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Understanding the colocalization coefficients


This article illustrates how the different colocalization coefficients can be interpreted.

The purpose of a CoLocalization coefficient is to characterize the degree of overlap between two channels in a microscopy image.

Details about the definitions of the coefficients commonly used in scientific literature can be found at Colocalization Theory; they are used in Huygens' Colocalization Analyzer.

To understand the meaning and implications of the different coefficients we will study in this article how their values are affected, and how to interpret them, on very simple 2D Two Channel test images.

A briefer description of each coefficient can be found at Colocalization Coefficients In Brief.

The images in this article were created not with the intention of being representative of any biological system, but to be intuitive data sets on which the mathematical definition of the coefficients would be applied directly. The purpose of this article is to illustrate how the coefficients behave mathematically, not biologically.




Effects of objects distribution and experimental conditions


Effect of the increasing spatial overlap


We start with two simple objects, one in each channel, having homogeneous intensities 100 against a background of 0. The objects are 2D squares of 10 × 10 pixels, on an image of 32 × 32 = 1024 pixels. First they do not overlap at all:

Image
Pearson's: -0.108,
Manders' coefficients:
overlap 0.000, k1 0.000, k2 0.000, M1 0.000, M2 0.000
Normalized k1, k2 coefficients: h1 0.000, h2 0.000


The Pearson's coefficient is negative signifying that the objects act repulsively: they 'repel' each other and wherever there is red signal the green one tends to be elsewhere.

Same objects, now partially overlapping (one quarter of the volume).

Image
Pearson's: 0.169,
Manders' coefficients:
overlap 0.250, k1 0.250, k2 0.250, M1 0.250, M2 0.250
Normalized k1, k2 coefficients: h1 0.250, h2 0.250


The same objects, with even more overlap (one half of their volume intersects). Let's call this situation A as a reference.

Image
Pearson's: 0.446,
Manders' coefficients:
overlap 0.500, k1 0.500, k2 0.500, M1 0.500, M2 0.500
Normalized k1, k2 coefficients: h1 0.500, h2 0.500


While Manders' coefficients seem to refer directly to the portion of overlapping portions of the two objects, the Pearson's coefficient has not such a straight forward interpretation. Its squared value (0.029 = 2.9% and 0.20 = 20% in the last two cases) represents the portion of the Variance of one channel that can be explained in terms of the variance of the other channel. See Pearsons Interpretation.

Effect of different channel gains


Starting from situation A, the red channel is now divided by two, lowering the intensity to 50.

Image
Pearson's: 0.446,
Manders' coefficients:
overlap 0.500, k1 1.000, k2 0.250, M1 0.500, M2 0.500
Normalized k1, k2 coefficients: h1 0.500, h2 0.500


All coefficients remain the same except k1 and k2, because these use not normalized values and therefore reflect relative intensity variations. Their normalized variants h1 and h2 are again insensitive to this.

Dividing the intensity of the red signal again by two (setting it to 25) the coefficients become:
Pearson's: 0.446,
Manders' coefficients:
overlap 0.500, k1 2.000, k2 0.125, m1 0.500, M2 0.500
Normalized k1, k2 coefficients: h1 0.500, h2 0.500


Notice how problematic k1 and k2 are due to this relative intensity sensitivity: they provide values difficult to interpret. Also notice that the fact that h1 and h2 use normalized pixel values for the computation doesn't mean that they are always smaller than one!!! They are in this case, but other simple cases can be imagined where they are not (see Hcoloc Coefficient). Because these h coefficients are difficult to interpret, they are not reported anymore by the latest versions of the Huygens Software. We stop discussing them here, and will introduce other new coefficients instead.

The Manders' M1 and M2 coefficients (which always have values between zero and one) tell us that half of the red signal colocalize with the green one, and vice versa. They are also invariant when intensities change relatively to each other.

Effect of variability


Let's take again the overlap of 50% from situation A, where the coefficients were as follows:
Pearson's: 0.446,
Manders' coefficients:
overlap 0.500, k1 0.500, k2 0.500, M1 0.500, M2 0.500


Let's now add some heterogeneous variations in the intensity of the green channel. The green object is not constant anymore, we add some "noise" to it (actually, using a Signal To Noise Ratio of 20), while the red object remains homogeneous:

Image
Pearson's: 0.445,
Manders' coefficients:
overlap 0.499, k1 0.497, k2 0.501, M1 0.500, M2 0.499


Let's add some more noise (another SNR of 10 on top of that):

Image
Pearson's: 0.437,
Manders' coefficients:
overlap 0.491, k1 0.495, k2 0.488, M1 0.500, M2 0.494


And yet another SNR = 5 on top of that:

Image
Pearson's: 0.425,
Manders' coefficients:
overlap 0.480, k1 0.510, k2 0.451, M1 0.500, M2 0.493


The Pearson's coefficient gets lower with more noise (or variability). Other coefficients apparently have a similar (M2) trend in the variations. k1 and k2 seem to change randomly. M1 is constant, always 0.5: half of the total red intensity (not necessarily the total red volume!!!) matches some green intensity. What is important here is to realize that all these coefficients are based on pixel intensities, and relative variations on local intensities will have a noticeable effect on them.

This makes much sense in fluorescence microscopy, where the acquired intensity reflects the concentration of fluorophores. A brighter region contains more tagged components, and this should have more weight when calculating the colocalization. These coefficients, being based on light intensities, reflect the amount of R and G components that are together in space, not how they distribute in space.

Still, you may see in the examples above just two overlapping objects because you don't care about these internal variations, and you would like to have a coefficient that simply quantifies the intersection volume (i.e. the number of pixels that colocalize). In this interpretation the acquired variability is just noise, it does not reflect any local distribution of components. None of these intensity-based coefficients fits in this interpretation (not even the so-called "overlap"). Look for example at the varying value of M2: it doesn't reflect anything about the volume occupied by the green signal, it says that about one half (~49%) of the green intensity matches some red intensity, and intensity means "amount of component", not "volume occupied by the component".

Introducing the binary intersection coefficient


To characterize the amount of overlapping volume the image is converted to a binary image (where pixels are just 1 or 0). Subsequently, the images are compared and the cases were both channels are '1', opposing are counted. This coefficient, referred to as "intersection" coefficient is available in the latest versions of the Huygens Software.

The intersection coefficients in Huygens report the following for all three cases above:
intersection: 0.333, i1: 0.500, i2: 0.500


The first coefficient indicates that, from all the volume occupied by the objects in the two channels, 33% is intersecting (this is the one third displayed in yellow).

The other two figures give information per channel: in both cases half (50%) of the objects' volume (image pixels, if you prefer looking at it like that) belong to the common intersecting volume.

Calculating these coefficients require setting a threshold to define what is an object and what is background. In practice a-specific staining or other sources of background signal will need to be excluded. This is certainly a source of subjectiveness, but as we show in the next section other colocalization coefficients are also affected by subjective choices.

In the case of noisy images, a simple threshold will produce grainy intersection volumes. To remedy this, pixels in a limited range around the threshold can be set to contribute less to the result.

Effect of the offset


Let's take again an intersection of 50% from situation A and apply to the green channel an offset of 50 (all the green pixel intensities are increased +50, thus the object is now 150 and there is a background of 50). The red object remains with an intensity of 100 with a zero background. This will be situation B.

Image
Pearson's: 0.446,
Manders' coefficients:
overlap 0.468, k1 1.000, k2 0.219, M1 1.000, M2 0.163
intersection 0.098, i1 1.000, i2 0.098


M1 = 1 and i1 = 1 tell us that all of the red intensity matches some green intensity. The opposite is not true: M2 = 0.163 tells us that only a small portion of the green signal colocalizes with the red one. The coeffient i2 = 0.098 tells us that 9.8% of the volume occupied by the green signal intersects with some red signal.

If the same offset is also applied to the red channel the colocalization "artificially" increases even more (let's call this situation C to refer to it later):

Image
Pearson's: 0.446,
Manders' coefficients:
overlap 0.890, k1 0.890, k2 0.890, M1 1.000, M2 1.000
intersection 1.000, i1 1.000, i2 1.000


Now also M2 = 1 and i2 = 1 show that all of the green intensity matches some red one. The term "some" is important here. The per-channel M coefficients do not care about the actual intensities of the pixels in the other channel: when inspecting the colocalization in one pixel with signal in certain channel they just look at the other channel to see if there is some intensity there (anything larger than zero!!!), but once that is granted, the pixel in the current channel is accounted.


Notice that the offsets applied to the objects do not affect the Pearson's coefficient: adding an offset to one or both channels doesn't change its value from 0.446.

On the other hand the offsets affect the Mander's coeffients noticeably. Any non-zero component is considered a relevant signal!!! A background level can be used to reduce the intensities before the calculation, to get rid of this offset. The image with the two offsets provides the realistic figures again if analyzed with the automatic background estimation in the Colocalization Analyzer (set to 52):
Pearson's: 0.446,
Manders' coefficients:
overlap 0.500, k1 0.500, k2 0.500, M1 0.500, M2 0.500
intersection: 0.333, i1: 0.500, i2: 0.500


In all the coefficients the provided background (either entered by the user or automatically estimated) is used to reduce all the intensities before the calculation. This happens in this analyzer even with the Pearson's coefficient, thus subtracting too large backgrounds may indeed have an effect also in this figure.

(To take into account the effects of the image offsets all the parameter backgrounds were set to zero for the calculation of the coefficients, except in the last case, where they were set to 52. Setting them to zero is done in the Colocalization Analyzer by setting the background fields at the bottom of the window both to zero. When using the Tcl Huygens command coloc the option -threshMode absolute was used with the default zero thresholds.)


Effect of adding non-colocalizing signal


Let's take the 50% matching objects from situation A again:

Image
Pearson's: 0.446,
Manders' coefficients:
overlap 0.500, k1 0.500, k2 0.500, M1 0.500, M2 0.500
intersection: 0.333, i1: 0.500, i2: 0.500


Now let's start adding other non-colocalizing objects in the green channel, and see what happens to the coefficients. Each new green square is one fourth in volume of the original objects. Situation A2:

Image
Pearson's: 0.380,
Manders' coefficients:
overlap 0.447, k1 0.500, k2 0.400, M1 0.500, M2 0.400
intersection 0.286, i1 0.500, i2 0.400


Let's add another object Situation A3:

Image
Pearson's: 0.329,
Manders' coefficients:
overlap 0.408, k1 0.500, k2 0.333, M1 0.500, M2 0.333
intersection 0.250, i1 0.500, i2 0.333


And yet another one Situation A4:

Image
Pearson's: 0.288,
Manders' coefficients:
overlap 0.378, k1 0.500, k2 0.286, M1 0.500, M2 0.286
intersection 0.222, i1 0.500, i2 0.286


We notice that many coefficients decrease this way, but not all of them: some remain constant. Because some coefficients separate the per-channel information, they reflect that the part of the red signal that colocalizes with the green one remains constant (50%) while a smaller and smaller portion of the green signal colocalizes with the red one.

Effect of reducing the analysis region


Reducing the analysis of your dataset to a limited region of interest has dramatic effects on the values of the coefficients, of course.

All colocalization parameters will increase if you limit the analysis to a region where the colocalization is high, discarding regions that would contribute to reduce the values.

Let's consider the following situation D for starters:

Image

A large red area against an empty background contains a few green spots in it. The green spots, therefore, completely colocalize with the red signal. The analysis of such a situation returns:
Pearson's: 0.156,
Manders' coefficients:
overlap 0.197, k1 0.039, k2 1.000, M1 0.039, M2 1.000
intersection 0.039, i1 0.039, i2 1.000


But now we crop the image before the analysis to get rid of the empty background. The effect of this is to reduce the Pearson's colocalization level, because that empty background in both channels was a region of high correlation (both channels had exactly the same signal, zero, and that is a high correlation!!!) The other coefficients remain constant because they intrinsically discard zeros anyway.

Image
Pearson's: 0.108,
Manders' coefficients:
overlap 0.197, k1 0.039, k2 1.000, M1 0.039, M2 1.000
intersection 0.039, i1 0.039, i2 1.000


But if you keep cropping the image and start getting rid of red signal, the thing changes:

Image
Pearson's: 0.000,
Manders' coefficients:
overlap 0.216, k1 0.047, k2 1.000, M1 0.047, M2 1.000
intersection 0.047, i1 0.047, i2 1.000


Pearson's coefficient is suddenly reported to be zero (actually, the red signal being completely uniform now, without any variability at all to compare with the green one, causes this coefficient to be undetermined) but the other coefficients accounting for the red colocalization are now higher. We have removed some red signal, and the remaining one has a higher degree of colocalization with the green one.

The tendency continues if we crop the image even more:

Image
Pearson's: 0.000,
Manders' coefficients:
overlap 0.474, k1 0.225, k2 1.000, M1 0.225, M2 1.000
intersection 0.225, i1 0.225, i2 1.000


The limiting case, were you crop the image so much that you reduce it to the exact size of one of the green blocks, would make all but Pearson's coefficients to be exactly one.

Effect of other experimental issues


See Blur And Noise Affect Colocalization.

Interpreting the coefficients


Distinguishing between similar situations


There is not a single colocalization coefficient that describes a situation completely: coefficients M on their own can not distinguish the situation C above from another one where the whole image is uniform in intensity in both channels, or yet another one where the two objects coincide without offset (total overlap in both the latter cases):

The two objects coincide completely. Here Pearson's is 1, because there are variations in the channel's intensities and they change together.

Image
Pearson's: 1.000,
Manders' coefficients:
overlap 1.000, k1 1.000, k2 1.000, M1 1.000, M2 1.000
intersection: 1.000, i1: 1.000, i2: 1.000


Now both channels have a constant intensity everywhere (the black margin doesn't belong to the image, is just a margin). The overlap is also 100%, but here Pearson's is undetermined (and shown as zero) because there is no co-variability that can be measured (all intensities are equal to the average).

Image
Pearson's: 0.000,
Manders' coefficients:
overlap 1.000, k1 1.000, k2 1.000, M1 1.000, M2 1.000
intersection: 1.000, i1: 1.000, i2: 1.000


Thus you need Pearson's coefficient to distinguish between these two extreme situations, and also between any of these and the 'intermediate' situation C above. The conclusion is that each coefficient represents something different, and all of them may be necessary to discriminate between different colocalization configurations.

Pearson's coefficient interpretation


This coefficient gives to a figure that quantifies how the variations in both channel go together, linked for whatever reason. The higher this figure, the more co-dependent both channels are. When the signal in one channel varies from place to place, the other one varies accordingly, increasing or decreasing in a similar ratio. In other words, this coefficient measures any linear dependencies between the two color channels.

Negative values close to -1 account for exclusion, co-variation in opposite ways: when one signal increases, the other one decreases accordingly.

See Pearsons Interpretation for more details.

Spearman coefficient interpretation


The Spearman coefficient can be interpreted as the Pearson's coefficient, with the addition that the spearman coefficient also measures other monotonic dependencies between the two channels, besides the linear one.

In fact, the Spearman coefficient is the Pearson coefficient, but based on intensity ranks, instead of intensity values.

See Pearsons Interpretation for more details.

Overlap coefficient interpretation


Here's a nice interpretation of the overlap coefficient: it gives approximately the ratio of intersecting volume to total volume (similarly to the intersection coefficient, but not exactly the same as we'll see). Let's use the volume of the smallest objects in the A2-A4? examples above as reference unit. One large object accounts for four small objects. In the last case (A4), for example you have one large red object (4 units), one large green object (4 units), and three green small objects (3 units). Total volume in both channels is 4 + 4 + 3 = 11.

Now take the intersecting volume. In the red channel, half of the large object (2 units) is in the intersection. In the green channel also only 2 units intersect. In total, 4 intersecting volume units. (Well, the intersection accounts twice because it belongs to both signals, and this is the difference with the intersection coefficient that only accounts it once).

The intersecting percentage is, with this criterium, 4 / 11 = 36.4% (compare with the overlap coefficient, 0.378).

Repeating a similar calculation for the other cases, we get A3: 40.0% (overlap 0.408), A2: 44.4% (overlap 0.447) and A: 50% (overlap 0.500).

As you can see, this intersection interpretation is only approximate, and it works well only when 1) the volume in each channel is more or less the same and 2) when the intensities are more or less constant and the images can be considered to be binary (black and white). On one side, the more and more objects we keep adding to one of the channels the more this interpretation of the overlap coefficient will fail. On the other hand, as already mentioned, all these coefficient actually consider pixel intensities and not "intersecting volumes", and when you have a real image with lots of different intensities in it this interpretation will also fail. That is why the intersection coefficients defined above, that are always easy to interpret, may prove useful.

Again, notice than any offset added to the image (in real life not only detection offset, also non-specific staining spread all over the channels that is not the signal you are interested in) will make the interpretation of the colocalization parameters difficult and should be removed as much as possible before the analysis. You can achieve this by applying backgrounds to be subtracted from the raw pixel intensities, or properly deconvolving the images.

The intersection coefficient is less sensible to intensity variations because it is based on a threshold. It can be directly interpreted as the portion of the total volume that is in a common intersection of the two channels' signals (and for the total volume account, the intersecting volume accounts only once). In the cases above, the intersecting volume successively goes from one-third of the total (33%) down to less than one-fourth (22%).

Manders M1 and M2 coefficients interpretation


Along all these examples, and specially in the last A-A4? series, we've noticed that the Manders' M1 and M2 coefficients provide the portion of the intensity in each channel that coincide with some intensity in the other channel. Only when the objects are homogeneous this represents the ratio of intersecting volume, but in any case considering each channel separately.

In all A-A4? cases half of the red pixels match some green pixel (M1 is always 0.5). But the opposite isn't true, because the more and more green signal we added, the ratio of green pixels matching red ones to the total number of green pixels decreases. This is reflected by the M2 coefficient.

In situation A, two out of four green volume units match red signal. M2 = 2/4 = 0.5 (50%). In situation A2, two out of five: M2 = 2/5 = 0.4 (40%). In situation A3 is two out of six: M2 = 2/6 = 0.333 (33%) and in A4 is two out of seven: M2 = 2/7 = 0.286 (29%).

We can therefore interpret, in these simple cases, the M coefficients as the percentage of pixels in one channel that intersect with some signal in other channel.

Still, as we saw above, they are in general also sensitive to local variations of intensity, and therefore this interpretation of the Manders' coefficients as volume of intersection only holds when the intensities are homogeneous. A more general interpretation doesn't account for "intersecting volume" but for the amount of the intensity in one channel that matches some intensity in the other channel.

Moreover, we have to remember from other paragraphs above that the M coefficients are sensitive to offsets in the signals, and that the user needs to establish what is signal and what is background when analyzing the data.

If your images have noticeable Photon Noise (variability not intrinsic to the objects but due to acquisition noise), DeConvolution is recommended for image restoration before the analysis.

Analogous to the M1 and M2 coefficients but less sensible to intensity variations (because they are based on thresholded values), the intersection i1 and i2 coefficients account for portions of the R and G channels separately that belong to an intersecting volume.

Intersection coefficients interpretation


These three figures are probably the most intuitive ones, because they simply account for colocalizing voxels.

The global coefficient i indicates the portion of voxels that are intersecting, from the total volume occupied by valid signal (above the threshold); you can multiply this figure by 100 and interpret as a percentage.

The other two figures i1 and i2 give similar information per channel: the number of voxels that intersect relative to the number of voxels with valid signal in that channel.

Definition of the colocalization coefficients


See Colocalization Theory.

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