- Research article
- Open Access
Characterization and simulation of cDNA microarray spots using a novel mathematical model
- Hye Young Kim†1,
- Seo Eun Lee†1,
- Min Jung Kim1,
- Jin Il Han1,
- Bo Kyung Kim1,
- Yong Sung Lee2,
- Young Seek Lee3 and
- Jin Hyuk Kim1Email author
© Kim et al; licensee BioMed Central Ltd. 2007
- Received: 29 March 2007
- Accepted: 20 December 2007
- Published: 20 December 2007
The quality of cDNA microarray data is crucial for expanding its application to other research areas, such as the study of gene regulatory networks. Despite the fact that a number of algorithms have been suggested to increase the accuracy of microarray gene expression data, it is necessary to obtain reliable microarray images by improving wet-lab experiments. As the first step of a cDNA microarray experiment, spotting cDNA probes is critical to determining the quality of spot images.
We developed a governing equation of cDNA deposition during evaporation of a drop in the microarray spotting process. The governing equation included four parameters: the surface site density on the support, the extrapolated equilibrium constant for the binding of cDNA molecules with surface sites on glass slides, the macromolecular interaction factor, and the volume constant of a drop of cDNA solution. We simulated cDNA deposition from the single model equation by varying the value of the parameters. The morphology of the resulting cDNA deposit can be classified into three types: a doughnut shape, a peak shape, and a volcano shape. The spot morphology can be changed into a flat shape by varying the experimental conditions while considering the parameters of the governing equation of cDNA deposition. The four parameters were estimated by fitting the governing equation to the real microarray images. With the results of the simulation and the parameter estimation, the phenomenon of the formation of cDNA deposits in each type was investigated.
This study explains how various spot shapes can exist and suggests which parameters are to be adjusted for obtaining a good spot. This system is able to explore the cDNA microarray spotting process in a predictable, manageable and descriptive manner. We hope it can provide a way to predict the incidents that can occur during a real cDNA microarray experiment, and produce useful data for several research applications involving cDNA microarrays.
- Contact Angle
- Contact Line
- Radial Profile
- Contact Radius
- Spot Signal
With the advance in technology for simultaneous acquisition of information on a number of genes' expression, the research area of bioinformatics is expanding into the reconstruction of a system composed of such genes. Within a decade, the technology of cDNA microarray experiments has been extensively developed, and its usage has exponentially increased. To improve the accuracy of microarray data, a number of techniques in post-processing microarray data were suggested from image analysis to statistical data processing. However, in spite of those efforts, the increase in accuracy of post-processing has a limit. After all, the improvement in wet-lab experiments, such as ensuring the conditions for sound spotting of cDNA probes, is necessary to obtain reliable microarray data. Most cDNA microarray images contain spots in various shapes, including doughnut-shaped spots, which consist of pixels of high intensity at the perimeter and those of low intensity in the central area. Such patterns of spot images are primarily due to the non-uniform distribution of cDNA molecules while the cDNA solution dries out during the microarray printing process.
Since microarray became widely used for obtaining high-throughput gene expression data, several methods for simulating the microarray experiment have been suggested [3–7]. Simulation is useful for designing and testing both the experiment and the analysis in real fields of study. This makes it possible to predict how the experiment will proceed or how well the analysis will work, before it is realized. In addition, it also provides synthetic data for analysis, when the realization is impossible due to a technical problem. Previous methods focused on mimicking images generated by real cDNA microarray experiments, and their parameters for determining the features of spots were based on certain probabilistic models. Wierling et al. classified spots into three categories: convex, crater-like, and cylindrical spots. Balagurunathan et al. simulated spots from simple circles and modified them by resizing radii, punching holes in the centre of spots, reshaping spots by removing chords, and enhancing intensities at the edge of spots. However, if it were not based on the investigation of physical and chemical nature, it would fail to correctly simulate and might confuse the experimenters with the results.
In this respect, it is important to study the formation of the variously shaped spots. The phenomenon of deposition of solute drop on the free surface has been studied [8–14]. Deegan et al. explained the formation of a ring-shaped stain when a coffee drop evaporates as a result of "pinning" of the contact line. Heim et al. noticed the importance of study of the deposition of DNA unspecifically bound on the microarray slide. However, microarray experiments are performed on slides specially manufactured for the purpose of increasing the sensitivity and the specificity of the results. A capping step to prevent non-specific adsorption on the support during hybridization and to decrease the background noise is often performed. To minimize the loss of cDNA by hybridization or washing out, the surface of a glass slide is usually coated with substrates . Therefore, it is necessary to study the characteristics of cDNA deposition by close investigation focused on the microarray experiment.
In this study, we generate a mathematical model of cDNA deposition during the microarray spotting process using the contact printing technology by bringing parameters from cDNA microarray experiments. This study analyzes cDNA microarray spot formation and elucidates parameters which affect the spot morphology. The study of the origin of various patterns in spot morphology suggests how to manufacture good microarray spots.
cDNA deposit model generation
where K E is the evaporation constant reflecting the conditions of the experiment, such as temperature, humidity, and atmospheric pressure, etc.
Several studies have investigated the morphological transition of a drop during its evaporation [17–20]. At the beginning of evaporation, the contact line is pinned, i.e., while the contact radius remains constant, the drop diminishes in volume as the contact angle and the drop height decrease (State I). When the contact angle and the drop height decrease to a certain level, the contact line is depinned, i.e., the contact radius starts to retract with the decrease of drop height and the contact angle becomes constant (State II).
where x0 is the surface site density on the support and a is the interaction factor for the non-ideal adsorption behaviour of macromolecules . If a = 0, the equation is the same as the Langmuir adsorption equation, and otherwise, it describes the non-ideal adsorption behaviours, such as repulsion (a > 0) and attraction (a < 0) between molecules [22, 23]. Therefore, the equilibrium constant for the binding reaction is Ke-ax, where K is the extrapolated equilibrium constant. The concentration of cDNA, ρ, rises during evaporation without reduction of the radius of the drop, and, therefore, the quantity of cDNA molecules, x, which bind with the surface sites in a unit area increases nonlinearly.
The solution of equation (8) is the density of cDNA deposit at the distance r from the centre, which can be solved by an ordinary differential equation solver with the four parameters, x0, K, a, and K V .
The peak-shaped spot was defined as a spot having the highest density at the central region and a declining density at the region farther from the centre (Figure 4b, left and middle). Because the peak-shaped spot can have an obscure boundary with the background, the edge of the spot can often be determined by the threshold. The histogram of the density of deposited cDNA was skewed to the right (Figure 4b, right). Therefore, the mean density of deposited cDNA is lower than the median value. Furthermore, the measure can vary with how large an area of spot is included.
We defined the peak-shaped spot having a small hole in its peak as the volcano-shaped spot because its 3D image resembles a volcano (Figure 4c, left and middle). The volcanic-shaped spot has characteristics of both the doughnut-shaped spot and the peak-shaped spot (Figure 4c, right).
Simulated cDNA deposit
Parameter estimation from real microarray images
The spot morphology varies with the size of the end of the spotting pin tip, the volume of sample delivered, the surface tension of the drop, the hydrophobicity of the surface of the slide, and so on. It has been proven that the increase in viscosity and contact angle of a micro-drop shrinks the initial size of the drop . Using the data and equation (2), we calculated the volume constants and the contact angles of the common micro-drops at the moment of being spotted on the glass slides, with the data of the spot diameters and the volumes of delivered sample provided by TeleChem International, Inc. . We found that the initial drop in state I has a wide variety in morphology, from a sphere-like shape with a small contact area (K V = 16.38, θ = 141.14°) to a flat spherical cap shape with a large contact area (K V = 0.46, θ = 32.01°).
subject to eq.(8)
Estimated values of the four parameters of the three types of spot shapes of Image I *
Estimated values of the four parameters of the three types of spot shapes of Image II *
Unlike the ring-shaped stain formed during the evaporation of a coffee drop, cDNA microarray images produce spots of various shapes, even in the same image. A solution to the coffee drop problem can partly explain the doughnut-shaped spot, which is one extreme of spot morphology in cDNA microarray images. Doughnut-shaped spots have frequently appeared in microarray images. In spite of the effort to prevent such spots from being generated, there is still an uneven density of signals in a single spot. In this study, we devised a generalized mathematical model that can manifest a wide spectrum of spot morphology.
The doughnut-shaped spot would be produced by experiment under the condition that a large quantity of cDNA is deposited early on, and so the concentration of the drop solution is quickly lowered. Such a situation can occur when the adsorption reaction is facilitated by a high surface site density, a large equilibrium constant of the binding reaction, and a low repulsion or attraction between cDNA molecules, or when the initial drop has a flat spherical cap shape. Unlike the doughnut-shaped spot, the peak-shaped spot would be produced under the condition that cDNA is not likely deposited early on, and the concentration of the drop solution continually increases due to the accumulated cDNA, and so the quantity of the deposited cDNA can be the maximum at the centre. Such a situation can occur when the adsorption reaction is impeded by a low surface site density, a small equilibrium constant of the binding reaction, and a high repulsion between cDNA molecules, or when the initial drop has a sphere-like shape.
It is known that the composition of buffer solution changes the spot morphology [30–33]. The optimal composition of buffer solution has been investigated by including detergent or betaine to 3 × SSC or 50% dimethyl-sulfoxide (DMSO). It is known that DMSO decreases the surface tension of drop solution during evaporation, while both 3 × SSC and 3 × SSC with betaine increase the surface tension . The decrease of surface tension is accompanied with a decrease in the contact angle and K V . Moreover, DMSO denatures DNA to be bound well with surface sites , which means the increases of the equilibrium constant for the binding reaction, Ke-ax. Putting them together, DMSO would decrease K V and a, and increase K. Figure 3 and 5 show that the doughnut shape is aggravated by increasing K and x0 and decreasing K V and a. As a result, DMSO would aggravate the doughnut shape. In addition, there is a study that the length and the sequence arrangement of DNA molecules would influence on the morphological organization of the deposit . The macromolecular interaction factor, a, would reflect the length and sequence arrangement of DNA molecules.
The extrapolated equilibrium constant for the binding reaction K also reflects the condition of the slide surface. While homemade poly-L-lysine-coated glass slides have been widely used, several commercial microarray slides are now preferred, such as the FMB cDNA slide (Full Moon Biosystems Inc.), ArrayIt SuperAmine and SuperAldehyde slides (TeleChem International, Inc.). It is known that CMT-GAPS™ slides (Corning) makes the spot morphology much more uniform by preventing doughnut shape formation . It can be assumed that the slides are manufactured to have the optimal binding efficiency. The surface site density, x0, affects the efficiency of cDNA binding. The increase in x0 facilitates the forward reaction of equation (4) and vice versa.
We have estimated parameters from real microarray images, but not all spot images were successful. It was practically impossible to fit our model equation to the doughnut-shaped spots, whose centre part has significantly high intensity. Table 1 and Table 2 show the mean squared deviation (MSD) between real and estimated data, as the goodness-of-fit measure. MSDs obtained from the doughnut-shaped spots were relatively larger than those of the other spots. We expect this problem originates from two major reasons: the effect of diffusion when the microarray slide passes through the hybridization process, and nonlinear transformation of spot signals to image intensity when the region out of dynamic range of the microarray scanner was used in converting spot signals. The effect of hybridization on the change in cDNA density is still controversial. Tran et al. claimed that DNA spotted on a glass slide diffuses during hybridization. However, this conflicts with the opinion of Pappaert et al., who claimed that the doughnut shape occurs not only from the uneven distribution of DNA deposition but also from the hybridization process. We observed a few microarray images that have concentric circles in the spots. Even though we did not include such phenomenon in this study, it is considered a result of the repetition of pinning and depinning of the contact line , and should be investigated further.
We developed a governing equation that can explain the dynamics of cDNA deposition during evaporation of a drop in the microarray spotting process. Experimental conditions were parameterized and included in the governing equation. The parameters determining the spot morphology were brought from the physical and chemical factors. This explains how various spot shapes can exist and suggests which parameters are to be adjusted for obtaining a good spot. This system is able to explore the cDNA microarray spotting process in a predictable, manageable and descriptive manner. We hope it provides a way to predict the incidents that can occur during a real cDNA microarray experiment, and produce useful data for several research applications involving cDNA microarrays.
When we consider the contact radius r and the contact angle θ, then
R = r cscθ
h = R(1 - cosθ) = r cscθ(1 - cosθ).
Tools for analysis and visualization
The ordinary differential equation, equation (8) was solved by applying the backward differentiation formula (BDF) method. cDNA deposits were simulated with a radius of 100 μ m, which were converted to images at a resolution of 10 μ m/pixel using LabVIEW and NI-Vision 8.2 (National Instrument, Inc.). The templates of spots were obtained by using the edge detection technique . BDF, CNO, and cubic Hermite spline interpolation were implemented with tools in LabVIEW.
All of the real microarray images were obtained from Stanford MicroArray Database . The data of the spot diameters and the volume of the initial drop of cDNA solution were obtained from TeleChem International, Inc. .
This work was supported by grants M10447010002-07N4701-00210 and M10641280004-06N4128-00410 from the national R&D project of MOST/KOSEF.
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