Microtubule Dynamics Assay.

Microtubule plus end growth and the GFP intensity of fission yeast EB1 (Mal3-GFP) in the plus end region were measured in flow cells assembled as described previously (39) using TIRF microscopy (19, 55). Alexa568-labeled microtubules were assembled from biotinylated, Alexa568-labeled and GMPCPP-stabilized seeds, which were attached to a functionalized glass coverslips via Neutravidin (Life Technologies). The free tubulin concentration was 10, 20, or 30 µM [of which 12.5% was labeled with Alexa568 (Life Technologies)]. For all experiments, Mal3-GFP was present at 200 nM. The final imaging buffer was 80 mM K-Pipes (pH 6.85; Sigma), 1 mM EGTA (Sigma), 1 mM DTT (Fischer), 5 mM 2-mercapothethanol (Sigma), 10 mM Na-ascorbate (Sigma), 90 mM KCl (Fischer), 1.5 mM GTP (Fermentas), 0.1% (vol/vol) methylcellulose (Sigma), 50 µg/mL β-casein (Sigma), 20 mM glucose (Fischer), 0.5 mg/mL glucose oxidase (Serva), 0.25 mg/mL catalase (Sigma), and 0.5 mM MgCl₂. A comparatively low concentration of MgCl₂ was used to obtain long uninterrupted growth episodes and good signal-to-noise ratios for Mal3-GFP, because high MgCl₂ concentrations decrease the affinity of Mal3 (24). For the lower tubulin concentrations, the final reaction mixture was supplemented with tubulin storage buffer (BRB 80) to maintain the same buffer environment for all conditions. Simultaneous imaging of the Alexa568 channel and the GFP channel was performed at 4 Hz with an exposure time of 100 ms per frame and identical laser settings for all conditions. All experiments were performed at 30 °C.

Automated Microtubule Tracking.

After image acquisition, the two fluorescent channels were aligned using a calibration grid (Compugraphics) as previously described (19) and drift-corrected using a custom macro written in ImageJ (https://imagej.nih.gov/ij/). For MSD analysis and fluctuation analysis of the microtubule growth, time traces of microtubule plus-end positions were extracted from the Alexa568 channel using a previously described MATLAB (The MathWorks) program, which has been shown to achieve subpixel resolution (19, 40). Cropped images of the microtubule’s end in the Alexa568 and GFP channel were also generated at each tracked end position. These images, 1 μm (axial) by 0.4 μm (lateral) in size, were aligned along the axis of the microtubule and centered 0.17 μm away from the tracked end position to capture the full intensity from the end. From the cropped images, time-averaged EB1-GFP fluorescence intensity profiles (comets) were generated for use in comet analysis. For the cap size fluctuation analysis, mean signal intensity values were extracted from the cropped images in the GFP and Alexa568 channels, resulting in time traces of intensity fluctuations at the microtubule end. These intensity values were background-subtracted assuming a flat background obtained from the mean of the pixel values falling in the range [Imin, Imax×0.2].

Fluorescence Background Subtraction from EB1-GFP Intensity Fluctuation Data.

The measured mean intensity at the end of the microtubule is a combination of the signal from the cap and a background signal from EB1-GFP in solution. The EB1-GFP intensity fluctuation data therefore required a second background subtraction. Theoretically the mean size of the cap is directly proportional to the mean growth speed of the microtubule. We therefore obtained an estimate for the background fluorescence signal by plotting the mean growth speed against the mean EB1-GFP intensity for each track and making a linear fit to the data, extrapolating back to the origin (Fig. S5A). The y-intercept of the fit represented the background fluorescence signal, which was subtracted from the measured intensities before analysis.

Processing Tracks and Selecting Growth Episodes.

The extracted end positions occasionally showed some artifacts due to tracking errors. To mitigate the effects of these errors the tracks were further processed; at each tracked microtubule end position xi at time ti a linear fit to the track in the 50-s window [ti−25 s,ti+25 s] was made to find an expected position x¯i, assuming constant growth within this window. The deviation xi−x¯i was then computed and the distribution of all of the deviations was found for each experimental condition. The end position xi was identified as a tracking error if its deviation from constant growth fell outside a threshold value corresponding to a deviation of 488 (4 σ), 894 (4 σ), and 1,136 (3 σ) tubulin dimers for the 10 μM, 20 μM, and 30 μM tubulin conditions, respectively. The threshold value was chosen to best identify the tracking errors observable by eye and each track was visually inspected to ensure that the procedure worked as expected. It was observed that tracking errors often occurred in close succession; hence, complete tracks or sections of tracks were deleted (Fig. S2A) until the number of tracking errors per track was less than three for each experimental condition. It was also observed that the vast majority of errors (90%) were in the negative direction, indicating that they were indeed experimental artifacts (Fig. S2B). The tracking errors left in the remaining data, which comprised 99%, 90%, and 86% of the original data for the three tubulin conditions, were then corrected by replacing xi with x¯i.

Short episodes of very slow growth were observed occasionally in the growth trajectories of the microtubules, reminiscent of “pauses” of growth (Fig. S2C). We considered these growth episodes to be deviations from steady-state behavior and not applicable to our analysis of steady-state growth. To identify these episodes a linear fit was made at each point xi in the window [ti−12.5 s,ti+12.5 s] and a velocity vi was extracted from the gradient of the fit. The 25-s window was chosen to be smaller than the average pause length (80 s) so that the beginning and end of a pause could be accurately identified. To capture the pauses seen by eye a threshold velocity was chosen, 65% of the mean velocity for the track, and if vi fell below this threshold xi was flagged as anomalous. Manual inspection of the tracks indicated that episodes exceeding 17.5 s, 15 s, and 12.5 s for the 10 μM, 20 μM, and 30 μM conditions, respectively, were apparently deviations from steady-state growth. These episodes were removed from the data and the remaining partial tracks were subsequently considered separately (Fig. S2C). Shorter episodes were identified as anomalous at higher tubulin concentration because these microtubules grow faster. Two, four, and five anomalous episodes were identified in the 10 μM, 20 μM, and 30 μM datasets, respectively, one of which was discounted as a false positive. The data now left for analysis were 98%, 88%, and 81% of the original data comprising 38, 25, and 17 tracks for the 10 μM, 20 μM, and 30 μM tubulin conditions with an average duration of 193 s, 196 s, and 148 s. The statistics of these procedures are summarized in Table S1.

The remaining trajectories and their corresponding EB1 intensity fluctuation data were considered as representing pure steady-state growth behavior recorded with the highest possible positional tracking precision our automated tracking routine can deliver.

MSD Analysis.

The growth trajectory of the microtubule represents a Brownian diffusion-with-drift process, where the “drift” is characterized by the mean growth speed of the microtubule and the “diffusion” characterizes growth fluctuations about the mean position. The mean growth speed, vg, is captured by the MD (23):〈L(t+τ)−L(t)〉=vgτ.[S1]The MSD provides additionally the diffusion constant, D, and the squared amplitude of the measurement noise, Γv2:〈(L(t+τ)−L(t))2〉−vg2τ2+2Dτ+2Γv2.[S2]The mean growth speed was obtained from a linear fit to the MD as a function of the time interval τ (Fig. 1D, Middle). The diffusion constant, D (reflecting the extent of the growth fluctuations), and the measurement noise, Γv (tracking precision), were determined from a linear fit to the variance of the displacement [i.e., the MSD minus the MD squared, as a function of the time interval τ (Fig. 1D, Bottom)]. From Eqs. S1 and S2 this givesMSD(τ)−MD(τ)2=2Dτ+2Γv2.[S3]

EB1 Comet Analysis.

Average EB1-GFP fluorescence intensity profiles (comets) of the growing microtubule end were generated to determine the average size of the EB cap and the maturation rate km. For each microtubule growth episode a time trace of the microtubule end position and a cropped image of the end were generated as described in Automated Microtubule Tracking. The instantaneous microtubule growth speed at each time point was calculated using a 10-s smoothing window, and the corresponding cropped images were then speed-sorted into quartiles. The lowest quartiles were discarded because they contained the greatest deviations from steady-state growth. For each remaining quartile the EB1-GFP comet profiles were extracted from the cropped images and averaged (19). The average comet profiles were then fitted with a convolved mathematical function assuming a monoexponential decay of the density of EB binding sites (figure 2 C and F in ref. 19) to obtain an estimate of km.

EB1 Binding/Unbinding Rates.

We have calculated the effect of the fast binding/unbinding turnover of EB1 molecules on the measured fluctuations of the total EB intensity. Under these growth conditions and this EB1 concentration we expect a total of ∼270 EB1 binding sites: B₀ (figure 5E in ref. 19). The EB1 binding/unbinding rates from single molecule measurements (figure 3B in ref. 19) are koff=3.4  s−1 and kon=0.12  nM−1 ⋅s−1=24 s−1 (at an EB1 concentration E0=200 nM). These rates are much greater than the maturation rate of binding sites (∼0.2  s−1), so changes in the number of binding sites can be neglected at steady state. Hence, the mean number of occupied binding sites is 〈EB〉=B0γ/1+γ, where γ=E0kon/koff, with a corresponding variance Var〈EB〉=〈EB〉/1+γ. For these experimental conditions, γ=7,〈EB〉=270 * 0.875 = 236 and Var〈EB〉=236/8=30 (i.e., this would lead to an SD of ∼2% mean signal and cannot explain the measured fluctuations). Furthermore, the characteristic correlation time of this system would be 1/E0kon+koff= 36 ms (i.e., less than the camera exposure time of 100 ms and hence very unlikely to be observed).

Bleaching Control Experiments.

Bleaching of EB1-GFP molecules in this assay type is expected to be insignificant due to the fast binding/unbinding at microtubule ends; furthermore, the total volume of the flow cell is large in comparison with the TIRF field. To ensure that this is also true in our experimental setting, we performed control experiments at the same experimental conditions as used elsewhere (10 μM free tubulin and 200 nM EB1-GFP). We prepared a sample and imaged it for 1,100 frames (typical movie length for all other data) and then imaged the same microscope sample for another 1,100 frames. For each movie, microtubule ends were tracked and the total EB intensities were calculated over the whole movie. The mean values of EB1 intensities are shown in Table S2: They have a similar range, regardless of whether the samples have been imaged before, indicating that bleaching of EB1-GFP does not significantly contribute to the obtained values. Data were obtained from six independent experiments (six samples). For two microscope slides a third movie was recorded to further demonstrate that bleaching is insignificant.

Generation of Velocity Time Series.

We generated velocity time series by calculating finite differences of the position time series at 2 Hz (half the image acquisition rate). Thus, two time series were generated for each track, one from even-numbered frames and one from odd-numbered frames. Velocities were not calculated at the smallest possible time difference to reduce the relative magnitude of the measurement noise. The two resulting time series were considered separately, consistent with the theoretical analysis that treats microtubule length increments (equivalently velocity fluctuations) as independent.

Correlation Analysis.

The ACF of a process x is defined as Cx(τ)=〈(x(t)−x¯)(x(t+τ)−x¯)〉, where x¯=〈x〉. The CCF characterizing the correlation between two processes x and y is defined as Cxy(τ)=〈(x(t)−x¯)(y(t+τ)−y¯)〉.

To calculate the ACF and CCF for a discrete time series x(t) we can write x(nT)=x[n], where T is the sampling interval and the time series runs from n=1 to N. The asymptotically unbiased sample autocovariance is given byC^x[m]= 1N−m∑i=1N−m(x[i+m]−x^)(x[i]−x^),[S4]wherex^=1N∑i=1Nx[i].[S5]For the relatively short time series studied here this estimator is biased toward zero. The bias of lowest order, O(1N), can be eliminated by splitting the time series into two halves and recombining their respective ACFs c^(1) and c^(2) in the following way (56):C^=2c^−12 (c^(1)+c^(2)).[S6]This bias correction was implemented using the sample autocovariance given in Eq. S4.

To calculate the cross-covariance of two fluctuating time series x[n] and y[n] we used the estimatorC^xy[m]= 1N−m∑i=1N−m(x[i+m]−x^)(y[i]−y^).[S7]

Analysis of Velocity ACFs and Comparison with MSD Analysis.

Mean velocity ACFs are plotted for each tubulin concentration in Fig. 2C. To fit these functions we used Eq. 5: Cv^′(τ)=λ a2(ka+kd)δτ0+λ2Γv2(2δτ0−δτΔt). An estimate for measurement noise was obtained from the magnitude of the correlation at the shortest time lag (Δt) and this was compared directly to the estimate obtained from MSD analysis. The amplitude of the velocity fluctuations was obtained from the magnitude of the correlation at zero lag and the previously obtained estimate of the measurement noise, and this was compared with the estimate of the fluctuation amplitude derived from MSD analysis because (setting Δt=1) σ_v² = 2D.

Analysis of EB1-GFP Autocorrelation Functions.

As a control we calculated the autocorrelation functions of the Alexa568-tubulin channel and found there was a slowly decaying correlation, revealing a source of correlated noise present in the fluorescence intensity measurements (Fig. 3D), which is likely due to thermal fluctuations of the microtubule (43). Cross-covariance analysis confirmed that the same external noise source was affecting the EB1-GFP intensity measurements (Fig. S4 A and B). The EB1-GFP autocorrelation functions therefore comprise a component from the EB1-GFP fluctuations and also a component from the correlated noise. To account for this we performed a global fit to the six curves in Fig. 3 C and D. A biexponential was fitted to the EB1-GFP autocorrelation functions (Fig. 3C) of the formCi(τ)=Aie−km, iτ+αAie−kc, iτ,[S8]where the subscript i=1,2,3 denotes the three experimental conditions. The first term corresponds to the cap size fluctuations (intrinsic noise) and the second term corresponds to the thermal fluctuations (correlated noise) with correlation time 1/kc (Fig. S4C). For each condition the decay rate kc,i was shared with the decay rate from a monoexponential fit to the corresponding Alexa568-tubulin autocorrelation function (Fig. 3D),Ci(τ)=Bie−kc, iτ.[S9]Because the relative strength of the EB1-GFP fluctuations and thermal fluctuations are expected to be the same independent of tubulin concentration the parameter α was shared globally between the three conditions. For both fits the first data point was excluded (because this point also contains a Gaussian component due to measurement noise) and the subsequent 24 data points were fitted, which was a fitting range determined from simulated data to give accurate results (Fig. S1). The extracted parameters were also found to be robust under a range of fitting lengths (22–28 data points), with an SD in results of between 3% and 6% for all parameters and experimental conditions.

Calculating Mean Cap Size and Cap Fluctuation Amplitude from Intensity Measurements.

From the time-averaged analysis (MSD and comet analysis) estimates of D,vg and km were obtained. We used these results to predict average values for the mean cap size, μ, and the squared cap size fluctuation amplitude, σ2, for each tubulin concentration because we haveμ=1kmvga and σ2=12km(2Da2+vga)[S10]from expressions given in the main text. The same properties, in fluorescence units rather than numbers of cap sites, can be calculated from the fluctuation analysis of EB1-GFP intensities. The mean EB1-GFP intensity is the mean cap size in fluorescence units and the intrinsic noise of the intensity fluctuations is the cap size fluctuation amplitude in fluorescence units. The intrinsic noise of the intensity fluctuations was found by multiplying the variance of the intensity measurements by the coefficient Ai in Eq. S8.

To compare the results of the time-averaged analysis and the fluctuation analysis we found a proportionality factor that related the fluorescence units to numbers of cap sites by performing a global fit (weighted least-squares minimization) to the values of μ and σ2 derived from the time-averaged analysis and the values of μ′ and σ2′ derived from the intensity measurements (Fig. S5B), where the dash denotes units of fluorescence intensity. We then converted the fluorescence intensity measurements into numbers of cap sites and made a direct comparison (Fig. 3 F and G).

ACF of the Cap Size Fluctuations.

To derive correlation functions it is useful to define the Fourier transform pairs of the ACF and the CCF, which are the power spectrum and cross-power spectrum, respectively, given by (57): Sx(ω)=〈x(ω)x∗(ω)〉, ω>0 and Sxy(ω)=〈x(ω) y∗(ω)〉, ω>0. The zero frequency components are removed by definition because the ACF and CCF are defined with respect to the mean. One can then obtain correlation functions by first finding power spectra and performing an inverse Fourier transform. We start with a deterministic equation for the evolution of the cap,n˙(t)= ka−kd−kmn(t).[S11]In the LNA to find an equation for the stochastic fluctuations, δn(t), we make the substitution n(t)=n¯+δn(t) where n¯=〈n(t)〉 and add a Gaussian white-noise term, ξc(t), to account for the stochasticity in the reactions:ddt(n¯+δn(t))= ka−kd−km(n¯+δn(t))+ξc(t)δn˙(t)= −km δn(t)+ξc(t). [S12]The second line is obtained by noting that n¯=μ=(ka−kd)/km (Eq. 1). With a Fourier transform Eq. S12 becomes−iωδn(ω)=−km δn(ω)+ξc(ω).[S13]The power spectrum, 〈|δn(ω)|2〉, is therefore given by〈|δn(ω)|2〉=〈|ξc(ω)|2〉km2+ω2.[S14]Because ξc(ω) is the sum of the noise due to the growth fluctuations, ξv(ω), and the noise due to the maturation process, ξm(ω), we can write ξc(ω)=1aξv(ω)+ξm(ω) and obtain〈|ξc(ω)|2〉=〈(1aξv(ω)+ξm(ω))(1aξv∗(ω)+ξm∗(ω))〉=1a2〈|ξv(ω)|2〉+〈|ξm(ω)|2〉[S15]because the cross-terms, that is, 〈ξv(ω)ξm∗(ω)〉, are zero as the noise sources are independent. From the main text we have〈ξv(t)ξv(t′)〉=(ka+kd) a2 δ(t−t′)[S16]and〈ξm(t) ξm(t′)〉=km〈n〉 δ(t−t′).[S17]With a Fourier transform of these expression one can obtain〈|ξv(ω)|2〉=a2(ka+kd)[S18]〈|ξm(ω)|2〉=km〈n〉[S19]and

From Eqs. S15, S18, and S19 we obtain an expression for the power spectrum in Eq. S14,Sc(ω)=2 kakm2+ω2.[S20]An inverse Fourier transform recovers the ACF of the cap size fluctuations given in the main text,Cc(τ)=kakme−kmτ.[S21]

Cross-Power Spectrum of Velocity and Cap Fluctuations.

A deterministic equation for the growth fluctuations is given byv(t)=(ka−kd)a.[S22]Substituting v(t)=v¯+δv(t) into Eq. S22 and adding the Gaussian white noise term ξv(t) we obtain an equation for the stochastic velocity fluctuations,δv(t)=ξv(t),[S23]where we have used v¯=vg and vg=(ka−kd)a (see the main text). With a Fourier transform this becomesδv(ω)=ξv(ω).[S24]The cross-power spectrum is given by 〈δv(ω)δn∗(ω)〉. Using Eqs. S13 and S24 we find〈δv(ω)δn∗(ω)〉=〈ξv(ω)ξc∗(ω)〉km+iω=〈ξv(ω)(1aξv∗(ω)+ξm∗(ω))〉km+iω=1a〈|ξv(ω)|2〉km+iω.[S25]Using Eq. S18 we obtain the cross-power spectrumSvc(ω)=(ka+kd)km+iωa.[S26]A Fourier transform of the above recovers the CCF (Eq. 4),Cvc(τ)=a(ka+kd)e−kmτU(τ)[S27]

ACF of Velocity Fluctuations in the Presence of Measurement Noise.

For a diffusion-with-drift process we can write an instantaneous velocity v(t)=vg+ ξv(t) (see the main text). We calculate the velocity fluctuations as v^(t)=(L(t+Δt)−L(t))/Δt. In this finite case the equivalent noise term ξ^v(t) has different properties; its covariance is given by 〈ξ^v(t)ξ^v(t′)〉=λa2(ka+kd)δtt′, where λ=1/Δt and δij is the Kronecker delta. In the presence of positional measurement noise the expression for v^(t) requires two additional terms accounting for the measurement of L(t) and L(t + Δt),v^(t)=vg+  ξ^v(t)+ γ^v(t+Δt)−γ^v(t)[S28]Each measurement error term, γ^v(t), is characterized by〈γ^v(t)γ^v(t′)〉=λ2Γv2δtt′[S29]where Γv2 is the squared amplitude of the positional measurement noise. From the definition of the ACF and defining v¯=〈v(t)〉,Cv^′(τ)=〈v^(t)v^(t+τ)〉−v¯2=〈(vg+ξ^v(t)+γ^v(t+Δt)−γ^v(t))(vg+ξ^v(t+τ)+γ^v(t+Δt+τ)−γ^v(t+τ))〉−v¯2.[S30]Expanding out the brackets givesCv^′(τ)=〈ξ^v(t) ξ^v(t+τ)〉+〈γ^v(t+Δt)γ^v(t+Δt+τ)〉−〈γ^v(t+Δt)γ^v(t+τ)〉+〈γ^v(t)γ^v(t+τ)〉,[S31]where we have used the fact that 〈ξv(t)γv(t′)〉=0 for all t, and v¯=vg. This can be written asCv′(τ)={〈ξ^v(t)2〉+〈γ^v(t+Δt)2〉+〈γ^v(t)2〉for τ=0−〈γ^v(t+Δt)2〉for τ=Δt0for τ>Δt,[S32]which can be written asCv′(τ)=λ(ka+kd)a2δτ0+λ2Γv2(2δτ0−δτΔt).[S33]

ACF of Cap Size Fluctuations in the Presence of Measurement Noise.

The ACF of the cap size fluctuation in the presence of measurement noise is simpler. The measurement noise γc(t) is characterized by〈γc(t)γc(t′)〉=Γc2δtt′[S34]Following a similar procedure as above the ACF is given byCc′(τ)=kakme−kmτ+Γc2δτ0[S35]

Simulation of the Cap Reaction Network.

A full stochastic simulation of the cap reaction network developed in Mathematica (Wolfram) from a previous study (24) was used to test the validity of our model and the robustness of our analysis to experimentally limiting factors. Twenty simulated tracks 200 s long were analyzed to replicate our experimental datasets. We tested our analysis under a range of simulated noise conditions (modeled as additive white Gaussian noise) covering what was measured experimentally. We also analyzed simulated data at 4 Hz and a faster rate of 10 Hz (Fig. S1). We found the simulation results fitted the theory well in all cases. Example simulation data and cross-correlation analysis results are shown in Fig. 4 A and B.