Reconstructing Generalized Logical Networks of Transcriptional Regulation in Mouse Brain from Temporal Gene Expression Data
© Mingzhou (Joe) Song et al. 2009
Received: 1 June 2008
Accepted: 12 December 2008
Published: 27 January 2009
Gene expression time course data can be used not only to detect differentially expressed genes but also to find temporal associations among genes. The problem of reconstructing generalized logical networks to account for temporal dependencies among genes and environmental stimuli from transcriptomic data is addressed. A network reconstruction algorithm was developed that uses statistical significance as a criterion for network selection to avoid false-positive interactions arising from pure chance. The multinomial hypothesis testing-based network reconstruction allows for explicit specification of the false-positive rate, unique from all extant network inference algorithms. The method is superior to dynamic Bayesian network modeling in a simulation study. Temporal gene expression data from the brains of alcohol-treated mice in an analysis of the molecular response to alcohol are used for modeling. Genes from major neuronal pathways are identified as putative components of the alcohol response mechanism. Nine of these genes have associations with alcohol reported in literature. Several other potentially relevant genes, compatible with independent results from literature mining, may play a role in the response to alcohol. Additional, previously unknown gene interactions were discovered that, subject to biological verification, may offer new clues in the search for the elusive molecular mechanisms of alcoholism.
The regulation of transcription occurring in an intriguingly complex biological system involves multiple interacting regulatory processes in gene regulatory networks (GRNs). Modeling transcriptional regulation requires algorithms that retain information about regulatory interactions. The generalized logical network (GLN) is a generative model that can be reconstructed from temporal trajectories, for example, from data collected in time-series studies of gene expression. Because these data capture information on temporal antecedence, the approach can be used to develop stronger hypotheses about casual relations among transcriptional events than one would be able to derive from mere correlation analyses. We designed a GLN reconstruction algorithm that differs from previous approaches because it makes use of hypothesis testing on the multinomial distribution to establish directed connections among genes. Our statistical approach allows explicit control of false positives by specifying a desirable alpha level, while other criteria used in network reconstruction, such as the Bayesian information criterion (BIC) used in dynamic Bayesian networks (DBNs) reconstruction and the coefficient of determination (COD) used in Boolean networks (BNs) reconstruction, do not explicitly enforce false-positive rate control.
GLNs also allow more aspects of systems to be studied than other network models by enabling (1) adaptive description for interactions among variables, (2) nonlinear interaction patterns, and (3) finite steady states, attractor basins, and state transition diagrams. The software CellNetAnalyzer  allows a user to draft a GLN from existing knowledge. Our method allows such networks to be reconstructed and derived solely from data-driven approaches. GLNs have the further advantage that they do not require parametric assumptions, unlike stochastic logical networks  which discretize differential equations based on strong assumptions. Additionally, our implementation of GLN modeling focuses on network reconstruction from temporal gene expression data, which can be used complementarily with network property analysis algorithms such as the network walking algorithm , and literature mining tools such as those reviewed in .
Temporal probabilistic networks. The dynamic Bayesian network (DBN) is an extension of Bayesian networks, which incorporates time transitions between Bayesian networks. A DBN describes temporal statistical dependencies among genes. DBNs have been successful in extracting probabilistic dependencies among genes in GRNs [5–7]. Certain DBNs can even be converted to probabilistic Boolean networks . However, DBN is an indirect tool to understand system dynamics since it does not explicitly describe temporal relations among entities in a functional form, while a GLN provides immediate functional relationships among variables.
Continuous dynamical system models. Differential equations in both deterministic [9, 10] and stochastic  formulations have been used to model interactions in GRNs in continuous time. The E-Cell Project [12, 13] uses differential equations to target knowledge-based reproduction, not data-driven reconstruction, of intracellular biochemical and molecular interactions within a single cell. The stochastic master equations relate state probabilities by differential equations, impractical for biological systems involving many variables because of the computational burden. Recent research has been focusing on improving the scalability of such models .
Discrete dynamical system models. The Boolean network (BN) [1, 15–18] and its Markovian  or probabilistic  extensions, where each variable takes the value of either 0 or 1, are 1st-order special cases of the GLN. The dichotomous nature of a BN seriously limits its capacity to discriminate quantitative differences among continuous random variables. As most biological networks are rarely binary, much information is lost. This can be crucial when such differences are more interesting than the mere information of presence (1) or absence (0). In addition, the coefficient of determination criterion used in BN reconstruction does not address the issue of model complexity and goodness of fit.
To summarize, these temporal probabilistic networks do not explicitly describe system dynamics. Continuous dynamical system models, computationally and data intensive and thus often not data driven, are also inconvenient for visualizing state transitions. BNs cannot capture subtle and nonlinear interactions. Details of these and various other major network reconstruction and modeling algorithms can be found in recent reviews [21, 22].
Temporal dependency may reflect causal interactions among processes in a dynamical system, but not always. System modeling may be further complicated by incomplete observations—a situation that is typical for biological experiments. For example, protein concentrations, post-translational protein modification states, and small molecular messengers are missing in a GRN developed entirely from transcriptome data. However, a consistent temporal dependency must arise from a causal interaction, even with incomplete observations. Therefore, statistically significant temporal dependencies among genes and environmental stimuli may still constitute a basis to establish causalities.
We reconstruct GLNs from trajectories of discrete random variables, the abundance of mRNAs, in order to uncover temporal dependencies among genes and environmental stimuli. Temporal dependencies among key genes in response to alcohol in mice are assessed through GLN modeling. The effects of alcohol on functions of gene products and the corresponding effect on gene expression are an active research area, particularly in the inflammatory and neural plasticity processes that result in lasting brain changes in response to alcohol. We believe that the GLN approach will provide highly relevant clues to discover biologically important gene interactions involved in the molecular mechanisms of brain changes in alcoholism. The resulting network model demonstrates the tremendous potential for GLN modeling to provide insight into the diverse molecular mechanisms underlying clinical phenomena such as alcoholism.
The paper is organized into eight sections. The GLN is defined in Section 2. A procedure is given in Section 3 to determine the statistical power of reconstructing a GLN given an experimental design. An algorithm for reconstruction of GLNs based on multinomial testing is described in Section 4. Comparisons of reconstruction accuracy between GLN and DBN modeling are made in Section 5. A microarray experiment for the influence of alcohol on mouse brain gene expression is recounted in Section 6. The GLN modeling result of the GRN in the mouse brain in response to alcohol is discussed in Section 7. Finally, conclusions and future work are given in Section 8.
2. The Generalized Logical Network
As a discrete-time and discrete-value dynamical system model, a GLN of nodes is a directed graph with a gtt attached to each node. Each abstract node can represent information about a molecule, a cell, a species, or a stimulus. The gtt allows a discrete variable to take more than two possible values and to reflect subtle but crucial changes, and encodes precisely the biological mechanisms that the nodes use to interact with each other.
With parents, the size of is , exponential in and posing a memory problem. The generalized logical decision diagram is a space efficient data structure to store a gtt by removing fictitious variables and redundancies, extending the binary decision diagram .
The following is an example showing the gtt of of 3 levels with two parents of 2 and 3 levels, respectively.
Synchronous th order GLNs allow modeling of variable time delays abundant in biological systems. Let be the initial states of a GLN. A trajectory of length is defined as . Our discussion is restricted to synchronous and first-order GLNs.
3. Statistical Power for GLN Reconstruction
Given the number of time points on a trajectory and the sample size per time point, one is statistically limited in detecting true interactions in a GLN beyond a certain network complexity by the statistical power. The gtts, distributions of each variable, sample size (number of replicas and time points), Type I error, and effect size together determine the statistical power. Power is independent of the computational approach used to reconstruct a GLN from observed trajectories. With estimation of statistical power, one can answer the question of whether the amount of data in the trajectory can statistically support any GLN for certain complexity at all.
4. GLN Reconstruction through Multinomial Tests
A GLN can be reconstructed from observed trajectories of a system under perturbed conditions. There are two important issues in GLN reconstruction. The first one is how to search efficiently for the best among feasible GLN candidates. This issue depends on how one handles the combinatorial computational cost, generally NP-hard, incurred by reconstructing a GLN. The second issue is how to determine the false-positive rate that the best candidate arises out of randomness caused by noise and sampling errors in a network where no nodes interact, recently gaining attention such as in BN fitting . Various criteria for goodness of fit have been used in reconstruction of a GLN from observed trajectories. Mutual information among variables has been employed in interaction graphs ; likelihood and BIC are used to determine network structure for Bayesian networks  and DBNs; the coefficient of determination has been used for BNs . These measures, however, do not control the false-positive rate directly.
By performing multinomial tests on the transition tables at each node, we are able to resolve simultaneously both issues above in one framework. The network topology inference reduces to selecting the parents for each node through multiple applications of the same multinomial test. The false-positive control is achieved by setting an -level, which can be adjusted for multiple comparisons, for the tests at each node, instead of always keeping a parent selection with the best value of criterion as in all other approaches mentioned above. Our criterion is the statistical significance of each test. Thus, we move forward from existing network topology inference approaches by assessing the probability of false-positive interactions arising by chance in GLN reconstruction.
The transition table of node
A -value can be computed for to indicate the statistical significance of a GLN model. The -value provides a means to tradeoff between goodness of fit and complexity. Therefore, GLN reconstruction is to find a GLN with the minimum -value. Since the statistics for the transition tables at each node are independent of each other, minimization of the overall -value reduces to minimizing the -values for individual transition tables at each node.
where is the maximum quantization level of all nodes.
Algorithm 1: Reconstruct-GLN (A collection of observed trajectories, -level, ).
For each node do
For to do
For each possible selection of parents do
Accumulate a transition table from given trajectories
Compute -value by performing multinomial test on the transition table
if -value is smaller than the current minimum -value for the current node then
minimum -value -value
Record the current transition table
Replace previous parents with the current selection of parents
Perform -value adjustment for multiple comparisons involved in parent selection
if the adjusted -value is less than the given -level then
Convert the transition table with the minimum -value to a gtt by maximum likelihood
estimation of multinomial parameters
Declare that the current node has no parents
Compute the overall -value for the reconstructed GLN
Return the reconstructed GLN, the associated -values for each node, and the overall -value
5. Accuracy of GLN versus DBN Reconstruction
As GLN modeling is proposed as a potential alternative to DBN modeling, it is important to assess the performance of GLN relative to DBN modeling in terms of their abilities to recover the topology of the underlying networks. We use Hamming distance, false positives, and false negatives to evaluate the difference between a reconstructed network and the original ground-truth network. The Hamming distance is defined by the total number of different directed edges between two networks of the same set of nodes. A false positive is an incidence of a directed edge in the reconstructed network but not in the original ground-truth network; a false negative is an incidence of a directed edge in the original network but not in the reconstructed network. The definitions imply that the Hamming distance is the sum of false positives and false negatives. We have chosen to use a simulated data set over a real biological data set, such as the yeast cell cycle gene expression data set, to do the performance evaluation. This is because many factors in a biological data set may contribute to the reconstruction performance in addition to the algorithm difference. For example, the ground truth GRN in yeast may not contain all active interactions; it may also include additional interactions that are inactive in the particular experiments. This makes the comparison of algorithm performance less certain. In a simulated example, one has control of all potential variations.
is often evaluated to balance maximum likelihood estimation with the number of parameters in each conditional distribution. In contrast, the statistic is used in GLN modeling, as opposed to the likelihood in DBN modeling; the tradeoff with model complexity in GLN modeling is incorporated into the degrees of freedom of the distribution, as opposed to the term in the BIC in DBN modeling. Additionally, GLN modeling allows the user to control false-positive rate by specifying the size for type I error, while DBN modeling does not facilitate such an option.
For each trajectory, we applied increasing levels of noise with . When , the noise is the strongest in terms of network topology reconstruction. When , it is the same as as far as the topology is concerned.
GLN modeling is built on statistical hypothesis testing, while DBN modeling on information theory. We are curious at a more theoretical level why the GLN reconstruction has shown a consistently superior performance over the DBN reconstruction in the simulation study. We plan to address this remaining issue in our future work.
6. Temporal Gene Expression in Mice Exposed to Alcohol
Thirty-five adult DBA/2J (D2) mice were housed on a 12:12 light:dark cycle and given food and water ad libitum. The mice were habituated for three days to i.p. injections of saline and on the forth day were injected with 20% alcohol in saline in a total dose of 4 g/kg. D2 mice are exquisitely sensitive to alcohol dependence, and at this dose show physical signs consistent with dependence from about 4–10 hours after injection. Brains were removed, and anterior cortex tissue was dissected at 2, 7, 12, and 24 hours following the alcohol injection with 7 biological replicates at each time point. All animals were housed and treated according to the National Institutes of Health guidelines for the use and care of laboratory animals  and an approved Institutional Animal Care and Use Committee protocol.
cDNA fragments, that had undergone PCR from clones, were printed on poly-L-lysine-coated (Sigma, Mo, USA) microscope slides (Erie Scientific, Portsmouth, NH, USA) using a custom-built robotic arrayer as described in . The clones were from several cDNA libraries, including ESTs cloned in the laboratory of S.E.B., Research Genetics/Invitrogen clone sets Brain Molecular Anatomy Project and Sequence Verified, and the National Institute on Aging (3) clone sets 7.4 K and 15 K. cDNA microarrays were hybridized using the 3DNA array 900 microarray labeling kit according to the manufacturer's protocol (Genisphere, Hatfield, Pa, USA). Total RNA samples were reverse transcribed, labeled with Cyanine-3 (Cy-3), and hybridized against a common reference RNA labeled with Cy-5. The common reference is whole-brain RNA extracted from 100 male B6 mice. All arrays contained the same reference RNA in the Cy-5 channel and were normalized by using within-print tips Lowess nonlinear normalization . Normalized array data were stored in the longhorn array database (LAD)  and then standardized by using the red channel (common reference RNA) as the baseline standard with software developed in the laboratory of S.E.B. (These PERL programs are available upon request.) Data were loaded into an in-house database used for sorting by various statistics.
7. GLN Modeling of Transcription Regulation in the Mouse Brain
We demonstrate a GRN reconstructed using GLN modeling from a microarray study of temporal gene expression microarrays in mouse brains following acute exposure to alcohol to uncover transcription interactions of involved genes. The microarray data were normalized, quantized, formed to trajectories, and used to reconstruct a GLN. We illustrate the significant interactions we identified, their agreement with the literature, as well as the dynamic behavior of the GRN in response to alcohol.
Through post hoc -tests, partial least squares, and one-way ANOVA (fixed effect only and without multiple testing correction) across time course analyses, a total of 392 differentially expressed genes were selected because they exhibit both temporal and alcohol related expression variation. Missing gene expression values were imputed using the R software package PAMR . Those genes not selected for inclusion do not have strong evidence from this experiment to be on any path from the alcohol node.
These selected genes were entered into the GLN model as candidate GLN components that connect to the alcohol treatment node through gene expression on a directed path.
The alcohol node is assigned based on the experimental condition: 1 for alcohol-injected samples and 0 for control samples. The quantization was implemented in Java and compiled to native code on SuSE Linux using the GCJ compiler. It took about 5 hours to finish the quantization on a 2.8 GHz Pentium dual-core processor computer with 4 GB RAM running SuSE Linux.
The -values and number of parents for each node in the generalized logical network
No. of parents
8. Conclusions and Future Work
Derived from a statistical property regarding the summation of independent chi-squares, our GLN reconstruction algorithm identifies significant dynamic associations among a subset of genes to a target gene by performing the multinomial test. Thus, we have offered a unique framework to reconstruct GLNs to characterize temporal interactions from time-course gene expression data. Results from our application of this technique to the study of alcohol's influence on gene expression in mouse brains reveal both consistently observed associations and novel hypotheses that remain an open problem for current biological investigation. Based on these results, there appears to be significant potential to inspect the temporal patterns in gene expression through GLN reconstruction. In this paper, we have demonstrated the value of GLN modeling for extracting the underlying causal interactions among genes involved in response to alcohol. Some of the inferences made on temporal dependencies corroborate present knowledge on gene regulation in mouse. The other inferences will be subject to more extensive in vivo biological verification.
Preselection of a subset of interesting genes to render a model computable is a challenge for GRN modeling from microarray data. Approaches which filter genes or gene-gene relations have been applied. While this leads to the improved signal in the data, it also introduces a problem of false-negative results, neglecting extensive information on highly relevant genes which exhibit subtle variation in the same temporal patterns as other connected genes. Rather than filtering based on statistical effects, one could develop GLN models from known pathways and evaluate how they respond and interact with pharmacological perturbations. This strategy can be implemented by reconstructing GLNs from GRNs established by literature mining such as Ingenuity Pathways Knowledge Base (size Ingenuity Systems, Redwood City, Calif, USA) and PathAssist (size JusticeTrax Inc., Mesa, Ariz, USA). This will possibly allow the modeling to begin at a more realistic starting point, and will reserve statistical power for the strong plausible relations that are previously reported.
A more diverse set of nodes can also be incorporated into the GLN modeling. The biological relevance of a reconstructed GLN can be substantially improved if simultaneous measurements of the proteome, the metabolome, and the transcriptome are available, without major modifications to the current algorithms. Once data are properly scaled, the method is highly generalizable and has significant potential for inferring temporal relations among widely diverse biological processes. The illustration of the validity of our results from a small time-course gene expression study indicates substantial potential for denser sampling, and for the incorporation of additional data representing other aspects of the neurobiological response to alcohol, including neurohormonal, physiological, and behavioral measures.
A previous version of this paper was presented at the 2nd Foundations of Systems Biology in Engineering at Stuttgart, Germany, in September 2007. M. Song, C. K. Lewis, and E. R. Lance were supported by the joint National Science Foundation (NSF)—Department of Energy (DOE) Faculty and Student Team program under Grant NSF HRD-0420407. M. Song was also supported in part by the National Research Initiative of the USDA Cooperative State Research, Education and Extension Service, Grant no. 2006-35504-17359, and a Grant no. 5U54CA132383 from the National Cancer Institute. R. K. Yordanova was supported by BISTI. M. A. Langston was supported in part by the National Institutes of Health (NIH) under Grants 1-P01-DA-015027-01, 5-U01-AA-013512, and 1-R01-MH-074460-01, by the DOE under the EPSCoR Laboratory Partnership Program, by the Australian Research Council, and by the European Commission under the Sixth Framework Program. Additionally, E. J. Chesler and M. A. Langston were supported by NIH/NIAAA INIA Bioinformatics Core and Pilot U01AA13499, U24AA13513; E. J. Chesler, M. A. Langston, and R. K. Yordanova by NICHD. S. E. Bergeson was supported by NIH Grants AA013182, AA013403, and AA013475.
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