Volume 8 Supplement 5
Articles selected from posters presented at the Tenth Annual International Conference on Research in Computational Biology
A stochastic differential equation model for transcriptional regulatory networks
 Adriana ClimescuHaulica^{1}Email author and
 Michelle D Quirk^{2}
DOI: 10.1186/147121058S5S4
© ClimescuHaulica and Quirk; licensee BioMed Central Ltd. 2007
Published: 24 May 2007
Abstract
Background
This work explores the quantitative characteristics of the local transcriptional regulatory network based on the availability of time dependent gene expression data sets.
The dynamics of the gene expression level are fitted via a stochastic differential equation model, yielding a set of specific regulators and their contribution.
Results
We show that a beta sigmoid function that keeps track of temporal parameters is a novel prototype of a regulatory function, with the effect of improving the performance of the profile prediction. The stochastic differential equation model follows well the dynamic of the gene expression levels.
Conclusion
When adapted to biological hypotheses and combined with a promoter analysis, the method proposed here leads to improved models of the transcriptional regulatory networks.
Background
The production of independent sets of time courses of microarray data [1–3], obtained for the most studied eukaryotic organism Saccharomyces cerevisiae, improved the knowledge on the relationship between genes through the transcriptional process in the cell. The mechanism of the gene expression regulation is not entirely known, yet progress has been made by combining in silico approaches with the analysis of experimental data. In particular, contributions from a qualitative analysis realized by the recognition of specific promoter sequences, binding sites, and transcription factors are enhanced by quantitative studies obtained from microarray gene expression data [4, 5]. The transcriptional regulatory network, built from thousands of genes, has a dynamical nature: the transcriptional program adapts itself to organismal development through the cell cycle, or as a response to changes in environment. In a systemic view the network architecture is potentially established by a qualitative analysis while quantitative methods address the main dynamical aspects – the network switches and the level of its parameters. This type of information may be obtained by processing gene expression data that keep track of the variations in the experimental conditions and temporal modifications suited for the understanding of a particular transcriptional behavior.
A mathematical model for the processing of time dependent gene expression data has been sought to describe the dynamical aspects of regulation and to estimate the level of contribution for each transcriptional regulator in a succession of events. In this work we strengthen the model proposed in [6], by means of a novel pattern for the regulatory function. This model uses a SDE to describe the dynamics of the target mRNA expression level that reflects the actual knowledge about the stochasticity in gene expression, in a biological framework [7]. The drift term of the SDE depends on the regulatory rate of the target gene. The noise term is modeled by a Brownian Motion process which accounts for the superposition of small random factors that arise dynamically. The regulation rate is obtained as a linear combination of the regulatory functions of specific elements of the network. We propose a beta sigmoid function as the prototype of the regulatory function, designed to keep track of the local temporal patterns of the target gene regulators.
Our analysis shows that the utilization of the beta sigmoid function enhances the results in [6] where sigmoid functions were considered. The comparison was made by applying the model to the same test data set as in [6], given by gene expression measurements of the mRNA levels of 6178 S. cerevisiae ORFs at 18 time points under the α factor synchronization method [1]. A candidate pool of potential regulators was constructed by joining transcription factors, cellcycle control factors and DNAbinding transcriptional regulators as found in the literature [1, 8]. We performed the same statistical analysis from [6] based on the maximum likelihood principle for the estimation of the model parameters. The AIC strategy was used for the selection of the best fitting combination of the pool regulators. With the addition of beta sigmoid pattern, the SDE model renders good prediction results even in the case of the previously worst fitted genes obtained by [6]. The procedure proposed herein may be well suited to quantify transcriptional regulatory networks, provided it is tailored to the characteristics of the input data set.
Results
 1.
the parameters of the goodness of fit: log likelihood (log L), AIC and QE of the predicted mRNA levels with respect to the observed values
 2.
the corresponding regulators with their regulatory effect expressed by the local network weights; positive weights correspond to activator genes and negative weights correspond to repressor genes.
List of genes reported as worst fitted in [6] and their prediction results from the SDE beta sigmoid model
Target  logL  AIC  QE  Best Fit 

YBR089W(NA)  3.13  10.27  16.56  YBR089W = 0.180 + 0.481 BAS1 
YDR285W(ZIP1)*  1.25  5.48  2.17  YDR285W = 0.253 + 0.258 GAT1 + 0.511 GCN4 + 0.202 FKH1 
YFR057W(NA)*  2.33  1.32  2.40  YFR057W = 0.057 + 0.405 GAL80 + 0.09129 IFH1 
YAL018C(NA)*  5.29  0.58  1.25  YAL018C = 0.256 + 1.985 GAL80 + 0.922 FKH2 + 1.983 IME1 + 0.624 HMS1 
YOR264W(DSE3)*  4.65  3.31  2.77  YOR264W = 0.265 + 0.205 CHA4 + 0.684 IME1 
YOL116W(MSN1)*  3.29  0.58  3.58  YOL116W = 0.045 + 0.229 HAL9 + 0.132 FZF1 
YGR269W(NA)**  12.41  14.83  0.48  YGR269W = 0.011 + 0.263 GZF3 + 0.313 CRZ1 + 0.383 DAL80 + 0.361 AZF1 
YOR383C(FIT3)**  10.07  9.73  0.81  YOR383C = 0.073 + 0.199 ARG81+ 0.325 GLN3 + 0.218IFH1 
YOR319W(HSH49)**  11.52  11.05  0.60  YOR319W = 0.033 + 0.337 CST6 + 0.170 CIN5 + 0.719 GAL80 + 0.191 IFH1 + 0.274 ACA1 
YKL001C(MET14)*  2.26  0.52  2.16  YKL001C = 0.05 + 0.141 MAC1 
YDL117W(CYK3)*  8.73  7.47  1.21  YDL117W = 0.463 + 0.577 AFT2 + 0.703 FKH2 + 0.177 HAP5 + 0.132 FAP7 
YKL185W(ASH1)**  10.01  10.02  0.52  YKL185W = 0.176 + 0.485 ASK10 + 0.241 DOT6 + 0.211 FAP7 + 0.163 HAP5 
YBR158W(AMN1)  0.53  9.06  6.40  YBR158W = 0.066 + 0.191 IFH1 + 0.392 CHA4 + 0.313 ABF1 
YBR108W(NA)*  9.23  8.47  0.98  YBR108W = 0.082 + 0.723 HMS1 + 1.512 GAL80 + 1.58 IME1 + 0.639 FKH2 
YAL020C(ATS1)**  9.84  11.69  0.61  YAL020C = 0.011 + 0.267 HAC1 + 0.534 GAL4 + 0.521 INO4 
YBR002C(RER2)  3.41  0.83  3.46  YBR002C = 0.020 + 0.239 FKH1 + 0.229 ABF1 
YCL040W(GLK1)*  6.78  7.56  3.13  YCL040W = 0.118 + 0.261 CST6 + 0.300 HMS1 
YNL018C(NA)*  5.15  4.31  1.43  YNL018C = 0.028 + 0.259 KRE33 + 0.182 CAD1 
YNL192W(CHS1)  1.64  2.71  3.45  YNL192W = 0.081 + 0.307 CHA4 + 0.182 ARG81 
YBR230C(NA)  0.74  6.51  2.20  YBR230C = 0.025 + 0.625 MAC1 + 0.694 HAP2 + 0.462 HOG1 
List of genes reported as worst fitted in [6] and their prediction results from the SDE sigmoid model
Target  logL  AIC  QE  Best Fit 

YBR089W(NA)  1.68  7.36  3.2  YBR089W = 0.166 + 0.367 HAA1 
YDR285W(ZIP1)  0.77  2.46  3.69  YDR285W = 0.191 + 0.368 INO2 
YFR057W(NA)  1.13  1.74  4.31  YFR057W = 0.098 + 0.188 GCN4 
YAL018C(NA)  1.52  2.96  1.79  YAL018C = 0.055 + 0.303 IME1 + 0.195 CRZ1 
YOR264W(DSE3)  2.26  0.52  5.56  YOR264W = 0.059 + 0.129 ARG80 
YOL116W(MSN1)  2.3  0.59  3.77  YOL116W = 0.092 + 0.193 HAL9 
YGR269W(NA)  2.4  0.81  5.19  YGR269W = 0.097 + 0.194 HMS1 
YOR383C(FIT3)  1.82  6.37  2.64  YOR383C = 0.367 + 0.287 ARG81 + 0.464 ECM22 + 0.412 GLN3 + 0.335 MAC1 
YOR319W(HSH49)  2.17  5.65  4.92  YOR319W = 0.83 + 1.13 CIN5 + 0.655 FHL1 + 0.354 DAL81 + 0.275 FKH1 
YKL001C(MET14)  2.58  1.16  4.34  YKL001C = 0.091 + 0.18 IME1 
YDL117W(CYK3)  2.59  1.18  4.35  YDL117W = 0.162 + 0.359 AFT2 
YKL185W(ASH1)  2.64  2.73  2.37  YKL185W = 0.150 + 0.407 ACE2 + 0.421 GAT1 + 0.302 INO2 
YBR158W(AMN1)  2.65  8.7  1.2  YBR158W = 0.139 + 0.926 KRE33 + 0.941 IME4 + 0.571 MAL13 + 0.264 GAT3 + 0.347 CBF1 + 0.285 AZF1 
YBR108W(NA)  2.66  1.33  2.85  YBR108W = 0.112 + 0.205 HAC1 
YAL020C(ATS1)  2.75  1.51  4.15  YAL020C = 0.133 + 0.256 ASK10 
YBR002C(RER2)  3.07  2.14  2.26  YBR002C = 0.101 + 0.2 HAP5 
YCL040W(GLK1)  3.09  2.18  3.18  YCL040W = 0.095 + 0.199 HAL9 
YNL018C(NA)  3.59  3.18  2.19  YNL018C = 0.078 + 0.154 ARG81 
YNL192W(CHS1)  3.21  1.57  2.13  YNL192W = 0.115 + 0.115 FZF1 + 0.306 DAL81 + 0.209 HMS2 
YBR230C(NA)  3.32  3.37  2.2  YBR230C = 0.52 + 0.484 MAC1 + 0.467 GZF3 + 0.374 INO4 + 0.244 EDS1 
Prediction results from the SDE beta sigmoid model for selected genes
Target  logL  AIC  QE  Best Fit 

YMR096W(SNZ1)  8.86  11.72  0.8  YMR096W = 0.069 + 0.330 HAP3 + 0.115 CIN5 
YNR025C(NA)  13.7  13.4  0.11  YNR025C = 0.033 + 0.556 ARG81 + 0.487 HSF1+ 0.195 FAP7 + 0.120 FKH1 + 0.319 DAL81 + 0.141 GCR2 
YPR200C(ARR2)  13.04  14.08  0.29  YPR200C = 0.00037 + 0.707 GAL4 + 0.369 INO4 + 0.364 HAP2 + 0.201 ABF1 + 0.129 FAP7 
YGR234W(YHB1)  15.27  24.53  0.46  YGR234W = 0.042 + 0.157 HIR1 + 0.139 ABF1 
YGR269W(NA)  12.42  14.84  0.48  YGR269W = 0.011 + 0.263 GZF3 + 0.313 CRZ1 + 0.383 DAL80 + 0.361 AZF1 
YGL150C(INO80)  16.71  21.43  0.21  YGL150C = 0.237 + 0.197 CST6 + 0.368 GAT3 + 0.169 KRE33 + 0.185 ABF1 + 0.122 CAD1 
YDR193W(NA)  10.67  13.35  0.48  YDR193W = 0.044 + 0.731 CST6 + 0.141 IFH1 + 0.185 DOT6 
YAL061W(NA)  21.24  28.47  0.02  YAL061W = 0.147 + 1.189 CST6 + 0.321 FKH1 + .369 IXR1+1.521 BYE1+.125 GAT3 +.165 ACA1 
YKL150W(MCR1)  12.29  16.57  0.41  YKL150W = 0.048 + 0.515 ACA1 + 0.222 HIR1 + 0.205 GAL80 
YDR515W(SLF1)  19.88  29.76  0.09  YDR515W = 0.087 + 2.080 CST6 + 0.190 IFH1 + 2.660 GTS1 + 0.956 FHL1 
Prediction results from the SDE sigmoid model corresponding to genes from Table 3
Target  logL  AIC  QE  Best Fit 

YMR096W(SNZ1)  7.27  8.54  6.16  YMR096W = 0.159 + 0.179 GCN4 + 0.174 HAA1 
YNR025C(NA)  3.8  1.61  5.42  YNR025C = 0.008 + 0.261 HMS1 + 0.278 ACA1 
YPR200C(ARR2)  3.97  3.94  5.28  YPR200C = 0.144 + 0.315 INO4 
YGR234W(YHB1)  11.08  18.17  5.28  YGR234W = 0.059 + 0.12 ARG81 
YGR269W(NA)  2.4  0.81  5.19  YGR269W = 0.097 + 0.194 HMS1 
YGL150C(INO80)  4.25  4.51  4.41  YGL150C = 0.082 + 0.168 GAT3 
YDR193W(NA)  6.22  0.44  4.45  YDR193W = 0.278 + 0.415 LEU3 + 0.166 GAL4 + 0.691 FAP7 + 0.293 CUP9 + 0.375 DAT1 
YAL061W(NA)  8.07  12.13  1.66  YAL061W = 0.087 + 0.191 CUP9 
YKL150W(MCR1)  7.93  9.86  3.84  YKL150W = 0.249 + 0.325 CBF1 + 0.175 HAA1 
YDR515W(SLF1)  5.94  0.12  2  YDR515W = 0.246 + 0.562 CIN5 + 0.347 CBF1 + 0.256 HIR1 + 0.453 HAP4 + 0.304 IFH1 

the SDE model can provide very good predictions of mRNA expression levels;

there exist genes for which the SDE model with beta sigmoid regulatory function gives a better prediction than the SDE model with sigmoid regulatory function.
A good quality of fitting of a particular gene allows the consideration of the regulators associated by the model for further investigation such as DNAbinding sites or promoter architectures. The quadratic error of prediction with beta sigmoid regulatory function is less than 0.5 for 1885 genes from the entire data set (see Additional file 1).
Discussion and conclusions
The global view of the regulatory network is a cascade model, with genes regulating genes regulating other genes at their turn [11]. The SDE model [6] revisited here addresses the network local connections, i.e. the strict neighborhood of one target gene. The drift term of the stochastic differential equation is given by the regulation rate which quantifies the local network architecture by a linear combination of regulatory functions of regulating genes. The choice of the regulatory function pattern is a central aspect of the model, since the fitting of the gene expression profiles is very sensitive with respect to the drift term of the SDE. This model has the ability to extract from a given set of potential regulators those that fit the target gene expression profile.
The prototype of regulatory function introduced in [6] has a sigmoid pattern, built on the statistical characteristics of mRNA expression levels – see Equation (14). By keeping track of the temporal pattern of regulation, we show that the prediction of target gene expression profiles is improved for 29% of genes tested. We propose a prototype of regulatory function supported by a beta sigmoid model, built on temporal parameters extracted from the expression profiles of the regulators – see Equation (13). The SDE method relies on the assumption that the best fit of the target expression profiles is informative for the identification of the regulators and of their contribution. Thus, for the study of a specific set of target genes, our prototype of regulatory function may give more accurate results and provides a switch for the model proposed in [6]. Conceptually the beta sigmoid model has the advantage to correspond to the biological process of regulation: the temporal window of the peak defined by the shape of the beta sigmoid function reflects features of the regulation mechanism.
The regulation of gene expression in eukaryotes is a complex phenomenon and various particularities from one type of gene to another may occur. Hence the regulatory pattern may vary from gene to gene [4]. This fact is revealed in our result which shows that there are genes for which we can choose the best model between the beta sigmoid and the sigmoid pattern while for other genes neither of them fits the data. Before reaching this conclusion one has to be aware about the limitation induced from the selection of the set of potential regulators since incomplete information at this level may deteriorate the results.
Further research on more complex and explicit regulatory functions are foreseen from the availability of data sets and studies on various experimental condition for the budding yeast (sporulation [3], diauxic shift, heat and cold shock, treatment with DTT, pheromone and DNAdamaging agents [12]). In this framework a challenging task could be the study of the existence of possible relationships between the type of regulation pattern and the gene specificity.
This work provides a second implementation of the algorithm based on the SDE model, enlarged with a new type of regulatory pattern. The predictions from the algorithm may be improved with better strategies for the selection of the candidate pool of regulators. Moreover, the algorithm is a potential tool for the investigation of the interactions between the regulators of a target gene, modeled with a drift term defined by a non linear combination of regulatory functions.
This study shows that the SDE framework constitutes a reliable tool for the analysis of the transcriptional regulatory networks, provided it is completed with a validation of the identified regulators by a promoter analysis.
Models and methods
SDE model of timecontinuous gene expression data
Let T denote a discrete set that corresponds to the time instants of the gene expression measurements. Consider two stochastic processes defined for a given target gene, (N_{ t })_{t∈T}and (X_{ t })_{t∈T}that model, respectively, the variation in time of the target gene amount of mRNA and the variation in time of the expression level of mRNA. Let $\mathcal{R}$ be the set of potential regulators for the target gene. Denote by g_{ t }the function that models the transcription rate of the target gene at time t
gt : $\mathcal{\text{P}}$($\mathcal{R}$) → $\mathcal{R}$_{+} (1)
where $\mathcal{\text{P}}$($\mathcal{R}$) is the set of all possible subsets of $\mathcal{R}$ and $\mathcal{R}$_{+} is the set of real positive numbers. Denote the real, positive mRNA degradation rate by λ.
The model proposed in [6] assumes that from time t to Δt the transcription and degradation process are given by
where (W_{ t })_{t∈T}is a Brownian Motion process that models the random error and σ is a positive scaling parameter. Consider infinitesimal time intervals, that is Δt → 0; from this it follows that the relation in Equation (2) becomes a stochastic differential equation
Since N_{ t }is proportional with the signal intensity S_{ t }, and X_{ t }= log(S_{ t } B) – where B is the background intensity – assume without loss of generality that
X_{ t }= log(N_{ t }) (4)
Thus, the chain rule of the stochastic calculus applies (Itô formula) and the SDE obtained for X_{ t }yields
Local regulatory network
Consider an increasing sequence of temporal values
T = {t_{0} <t_{1} <...<t_{ n }} (6)
Let m be the cardinality of the set $\mathcal{R}$ and let ${X}_{t}^{i}$ be the mRNA expression level of the ith regulator from the set $\mathcal{R}$, measured at time t ∈ T. Denote
where F_{ i }denotes the regulatory functions of the potential regulators from $\mathcal{R}$. The constants c_{0}, c_{1},...,c_{ m }are the learning parameters of the network; they modulate the network behavior and carry information about the local regulatory process.
Beta sigmoid pattern of regulation
The regulatory function is a key element of the model and fits the quantitative pattern with a specific regulator that acts on the mRNA expression of the target gene.
Our work revealed a prototype of the regulatory function based on the beta sigmoid function, given by
where I_{ A }is the indicator function of the set A (I_{ A }(x) = 1 if x ∈ A and I_{ A }(x) = 0 if x ∉ A) and
is the sigmoid function; μ_{ i }and σ_{ i }are the mean and deviation of ${\underset{\xaf}{X}}^{i}$, the prototype of the regulatory function from [6].
The learning in the local network is driven by the SDE
where $\tilde{c}$_{0} = c_{0}  λ  σ^{2}/2. The network weights c_{1},...,c_{ m }carry information in both their magnitude and sign: positive values correspond to regulators with activation, and negative values correspond to repression.
Statistical analysis
 1.
the set of m regulators (model selection);
 2.
their corresponding parameters σ and the set {$\tilde{c}$_{0}, c_{1},...,c_{ m }} of parameters estimation;
with the best fit with respect to Equation (15). The beta sigmoid as regulatory function adds supplementary parameters ${t}_{s}^{i}$, ${t}_{m}^{i}$ and x_{ max }to the model. These parameters are estimated from the corresponding time course mRNA levels according to their definitions given in Equation (10) and employed in the computation of the estimators of σ and $\underset{\xaf}{c}$:
For the evaluation of the impact of the beta sigmoid regulatory function model, the network weights are estimated from gene expression data with the standard statistical procedure described in detail in [6]. Equation (15) is considered in discrete form for each time interval [t_{ j }, t_{j+1}], j = {1, 2,...,n} that corresponds to time measurements. The estimators of σ and {$\tilde{c}$_{0}, c_{1},...,c_{ m }} are obtained maximizing the loglikelihood function log L (ML approach [13]) of the ndimensional random vector with elements
The computation of log L uses basic properties of Brownian Motion: the increments ${W}_{{t}_{j+1}}{W}_{{t}_{j}}$ are pairwise independent and each increment is normally distributed, with zero mean and standard deviation given by $\sqrt{{t}_{j+1}{t}_{j}}$.
The criteria used for the selection of the regulators is AIC [14]. Between any two combinations of regulators, the best combination is that for which the AIC of the regulators has the smallest value. The computation of AIC follows from
AIC = 2$\widehat{\mathrm{log}L}$ + 2(m + 1)
where $\widehat{\mathrm{log}L}$ is the estimator of log L and is obtained from the functional invariance property of the maximum likelihood estimators $\widehat{\sigma}$ and $\widehat{\underset{\xaf}{c}}$, i.e.,
Let $\mathscr{H}$ denote the set formed by a candidate pool of regulators of the target gene; denote by $\mathscr{H}$ the cardinality of $\mathscr{H}$. Ideally, ML and AIC procedures shall be performed on each combination of regulators from $\mathscr{H}$. Since the number of all possible combinations of regulators is ${2}^{\left\mathscr{H}\right}$, an enumeration algorithm for those sets will explode quickly. The heuristic procedure used is the forward selection strategy [15]. At first the regulator with the biggest loglikelihood with respect to the target gene is selected. A new regulator is added if it will increase the AIC more than any other single regulator outside the current combination. The actual implementation stops for a combination of maximum 10 regulators, exactly as done in [6]. Under these conditions the performance of the algorithm we propose is expressed by an order of magnitude equal to O(nm^{2}). In practice this is a slight enhancement compared to the algorithm proposed in [6] for which the order of magnitude equals O(n^{2}m^{2}) – since for actual experimental data the number of time courses n is quite small. The difference in the performance of the two algorithms comes from the fact that the search of the maximum is less costly than the computation of the statistical parameters for a data set.
Availability
The method was implemented in R 2.2.1 (R Development Core Team, http://www.rproject.org/). The source code is available upon request.
List of abbreviations used
 SDE:

Stochastic Differential Equation
 AIC:

Akaike Information Criteria
 ML:

Maximum Likelihood
 QE:

Quadratic Error
Declarations
Acknowledgements
ACH is grateful for support from Laboratoire IGS CNRS Marseille where this work has been initiated. The authors thanks Dr. Karsten Suhre and Dr. Yves Vandenbrouck for useful discussions.
This article has been published as part of BMC Bioinformatics Volume 8, Supplement 5, 2007: Articles selected from posters presented at the Tenth Annual International Conference on Research in Computational Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/14712105/8?issue=S5.
Authors’ Affiliations
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