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Interactions between objects inside living cells are often investigated by looking for colocalization between fluorescence microscopy images that are recorded in separate colours corresponding to the fluorescent label of each object. The fundamental limitation of this approach in the case of dynamic objects is that coincidental colocalization cannot be distinguished from true interaction. Instead, correlation between motion trajectories obtained by dual colour single particle tracking provides a much stronger indication of interaction. However, frequently occurring phenomena in living cells, such as immobile phases or transient interactions, can limit the correlation to small parts of the trajectories. The method presented here, developed for the detection of interaction, is based on the correlation inside a window that is scanned along the trajectories, covering different subsets of the positions. This scanning window method was validated by simulations and, as an experimental proof of concept, it was applied to the investigation of the intracellular trafficking of polymeric gene complexes by endosomes in living retinal pigment epithelium cells, which is of interest to ocular gene therapy.

In the field of gene therapy, a lot of effort goes into the development of nanomedicines, with a size in the order of 100 nm, for the delivery of therapeutic nucleic acids to target cells [

The most common way of investigating interactions in multi-colour images is by comparing pixel values between colours, for which different quantification methods exist [

In live-cell imaging, or any other application that involves dynamic events, the objects of interest, such as proteins or organelles, might be mobile. Two objects that are moving past each other by coincidence could, therefore, be identified as being colocalized by either the pixel or object based methods. This can be especially problematic in case of very dense object populations. One potential solution is to perform dual colour Image Cross-Correlation Spectroscopy (ICCS) [

However, an objective measure for the correlation threshold has not been determined. Also, as the published correlation method is based on calculating the correlation between complete trajectories, it performs suboptimal in case trajectories are not completely correlated. For instance, intracellular motion can exhibit variable mobility, including immobile phases that inherently do not correlate. Another example is photobleaching of fluorescent labels, which degrades the localization precision in the trajectories, in turn affecting their correlation. There is also the possibility of transient interactions that take place during only a short time span, restricting the correlation to only a part of the trajectories. If the uncorrelated part of the trajectories in these situations is sufficiently large, the correlation determined from all positions in the trajectories will not exceed the correlation threshold, despite (transient) interaction being present. A method that can identify correlation in smaller segments of the trajectories with an objectively determined correlation threshold is, therefore, required. Here, such a method is presented, based on a scanning window approach in which the correlation is calculated over a limited number of positions within the trajectories. The optimal size of the window and the correlation threshold value are selected according to criteria that account for the localization precision in the trajectories and the mobility of the objects. The scanning window method is verified by simulations and applied to investigate the intracellular trafficking of polymeric gene complexes inside endosomes of living cells.

As mentioned in the Introduction, we have recently proposed a new approach to identify interaction [_{i}_{A}_{i}_{A}_{i}_{B}_{i}_{B}_{i}

with <_{A}_{B}_{A}_{B}_{A}_{i}_{B}_{i}_{o} as the overlay precision with which both colour images are aligned, which can be calculated as the standard deviation of the differences between identical positions in the images after overlay [

The effect of σ_{A}_{B}_{o} on the correlation ρ between the trajectories is illustrated in _{min}, defined as the minimum correlation that is expected in case of correlated trajectories, can be imposed. As will be explained below, the ρ_{min} threshold value depends on σ_{A}_{B}_{o}, as well as on other trajectory properties.

For a certain localization and overlay precision, the correlation threshold ρ_{min} is defined as the minimum value of the correlation coefficient of interacting objects with a _{A}_{B}_{o} = 0. For Brownian or linear motion, which is common in live-cell imaging, it can be shown that the expected correlation ρ between trajectories with

where

The expected value of the observed correlation ρ is thus identical for all trajectory pairs with _{min} can be used for all these trajectories.

It can be shown that the same applies to the general and more realistic case of σ_{A}_{B}_{o} ≠ 0 (see

where var(_{A}_{B}

In many circumstances, such as live-cell imaging, objects usually exhibit a variable mobility. When a certain part of the trajectories exhibits low mobility, the local mean step length _{A}_{B}

One obvious solution to this problem lies in identifying correlation in smaller parts of the trajectories to which the framework of Section 2.2 can be applied. This idea leads to the scanning window method, as illustrated in _{min} for that window, the objects are considered to be interacting in that window. The threshold ρ_{min} depends on the size of the window and the local relative localization error

This raises the important question of what is the optimal window size. On the one hand, the window should be as small as possible in order to have the best temporal resolution and to ensure that the variation in relative localization error is minimal. On the other hand, the window should include a sufficient number of positions in order to detect correlation with sufficient statistical significance. Consider correlated trajectories and define _{min}, this probability ^{3}. A probability of more than 0.99 is achieved by

The values of the correlation thresholds ρ_{min} (see Section 2.2) and the values of the probabilities _{w}^{6} in all cases (e.g., for ^{5}). The type of motion was chosen to be Brownian motion, since it is common on a microscopic scale, and because unrelated Brownian trajectories on average do not exhibit correlation. The diffusion coefficient was taken to be ^{2}/s and the time interval between subsequent positions was τ = 0.1 s, resulting in a one-dimensional mean step of

Subsequently, the correlation ρ between both trajectories is calculated, using the Matlab function _{w,p}_{w,p}_{w}_{min} for a given _{min} do not always increase with the trajectory length _{min} increase with

The main input for the scanning window method consists of the trajectory _{A}_{i}_{A}_{i}_{B}_{i}_{B}_{i}_{i}_{A}_{B}_{o} between the images (see Section 5.4 for information on how these values can be determined experimentally).

Consider first the _{A}_{i}_{B}_{i}_{1} to _{3}. The relative localization error _{1} to _{4}. In the same manner, the probability

Having determined the optimal window size _{min} can be determined from _{min}, all

This procedure is repeated, starting at the next positions _{A}_{2}) and _{B}_{2}). The _{A}_{l−}_{2}) and _{B}_{l−}_{2}) have been reached. Except near the start and the end of the trajectory, the positions are evaluated at least three times by different windows. Therefore, for each position there are at least three binary values, indicating that correlation was found or not within a particular window. If correlation was found at least one time, the position is flagged as being correlated. This results in a list of binary values that identify the positions where the scan found correlation (see _{A}_{l}_{B}_{l}

The identical scanning window procedure as described above is applied to the

The performance of the scanning window method was verified with simulated pairs of two-dimensional Brownian motion trajectories, as explained in Section 5.1. Brownian motion was chosen, not only because it is common on a microscopic scale, but also because random Brownian motion trajectories are not expected to be correlated. A number of different situations were considered (see

The situation of complete interaction was investigated for a diffusion coefficient ^{2}/s. The results are shown in

As shown in _{A}_{B}_{o} = 0. At almost all positions, the colocalization method finds interaction 81% of the time, for both relative localization errors

Similarly, it was tested if the scanning window method can correctly detect the absence of interaction. This was investigated for a diffusion coefficient ^{2}/s, the results of which are shown in

Simulations were also carried out to evaluate the performance of the scanning window method in more complicated situations (representing the ones shown in ^{2}/s at positions 1–10 and ^{2}/s at positions 11–20. This results in a corresponding local relative localization error

Complete interaction was also investigated with a variable localization precision σ = 4.47 nm at positions 1–10 and σ = 44.7 nm at positions 11–20. This results in a corresponding local relative localization error ^{2}/s was constant at all positions. The scanning window method finds 100% of the time interaction at most positions, as shown in

Variable interaction was the last situation that was investigated, with the objects only interacting at positions 1–10 and not interacting at positions 11–20. The results are shown in ^{2}/s and σ = 4.47 nm). Comparison to

The simulations show that the scanning window method is capable of reliably identifying interaction, independent of the relative localization error. Even when parts of the trajectories are not correlated because of transient interactions, or exhibit low correlation because of a large local relative localization error, the scanning window method is still able to detect interaction when it takes place. An important benefit compared to the object based colocalization method is that the scanning window method is significantly less sensitive for false negatives that cannot be avoided by object based colocalization. Furthermore, it is much less sensitive to false positives in case of coincidental colocalization.

In pharmaceutical research, nanomedicines such as polymeric gene complexes (polyplexes) are being developed for the delivery of therapeutic nucleic acids to target cells, such as retinal pigment epithelium (RPE) cells in the context of ocular gene therapy [

The scanning window method is, therefore, expected to perform better in the investigation of intracellular trafficking of nanomedicines than the full trajectory method, since it inherently is capable of detecting interaction in small segments of trajectories. First, as a negative control, dual colour SPT measurements of a mixture of non-interacting yellow-green and dark red fluorescently labelled 0.1 μm diameter beads undergoing free diffusion were analysed with the scanning window method to verify that no interactions are detected (see

Because of the variable localization precision and mobility in most trajectories in the live-cell dual colour SPT data, the relative localization error

Visual inspection of the trajectory pairs where the scanning window method only finds correlation in a part of the trajectories, suggests that this is mostly caused by either a low mobility or low localization precision in the other part of the trajectories (cfr.

We have recently reported correlation between entire trajectories as a measure for the interaction between two dynamic species that is less prone to false positives and false negatives than classic object based colocalization [_{min} for the optimal window depends in turn on both the window size and the local relative localization error _{min} can be determined from

The scanning window method was validated with simulated trajectory pairs (see Section 3.1). It was shown that the method is able to accurately identify interaction, independent of the relative localization error

The performance of the scanning window method was also tested with simulated trajectory pairs that represent more complicated behaviour. In case of interaction along the entire trajectory, but with a changing diffusion coefficient, the scanning window method is still able to detect the interaction (see

As a comparison, the same simulated data was also analysed with an earlier reported object based colocalization method that makes use of a maximum distance to decide whether or not there is interaction at a particular position [

As a proof of concept, the scanning window method was applied to the trajectories of polyplexes and endosomes inside living cells, obtained by dual colour SPT experiments (see Section 3.2). When interaction was found in at least one window, the polyplex was considered to be residing in, or at least interacting with, the endosome. Compared to the previously published full trajectory method [

Comparison of the scanning window method with the full trajectory method shows that the latter method misses at least half of the interactions (see

The scanning window method could be tested on other types of motion besides diffusion, and _{min} could be adjusted if required. In the specific case that the objects are undergoing different types of motion, trajectory analysis could first be applied to determine the trajectory segments that correspond to these types of motion [

The scanning window method was validated by simulations in Matlab. Different sets of 1000 pairs of two-dimensional Brownian motion trajectories with length ^{2}/s, resulting in _{o} = 0. In one set, the localization precision was different in the first and second half of the trajectories. In another set, the diffusion coefficient was different in the two trajectory halves, both leading to local relative localization errors in the windows that are variable. The different conditions of each set of simulated trajectories are listed in

The preparation of the sample for the live-cell dual colour SPT experiments is described in detail elsewhere [_{2}. The pGL4.13 plasmid was labelled with Cy5 using the Label IT Nucleic Acid Labeling Kit (Mirus Bio Corporation, Madison, WI, USA) according to the manufacturer’s instructions at a 1:2 (

The dual colour SPT experiments were carried out on a custom-built laser widefield epi-fluorescence microscope set-up that is described elsewhere in detail [_{2}, and 100% humidity.

Movies of 60 seconds were recorded on different time points at a speed of 2 frames per second and with an image acquisition time of 30 ms. For each movie, a different cell was selected for imaging in order to minimize photobleaching and phototoxicity, and to obtain information on a large population of cells. Cells were chosen, based on a relatively low expression level of eGFP-constructs to minimize the possibility of a disturbed cell functioning.

After recording the movies, the images in the two different colours (_{c} that is expected on average [

where ^{2} describes the variance of the local photon background,

where _{0} = 150 nm is the standard deviation of the Gaussian approximation of the spot in case the object is stationary and located in the focal plane, and _{0} is defined by:

with λ ≈ 550 nm the wavelength of light and ^{2} and _{o} = 3 nm for all movies by an experimental procedure as reported before [

The scanning window method is applied to each pair of trajectories, as explained in Section 2.5. To restrict the calculation time, trajectory pairs that cannot realistically correspond to interacting objects are not considered,

We have developed the scanning window method for measuring the interaction between moving objects in dual colour microscope time-lapse images. Employing a scanning window along two trajectories in which the correlation between the positions is calculated, not only spatial but also temporal information about the interaction becomes available. The scanning window method was validated with simulations and applied to the trajectories of endosomes and polymeric gene nanoparticles in living cells. Interaction was more reliably found with the scanning window method than by simple correlation analysis over the entire trajectory at once, which in turn was already proven to perform more reliably than the classic object-based approach. The additional temporal information thus allows a more sensitive estimation of the interactions between objects, and moreover, provides a means to detect transient interaction events.

Hendrik Deschout would like to acknowledge the financial support of the Agency for Innovation by Science and Technology in Belgium. Financial support by the Ghent University Special Research Fund and the Fund for Scientific Research Flanders (FWO, Belgium) is acknowledged with gratitude.

Consider a one-dimensional trajectory _{A}_{B}

The numerator is called the covariance and is defined as:

where E[

Assume now that the observed trajectories _{A}_{B}_{A}_{B}

with δ_{A}_{B}_{A}_{B}_{o}, which is called the overlay precision. For mathematical convenience, it is therefore assumed that δ_{A}_{B}_{A}_{B}

The variance of _{A}_{B}

The Pearson correlation ρ between the observed trajectories _{A}_{B}

Consider now the special situation σ_{A}_{B}

Both correlations will be equal if the following condition for σ is fulfilled:

This is a quadratic equation in σ^{2}, with solution:

Using _{A}_{B}

This expression is more useful than _{A}_{B}

In reality, the complete trajectories _{A}_{B}_{A}_{i}_{B}_{i}_{i}

with <_{A}_{B}_{A}_{B}

Consider a one-dimensional trajectory _{A}_{B}

The observed trajectories _{A}_{B}

with δ_{A}_{B}_{A}_{B}

According to

This allows to rewrite

The correlation between observed trajectories of interacting objects is thus completely determined by the ratio of σ^{2}/var(

The mean step in the trajectory over a time interval τ <

Combining

Another example is linear motion with velocity

The (mean) step in the trajectory over a time interval τ <

Combining

Linear and Brownian motion thus give rise to the following relationship between trajectory variance and mean step:

where

In other words, for a certain ratio

However, the mean step _{A}_{i}_{B}_{i}_{i}_{i}_{i}_{−1} (

These are estimations of the mean steps defined in

In case of low localization precision or low mobility, the relative localization error

Two sets of 1000 pairs of two-dimensional Brownian motion trajectories with length ^{2}/s, resulting in _{o} = 0. The scanning window method is applied to each pair of simulated trajectories.

The results in the situation of complete interaction are shown in

As shown in

Similarly, it was tested if the scanning window method can correctly detect the absence of interaction, the results of which are shown in

For large relative localization errors, the performance of the scanning window method can thus be affected, leading to a somewhat higher probability to detect false positives. In this situation, the results of the scanning window method should thus be interpreted with care.

Dual colour SPT measurements were performed on a mixture of yellow-green and dark red fluorescently labelled 0.1 μm diameter beads (FluoSpheres, Molecular Probes, Gent, Belgium). Afterwards, the scanning window method was used to search for interaction between the bead trajectories. This provides a negative control, since no interaction is expected between the yellow-green and dark red beads.

The microscope sample was prepared by diluting the bead mixture in water and applying 5 mL between a microscope slide and a cover slip with a double-sided adhesive spacer of 120 μm thickness (Secure-Seal Spacer, Molecular Probes, Bleiswijk, The Netherlands) in between. The dual colour SPT experiments were carried out on a custom-built laser widefield epi-fluorescence microscope set-up that is described elsewhere in detail [

The scanning window method was applied to all possible pairs of trajectories. Note that no restriction was imposed on the distance between the trajectories, which is possible with the scanning window method because correlation is translation independent. Using the scanning window method, a pair of trajectories was considered to interact when interaction was found in at least one window. The average percentage of trajectory pairs in a dual colour SPT experiment that are found to interact by the scanning window is shown in

Validation simulations for interaction and no interaction in case of a large relative localization error. The percentage of 1000 pairs of simulated Brownian motion trajectories where the scanning window method has found interaction is shown for each position along the trajectories (black line), in case of (^{2}/s, and a time interval τ = 0.1 s between successive positions. The localization precision was chosen σ = 223.6 nm, corresponding to a relative localization error of

The scanning window method applied to dual colour SPT measurements on a mixture of yellow-green and dark red 0.1 μm diameter beads diffusing in water as an experimental negative control. (

The authors declare no conflict of interest.

The effect of the localization and overlay precision on the observed trajectories of interacting objects. The localization precision σ_{A}_{B}_{o} between the images is defined as the standard deviation of the differences (dotted lines) between identical positions in the images after overlay.

The effect of a time dependent mobility, a time dependent localization precision, or a time dependent interaction on the observed trajectories of interacting objects. The localization precision of the positions in the observed trajectory

An illustration of the scanning window method. A trajectory _{min} of the window (see

Validation simulations for interaction and no interaction. The percentage of 1000 pairs of simulated Brownian motion trajectories where the scanning window method has found interaction is shown for each position along the trajectories (black line), in case of (^{2}/s, and a time interval τ = 0.1 s between successive positions. The localization precision was chosen σ = 4.47 nm or σ = 44.7 nm, corresponding to a relative localization error of

Validation simulations for variable diffusion coefficient, variable localization precision, and variable interaction. The percentage of 1000 pairs of simulated Brownian motion trajectories where the scanning window method has found interaction is shown for each position along the trajectories (black line), in case of (^{2}/s at positions 1–10 and ^{2}/s at positions 11–20; (^{2}/s, and a localization precision σ = 4.47 nm at positions 1–10 and σ = 44.7 nm at positions 11–20; (^{2}/s, a localization precision σ = 4.47 nm, and interaction at positions 1–10 and no interaction at positions 11–20. All simulated trajectories had a length

Interactions between endosomes and polyplexes measured by the scanning window method. (

An example of transient interaction detected by the scanning window method. (

The probability

Simulated values of the probability | ||||||
---|---|---|---|---|---|---|

| ||||||

… | ||||||

0.97177 | 0.99992 | 1 | 1 | 1 | ||

0.90433 | 0.99872 | 1 | 1 | 1 | ||

0.82096 | 0.99532 | 0.99980 | 1 | 1 | ||

0.73582 | 0.99020 | 0.99950 | 1 | 1 | ||

0.65341 | 0.98112 | 0.99815 | 0.99982 | 1 | ||

0.04623 | 0.07992 | 0.12200 | 0.19284 | 1 |

The correlation threshold ρ_{min} is the minimum statistically significant correlation in a window with length

Simulated values of the correlation threshold ρ_{min} | ||||||
---|---|---|---|---|---|---|

| ||||||

… | ||||||

0.99693 | 0.95043 | 0.97095 | 0.99242 | 0.99998 | ||

0.99692 | 0.95013 | 0.88554 | 0.89819 | 0.9999 | ||

0.99692 | 0.95003 | 0.88114 | 0.91113 | 0.99972 | ||

0.99692 | 0.95004 | 0.88418 | 0.87069 | 0.99965 | ||

0.99692 | 0.95001 | 0.87854 | 0.81552 | 0.99925 | ||

0.99693 | 0.95002 | 0.87836 | 0.81141 | 0.77525 |

The conditions for each set of simulated trajectory pairs for the validation of the scanning window method. Each set consists of 1000 pairs of Brownian motion trajectories with trajectory length

Trajectory parameters used for the validation simulations | |||||
---|---|---|---|---|---|

| |||||

Situation | Position | Interaction | ^{2}/s) |
σ (nm) | |

interaction, |
1–20 | yes | 1 | 0.447 | 4.47 |

interaction, |
1–20 | yes | 1 | 0.447 | 44.7 |

no interaction, |
1–20 | no | 1 | 0.447 | 4.47 |

no interaction, |
1–20 | no | 1 | 0.447 | 44.7 |

interaction, variable |
1–10 | yes | 1 | 0.447 | 4.47 |

11–20 | yes | 0.01 | 0.0447 | 4.47 | |

interaction, variable σ | 1–10 | yes | 1 | 0.447 | 4.47 |

11–20 | yes | 1 | 0.447 | 44.7 | |

variable interaction | 1–10 | yes | 1 | 0.447 | 4.47 |

11–20 | no | 1 | 0.447 | 4.47 |

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