BACKGROUND
signaling and regulatory cascades establish the bridge between environmental stimuli and cellular responses, and represent a key aspect of cellular adaptation to different environmental conditions.
cells can sense several stimuli, both internally and externally, and the received information will subsequently be propagated through a cascade of physico-chemical signals.
the ultimate recipients of these signals will determine how the cell responds, by acting at different regulatory levels .
contrary to metabolic networks, most signaling and regulatory pathways are relatively poorly studied and signaling properties of a protein cannot be easily derived from its gene sequence  <cit> .
moreover, signal transduction networks operate over a wide range of time-scales, and due to the presence of feedback loops and cross talk it is difficult to discern how concurrent signals are processed.
thus, methods to analyze and model signal transduction and regulatory circuits are of prime importance in biology, medicine and cell engineering, since they can bring insights into the mechanistic events underlying complex cellular behavior.
the availability of models for signaling and regulatory cascades represents an opportunity to expand the search space when looking for intervention targets that may lead to desired phenotypes – e.g.
when looking for better drug targets in medicine, designing novel regulatory circuits in synthetic biology or finding regulatory targets that can release metabolic control in metabolic engineering.
most eukaryotic cells, including many yeasts and humans, can sense the availability of carbon sources in their surroundings and, in the presence of their favorite sugar , trigger a cascade of signals that will repress the utilization of less-favorite sugars as well as the function of different catabolic routes  <cit> .
this phenomenon is commonly termed carbon-catabolite or glucose repression.
because of its role in nutrient sensing and its industrial impact on the simultaneous utilization of different carbon sources, glucose repression has been a model system for studying signaling and regulation.
in particular, glucose repression in the yeast saccharomyces cerevisiae has been extensively studied and two main signaling pathways have been identified: a repression pathway, mediated through the protein kinase snf <dig> and the transcription factor mig <dig>  and a glucose induction pathway, mediated through the membrane receptors snf <dig> and rgt <dig> and the transcription factor rgt <dig> .
growing evidence suggests the existence of extensive cross talking between these two pathways.
figure  <dig> summarizes key aspects of the system.
besides the role of glucose repression on the utilization of alternative carbon sources, glucose repression in yeast leads to the transcriptional shutdown of genes related to respiration, mitochondrial activities, and gluconeogenesis  <cit> .
this transcriptional behavior causes the wild-type yeast to exert respiro-fermentative metabolism during growth on excess glucose, redirecting carbon towards by-products of metabolism such as ethanol, acetate and glycerol, at the cost of biomass formation.
despite being an extensively studied system, knowledge on yeast glucose repression is still far from complete and key questions remain, including: what exactly triggers the cascade signal?
how to differentiate between causes and consequences?
how does the knowledge derived from phenotypic observations relate to mechanistic events?
how does the current knowledge on glucose repression fit with available high-throughput data?
in order to attempt to bring insights into these questions, we aim here at creating a mechanistic, semi-quantitative model of glucose repression signaling cascades and genetic regulatory circuits in yeast.
modeling approaches of different levels of abstraction have been proposed to analyze and simulate signal transduction and regulatory networks, ranging from purely topological to kinetic models.
while attractive in principle, quantitative kinetic models based on odes are hampered by difficulties in determining the necessary parameters and kinetic equations.
at the other extreme, strictly descriptive models have also been reported  <cit> , in which the precision of the formalism proposed, based on process engineering, establish a clear and unique qualitative representation of the network interactions.
somewhere in between lie more semi-quantitative and qualitative approaches that require the topological description of signaling interactions and make use of well-established mathematical frameworks to analyze network structure and functionality.
such methods include  stoichiometric representation and extreme pathway analysis  <cit> ,  boolean  and bayesian  representation of interactions  <cit> ,  logical hypergraph representation and logical steady state analyses  <cit> , and  petri nets graph representation and simulations  <cit> .
all these methods describe signal flow qualitatively within a mathematical formalism and without loss of information on the network topology, allowing insightful computations on network structure such as evaluating the degree of cross talk, determining all possible elementary 'flux' modes and calculating the number of theoretically possible positive and negative feedback loops.
moreover, logical hypergraph analyses and petri net models also have the potential to be used for semi-quantitative simulation of network behavior, since they allow simple predictions of the state of a system in response to different stimuli.
in this work, we reconstructed the signaling and transcriptional regulatory network of glucose repression in s. cerevisiae based on established knowledge reported in the literature.
we converted this information into a logical hypergraph, and performed structural and functional analyses on the network following the framework proposed by klamt and co-workers  <cit> .
next, we performed logical steady state analyses to compute the state of all nodes in the system under all possible environmental conditions , and for all different single gene deletions and some double gene deletions.
furthermore, we developed a framework to evaluate model predictions by comparing changes in the state of the regulatory layer against changes in gene expression data .
based on the results from the model evaluation, we identified main errors and discuss possible sources of discrepancies, as well as the inherent limitations to boolean modeling.
our results point towards the existence of incoherencies between high-throughput data and literature-based knowledge related with glucose repression.
to our knowledge, this represents the first attempt to mechanistically and semi-quantitatively model glucose repression signaling and regulatory pathways in the yeast s. cerevisiae.
RESULTS
reconstruction of the signaling/regulatory network and model setup
glucose repression signaling and regulatory network was reconstructed from low-throughput data reported in peer-reviewed publications.
information was gathered based on biochemical studies and physiological observations, and it was included in a collection database containing:  list of proteins with sensor, signaling or transcription factor functions found to be related to glucose repression;  list of genes known to be transcriptionally regulated by glucose repression related transcription factors;  type of regulation exerted on each of the previous species by metabolites and/or regulatory proteins.
the reconstructed network accounts for  <dig> species  and  <dig> interactions, which cover most of the current knowledge on the mig1/snf <dig> and snf3/rgt <dig> pathways, as well as galactose and maltose regulatory systems.
transcription factors included are rgt <dig>  mig <dig>  mig <dig>  mig <dig>  sip <dig>  cat <dig>  malr and gal <dig>  regulatory targets include genes encoding hexose transporters and enzymes involved in maltose catabolism, gluconeogenesis and the leloir pathway.
the complete list of species and interactions considered is given as supplementary material .
thereafter, the reconstructed signaling/regulatory network was converted into a logical hypergraph , representing all interactions in a logical manner , according to the framework proposed by klamt et al.
<cit>  to model signaling networks.
the conversion of signaling and regulatory interactions into boolean functions was based on described functions reported in the literature .
the resulting hypergraph consists of  <dig> nodes  and  <dig> hyperedges, and represents a logical model for glucose repression signaling and regulatory pathways.
for ease of visualization, we have depicted the hypegraph into four separate sub-networks, each representing in more detail a different system: mig1/snf <dig> and snf3/rgt <dig> pathways , galactose regulation , maltose regulation  and the sip4/cat <dig> regulatory system .
the model takes as input the availability of carbon sources  and outputs the logical steady state of the network.
in our model, nodes can assume one of two logical states,  <dig> or  <dig>  corresponding to on or off, or in more subtle instances, higher or lower activity.
for protein nodes, this can most simply be interpreted as a protein being active  or inactive , whereas in the case of gene nodes, it can be seen as a gene being expressed  or not .
we used the model to analyze structural characteristics of the network and to compute logical steady states of all nodes in the network.
in particular, we simulated how gene transcripts change their logical state in response to perturbations , and evaluated the predictions by using available transcriptional datasets for different carbon source conditions and yeast deletion mutants.
during logical states simulations, although most nodes were left unconstrained, a few nodes were assigned a default value of  <dig>  e.g., grr <dig> was set to a fix state of  <dig> since its regulation is not considered in the model .
also, in the case of genes constitutively expressed at a basal level such as malt, encoding a maltose transporter, the node state was set to a fix value of  <dig>
structural and functional analyses of the network
logical steady state analyses were performed for all combinations of sugar availability for the wild type, all single gene knockout mutants , and three double gene deletion mutants of interest, in a total of  <dig> simulations .
we notice that most of the gene nodes change their logical state in over 10% of the simulations, but only few  change in more than 15% of the simulations.
the predictions for the wild type along with a subset of the deletion mutants for which transcriptome data was available were analyzed and used to evaluate the model, as will be discussed below.
the capability to make semi-quantitative predictions of gene expression levels and protein activity for any combination of gene deletions and nutrient conditions is a key feature of the model.
the reconstructed network contains  <dig> negative and  <dig> positive feedback loops, which is indicative of the high degree of crosstalk between pathways and complex cause-effect relationships.
this is further supported by the dependency matrix of the network , which is based solely on the underlying network interaction graph without information on boolean functions.
in the matrix, yellow elements represent an ambivalent relationship between an ordered pair of species , where i and j represents the column and the row number, respectively, in the sense that both activating and inhibiting paths exist from i to j. dark or light green  elements, dij, indicate that species i is a total or non-total activator  of species j, respectively, i.e.
only activating  paths from i to j exist and feedback loops are either absent  or present  – see figure  <dig> legend for details.
examining the dependency matrix, the large number of ambivalent relationships  as well as the prevalence of negative feedback loops in signaling paths  is noteworthy.
it underscores the difficulties in making predictions based on intuitive approaches and emphasizes the need for a logical modeling framework.
the matrix also reveals the high degree of crosstalk between the mig1/snf <dig> and snf3/rgt <dig> pathways, since it is quickly noted that the signaling proteins in one pathway generally affect proteins in the other pathway.
model evaluation
in order to evaluate the capability of the logical model to predict differential gene expression, we performed logical steady state analysis of the glucose repression regulatory response for five different gene knockouts, and compared the results with available whole-genome gene expression from dna-microarrays.
we used data from the yeast mutants Δrgt <dig>  Δmig <dig>  Δmig1Δmig <dig>  Δsnf1Δsnf <dig>  Δgrr <dig> and their isogenic reference strains.
this type of analysis not only gives an indication of the model's predictive strength, but also hints at possible errors in the model  in the cases where discrepancies between model and observation occur.
the best we can hope to achieve with a boolean model is a correct prediction of the sign of the change in gene expression, i.e., the model prediction yimod should equal the experimental observation yiexp when evaluating change in expression of gene i following a knockout or change in conditions.
in order to assess what experimental results should be regarded as a change, it is necessary to make an interpretation of the gene expression data that allows a comparison with the binary outputs of the boolean model.
intuitively, the experimental change should be relatively large and statistically significant in order to be reflected by the discrete boolean model.
therefore, we established a fold-change threshold  and a student's t-test p-value cut-off  for all pairwise gene expression comparisons between a deletion mutant and its isogenic reference strain.
all genes with p-value < α and fc ≥ fcmin  were assigned with a value of yiexp =  <dig> , and  <dig> otherwise.
such conversion of gene expression into discrete boolean values based on a somewhat subjective threshold may yield a number of type- <dig>  and type- <dig>  errors.
the hereby identified experimental variation, yiexp, was then compared with the model prediction, yimod, for each gene i. yimod was determined by the difference in boolean output for gene i between the mutant and the reference state , at a defined external condition.
thus, yimod can assume the values - <dig>   <dig> or  <dig>  corresponding to a decrease, no change or increase in gene expression on transcript i in the mutant, respectively.
model prediction capabilities were evaluated based on the difference |yimod - yiexp|, with a value of  <dig> meaning a correct prediction, a value of  <dig> implying a small error, and a value of  <dig> indicating a large error .
a summary of the results from the comparison between model prediction and experimental up- and down-regulation for all five different knockouts evaluated is shown in table  <dig>  in the following, we discuss more thoroughly the results for Δrgt <dig>  and use this to analyze common reasons for discrepancies in all the knockouts.
the remaining comparisons are briefly commented afterwards.
in parentheses are the numbers when dubious genes are not included in the computations .
evaluation of Δrgt <dig> mutant
transcriptome data for the yeast Δrgt <dig> mutant and its isogenic reference strain is available from  <cit>  during shake flasks cultivations using galactose as the single carbon source.
sampling was made in the mid-exponential phase, where pseudo-steady-state can be assumed .
we converted the gene expression data from this study into yiexp according to the procedure described above, and compared it with the simulation results yimod .
yimod are derived from logical steady state analyses of the logical model assuming galactose present and all other carbon sources absent.
the logical steady states were first determined for the original model without further constrains .
afterwards, in order to simulate the rgt <dig> gene deletion, the node rgt <dig> of the hypergraph was set to zero, and a new logical steady state analysis was performed .
yimod is given by the difference between xi,rgt <dig> and xi,wt.
in general, the model predictions are very good for the Δrgt <dig> mutant, with 82% true predictions, reflecting the fact that rgt <dig> along with its regulators has been extensively studied  <cit> .
only for  <dig> out of the  <dig> genes evaluated, the experimental fold changes do not correspond to model prediction.
the six genes are snf <dig>  mth <dig>  mig <dig>  mal <dig> , suc <dig> and hxt <dig>  in the following, the causes of these discrepancies are investigated.
analysis of discrepancies in snf <dig>  snf <dig> encodes a high-affinity glucose sensor located in the plasma membrane.
gene expression data shows no differential expression of snf <dig> in the Δrgt <dig> mutant .
however, in the model this gene is found to be down-regulated in the mutant relative to the wild type.
the most likely explanation for this discrepancy lies in the logical equation for snf <dig> applied in the model:
snf <dig> = not  =  and
in the model, rgt <dig> deletion leads to an active mig <dig> and consequently repression of snf <dig>  the most plausible explanation for this is that the model overestimates the importance of mig <dig> and that, in reality, the presence of active mig <dig> by itself is not enough to prevent snf <dig> transcription.
nevertheless, this explanation cannot be accepted out of hand: the dependency matrix reveals that rgt <dig> is an ambivalent factor with regard to snf <dig>  i.e., both repressing and activating paths from rgt <dig> to snf <dig> exist.
it is thus possible that mig <dig> is indeed a significant repressor of snf <dig>  but that other important, repressing pathways from rgt <dig> to snf <dig> are also deactivated by the rgt <dig> deletion.
a closer inspection of the signaling paths from rgt <dig> to snf <dig> in the hypergraph reveals that, from the  <dig> existing possible paths,  <dig> are repressing paths of the form rgt <dig> -> rgt <dig> -| mig2/mig <dig> -| mig <dig> -| snf <dig>  this means that rgt <dig> represses mig <dig> and mig <dig>  both of which repress mig <dig>  which then represses snf <dig> .
it seems unlikely that these  <dig> signal flux modes play significant physiological roles since  no significant role of mig <dig> has ever been found  <cit> , and  mig <dig> can at most serve to attenuate mig <dig> expression since the two proteins are active at basically the same conditions.
analysis of discrepancies in mth <dig>  mth <dig> is a signaling protein intermediate between the membrane sensors snf3/rgt <dig> and the transcription factor rgt <dig>  <cit> .
gene expression of mth <dig> is found to be up-regulated in the Δrgt <dig> mutant, whereas the model predicts the expression level of mth <dig> to be unchanged.
despite the fact that rgt <dig> repression of mth <dig> is reported in the literature   <cit> , the repression of mth <dig> by rgt <dig> is ignored for logical steady state calculations for two reasons.
firstly, kaniak et al.
state that this transcriptional repression is weak  transcriptional repressor of mig <dig>  the discrepancy for hxt <dig> seems to have similar reasons.
analysis of discrepancies in mal <dig> .
expression of mal <dig>  encoding a mal regulator, was found to be experimentally repressed in the mutant, while it remained unchanged in the model.
the three genes required for maltose metabolism are mapped in various mal loci, of which  <dig> are currently known  <cit> .
in the model, no distinction is made between the different complex loci.
since this is a recurrent discrepancy in all evaluations performed, we present a general discussion on the mal regulatory system later in the discussion section.
analysis of discrepancies in suc <dig>  expression of suc <dig>  encoding for invertase , was not found to be significantly different between the Δrgt <dig> mutant and wild-type, whereas the model predicts a decrease in gene expression in the mutant.
this discrepancy illustrates both the difficulties in choosing the correct logical equation for a specific species based on literature and in converting a continuous reality to a binary model.
in the model it is assumed that both mig <dig> and mig <dig> should be absent in order for suc <dig> to be expressed at high levels ), cf.
lutfiyya et al.
who found that the single deletions had relatively low impact on suc <dig> expression level, whereas Δmig1Δmig <dig> double deletion had great effect  <cit> .
contrary to this, klein et al.
found a large increase in suc <dig> expression in a Δmig <dig> strain and further increase by additional disruption of mig <dig>  <cit> .
the klein et al.
observations could have been implemented in the model via a boolean function where the absence of either one of the two repressors induces expression of suc <dig> ) or, alternatively, by simply ignoring mig <dig> in the equations.
while implementing either of the alternatives would have led to correct model prediction for the knockout , the actual output for suc <dig> in the wild-type would have been the same regardless of the chosen equation .
nevertheless, the ambiguity in the literature combined with the difficulty of imposing a discrete model on a continuous reality made the choice of logical equation extremely hard in this case.
the model apparently mistakenly predicts a decrease in suc <dig> expression, because rgt <dig> deletion causes mig <dig> to be expressed  which then leads to repression of suc <dig> with the chosen boolean equations.
based on this evaluation, and assuming that a type- <dig> error has not occurred, it therefore seems reasonable to say that the model overestimates the importance of mig <dig> in the regulation of suc <dig>  alternatively, it is possible that mig <dig> transcription is induced, but that the mig <dig> protein is also post-translationally regulated, something that is not described in the model or, to the best of our knowledge, in the literature.
this could mean that mig <dig> protein activity is not increased by the deletion of rgt <dig>  despite the eventual increase in transcript levels.
evaluation of Δmig <dig> and Δmig1Δmig <dig> mutants
transcriptome data for Δmig <dig> and Δmig1Δmig <dig> mutants, and their isogenic reference strain was available from  <cit>  during aerobic batch cultivations using  <dig> g/l glucose as single carbon source.
samples were taken in the mid-exponential phase at a residual glucose concentration of  <dig> g/l.
therefore, experimental observations were evaluated against model predictions with only glucose present.
for both knockouts, the percentage of true predictions is of 63% .
when examining the discrepancies between experimental observations and model predictions, it is particularly noticeable that model predictions for mal and gal regulatory genes are almost always wrong.
in particular, in the Δmig <dig> case both mal <dig> and gal <dig> evaluation produces a large error , meaning that the experimental direction of regulation is opposite to that predicted by the model.
whereas in the case of mal genes this is one of several discrepancies observed in this work , in the case of gal <dig> we find the experimental gene expression data from  <cit>  to directly contradict the results of a northern blot analysis  <cit> .
while the former study showed a decrease in gal <dig> levels upon deletion of mig <dig>  the second showed more than 6-fold increase in the levels of gal <dig> transcripts in a Δmig <dig> mutant, in a medium with the same glucose concentration .
evaluation of Δgrr <dig> mutant
transcriptome data for Δgrr <dig> and its isogenic reference strain were also available from  <cit> , at conditions corresponding to high extracellular glucose concentration.
evaluation of model predictions for Δgrr <dig> yielded the poorest results, with only 39% of true predictions .
this is probably due to the fact that, despite being an important player in a large number of cellular activities such as cell morphology, heavy metal tolerance, osmotic stress and nitrogen starvation  <cit> , grr <dig> is in the model only represented as a simple regulator acting upstream of rgt <dig>  the highly pleiotropic nature of grr <dig> is revealed in the dna-microarray data, where  <dig> out of  <dig> evaluated genes are found to be differentially expressed .
the low percentage of correct predictions illustrates the danger of including species without properly accounting for all key functions.
we notice that the model even fails to efficiently predict the alteration of gene expression for targets of rgt <dig> , which, according to the model, should act downstream of grr <dig>  however, given the scale of the perturbations caused by grr <dig> deletion, it is difficult to say whether these discrepancies are caused by faults in the current hypotheses underlying the model or by secondary effects not taken into consideration, such as altered metabolism.
evaluation of Δsnf1Δsnf <dig> mutant
transcriptome data for Δsnf1Δsnf <dig> and its isogenic reference strain were available from  <cit> , during aerobic continuous cultivations at dilution rate  <dig>  h- <dig> and glucose  as single carbon source.
under these conditions, the residual glucose concentration inside the fermentor is very low, and typically no glucose repression is observed.
this behavior is, to some extent, similar to what happens in the absence of glucose.
thus, experimental observations were compared with model predictions for the case where all sugars were absent.
the percentage of true predictions was only 50% .
a surprising discrepancy was observed for the expression levels of cat <dig> and sip <dig>  which were predicted to decrease, but are found to be experimentally unchanged .
nevertheless, the prediction of a down-regulatory effect on cat <dig> and sip <dig> gene targets  is confirmed experimentally.
while the observed changes for cat <dig> and sip <dig> are not statistically significant, they are nevertheless in the predicted direction, and the occurrence of type- <dig> errors is therefore likely.
an additional and likely explanation is that the effect of the snf1-snf <dig> complex on repression by sip <dig> and cat <dig> is, to a larger extent, mediated via direct posttranslational regulation rather than via indirect, mig <dig> mediated transcriptional regulation.
global evaluation of predictive power
boolean models lie at the boundary between qualitative and quantitative models.
for the present model of glucose repression, testing current hypotheses is at least as important as making predictions.
these goals, however, are connected in the sense that one way in which inconsistencies in the model  are revealed is through failure to make predictions.
therefore, evaluating the predictive power of the model is an important task.
above, we looked into the capacity of the model to predict differential gene expression for individual knockouts, and observed that the frequencies of correct predictions varied between the different knockouts .
what remains to be demonstrated is that the correct predictions were not obtained by chance.
for the simplest random model, the probabilities that the expression of a gene is predicted to be unchanged, up-regulated or down-regulated, respectively, would all equal 1/ <dig>  overall, the average probability of having the correct prediction in each situation would also be 1/ <dig>  table  <dig> shows that, for all knockouts, the frequency of correct predictions exceeds 33%.
we tested the significance of this occurrence by applying a binomial distribution test and p <dig> =  <dig>   to all predictions .
overall and for most knockouts, the very low p-values unambiguously show that the results were not obtained due to chance.
however, for Δgrr <dig> it cannot be shown at 95% confidence level that the prediction rate is significantly better than with a random model.
h0: x/n = 1/3
p-values for the statistical evaluation of the null hypothesis, h <dig>  tested for whether the observed level of success was due to chance, i.e.
that the average proportion of correct predictions  is 1/ <dig>  against the alternative hypothesis that the model yields a higher proportion of true predictions.
the test is based on the binomial distribution
next, we applied the same distribution to test the significance of the fraction of large errors observed  in the subset of cases where differential expression was observed experimentally as well as in the model.
since in this case there would be 50% chance of a correct prediction if nothing was known a priori, we use p <dig> =  <dig> .
out of  <dig> cases, only  <dig> cases of opposite signs predictions are observed.
this is significant with a p-value less than 10- <dig>  the three encountered large errors were in two predictions for malr  and one prediction for gal <dig> in the Δmig <dig> mutant .
discussion
the goal of our boolean network modeling and analyses was dual.
first, we wanted to use the model to test how the underlying biological hypotheses  found in literature fit the observations from genome-wide gene expression studies of different signaling knockout mutants.
we proposed a framework to compare simulation results with experimental observations and looked for discrepancies between the two.
these discrepancies hinted at the identification of different types of errors, as discussed below.
second, we wanted to evaluate the model's predictive strength and investigate to what extent we could use it to simulate transcriptional responses upon deletion of various components of the glucose repression cascade.
such a model can eventually be combined with genome-scale stoichiometric metabolic models to further constrain the solution space during optimization problems .
gene expression data in the form of discrete boolean on/off information has been previously used to constrain fluxes  in stoichiometric metabolic models, adding a layer of transcriptional regulation into this type of models  <cit> .
our approach goes even further by also including signaling information, opening new doors to search for targets that can release metabolic control at different regulatory levels.
glucose repression is a complex and intertwined regulatory system, with extensive cross-talk among pathways, feedback loops and different levels of regulation responding at different time scales.
this makes it difficult to decide the logical rules describing some of the species and their influence, in particular species that are heavily regulated  or those with extensive pleiotropic effects .
noticeably, we explicitly decided not to include hexokinase- <dig>  in the model, despite hxk <dig> being hinted to be a key regulator in the snf1/mig <dig> pathway .
this decision was based on observations that changes in the activity of hxk <dig> lead to an array of effects, namely altered metabolism, which may indirectly trigger other regulatory responses  <cit> .
thus, it becomes difficult to describe the role of hxk <dig> in a model of moderate size like ours.
more knowledge on the exact signaling and regulatory roles of hxk <dig> in glucose repression will be necessary before it can be included in our model.
we performed logical steady state analyses for the wildtype, all single gene deletions and three double gene deletions under all combinations of sugar availability, and observed that most nodes change their logical steady state in more than 10% of the gene deletion simulations.
thereafter, we evaluated the model predictions against available gene expression data by comparing changes in transcript levels  with simulations of the logical steady state of the model.
determination of the overall true prediction rate for the analyzed knockouts shows that Δrgt <dig> yielded the best results , while true predictions for Δgrr <dig> were the weakest .
the highest prediction rate found for Δrgt <dig> is probably related with the fact that many of the regulatory mechanisms included in our model are originated from transcriptional studies on the role of rgt <dig>  conversely, the very bad prediction capabilities for Δgrr <dig> are likely related with the pleiotropic role of grr <dig> in nutrient sensing, and the fact that our model does not account for all the regulatory effects associated with grr <dig>  somewhere in between, true predictions for Δmig <dig>  Δmig1Δmig <dig> and Δsnf1Δsnf <dig> lay in the range 50%–63%.
these prediction rates can be improved if we do not account for genes with dubious regulation.
namely, the genes hxt <dig>  hxt <dig>  ygl157w, ykr075c, yor062c, ynl234w and the mal loci were included in the model even tough little is known about their regulation.
these genes are presumably regulated by one or two of the transcription factors in the model via a hypothesized mechanism , but may very well also be regulated via other mechanisms, possibly involving other regulators.
nevertheless, their inclusion in the model allows us to see to what extent their expression is explained by the proposed mechanism.
we observed that, if these dubiously regulated genes were not included for model evaluation, the rate of correct predictions increased markedly in the case of Δmig <dig>  Δmig1Δmig <dig> and Δsnf1Δsnf <dig> .
moreover, if we exclude the genes with dubious regulation mentioned above as well as the results from the highly pleiotropic knockout of grr <dig>  we observe an improvement in the overall success rate from 60% to 71%.
this suggests that the proposed mechanisms are probably incomplete for these genes.
overall, analyses of the discrepancies between model predictions and transcriptome data hints at four main sources of errors:  errors arising from imprecise conversion of knowledge into logical representation,  errors inherent to the boolean formalism,  errors arising from the discrete evaluation of experimental gene expression data, and  situations where high-throughput data goes against literature-based knowledge.
errors arising from imprecise conversion of knowledge into logical representation.
many of the discrepancies found in the model are originated from biological ambiguities or from difficulties in translating biological behavior into logical rules.
for example, during evaluation of the Δrgt <dig> results we saw that it was not trivial to describe in boolean terms the regulation of snf <dig> and suc <dig> by mig <dig> and mig <dig>  in both cases, the error seems to arise from an over-estimation of the importance of mig <dig> repression.
although mig <dig> and mig <dig> are believed to have somewhat redundant roles as repressors, the boolean formalism does not make it easy to distinguish different levels of regulation.
most notoriously, the model fails to predict the response of the mal genes .
in s. cerevisae the mal genes, that are required for utilization of maltose as carbon source, co-locate in telomere-associated mal loci.
although there are five different mal loci identified  <cit>  there is large variations in terms of presence and activity of these loci in different strains  <cit> .
moreover, experimental studies suggest that different mal loci are regulated by distinct regulatory mechanisms, and, in particular, the mal <dig> locus has been reported to be regulated differently than the mal <dig>  mal <dig> and mal <dig> loci  <cit> .
in the present work, network reconstruction of the mal system was based on investigations of the mal <dig> locus  <cit> .
however, during model evaluation we made no distinction between different alleles – e.g.
malr is used to represent all mal activator encoding genes, i.e.
mal <dig>  mal <dig>  mal <dig>  mal <dig> and mal <dig> .
this generalization may be a major reason for the very low prediction rates.
errors inherent to the boolean formalism.
another source of discrepancies is the limited nature of the binary boolean model itself.
in some cases, very steep response curves for gene expression and protein activities are observed, corresponding well with the binary nature of the boolean model.
however, the boolean formalism lacks the capacity to describe a continuous reality that cannot be represented in an on/off manner.
for example, it is impossible to distinguish between absence, low levels and high levels of glucose, three different conditions that trigger different regulatory responses.
thus, a discretionary approximation that conveniently explains the biological context has to be made .
in other instances, the boolean formalism may not be sensitive enough to represent different levels of expression, such as in the case of regulation of gal <dig> by mig <dig>  gal <dig> and gal <dig>  <cit> .
furthermore, when considering inducible proteins that are expressed at a basal level, a value of zero may indicate presence at basal level rather than total absence.
in such instances, defining whether a gene is being expressed  or not  is somewhat subjective.
a further limitation of the boolean formalism, particularly when focusing on logical steady state analysis with no distinction between processes of different time scales, is the difficulty of incorporating negative feedback loops.
as discussed above there are difficulties in representing the negative feedback loop regulating mth <dig> expression, and it is also impossible to represent mig <dig> repression of the mig <dig> gene.
such auto-repression cannot be included by an and-relationship, since mig <dig>  would then never be active in a logical steady state.
errors arising from the discrete evaluation of experimental gene expression data.
conversion of gene expression changes into discrete boolean values is a simplification, which is presumably prone to errors.
nevertheless, we used commonly accepted thresholds of fold-change and significance to decide whether a gene is changing its expression.
we have also checked whether choosing different α and fcmin greatly impacts model evaluation results , and observed that it does not.
we have identified a number of discrepancies likely to be due to type- <dig> errors when assigning experimental variation , which show a regulatory change in the expected direction although statistically is not significant.
however, one must look into these errors carefully, especially when the significance of the experimental change is calculated based on a high number of replicates, since this may hint at an error in the underlying hypothesis instead of a type- <dig> gene expression error.
situations where high-throughput data goes against literature-based knowledge.
large errors found in the model evaluation  indicate situations where model prediction and observed changes of gene expression have opposite signs.
this type of error represents a situation where the hypothesis underlying the logical model needs to be reconsidered.
we have found this type of error only  <dig> times .
in the case of the mal genes, we have already discussed probable sources of the wrong predictions, namely the incorrect assumption about the regulatory mechanisms controlling the expression of the mal loci.
for the gal <dig> gene, we have observed that gene expression from the Δmig <dig> transcriptome study directly contradicts other studies.
although more careful analysis may be advisable, it is likely this result arises from the microarray experiment itself, either due to a problem with the array hybridization or with the normalization method used at probe-sets level.
CONCLUSIONS
overall, the boolean model showed some potential as a predictive model.
the overall success rate  is promising.
the observed errors are most likely due to a combination of lack of knowledge on the glucose regulatory network and simplifications required by the boolean formalism.
in this regard it should be noted that the deterministic and binary approximation to reality inherent in the boolean formalism demands careful interpretation of model outputs and limits the overall success rate, which may be achieved.
even though there is much information on glucose repression in yeast, it is clear from our analysis that there are still connections and parts that are missing.
since the model is set up rigorously based on data from the literature, these inconsistencies seem to be caused by a combination of contradictions in reported experimental results and perhaps due to incorrect insights about the network topology.
future efforts in modeling of glucose repression may need to take into consideration the uncertainties concerning the connectivity of the regulatory network, as well as network dynamics.
methods to discern these regulatory uncertainties , moreover have the potential to be used for reverse engineering of signaling and regulatory networks.
nevertheless, the model presented here represents a condensed way of organizing regulatory information on glucose repression, and strongly facilitates integration and evaluation of new hypotheses.
it can also serve as the basis for further efforts in modeling glucose repression signaling and regulatory pathways using probabilistic and/or dynamic approaches.
the model presented here is thus an important step towards a holistic understanding of glucose repression in saccharomyces cerevisiae, and the model may further be used for design of new experiments that can lead to a better understanding of this complex regulatory system.

 signaling and regulatory cascades establish the bridge between environmental stimuli and cellular responses, and represent a key aspect of cellular adaptation to different environmental conditions. cells can sense several stimuli, both internally and externally, and the received information will subsequently be propagated through a cascade of physico-chemical signals. the ultimate recipients of these signals will determine how the cell responds, by acting at different regulatory levels. contrary to metabolic networks, most signaling and regulatory pathways are relatively poorly studied and signaling properties of a protein cannot be easily derived from its gene sequence. moreover, signal transduction networks operate over a wide range of time-scales, and due to the presence of feedback loops and cross talk it is difficult to discern how concurrent signals are processed. thus, methods to analyze and model signal transduction and regulatory circuits are of prime importance in biology, medicine and cell engineering, since they can bring insights into the mechanistic events underlying complex cellular behavior. the availability of models for signaling and regulatory cascades represents an opportunity to expand the search space when looking for intervention targets that may lead to desired phenotypes – e.g. when looking for better drug targets in medicine, designing novel regulatory circuits in synthetic biology or finding regulatory targets that can release metabolic control in metabolic engineering. most eukaryotic cells, including many yeasts and humans, can sense the availability of carbon sources in their surroundings and, in the presence of their favorite sugar, trigger a cascade of signals that will repress the utilization of less-favorite sugars as well as the function of different catabolic routes. this phenomenon is commonly termed carbon-catabolite or glucose repression. because of its role in nutrient sensing and its industrial impact on the simultaneous utilization of different carbon sources, glucose repression has been a model system for studying signaling and regulation. in particular, glucose repression in the yeast saccharomyces cerevisiae has been extensively studied and two main signaling pathways have been identified: a repression pathway, mediated through the protein kinase snf and the transcription factor mig and a glucose induction pathway, mediated through the membrane receptors snf and rgt and the transcription factor rgt. growing evidence suggests the existence of extensive cross talking between these two pathways. figure summarizes key aspects of the system. besides the role of glucose repression on the utilization of alternative carbon sources, glucose repression in yeast leads to the transcriptional shutdown of genes related to respiration, mitochondrial activities, and gluconeogenesis. this transcriptional behavior causes the wild-type yeast to exert respiro-fermentative metabolism during growth on excess glucose, redirecting carbon towards by-products of metabolism such as ethanol, acetate and glycerol, at the cost of biomass formation. despite being an extensively studied system, knowledge on yeast glucose repression is still far from complete and key questions remain, including: what exactly triggers the cascade signal? how to differentiate between causes and consequences? how does the knowledge derived from phenotypic observations relate to mechanistic events? how does the current knowledge on glucose repression fit with available high-throughput data? in order to attempt to bring insights into these questions, we aim here at creating a mechanistic, semi-quantitative model of glucose repression signaling cascades and genetic regulatory circuits in yeast. modeling approaches of different levels of abstraction have been proposed to analyze and simulate signal transduction and regulatory networks, ranging from purely topological to kinetic models. while attractive in principle, quantitative kinetic models based on odes are hampered by difficulties in determining the necessary parameters and kinetic equations. at the other extreme, strictly descriptive models have also been reported, in which the precision of the formalism proposed, based on process engineering, establish a clear and unique qualitative representation of the network interactions. somewhere in between lie more semi-quantitative and qualitative approaches that require the topological description of signaling interactions and make use of well-established mathematical frameworks to analyze network structure and functionality. such methods include stoichiometric representation and extreme pathway analysis, boolean and bayesian representation of interactions, logical hypergraph representation and logical steady state analyses, and petri nets graph representation and simulations. all these methods describe signal flow qualitatively within a mathematical formalism and without loss of information on the network topology, allowing insightful computations on network structure such as evaluating the degree of cross talk, determining all possible elementary'flux' modes and calculating the number of theoretically possible positive and negative feedback loops. moreover, logical hypergraph analyses and petri net models also have the potential to be used for semi-quantitative simulation of network behavior, since they allow simple predictions of the state of a system in response to different stimuli. in this work, we reconstructed the signaling and transcriptional regulatory network of glucose repression in s. cerevisiae based on established knowledge reported in the literature. we converted this information into a logical hypergraph, and performed structural and functional analyses on the network following the framework proposed by klamt and co-workers. next, we performed logical steady state analyses to compute the state of all nodes in the system under all possible environmental conditions, and for all different single gene deletions and some double gene deletions. furthermore, we developed a framework to evaluate model predictions by comparing changes in the state of the regulatory layer against changes in gene expression data. based on the results from the model evaluation, we identified main errors and discuss possible sources of discrepancies, as well as the inherent limitations to boolean modeling. our results point towards the existence of incoherencies between high-throughput data and literature-based knowledge related with glucose repression. to our knowledge, this represents the first attempt to mechanistically and semi-quantitatively model glucose repression signaling and regulatory pathways in the yeast s. cerevisiae.  reconstruction of the signaling/regulatory network and model setup glucose repression signaling and regulatory network was reconstructed from low-throughput data reported in peer-reviewed publications. information was gathered based on biochemical studies and physiological observations, and it was included in a collection database containing: list of proteins with sensor, signaling or transcription factor functions found to be related to glucose repression; list of genes known to be transcriptionally regulated by glucose repression related transcription factors; type of regulation exerted on each of the previous species by metabolites and/or regulatory proteins. the reconstructed network accounts for species and interactions, which cover most of the current knowledge on the mig1/snf and snf3/rgt pathways, as well as galactose and maltose regulatory systems. transcription factors included are rgt mig mig mig sip cat malr and gal regulatory targets include genes encoding hexose transporters and enzymes involved in maltose catabolism, gluconeogenesis and the leloir pathway. the complete list of species and interactions considered is given as supplementary material. thereafter, the reconstructed signaling/regulatory network was converted into a logical hypergraph, representing all interactions in a logical manner, according to the framework proposed by klamt et al. to model signaling networks. the conversion of signaling and regulatory interactions into boolean functions was based on described functions reported in the literature. the resulting hypergraph consists of nodes and hyperedges, and represents a logical model for glucose repression signaling and regulatory pathways. for ease of visualization, we have depicted the hypegraph into four separate sub-networks, each representing in more detail a different system: mig1/snf and snf3/rgt pathways, galactose regulation, maltose regulation and the sip4/cat regulatory system. the model takes as input the availability of carbon sources and outputs the logical steady state of the network. in our model, nodes can assume one of two logical states, or corresponding to on or off, or in more subtle instances, higher or lower activity. for protein nodes, this can most simply be interpreted as a protein being active or inactive, whereas in the case of gene nodes, it can be seen as a gene being expressed or not. we used the model to analyze structural characteristics of the network and to compute logical steady states of all nodes in the network. in particular, we simulated how gene transcripts change their logical state in response to perturbations, and evaluated the predictions by using available transcriptional datasets for different carbon source conditions and yeast deletion mutants. during logical states simulations, although most nodes were left unconstrained, a few nodes were assigned a default value of e.g., grr was set to a fix state of since its regulation is not considered in the model. also, in the case of genes constitutively expressed at a basal level such as malt, encoding a maltose transporter, the node state was set to a fix value of structural and functional analyses of the network logical steady state analyses were performed for all combinations of sugar availability for the wild type, all single gene knockout mutants, and three double gene deletion mutants of interest, in a total of simulations. we notice that most of the gene nodes change their logical state in over 10% of the simulations, but only few change in more than 15% of the simulations. the predictions for the wild type along with a subset of the deletion mutants for which transcriptome data was available were analyzed and used to evaluate the model, as will be discussed below. the capability to make semi-quantitative predictions of gene expression levels and protein activity for any combination of gene deletions and nutrient conditions is a key feature of the model. the reconstructed network contains negative and positive feedback loops, which is indicative of the high degree of crosstalk between pathways and complex cause-effect relationships. this is further supported by the dependency matrix of the network, which is based solely on the underlying network interaction graph without information on boolean functions. in the matrix, yellow elements represent an ambivalent relationship between an ordered pair of species, where i and j represents the column and the row number, respectively, in the sense that both activating and inhibiting paths exist from i to j. dark or light green elements, dij, indicate that species i is a total or non-total activator of species j, respectively, i.e. only activating paths from i to j exist and feedback loops are either absent or present – see figure legend for details. examining the dependency matrix, the large number of ambivalent relationships as well as the prevalence of negative feedback loops in signaling paths is noteworthy. it underscores the difficulties in making predictions based on intuitive approaches and emphasizes the need for a logical modeling framework. the matrix also reveals the high degree of crosstalk between the mig1/snf and snf3/rgt pathways, since it is quickly noted that the signaling proteins in one pathway generally affect proteins in the other pathway. model evaluation in order to evaluate the capability of the logical model to predict differential gene expression, we performed logical steady state analysis of the glucose repression regulatory response for five different gene knockouts, and compared the results with available whole-genome gene expression from dna-microarrays. we used data from the yeast mutants Δrgt Δmig Δmig1Δmig Δsnf1Δsnf Δgrr and their isogenic reference strains. this type of analysis not only gives an indication of the model's predictive strength, but also hints at possible errors in the model in the cases where discrepancies between model and observation occur. the best we can hope to achieve with a boolean model is a correct prediction of the sign of the change in gene expression, i.e., the model prediction yimod should equal the experimental observation yiexp when evaluating change in expression of gene i following a knockout or change in conditions. in order to assess what experimental results should be regarded as a change, it is necessary to make an interpretation of the gene expression data that allows a comparison with the binary outputs of the boolean model. intuitively, the experimental change should be relatively large and statistically significant in order to be reflected by the discrete boolean model. therefore, we established a fold-change threshold and a student's t-test p-value cut-off for all pairwise gene expression comparisons between a deletion mutant and its isogenic reference strain. all genes with p-value < α and fc ≥ fcmin were assigned with a value of yiexp =, and otherwise. such conversion of gene expression into discrete boolean values based on a somewhat subjective threshold may yield a number of type- and type- errors. the hereby identified experimental variation, yiexp, was then compared with the model prediction, yimod, for each gene i. yimod was determined by the difference in boolean output for gene i between the mutant and the reference state, at a defined external condition. thus, yimod can assume the values - or corresponding to a decrease, no change or increase in gene expression on transcript i in the mutant, respectively. model prediction capabilities were evaluated based on the difference |yimod - yiexp|, with a value of meaning a correct prediction, a value of implying a small error, and a value of indicating a large error. a summary of the results from the comparison between model prediction and experimental up- and down-regulation for all five different knockouts evaluated is shown in table in the following, we discuss more thoroughly the results for Δrgt and use this to analyze common reasons for discrepancies in all the knockouts. the remaining comparisons are briefly commented afterwards. in parentheses are the numbers when dubious genes are not included in the computations. evaluation of Δrgt mutant transcriptome data for the yeast Δrgt mutant and its isogenic reference strain is available from during shake flasks cultivations using galactose as the single carbon source. sampling was made in the mid-exponential phase, where pseudo-steady-state can be assumed. we converted the gene expression data from this study into yiexp according to the procedure described above, and compared it with the simulation results yimod. yimod are derived from logical steady state analyses of the logical model assuming galactose present and all other carbon sources absent. the logical steady states were first determined for the original model without further constrains. afterwards, in order to simulate the rgt gene deletion, the node rgt of the hypergraph was set to zero, and a new logical steady state analysis was performed. yimod is given by the difference between xi,rgt and xi,wt. in general, the model predictions are very good for the Δrgt mutant, with 82% true predictions, reflecting the fact that rgt along with its regulators has been extensively studied. only for out of the genes evaluated, the experimental fold changes do not correspond to model prediction. the six genes are snf mth mig mal, suc and hxt in the following, the causes of these discrepancies are investigated. analysis of discrepancies in snf snf encodes a high-affinity glucose sensor located in the plasma membrane. gene expression data shows no differential expression of snf in the Δrgt mutant. however, in the model this gene is found to be down-regulated in the mutant relative to the wild type. the most likely explanation for this discrepancy lies in the logical equation for snf applied in the model: snf = not = and in the model, rgt deletion leads to an active mig and consequently repression of snf the most plausible explanation for this is that the model overestimates the importance of mig and that, in reality, the presence of active mig by itself is not enough to prevent snf transcription. nevertheless, this explanation cannot be accepted out of hand: the dependency matrix reveals that rgt is an ambivalent factor with regard to snf i.e., both repressing and activating paths from rgt to snf exist. it is thus possible that mig is indeed a significant repressor of snf but that other important, repressing pathways from rgt to snf are also deactivated by the rgt deletion. a closer inspection of the signaling paths from rgt to snf in the hypergraph reveals that, from the existing possible paths, are repressing paths of the form rgt -> rgt -| mig2/mig -| mig -| snf this means that rgt represses mig and mig both of which repress mig which then represses snf. it seems unlikely that these signal flux modes play significant physiological roles since no significant role of mig has ever been found, and mig can at most serve to attenuate mig expression since the two proteins are active at basically the same conditions. analysis of discrepancies in mth mth is a signaling protein intermediate between the membrane sensors snf3/rgt and the transcription factor rgt. gene expression of mth is found to be up-regulated in the Δrgt mutant, whereas the model predicts the expression level of mth to be unchanged. despite the fact that rgt repression of mth is reported in the literature, the repression of mth by rgt is ignored for logical steady state calculations for two reasons. firstly, kaniak et al. state that this transcriptional repression is weak transcriptional repressor of mig the discrepancy for hxt seems to have similar reasons. analysis of discrepancies in mal. expression of mal encoding a mal regulator, was found to be experimentally repressed in the mutant, while it remained unchanged in the model. the three genes required for maltose metabolism are mapped in various mal loci, of which are currently known. in the model, no distinction is made between the different complex loci. since this is a recurrent discrepancy in all evaluations performed, we present a general discussion on the mal regulatory system later in the discussion section. analysis of discrepancies in suc expression of suc encoding for invertase, was not found to be significantly different between the Δrgt mutant and wild-type, whereas the model predicts a decrease in gene expression in the mutant. this discrepancy illustrates both the difficulties in choosing the correct logical equation for a specific species based on literature and in converting a continuous reality to a binary model. in the model it is assumed that both mig and mig should be absent in order for suc to be expressed at high levels ), cf. lutfiyya et al. who found that the single deletions had relatively low impact on suc expression level, whereas Δmig1Δmig double deletion had great effect. contrary to this, klein et al. found a large increase in suc expression in a Δmig strain and further increase by additional disruption of mig. the klein et al. observations could have been implemented in the model via a boolean function where the absence of either one of the two repressors induces expression of suc ) or, alternatively, by simply ignoring mig in the equations. while implementing either of the alternatives would have led to correct model prediction for the knockout, the actual output for suc in the wild-type would have been the same regardless of the chosen equation. nevertheless, the ambiguity in the literature combined with the difficulty of imposing a discrete model on a continuous reality made the choice of logical equation extremely hard in this case. the model apparently mistakenly predicts a decrease in suc expression, because rgt deletion causes mig to be expressed which then leads to repression of suc with the chosen boolean equations. based on this evaluation, and assuming that a type- error has not occurred, it therefore seems reasonable to say that the model overestimates the importance of mig in the regulation of suc alternatively, it is possible that mig transcription is induced, but that the mig protein is also post-translationally regulated, something that is not described in the model or, to the best of our knowledge, in the literature. this could mean that mig protein activity is not increased by the deletion of rgt despite the eventual increase in transcript levels. evaluation of Δmig and Δmig1Δmig mutants transcriptome data for Δmig and Δmig1Δmig mutants, and their isogenic reference strain was available from during aerobic batch cultivations using g/l glucose as single carbon source. samples were taken in the mid-exponential phase at a residual glucose concentration of g/l. therefore, experimental observations were evaluated against model predictions with only glucose present. for both knockouts, the percentage of true predictions is of 63%. when examining the discrepancies between experimental observations and model predictions, it is particularly noticeable that model predictions for mal and gal regulatory genes are almost always wrong. in particular, in the Δmig case both mal and gal evaluation produces a large error, meaning that the experimental direction of regulation is opposite to that predicted by the model. whereas in the case of mal genes this is one of several discrepancies observed in this work, in the case of gal we find the experimental gene expression data from to directly contradict the results of a northern blot analysis. while the former study showed a decrease in gal levels upon deletion of mig the second showed more than 6-fold increase in the levels of gal transcripts in a Δmig mutant, in a medium with the same glucose concentration. evaluation of Δgrr mutant transcriptome data for Δgrr and its isogenic reference strain were also available from, at conditions corresponding to high extracellular glucose concentration. evaluation of model predictions for Δgrr yielded the poorest results, with only 39% of true predictions. this is probably due to the fact that, despite being an important player in a large number of cellular activities such as cell morphology, heavy metal tolerance, osmotic stress and nitrogen starvation, grr is in the model only represented as a simple regulator acting upstream of rgt the highly pleiotropic nature of grr is revealed in the dna-microarray data, where out of evaluated genes are found to be differentially expressed. the low percentage of correct predictions illustrates the danger of including species without properly accounting for all key functions. we notice that the model even fails to efficiently predict the alteration of gene expression for targets of rgt, which, according to the model, should act downstream of grr however, given the scale of the perturbations caused by grr deletion, it is difficult to say whether these discrepancies are caused by faults in the current hypotheses underlying the model or by secondary effects not taken into consideration, such as altered metabolism. evaluation of Δsnf1Δsnf mutant transcriptome data for Δsnf1Δsnf and its isogenic reference strain were available from, during aerobic continuous cultivations at dilution rate h- and glucose as single carbon source. under these conditions, the residual glucose concentration inside the fermentor is very low, and typically no glucose repression is observed. this behavior is, to some extent, similar to what happens in the absence of glucose. thus, experimental observations were compared with model predictions for the case where all sugars were absent. the percentage of true predictions was only 50%. a surprising discrepancy was observed for the expression levels of cat and sip which were predicted to decrease, but are found to be experimentally unchanged. nevertheless, the prediction of a down-regulatory effect on cat and sip gene targets is confirmed experimentally. while the observed changes for cat and sip are not statistically significant, they are nevertheless in the predicted direction, and the occurrence of type- errors is therefore likely. an additional and likely explanation is that the effect of the snf1-snf complex on repression by sip and cat is, to a larger extent, mediated via direct posttranslational regulation rather than via indirect, mig mediated transcriptional regulation. global evaluation of predictive power boolean models lie at the boundary between qualitative and quantitative models. for the present model of glucose repression, testing current hypotheses is at least as important as making predictions. these goals, however, are connected in the sense that one way in which inconsistencies in the model are revealed is through failure to make predictions. therefore, evaluating the predictive power of the model is an important task. above, we looked into the capacity of the model to predict differential gene expression for individual knockouts, and observed that the frequencies of correct predictions varied between the different knockouts. what remains to be demonstrated is that the correct predictions were not obtained by chance. for the simplest random model, the probabilities that the expression of a gene is predicted to be unchanged, up-regulated or down-regulated, respectively, would all equal 1/ overall, the average probability of having the correct prediction in each situation would also be 1/ table shows that, for all knockouts, the frequency of correct predictions exceeds 33%. we tested the significance of this occurrence by applying a binomial distribution test and p = to all predictions. overall and for most knockouts, the very low p-values unambiguously show that the results were not obtained due to chance. however, for Δgrr it cannot be shown at 95% confidence level that the prediction rate is significantly better than with a random model. h0: x/n = 1/3 p-values for the statistical evaluation of the null hypothesis, h tested for whether the observed level of success was due to chance, i.e. that the average proportion of correct predictions is 1/ against the alternative hypothesis that the model yields a higher proportion of true predictions. the test is based on the binomial distribution next, we applied the same distribution to test the significance of the fraction of large errors observed in the subset of cases where differential expression was observed experimentally as well as in the model. since in this case there would be 50% chance of a correct prediction if nothing was known a priori, we use p =. out of cases, only cases of opposite signs predictions are observed. this is significant with a p-value less than 10- the three encountered large errors were in two predictions for malr and one prediction for gal in the Δmig mutant. discussion the goal of our boolean network modeling and analyses was dual. first, we wanted to use the model to test how the underlying biological hypotheses found in literature fit the observations from genome-wide gene expression studies of different signaling knockout mutants. we proposed a framework to compare simulation results with experimental observations and looked for discrepancies between the two. these discrepancies hinted at the identification of different types of errors, as discussed below. second, we wanted to evaluate the model's predictive strength and investigate to what extent we could use it to simulate transcriptional responses upon deletion of various components of the glucose repression cascade. such a model can eventually be combined with genome-scale stoichiometric metabolic models to further constrain the solution space during optimization problems. gene expression data in the form of discrete boolean on/off information has been previously used to constrain fluxes in stoichiometric metabolic models, adding a layer of transcriptional regulation into this type of models. our approach goes even further by also including signaling information, opening new doors to search for targets that can release metabolic control at different regulatory levels. glucose repression is a complex and intertwined regulatory system, with extensive cross-talk among pathways, feedback loops and different levels of regulation responding at different time scales. this makes it difficult to decide the logical rules describing some of the species and their influence, in particular species that are heavily regulated or those with extensive pleiotropic effects. noticeably, we explicitly decided not to include hexokinase- in the model, despite hxk being hinted to be a key regulator in the snf1/mig pathway. this decision was based on observations that changes in the activity of hxk lead to an array of effects, namely altered metabolism, which may indirectly trigger other regulatory responses. thus, it becomes difficult to describe the role of hxk in a model of moderate size like ours. more knowledge on the exact signaling and regulatory roles of hxk in glucose repression will be necessary before it can be included in our model. we performed logical steady state analyses for the wildtype, all single gene deletions and three double gene deletions under all combinations of sugar availability, and observed that most nodes change their logical steady state in more than 10% of the gene deletion simulations. thereafter, we evaluated the model predictions against available gene expression data by comparing changes in transcript levels with simulations of the logical steady state of the model. determination of the overall true prediction rate for the analyzed knockouts shows that Δrgt yielded the best results, while true predictions for Δgrr were the weakest. the highest prediction rate found for Δrgt is probably related with the fact that many of the regulatory mechanisms included in our model are originated from transcriptional studies on the role of rgt conversely, the very bad prediction capabilities for Δgrr are likely related with the pleiotropic role of grr in nutrient sensing, and the fact that our model does not account for all the regulatory effects associated with grr somewhere in between, true predictions for Δmig Δmig1Δmig and Δsnf1Δsnf lay in the range 50%–63%. these prediction rates can be improved if we do not account for genes with dubious regulation. namely, the genes hxt hxt ygl157w, ykr075c, yor062c, ynl234w and the mal loci were included in the model even tough little is known about their regulation. these genes are presumably regulated by one or two of the transcription factors in the model via a hypothesized mechanism, but may very well also be regulated via other mechanisms, possibly involving other regulators. nevertheless, their inclusion in the model allows us to see to what extent their expression is explained by the proposed mechanism. we observed that, if these dubiously regulated genes were not included for model evaluation, the rate of correct predictions increased markedly in the case of Δmig Δmig1Δmig and Δsnf1Δsnf. moreover, if we exclude the genes with dubious regulation mentioned above as well as the results from the highly pleiotropic knockout of grr we observe an improvement in the overall success rate from 60% to 71%. this suggests that the proposed mechanisms are probably incomplete for these genes. overall, analyses of the discrepancies between model predictions and transcriptome data hints at four main sources of errors: errors arising from imprecise conversion of knowledge into logical representation, errors inherent to the boolean formalism, errors arising from the discrete evaluation of experimental gene expression data, and situations where high-throughput data goes against literature-based knowledge. errors arising from imprecise conversion of knowledge into logical representation. many of the discrepancies found in the model are originated from biological ambiguities or from difficulties in translating biological behavior into logical rules. for example, during evaluation of the Δrgt results we saw that it was not trivial to describe in boolean terms the regulation of snf and suc by mig and mig in both cases, the error seems to arise from an over-estimation of the importance of mig repression. although mig and mig are believed to have somewhat redundant roles as repressors, the boolean formalism does not make it easy to distinguish different levels of regulation. most notoriously, the model fails to predict the response of the mal genes. in s. cerevisae the mal genes, that are required for utilization of maltose as carbon source, co-locate in telomere-associated mal loci. although there are five different mal loci identified there is large variations in terms of presence and activity of these loci in different strains. moreover, experimental studies suggest that different mal loci are regulated by distinct regulatory mechanisms, and, in particular, the mal locus has been reported to be regulated differently than the mal mal and mal loci. in the present work, network reconstruction of the mal system was based on investigations of the mal locus. however, during model evaluation we made no distinction between different alleles – e.g. malr is used to represent all mal activator encoding genes, i.e. mal mal mal mal and mal. this generalization may be a major reason for the very low prediction rates. errors inherent to the boolean formalism. another source of discrepancies is the limited nature of the binary boolean model itself. in some cases, very steep response curves for gene expression and protein activities are observed, corresponding well with the binary nature of the boolean model. however, the boolean formalism lacks the capacity to describe a continuous reality that cannot be represented in an on/off manner. for example, it is impossible to distinguish between absence, low levels and high levels of glucose, three different conditions that trigger different regulatory responses. thus, a discretionary approximation that conveniently explains the biological context has to be made. in other instances, the boolean formalism may not be sensitive enough to represent different levels of expression, such as in the case of regulation of gal by mig gal and gal. furthermore, when considering inducible proteins that are expressed at a basal level, a value of zero may indicate presence at basal level rather than total absence. in such instances, defining whether a gene is being expressed or not is somewhat subjective. a further limitation of the boolean formalism, particularly when focusing on logical steady state analysis with no distinction between processes of different time scales, is the difficulty of incorporating negative feedback loops. as discussed above there are difficulties in representing the negative feedback loop regulating mth expression, and it is also impossible to represent mig repression of the mig gene. such auto-repression cannot be included by an and-relationship, since mig would then never be active in a logical steady state. errors arising from the discrete evaluation of experimental gene expression data. conversion of gene expression changes into discrete boolean values is a simplification, which is presumably prone to errors. nevertheless, we used commonly accepted thresholds of fold-change and significance to decide whether a gene is changing its expression. we have also checked whether choosing different α and fcmin greatly impacts model evaluation results, and observed that it does not. we have identified a number of discrepancies likely to be due to type- errors when assigning experimental variation, which show a regulatory change in the expected direction although statistically is not significant. however, one must look into these errors carefully, especially when the significance of the experimental change is calculated based on a high number of replicates, since this may hint at an error in the underlying hypothesis instead of a type- gene expression error. situations where high-throughput data goes against literature-based knowledge. large errors found in the model evaluation indicate situations where model prediction and observed changes of gene expression have opposite signs. this type of error represents a situation where the hypothesis underlying the logical model needs to be reconsidered. we have found this type of error only times. in the case of the mal genes, we have already discussed probable sources of the wrong predictions, namely the incorrect assumption about the regulatory mechanisms controlling the expression of the mal loci. for the gal gene, we have observed that gene expression from the Δmig transcriptome study directly contradicts other studies. although more careful analysis may be advisable, it is likely this result arises from the microarray experiment itself, either due to a problem with the array hybridization or with the normalization method used at probe-sets level.  overall, the boolean model showed some potential as a predictive model. the overall success rate is promising. the observed errors are most likely due to a combination of lack of knowledge on the glucose regulatory network and simplifications required by the boolean formalism. in this regard it should be noted that the deterministic and binary approximation to reality inherent in the boolean formalism demands careful interpretation of model outputs and limits the overall success rate, which may be achieved. even though there is much information on glucose repression in yeast, it is clear from our analysis that there are still connections and parts that are missing. since the model is set up rigorously based on data from the literature, these inconsistencies seem to be caused by a combination of contradictions in reported experimental results and perhaps due to incorrect insights about the network topology. future efforts in modeling of glucose repression may need to take into consideration the uncertainties concerning the connectivity of the regulatory network, as well as network dynamics. methods to discern these regulatory uncertainties, moreover have the potential to be used for reverse engineering of signaling and regulatory networks. nevertheless, the model presented here represents a condensed way of organizing regulatory information on glucose repression, and strongly facilitates integration and evaluation of new hypotheses. it can also serve as the basis for further efforts in modeling glucose repression signaling and regulatory pathways using probabilistic and/or dynamic approaches. the model presented here is thus an important step towards a holistic understanding of glucose repression in saccharomyces cerevisiae, and the model may further be used for design of new experiments that can lead to a better understanding of this complex regulatory system. 
in the yeast saccharomyces cerevisiae, the presence of high levels of glucose leads to an array of down-regulatory effects known as glucose repression. we established a logical model of yeast glucose repression, formalized as a hypergraph. though the binary nature of logical models entails inherent limitations, our model constitutes a primary tool for storing regulatory knowledge, searching for incoherencies in hypotheses and evaluating the effect of deleting regulatory elements involved in glucose repression. this process is complex due to the presence of feedback loops and crosstalk between different pathways, complicating the use of intuitive approaches to analyze the system. overall, we observed 71% true predictions, and analyzed sources of errors and discrepancies for the remaining. we computed the logical steady states of all nodes in the network in order to simulate wildtype and deletion mutant responses to different sugar availabilities. evaluation of the model predictive power was achieved by comparing changes in the logical state of gene nodes with transcriptome data. the model was constructed based on verified regulatory interactions and it includes gene transcripts, proteins, metabolites and hyperedges.
 in the yeast saccharomyces cerevisiae, the presence of high levels of glucose leads to an array of down-regulatory effects known as glucose repression. this process is complex due to the presence of feedback loops and crosstalk between different pathways, complicating the use of intuitive approaches to analyze the system.  we established a logical model of yeast glucose repression, formalized as a hypergraph. the model was constructed based on verified regulatory interactions and it includes gene transcripts, proteins, metabolites and hyperedges. we computed the logical steady states of all nodes in the network in order to simulate wildtype and deletion mutant responses to different sugar availabilities. evaluation of the model predictive power was achieved by comparing changes in the logical state of gene nodes with transcriptome data. overall, we observed 71% true predictions, and analyzed sources of errors and discrepancies for the remaining.  though the binary nature of logical models entails inherent limitations, our model constitutes a primary tool for storing regulatory knowledge, searching for incoherencies in hypotheses and evaluating the effect of deleting regulatory elements involved in glucose repression. 