# Kinetic-Analysis

## Kinetic Analysis of TIRF Time Series Image of Cargo and Compartment

Among key endosomal components implicated in early endosome fusion are proteins such as early endosome antigen (EEA1), adaptor protein, phosphotyrosine interaction (APPL1), and Rabenosyn-5. To determine the role of endosomes containing these proteins, we imaged each one simultaneously with clathrin and Tf, using multi-color Total Internal Reflection Fluorescence Microscopy (TIRF-M).

### Flux of Cargo from the Plasma Membrane Through the Endosomal Compartment System

To visualize the movement of Tf from clathrin-enriched plasma membrane regions into the endosomal system, Tf-DyLight-649 was added to cells co-transfected with TagRFP-T-clathrin and EGFP-tagged Rabenosyn-5, APPL1, or EEA1. Cells were imaged at 1 Hz (one frame/s) continuously for 15 min: 5 min before the addition, 5 min in the presence, and 5 min after the removal of Tf-DyLight-649. These complex time-lapse image sets were analyzed by systematically identifying all individual endosomal structures within the images and measuring the relative amount of Tf specifically associated with them at each time point.

### The Algorithm

- Images of TagRFP-T-clathrin are convolved using a difference of Gaussians (DOG) filter to eliminate the diffuse signal originating from out-of-focus or auto fluorescence.
- The positions (x,y) of local 2-D maxima (pixels with intensity > all 8 neighbors) in the filter images that exceed a threshold are identified and mapped onto the unprocessed clathrin images.
- The mean intenisty of the 5 × 5 pixel square (25 pixels) surrounding each (x,y) position [i.e. clathrin(in)] is recorded. The mean intensity of the one-pixel-wide frame (24 pixels) surrounding the square, i.e. clathrin(out), is subtracted thus removing the local background: clathrin(in] - clathrin(out).
- The (x,y) positions of the maximum-intensity pixels (MIPs) derived from the clathrin image are mapped onto the corresponding unprocessed Tf image and the same calculation is performed. Thus, Tf has a positive value only if the signal detected within the 5 × 5 pixel region is higher than that of the immediate surrounding region.
- The ratios of the values obtained r(x,y,t)=[Tf(in(x,y)) − Tf(out(x,y))/clathrin(in(x,y)) − clathrin(out(x,y))] for each region then is calculated, and the mean R(t) of all regions in the image is plotted for each time point.

In the same cell, the trafficking of Tf into Rabenosyn-5–containing endosomes is measured similarly. The R(t) data are fit with a kinetic model, described in detail in the SI Appendix, “Curve Fits to Ratio Data” section, that incorporates known constants for Tf binding to the TfR, the concentration of free ligand, and the amount of Tf associated with clathrin. Tf associates rapidly and saturably with clathrin-containing regions, displaying kinetic constants consistent with binding to the TfR (SI Appendix, Table S1) (30–32). Interestingly, Tf is detected almost immediately within Rabenosyn-5–enriched regions, with virtually no time elapsing between the detection of Tf in clathrin coated regions and its detection in Rabenosyn-5 endosomes, and continues to accumulate in these endosomes as long as free Tf is present. When Tf is removed from the medium, it disappears rapidly from clathrin-enriched regions because of transfer into the endosomal pathway (SI Appendix, Table S1). Strikingly, the rate of exit of Tf from clathrin-containing membrane regions (0.0053 ± 0.002/s, mean ± SEM, n = 4), is indistinguishable from its rate of entry into Rabenosyn-5–enriched endosomes (0.0037 ± 0.002/s, mean ± SEM, n = 4) (SI Appendix, Table S1), implying that Tf internalized via clathrin-coated pits could be delivered almost directly into Rabenosyn-5–containing endosomes.

### Segmentation of vesicles from background

To generate images in which vesicles or other fluorescence structures can be analyzed without interference from diffuse fluorescence, images were then convolved with a difference of Gaussians (DOG) filter consisting of 1) a small, two dimensional, Gaussian spot with unit area (sigma = 150 nm) that acted as a vesicle matched detector, i.e. an approximation to a near-diffraction limited spot, and 2) a larger, inverted, two dimensional Gaussian (sigma = 300 nm) with negative unit area that estimated and subtracted the local background. The Gaussian smoothed images were visually thresholded (global threshold) to select for pixels belonging to objects (e.g. vesicles) and eliminate areas devoid of signal (but containing noise).

### Transferrin Trafficking

For analyzing the trafficking of transferrin, the thresholded-DOG time series image of an endocytic marker (clathrin, rabenosyn-5, APPL, EEA1) was analyzed to identify individual objects (spots, vesicles) and their central (x,y,t) positions in every time point, by finding all 2-dimensional intensity maxima (pixels brighter than their 8 neighbors) and thresholding the maxima to eliminate spurious peaks. The average local fluorescence within a 5x5 pixel region centered at each (x,y,t) position was obtained in both the running-averaged, background-subtracted endocytic marker image and the corresponding transferrin image. From this value, the average fluorescence of a 1 pixel wide frame surrounding the 5 x 5 pixel region was subtracted (e.g. Figure 1b). The fluorescence ratio of transferrin to endocytic marker was calculated for each object position, and the mean ratio and standard error for each time point was calculated (there were typically 100s of objects in each time point) and plotted (e.g. Figure 1c and 3b). The fluorescence ratio was used as an estimate of the (relative) transferrin concentration associated with each object – both signals should be roughly in some proportion to the amount of surface membrane – while also correcting for the exponential decrease in fluorescence brightness with increasing depth in TIRF. Additionally, the kinetics of trafficking of transferrin through clathrin, rabenosyn-5 and APPL1 were modeled, and the models fit to the ratio time course data to determine rates of entry/filling and exiting/emptying of transferrin for each endocytic marker.

### Curve Fits to Ratio Data

The Tf/Clathrin ratio data were fit with a simple kinetic model:

koff Tf + clathrin ↔ Tf-clathrin → ? kon kempty

The TfR is modeled as having two binding sites for Tf, one low affinity and one high affinity. TfRs are assumed to be in instantaneous, steady-state colocalization with clathrin, independent of binding Tf, therefore clathrin is a proxy for TfR. Tf/Clathrin is [Tf-Clathrin] (times a scale factor).The extracellular [Tf] is stepped from zero to a constant concentration, and the increase in the total Tf-TfR, and therefore Tf-clathrin, is proportional to 1-e-k·t where k is the sum of the on and off rates for Tf plus a rate kempty, the rate at which Tf apparently disassociates from clathrin objects. This may be because Tf in fact disassociates from a clathrin object, or a Tf-clathrin object disappears from the TIRF imaging zone (and is replaced by a clathrin-alone object, preserving the steady state). Conversely when the [Tf] is stepped back to zero, the Tf-clathrin decreases as e-k·t where kon·[Tf] is now zero since [Tf] is zero. The two Tf binding sites are assumed to be independent, so the equations for the ratio, R(t)clathrin are

R(t≤t_add)clathrin = 0 R(t_add<t≤t_wash)clathrin = A·{(1-exp(-k1·t)) + (1-exp(-k2·t))}/2 R(t>t_wash)clathrin = R(t_wash)clathrin·{exp(-k1·t) + exp(-k2·t)}/2

where A is an arbitrary constant to match the ratio data amplitude, and

k1 = k1on·[Tf] + k1off + kempty k2 = k2on·[Tf] + k2off + kempty

The equations for R(t)clathrin were fit to the ratio data using a Levenberg–Marquardt least-squares fit algorithm, with t_add, t_wash, [Tf], A, and kempty as parameters. [Tf] is either a constant (t_add< t < t_wash) or 0. Tf binding rate constants used are (Giannetti and Bjorkman, 2004)

k1on=1.0·105 M^-1 s^-1 k1off=3.2·10-3 s^-1 k2on=8.3·105 M^-1 s^-1 k2off=1.0·10-3 s^-1

The Tf-Rbsn (or Tf-APPL) ratio data were fit with a similar model, except that for the association of Tf with rabenosyn-5, we assume that the Tf must pass through the clathrin pathway to be available to taken up in rabenosyn-5 vesicles

Tf-clathrin + Rbsn → clathrin + Tf-Rbsn → ? kfill kempty where kfill = kon·[Tf-clathrin]·[Rbsn]

This yields the differential equations for the change in rabenosyn-5

d[Rbsn]/dt = kempty·[Tf-Rbsn] – kon·[Rbsn]·[Tf-clathrin] d[Tf-Rbsn]/dt = kon·[Rbsn]·[Tf-clathrin] - kempty·[Tf-Rbsn]

[Rbsn]+[Tf-Rbsn] is assumed to be constant. We used R(t)clathrin (see above) as the [Tf-clathrin] driving the association with rabenosyn-5 at time t yielding

d[Rbsn]/dt = kempty·[Tf-Rbsn] – kon·[Rbsn]·R(t)clathrin d[Tf-Rbsn]/dt = kon·[Rbsn]·R(t)clathrin – kempty·[Tf-Rbsn]

[Tf-Rbsn](t) is related to the observed ratio by an unknown scale factor, i.e.

R(t)rabenosyn = B·[Tf-Rbsn](t).

The system of ordinary differential equations was numerically integrated (for a given B, kon and kempty) using the ODE function of Scilab (http://www.scilab.org/) producing [Tf-Rbsn], and hence R(t)rabenosyn . Initial conditions were [Rbsn] =1 and [Tf-Rbsn]=0. This function was fit to the Tf-Rbsn ratio data using the L-M method, with B, kon and kempty as parameters (i.e., the ODE solver

### Programs

### Abbreviations

Tf transferrin TfR transferrin receptor Rbsn rabenosyn-5 kon, koff exponential time constants [] concentration

### References

Rabenosyn-5 defines the fate of the transferrin receptor following clathrin-mediated endocytosis.Navaroli DM, Bellvé KD, Standley C, Lifshitz LM, Cardia J, Lambright D, Leonard D, Fogarty KE, Corvera S. Natl Acad Sci U S A. 2012 Feb 21;109(8):E471-80. doi: 10.1073/pnas.1115495109. Epub 2012 Jan 30.