MATCHING TWO-DIMENSIONAL GEL ELECTROPHORESIS’

SPOTS

Ant´onio dos Anjos

, Bjarne Kjær Ersbøll

, Faroq AL-Tam

and Hamid Reza Shahbazkia

Department of Electronic and Informatics Engineering, University of Algarve, Faro, Portugal

Department of Informatics and Mathematical Modelling, Technical University of Denmark, Kgs. Lyngby, Denmark

Keywords:

2-D Electrophoresis, Matching, Registration.

Abstract:

This paper describes an approach for matching Two-Dimensional Electrophoresis (2-DE) gels’ spots, involv-

ing the use of image registration. The number of false positive matches produced by the proposed approach

is small, when compared to academic and commercial state-of-the-art approaches. This article contributes to

solving one of the greatest bottlenecks in the 2-DE analysis pipeline.

1 INTRODUCTION

In the context of Proteomics and Genomics, dif-

ferential analysis implies that two or more Two-

Dimensional Electrophoresis (2-DE) gel images are

compared in order to ﬁnd the differences in terms of

protein or DNA expression. Although it is possible

to superpose two images and try to ﬁnd their differ-

ences, it may not be the best approach. Distortions

are present in the majority of the gels (Figure 1). This

way, when two different images are superposed, the

spots in one image will hardly be aligned with the

spots in the other image (Figure 1(c)). Essentially be-

cause these distortions are mostly non-rigid, the prob-

lem of ﬁnding a correspondence between spots be-

comes an extremely difﬁcult task.

State-of-the-art software packages are known for

producing too many false positive matches (Rosen-

gren et al., 2003; Srinark and Kambhamettu, 2008).

The correction of false positive matches is an ex-

tremely time consuming task, being one of the main

bottlenecks in the analysis of 2-DE gels.

The problem of matching two sets of spots can be

put as a bipartite matching problem. However, the

following section shows that bipartite matching algo-

rithms may not be enough to achieve an acceptable

matching solution. Thus, other approaches will also

be considered, namely approaches that involve Image

Registration.

(a) Static image. (b) Deformed image.

Figure 1: Distortion between 2-DE gels. Figures 1(a) and

1(b) represent approximately the same protein expression,

but are clearly different due to deformations.

2 BIPARTITE MATCHING

Matching protein spots between gels can be formu-

ated as a bipartite matching problem. According

to graph theory, a graph G = (V, E) with vertices

V and edges E is bipartite if there exists a partition

V = X ∪Y with X ∩Y =

0 and E ⊂ X ×Y. A match-

ing is a subset M ⊂ E such that ∀v ∈ V there is at most

one edge in M that is incident on v. The size of the

matching is given by the number of edges in M, and

is deﬁned as |M|. A perfect matching is a matching

M where all the vertices are connected to one other

111

dos Anjos A., Kjær Ersbøll B., AL-Tam F. and Reza Shahbazkia H..

MATCHING TWO-DIMENSIONAL GEL ELECTROPHORESIS’ SPOTS.

DOI: 10.5220/0003702401110117

In Proceedings of the International Conference on Bioinformatics Models, Methods and Algorithms (BIOINFORMATICS-2012), pages 111-117

ISBN: 978-989-8425-90-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

vertex at most.

When the goal is to ﬁnd the best matching be-

tween vertices, the problem can be posed as an as-

signment problem where each edge e is assigned a

weight w(e). The maximum-weight matching satis-

ﬁes the condition that for any other M

′

, W(M

′

) ≤

W(M), where W(M) =

∑

e∈M

w(e). One of the most

famous and reliable optimization algorithms that can

be applied to ﬁnd the maximum-weight in a bipar-

tite matching is the Kuhn-Munkres algorithm (Kuhn,

1955), also called Hungarian Algorithm (HA).

Considering the spots in two different 2-DE gels,

the goal is to match each corresponding spot in gel X

to every other spot in gel Y. The spots are the ver-

tices of the graph. A weighing function w(e) based

on the similarity between each inter-gel spot has to be

deﬁned.

It is proven in (Kuhn, 1955) that a maximum-

weight matching is a perfect matching. This requires

that each of the parties of the graph has the same num-

ber of vertices. 2-DE gels are unlikely to express the

same exact number of spots, even if the feature de-

tection step is perfect. This is easily solved by adding

the number of missing vertices to the set that has them

missing, and assigning an extremely high weight to

connect to those vertices.

The complexity of the Hungarian Algorithm is

O(V

). However, considering that manually correct-

ing the results of a poor matching process between

two 2-DE gel images may take several hours, the

O(V

) cost in the HA is a minor drawback.

In order to extract the features (i.e. protein spots),

the approach proposed in (dos Anjos et al., 2011) is

used.

If each vertex v

in the ﬁrst party identiﬁes the

location of a spot in the ﬁrst image, and each ver-

tex v

identiﬁes the location of a spot in the sec-

ond image, then, a cost function for each edge

e could be the Euclidean Distance: w(v

) =

− v

)

+ (v

− v

)

. This weighing function

produces a perfect match between Figures 2(a) and

2(b), however, if only one outlier is introduced, only

≈ 60% of the spots are correctly matched. Therefore,

a more robust weighing function that includes more

information from the spots, namely the differences of

distances d, areas a, volumes vo, circularities c, ec-

centricities ec, and orientations o, has to be deﬁned:

w(v

) = α

k + α



− a



+α



− vo



+ α



− c



+α



− ec



+ α



− o



(1)

where the alphas are the respective weights of each

of the normalized components in w.

(a) Synthetic blob image. (b) Distorted image.

Figure 2: Difference between synthetic images. A large

distortion between images is present.

(a) Static 2-DE gel image. (b) Deformed 2-DE gel image.

Figure 3: 2-DE gels images and difference image.

Applying the HA with this weighing function

(Equation (1)), to Figures 2(a) and 2(b), produces a

match without any errors. Moreover, the outlier is

correctly identiﬁed. As reference, the weights were

set to: α

= 0.65;α

= 1.0;α

= 1.5. The rest of the

coefﬁcients were set to zero.

Figure 3(a) presents the image of one highly pop-

ulated 2-DE gel. The same image is presented in Fig-

ure 3(b) with an arbitrary elastic deformation. It is

possible to observe in Figure 3(c) that the gels’ spots

are not aligned.

Using as weights: α

= 2.5, α

= 1.5, α

= 2.5, at

least, 103 spots were correctly matched, among a total

of 476 detected. This was manually veriﬁed. Not only

were the relatively close positioned spots matched but

also the relatively far positioned ones. Nevertheless,

the HA has limitations when the distortions in the im-

BIOINFORMATICS 2012 - International Conference on Bioinformatics Models, Methods and Algorithms

112

ages are extremely high. Additionally, the distortions

present in the 2-DE images are mostly local and non-

rigid, making it extremely difﬁcult to ﬁnd an optimum

valid match. If these distortions are corrected, the

number of matches may increase. This takes us to

a different ﬁeld: Image Registration – which is dis-

cussed in the next section.

3 IMAGE REGISTRATION

The goal of image registration is to ﬁnd a geometric,

or elastic transformationthat makes one image similar

to the other. Putting it more formally, the problem is

the following: Given two images S(i) and D(i), where

S is called the static image and D the deformed (also

called reference and match, respectively), the goal is

to ﬁnd a transform T such that:

W = D(T(i)) ≈ S(i) (2)

where W is the warped image.

In differential analysis the objective is to identify

the differences and to quantify them, therefore, it may

be dangerous to register one image into the other us-

ing some image registration methods. For example,

Modersitzky (Modersitzki, 2004) completely warps a

ball into a C shape using a ﬂuid registration method.

3.1 Geometric Transform

Non-rigid transforms, also called deformable and

elastic, allow relatively complex distortions in the im-

age. These include stretching and curving of the im-

age.

Afﬁne transform, the most general transform that

preserves parallelism between lines, is a non-rigid

transform and is represented in homogeneous coor-

dinates by:





′









0 0 1













(3)

where θ

and θ

are the translations in x and y direc-

tions respectively.

The aforementioned transform preserves the

straightness of the lines. Curved transforms, as the

name points out, do not behave in a similar way.

These include the polynomial transformations that

can be expressed by:

′

∑

p−i

and y

′

∑

p−i

(4)

where p is the order of the polynomial transforma-

tion. Quadratic and cubic polynomial transforms are

the most used since they allow ﬂexible enough trans-

formations, and higher order polynomials tend to get

unstable, specially at the borders of the images. Ad-

ditionally, very small perturbations may lead to very

unstable transformations in the image with a polyno-

mial of higher order.

4 REGISTRATION WITH

KNOWN CORRESPONDENCES

As previously seen, just by using the direct applica-

tion of the HA to the weighed distances between cor-

respondent spots in different gels, there was a sub-

stantial amount of correct spot matches between the

gels. Image registration can be used to improve these

results.

The idea is to select thebest matches and use those

matches to create a warping transform that hopefully

will drag the other misplaced and mismatched spots

to a corrected position in the gel image.

The proposed approach is the following:

1. Detect features;

2. Calculate the cost matrix;

3. Match using the Hungarian Algorithm;

4. Iterate until stability:

(a) Select only very good markers (≤ threshold);

(b) Warp;

(d) Rematch;

In this approach, the image is not really being

warped at each step. Because features have already

been detected, only the coordinates of the center of

the spots are recalculated, or warped. The objective

is to match the protein spots in one gel image to the

spots in the other gel image, instead of the whole im-

age.

Markers are detected and selected and the coefﬁ-

cients for a polynomial transform (Equation (4)) are

calculated. The degree of the polynomial transform

is limited by the number of detected points, there-

fore, higher order polynomial transforms were pre-

ferred until a maximum degree of 3. The intensity

levels resulting from the warping are simply ignored

since the objective here is just to correct the positions

of the spots.

The weights used in this matching were the same

as the previous ones: α

= 2.5; α

= 1.5; α

= 2.5.

Two new parameters are additionally used, namely t

and t

, that deﬁne a threshold for the maximum al-

lowed distance for a spot to be accepted as a good

MATCHING TWO-DIMENSIONAL GEL ELECTROPHORESIS' SPOTS

113

Figure 4: Matched with iterative registration. Blue numbers

represent reliable matches.

match, and the maximum allowed cost for a match to

be considered valid, respectively. The values set for

these thresholds were: t

= 0.09;t

= 0.1.

By using the presented approach, we can get a

minimum of 255 correct matches as shown in Figure

4, in contrast to 103 in the previous attempt. This

represents an increase of more than 100%. The blue

markers are the matches accepted as having an accept-

able quality. The green markers are markers that are

not considered reliable for being at a relatively far Eu-

clidean distance from the match, although many can

also be found as correct matches.

Although there is a great improvement, the results

obtained from matching “naturally” deformed images

is far from acceptable. A more advanced approach is

presented in the next section as an attempt to improve

these results.

5 REGISTRATION WITHOUT

KNOWN CORRESPONDENCES

For two images, it may be acceptable to ask the user

to manually place landmarks in order to allow the

calculation of the transformation. For large amounts

of images, that is not a suitable approach. For the

global misalignment correction, an automatic para-

metric afﬁne transformation was chosen. Also, in

order to correct local distortions, non-parametric de-

formable registration is considered in this section.

5.1 Correction of Global Distortions

When a 2-DE gel is placed in the scanner, the posi-

tion of the gel will hardly be the same as the previ-

ously scanned gels. Rotation and translation are the

most common differences between gels, and these

are global deformations. Before any other attempt

of matching relative nearby spots between gels, these

global distortions should be corrected.

Parametric image registration deals with the prob-

lem of ﬁnding the transform’s parameters, in order to

minimize a similarity measure or error function.

Let us start by choosing the SSD as the similarity

measure. Thus, the problem is to ﬁnd the parameters

Θ that minimize the following error function:

E(S, D(T(Θ))) =

∑

(S(~x

) − D(T(Θ,~x

)))

(5)

where Θ is a set of K parameters Θ =

{θ

,θ

,. . . ,θ

} belonging to the trans-

form that warps D. E will stabilize and, therefore,

should be minimum when

∂E

∂θ

= 0, for all k.

The error can be minimized by either Gradient De-

scent, Gauss-Newton, or Levenberg-Marquardt opti-

mization (Levenberg, 1944; Marquardt, 1963). Since

the latter has become a standard technique for nonlin-

ear least-squares problems and can be thought of as a

combination of Gradient Descent and Gauss-Newton,

this is the method used in the proposed approach.

5.1.1 Tests

Figures 5(a) and 5(b) show the resulting difference

images of automatically registering images in Figure

2(b) to 2(a), and 1(b) to 1(a), respectively. Compare

Figure 5(a) with Figure 2(c). Also, compare Figure

5(b) with Figure 1(c). It is obvious that the global reg-

istration method corrected the global afﬁne distortions

on both images. One of the great advantages of cal-

culating the distortion as global is that the number of

parameters that have to be estimated for the transfor-

mation is relatively small. The disadvantage is that lo-

cal distortions are not corrected, and 2-DE gel images

are also proliﬁc in these kind of distortions. There-

fore, these resulting global transforms can be used as

a very good initial seed for a registration method that

handles local distortions.

5.2 Correction of Local Distortions

In order to take care of local distortions, automatic

non-parametric image registration was chosen since

parametric transformations do not perform well for

local distortions (Goshtasby, 1986). Image registra-

tion without known correspondences is an ill-posed

problem. A mathematical problem is ill-posed if its

BIOINFORMATICS 2012 - International Conference on Bioinformatics Models, Methods and Algorithms

114

(a) Synthetic image. (b) 2-DE image.

Figure 5: Differences after afﬁne registration.

solution is not unique, does not exist, or does not de-

pend continuously on the data (Hadamard, 1902).

An approach to solve ill-posed problems is based

in the regularization theory. Generally, it involves

the inclusion of constraints on the possible solutions.

Constraints are, usually, imposed by a regularizer.

The registration method presented in the previoussec-

tion did not need an explicit inclusion of any kind of

regularizer, since the solutions were implicitly regu-

larized by properties of the space. The most common

constraint used in the regularization of image process-

ing problems is smoothness. Regularization theory

involves the use of two terms:

• the error term E;

• the regularization term R.

The error term is usually the sum of squared differ-

ences, rewritten as follows:

SSD

(S, D,T) =

∑

(S(~x

) − D(T(~x

)))

(6)

and the regularization term:

R(T) = k∇Tk

(7)

The regularizer can be seen as a term that will prevent

abnormal transformations from occurring or, in other

words, a term that penalizes certain exaggerations in

the transformation.

Consequently, the total error to be minimized can

be expressed as:

(S, D,T) =

∑

E(S, D, T) + λR(T) (8)

with λ being a value that controls the trade-off be-

tween data ﬁtness and transformation smoothness.

If the transformation caused by T in ~x is repre-

sented by u(~x), or T(~x) =~x+ u(~x), then the problem

can be posed in a variational way. Therefore, the ob-

jective now is to ﬁnd the u that minimizes E(S,D, u).

In other words, we want to minimize the following

error function:

(S, D,u) =

∑

E(S, D, u) + λR(u) (9)

When E

is minimized, then u satisﬁes the Euler-

Lagrange equation:

∇(E(S, D, u)) + λ∇R(u) = 0. (10)

An approach presented in (Knut Conradsen, 1992)

tries to ﬁnd the best u using an approximation to the

cross-correlation as matching criteria. In the cited

source, an algorithm to match two 2-DE gel images

of 512 × 512 pixels is presented. The method pro-

posed by the author estimates a transformation that

maps one 2-DE gel image to the other by using the

intensity of the pixels as main criteria. For each of the

pixels in one gel image, a disparity vector is estimated

or, in other words, the distance between coordinates

of the pixel in one gel and the best candidate position

on the other is calculated. The best candidate is iden-

tiﬁed by the local arrangement of gray levels for the

neighbor pixel in the other gel, and the matching is

done by cross-correlating the two gels. The different

maxima of the cross-correlation function will produce

the disparity vectors. These vectors are used to warp

the second gel so that it matches the ﬁrst as well as

possible.

A median smoothing is applied at each level of the

pyramid, and together with the scaling process, these

act as a regularizer.

6 REGISTRATION-BASED

MATCHING

Some modiﬁcations were made to the algorithm pre-

sented in (Knut Conradsen, 1992). First, it was

adapted to work with any image size. As some ar-

tifacts could be seen at the end of the registration

process, another improvement consists in the use of

the original image and the cumulative displacement

matrices when warping at the intermediary steps, in-

stead of always using the same warped image with

the intermediate matrices. Also, cubic B-Spline in-

terpolation replaced cubic interpolation. Images with

more complex distortions were not being correctly

registered, therefore, a parameter was introduced in

order to allow more iterations at each level of the

pyramid. These changes introduced considerable im-

provements at the cost of more computational com-

plexity.

In the proposed approach, the intention is to ﬁnd

correct positions for each spot in order to allow a bet-

ter matching. Therefore, what matters are the trans-

forms and not the warped images.

We proposed the following approach:

1. Normalize the images;

MATCHING TWO-DIMENSIONAL GEL ELECTROPHORESIS' SPOTS

115

2. Global Afﬁne Registration;

3. Full image registration using the previous calcu-

lated transform as seed;

4. Detect features in the original images;

5. Correct the spots’ coordinates using the transform

calculated in step 3;

6. Match features with corrected coordinates;

After the results of the afﬁne global and full reg-

istrations, the features in the original images are de-

tected using the approach proposed in (dos Anjos

et al., 2011). As mentioned before, position, area,

volume and other descriptors are extracted from the

spots. Nevertheless, only the coordinates of the de-

formed image are corrected by using the total dis-

placement ﬁeld calculated from the image registration

process.

The total displacement u

(~x) for the position of

each detected spot is the following:

(~x) =~x− ((A~x+ b) + u(~x)) (11)

where (A~x + b) is the afﬁne transform calculated as

described in Section 5.1, and u(~x) is the displacement

for each coordinate as calculated in (Knut Conradsen,

1992) with the proposed improvements.

This way, position can be given a much higher

weight in the cost function as it is supposed to be more

reliable. Another improvement, relatively to the pre-

vious presented approach is the replacement of the de-

tected volume by the volume of the spot weighed by

a Gaussian centered at the spot. Therefore, instead of

relying only on the detected volume, one relies on the

volume and on the context deﬁned by the spot’s im-

mediate surroundings. Therefore, even having lower

representation, the information surrounding the spot

is included, contributing for a better match. Addition-

ally, a threshold t

was created in order to immedi-

ately exclude the possibility of matches that are at a

very far position from the correspondingspot. Finally,

spot matches not respecting this threshold are set to ∞.

Only then, the cost matrix is processed by the HA.

7 RESULTS

All the parameters were the same for the complete set

of images. Distance and (new) volume, for the weigh-

ing function were set to: α

= 1.0; α

vo2

= 1.0. All the

other weight coefﬁcients were set to zero. Moreover,

= 0.35;t

= 35, where t

is not normal-

ized as it is used during the construction of the cost

matrix.

As it will be demonstrated, one of the great advan-

tages of this approach is that it is very difﬁcult to ﬁnd

(a) 200 spots found. (b) 217 spots found.

Figure 6: Figures 1(a) and 1(b). Valid matches: 155.

a false match between the markers presented as valid

markers, meaning that the presence of false matches

is extremely low, or non-existent. Only markers pre-

sented in blue are matches that respect all the deﬁned

thresholds. Green markers have found a match in the

HA but do not respect either t

or t

. Finally, the red

markers did not ﬁnd any match in the HA, being con-

sidered as outliers.

Figure 6 shows the result of the suggested match-

ing process used to match the spots of Figures 1(a)

and 1(b).

A very popular approach used in feature matching

is Shape Context (Belongie and Malik, 2000). It ba-

sically consists of analyzing the spacial relationship

between points.

Shape contexts use mainly four parameters. The

ﬁrst parameter deﬁnes the number of radial bins for

the creation of the histograms (set to 10), the second is

the number of theta bins that deﬁnes how many slices

should the histograms be divided in (set to 24), and

the minimum and maximum widths of the bins (set to

1 and 100, respectively).

Table 1 presents a comparison of the results of the

matches using the proposed approach and matching

with shape contexts.

It is clear that the proposed approach is doing a

better job than the shape context approach. The num-

ber of false matched spots is extremely low when us-

ing the proposed approach. This is highly desirable

because it avoids the necessity of a laborious manual

intervention to correct the results.

PDQuest is considered as one of the best software

packages in matching 2-DE gel images (Rosengren

et al., 2003). Despite not directly comparable with

PDQuest, or with any other commercial software, be-

cause the detected spots are not exactly the same as

the detected by our approach, results from matching

the same images used in Table 1 are presented in Ta-

ble 2 using PDQuest Version 8.0.1 build 055.

Although the number of false matched spots is

BIOINFORMATICS 2012 - International Conference on Bioinformatics Models, Methods and Algorithms

116

Table 1: Comparison between the detection of the proposed approach and shape context approach.

Spot Det Proposed Shape Context

Stat Def Mat Min % F.P. % F.P. Mat Min % F.P. % F.P.

341 374 238 69.79% 0 0.00% 272 79.77% ≥ 35 ≥ 10.26%

200 217 155 77.50% 0 0.00% 130 65.00% ≥ 20 ≥ 10.00%

97 114 56 57.73% 1 1.03% 52 53.61% ≥ 15 ≥ 15.46%

49 44 32 72.73% 1 2.27% 36 81.82% ≥ 12 ≥ 27.27%

422 398 312 78.39% 2 0.50% 315 79.15% ≥ 30 ≥ 07.54%

160 207 79 49.38% 2 1.25% 118 73.75% ≥ 20 ≥ 12.50%

160 195 41 25.63% 7 4.38% 48 30.00% ≥ 07 ≥ 04.38%

160 260 104 65.00% 1 0.63% 117 73.13% ≥ 10 ≥ 06.25%

160 278 92 57.50% 0 0.00% 112 70.00% ≥ 15 ≥ 09.38%

160 249 54 33.75% 6 3.75% 88 55.00% ≥ 10 ≥ 06.25%

Avg: 58.74% Med: 1.37% Avg: 66.12% Med: ≥ 13.99%

Table 2: Matching with PDQuest.

Spot Det PDQuest

Stat Def Mat Min % F.P. % F.P.

154 331 123 79.87% 0 0.00%

174 180 137 78.74% 7 4.02%

91 58 52 89.66% 4 6.90%

45 31 29 93.55% 1 3.23%

332 293 209 71.33% 5 1.71%

137 154 78 56.93% 3 2.19%

137 146 60 43.80% 8 5.84%

137 175 95 69.34% 0 0.00%

137 214 100 72.99% 0 0.00%

137 249 58 42.34% 4 2.92%

Avg: 69.85% Med: 3.65%

also low, when compared to the proposed approach,

it is possible to conclude that a lot of post-processing

will be needed. First, many undetected valid spots

have to be manually added and, then, they need to be

re-matched. Nevertheless, the matching is also very

good.

In summary, an effective spot matching approach

is proposed in this article. These results are outper-

forming the results obtained from importantacademic

and commercial approaches.

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