments so that the bead velocity, which is proportional to the drag force, could be regarded to be constant in each segment (Figure 2A). The displacement was calculated in each segment with a fixed time interval Δt as ΔX=X(t+Δt)-X(t) (Figure 2B). The probability distribution of P(ΔX) was fitted with a Gaussian function (Figure 2C) to calculate the right-hand term of equation (1)
As expected theoretically, χ showed dependence on the time interval of the analysis Δt with convergence at Δt≥20 ms (Figure 2D) (see supplementary text for detail). The value of χ at Δ t=30 ms was used as the converged value χ*. This is because equation (1) is a macroscopic relation valid for the time scales of the fluctuation that dominate the drag force. Thus, equation (1) should be rewritten as which we tested by plotting the converged value χ*, against F (Figure 2E). Here, the force F was measured independently by using Stokes law F =Γν. The friction coefficient Γ was determined as 2.9 ± 0.4 μN s/m (mean ± SEM, n=4) from the power spectrum density of the same bead at the trap center of the optical tweezers before the run (Bormuth, Varga, Howard, & Schäffer, 2009) (Figure 2-figure supplement 1 B, C). This value is consistent with the previous reports (Bormuth et al., 2009; Okada & Hirokawa, 1999; Mark J. Schnitzer & Block, 1997). Here, the Gaussian distribution of ΔX gives the estimate for the diffusion coefficient D as 0.003 μm2/s. This value is twice larger than expected from the Einstein relation D=kBT/Γ with the parameter values for Γ =2.9 μN s/m and T=300 K, which shows the violation of this relation only valid for the equilibrium systems, and experimentally supports the necessity to extend FDT by introducing the effective temperature as in equations (1) and (1’).
As shown in Figure 2E, the data points aligned linearly, as expected from equation (1’). The slope of the plot F/χ* gives the effective temperature as kBTeff. Here, it should be noted that the data points include both beads driven by a single kinesin molecule (closed square, stall force < 12 pN) and by two kinesin molecules (open square, stall force >12 pN). Still the values for the effective temperature were scattered around Teff=650K, indicating that the effective temperature was insensitive to the number of the active motors on the bead, velocity or force.
Fluctuation measurement of axonally transported vesicles in vivo
Next, we examined the validity of FT with a real biological system, the axonal transport of endosomes in supracervical ganglion (SCG) neurons. The endosomes were selectively stained with a membrane-staining dye, DiI (Figure 3A). As established previously, most endosomes show linear movement along the axon anterogradely (to the axon terminal) or retrogradely (to the cell body). Although they sometimes show stochastic switching of the velocity or reversal of the direction, the fluctuation analyzed here is the fluctuation of the displacement around a constant velocity. Thus, the segment of unidirectional movement of constant velocity was chosen for the analysis (boxed regions in Figure 3B).
The movement of the endosome was recorded at the frame rate of 98 frames per sec. The position of the endosome was determined as the centroid of the fluorescent spot with the accuracy of 8 nm (see Methods for details). The degree of fluctuation χ (equation (2)) was calculated from the Gaussian distribution P(ΔX) (Figure 3C) of the displacement during the time interval Δt, namely ΔX(t)= ΔX(t+Δt)- ΔX(t) as described above (see Methods for details, and Figure 3-figure supplement 1 for the evaluations of the errors). Here we note that the assumption of white Gaussian noise in X(t) (supplementary text) was checked by the Gaussian forms of P(ΔX) (Figure 3C) and the power spectrum density S(v) of the position X(t) (Figure 3D).
The degree of fluctuation χ thus calculated showed convergence (χ*) but with a relaxation time two times longer than that of the in vitro kinesin-coated beads (Figure 2D, 3E, 3-figure supplement 2A). The relaxation time did show dependency on the enzymatic turnover rate (Figure 3-figure supplement 2B, C), but the relaxation time was much longer than the enzymatic cycle time (~10 ms/molecule with saturating ATP for kinesin). The microenvironment around the vesicle, especially its viscoelastic nature, would affect the relaxation time as well.
Validation of FT with axonally transported vesicles in vivo
To validate FT (equation (1’)), we first searched for the traces that contain two successive constant velocity segments (~2 s duration for each segment) with different velocities as shown in Figure 3B, because such traces would enable us to test FT without further estimation of the parameter values.
For each vesicle, its size or surrounding environment will not change substantially during the few seconds of the run event. Then, the friction efficient Γ would be same for both before and after the velocity change. The drag forces in the two velocity segments are written as F1=Γv1 and F2=Γv2, respectively, given by the Stokes law. If FT (equation (1’)) holds with the same Teff value for both segments, F1=kBTeff χ1* and F2=kBTeff χ2*. Their ratio thus gives unity,
The results (Figure 4A) were consistent with this relation, suggesting that the effective temperature would take same or similar value for each vesicle during the few seconds of the run event.
As the second validation, we tried to examine the relationship between the drag force and the fluctuation among different vesicles. The measurement of the drag force is generally difficult because the friction coefficient Γ is difficult to estimate for the moving vesicles. However, Γcan be estimated for a tethered vesicle from the power spectrum of its Brownian movement. We, therefore, searched for the trajectories (14 from > 200 traces) with a constant velocity long run (~ 2 s in duration) followed by a long pause (no net movement > 3 s) (Figure 4-figure supplement 1A). The latter, pause segment, was used to estimate the friction coefficient Γ (Figure 4-figure supplement 1B, C). This value was applied to the former, running segment, because the friction coefficient, which is determined by the size of the endosome and its environment, will not change much during the single run event. The drag force was thus calculated from the Stokes law: F=Γv. As shown in Figure 4B-D, there were no clear relationship between the velocity and the fluctuation χ* (Figure 4B), but the drag force was proportional to the fluctuation χ* (Figure 4C). This is because the friction coefficient varies among vesicles. The plot in Figure 4D shows that Γ is roughly proportional to the vesicle size, though the proportional constant or the effective viscosity was ~1000 times larger than water as reported previously in non-neuronal cells (Wirtz, 2009). It should also be noted here that the Einstein relation D= kBT/Γ is violated. The diffusion coefficient D was estimated as ~0.01 μm2/s from the Gaussian distribution of ΔX, and this value was more than 10 times larger than expected from the Einstein relation with the values for Γ =~10 μN s/m and T=300 K. The extent of the violation was larger for the in vivo organelle transport than for the in vitro bead movement discussed in the previous section, which would reflect the complex interactions in the in vivo environment.
More importantly, the proportional relation between the drag force and the fluctuation χ* in Figure 4C implies that FT holds for these different vesicles in different neurons (n=14) with the same or similar value of proportional coefficient, or the effective temperature. There were no statistically significant differences between the effective temperature values between the anterograde and retrograde endosomes. These results would imply that the effective temperature would reflect the cellular environments surrounding the endosome rather than the number or other properties (e.g. kinesin or dynein) of the motor protein that conveys the vesicle.
The slope of the best fit line gives the estimate of this global effective temperature as Teff=4200 K. This does not literally mean that the temperature of the cytoplasm is 4200 K. Although the exact physical meaning of the effective temperature is still controversial (Kumiko Hayashi & Sasa, 2004), it is often observed Teff > T in non-equilibrium systems (Cugliandolo, 2011). The most plausible interpretation would be that the fluctuation process(es) that dominantly determine the drag force are actively driven by using energy 14 times higher than the thermal energy, which might reflect the recently reported active stirring of the cytoplasm by motor proteins (Fakhri et al., 2014). Here, the free energy obtained by single ATP hydrolysis is about 20 kBT, which would give a reference for the energy scale of active processes in living cells.
Drag force measurement using FT for anterogradely transported endosomes in axon
Using the value of Teff=4200 K calibrated for endosomes (n=14) described above, we analyzed the remaining traces for the anterogradely transported endosomes with the segments of constant velocity that lasted for longer than 3 seconds. The measure of the fluctuation χ was then calculated for the constant velocity region for each endosome for various intervals Δt from 10 ms to 100 ms, which confirmed that the time constant for the convergence was around Δt=50 ms (Figure 3E, 5A). We, therefore, analyzed the remaining shorter traces which had segments of constant velocity that lasted for about 2 seconds, with the interval Δt up to 50 ms.
As summarized in Figure 5A, B, the plots (79 runs) appeared to be clustered into several groups. The histogram of the χ values at Δt=50 ms, the proxy for the convergent value χ*, showed discrete distribution, which was statistically confirmed by the k-means clustering (Method). Because χ* is approximately proportional to force F (equation (1’)), this discrete distribution of χ* is most likely to reflect the force producing unit (FPU) in this system. At the same time, it also implies that the proportionality constant between the force and the fluctuation, or the effective temperature, does not vary much among these 79 vesicles from 76 neurons. Otherwise, the discrete force distribution by the presence of FPU would be obscured by the distribution of the effective temperatures.
By applying the global effective temperature value of 4200K (Figure 4C), the peak positions of the force distribution were multiples of about 5 pN. This value agrees well with the reported value for the force production by a single kinesin in vitro (M J Schnitzer, Visscher, & Block, 2000), suggesting that a single kinesin molecule would correspond to a single FPU in the anterogradely transported endosomes. There was a weak tendency toward more FPUs on larger endosomes (Figure 5B). The number of FPUs would thus not be tightly regulated to compensate for the greater drag resistance for the larger vesicles, but would simply reflect the geometric constraints that larger vesicles have more space to accommodate additional FPUs.
It should be noted here that the velocity distribution does not show clear peaks (Figure 5-figure supplement 1). If the friction coefficient Γ were constant among different vesicles, the velocity would also distribute discretely as reported with frog melanosomes (Levi, Serpinskaya, Gratton, & Gelfand, 2006), because the velocity should be proportional to the drag force as v=F/Γ For the endosomes in neurons, however, the discrete distribution of force was obscured because Γ varied much among different vesicles reflecting the large variance of the vesicle size (Figure 4D).
Force-velocity relations for anterogradely transported endosomes in axon
Assuming that no external forces are exerted on the endosomes transported in the axon, the drag force should be balanced with the force produced by the motor protein, most likely kinesin (Figure 1). Thus, the measured values for the force and the velocity should scatter along the force-velocity relation curve for the kinesin motor property.
For the anterogradely transported endosomes, one kinesin dimer molecule would most likely correspond to the anterograde FPU, because the previous biochemical measurement reported that only 1-4 kinesin dimers are bound to vesicles (Adam G Hendricks et al., 2010). Based on this assumption, we have compared the mechanical properties of kinesin in vitro and the force-velocity relation of the anterogradely transported endosome. The force-velocity relations were plotted (Figure 5C) along with the phenomenological model that describes the in vitro results of kinesin (M J Schnitzer et al., 2000). The force ranges for 1, 2, 3 and 4 kinesin molecules were consistent with the in vivo results, but the velocity was much faster in vivo. Better fitting was achieved by increasing the enzymatic turnover rate (Figure 5D), which might suggest the acceleration of the enzymatic reaction by the macromolecular crowding in the cytoplasm (Ellis, 2001).
Measurement and analysis of the force for retrogradely transported endosomes in axon
The traces for the retrogradely transported endosomes were similarly analyzed. Traces with the segments of constant velocity that lasted for longer than 3 seconds were first examined to confirm the time constant for the convergence as ~ 50 ms, and then analyzed the remaining shorter traces which had segments of constant velocity that lasted for about 2 seconds (Figure 6A). The plots (119 runs) were also clustered into several groups. The discrete distribution of the χ values at Δt=50 ms (the proxy for the convergent value χ*) was statistically confirmed by the k-means clustering. Thus, the presence of FPU was also demonstrated with retrogradely transported endosomes. The number of FPUs showed a weak positive correlation to the endosome size (Figure 6B) as observed with the anterogradely transported endosomes (Figure 5B), which is consistent with the geometric constraints model (Figure 5B, inset) for the regulation of the number of FPUs on the endosome. Note that the discrete distribution of χ* also gives another experimental support that the effective temperature does not vary much among these 119 vesicles from 112 neurons.
By applying the global effective temperature value of 4200K (Figure 4C), the peak positions of the force distribution were multiples of about 6 pN, similar to those with the anterogradely transported endosomes.
Although the force-velocity relations for dynein are still controversial, the phenomenological model (M J Schnitzer et al., 2000) for kinesin used in Figure 5C is a general phenomenological model which can be applied to a wide range of processive motor proteins including dynein by adjusting the parameters. We have, therefore, fitted the force-velocity relations of the retrograde vesicles (Figure 6C) to the same model. The maximum velocity was increased similarly with kinesin.
The maximum force of the best fit curve was around 10 pN and inconsistent with most in vitro studies that report only 1 pN for a single dynein molecule (A. G. Hendricks et al., 2012; Mallik et al., 2004; Rai et al., 2013). Several groups independently reported that dynein can be activated to produce maximum force around 5 pN (Belyy et al., 2016; Nicholas et al., 2015; Toba et al., 2006), and there might be a mechanism to increase the maximum force of dynein to 10 pN in the cytoplasm. Considering that the average number of dynein molecules on a single vesicle (Adam G Hendricks et al., 2010) is two times larger than the average number of retrograde FPUs in this study, two dimers of dynein along with dynactin or other regulatory protein(s) might serve as a single FPU, as was recently reported for the cooperative, collective force production by multiple dynein molecules (Torisawa et al., 2014). Alternatively, only half of the dynein molecules on the endosome are activated to produce force up to 10 pN. Thus, the results obtained with this measurement suggested that our understanding is still limited on retrograde transport mechanism supported by dynein.
Discussion
In this study, we have demonstrated that FT is practically useful for the non-invasive force measurement in vivo. Theoretically, the proportionality constant between the entropy production and the fluctuation, or the effective temperature, is affected by many complex interactions in real biological systems and is not necessarily equal to the real temperature of the system (Cugliandolo, 2011; Zimmermann & Seifert, 2012). It is not clear a priori whether it takes same or similar values even in a single cell. However, as we have shown experimentally, the variance of the effective temperature Teff (10% CV) was much smaller than that of the friction coefficient Γ (100% CV) for the axonally transported endosomes (Figure 4B, C). Hence, the number of the FPUs on the vesicle was reflected only to the fluctuation χ* as a discrete distribution with distinct peaks (Figure 5A, B), but not to the velocity v (Figure 5–figure supplement 1). The large variance of Γ reflects the various sizes of the axonal endosomes (Figure 4D), and there was a weak positive correlation between the number of the FPUs and the endosome size (Figure 5B, 6B).
Although the physical or mechanistic details behind the small variance of Teff remains unclear, it implies that calibration for Teff is not necessary for each vesicle or neuron, but a single result of calibration can be applied to other vesicles. In this study, a single value of Teff (4200K) can be applied to the ~200 vesicles analyzed here. The force value ~10 pN, thus determined, is consistent with previous in vivo measurements in macrophages (A. G. Hendricks et al., 2012). This force value, along with the other parameter values (the diameter of endosome 2r =500 nm, and the velocity v=2 μm/s) gives the effective viscosity ηeff as ~1000 cP by using the relation F=6πηeffr v. This estimate is 1000x higher than water, but is consistent with the microrheology measurement in the cytoplasm with traces of similar sizes to the endosomes (Wirtz, 2009).
The maximum force or the stall force in vivo estimated from the force-velocity relation was also consistent with the results with purified or recombinant kinesin measured in vitro (M J Schnitzer et al., 2000). However, the force velocity relation in vivo showed large discrepancy from the in vitro one, not in the force but in the velocity (Figure 5C). Here, the velocity was directly measured from the trajectory. Therefore, the discrepancy is not likely to be caused by any potential ambiguities in the force estimation by FT. Since 4-fold increase in the velocity gives better fitting of the in vitro relations to the in vivo results (Figure 5D), most naïve interpretation of the results would be that kinesin will move 4 times faster in vivo. However, the in vivo velocity of constitutively active truncated kinesin-1 dimer does not differ much from the in vitro velocity. There might be some regulatory mechanisms to accelerate the velocity by the scaffold protein that anchors kinesin to the cargo vesicle as suggested from our previous analysis of the APP-transport vesicles (Chiba et al., 2014). Similarly, the force-velocity relation of the retrograde endosomes also suggested the presence of some cooperative, collective force production mechanisms for multiple dynein molecules in vivo. These issues were first realized by the drag force measurement in vivo, and are good examples to show the importance of the force measurements in the future studies.
In summary, we have established a FT-based method to estimate the drag force exerted on the vesicles transported in living cells by analyzing only their movement. Unlike other existing methods for force measurement, it is fully passive and non-invasive. We used vital staining with a fluorescent dye for a selective visualization of endosomes, but DIC or phase-contrast image of unstained samples can be used as well. Thus, this non-invasive method would serve as a powerful and versatile tool for basic research in the field of intracellular transport, as well as some potential applications for the examination of the molecular motor functions in clinical samples. Furthermore, the violation of FDT has been reported in various biological systems both in vitro and in vivo such as actomyosin (Mizuno et al., 2007), DNA hairpins (Dieterich et al., 2015) and red blood cells (Turlier et al., 2016). With the idea of effective temperature, as a violation factor of the fluctuation-dissipation theorem, other fluctuation theories of nonequilibrium statistical mechanics other than the FT used in this paper, such as the Jarzynski equality (Jarzynski, 1997) and the Crooks fluctuation theorem (Crooks, 1999) may connect the non-equilibrium fluctuation to the energetic properties of systems. Thus, our method can be expanded to other mechanical processes in living cells, and would provide a basis for the versatile general method for the non-invasive in vivo mechanical measurements.
Materials and Methods
Reagents
All reagents were purchased from Wako or Sigma-Aldrich, unless otherwise stated.
Purification of kinesin for in vitro assay
As described previously (Okada & Hirokawa, 1999), a constitutive active dimer construct of mouse kinesin-1 (KIF5C 1-560aa) was subcloned into a plasmid vector pET21B (EMD biosciences). An in vivo biotinylation tag, BCCP (biotin carbonyl carrier protein, Promega) was inserted to the C-terminus of the construct. The plasmid was introduced into bacterial cell BL21(DE3)RIL (Agilent). The transformant was cultured with 2x YT medium supplemented with 30 mM phosphate buffer (pH 7.4) at 37 °C to mid-log phase (OD600=1.0). The culture was cooled down to 23 °C, and the protein expression was induced by adding 0.1 mM (final concentration) isopropyl β-D-1-thiogalactopyranoside (IPTG). The bacterial cells were collected 5 hrs after induction, and rinsed with ice-cold phosphate-buffered saline (PBS) supplemented with phenylmethyl-sulfonyl fluoride (PMSF).
Then, the bacterial cells were resuspended with five volumes of buffer A (HEPES 50 mM, potassium acetate 500 mM, magnesium acetate 5 mM, imidazole 10 mM, pH 7.4 adjusted with KOH) supplemented with ATP 0.1 mM and the following protease inhibitors: Pefabloc SC 1 mM, Leupeptin 20 μM, Pepstatin A 10 μM, Na-p-tosyl-L-arginine methyl ester (TAME) 1 mM. The bacterial cell wall was solubilized with lysozyme (2 mg/ml). DNase I (10 μg/ml) was added to reduce viscosity by the bacterial genomic DNA. Then bacterial cells were broken by sonication.
The soluble protein was recovered by centrifugation at 20,000 xg for 30 min, and was applied to the immobilized metal affinity chromatography column TALON (Takara). The protein was eluted with buffer B (PIPES 20 mM, imidazole 250 mM, magnesium sulfate 2 mM, EGTA 1 mM) supplemented with ATP 0.1 mM and protease inhibitors. The peak fractions were pooled and stored at −80°C after snap-freezing in liquid nitrogen.
Preparation of microtubules
Tubulin was purified by the high-molarity PIPES buffer method (Castoldi & Popov, 2003) with modifications (Yajima et al., 2012). Porcine brains were cleaned by removing meninges, blood clots and vessels in washing buffer (PIPES 50 mM, PMSF 5 mM, pH 6.8). They were homogenized in a pre-chilled Waring blender with PEM buffer (PIPES 100 mM, EGTA 1 mM, MgCl2 1 mM, pH 6.8 adjusted with KOH) supplemented with PMSF 0.5 mM, Leupeptin 2 μM and DTT 0.5 mM. After clarification with centrifugation at 15,200 xg, 60 min, microtubules were polymerized by warming the supernatant to 37 °C after supplementation with MgATP 1 mM, MgGTP 0.5 mM and glycerol. The polymerized microtubules were collected by ultracentrifugation at 100,000 xg 37 °C. Then, they were depolymerized in ice-cold P1000EM buffer (PIPES 1,000 mM, EGTA 1 mM, MgCl2 1 mM, pH 6.8 adjusted with KOH) at 0 °C. The supernatant was collected by ultracentrifugation at 100,000 xg, 4 °C. The polymerization and depolymerization cycles were repeated four times, and the final supernatant was pooled and stored in liquid nitrogen.
TMR (tetramethyl rhodamine)-labeled microtubules were prepared as follows. Microtubules were polymerized in PEM buffer supplemented with 1 mM GTP at 37 °C. Then, 5-(and-6)-Carboxytetramethylrhodamine, Succinimidyl Ester (Life Technologies) was added at 5-10 molar excess. Labeled microtubules were separated from free dye by ultracentrifugation through glycerol cushion, and were resuspended with ice-cold PEM buffer. The microtubules were depolymerized by cooling down the solution to 0°C, and the supernatant was collected by ultracentrifugation at 100,000 xg, 4 °C. The labeling efficiency was measured spectroscopically, and stored in liquid nitrogen.
Bead assay
For the bead assay, the carboxy-modified fluorescent 0.5 μm latex bead (Life Technologies) was biotinylated with (+)-biotinyl-3,6,9,-trioxaundecanediamine (Amine-PEG3-biotin, Pierce) using condensation agent DMT-MM(4-(4,6-dimethoxy-1,3,5-triazin-2-yl)-4-methyl-morpholinium). The purified recombinant kinesin dimer was immobilized on the bead surface via streptavidin (Sigma) in assay buffer (PIPES 80 mM, magnesium acetate 5 mM, EGTA 1 mM, ATP 2 mM, casein 0.5 mg/ml, taxol 10 μM, β-mercaptoethanol 10 mM, catalase 0.1 mg/ml, glucose 10 mM, glucose oxidase 0.05 mg/ml, pH 6.8). Diluted, TMR-labeled microtubules were absorbed to the glass surface of the flow cell chamber, and the remaining surface was coated with a biocompatible polymer Lipidure-BL-103 (NOF). Then, the kinesin-coated beads were injected into the chamber. The optical tweezers instrument is based on the inverted microscope IX2 (Olympus). The beam of near infra-red laser (1064 nm BL-106C, Spectra-Physics) was collimated to fill the back aperture of the objective lens (PlanApo 60x/1.40, Olympus). The bead trapped at the focus was illuminated with green laser (532 nm, 400 mW, Genesis CX, Coherent), and its image was projected to EMCCD camera iXon DU-860D-CS0-#BV (Andor). The images were recorded at the speed of 400 frames per second at 22 °C. The constant velocity segments (n=45) used in the analysis (Figure 2E) were cut from 31 runs from 5 different bead assays.
Primary culture of neurons
Superior cervical ganglions (SCGs) isolated from 3 weeks old ICR mice (male) were enzymatically treated in 0.5 % trypsin (Sigma) followed by 2 hr treatment with 0.5 % collagenase (Worthington). Dissociated cells were rinsed with DMEM/F12 containing 10 % heat inactivated bovine serum (Life Technologies), and plated onto Matrigel (BD-Biosciences)-coated glass-bottom dish (Matsunami). The neurons were cultured for two to four days with a DMEM/F12 medium supplemented with 10 % heat inactivated bovine serum and 200 ng/ml 2.5S nerve growth factor. All the animal experiments were conducted in compliance with the protocol which was approved by Institutional Animal Care and Use Committee, Tohoku University.
Observation of endosomes and image analysis
The neurons were stained for 10 min with 100 nM DiI (Life Technologies), and then observed with a fluorescent microscope (IX71, Olympus) equipped with a heating plate (CU-201, Live Cell Instrument). The images of the motile endosomes were obtained with a 100x objective lens (UPlanFL 100x/1.3, Olympus) and an EMCCD camera LucaS (Andor) at 98 frames per second at 37 °C. The center position of each endosome was determined from the recorded image using ImageJ (Rasband, 1997), and the displacement from the position in the first frame was calculated for each frame. Here we focused on the displacement along the direction of the motion X(t). The data were collected from 34 preparations (culture dishes). 79 endosomes from 76 different cells for anterograde, and 119 endosomes from 112 different cells for retrograde were investigated. The cells for observation were chosen randomly after visual inspection, and the trajectories with longer than 2 second constant velocity run(s) were selected for the analyses. The constant velocity segment was selected by fitting the trajectory with a constant velocity movement, so that the residual does not exceed the variance perpendicular to the movement. The accuracy of the position measurement was verified with fluorescent beads with a similar size and fluorescent intensity to the endosomes (300 nm latex bead, Polyscience). The standard deviation of the position of the bead tightly attached to the glass surface was 8.4±0.4 nm (5 different beads in 2 independent preparations), which is much smaller than the displacement between frames ΔX (=X(t+Δt)-X(t)) analyzed in this study and would affect the accuracy of the fluctuation measurement by less than 10% CV, within the range of the estimation errors in the fluctuation (Figure 3–figure supplement 1).
Preparation of permeabilized and reactivated neuron
For some experiments, the plasma membrane of the neuron was permeabilized to control the cytoplasmic ATP concentration (Okada, Sato-Yoshitake, & Hirokawa, 1995). Clarified brain homogenate was used to compensate for the loss of cytoplasmic components after membrane permeabilization. Mouse brain was cleaned in ice-cold PBS and homogenized with three volumes of KHMgE buffer (K-acetate 115 mM, Hepes 20 mM, MgCl2 1 mM, EGTA 1 mM, pH 7.4) supplemented with protease inhibitor cocktail (Complete EDTA-free, Roche). The homogenate was clarified by the successive centrifugation at 1,000 xg 10min and 100,000 xg 1 hr. The brain cytosol thus prepared was aliquoted and snap frozen with liquid nitrogen. The assay buffer was prepared just before use by mixing the brain cytosol with equal volume of KHMgE buffer supplemented with ATP regeneration system (0.125 mM or 0.0125 mM ATP, 10 mM creatine phosphate, 8 U/ml creatine phosphokinase), protease inhibitor cocktail (Complete EDTA-free) and 5 mM beta-mercapto ethanol. The neurons were first rinsed with KHMgE, then followed by 8 min incubation with the assay buffer containing 0.01 mg/ml digitonin. Fluorescent dextran (VECTOR) was used to examine the membrane permeabilization after digitonin treatment. The data were collected from 11 preparations (culture dishes) for [ATP]=125 μM (62 endosomes from 53 cells) and 6 preparations (culture dishes) for [ATP]=12.5 μM (30 endosomes from 23 cells).
Measurement of the friction coefficient from the power spectrum
The power spectrum density, S(v), of the bead position attached but not moving along the microtubule is known to be proportional to the Lorentzian function 1/(1+(Γ/κeff v)2) where v is frequency and Keff is an effective stiffness (Bormuth et al., 2009). Here, Keff can be estimated from the equilibrium probability distribution of the bead position:
PB(X) (=(2π kBT/Keff)0.5exp(-KeffX2/2kBT)). The same relations can be applied to the pausing motion of the endosomes. Therefore, we first examined the distribution of X(t) and fitted with a Gaussian distribution to estimate Keff. Then, the power spectrum density of X(t) was calculated as where v is a frequency and τs=Nw/98 s. Note that the recording rate was 98 fps, Nw is the window size, and 〈·〉 denotes the time average over the trajectory investigated. The Lorentzian function was fitted to this spectrum to estimate Γ. The estimated values of Γ had error of approximately 20% (Figure 2-figure supplement 1C, Figure 4-figure supplement 1C).
Analysis of fluctuation using FT
The value of χ is defined as from the distribution, P(ΔX), of the displacement ΔX =X(t+Δt)−X(t). Since the noise was confirmed to be Gaussian (Figure 3C, D), P(ΔX) was fitted with a Gaussian function where the fitting parameters a and b correspond to the variance and the mean of the distribution. By substituting equation (M4) to equation (M3),
Thus, χ was calculated as 2b/a for each P(ΔX) for various interval Δt from 10 ms to 100 ms. The values for a and b were also estimated from the cumulative Gaussian distribution of ΔX, and directly as the sample variance (a=<(ΔX-<ΔX>)2>) and the average (b=<ΔX>). These two estimations provided the same values of a and b within the error of χ (Figure 3-figure supplement 1). The converged value χ was determined by plotting χ against Δt as shown in Figure 2D and 3E (χ* =χ at 30 ms in Figure 2, and χ* =χ at 51 ms in Figure 4-6.)
Teff was estimated from the trajectories of endosomes that paused just before or after the run (Figure 4-figure supplement 1). For these trajectories (n=14), the drag coefficient Γ was estimated from the power spectrum during the pause period as described in the previous section, and the fluctuation χ* was determined from the constant velocity segment. Then, Teff was estimated from the following relation: as shown in Figure 4C. The Teff value (=4200K) thus determined was used for all other endosomes analyzed in this paper.
k-means clustering
The χ-Δt plots for the anterograde and retrograde endosomes (Figure 5A, 6A) were classified statistically by using a k-means clustering method using a program package R with a library “cluster”. First, the number of clusters k was determined by calculating AIC (Akaike′s Information Criterion) for χ-Δt plots. In the case of the anterograde χ-Δt plots (Figure 5A), AIC values were −122.1 for k=2, −134.6 for k=3, −140.2 for k=4, and −134.4 for k=5. In the case of the retrograde χ-Δt plots (Figure 6A), AIC values were −232.3 for k=2, −246.6 for k=3, −247.0 for k=4 and −245.3 for k=5. From these AIC values along with the Gap statistics, the most probable values of k were determined as k =4 for anterograde and k=3 for retrograde, respectively. The initial value for the k-th cluster center trajectory was chosen as the k-th peak value of χ* (the arrows in Figure 5A, 6A). Each trajectory of χ was classified to the cluster based on the mean square deviation from Then was renewed as the mean of the trajectories classified to that cluster. This procedure was repeated until convergence.
Analysis of the force-velocity relation
We adopted the widely-accepted model of kinesin-1 (M J Schnitzer et al., 2000) for the analysis of the measured force-velocity relations. In this model, velocity is expressed as the function of ATP concentration as where d is the step size (=8 nm). kcat and kb are the catalytic turnover rate and the apparent second order rate constant for ATP binding (=ratio of kcat and Michaelis-Menten constant km). The load (F) dependencies are introduced as
The values for the parameters kcat and kb were taken from the reported in vitro results (M J Schnitzer et al., 2000): qcat=0.0062, qb=0.04.
The ATP concentration in the cytoplasm is around 4 mM. Since it is much higher than Km which is around 50 μM for both kinesin and dynein, the inaccuracy in the ATP concentration does not affect the results. was determined from the maximum velocity of endosome observed in our experiments (3.9 μm/s and 4.1 μm/s for anterograde and retrograde runs, respectively). Namely, (488 /s and 515 /s for anterograde and retrograde runs, respectively). For the analysis of anterograde runs, the characteristic distance δ was set to 3.8 nm, which is the value reported with kinesin-1 (M J Schnitzer et al., 2000). For the retrograde data sets, the value δ was estimated to be 3.6 nm by fitting the model.
Supplemental Text for the Theory section
For the bead moving at a constant speed (v) in vitro, under highly damped conditions, the motion of center position X(t) can be described by the Langevin equation: where the second term on the right-hand side represents the stochastic collisions with solvent molecules, where ξ is Gaussian noise with 〈ξ(t)ξ(t’)) = δ(t-t−〉, where < > denotes the time average over the time course, and D is a diffusion coefficient. It is well known, in near equilibrium, that D in equation (S1) satisfies the fluctuation-dissipation theorem (Howard, 2001), D=ÆbT/j where γ is the friction coefficient of the bead, kB is the Boltzmann constant, and T is the temperature of the environment. Using this relation, equation (1) is rewritten as where F is a force defined as F =γν.
In the case of a cargo vesicle transport in cells, these relations cannot be applied directly. The cytoplasm in the living cell could be far from equilibrium and the fluctuation-dissipation theorem would be broken (Turlier et al., 2016). There can be more interactions between the vesicle with the surrounding environments than the simple collisions with solvents. The vesicles can be temporarily tethered to cytoskeletal filaments or interacting with other vesicles that are also driven by other molecular motors. Even in such complex situations, the friction force of the vesicle can be regarded to be proportional to the velocity unless the movement is too fast. This apparent friction coefficient Γ thus includes the interaction effect between the cargo and surrounding environment, so that it would be larger than γ=6πητ as assumed in the classical Stokes formula (r: the radius of the cargo, η: the viscosity of the cytosol). This assumption was already examined experimentally by tracking the trajectories of the passive tracer beads in the cytoplasm (Wirtz, 2009).
Similarly, the second term on the right-hand side in equation (S2) should be revised. Since the assumption for Gaussian noise is based on the assumption that the noise is caused by large numbers of independent interactions, it is not dependent on whether the system is close to the thermal equilibrium or not. In fact, the probability distribution of the fluctuation ΔX(t) measured in this study was Gaussian, and its power spectrum was inversely proportional to the square of the frequency (Fig 3C, D). These results experimentally confirmed the assumption of the white Gaussian noise. For the amplitude of the Gaussian noise, however, the fluctuation-dissipation theorem can no longer be assumed. Hence, we replace the equilibrium temperature T with an empirical parameter Teff.
With these modifications, the movement of a vesicle in the cytoplasm can be modeled as (Figure 1)
From the stochastic model (S3), the following fluctuation theorem is derived (Kumiko Hayashi et al., 2010):
In this paper, the quantity on the right hand side is called as the degree of fluctuation χ:
Then, equation (S5) becomes
Here, it should be noted that equation (S3) is a phenomenological macroscopic model. Microscopically, the motion of the vesicle, for example, will be described by a more complex model like where is thermal noise acting on the bead, Um is an interaction between vesicle and motors, Uother is the other interactions acting on the vesicle. Equation (S3) corresponds to the coarse-grained model of equation (S7), and the microscopic effect of Um and Uother contributes to the effective viscosity Γ, effective noise and the drag force F. Note that this decomposition of energy has been theoretically studied (Kumiko Hayashi & Sasa, 2005). Thus the effective temperature Teff is dependent on the microscopic hidden variables (Seifert, 2005; Zimmermann & Seifert, 2012).
Equation (S6) holds for the time scale long enough to be compared with the characteristic time scales for the microscopic interactions Um and Uother, among which the enzymatic reaction of motor protein will be the slowest process with around 10 ms cycle time. Hence, the right-hand side of equation (S6), χ will show a relaxation behavior against the time interval Δt with the analysis for ΔX. The relaxation time constant τ is expected to show dependence on the cycle time of the motor protein, which was confirmed experimentally in the following sections. Therefore, equation (S6) is modified as follows to reflect that the converged value of χ*, instead of the transient values of χ, should be used for the estimation of the drag force:
Acknowledgments
We thank N. Sawairi and M. Tomishige for initial stages of experiments; S. Xu, J. Asada, M. Komeno, M. Kakiuchi and K. Ito for their technical and secretarial assistance; W. Kylius for editing the manuscript.