diff --git "a/deduped/dedup_0501.jsonl" "b/deduped/dedup_0501.jsonl" new file mode 100644--- /dev/null +++ "b/deduped/dedup_0501.jsonl" @@ -0,0 +1,504 @@ +{"text": "Modern developmental biology relies heavily on the analysis of embryonic gene expression patterns. Investigators manually inspect hundreds or thousands of expression patterns to identify those that are spatially similar and to ultimately infer potential gene interactions. However, the rapid accumulation of gene expression pattern data over the last two decades, facilitated by high-throughput techniques, has produced a need for the development of efficient approaches for direct comparison of images, rather than their textual descriptions, to identify spatially similar expression patterns.The effectiveness of the Binary Feature Vector (BFV) and Invariant Moment Vector (IMV) based digital representations of the gene expression patterns in finding biologically meaningful patterns was compared for a small (226 images) and a large (1819 images) dataset. For each dataset, an ordered list of images, with respect to a query image, was generated to identify overlapping and similar gene expression patterns, in a manner comparable to what a developmental biologist might do. The results showed that the BFV representation consistently outperforms the IMV representation in finding biologically meaningful matches when spatial overlap of the gene expression pattern and the genes involved are considered. Furthermore, we explored the value of conducting image-content based searches in a dataset where individual expression components (or domains) of multi-domain expression patterns were also included separately. We found that this technique improves performance of both IMV and BFV based searches.We conclude that the BFV representation consistently produces a more extensive and better list of biologically useful patterns than the IMV representation. The high quality of results obtained scales well as the search database becomes larger, which encourages efforts to build automated image query and retrieval systems for spatial gene expression patterns. Drosophila melanogaster (the fruit fly) has been a canonical model animal for understanding this developmental process in the laboratory. The raw data from experiments consist of photographs of the Drosophila embryo showing a particular gene expression pattern revealed by a gene-specific probe in wildtype and mutant backgrounds. Manual, visual comparison of these spatial gene expressions is usually carried out to identify overlaps in gene expression and to infer interactions . As a ra genome . In addiWith this rapid increase in the amount of available primary gene expression images, searchable textual descriptions of images have become available ,10,11. HWe previously proposed a binary coded bit stream pattern to represent gene expression pattern images . In thisIn this paper, we explore how a more sophisticated Invariant Moment Vectors based dPreviously, we had examined the performance of the BFV representation for a limited dataset of early stage images . Here weDuring these investigations, we also developed another measure of image-to-image similarity for the BFV representation. This measure is aimed at finding images that contain as much of the query image expression pattern as possible, but without penalizing for the presence of any expression outside the overlap region in the target image. In addition, we examined whether partitioning a multi-domain expression pattern into multiple BFV representations, each containing only one domain, yields a better result set.Recently, Peng and Myers have proAn image database of 226 gene expression pattern images was initially generated using data from the literature -29. All In order to present comprehensible result sets in this paper, we have primarily discussed the findings from the dataset of 226 and provided information on how those queries scaled when they were conducted for the larger dataset. In general, our focus was to show the retrieval of biologically significant matches based on both the visual overlap of the spatial gene expression pattern and the genes associated with the pattern retrieved.\u03c61 through \u03c67), and binary feature representations were stored in a database. We also calculated and stored the expression area (the count of the number of 1's in the binary feature represented image), the X and Y coordinates of the centroid , and the principal angle (\u03b8) for each extracted pattern.Each image was standardized and the binary expression pattern extracted following the procedures described previously . These eSS, SC) based on the BFV representation (See equations 2 and 3 in Methods). SS is designed to find gene expression patterns with overall similarity to the query image, whereas SC is for finding images that contain as much of the query image expression pattern as possible without penalizing for the presence of any expression outside the overlap region in the target image. For a given pair of gene expression patterns (A and B), SS is the same irrespective of which image in the pair is the query image. That is, SS = SS . This is not so for SC, because SC measures how much of the query gene expression pattern is contained in the image. Therefore, SC \u2260 SC.To quantify the similarity of gene expressions in two images, we computed two measures . Results from D\u03c6 should be compared to that from SS, as both of these measurements do not depend on the reference image, i.e., D\u03c6 = D\u03c6 and, also they capture overall similarity or dissimilarity.For IMV representation, we computed one dissimilarity measure gene in a wildtype embryo. The BESTi-matches based on the SS measure for the representations are given in Figure sloppy paired genes (slp1 and slp2) in a variety of genetic backgrounds or in combination with a head gap gene orthodentical (otd); all of these genes are essential for the pattern formation in Drosophila head development or m or mcis-Separating multi-domain gene expression patterns into individual components was straightforward; we simply generated multiple images from the same initial image and included them in the target dataset. This resulted in 192 additional images in the database all of which were components of the initial gene expression patterns. The images were separated into expression regions horizontally and/or vertically depending on the gene expression. For this new set of images, the IMV as well as BFV representations were re-calculated and the BESTi query constructed as above.SS and IMV queries for this data set are given in Figures tailless (tll), which is known to interact with slp1 in defining the embryonic head [race (related to angiotensin converting enzyme), sog (short gastrulation) and eve (even-skipped) due to enhanced race expression in the anterior domain caused by a transgenic construct causing ectopic expression of sog [Results from BFV-nic head , and witn of sog . TherefoSS, SC and D\u03c6 in finding BESTi matches for a query pattern with multiple regions of expression mutant background, and a middle stripe due to misexpressed sog using an eve stripe-2 enhancer [Figure SS finds many images from the same paper as well as some images from other research articles with similar expression patterns. The results correctly include expression pattern of eve value calculated does not show a significant difference in the early stage embryos used in this study. The results using the SC based search are given in Figure SS results. However, as expected, there are significant differences between the two searches.When t Figure . HoweverSS as the search criterion. These searches are based on the complete expression . This is referred to as the Binary Sequence Vector (BSV) in . In otheDE, between the binary feature vectors of every possible pair of images in the dataset. DE was introduced in [The expression patterns are ordered by evaluating a set of difference values, duced in and is fDE = Count(A XOR B)/Count(A OR B) \u00a0\u00a0\u00a0 (1)Count(A XOR B) corresponds to the number of pixels not spatially common to the two images and the term Count(A OR B) provides the normalizing factor, as it refers to the total number of stained pixels (expression area) depicted in either of the two images being compared. For simplicity, we use the one's complement of DE, as a measure of similarity of gene expression patterns between two images, SS, is given by the equationThe term SS = (1 - DE). \u00a0\u00a0\u00a0 (2)SS quantifies the amount of similarity based on the overlap between two expression patterns. SS is equal to 1 when the two expression patterns are identical (DE = 0).SC quantifies the amount of similarity based on the containment of one expression pattern in the other given byWe introduce a new similarity measure in this paper that does not penalize for any non-overlapping region. The measure SC = Count(A AND B)/Count (A) \u00a0\u00a0\u00a0 (3)i.e., there is complete overlap (with respect to the query image) SC is equal to 1. Note that, SC \u2260 SC, because the denominator corresponds to the gene expression area of the query image.If the entire query image is contained within the result set images found in the database, Some methodologies of image analysis produce numeric descriptors that compensate for variations of scale, translation and rotation. In the following section, we describe the invariant moment analysis of gene expression data. Invariant moment calculations have been used in optical character recognition and other applications for many years .To calculate these invariant moment descriptors the standardized binary image is conveMpq = \u222cxp yq fdxdy, \u00a0\u00a0\u00a0 (4)Mpq is the two-dimensional moment of the function of the gene expression pattern, f. The order of the moment is defined as (p + q), where both p and q are positive natural numbers. When implemented in a digital or discrete form this equation becomeswhere and which are the coordinates of the center of gravity, centroid, of the area showing expression. They are calculated asWe then normalize for image translation using Discrete representations of the central moments are then defined as follows:A further normalization for variations in scale can be implemented using the formula, is the normalization factor. From the central moments, the following values are calculated:and \u03c67 is a skew invariant to distinguish mirror images. In the above, \u03c61 and \u03c62 are second order moments and \u03c63 through \u03c67 are third order moments. \u03c61 (the sum of the second order moments) may be thought of as the \"spread\" of the gene expression pattern; whereas the square root of \u03c62 (the difference of the second order moments) may be interpreted as the \"slenderness\" of the pattern. Moments \u03c63 through \u03c67 do not have any direct physical meaning, but include the spatial frequencies and ranges of the image.where f is the angular displacement of the minimum rotational inertia line that passes through the centroid and is given as:In order to provide a discriminator for image inversion (and rotation), sometimes called the \"6\", \"9\" problem, it has been suggested ,42 that \u03b8. It is calculated knowing that the moment of inertia of f around the line is a line through with slope \u03b8. We can find the \u03b8 value at which the momentum is minimum by differentiating this equation with respect to \u03b8 and setting the results equal to zero. This produces the following equation:The slope of the principal axis is called the principal angle \u03b8| < 45\u00b0 one can distinguish the \"6\" from the \"9\" and rotationally similar gene expression patterns.Using the condition |\u03c61 through \u03c67) and combinations of these moments. For example, if the first two invariant moments are used, thenIn invariant moment analysis, our initial method of image comparison calculates the Euclidean distance between the images using all moments (Dij, between a pair of images i and j where i, j = 1, 2,...n is given byand the distance D\u03c6, between any two images is calculated asThis can be expanded to use all of the moment variables. Here, the Euclidean distance, i and q designate images whose distance is being calculated and j designates the parameters used in the distance calculation and j = 1, 2, ..., 7. This assumes that all moments have the same dimensions or that they are dimensionless.where \u03c61 has the dimension of distance squared, while \u03c62 has the dimension of the fourth power of distance, thus requiring the square root function to equalize dimensions for comparable distance calculation purposes. In general, the greater number of invariant moments used in the distance calculation, the more selective the ranking. We have also allowed for the use of the centroids and principal angle as a means of list limiting.Using this method, it is possible to rank each of the images in order of their similarity based on, for example, the first two invariant moments that have clear-cut physical meanings. Expansion to include additional moments or parameters can be performed in a number of ways. It is possible to add additional parameters to the distance calculation making sure that each of the parameters has the same dimension. For example, SK originally conceived the project, developed the image distance measures based on the BFV representation, wrote an early version of the manuscript, and edited it until the final version. RG was responsible for writing new and using pre-existing programs to perform the image distance and parameter calculations, helped prepare the figures, searched the literature for gene expression data, maintained the database of gene expression pattern images, and helped in writing the manuscript. BVE provided the IMV method description, managed the day-to-day activities in the project, and did significant editing to produce the manuscript in the desired format for the journal. SP originally proposed the use of invariant moment vectors for biological image analysis, contributed significantly for the image distance and parameter calculations and provided critical feedback during the later stages of revision."} +{"text": "Dense time series of metabolite concentrations or of the expression patterns of proteins may be available in the near future as a result of the rapid development of novel, high-throughput experimental techniques. Such time series implicitly contain valuable information about the connectivity and regulatory structure of the underlying metabolic or proteomic networks. The extraction of this information is a challenging task because it usually requires nonlinear estimation methods that involve iterative search algorithms. Priming these algorithms with high-quality initial guesses can greatly accelerate the search process. In this article, we propose to obtain such guesses by preprocessing the temporal profile data and fitting them preliminarily by multivariate linear regression.The results of a small-scale analysis indicate that the regression coefficients reflect the connectivity of the network quite well. Using the mathematical modeling framework of Biochemical Systems Theory (BST), we also show that the regression coefficients may be translated into constraints on the parameter values of the nonlinear BST model, thereby reducing the parameter search space considerably.The proposed method provides a good approach for obtaining a preliminary network structure from dense time series. This will be more valuable as the systems become larger, because preprocessing and effective priming can significantly limit the search space of parameters defining the network connectivity, thereby facilitating the nonlinear estimation task. The intriguing aspect of profiles is that they implicitly contain information about the dynamics and regulation of the pathway or network from which the data were obtained. The challenge for the mathematical modeler is thus to develop methods that extract this information and lead to insights about the underlying pathway or network.The rapid development of experimental tools like nuclear magnetic resonance (NMR), mass spectrometry (MS), tissue array analysis, phosphorylation of protein kinases, and fluorescence labeling combined with autoradiography on two-dimensional gels promises unprecedented, powerful strategies for the identification of the structure of metabolic and proteomic networks. What is common to these techniques is that they allow simultaneous measurements of multiple metabolites or proteins. At present, these types of measurements are in their infancy and typically limited to snapshots of many metabolites at one time point , to swith NMR ,4, 2-d gwith NMR or protewith NMR ), or to with NMR that peret al. [i.e., the maximum deviation from steady state) as well as the slopes of the traces at the initial phase of the response. Torralba et al. [in vitro glycolytic system. Similarly, by studying a large number of perturbations, Samoilov et al. [In simple cases, the extraction of information can be accomplished to some degree by direct observation and interpretation of the shape of profiles. For instance, assuming a pulse perturbation from a stable steady state, Vance et al. present a et al. recentlyv et al. showed tFor larger and more complex systems, simple inspection of peaks and initial slopes is not feasible. Instead, the extraction of information from profiles requires two components. One is of a mathematical nature and consists of the need for a model structure that is believed to have the capability of capturing the dynamics of the underlying network structure with sufficient accuracy. The second is computational and consists of fitting this model to the observed data. Given these two components along with profile data, the inference of a network is in principle a regression problem, where the aim is minimization of the distance between the model and the data. If a linear model is deemed appropriate for the given data, this process is indeed trivial, because it simply requires multivariate linear regression, which is straightforward even in high-dimensional cases. However, linear models are seldom valid as representations of biological data, and the alternative of a nonlinear model poses several taxing challenges.canonical forms, which are nonlinear structures that conceptually resemble the unalterable linear systems models, but are nonlinear. Canonical models have in common that they always have the same mathematical structure, no matter what the application area is. They also have a number of desirable features, which include the ability to capture a wide variety of behaviors, minimal requirements for a priori information, clearly defined relationships between network characteristics and parameters, and greatly enhanced facility for customized analysis.First, in contrast to linear models, there are infinite possibilities for nonlinear model structures. In specific cases, the subject area from which the data were obtained may suggest particular models, such as a logistic function for bacterial growth, but in a generic sense there are hardly any guidelines that would help with model selection. One strategy for tackling this problem is the use of The best-known examples of nonlinear canonical forms are Lotka-Volterra models , their The strict focus on two-component interactions in LV models has substantial mathematical advantages, but it has proven less convenient for the representation of metabolic pathways, where individual reaction steps depend on the substrate, but not necessarily on the product of the reaction, or are affected by more than two variables. A simple example of the latter is a bi-substrate reaction that also depends on enzyme activity, a co-factor and possibly on inhibition or modulation by some other metabolite in the system. These types of processes have been modeled very successfully with GMA and S-systems. Between these two forms, the S-system representation has unique advantages for system identification from profiles, as was shown elsewhere -24 and wcf. [The inference of a nonlinear model structure from experimental data is in principle a straightforward \"inverse problem\" that should be solvable with a regression method that minimizes the residual error between model and data. In practice, however, this process is everything but trivial (cf. ) as it acf. , similarcf. , mutual cf. , and gencf. . An indicf. . The alget al. [Nonlinear estimation methods have been studied for a long time, and while computational and algorithmic efficiency will continue to increase, the combinatorial explosion of the number of parameters in systems with increasingly more variables mandates that identification tasks be made easier if larger systems are to be identified. One important possibility, which we pursue here, is to prime the iterative search with high-quality starting conditions that are better than na\u00efve defaults. Clearly, if it is possible to identify parameter guesses that are relatively close to the true, yet unknown solution, the algorithm is less likely to get trapped in suboptimal local minima. We are proposing here to obtain such initial guesses by preprocessing the temporal profile data and fitting them preliminarily by straightforward multivariate linear regression. The underlying assumption is that the structural and regulatory connectivity of the network will be reflected, at least qualitatively, in the regression coefficients. D'haeseleer et al. exploredet al. [et al. [e.g., [et al. [Several other groups have recently begun to target network identification tasks with rather diverse strategies. Chevalier et al. and Diazet al. ,34 propo [et al. recently. [e.g., ,36) that. [e.g., used a n [et al. used neun species can often be represented by a system of nonlinear differential equations of the generic formThe behavior of a biochemical network with X is a vector of variables Xi, i = 1, ..., n, f is a vector of nonlinear functions fi, and \u03bc is a set of parameters. If the mathematical structure of the functions fi is known, the identification of the network consists of the numerical estimation of \u03bc. In addition to the challenges associated with nonlinear searches mentioned above, this estimation requires numerical integration of the differential equations in (1) at every step of the search. This is a costly process, requiring in excess of 95% of the total search time; if the differential equations are stiff, this percentage approaches 100% [n differential equations, and if measurements are available at N time points, the decoupling leads to n \u00d7 N algebraic equations of the formwhere hes 100% . A simplhes 100% ,17. Thus\u03bcij are the \"unknowns\" that need to be identified.It may be surprising at first that it is valid to decouple the tightly coupled system of nonlinear differential equations. Indeed, this is only justified for the purpose of parameter estimation, where the decoupled algebraic equations simply provide numerical values of variables (metabolites or proteins) and slopes at a finite set of discrete time points. The experimental measurements thus serve as the \"data points,\" while the parameters a priori not trivial. However, we have recently shown [The quality of this decoupling approach is largely dependent on an efficient and accurate estimation of slopes from the data. Since the data must be expected to contain noise, this estimation is ly shown ,39 that f in Eq. (1) about one or several reference states. As long as the system stays close to the given reference state(s), this linearization is a suitable and valid approximation. We consider four options: (I) linearization of absolute deviations from steady state; (II) linearization of relative deviations from steady state; (III) piecewise linearization; and (IV) Lotka-Volterra linearization.The smoothing and decoupling approach reduces the cost of finding a numerical solution of the estimation task considerably. Nonetheless, algorithmic issues associated with local minima and the lack of convergence persist and can only be ameliorated with good initial guesses. To this end, we linearize the model zi = Xi - Xir, where Xir denotes the value at a reference state of choice. If the reference state is chosen at a stable steady state, the first-order Taylor-approximation is given byOption (I) is based on deviations of the type A is the n \u00d7 n Jacobian with elements aij = (dfi / dXj) calculated at Xr .where Xr cf. -34). If . If A isui = zi/Xir. At a steady state, this yields the linear systemFor option II, we define a new variable A' is an n \u00d7 n matrix in which a'ij = (Xjr / Xir)\u00b7aij.where f, a perturbation of this magnitude may already lead to appreciable approximation errors. While this is a valid argument, it must be kept in mind that the purpose of this priming step is simply to detect the topological structure of connectivity and not necessarily to estimate precise values of interaction parameters. Simulations (see below) seem to indicate that this detection is indeed feasible in many cases, even if the deviations are relatively large.A general concern regarding linearization procedures is the range of validity of sufficiently accurate representation, which is impossible to define generically. From an experimental point of view, the perturbations from steady state must be large enough to yield measurable responses. This may require that they be at the order of 10% or more. Depending on the nonlinearities in ai0, which is equal to fi(Xr). The choice of subsets and operating points offers further options. In the analysis below, we use the locations of extreme values (maximum deviation from steady state) of the variables as the breakpoints between different subsets. Thus, a variable with a maximum and a later minimum has its time course divided into three subsets.In order to overcome the limitation of small perturbations, a piecewise linear regression (option III) may be a superior alternative. In this case, we subdivide the dataset into appropriate time intervals and linearize the system around a chosen state within each subset. Most reference states are now different from the steady state, with the consequence that Eq. (3) has a constant term Xi and Xj is assumed to be proportional to the product XiXj [Xi isThe fourth alternative (option IV) is a Lotka-Volterra linearization. In a Lotka-Volterra model, the interaction between two species uct XiXj . FurtherXi, which is usually valid in biochemical and proteomic systems, because all quantities of interest are non-zero. Thus, the differentials are again replaced by estimated slopes, the slopes are divided by the corresponding variable at each time point, and fitting the nonlinear LV model to the time profiles becomes a matter of linear regression that does not even require the choice of a reference state. The quality of this procedure is thus solely dependent on the quality of the data and ability of the LV model to capture the dynamics of the observed network. It is known that thatXi, i.e., vij's and wi's, or \u03b1ij's). The response variable is the rate of change of a metabolite, while the predictors are the concentrations of each metabolite in the network. The different linearization models (I-IV) differ in the transformations of the original datasets, which are summarized in Table yi = /Xir, and the predictor variables are transformed as xi = (Xi - Xir)/Xir.No matter which option is chosen, the next step of the analysis consists of subjecting all measured time traces to multivariate linear regression and solving for the regression coefficients (Xj affects the dynamics (slope) of another metabolite Xi. In particular, a coefficient that is zero or close to zero signals that there is no significant effect of Xj on the slope of Xi. By the same token, a coefficient that is significantly different from zero suggests the presence of an effect, and its value tends to reflect the strength and direction of the interaction. In either case, the coefficients computed from the linear regression provide valuable insight into the connectivity of the network. Furthermore, the estimated coefficients provide constraints on the parameter values of the desired nonlinear model f. Indeed, if f consists of an S-system model, the coefficients estimated from the regression can be converted into combinations of S-system parameters, as is demonstrated in the following theoretical section and illustrated later with a specific example.The result of the regression is a matrix of coefficients that indicate to what degree a metabolite f in Eq. (1) if this model has the form of an S-system. To determine the relationships between the regression coefficients and the parameters of the S-system, it is convenient to work backwards by computing the different types of linearizations discussed before for the particular case of S-system models. This derivation is simply a matter of applying Taylor's theorem.The regression analysis yields coefficients that offer information on the connectivity of the network of interest. It also provides clues about the parameter values of the underlying nonlinear network model In the S-system formalism, the rate of change in each pool (variable) is represented as the difference between influx into the pool and efflux out of the pool. Each term is approximated by a product of power-law functions, so that the generic form of any S-system model isn is the number of state variables [gij and hij are called kinetic orders and describe the quantitative effect of Xj on the production or degradation of Xi, respectively. A kinetic order of zero implies that the corresponding variable Xj does not have an effect on Xi. If the kinetic order is positive, the effect is activating or augmenting, and if it is negative, the effect is inhibiting. The multipliers \u03b1 i and \u03b2 i are rate constants that quantify the turnover rate of the production or degradation, respectively.where ariables ,14. The zi = Xi - Xis, where the subscript s denotes the value of the variable at steady state, then leads directly toIf the Taylor linearization is performed at a steady state, the production term of the S-system model equals the degradation term. The absolute deviation of the first option, wherecij = gij - hij,cf. [Fij are always non-negative, while cij may be either positive or negative depending on the relationship between Xi and Xj. A common scenario is that a variable Xj influences either the production or degradation of variable Xi, but not both. In this case, a positive (negative) cij implies activation (inhibition) of production or inhibition (activation) of degradation. The special case of cij = 0 permits two possible interpretations: 1) gij = hij = 0, which implies that Xj has no effect on either production or degradation of Xi; or 2) gij = hij \u2260 0, which means that Xj has the same effect on both production and degradation of Xi. The former case is the more likely, but there are examples where the latter may be true as well, and this is indeed the case in the small gene network in Figure (cf. ). The soaij in Eq. (3) corresponds to the product of Fij and cij:Comparing the expression in Eq. (6) with the linear regression results, one sees immediately that each coefficient aij = Fijcij. \u00a0\u00a0\u00a0 (7)aij have been estimated, the parameters of the corresponding S-system are constrained \u2013 though not fully determined \u2013 by Eq. (7). In particular, Eq. (7) does not allow a distinction between various combinations of gij and hij, as long as the two have the same difference. For instance, re-interpreting the regression coefficients as S-system parameters does not differentiate between the overall absence of effect of Xj on Xi (gij = hij = 0) and the same effect of Xj on both the production and degradation of Xi (gij = hij \u2260 0). This observation is related to the observation of Sorribas and Cascante [Thus, once the regression has been performed and the coefficients Cascante that steui = (Xi - Xis) / Xis, in option II, are assessed in an analogous fashion. In this case one obtainsRelative deviations from steady state, wherecij = gij - hij,Fi are positive, while cij may be either positive or negative.. Again, The piecewise linear model for an S-system is easily derived as well. It is given asXjr denotes the value of the variable at the reference state. This case also includes the situation of a single approximation, which however is not necessarily based on a steady-state operating point.where Xi and then linearizing around an operating point. The resulting expressions become especially simple if this point is chosen as the steady state. In this case, the relationship between the parameters of the LV system and the S-system areIn the case of the Lotka-Volterra linearization, the correspondence between computed regression coefficients and S-system parameters is determined most easily by dividing the S-system equations by the corresponding cij = gij - hij.where et al. [We applied the methods described in the previous sections to simulated time profiles obtained from the small gene network in Figure et al. used it et al. ,39.et al. [To generate time profiles, the system was implemented with the parameter values published by Hlavacek and Savageau , and as et al. , the modet al. [i.e., the maximum deviations from steady state) of the various variables both in time and size is in accordance with their \"topological distance\" from the perturbed variable, and variables not directly affected by the perturbed variable have zero initial slopes. As an example, the effect of a perturbation in X3 is shown in Figure X1 and X4 reaching their maximal deviation from steady state before X2 and X5, suggesting that X1 and X4 precede X2 and X5 in the pathway. The value of the initial slope is different from zero for X1 and X4, implying that these variables are directly affected by X3, whereas X2 and X5 have zero initial slopes suggesting that their responses are mediated through other variables.Quasi as a pre-analysis, we examined the guidelines proposed by Vance et al. . Indeed,X2 on X3. This result is not surprising, because the effect of X2 is the same on both the production and degradation of X3, which leads to cancellation. It is noted that this analysis does not necessarily distinguish between transfer of mass and a positive modulation, because both result in a positive effect on a variable. In a realistic situation, biological knowledge may exclude one of the two options, as in this case, where modulation is the only possibility for the effect of X3 on both X1 and X4, because the former is a protein and the latter are RNA transcripts. For the mathematical model in the S-system form, this is not an issue, as both types of influence are included in the equations in the same way (as a positive kinetic order).Maximal information about the network is obtained when every variable is perturbed sequentially. Experimentally, such perturbations could be implemented with modern methods of RNA interference or, for While Vance's method works well in this simple noise-free system, it is not scalable to larger and more complex systems. The next step of our analysis is therefore regression according to the four options presented above and with a number of simulated datasets of the gene network that differ in the variable to be perturbed and the size of the perturbation. Because the illustration here uses a known model and artificial data, it is easy to compute the true regression coefficients through differentiation of the S-system model. These coefficients can be used as a reference for comparisons with coefficients computed from the entire time traces, which mimics the estimation process for (smoothed) actual data.The results for three of the options can be summarized in the following three points, while the piecewise linear model will be discussed afterwards.The network connectivity is reflected in the values of the regression coefficients. The values of the estimated coefficients provide strong indication as to which variables have a significant influence on the dynamics of other variables. A comparison between computed and estimated coefficients is shown in Table a12 and a24) are not estimated as exactly zero, but their values are at least one order of magnitude smaller than the coefficients that are in actuality not zero. Table X3 and X4. A possible explanation for X3 is that the effect of X2 is present in the non-linear system, but not in the linear system, and thus the behavior of X3 must be explained by the other variables. Overall, of the 25 theoretically possible connections, 76% are correctly identified, while 24 % are false positives.(1) The different linear models give the same results. A comparison of the results of the three models reveals that the values of the regression coefficients are very similar The greater the perturbation, the less accurate is the estimation of the regression coefficients. The deviation between the estimated and computed coefficients increases as the size of the perturbation increases (see Table (3) t = 0 to the time of the first extreme value for a given variable . For the perturbed variable the first limit point was given by the smallest of the limit points of the other variables. The second interval contained the data points from the first to the second extreme value (a minimum), while the third interval included the remaining data points. The midpoint of each interval was taken to be the reference state. The result of the piecewise linear regression for a 5% deviation in X3 is given in Table X3 in the two last subsets reflect the variable's connectivity to a much greater extent than the other linearization approaches. As the reference state is different from the steady state, the effect of X2 is present in the linear system as well, and thus there is no compensation through the other variables. Another benefit is that the piecewise model tolerates larger perturbations. Even for a two-fold perturbation, the fraction of correctly identified coefficients in the last subset is 84%.The piecewise linear model was obtained by dividing the whole dataset into three smaller subsets for each variable. The first interval contained the data points from X3 is identified correctly from comparing the six models. The classification of the remaining four connections varies greatly among the different models, and it is therefore impossible to deduce a type of interaction with sufficient reliability.If we compare the results of all four linearized models, the degree of similarity may provide a measure of how reliable the estimated coefficients are, assuming that an interaction identified in all models is more reliable than an interaction identified in only one or few of the models. Considering the piecewise linear model as three models, yielding a total of 6 models from one dataset, one may thus determine the most likely connectivity for the small gene network. The result is presented in Table X1. Table X1, X3 and X5. If so, the linear model in Eq. (8) suggests the following:In addition to reflecting the connectivity, the coefficients provide likely parameter ranges or likely constraints on parameter values of the true model. As an example, consider variable and the regression coefficients (aij) are taken from the model in Eq. (4). The values of the variables at steady state are known. Because the kinetic orders may be positive or negative and the cij may result from different combinations of gij's and hij's, it is not possible to deduce directly which exponent is greater than the other. However, in many cases one may have additional information on the system, which further limits the degrees of freedom . In addIdentifying the structure of metabolic or proteomic networks from time series is a task that most likely will require large, parallelized computational effort. The search space for the algorithms is typically of high dimension and unknown structure and very often contains numerous local minima. This generic and frequent problem may be ameliorated if the search algorithm is provided with good initial guesses and/or constraints on admissible parameter values. Here, we have shown that linear regression may provide such information directly from the types of data to be expected from future experiments. For illustrative purposes, we used artificial data from a known network, but all methods are directly applicable to actual profile data and scaleable to large systems.The coefficients estimated from the different regressions reflect the effect of one variable on another surprisingly well and thus provide a simple fashion of prescreening the connectivity of the network. In addition, the estimated coefficients provide constraints on the parameter values, if the alleged nonlinear model has the form of an S-system. To explore the pre-assessment of data as fully as feasible, we studied four linearization strategies: using an absolute deviation from steady state; a relative deviation from steady state; piecewise linearization; and Lotka-Volterra linearization. Interestingly, all models gave qualitatively similar results for the analyzed example, and this degree of similarity may provide a measure of how reliable the identified connections are. Specifically, of the 25 possible connections in the small gene network studied, 19 were identified correctly in at least 83 % of the regression analyses.A concern of any linearization approach is the validity of the linear approximation. However, as long as the perturbation from steady state remains relatively small, the estimated linear model is likely to be a good fit of the actual nonlinear model, at least qualitatively. This limitation may furthermore be alleviated by fitting the profile data in a piecewise linear fashion. As most reference states in this case are different from the steady state, this strategy has the added benefit that more of the true relationships within the nonlinear model are likely to be preserved. As an alternative, one could explore the performance of the so-called \"log-linear\" model, which is linear in log-transformed variables .The Lotka-Volterra linearization did not perform as well as expected with regard to large perturbations. This may be a consequence of the particular example, which was originally in S-system form rather than in a form more conducive to the LV structure, which emphasizes interactions between pairs of variables. Since it is easy to perform the LV analysis along with the other regressions discussed here, it may be advisable to execute all four analyses.The illustrative model used for testing the procedure consisted of a relatively small system with only five variables and relatively few interactions. Nonetheless, one should recall that this very system required substantial identification time in a direct estimation approach . In ordeNone declared.SRV performed the analysis and prepared the results. JS developed and implemented the neural network for computation of slopes. EOV developed the basic ideas and directed the project.universal function which smoothes the data with predetermined precision and also allows the straightforward computation of slopes that can be used for network identification purposes. This appendix briefly outlines the procedure; details can be found in Almeida [It was recently shown that good parameter estimates of S-system models from metabolic profiles might be obtained by training an artificial neural network (ANN) directly with the experimental data. The result of this training is a so-called Almeida and Voit Almeida . The ANN Almeida .Noise and sample size do not have a devastating effect on the results of the ANN-method, as long as the true trend is well represented . In factThe use of the entire time course is in stark contrast to earlier methods of parameter estimation and structure identification in metabolic networks. Mendes and Kell applied et al. [Chevalier and co-workers first fiet al. suggestez(tk + 1) = z(tk)exp(h\u00b7A), \u00a0\u00a0\u00a0 (A1)h is the step size. The problem is thereby reduced to a mulitilinear regression in which the matrix \u03a6 = exp(h\u00b7A) is the output. Instead of estimating the slopes, they obtain the Jacobian directly by expanded in its Taylor-series. This approach yields a faster convergence to the elements of the Jacobian than the one suggested by Chevalier et al. [where r et al. , but theOur approach takes advantage of the entire time course and is therefore less sensitive to the particularities of assessing a system at a single point. The ANN itself does not provide much insight, because it is strictly a black-box model, but it is a valuable tool for controlling problems that are germane to any data analysis, namely noise, measurement inaccuracies, and missing data."} +{"text": "Cluster analyses are used to analyze microarray time-course data for gene discovery and pattern recognition. However, in general, these methods do not take advantage of the fact that time is a continuous variable, and existing clustering methods often group biologically unrelated genes together.We propose a quadratic regression method for identification of differentially expressed genes and classification of genes based on their temporal expression profiles for non-cyclic short time-course microarray data. This method treats time as a continuous variable, therefore preserves actual time information. We applied this method to a microarray time-course study of gene expression at short time intervals following deafferentation of olfactory receptor neurons. Nine regression patterns have been identified and shown to fit gene expression profiles better than k-means clusters. EASE analysis identified over-represented functional groups in each regression pattern and each k-means cluster, which further demonstrated that the regression method provided more biologically meaningful classifications of gene expression profiles than the k-means clustering method. Comparison with Peddada et al.'s order-restricted inference method showed that our method provides a different perspective on the temporal gene profiles. Reliability study indicates that regression patterns have the highest reliabilities.Our results demonstrate that the proposed quadratic regression method improves gene discovery and pattern recognition for non-cyclic short time-course microarray data. With a freely accessible Excel macro, investigators can readily apply this method to their microarray data. Microarray time-course experiments allow researchers to explore the temporal expression profiles for thousands of genes simultaneously. The premise for pattern analysis is that genes sharing similar expression profiles might be functionally related or co-regulated . Due to In microarray time-course studies, time dependency of gene expression levels is usually of primary interest. Since time can affect the gene expression levels, it is important to preserve time information in time-course data analysis. However, most methods for analyzing microarray time-course data treat time as a nominal variable rather than a continuous variable, and thus ignore the actual times at which these points were sampled. Peddada et al. (2003) proposed a method for gene selection and clustering using order-restricted inference, which preserves the ordering of time but treats time as nominal . Recentl[In this paper, we propose a model-based approach, step down quadratic regression, for gene identification and pattern recognition in non-cyclic short time-course microarray data. This approach takes into account time information because time is treated as a continuous variable. It is performed by initially fitting a quadratic regression model to each gene; a linear regression model will be fit to the gene if the quadratic term is determined to have no statistically significant relationship with time. Significance of gene differential expression and classification of gene expression patterns can be determined based on relevant F-statistics and least squares estimates. Major advantages of our approach are that it not only preserves the ordering of time but also utilizes the actual times at which they were sampled; it identifies differentially expressed genes and classifies these genes based on their temporal expression profiles; and the temporal expression patterns discovered are readily understandable and biologically meaningful. A free Excel macro for applying this method is available at . The pro. Biologi showed t and regrth gene:We propose a step-down quadratic regression method for gene discovery and pattern recognition for non-cyclic short time-course microarray experiment. The first step is to fit the following quadratic regression model to the jyij = \u03b2j 0+ \u03b2j1x + \u03b2j2x2 + \u03b5ij \u00a0\u00a0\u00a0 (1)yij denotes the expression of the jth gene at the ith replication, x denotes time, \u03b2j 0is the mean expression of the jth gene at x = 0, \u03b2j 1is the linear effect parameter of the jth gene, \u03b2j 2is the quadratic effect parameter of the jth gene, and, \u03b5ij is the random error associated with the expression of the jth gene at the ith replication and is assumed to be independently distributed normal with mean 0 and variance . Two levels of significance, \u03b10 and \u03b11, need to be pre-specified, where \u03b10 to is recommended to be small to reduce the false positive rate in the gene discovery and \u03b11 less stringent to control pattern classification. \u03b10 could be chosen using various multiple testing p-value adjustment procedures, for example, False Discovery Rate (FDR) [where te (FDR) . The tem\u03b10, the jth gene is considered to have no significant differential expression over time. The expression pattern of the gene is \"flat\".1. If overall model (1) p-value >\u03b10, the jth gene will be considered to have significant differential expression over time. The patterns are then determined based on the p-values obtained from F tests p-value \u2264 p-value \u03b11 and p-value of linear effect \u2264 \u03b11, the jth gene is considered to be significant in both the quadratic and linear terms. The expression pattern of the gene is \"quadratic-linear\".a. If both p-value of quadratic effect \u2264 \u03b11 and p-value of linear effect >\u03b11, the jth gene is considered to be significant only in the quadratic term. The expression pattern of the gene is \"quadratic\".b. If p-value of quadratic effect \u2264 >\u03b11, the jth gene is considered to be non-significant in the quadratic term. The quadratic term will be dropped and a linear regression model will be fitted to the gene:c. If p-value of quadratic effect yij = \u03b2j 0+ \u03b2j1x + \u03b5ij \u00a0\u00a0\u00a0 (2)From fitting model (2),\u03b11, the jth gene is considered to be significant in the linear term. The expression pattern of the gene is \"linear\".\u2022 If p-value of linear effect \u2264 >\u03b11, the jth gene is considered to be non-significant in the linear term. The expression pattern of the gene is \"flat\".\u2022 If p-value of linear effect and and the predicted signals , increases faster then slower , or increases slower then faster . Peddada et al.'s UD2 profile contains genes that are first up-regulated then down-regulated with maximum at the second time points, which could be classified as regression pattern QLCD in general , but it could also be classified as LD if the expression levels of all time points are close to a line ; or classified as QC if the expression profile is close to quadratic ; or classified as QLCU if the expression levels of last 4 time points are much closer than those of the first time point. Similarly, Peddada et al.'s UD3 profile could be classified as regression patterns QC, QLCU, and QLCD .Peddada et al.'s method was appll Figure , Oazin, e Figure , Grik5; c Figure , Ubl1; oANOVA-protected k-means clustering was applied to the expression signals of 3834 present genes. Out of 3834 present genes, 770 were identified to be differentially expressed over time by one way ANOVA . These 770 genes were used for classification by k-means clustering with k = 9 and the distance measure being Pearson correlation coefficient . The similarity of the temporal expression profiles in Figure Clu; and D17H6S56E-5); similarly see Figure Sfpi1; and b, Anxa2). Once again, the regression method provides better classification. Figure Psmb6; Adora2b). Adora2b clearly starts differential expression later than Psmb6 , these two genes show similar upward regulation. These two genes were classified into the same regression group, but in different k-means clusters. Based on the above analysis, our regression method is demonstrated to be more appropriate for the classification of temporal gene expression profiles than k-means method.In order to make the regression patterns comparable with the k-means clusters, the quadratic regression method was applied to the 770 ANOVA significant genes. Table To further explore the effectiveness of the regression method on gene classification, EASE software was used to examine the potential relationship between the biological functions of the genes and their expression patterns . EASE caKerr and Churchill (2001) introduced a bootstrap technique to assess the stability of clustering results . We appl\u03b10 = 0.01 and the numbers of significant genes in each of the 50 data were obtained. The average proportion of significance is 1.01% with standard deviation 0.01%. This demonstrates that the false positive rate of the regression method is accurate because 1% of 10000 genes would be expected to be significant at 0.01 level by chance. The false positive rates of the regression patterns LU, LD, QC, QV are all approximately equal to 1/6 of the average false positives, and those of QLCU, QLCD, QLVU, and QLVD are all approximately equal to 1/12 of the average false positives.We investigated the false positive rate (gene specific) of our method via a simulation study. The data were generated randomly from N, containing expression signals of 10000 \"null\" genes , with 5 time points and 3 replications per time point per gene. 50 of such data were generated. The regression approach was applied to each gene in each simulated data at th-order polynomial regression model to this data . The model with 4th-order polynomial will work similarly to connecting the mean at each time point, therefore will provide a good fit to the data with smallest R2 and minimum Mean Squared Error, compared with lower-order polynomials. However, the purpose of pattern analysis is to cluster the data instead of fitting models, so the quadratic fit is useful even though the goodness of fit may not be great. Also, the use of high-order polynomials (higher than the second-order) should be avoided if possible [The proposed step-down quadratic regression method is an effective statistical approach for gene discovery and pattern recognition. It utilizes the actual time information, and provides biologically meaningful classification of temporal gene expression profiles. Furthermore, it does not require replication at each time point, which ANOVA-type methods do require. Also, this method can identify genes with subtle changes over time and therefore discover genes that might be undetectable by other methods, eg, ANOVA-type methods. However, there are several limitations to this method. Firstly, it is designed to fit time-course data with a small number of time points. We recommend this method when there are 4 to 10 time points in the data. For an experiment with more time points, spline-type methods ,12 couldpossible , particu\u03b10 to increase the stability of regression patterns. \u03b10 could be reduced using various multiple testing p-value adjustment procedures, for example, Westfall and Young's step down method [\u03b1): let p(1)