On graphical and numerical characterization of proteomics maps

We outlined a mathematical approach suitable for characterization of experimental data given by 2-D densitograms. In particular we consider numerical characterization of proteomics maps. The basis of our approach is to order "spots" of a 2-D map and assign them unique labels (that in general will depend on the criteria used for ordering). In this way a map is "translated" into a sequence. In the next step one associates with the generated sequence a geometrical path and views such a path as a mathematical object that needs characterization. We have ordered spots representing proteins in 2-D gel plates according to their relative intensities which results in a zigzag path that produces a complicated "fingerprint" pattern. Mathematical characterization of zigzag pattern follows similar mathematical characterizations of embedded patterns based on matrices, the elements of which are given as quotients of Euclidean distance between spots and the distance along the zigzag path. The leading eigenvalue of constructed matrices is taken to represent characterization of the original 2-D map. Comparison of different 2-D maps (simulated by using random generator) allows one to construct partial order, which although qualitative in nature gives some insight into perturbation induced by foreign agents to the proteome of the control cell.     

Canonical labels for protein spots of proteomics maps

We consider the problem of canonical labeling for a class of maps, which include proteomics maps, which consist of a set of vertices or protein spots. If this problem is solved and followed, different laboratories studying proteomics maps will arrive at the same numbering of spots, which would facilitate comparisons of data from different sources. In addition, the proposed canonical numberings of protein spots would allow compiling a catalog of proteomics maps just as canonical labeling allows making catalogs graphs, or molecules, and other canonically labeled systems, which would make search for similar sets of maps very efficient. We approach the problem by modifying the algorithm of Jeffrey for graphical representation of DNA based on the chaos game. Graphical representation of DNA as a chaos game map has an important property in that this representation allows one to assign sequential labels to spots in a DNA map. We have modified the approach for sequential labeling of chaos game map representations to graphical representation of any tabular data, such as listing of (x, y) coordinates of protein spots of proteomics maps.     

A comparative study of proteomics maps using graph theoretical biodescriptors

This paper reports the development of new methods for mathematical characterization of effects of different toxic agents on the cellular proteome. We describe numerical characterization of proteomics maps based on mathematical invariants. A graph is first associated with a proteomics map by considering partial ordering of spots on 2-D gels by ordering proteins with respect to the mass and the charge, the two properties by which proteins are separated. The graph is then embedded over the map, and several graph theoretical invariants have been constructed. In particular we consider invariants that can be extracted from the Euclidean distance-adjacency matrix of the embedded graph, in which only Euclidean distances between adjacent vertices of a graph are considered. The approach is illustrated using proteomics patterns of normal liver cells of rats and those derived from liver cells of animals exposed to four peroxisome proliferators. In contrast to direct comparison of spot abundance our approach incorporates information on spots locations. The difference between the two approaches is that in the first case only changes in abundances are considered as a measure of perturbation of the proteome map, but in the second case not only the charge but also the mass of proteins are used for ordering protein spots.     

On characterizing proteomics maps by using weighted Voronoi maps

In contrast to the standard construction of Voronoi regions, in which the boundaries between different regions are at equal distance from the given points, we consider the construction of modified Voronoi regions obtained by giving greater weights to spots reported to have higher abundance. Specifically we are interested in applying this approach to 2-D proteomics maps and their numerical characterization. As will be seen, the boundaries of the weighted Voronoi regions are sensitive to the relative abundances of the protein spots and thus the abundances of protein spots, the z component of the (x, y, z) triplet, are automatically incorporated in the numerical analysis of the adjacency matrix, rather than used to augment the adjacency matrix as non-zero diagonal matrix elements. The outlined approach is general and it may be of interest for numerical analyses of other maps that are defined by triplets (x, y, z) as input information.     

On the dependence of a characterization of proteomics maps on the number of protein spots considered

We have reexamined the numerical characterization of proteomics maps based on the construction of novel distance matrices associated with the nearest neighbor graph for the protein spots. In particular we consider dependence of a characterization of proteomics map on the number of proteins considered in the analysis. We examined a collection of proteomics maps in which we approximately doubled the number of spots to be used for quantitative analysis, considering cases of maps having 30, 50, 100, 250, 500, and 1054 protein spots. For each case we have compared the similarity-dissimilarity results for five proteomics maps of rat liver cells associated with the control group and four proliferators administrated by intraperitoneal injection. We found that proteins maps based on a set of about the 250 most abundant proteins spots suffice for a satisfactory numerical characterization of such maps.     

Novel characterization of proteomics maps by sequential neighborhoods of protein spots

We consider a characterization of proteomics maps based on an alternative kind of neighborhood graphs for the protein spots on 2-D gel. The novel approach considers for every protein spot only the nearest neighborhood consisting of protein spots of higher abundance. The approach has the simplicity and advantages of the recently introduced characterization of proteome maps based on considering the nearest neighborhoods of protein spots, but it also has important additional desirable computational features. The characterization of the nearest neighborhood graphs of 2-D gel proteomics maps is sensitive to the number of spots considered and may lead to changes in the degree of similarity of different maps when the number of points has been changed, thus imposing restrictions on the protocol used for comparison of maps. The novel approach presented in this work is less sensitive to the number of points used in the analysis because graphs are constructed in a stepwise process in which the role of more distant neighbors has been diminished by linking a new spot to the nearest spot that has been already part of the neighborhood graph. In this way a graph with N + 1 spots is obtained from the graph on N spots by adding a single new link, while in the case of the nearest neighborhood graphs adding a new spot introduces novel neighborhoods and generally results in a graph that may differ significantly from the neighborhood graph on N points.     

On 3-D graphical representation of DNA primary sequences and their numerical characterization

In this article we (1) outline the construction of a 3-D "graphical" representation of DNA primary sequences, illustrated on a portion of the human beta globin gene; (2) describe a particular scheme that transforms the above 3-D spatial representation of DNA into a numerical matrix representation; (3) illustrate construction of matrix invariants for DNA sequences; and (4) suggest a data reduction based on statistical analysis of matrix invariants generated for DNA. Each of the four contributions represents a novel development that we hope will facilitate comparative studies of DNA and open new directions for representation and characterization of DNA primary sequences.     

Novel matrix invariants for characterization of changes of proteomics maps

Previous studies on mathematical characterization of proteomics maps by sets of map invariants were based on the construction of a set of distance-related matrices obtained by matrix multiplication of a single matrix by itself. Here we consider an alternative characterization of proteomics maps based on a set of matrices characterizing local features of an embedded zigzag curve over the map. It is shown that novel invariants can well characterize proteomics maps. Advantages of the novel approach are discussed.     

Representation of proteins as walks in 20-D space

A novel representation of proteins was introduced. It is independent of arbitrary decisions with respect to the choice of labels to be assigned to the 20 natural amino acids. The approach is based on an assignment of 20 unit vectors in 20-dimensional vector space to the 20 natural amino acids. Proteins are then represented by a walk, that is, a sequence of steps in the 20-dimensional space analogous to a walk in the (x, y) plane in the case of binary strings. A straightforward numerical characterization of proteins is obtained from the distance matrix associated with the walk representing the protein in 20-dimensional space combining the information on the Euclidean distance between various amino acids in protein sequence. The Line Distance matrix offers additional numerical characterization of proteins, while the lengths of steps of the walk in 20-D space allow construction of a "protein profile," which represents distribution of average lengths of the steps and their powers.     

Novel matrix invariants for characterization of changes of proteomics maps

Previous studies on mathematical characterization of proteomics maps by sets of map invariants were based on the construction of a set of distance-related matrices obtained by matrix multiplication of a single matrix by itself. Here we consider an alternative characterization of proteomics maps based on a set of matrices characterizing local features of an embedded zigzag curve over the map. It is shown that novel invariants can well characterize proteomics maps. Advantages of the novel approach are discussed.     

On the similarity of DNA primary sequences

We consider numerical characterization of graphical representations of DNA primary sequences. In particular we consider graphical representation of DNA of beta-globins of several species, including human, on the basis of the approach of A. Nandy in which nucleic bases are associated with a walk over integral points of a Cartesian x, y-coordinate system. With a so-generated graphical representation of DNA, we associate a distance/distance matrix, the elements of which are given by the quotient of the Euclidean and the graph theoretical distances, that is, through the space and through the bond distances for pairs of bases of graphical representation of DNA. We use eigenvalues of so-constructed matrices to characterize individual DNA sequences. The eigenvalues are used to construct numerical sequences, which are subsequently used for similarity/dissimilarity analysis. The results of such analysis have been compared and combined with similarity tables based on the frequency of occurrence of pairs of bases.     

2-D Graphical representation of proteins based on physico-chemical properties of amino acids

A 2-D graphical representation of proteins based on 2-D map of amino acids is outlined. The Amino Acid map was obtained by constructing the partial order on a selected pair of physico-chemical properties of amino acids. The plot of the difference between the (x, y) coordinates of two graphical representations of proteins allows a visual inspection of protein alignment. The approach is illustrated on segments of a protein of the yeast Saccharomyces cerevisiae.     

A turning point in proteome analysis: sample prefractionation via multicompartment electrolyzers with isoelectric membranes

Sample prefractionation, as obtained via multicompartment electrolyzers with isoelectric membranes, greatly enhanced the load ability, resolution and detection sensitivity of two-dimensional (2-D) maps in proteome analysis. This was demonstrated with different samples. In an Escherichia coli total cell extract, analysis by a 2-D map run in a pH 4-5 gradient showed many more spots when prefractionated, as compared with standard maps available in databases such as SWISS-2DPAGE. Analysis of human plasma in the pH 3-6 range showed an increase in the number of highly acidic proteins in the fractionated sample compared to whole plasma. With both samples no protein precipitation or smears occurred and much larger sample amounts could be loaded upon prefractionation, so that a large number of spots could be visualized by Coomassie staining, which is fully compatible with subsequent matrix assisted laser desorption/ionization-time of flight (MALDI-TOF) analysis.     

A naturally logical representation for DNA primary sequences

In this paper, we (1) introduce a logical representation (LR) for DNA primary sequences; (2) show relations between LR and some other representations including the characteristic sequences of a DNA sequence, Randic's 2-D, 4-D representations, and Z-curve (a 3-D graphical representation); and (3) outline the constructions of the S/S matrix specific for a logical sequence and its 2*2 condensed matrix.     

Two dimensional electrophoretic analysis of human tears: collection method in dry eye syndrome

Tear proteomics, by 2-DE, can give a fingerprint of the protein profile, which is well suited in clinical proteomics for biomarker identification and in diagnostics. The mode of tear collection can influence the representation of the proteins in the tear and therefore it is important to use the appropriate method. In this study, capillary and Schirmer mode of tear collection was done in the healthy controls and the Schirmer method was validated in dry eye syndrome conditions. 2-D PAGE of normal and dry eye tear was performed using pH 3-10 linear IPG strips followed by 13% SDS-PAGE. The spot intensity was analyzed by the PD quest software. The two methods were compared using Bland-Altman statistical tool. The 2-D map of capillary and Schirmer tear showed 147 ± 8 spots and 145 ± 7 spots respectively. Both the collection methods were in agreement with each other and were comparable. Dry eye tear protein showed differential expression of proteins as observed in 25-35 kDa region. One of the significantly reduced protein was identified as proline-rich 4 protein. Schirmer method of tear collection is reliable in patients with dry eye, which can display the differential protein expression and help in biomarker identification.     

A new integrated statistical approach to the diagnostic use of two-dimensional maps

Two-dimensional (2-D) electrophoresis is a very useful technique for the analysis of proteins in biological tissues. The complexity of the 2-D maps obtained causes many difficulties in the comparison of different samples. A new method is proposed for comparing different 2-D maps, based on five steps: (i) the digitalisation of the image; (ii) the transformation of the digitalised map in a fuzzy entity, in order to consider the variability of the 2-D electrophoretic separation; (iii) the calculation of a similarity index for each pair of maps; (iv) the analysis by multidimensional scaling of the previously obtained similarity matrix; (v) the analysis by classification or cluster analysis techniques of the resulting map co-ordinates. The method adopted was first tested on some simulated samples in order to evaluate its sensitivity to small changes in the spots position and size. The optimal setting of the method parameters was also investigated. Finally, the method was successfully applied to a series of real samples corresponding to the electrophoretic bidimensional analysis of sera from normal and nicotine-treated rats. Multidimensional scaling allowed the separation of the two classes of samples without any misclassification.     

On a 3-D representation of DNA primary sequences

We introduce a graphical representation of DNA primary sequences by taking four special vectors in a 3-D space to represent the four nucleic acid bases in DNA sequences, so that a DNA primary sequence is denoted in a 3-D space by a successive vector sequence which is a directed walk on the space. It is demonstrated that this representation has no overlap and intersection and allows numerical characterization.     

Spot overlapping in two-dimensional polyacrylamide gel electrophoresis separations: a statistical study of complex protein maps

A statistical approach able to extract the information contained in a two-dimenisional polyacrylamide gel electrophoresis (2-D PAGE) separation is here reported. The method is based on the quantitative theory of peak overlapping, a procedure previously developed by the authors and here extended to 2-D separations. The whole map is divided into many strips in order to obtain 1-D separations on which the statistic procedure is applied: the developed algorithms, on the basis of spot experimental data (intensity and spatial coordinates) permit to estimate the intrinsic number of components and to single out the specific order present in spot positions. The procedure was validated on computer-simulated maps. Its applicability to real samples was tested on maps obtained from literature sources. The following important information on protein mixtures can be extracted: (i) the number of proteins can be accurately estimated, on the basis of the spatial coordinates and intensities of spots detected in the 2-D PAGE map; (ii) the model describing distribution of interdistance between adjacent spots can be identified in both the separation dimensions; (iii) the presence of repeated interdistances in spot positions in the maps can be easily singled out: these regularities suggest specific protein modifications.     

Spot overlapping in two-dimensional polyacrylamide gel electrophoresis maps: relevance to proteomics

Proteomics requires a large-scale, simultaneous separation of proteins from a mixture, assessment of the relative abundance of these molecules, and identification and characterization of each component. In 2-D PAGE separations, the best method of choice for protein analysis, separation of all the proteins present in the sample is still far to be achieved and comigrating proteins in the same spot are in general present. A statistical estimation of the degree of spot overlapping present in a 2-D PAGE separation is here described: for different conditions of spot overcrowding in the map, the degree of overlapping can be quantified in terms of purity degree of each spot or percentage of proteins that will appear in the map as a single spot. A computer simulation approach is described: it is based on the protein separation pattern present in the experimental maps. The results thus obtained are compared to a theoretical model (statistical degree of peak overlapping model) based on random spot position. The described procedures were applied to an experimental reference map of human plasma. The severity of spot overlapping in 2-D PAGE maps is estimated and the influence of different experimental conditions (strip dimension, detector system performance, pI range) is discussed. These informations are useful to quantitatively estimate the degree of error associated with identification and quantitation of each protein and to set-up experimental conditions which will increase resolution and greatly decrease the probability of spot overlapping.